id	sid	tid	token	lemma	pos
ap-9037	1	1	acta	acta	PROPN
ap-9037	1	2	polytechnica	polytechnica	PROPN
ap-9037	1	3	https://doi.org/10.14311/ap.2023.63.0273	https://doi.org/10.14311/ap.2023.63.0273	PROPN
ap-9037	1	4	acta	acta	PROPN
ap-9037	1	5	polytechnica	polytechnica	PROPN
ap-9037	1	6	63(4):273–292	63(4):273–292	PROPN
ap-9037	1	7	,	,	PUNCT
ap-9037	1	8	2023	2023	NUM
ap-9037	1	9	©	©	ADP
ap-9037	1	10	2023	2023	NUM
ap-9037	1	11	the	the	DET
ap-9037	1	12	author(s	author(s	NOUN
ap-9037	1	13	)	)	PUNCT
ap-9037	1	14	.	.	PUNCT
ap-9037	2	1	licensed	license	VERB
ap-9037	2	2	under	under	ADP
ap-9037	2	3	a	a	DET
ap-9037	2	4	cc	cc	NOUN
ap-9037	2	5	-	-	PUNCT
ap-9037	2	6	by	by	ADP
ap-9037	2	7	4.0	4.0	NUM
ap-9037	2	8	licence	licence	NOUN
ap-9037	2	9	published	publish	VERB
ap-9037	2	10	by	by	ADP
ap-9037	2	11	the	the	DET
ap-9037	2	12	czech	czech	PROPN
ap-9037	2	13	technical	technical	PROPN
ap-9037	2	14	university	university	PROPN
ap-9037	2	15	in	in	ADP
ap-9037	2	16	prague	prague	PROPN
ap-9037	2	17	construction	construction	NOUN
ap-9037	2	18	of	of	ADP
ap-9037	2	19	angular	angular	ADJ
ap-9037	2	20	-	-	PUNCT
ap-9037	2	21	dependent	dependent	ADJ
ap-9037	2	22	potentials	potential	NOUN
ap-9037	2	23	from	from	ADP
ap-9037	2	24	trigonometric	trigonometric	ADJ
ap-9037	2	25	pöschl	pöschl	NOUN
ap-9037	2	26	-	-	PUNCT
ap-9037	2	27	teller	teller	NOUN
ap-9037	2	28	systems	system	NOUN
ap-9037	2	29	within	within	ADP
ap-9037	2	30	the	the	DET
ap-9037	2	31	dunkl	dunkl	NOUN
ap-9037	2	32	formalism	formalism	NOUN
ap-9037	2	33	axel	axel	PROPN
ap-9037	2	34	schulze	schulze	PROPN
ap-9037	2	35	-	-	PUNCT
ap-9037	2	36	halberg	halberg	PROPN
ap-9037	2	37	indiana	indiana	PROPN
ap-9037	2	38	university	university	PROPN
ap-9037	2	39	northwest	northwest	NOUN
ap-9037	2	40	,	,	PUNCT
ap-9037	2	41	department	department	NOUN
ap-9037	2	42	of	of	ADP
ap-9037	2	43	mathematics	mathematic	NOUN
ap-9037	2	44	and	and	CCONJ
ap-9037	2	45	acturial	acturial	ADJ
ap-9037	2	46	science	science	NOUN
ap-9037	2	47	and	and	CCONJ
ap-9037	2	48	department	department	NOUN
ap-9037	2	49	of	of	ADP
ap-9037	2	50	physics	physics	PROPN
ap-9037	2	51	,	,	PUNCT
ap-9037	2	52	3400	3400	NUM
ap-9037	2	53	broadway	broadway	NOUN
ap-9037	2	54	,	,	PUNCT
ap-9037	2	55	gary	gary	PROPN
ap-9037	2	56	in	in	ADP
ap-9037	2	57	46408	46408	NUM
ap-9037	2	58	,	,	PUNCT
ap-9037	2	59	united	united	PROPN
ap-9037	2	60	states	states	PROPN
ap-9037	2	61	of	of	ADP
ap-9037	2	62	america	america	PROPN
ap-9037	2	63	correspondence	correspondence	NOUN
ap-9037	2	64	:	:	PUNCT
ap-9037	2	65	axgeschu@iun.edu	axgeschu@iun.edu	ADV
ap-9037	3	1	abstract	abstract	ADJ
ap-9037	3	2	.	.	PUNCT
ap-9037	4	1	we	we	PRON
ap-9037	4	2	generate	generate	VERB
ap-9037	4	3	solvable	solvable	ADJ
ap-9037	4	4	cases	case	NOUN
ap-9037	4	5	of	of	ADP
ap-9037	4	6	the	the	DET
ap-9037	4	7	two	two	NUM
ap-9037	4	8	angular	angular	ADJ
ap-9037	4	9	equations	equation	NOUN
ap-9037	4	10	resulting	result	VERB
ap-9037	4	11	from	from	ADP
ap-9037	4	12	variable	variable	ADJ
ap-9037	4	13	separation	separation	NOUN
ap-9037	4	14	in	in	ADP
ap-9037	4	15	the	the	DET
ap-9037	4	16	three	three	NUM
ap-9037	4	17	-	-	PUNCT
ap-9037	4	18	dimensional	dimensional	ADJ
ap-9037	4	19	dunkl	dunkl	NOUN
ap-9037	4	20	-	-	PUNCT
ap-9037	4	21	schrödinger	schrödinger	NOUN
ap-9037	4	22	equation	equation	NOUN
ap-9037	4	23	expressed	express	VERB
ap-9037	4	24	in	in	ADP
ap-9037	4	25	spherical	spherical	ADJ
ap-9037	4	26	coordinates	coordinate	NOUN
ap-9037	4	27	.	.	PUNCT
ap-9037	5	1	it	it	PRON
ap-9037	5	2	is	be	AUX
ap-9037	5	3	shown	show	VERB
ap-9037	5	4	that	that	SCONJ
ap-9037	5	5	the	the	DET
ap-9037	5	6	dunkl	dunkl	NOUN
ap-9037	5	7	formalism	formalism	NOUN
ap-9037	5	8	interrelates	interrelate	VERB
ap-9037	5	9	these	these	DET
ap-9037	5	10	angular	angular	ADJ
ap-9037	5	11	equations	equation	NOUN
ap-9037	5	12	with	with	ADP
ap-9037	5	13	trigonometric	trigonometric	ADJ
ap-9037	5	14	pöschl	pöschl	NOUN
ap-9037	5	15	-	-	PUNCT
ap-9037	5	16	teller	teller	NOUN
ap-9037	5	17	systems	system	NOUN
ap-9037	5	18	.	.	PUNCT
ap-9037	6	1	based	base	VERB
ap-9037	6	2	on	on	ADP
ap-9037	6	3	this	this	DET
ap-9037	6	4	interrelation	interrelation	NOUN
ap-9037	6	5	,	,	PUNCT
ap-9037	6	6	we	we	PRON
ap-9037	6	7	use	use	VERB
ap-9037	6	8	point	point	NOUN
ap-9037	6	9	transformations	transformation	NOUN
ap-9037	6	10	and	and	CCONJ
ap-9037	6	11	darboux	darboux	VERB
ap-9037	6	12	-	-	PUNCT
ap-9037	6	13	crum	crum	NOUN
ap-9037	6	14	transformations	transformation	NOUN
ap-9037	6	15	to	to	PART
ap-9037	6	16	construct	construct	VERB
ap-9037	6	17	new	new	ADJ
ap-9037	6	18	solvable	solvable	ADJ
ap-9037	6	19	cases	case	NOUN
ap-9037	6	20	of	of	ADP
ap-9037	6	21	the	the	DET
ap-9037	6	22	angular	angular	ADJ
ap-9037	6	23	equations	equation	NOUN
ap-9037	6	24	.	.	PUNCT
ap-9037	7	1	instead	instead	ADV
ap-9037	7	2	of	of	ADP
ap-9037	7	3	the	the	DET
ap-9037	7	4	stationary	stationary	ADJ
ap-9037	7	5	energy	energy	NOUN
ap-9037	7	6	,	,	PUNCT
ap-9037	7	7	we	we	PRON
ap-9037	7	8	use	use	VERB
ap-9037	7	9	the	the	DET
ap-9037	7	10	constants	constant	NOUN
ap-9037	7	11	due	due	ADP
ap-9037	7	12	to	to	ADP
ap-9037	7	13	the	the	DET
ap-9037	7	14	separation	separation	NOUN
ap-9037	7	15	of	of	ADP
ap-9037	7	16	variables	variable	NOUN
ap-9037	7	17	as	as	ADP
ap-9037	7	18	transformation	transformation	NOUN
ap-9037	7	19	parameters	parameter	NOUN
ap-9037	7	20	for	for	ADP
ap-9037	7	21	our	our	PRON
ap-9037	7	22	darboux	darboux	ADJ
ap-9037	7	23	-	-	PUNCT
ap-9037	7	24	crum	crum	NOUN
ap-9037	7	25	transformations	transformation	NOUN
ap-9037	7	26	.	.	PUNCT
ap-9037	8	1	keywords	keyword	NOUN
ap-9037	8	2	:	:	PUNCT
ap-9037	8	3	dunkl	dunkl	NOUN
ap-9037	8	4	operator	operator	NOUN
ap-9037	8	5	,	,	PUNCT
ap-9037	8	6	schrödinger	schrödinger	ADJ
ap-9037	8	7	equation	equation	NOUN
ap-9037	8	8	,	,	PUNCT
ap-9037	8	9	trigonometric	trigonometric	ADJ
ap-9037	8	10	pöschl	pöschl	NOUN
ap-9037	8	11	-	-	PUNCT
ap-9037	8	12	teller	teller	NOUN
ap-9037	8	13	potential	potential	NOUN
ap-9037	8	14	,	,	PUNCT
ap-9037	8	15	angular	angular	ADJ
ap-9037	8	16	equation	equation	NOUN
ap-9037	8	17	,	,	PUNCT
ap-9037	8	18	darboux	darboux	NOUN
ap-9037	8	19	-	-	PUNCT
ap-9037	8	20	crum	crum	NOUN
ap-9037	8	21	transformation	transformation	NOUN
ap-9037	8	22	.	.	PUNCT
ap-9037	9	1	1	1	X
ap-9037	9	2	.	.	X
ap-9037	9	3	introduction	introduction	NOUN
ap-9037	9	4	dunkl	dunkl	PROPN
ap-9037	9	5	operators	operator	NOUN
ap-9037	9	6	are	be	AUX
ap-9037	9	7	commuting	commute	VERB
ap-9037	9	8	differential	differential	ADJ
ap-9037	9	9	-	-	PUNCT
ap-9037	9	10	difference	difference	NOUN
ap-9037	9	11	operators	operator	NOUN
ap-9037	9	12	associated	associate	VERB
ap-9037	9	13	with	with	ADP
ap-9037	9	14	reflection	reflection	NOUN
ap-9037	9	15	groups	group	NOUN
ap-9037	9	16	[	[	X
ap-9037	9	17	1	1	X
ap-9037	9	18	]	]	PUNCT
ap-9037	9	19	that	that	PRON
ap-9037	9	20	play	play	VERB
ap-9037	9	21	an	an	DET
ap-9037	9	22	important	important	ADJ
ap-9037	9	23	role	role	NOUN
ap-9037	9	24	in	in	ADP
ap-9037	9	25	various	various	ADJ
ap-9037	9	26	areas	area	NOUN
ap-9037	9	27	of	of	ADP
ap-9037	9	28	mathematics	mathematic	NOUN
ap-9037	9	29	and	and	CCONJ
ap-9037	9	30	the	the	DET
ap-9037	9	31	natural	natural	ADJ
ap-9037	9	32	sciences	science	NOUN
ap-9037	9	33	.	.	PUNCT
ap-9037	10	1	for	for	ADP
ap-9037	10	2	example	example	NOUN
ap-9037	10	3	,	,	PUNCT
ap-9037	10	4	these	these	DET
ap-9037	10	5	operators	operator	NOUN
ap-9037	10	6	are	be	AUX
ap-9037	10	7	closely	closely	ADV
ap-9037	10	8	related	relate	VERB
ap-9037	10	9	to	to	ADP
ap-9037	10	10	integrable	integrable	ADJ
ap-9037	10	11	systems	system	NOUN
ap-9037	10	12	of	of	ADP
ap-9037	10	13	calogero	calogero	PROPN
ap-9037	10	14	-	-	PUNCT
ap-9037	10	15	moser	moser	PROPN
ap-9037	10	16	-	-	PUNCT
ap-9037	10	17	sutherland	sutherland	PROPN
ap-9037	10	18	type	type	NOUN
ap-9037	10	19	[	[	X
ap-9037	10	20	2–4	2–4	NUM
ap-9037	10	21	]	]	PUNCT
ap-9037	10	22	,	,	PUNCT
ap-9037	10	23	they	they	PRON
ap-9037	10	24	are	be	AUX
ap-9037	10	25	applied	apply	VERB
ap-9037	10	26	in	in	ADP
ap-9037	10	27	the	the	DET
ap-9037	10	28	construction	construction	NOUN
ap-9037	10	29	of	of	ADP
ap-9037	10	30	angular	angular	ADJ
ap-9037	10	31	momentum	momentum	NOUN
ap-9037	10	32	algebras	algebra	NOUN
ap-9037	11	1	[	[	X
ap-9037	11	2	5	5	NUM
ap-9037	11	3	,	,	PUNCT
ap-9037	11	4	6	6	NUM
ap-9037	11	5	]	]	PUNCT
ap-9037	11	6	,	,	PUNCT
ap-9037	11	7	for	for	ADP
ap-9037	11	8	generalised	generalised	ADJ
ap-9037	11	9	fourier	fourier	ADJ
ap-9037	11	10	analysis	analysis	NOUN
ap-9037	11	11	related	relate	VERB
ap-9037	11	12	to	to	ADP
ap-9037	11	13	root	root	NOUN
ap-9037	11	14	systems	system	NOUN
ap-9037	11	15	[	[	X
ap-9037	11	16	7	7	NUM
ap-9037	11	17	]	]	PUNCT
ap-9037	11	18	,	,	PUNCT
ap-9037	11	19	in	in	ADP
ap-9037	11	20	nonlinear	nonlinear	ADJ
ap-9037	11	21	models	model	NOUN
ap-9037	11	22	[	[	X
ap-9037	11	23	8	8	NUM
ap-9037	11	24	]	]	PUNCT
ap-9037	11	25	,	,	PUNCT
ap-9037	11	26	and	and	CCONJ
ap-9037	11	27	in	in	ADP
ap-9037	11	28	the	the	DET
ap-9037	11	29	generalisation	generalisation	NOUN
ap-9037	11	30	of	of	ADP
ap-9037	11	31	classical	classical	ADJ
ap-9037	11	32	orthogonal	orthogonal	ADJ
ap-9037	11	33	polynomials	polynomial	NOUN
ap-9037	11	34	through	through	ADP
ap-9037	11	35	reflection	reflection	NOUN
ap-9037	11	36	groups	group	NOUN
ap-9037	12	1	[	[	X
ap-9037	12	2	9	9	NUM
ap-9037	12	3	]	]	PUNCT
ap-9037	12	4	.	.	PUNCT
ap-9037	13	1	a	a	DET
ap-9037	13	2	further	further	ADJ
ap-9037	13	3	field	field	NOUN
ap-9037	13	4	of	of	ADP
ap-9037	13	5	application	application	NOUN
ap-9037	13	6	for	for	ADP
ap-9037	13	7	dunkl	dunkl	NOUN
ap-9037	13	8	operators	operator	NOUN
ap-9037	13	9	that	that	PRON
ap-9037	13	10	has	have	AUX
ap-9037	13	11	been	be	AUX
ap-9037	13	12	studied	study	VERB
ap-9037	13	13	extensively	extensively	ADV
ap-9037	13	14	in	in	ADP
ap-9037	13	15	recent	recent	ADJ
ap-9037	13	16	years	year	NOUN
ap-9037	13	17	,	,	PUNCT
ap-9037	13	18	is	be	AUX
ap-9037	13	19	the	the	DET
ap-9037	13	20	construction	construction	NOUN
ap-9037	13	21	of	of	ADP
ap-9037	13	22	deformed	deform	VERB
ap-9037	13	23	quantum	quantum	ADJ
ap-9037	13	24	theories	theory	NOUN
ap-9037	13	25	.	.	PUNCT
ap-9037	14	1	such	such	ADJ
ap-9037	14	2	deformations	deformation	NOUN
ap-9037	14	3	are	be	AUX
ap-9037	14	4	obtained	obtain	VERB
ap-9037	14	5	by	by	ADP
ap-9037	14	6	replacing	replace	VERB
ap-9037	14	7	the	the	DET
ap-9037	14	8	conventional	conventional	ADJ
ap-9037	14	9	derivatives	derivative	NOUN
ap-9037	14	10	,	,	PUNCT
ap-9037	14	11	as	as	SCONJ
ap-9037	14	12	they	they	PRON
ap-9037	14	13	appear	appear	VERB
ap-9037	14	14	for	for	ADP
ap-9037	14	15	example	example	NOUN
ap-9037	14	16	in	in	ADP
ap-9037	14	17	quantum	quantum	ADJ
ap-9037	14	18	momentum	momentum	NOUN
ap-9037	14	19	operators	operator	NOUN
ap-9037	14	20	,	,	PUNCT
ap-9037	14	21	through	through	ADP
ap-9037	14	22	dunkl	dunkl	PROPN
ap-9037	14	23	operators	operator	NOUN
ap-9037	15	1	[	[	X
ap-9037	15	2	10	10	NUM
ap-9037	15	3	]	]	PUNCT
ap-9037	15	4	.	.	PUNCT
ap-9037	16	1	by	by	ADP
ap-9037	16	2	means	mean	NOUN
ap-9037	16	3	of	of	ADP
ap-9037	16	4	this	this	DET
ap-9037	16	5	process	process	NOUN
ap-9037	16	6	,	,	PUNCT
ap-9037	16	7	we	we	PRON
ap-9037	16	8	can	can	AUX
ap-9037	16	9	construct	construct	VERB
ap-9037	16	10	a	a	DET
ap-9037	16	11	large	large	ADJ
ap-9037	16	12	variety	variety	NOUN
ap-9037	16	13	of	of	ADP
ap-9037	16	14	relativistic	relativistic	ADJ
ap-9037	16	15	and	and	CCONJ
ap-9037	16	16	nonrelativistic	nonrelativistic	ADJ
ap-9037	16	17	quantum	quantum	NOUN
ap-9037	16	18	systems	system	NOUN
ap-9037	16	19	within	within	ADP
ap-9037	16	20	the	the	DET
ap-9037	16	21	dunkl	dunkl	NOUN
ap-9037	16	22	context	context	NOUN
ap-9037	16	23	.	.	PUNCT
ap-9037	17	1	particular	particular	ADJ
ap-9037	17	2	cases	case	NOUN
ap-9037	17	3	that	that	PRON
ap-9037	17	4	have	have	AUX
ap-9037	17	5	recently	recently	ADV
ap-9037	17	6	been	be	AUX
ap-9037	17	7	studied	study	VERB
ap-9037	17	8	include	include	VERB
ap-9037	17	9	the	the	DET
ap-9037	17	10	harmonic	harmonic	ADJ
ap-9037	17	11	oscillator	oscillator	NOUN
ap-9037	17	12	in	in	ADP
ap-9037	17	13	the	the	DET
ap-9037	17	14	two	two	NUM
ap-9037	17	15	-	-	PUNCT
ap-9037	17	16	dimensional	dimensional	ADJ
ap-9037	17	17	plane	plane	NOUN
ap-9037	17	18	[	[	X
ap-9037	17	19	11	11	NUM
ap-9037	17	20	,	,	PUNCT
ap-9037	17	21	12	12	NUM
ap-9037	17	22	]	]	PUNCT
ap-9037	17	23	,	,	PUNCT
ap-9037	17	24	the	the	DET
ap-9037	17	25	planar	planar	ADJ
ap-9037	17	26	coulomb	coulomb	NOUN
ap-9037	17	27	system	system	NOUN
ap-9037	17	28	[	[	X
ap-9037	17	29	13	13	NUM
ap-9037	17	30	]	]	PUNCT
ap-9037	17	31	,	,	PUNCT
ap-9037	17	32	the	the	DET
ap-9037	17	33	klein	klein	PROPN
ap-9037	17	34	-	-	PUNCT
ap-9037	17	35	gordon	gordon	PROPN
ap-9037	17	36	equation	equation	NOUN
ap-9037	17	37	for	for	ADP
ap-9037	17	38	several	several	ADJ
ap-9037	17	39	interactions	interaction	NOUN
ap-9037	17	40	[	[	X
ap-9037	17	41	14	14	NUM
ap-9037	17	42	]	]	X
ap-9037	17	43	,	,	PUNCT
ap-9037	17	44	rational	rational	ADJ
ap-9037	17	45	extensions	extension	NOUN
ap-9037	17	46	of	of	ADP
ap-9037	17	47	the	the	DET
ap-9037	17	48	dunkl	dunkl	NOUN
ap-9037	17	49	oscillator	oscillator	NOUN
ap-9037	18	1	[	[	X
ap-9037	18	2	15	15	NUM
ap-9037	18	3	]	]	PUNCT
ap-9037	18	4	,	,	PUNCT
ap-9037	18	5	the	the	DET
ap-9037	18	6	relativistic	relativistic	ADJ
ap-9037	18	7	harmonic	harmonic	ADJ
ap-9037	18	8	oscillator	oscillator	NOUN
ap-9037	18	9	[	[	X
ap-9037	18	10	16	16	NUM
ap-9037	18	11	,	,	PUNCT
ap-9037	18	12	17	17	NUM
ap-9037	18	13	]	]	PUNCT
ap-9037	18	14	,	,	PUNCT
ap-9037	18	15	and	and	CCONJ
ap-9037	18	16	three	three	NUM
ap-9037	18	17	-	-	PUNCT
ap-9037	18	18	dimensional	dimensional	ADJ
ap-9037	18	19	systems	system	NOUN
ap-9037	18	20	that	that	PRON
ap-9037	18	21	allow	allow	VERB
ap-9037	18	22	for	for	ADP
ap-9037	18	23	separation	separation	NOUN
ap-9037	18	24	of	of	ADP
ap-9037	18	25	variables	variable	NOUN
ap-9037	18	26	in	in	ADP
ap-9037	18	27	the	the	DET
ap-9037	18	28	schrödinger	schrödinger	ADJ
ap-9037	18	29	equation	equation	NOUN
ap-9037	18	30	[	[	X
ap-9037	18	31	18	18	NUM
ap-9037	18	32	]	]	PUNCT
ap-9037	18	33	.	.	PUNCT
ap-9037	19	1	in	in	ADP
ap-9037	19	2	the	the	DET
ap-9037	19	3	present	present	ADJ
ap-9037	19	4	work	work	NOUN
ap-9037	19	5	,	,	PUNCT
ap-9037	19	6	we	we	PRON
ap-9037	19	7	focus	focus	VERB
ap-9037	19	8	on	on	ADP
ap-9037	19	9	the	the	DET
ap-9037	19	10	latter	latter	ADJ
ap-9037	19	11	type	type	NOUN
ap-9037	19	12	of	of	ADP
ap-9037	19	13	systems	system	NOUN
ap-9037	19	14	.	.	PUNCT
ap-9037	20	1	more	more	ADV
ap-9037	20	2	precisely	precisely	ADV
ap-9037	20	3	,	,	PUNCT
ap-9037	20	4	we	we	PRON
ap-9037	20	5	consider	consider	VERB
ap-9037	20	6	dunkl	dunkl	NOUN
ap-9037	20	7	-	-	PUNCT
ap-9037	20	8	schrödinger	schrödinger	ADJ
ap-9037	20	9	equations	equation	NOUN
ap-9037	20	10	that	that	PRON
ap-9037	20	11	allow	allow	VERB
ap-9037	20	12	for	for	ADP
ap-9037	20	13	separation	separation	NOUN
ap-9037	20	14	of	of	ADP
ap-9037	20	15	variables	variable	NOUN
ap-9037	20	16	in	in	ADP
ap-9037	20	17	spherical	spherical	ADJ
ap-9037	20	18	coordinates	coordinate	NOUN
ap-9037	20	19	.	.	PUNCT
ap-9037	21	1	while	while	SCONJ
ap-9037	21	2	the	the	DET
ap-9037	21	3	radial	radial	ADJ
ap-9037	21	4	equation	equation	NOUN
ap-9037	21	5	maintains	maintain	VERB
ap-9037	21	6	its	its	PRON
ap-9037	21	7	form	form	NOUN
ap-9037	21	8	within	within	ADP
ap-9037	21	9	the	the	DET
ap-9037	21	10	dunkl	dunkl	NOUN
ap-9037	21	11	formalism	formalism	NOUN
ap-9037	21	12	,	,	PUNCT
ap-9037	21	13	the	the	DET
ap-9037	21	14	two	two	NUM
ap-9037	21	15	angular	angular	ADJ
ap-9037	21	16	equations	equation	NOUN
ap-9037	21	17	are	be	AUX
ap-9037	21	18	different	different	ADJ
ap-9037	21	19	from	from	ADP
ap-9037	21	20	their	their	PRON
ap-9037	21	21	conventional	conventional	ADJ
ap-9037	21	22	counterparts	counterpart	NOUN
ap-9037	21	23	.	.	PUNCT
ap-9037	22	1	in	in	ADP
ap-9037	22	2	particular	particular	ADJ
ap-9037	22	3	,	,	PUNCT
ap-9037	22	4	both	both	PRON
ap-9037	22	5	can	can	AUX
ap-9037	22	6	be	be	AUX
ap-9037	22	7	taken	take	VERB
ap-9037	22	8	to	to	ADP
ap-9037	22	9	a	a	DET
ap-9037	22	10	form	form	NOUN
ap-9037	22	11	that	that	PRON
ap-9037	22	12	resembles	resemble	VERB
ap-9037	22	13	the	the	DET
ap-9037	22	14	conventional	conventional	ADJ
ap-9037	22	15	schrödinger	schrödinger	ADJ
ap-9037	22	16	equation	equation	NOUN
ap-9037	22	17	for	for	ADP
ap-9037	22	18	a	a	DET
ap-9037	22	19	generalised	generalise	VERB
ap-9037	22	20	trigonometric	trigonometric	ADJ
ap-9037	22	21	pöschl	pöschl	NOUN
ap-9037	22	22	-	-	PUNCT
ap-9037	22	23	teller	teller	NOUN
ap-9037	22	24	potential	potential	NOUN
ap-9037	22	25	[	[	X
ap-9037	22	26	19	19	NUM
ap-9037	22	27	]	]	PUNCT
ap-9037	22	28	.	.	PUNCT
ap-9037	23	1	therefore	therefore	ADV
ap-9037	23	2	,	,	PUNCT
ap-9037	23	3	existing	exist	VERB
ap-9037	23	4	results	result	NOUN
ap-9037	23	5	and	and	CCONJ
ap-9037	23	6	methods	method	NOUN
ap-9037	23	7	related	relate	VERB
ap-9037	23	8	to	to	ADP
ap-9037	23	9	such	such	ADJ
ap-9037	23	10	potentials	potential	NOUN
ap-9037	23	11	can	can	AUX
ap-9037	23	12	be	be	AUX
ap-9037	23	13	used	use	VERB
ap-9037	23	14	directly	directly	ADV
ap-9037	23	15	for	for	ADP
ap-9037	23	16	constructing	construct	VERB
ap-9037	23	17	new	new	ADJ
ap-9037	23	18	solvable	solvable	ADJ
ap-9037	23	19	cases	case	NOUN
ap-9037	23	20	of	of	ADP
ap-9037	23	21	both	both	CCONJ
ap-9037	23	22	angular	angular	ADJ
ap-9037	23	23	equations	equation	NOUN
ap-9037	23	24	,	,	PUNCT
ap-9037	23	25	leading	lead	VERB
ap-9037	23	26	to	to	ADP
ap-9037	23	27	an	an	DET
ap-9037	23	28	overall	overall	ADJ
ap-9037	23	29	three	three	NUM
ap-9037	23	30	-	-	PUNCT
ap-9037	23	31	dimensional	dimensional	ADJ
ap-9037	23	32	potential	potential	NOUN
ap-9037	23	33	.	.	PUNCT
ap-9037	24	1	while	while	SCONJ
ap-9037	24	2	solutions	solution	NOUN
ap-9037	24	3	for	for	ADP
ap-9037	24	4	a	a	DET
ap-9037	24	5	particular	particular	ADJ
ap-9037	24	6	class	class	NOUN
ap-9037	24	7	of	of	ADP
ap-9037	24	8	potentials	potential	NOUN
ap-9037	24	9	were	be	AUX
ap-9037	24	10	obtained	obtain	VERB
ap-9037	24	11	previously	previously	ADV
ap-9037	24	12	[	[	X
ap-9037	24	13	20	20	NUM
ap-9037	24	14	]	]	X
ap-9037	24	15	,	,	PUNCT
ap-9037	24	16	here	here	ADV
ap-9037	24	17	,	,	PUNCT
ap-9037	24	18	our	our	PRON
ap-9037	24	19	focus	focus	NOUN
ap-9037	24	20	is	be	AUX
ap-9037	24	21	more	more	ADV
ap-9037	24	22	general	general	ADJ
ap-9037	24	23	.	.	PUNCT
ap-9037	25	1	the	the	DET
ap-9037	25	2	purpose	purpose	NOUN
ap-9037	25	3	of	of	ADP
ap-9037	25	4	the	the	DET
ap-9037	25	5	present	present	ADJ
ap-9037	25	6	work	work	NOUN
ap-9037	25	7	is	be	AUX
ap-9037	25	8	to	to	PART
ap-9037	25	9	construct	construct	VERB
ap-9037	25	10	new	new	ADJ
ap-9037	25	11	solutions	solution	NOUN
ap-9037	25	12	to	to	ADP
ap-9037	25	13	the	the	DET
ap-9037	25	14	angular	angular	ADJ
ap-9037	25	15	equations	equation	NOUN
ap-9037	25	16	by	by	ADP
ap-9037	25	17	incorporating	incorporate	VERB
ap-9037	25	18	existing	exist	VERB
ap-9037	25	19	results	result	NOUN
ap-9037	25	20	,	,	PUNCT
ap-9037	25	21	and	and	CCONJ
ap-9037	25	22	by	by	ADP
ap-9037	25	23	adapting	adapt	VERB
ap-9037	25	24	and	and	CCONJ
ap-9037	25	25	applying	apply	VERB
ap-9037	25	26	the	the	DET
ap-9037	25	27	darboux	darboux	ADJ
ap-9037	25	28	-	-	PUNCT
ap-9037	25	29	crum	crum	NOUN
ap-9037	25	30	transformation	transformation	NOUN
ap-9037	25	31	to	to	ADP
ap-9037	25	32	our	our	PRON
ap-9037	25	33	equations	equation	NOUN
ap-9037	25	34	.	.	PUNCT
ap-9037	26	1	the	the	DET
ap-9037	26	2	article	article	NOUN
ap-9037	26	3	is	be	AUX
ap-9037	26	4	organized	organize	VERB
ap-9037	26	5	as	as	SCONJ
ap-9037	26	6	follows	follow	VERB
ap-9037	26	7	.	.	PUNCT
ap-9037	27	1	in	in	ADP
ap-9037	27	2	section	section	NOUN
ap-9037	27	3	2	2	NUM
ap-9037	27	4	,	,	PUNCT
ap-9037	27	5	we	we	PRON
ap-9037	27	6	state	state	VERB
ap-9037	27	7	the	the	DET
ap-9037	27	8	known	know	VERB
ap-9037	27	9	general	general	ADJ
ap-9037	27	10	solution	solution	NOUN
ap-9037	27	11	of	of	ADP
ap-9037	27	12	the	the	DET
ap-9037	27	13	schrödinger	schrödinger	ADJ
ap-9037	27	14	equation	equation	NOUN
ap-9037	27	15	with	with	ADP
ap-9037	27	16	trigonometric	trigonometric	ADJ
ap-9037	27	17	pöschl	pöschl	NOUN
ap-9037	27	18	-	-	PUNCT
ap-9037	27	19	teller	teller	NOUN
ap-9037	27	20	potential	potential	NOUN
ap-9037	27	21	.	.	PUNCT
ap-9037	28	1	since	since	SCONJ
ap-9037	28	2	this	this	DET
ap-9037	28	3	equation	equation	NOUN
ap-9037	28	4	have	have	VERB
ap-9037	28	5	the	the	DET
ap-9037	28	6	same	same	ADJ
ap-9037	28	7	form	form	NOUN
ap-9037	28	8	as	as	ADP
ap-9037	28	9	the	the	DET
ap-9037	28	10	angular	angular	ADJ
ap-9037	28	11	equations	equation	NOUN
ap-9037	28	12	resulting	result	VERB
ap-9037	28	13	from	from	ADP
ap-9037	28	14	the	the	DET
ap-9037	28	15	variable	variable	ADJ
ap-9037	28	16	separation	separation	NOUN
ap-9037	28	17	in	in	ADP
ap-9037	28	18	the	the	DET
ap-9037	28	19	three	three	NUM
ap-9037	28	20	-	-	PUNCT
ap-9037	28	21	dimensional	dimensional	ADJ
ap-9037	28	22	scenario	scenario	NOUN
ap-9037	28	23	,	,	PUNCT
ap-9037	28	24	the	the	DET
ap-9037	28	25	solution	solution	NOUN
ap-9037	28	26	presented	present	VERB
ap-9037	28	27	in	in	ADP
ap-9037	28	28	section	section	NOUN
ap-9037	28	29	2	2	NUM
ap-9037	28	30	can	can	AUX
ap-9037	28	31	be	be	AUX
ap-9037	28	32	adapted	adapt	VERB
ap-9037	28	33	to	to	ADP
ap-9037	28	34	those	those	DET
ap-9037	28	35	angular	angular	ADJ
ap-9037	28	36	equations	equation	NOUN
ap-9037	28	37	in	in	ADP
ap-9037	28	38	a	a	DET
ap-9037	28	39	straightforward	straightforward	ADJ
ap-9037	28	40	way	way	NOUN
ap-9037	28	41	.	.	PUNCT
ap-9037	29	1	besides	besides	SCONJ
ap-9037	29	2	this	this	DET
ap-9037	29	3	topic	topic	NOUN
ap-9037	29	4	,	,	PUNCT
ap-9037	29	5	section	section	NOUN
ap-9037	29	6	2	2	NUM
ap-9037	29	7	also	also	ADV
ap-9037	29	8	briefly	briefly	ADV
ap-9037	29	9	reviews	review	VERB
ap-9037	29	10	the	the	DET
ap-9037	29	11	darboux	darboux	ADJ
ap-9037	29	12	-	-	PUNCT
ap-9037	29	13	crum	crum	NOUN
ap-9037	29	14	transformation	transformation	NOUN
ap-9037	29	15	that	that	PRON
ap-9037	29	16	will	will	AUX
ap-9037	29	17	be	be	AUX
ap-9037	29	18	applied	apply	VERB
ap-9037	29	19	to	to	ADP
ap-9037	29	20	the	the	DET
ap-9037	29	21	above	above	ADJ
ap-9037	29	22	mentioned	mention	VERB
ap-9037	29	23	angular	angular	ADJ
ap-9037	29	24	equations	equation	NOUN
ap-9037	29	25	.	.	PUNCT
ap-9037	30	1	in	in	ADP
ap-9037	30	2	section	section	NOUN
ap-9037	30	3	3	3	NUM
ap-9037	30	4	,	,	PUNCT
ap-9037	30	5	we	we	PRON
ap-9037	30	6	begin	begin	VERB
ap-9037	30	7	to	to	PART
ap-9037	30	8	develop	develop	VERB
ap-9037	30	9	the	the	DET
ap-9037	30	10	method	method	NOUN
ap-9037	30	11	used	use	VERB
ap-9037	30	12	in	in	ADP
ap-9037	30	13	this	this	DET
ap-9037	30	14	work	work	NOUN
ap-9037	30	15	by	by	ADP
ap-9037	30	16	defining	define	VERB
ap-9037	30	17	the	the	DET
ap-9037	30	18	hamiltonian	hamiltonian	NOUN
ap-9037	30	19	within	within	ADP
ap-9037	30	20	the	the	DET
ap-9037	30	21	dunkl	dunkl	NOUN
ap-9037	30	22	formalism	formalism	NOUN
ap-9037	30	23	and	and	CCONJ
ap-9037	30	24	by	by	ADP
ap-9037	30	25	generating	generate	VERB
ap-9037	30	26	the	the	DET
ap-9037	30	27	associated	associated	ADJ
ap-9037	30	28	schrödinger	schrödinger	ADJ
ap-9037	30	29	equation	equation	NOUN
ap-9037	30	30	in	in	ADP
ap-9037	30	31	spherical	spherical	ADJ
ap-9037	30	32	coordinates	coordinate	NOUN
ap-9037	30	33	.	.	PUNCT
ap-9037	31	1	the	the	DET
ap-9037	31	2	separation	separation	NOUN
ap-9037	31	3	of	of	ADP
ap-9037	31	4	variables	variable	NOUN
ap-9037	31	5	then	then	ADV
ap-9037	31	6	results	result	VERB
ap-9037	31	7	,	,	PUNCT
ap-9037	31	8	as	as	ADP
ap-9037	31	9	usual	usual	ADJ
ap-9037	31	10	,	,	PUNCT
ap-9037	31	11	in	in	ADP
ap-9037	31	12	a	a	DET
ap-9037	31	13	radial	radial	ADJ
ap-9037	31	14	equation	equation	NOUN
ap-9037	31	15	and	and	CCONJ
ap-9037	31	16	two	two	NUM
ap-9037	31	17	angular	angular	ADJ
ap-9037	31	18	equations	equation	NOUN
ap-9037	31	19	.	.	PUNCT
ap-9037	32	1	in	in	ADP
ap-9037	32	2	section	section	NOUN
ap-9037	32	3	4	4	NUM
ap-9037	32	4	,	,	PUNCT
ap-9037	32	5	one	one	NUM
ap-9037	32	6	of	of	ADP
ap-9037	32	7	these	these	DET
ap-9037	32	8	angular	angular	ADJ
ap-9037	32	9	equations	equation	NOUN
ap-9037	32	10	(	(	PUNCT
ap-9037	32	11	the	the	DET
ap-9037	32	12	polar	polar	ADJ
ap-9037	32	13	equation	equation	NOUN
ap-9037	32	14	)	)	PUNCT
ap-9037	32	15	is	be	AUX
ap-9037	32	16	shown	show	VERB
ap-9037	32	17	to	to	PART
ap-9037	32	18	match	match	VERB
ap-9037	32	19	the	the	DET
ap-9037	32	20	form	form	NOUN
ap-9037	32	21	of	of	ADP
ap-9037	32	22	the	the	DET
ap-9037	32	23	equation	equation	NOUN
ap-9037	32	24	that	that	PRON
ap-9037	32	25	was	be	AUX
ap-9037	32	26	reviewed	review	VERB
ap-9037	32	27	in	in	ADP
ap-9037	32	28	section	section	NOUN
ap-9037	32	29	2	2	NUM
ap-9037	32	30	.	.	PUNCT
ap-9037	32	31	by	by	ADP
ap-9037	32	32	means	mean	NOUN
ap-9037	32	33	of	of	ADP
ap-9037	32	34	this	this	DET
ap-9037	32	35	interrelation	interrelation	NOUN
ap-9037	32	36	,	,	PUNCT
ap-9037	32	37	solutions	solution	NOUN
ap-9037	32	38	of	of	ADP
ap-9037	32	39	the	the	DET
ap-9037	32	40	polar	polar	ADJ
ap-9037	32	41	equation	equation	NOUN
ap-9037	32	42	is	be	AUX
ap-9037	32	43	constructed	construct	VERB
ap-9037	32	44	.	.	PUNCT
ap-9037	33	1	these	these	DET
ap-9037	33	2	solutions	solution	NOUN
ap-9037	33	3	are	be	AUX
ap-9037	33	4	subsequently	subsequently	ADV
ap-9037	33	5	generalised	generalise	VERB
ap-9037	33	6	by	by	ADP
ap-9037	33	7	means	mean	NOUN
ap-9037	33	8	of	of	ADP
ap-9037	33	9	darboux	darboux	VERB
ap-9037	33	10	-	-	PUNCT
ap-9037	33	11	crum	crum	NOUN
ap-9037	33	12	transformations	transformation	NOUN
ap-9037	33	13	,	,	PUNCT
ap-9037	33	14	as	as	SCONJ
ap-9037	33	15	introduced	introduce	VERB
ap-9037	33	16	in	in	ADP
ap-9037	33	17	section	section	NOUN
ap-9037	33	18	2	2	NUM
ap-9037	33	19	.	.	PUNCT
ap-9037	33	20	section	section	NOUN
ap-9037	33	21	5	5	NUM
ap-9037	33	22	essentially	essentially	ADV
ap-9037	33	23	repeats	repeat	VERB
ap-9037	33	24	the	the	DET
ap-9037	33	25	process	process	NOUN
ap-9037	33	26	from	from	ADP
ap-9037	33	27	section	section	NOUN
ap-9037	33	28	4	4	NUM
ap-9037	33	29	,	,	PUNCT
ap-9037	33	30	but	but	CCONJ
ap-9037	33	31	this	this	DET
ap-9037	33	32	time	time	NOUN
ap-9037	33	33	applied	apply	VERB
ap-9037	33	34	to	to	ADP
ap-9037	33	35	the	the	DET
ap-9037	33	36	remaining	remain	VERB
ap-9037	33	37	angular	angular	ADJ
ap-9037	33	38	equation	equation	NOUN
ap-9037	33	39	(	(	PUNCT
ap-9037	33	40	the	the	DET
ap-9037	33	41	azimuthal	azimuthal	ADJ
ap-9037	33	42	equation	equation	NOUN
ap-9037	33	43	)	)	PUNCT
ap-9037	33	44	.	.	PUNCT
ap-9037	34	1	273	273	NUM
ap-9037	34	2	https://doi.org/10.14311/ap.2023.63.0273	https://doi.org/10.14311/ap.2023.63.0273	PROPN
ap-9037	34	3	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
ap-9037	34	4	https://www.cvut.cz/en	https://www.cvut.cz/en	PROPN
ap-9037	34	5	axel	axel	VERB
ap-9037	34	6	schulze	schulze	NOUN
ap-9037	34	7	-	-	PUNCT
ap-9037	34	8	halberg	halberg	PROPN
ap-9037	34	9	acta	acta	PROPN
ap-9037	34	10	polytechnica	polytechnica	PROPN
ap-9037	34	11	2	2	NUM
ap-9037	34	12	.	.	PUNCT
ap-9037	34	13	preliminaries	preliminary	NOUN
ap-9037	34	14	in	in	ADP
ap-9037	34	15	order	order	NOUN
ap-9037	34	16	to	to	PART
ap-9037	34	17	make	make	VERB
ap-9037	34	18	this	this	DET
ap-9037	34	19	article	article	NOUN
ap-9037	34	20	self	self	NOUN
ap-9037	34	21	-	-	PUNCT
ap-9037	34	22	contained	contain	VERB
ap-9037	34	23	,	,	PUNCT
ap-9037	34	24	we	we	PRON
ap-9037	34	25	will	will	AUX
ap-9037	34	26	now	now	ADV
ap-9037	34	27	summarise	summarise	VERB
ap-9037	34	28	the	the	DET
ap-9037	34	29	results	result	NOUN
ap-9037	34	30	on	on	ADP
ap-9037	34	31	the	the	DET
ap-9037	34	32	trigonometric	trigonometric	ADJ
ap-9037	34	33	version	version	NOUN
ap-9037	34	34	of	of	ADP
ap-9037	34	35	the	the	DET
ap-9037	34	36	pöschl	pöschl	NOUN
ap-9037	34	37	-	-	PUNCT
ap-9037	34	38	teller	teller	NOUN
ap-9037	34	39	potential	potential	NOUN
ap-9037	34	40	,	,	PUNCT
ap-9037	34	41	and	and	CCONJ
ap-9037	34	42	on	on	ADP
ap-9037	34	43	the	the	DET
ap-9037	34	44	closed	closed	ADJ
ap-9037	34	45	-	-	PUNCT
ap-9037	34	46	form	form	NOUN
ap-9037	34	47	solution	solution	NOUN
ap-9037	34	48	of	of	ADP
ap-9037	34	49	the	the	DET
ap-9037	34	50	associated	associated	ADJ
ap-9037	34	51	schrödinger	schrödinger	ADJ
ap-9037	34	52	equation	equation	NOUN
ap-9037	34	53	.	.	PUNCT
ap-9037	35	1	furthermore	furthermore	ADV
ap-9037	35	2	,	,	PUNCT
ap-9037	35	3	we	we	PRON
ap-9037	35	4	briefly	briefly	ADV
ap-9037	35	5	review	review	VERB
ap-9037	35	6	the	the	DET
ap-9037	35	7	two	two	NUM
ap-9037	35	8	algorithms	algorithm	NOUN
ap-9037	35	9	of	of	ADP
ap-9037	35	10	the	the	DET
ap-9037	35	11	darboux	darboux	ADJ
ap-9037	35	12	-	-	PUNCT
ap-9037	35	13	crum	crum	NOUN
ap-9037	35	14	transformation	transformation	NOUN
ap-9037	35	15	.	.	PUNCT
ap-9037	36	1	details	detail	NOUN
ap-9037	36	2	on	on	ADP
ap-9037	36	3	these	these	DET
ap-9037	36	4	topics	topic	NOUN
ap-9037	36	5	can	can	AUX
ap-9037	36	6	be	be	AUX
ap-9037	36	7	found	find	VERB
ap-9037	36	8	in	in	ADP
ap-9037	36	9	[	[	X
ap-9037	36	10	19	19	NUM
ap-9037	36	11	,	,	PUNCT
ap-9037	36	12	21	21	NUM
ap-9037	36	13	]	]	PUNCT
ap-9037	36	14	,	,	PUNCT
ap-9037	36	15	and	and	CCONJ
ap-9037	36	16	references	reference	NOUN
ap-9037	36	17	therein	therein	ADV
ap-9037	36	18	.	.	PUNCT
ap-9037	37	1	2.1	2.1	NUM
ap-9037	37	2	.	.	PUNCT
ap-9037	38	1	the	the	DET
ap-9037	38	2	trigonometric	trigonometric	ADJ
ap-9037	38	3	pöschl	pöschl	NOUN
ap-9037	38	4	-	-	PUNCT
ap-9037	38	5	teller	teller	NOUN
ap-9037	38	6	system	system	NOUN
ap-9037	38	7	the	the	DET
ap-9037	38	8	stationary	stationary	ADJ
ap-9037	38	9	schrödinger	schrödinger	ADJ
ap-9037	38	10	equation	equation	NOUN
ap-9037	38	11	for	for	ADP
ap-9037	38	12	the	the	DET
ap-9037	38	13	trigonometric	trigonometric	ADJ
ap-9037	38	14	pöschl	pöschl	NOUN
ap-9037	38	15	-	-	PUNCT
ap-9037	38	16	teller	teller	NOUN
ap-9037	38	17	potential	potential	NOUN
ap-9037	38	18	can	can	AUX
ap-9037	38	19	be	be	AUX
ap-9037	38	20	written	write	VERB
ap-9037	38	21	in	in	ADP
ap-9037	38	22	the	the	DET
ap-9037	38	23	form	form	NOUN
ap-9037	38	24	:	:	PUNCT
ap-9037	38	25	ψ′′(x	ψ′′(x	NOUN
ap-9037	38	26	)	)	PUNCT
ap-9037	39	1	+	+	CCONJ
ap-9037	39	2	[	[	PUNCT
ap-9037	39	3	a	a	DET
ap-9037	39	4	+	+	NOUN
ap-9037	39	5	b	b	NOUN
ap-9037	39	6	sin(x)2	sin(x)2	NOUN
ap-9037	39	7	+	+	CCONJ
ap-9037	39	8	c	c	NOUN
ap-9037	39	9	cos(x)2	cos(x)2	NOUN
ap-9037	39	10	]	]	PUNCT
ap-9037	39	11	ψ(x	ψ(x	NOUN
ap-9037	39	12	)	)	PUNCT
ap-9037	40	1	=	=	SYM
ap-9037	40	2	0	0	NUM
ap-9037	40	3	,	,	PUNCT
ap-9037	40	4	(	(	PUNCT
ap-9037	40	5	1	1	X
ap-9037	40	6	)	)	PUNCT
ap-9037	40	7	introducing	introduce	VERB
ap-9037	40	8	real	real	ADV
ap-9037	40	9	-	-	PUNCT
ap-9037	40	10	valued	value	VERB
ap-9037	40	11	constants	constant	NOUN
ap-9037	40	12	a	a	DET
ap-9037	40	13	,	,	PUNCT
ap-9037	40	14	b	b	NOUN
ap-9037	40	15	,	,	PUNCT
ap-9037	40	16	and	and	CCONJ
ap-9037	40	17	c.	c.	PROPN
ap-9037	40	18	the	the	DET
ap-9037	40	19	general	general	ADJ
ap-9037	40	20	solution	solution	NOUN
ap-9037	40	21	of	of	ADP
ap-9037	40	22	(	(	PUNCT
ap-9037	40	23	1	1	NUM
ap-9037	40	24	)	)	PUNCT
ap-9037	40	25	is	be	AUX
ap-9037	40	26	given	give	VERB
ap-9037	40	27	by	by	ADP
ap-9037	40	28	:	:	PUNCT
ap-9037	40	29	ψ(x	ψ(x	NUM
ap-9037	40	30	)	)	PUNCT
ap-9037	40	31	=	=	PROPN
ap-9037	40	32	c1	c1	PROPN
ap-9037	40	33	sin(x	sin(x	PROPN
ap-9037	40	34	)	)	PUNCT
ap-9037	40	35	1	1	NUM
ap-9037	40	36	2	2	NUM
ap-9037	40	37	−	−	NOUN
ap-9037	40	38	1	1	NUM
ap-9037	40	39	2	2	NUM
ap-9037	40	40	√	√	NUM
ap-9037	40	41	1−4b	1−4b	NUM
ap-9037	40	42	cos(x	cos(x	PROPN
ap-9037	40	43	)	)	PUNCT
ap-9037	40	44	1	1	NUM
ap-9037	40	45	2	2	NUM
ap-9037	40	46	+	+	CCONJ
ap-9037	40	47	1	1	NUM
ap-9037	40	48	2	2	NUM
ap-9037	40	49	√	√	NUM
ap-9037	40	50	1−4c	1−4c	NUM
ap-9037	40	51	×	×	NOUN
ap-9037	40	52	2f1	2f1	NUM
ap-9037	40	53	[	[	PUNCT
ap-9037	40	54	1	1	NUM
ap-9037	40	55	2	2	NUM
ap-9037	40	56	−	−	NOUN
ap-9037	40	57	√	√	NUM
ap-9037	40	58	a	a	DET
ap-9037	40	59	2	2	NUM
ap-9037	40	60	+	+	NUM
ap-9037	40	61	k−	k−	PROPN
ap-9037	40	62	,	,	PUNCT
ap-9037	40	63	1	1	NUM
ap-9037	40	64	2	2	NUM
ap-9037	40	65	+	+	CCONJ
ap-9037	40	66	√	√	DET
ap-9037	40	67	a	a	DET
ap-9037	40	68	2	2	NUM
ap-9037	40	69	+	+	NUM
ap-9037	40	70	k−	k−	PROPN
ap-9037	40	71	,	,	PUNCT
ap-9037	40	72	1	1	NUM
ap-9037	40	73	−	−	NOUN
ap-9037	40	74	√	√	NUM
ap-9037	40	75	1	1	NUM
ap-9037	40	76	−	−	PROPN
ap-9037	40	77	4b	4b	X
ap-9037	40	78	2	2	NUM
ap-9037	40	79	,	,	PUNCT
ap-9037	40	80	sin(x)2	sin(x)2	VERB
ap-9037	40	81	]	]	PUNCT
ap-9037	41	1	+	+	CCONJ
ap-9037	41	2	c2	c2	PROPN
ap-9037	41	3	sin(x	sin(x	PROPN
ap-9037	41	4	)	)	PUNCT
ap-9037	41	5	1	1	NUM
ap-9037	41	6	2	2	NUM
ap-9037	41	7	+	+	CCONJ
ap-9037	41	8	1	1	NUM
ap-9037	41	9	2	2	NUM
ap-9037	41	10	√	√	NUM
ap-9037	41	11	1−4b	1−4b	NUM
ap-9037	41	12	cos(x	cos(x	PROPN
ap-9037	41	13	)	)	PUNCT
ap-9037	41	14	1	1	NUM
ap-9037	41	15	2	2	NUM
ap-9037	41	16	+	+	CCONJ
ap-9037	41	17	1	1	NUM
ap-9037	41	18	2	2	NUM
ap-9037	41	19	√	√	NUM
ap-9037	41	20	1−4c	1−4c	NUM
ap-9037	41	21	(	(	PUNCT
ap-9037	41	22	2	2	NUM
ap-9037	41	23	)	)	PUNCT
ap-9037	41	24	×	×	NOUN
ap-9037	41	25	2f1	2f1	NUM
ap-9037	41	26	[	[	PUNCT
ap-9037	41	27	1	1	NUM
ap-9037	41	28	2	2	NUM
ap-9037	41	29	−	−	NOUN
ap-9037	41	30	√	√	NUM
ap-9037	41	31	a	a	DET
ap-9037	41	32	2	2	NUM
ap-9037	41	33	+	+	CCONJ
ap-9037	41	34	k+	k+	NOUN
ap-9037	41	35	,	,	PUNCT
ap-9037	41	36	1	1	NUM
ap-9037	41	37	2	2	NUM
ap-9037	41	38	+	+	CCONJ
ap-9037	41	39	√	√	DET
ap-9037	41	40	a	a	DET
ap-9037	41	41	2	2	NUM
ap-9037	41	42	+	+	CCONJ
ap-9037	41	43	k+	k+	NOUN
ap-9037	41	44	,	,	PUNCT
ap-9037	41	45	1	1	NUM
ap-9037	41	46	+	+	CCONJ
ap-9037	41	47	√	√	NUM
ap-9037	41	48	1	1	NUM
ap-9037	41	49	−	−	PROPN
ap-9037	41	50	4b	4b	X
ap-9037	41	51	2	2	NUM
ap-9037	41	52	,	,	PUNCT
ap-9037	41	53	sin(x)2	sin(x)2	VERB
ap-9037	41	54	]	]	PUNCT
ap-9037	41	55	,	,	PUNCT
ap-9037	41	56	where	where	SCONJ
ap-9037	41	57	c1	c1	PROPN
ap-9037	41	58	,	,	PUNCT
ap-9037	41	59	c2	c2	PROPN
ap-9037	41	60	denote	denote	VERB
ap-9037	41	61	the	the	DET
ap-9037	41	62	arbitrary	arbitrary	ADJ
ap-9037	41	63	linear	linear	ADJ
ap-9037	41	64	factors	factor	NOUN
ap-9037	41	65	,	,	PUNCT
ap-9037	41	66	2f1	2f1	PRON
ap-9037	41	67	represents	represent	VERB
ap-9037	41	68	the	the	DET
ap-9037	41	69	hypergeometric	hypergeometric	ADJ
ap-9037	41	70	function	function	NOUN
ap-9037	41	71	[	[	X
ap-9037	41	72	22	22	NUM
ap-9037	41	73	]	]	PUNCT
ap-9037	41	74	,	,	PUNCT
ap-9037	41	75	and	and	CCONJ
ap-9037	41	76	the	the	DET
ap-9037	41	77	abbreviation	abbreviation	NOUN
ap-9037	41	78	k	k	PROPN
ap-9037	41	79	that	that	PRON
ap-9037	41	80	stands	stand	VERB
ap-9037	41	81	for	for	ADP
ap-9037	41	82	:	:	PUNCT
ap-9037	41	83	k±	k±	PROPN
ap-9037	41	84	=	=	PUNCT
ap-9037	41	85	±	±	NOUN
ap-9037	41	86	√	√	NOUN
ap-9037	41	87	1	1	NUM
ap-9037	41	88	−	−	PROPN
ap-9037	41	89	4b	4b	NOUN
ap-9037	41	90	+	+	CCONJ
ap-9037	41	91	√	√	NUM
ap-9037	41	92	1	1	NUM
ap-9037	41	93	−	−	PROPN
ap-9037	41	94	4c	4c	NOUN
ap-9037	41	95	4	4	NUM
ap-9037	41	96	.	.	PUNCT
ap-9037	42	1	(	(	PUNCT
ap-9037	42	2	3	3	X
ap-9037	42	3	)	)	PUNCT
ap-9037	42	4	note	note	NOUN
ap-9037	42	5	that	that	SCONJ
ap-9037	42	6	the	the	DET
ap-9037	42	7	parameters	parameter	NOUN
ap-9037	42	8	entering	enter	VERB
ap-9037	42	9	in	in	ADP
ap-9037	42	10	the	the	DET
ap-9037	42	11	general	general	ADJ
ap-9037	42	12	solution	solution	NOUN
ap-9037	42	13	(	(	PUNCT
ap-9037	42	14	2	2	X
ap-9037	42	15	)	)	PUNCT
ap-9037	42	16	are	be	AUX
ap-9037	42	17	subject	subject	ADJ
ap-9037	42	18	to	to	ADP
ap-9037	42	19	restrictions	restriction	NOUN
ap-9037	42	20	,	,	PUNCT
ap-9037	42	21	depending	depend	VERB
ap-9037	42	22	on	on	ADP
ap-9037	42	23	the	the	DET
ap-9037	42	24	domain	domain	NOUN
ap-9037	42	25	of	of	ADP
ap-9037	42	26	equation	equation	NOUN
ap-9037	42	27	(	(	PUNCT
ap-9037	42	28	1	1	NUM
ap-9037	42	29	)	)	PUNCT
ap-9037	42	30	,	,	PUNCT
ap-9037	42	31	and	and	CCONJ
ap-9037	42	32	on	on	ADP
ap-9037	42	33	the	the	DET
ap-9037	42	34	boundary	boundary	ADJ
ap-9037	42	35	conditions	condition	NOUN
ap-9037	42	36	imposed	impose	VERB
ap-9037	42	37	on	on	ADP
ap-9037	42	38	it	it	PRON
ap-9037	42	39	.	.	PUNCT
ap-9037	43	1	2.2	2.2	NUM
ap-9037	43	2	.	.	PUNCT
ap-9037	44	1	the	the	DET
ap-9037	44	2	darboux	darboux	VERB
ap-9037	44	3	-	-	PUNCT
ap-9037	44	4	crum	crum	NOUN
ap-9037	44	5	transformation	transformation	NOUN
ap-9037	44	6	the	the	DET
ap-9037	44	7	purpose	purpose	NOUN
ap-9037	44	8	of	of	ADP
ap-9037	44	9	the	the	DET
ap-9037	44	10	darboux	darboux	ADJ
ap-9037	44	11	-	-	PUNCT
ap-9037	44	12	crum	crum	NOUN
ap-9037	44	13	transformation	transformation	NOUN
ap-9037	44	14	is	be	AUX
ap-9037	44	15	to	to	PART
ap-9037	44	16	provide	provide	VERB
ap-9037	44	17	an	an	DET
ap-9037	44	18	interrelation	interrelation	NOUN
ap-9037	44	19	between	between	ADP
ap-9037	44	20	the	the	DET
ap-9037	44	21	solutions	solution	NOUN
ap-9037	44	22	of	of	ADP
ap-9037	44	23	the	the	DET
ap-9037	44	24	following	follow	VERB
ap-9037	44	25	partner	partner	NOUN
ap-9037	44	26	equations	equation	NOUN
ap-9037	44	27	:	:	PUNCT
ap-9037	44	28	ψ′′(x	ψ′′(x	NOUN
ap-9037	44	29	)	)	PUNCT
ap-9037	45	1	+	+	CCONJ
ap-9037	45	2	[	[	PUNCT
ap-9037	45	3	e	e	NOUN
ap-9037	45	4	−	−	PROPN
ap-9037	45	5	u(x	u(x	PROPN
ap-9037	45	6	)	)	PUNCT
ap-9037	45	7	]	]	PUNCT
ap-9037	46	1	ψ(x	ψ(x	X
ap-9037	46	2	)	)	PUNCT
ap-9037	46	3	=	=	SYM
ap-9037	46	4	0	0	NUM
ap-9037	46	5	,	,	PUNCT
ap-9037	46	6	(	(	PUNCT
ap-9037	46	7	4	4	X
ap-9037	46	8	)	)	PUNCT
ap-9037	46	9	ψ̂′′(x	ψ̂′′(x	NOUN
ap-9037	46	10	)	)	PUNCT
ap-9037	47	1	+	+	CCONJ
ap-9037	47	2	[	[	PUNCT
ap-9037	47	3	e	e	X
ap-9037	47	4	−	−	NOUN
ap-9037	47	5	û(x	û(x	ADP
ap-9037	47	6	)	)	PUNCT
ap-9037	47	7	]	]	PUNCT
ap-9037	47	8	ψ̂(x	ψ̂(x	X
ap-9037	47	9	)	)	PUNCT
ap-9037	47	10	=	=	SYM
ap-9037	47	11	0	0	X
ap-9037	47	12	.	.	PUNCT
ap-9037	48	1	(	(	PUNCT
ap-9037	48	2	5	5	NUM
ap-9037	48	3	)	)	PUNCT
ap-9037	48	4	here	here	ADV
ap-9037	48	5	,	,	PUNCT
ap-9037	48	6	e	e	PROPN
ap-9037	48	7	denotes	denote	VERB
ap-9037	48	8	the	the	DET
ap-9037	48	9	stationary	stationary	ADJ
ap-9037	48	10	energy	energy	NOUN
ap-9037	48	11	,	,	PUNCT
ap-9037	48	12	and	and	CCONJ
ap-9037	48	13	u	u	NOUN
ap-9037	48	14	,	,	PUNCT
ap-9037	48	15	û	û	NUM
ap-9037	48	16	stand	stand	VERB
ap-9037	48	17	for	for	ADP
ap-9037	48	18	the	the	DET
ap-9037	48	19	respective	respective	ADJ
ap-9037	48	20	potentials	potential	NOUN
ap-9037	48	21	.	.	PUNCT
ap-9037	49	1	these	these	DET
ap-9037	49	2	potentials	potential	NOUN
ap-9037	49	3	,	,	PUNCT
ap-9037	49	4	as	as	ADV
ap-9037	49	5	well	well	ADV
ap-9037	49	6	as	as	ADP
ap-9037	49	7	the	the	DET
ap-9037	49	8	corresponding	correspond	VERB
ap-9037	49	9	solutions	solution	NOUN
ap-9037	49	10	ψ	ψ	NOUN
ap-9037	49	11	and	and	CCONJ
ap-9037	49	12	ψ̂	ψ̂	PROPN
ap-9037	49	13	of	of	ADP
ap-9037	49	14	the	the	DET
ap-9037	49	15	two	two	NUM
ap-9037	49	16	equations	equation	NOUN
ap-9037	49	17	,	,	PUNCT
ap-9037	49	18	can	can	AUX
ap-9037	49	19	be	be	AUX
ap-9037	49	20	linked	link	VERB
ap-9037	49	21	by	by	ADP
ap-9037	49	22	the	the	DET
ap-9037	49	23	darboux	darboux	VERB
ap-9037	49	24	-	-	PUNCT
ap-9037	49	25	crum	crum	NOUN
ap-9037	49	26	transformation	transformation	NOUN
ap-9037	49	27	,	,	PUNCT
ap-9037	49	28	where	where	SCONJ
ap-9037	49	29	we	we	PRON
ap-9037	49	30	need	need	VERB
ap-9037	49	31	to	to	PART
ap-9037	49	32	distinguish	distinguish	VERB
ap-9037	49	33	two	two	NUM
ap-9037	49	34	algorithms	algorithm	NOUN
ap-9037	49	35	.	.	PUNCT
ap-9037	50	1	•	•	NOUN
ap-9037	50	2	standard	standard	ADJ
ap-9037	50	3	algorithm	algorithm	NOUN
ap-9037	50	4	:	:	PUNCT
ap-9037	50	5	let	let	VERB
ap-9037	50	6	the	the	DET
ap-9037	50	7	transformation	transformation	NOUN
ap-9037	50	8	functions	function	NOUN
ap-9037	50	9	ψ1	ψ1	NOUN
ap-9037	50	10	,	,	PUNCT
ap-9037	50	11	.	.	PUNCT
ap-9037	50	12	.	.	PUNCT
ap-9037	51	1	.	.	PUNCT
ap-9037	52	1	,	,	PUNCT
ap-9037	52	2	ψn	ψn	VERB
ap-9037	52	3	be	be	AUX
ap-9037	52	4	solutions	solution	NOUN
ap-9037	52	5	of	of	ADP
ap-9037	52	6	:	:	PUNCT
ap-9037	52	7	ψ′′	ψ′′	PROPN
ap-9037	52	8	j	j	PROPN
ap-9037	52	9	(	(	PUNCT
ap-9037	52	10	x	x	X
ap-9037	52	11	)	)	PUNCT
ap-9037	53	1	+	+	CCONJ
ap-9037	53	2	[	[	PUNCT
ap-9037	53	3	ej	ej	PRON
ap-9037	53	4	−	−	PROPN
ap-9037	53	5	u(x	u(x	PROPN
ap-9037	53	6	)	)	PUNCT
ap-9037	53	7	]	]	PUNCT
ap-9037	53	8	ψj(x	ψj(x	PUNCT
ap-9037	53	9	)	)	PUNCT
ap-9037	53	10	=	=	SYM
ap-9037	54	1	0	0	NUM
ap-9037	54	2	,	,	PUNCT
ap-9037	54	3	j	j	PROPN
ap-9037	54	4	=	=	SYM
ap-9037	54	5	1	1	NUM
ap-9037	54	6	,	,	PUNCT
ap-9037	54	7	2	2	NUM
ap-9037	54	8	,	,	PUNCT
ap-9037	54	9	.	.	PUNCT
ap-9037	54	10	.	.	PUNCT
ap-9037	54	11	.	.	PUNCT
ap-9037	54	12	,	,	PUNCT
ap-9037	55	1	n	n	X
ap-9037	55	2	,	,	PUNCT
ap-9037	55	3	where	where	SCONJ
ap-9037	55	4	the	the	DET
ap-9037	55	5	constants	constant	NOUN
ap-9037	55	6	e1	e1	VERB
ap-9037	55	7	,	,	PUNCT
ap-9037	55	8	.	.	PUNCT
ap-9037	55	9	.	.	PUNCT
ap-9037	55	10	.	.	PUNCT
ap-9037	56	1	,	,	PUNCT
ap-9037	56	2	en	en	X
ap-9037	56	3	are	be	AUX
ap-9037	56	4	usually	usually	ADV
ap-9037	56	5	referred	refer	VERB
ap-9037	56	6	to	to	ADP
ap-9037	56	7	as	as	ADP
ap-9037	56	8	factorization	factorization	NOUN
ap-9037	56	9	energies	energy	NOUN
ap-9037	56	10	or	or	CCONJ
ap-9037	56	11	transformation	transformation	NOUN
ap-9037	56	12	energies	energy	NOUN
ap-9037	56	13	.	.	PUNCT
ap-9037	57	1	these	these	DET
ap-9037	57	2	names	name	NOUN
ap-9037	57	3	stem	stem	VERB
ap-9037	57	4	from	from	ADP
ap-9037	57	5	the	the	DET
ap-9037	57	6	fact	fact	NOUN
ap-9037	57	7	that	that	SCONJ
ap-9037	57	8	for	for	ADP
ap-9037	57	9	schrödinger	schrödinger	ADJ
ap-9037	57	10	equations	equation	NOUN
ap-9037	57	11	,	,	PUNCT
ap-9037	57	12	these	these	DET
ap-9037	57	13	constants	constant	NOUN
ap-9037	57	14	formally	formally	ADV
ap-9037	57	15	represent	represent	VERB
ap-9037	57	16	energies	energy	NOUN
ap-9037	57	17	.	.	PUNCT
ap-9037	58	1	however	however	ADV
ap-9037	58	2	,	,	PUNCT
ap-9037	58	3	in	in	ADP
ap-9037	58	4	the	the	DET
ap-9037	58	5	present	present	ADJ
ap-9037	58	6	work	work	NOUN
ap-9037	58	7	,	,	PUNCT
ap-9037	58	8	we	we	PRON
ap-9037	58	9	will	will	AUX
ap-9037	58	10	apply	apply	VERB
ap-9037	58	11	the	the	DET
ap-9037	58	12	darboux	darboux	ADJ
ap-9037	58	13	-	-	PUNCT
ap-9037	58	14	crum	crum	NOUN
ap-9037	58	15	transformation	transformation	NOUN
ap-9037	58	16	to	to	ADP
ap-9037	58	17	equations	equation	NOUN
ap-9037	58	18	that	that	PRON
ap-9037	58	19	have	have	VERB
ap-9037	58	20	the	the	DET
ap-9037	58	21	same	same	ADJ
ap-9037	58	22	form	form	NOUN
ap-9037	58	23	as	as	ADP
ap-9037	58	24	(	(	PUNCT
ap-9037	58	25	4	4	NUM
ap-9037	58	26	)	)	PUNCT
ap-9037	58	27	,	,	PUNCT
ap-9037	58	28	but	but	CCONJ
ap-9037	58	29	are	be	AUX
ap-9037	58	30	not	not	PART
ap-9037	58	31	schrödinger	schrödinger	ADJ
ap-9037	58	32	equations	equation	NOUN
ap-9037	58	33	,	,	PUNCT
ap-9037	58	34	so	so	SCONJ
ap-9037	58	35	that	that	SCONJ
ap-9037	58	36	the	the	DET
ap-9037	58	37	constants	constant	NOUN
ap-9037	58	38	e1	e1	PROPN
ap-9037	58	39	,	,	PUNCT
ap-9037	58	40	.	.	PUNCT
ap-9037	58	41	.	.	PUNCT
ap-9037	58	42	.	.	PUNCT
ap-9037	59	1	,	,	PUNCT
ap-9037	59	2	en	en	INTJ
ap-9037	59	3	do	do	AUX
ap-9037	59	4	not	not	PART
ap-9037	59	5	represent	represent	VERB
ap-9037	59	6	energies	energy	NOUN
ap-9037	59	7	.	.	PUNCT
ap-9037	60	1	for	for	ADP
ap-9037	60	2	this	this	DET
ap-9037	60	3	reason	reason	NOUN
ap-9037	60	4	we	we	PRON
ap-9037	60	5	will	will	AUX
ap-9037	60	6	refer	refer	VERB
ap-9037	60	7	to	to	ADP
ap-9037	60	8	them	they	PRON
ap-9037	60	9	as	as	ADP
ap-9037	60	10	transformation	transformation	NOUN
ap-9037	60	11	parameters	parameter	NOUN
ap-9037	60	12	throughout	throughout	ADP
ap-9037	60	13	this	this	DET
ap-9037	60	14	work	work	NOUN
ap-9037	60	15	.	.	PUNCT
ap-9037	61	1	further	further	ADJ
ap-9037	61	2	details	detail	NOUN
ap-9037	61	3	will	will	AUX
ap-9037	61	4	be	be	AUX
ap-9037	61	5	discussed	discuss	VERB
ap-9037	61	6	below	below	ADV
ap-9037	61	7	at	at	ADP
ap-9037	61	8	the	the	DET
ap-9037	61	9	beginning	beginning	NOUN
ap-9037	61	10	of	of	ADP
ap-9037	61	11	section	section	NOUN
ap-9037	61	12	4.2	4.2	NUM
ap-9037	61	13	.	.	PUNCT
ap-9037	62	1	now	now	ADV
ap-9037	62	2	,	,	PUNCT
ap-9037	62	3	a	a	DET
ap-9037	62	4	solution	solution	NOUN
ap-9037	62	5	of	of	ADP
ap-9037	62	6	equation	equation	NOUN
ap-9037	62	7	(	(	PUNCT
ap-9037	62	8	5	5	NUM
ap-9037	62	9	)	)	PUNCT
ap-9037	62	10	is	be	AUX
ap-9037	62	11	given	give	VERB
ap-9037	62	12	by	by	ADP
ap-9037	62	13	:	:	PUNCT
ap-9037	62	14	ψ̂(x	ψ̂(x	NUM
ap-9037	62	15	)	)	PUNCT
ap-9037	62	16	=	=	PUNCT
ap-9037	62	17	wψ1,ψ2,	wψ1,ψ2,	NOUN
ap-9037	62	18	...	...	PUNCT
ap-9037	62	19	,ψn	,ψn	PUNCT
ap-9037	62	20	,	,	PUNCT
ap-9037	62	21	ψ(x	ψ(x	NOUN
ap-9037	62	22	)	)	PUNCT
ap-9037	62	23	wψ1,ψ2,	wψ1,ψ2,	NOUN
ap-9037	62	24	...	...	PUNCT
ap-9037	62	25	,ψn(x	,ψn(x	PUNCT
ap-9037	62	26	)	)	PUNCT
ap-9037	62	27	,	,	PUNCT
ap-9037	62	28	(	(	PUNCT
ap-9037	62	29	6	6	NUM
ap-9037	62	30	)	)	PUNCT
ap-9037	62	31	provided	provide	VERB
ap-9037	62	32	the	the	DET
ap-9037	62	33	potential	potential	NOUN
ap-9037	62	34	û	û	NUM
ap-9037	62	35	is	be	AUX
ap-9037	62	36	constrained	constrain	VERB
ap-9037	62	37	with	with	ADP
ap-9037	62	38	its	its	PRON
ap-9037	62	39	counterpart	counterpart	NOUN
ap-9037	62	40	u	u	NOUN
ap-9037	62	41	as	as	ADP
ap-9037	62	42	:	:	PUNCT
ap-9037	62	43	û(x	û(x	X
ap-9037	62	44	)	)	PUNCT
ap-9037	62	45	=	=	SYM
ap-9037	62	46	u(x	u(x	PROPN
ap-9037	62	47	)	)	PUNCT
ap-9037	62	48	−	−	PROPN
ap-9037	62	49	2	2	NUM
ap-9037	62	50	d2	d2	PROPN
ap-9037	62	51	dx2	dx2	PROPN
ap-9037	62	52	log	log	NOUN
ap-9037	63	1	[	[	X
ap-9037	63	2	wψ1,ψ2,	wψ1,ψ2,	NOUN
ap-9037	63	3	...	...	PUNCT
ap-9037	63	4	,ψn	,ψn	PUNCT
ap-9037	63	5	(	(	PUNCT
ap-9037	63	6	x	x	NOUN
ap-9037	63	7	)	)	PUNCT
ap-9037	63	8	]	]	PUNCT
ap-9037	63	9	.	.	PUNCT
ap-9037	64	1	(	(	PUNCT
ap-9037	64	2	7	7	X
ap-9037	64	3	)	)	PUNCT
ap-9037	64	4	note	note	NOUN
ap-9037	64	5	that	that	SCONJ
ap-9037	64	6	w	w	X
ap-9037	64	7	in	in	ADP
ap-9037	64	8	(	(	PUNCT
ap-9037	64	9	6	6	NUM
ap-9037	64	10	)	)	PUNCT
ap-9037	64	11	and	and	CCONJ
ap-9037	64	12	(	(	PUNCT
ap-9037	64	13	7	7	X
ap-9037	64	14	)	)	PUNCT
ap-9037	64	15	denotes	denote	VERB
ap-9037	64	16	the	the	DET
ap-9037	64	17	wronskian	wronskian	NOUN
ap-9037	64	18	with	with	ADP
ap-9037	64	19	respect	respect	NOUN
ap-9037	64	20	to	to	ADP
ap-9037	64	21	the	the	DET
ap-9037	64	22	functions	function	NOUN
ap-9037	64	23	in	in	ADP
ap-9037	64	24	its	its	PRON
ap-9037	64	25	index	index	NOUN
ap-9037	64	26	.	.	PUNCT
ap-9037	65	1	274	274	NUM
ap-9037	65	2	vol	vol	NOUN
ap-9037	65	3	.	.	PUNCT
ap-9037	65	4	63	63	NUM
ap-9037	65	5	no	no	NOUN
ap-9037	65	6	.	.	PUNCT
ap-9037	66	1	4/2023	4/2023	NOUN
ap-9037	66	2	construction	construction	NOUN
ap-9037	66	3	of	of	ADP
ap-9037	66	4	angular	angular	ADJ
ap-9037	66	5	-	-	PUNCT
ap-9037	66	6	dependent	dependent	ADJ
ap-9037	66	7	potentials	potential	VERB
ap-9037	66	8	•	•	VERB
ap-9037	66	9	confluent	confluent	ADJ
ap-9037	66	10	algorithm	algorithm	NOUN
ap-9037	66	11	:	:	PUNCT
ap-9037	66	12	let	let	VERB
ap-9037	66	13	the	the	DET
ap-9037	66	14	transformation	transformation	NOUN
ap-9037	66	15	functions	function	NOUN
ap-9037	66	16	ψ1	ψ1	NOUN
ap-9037	66	17	,	,	PUNCT
ap-9037	66	18	.	.	PUNCT
ap-9037	66	19	.	.	PUNCT
ap-9037	67	1	.	.	PUNCT
ap-9037	68	1	,	,	PUNCT
ap-9037	68	2	ψn	ψn	PUNCT
ap-9037	68	3	be	be	AUX
ap-9037	68	4	the	the	DET
ap-9037	68	5	solutions	solution	NOUN
ap-9037	68	6	of	of	ADP
ap-9037	68	7	the	the	DET
ap-9037	68	8	system	system	NOUN
ap-9037	68	9	:	:	PUNCT
ap-9037	68	10	ψ′′	ψ′′	NOUN
ap-9037	68	11	1(x	1(x	NUM
ap-9037	68	12	)	)	PUNCT
ap-9037	69	1	+	+	CCONJ
ap-9037	69	2	[	[	PUNCT
ap-9037	69	3	e1	e1	NOUN
ap-9037	69	4	−	−	NOUN
ap-9037	69	5	u(x	u(x	NOUN
ap-9037	69	6	)	)	PUNCT
ap-9037	69	7	]	]	PUNCT
ap-9037	70	1	ψ1(x	ψ1(x	X
ap-9037	70	2	)	)	PUNCT
ap-9037	70	3	=	=	SYM
ap-9037	70	4	0	0	NUM
ap-9037	70	5	,	,	PUNCT
ap-9037	70	6	(	(	PUNCT
ap-9037	70	7	8)	8)	NUM
ap-9037	70	8	ψ′′	ψ′′	X
ap-9037	70	9	j	j	PROPN
ap-9037	70	10	(	(	PUNCT
ap-9037	70	11	x	x	X
ap-9037	70	12	)	)	PUNCT
ap-9037	71	1	+	+	CCONJ
ap-9037	71	2	[	[	PUNCT
ap-9037	71	3	e1	e1	NOUN
ap-9037	71	4	−	−	NOUN
ap-9037	71	5	u(x	u(x	NOUN
ap-9037	71	6	)	)	PUNCT
ap-9037	71	7	]	]	PUNCT
ap-9037	71	8	ψj(x	ψj(x	X
ap-9037	71	9	)	)	PUNCT
ap-9037	71	10	=	=	SYM
ap-9037	71	11	−ψj−1(x	−ψj−1(x	NUM
ap-9037	71	12	)	)	PUNCT
ap-9037	71	13	,	,	PUNCT
ap-9037	71	14	j	j	PROPN
ap-9037	71	15	=	=	SYM
ap-9037	71	16	2	2	NUM
ap-9037	71	17	,	,	PUNCT
ap-9037	71	18	.	.	PUNCT
ap-9037	71	19	.	.	PUNCT
ap-9037	72	1	.	.	PUNCT
ap-9037	73	1	,	,	PUNCT
ap-9037	73	2	n	n	X
ap-9037	73	3	,	,	PUNCT
ap-9037	73	4	(	(	PUNCT
ap-9037	73	5	9	9	X
ap-9037	73	6	)	)	PUNCT
ap-9037	73	7	observe	observe	VERB
ap-9037	73	8	that	that	SCONJ
ap-9037	73	9	there	there	PRON
ap-9037	73	10	is	be	VERB
ap-9037	73	11	only	only	ADV
ap-9037	73	12	a	a	DET
ap-9037	73	13	single	single	ADJ
ap-9037	73	14	transformation	transformation	NOUN
ap-9037	73	15	parameter	parameter	NOUN
ap-9037	73	16	e1	e1	PROPN
ap-9037	73	17	.	.	PUNCT
ap-9037	74	1	for	for	ADP
ap-9037	74	2	example	example	NOUN
ap-9037	74	3	,	,	PUNCT
ap-9037	74	4	in	in	ADP
ap-9037	74	5	the	the	DET
ap-9037	74	6	case	case	NOUN
ap-9037	74	7	n	n	NOUN
ap-9037	74	8	=	=	SYM
ap-9037	74	9	2	2	NUM
ap-9037	74	10	(	(	PUNCT
ap-9037	74	11	second	second	ADJ
ap-9037	74	12	-	-	PUNCT
ap-9037	74	13	order	order	NOUN
ap-9037	74	14	transformation	transformation	NOUN
ap-9037	74	15	)	)	PUNCT
ap-9037	74	16	,	,	PUNCT
ap-9037	74	17	the	the	DET
ap-9037	74	18	above	above	ADJ
ap-9037	74	19	system	system	NOUN
ap-9037	74	20	of	of	ADP
ap-9037	74	21	equations	equation	NOUN
ap-9037	74	22	reads	read	VERB
ap-9037	74	23	:	:	PUNCT
ap-9037	74	24	ψ′′	ψ′′	NOUN
ap-9037	74	25	1(x	1(x	NUM
ap-9037	74	26	)	)	PUNCT
ap-9037	75	1	+	+	CCONJ
ap-9037	76	1	[	[	PUNCT
ap-9037	76	2	e1	e1	NOUN
ap-9037	76	3	−	−	NOUN
ap-9037	76	4	u(x	u(x	NOUN
ap-9037	76	5	)	)	PUNCT
ap-9037	76	6	]	]	PUNCT
ap-9037	77	1	ψ1(x	ψ1(x	X
ap-9037	77	2	)	)	PUNCT
ap-9037	77	3	=	=	SYM
ap-9037	77	4	0	0	NUM
ap-9037	77	5	,	,	PUNCT
ap-9037	77	6	ψ′′	ψ′′	PROPN
ap-9037	77	7	2(x	2(x	NUM
ap-9037	77	8	)	)	PUNCT
ap-9037	78	1	+	+	CCONJ
ap-9037	78	2	[	[	PUNCT
ap-9037	78	3	e1	e1	NOUN
ap-9037	78	4	−	−	NOUN
ap-9037	78	5	u(x	u(x	NOUN
ap-9037	78	6	)	)	PUNCT
ap-9037	78	7	]	]	PUNCT
ap-9037	79	1	ψ2(x	ψ2(x	NOUN
ap-9037	79	2	)	)	PUNCT
ap-9037	79	3	=	=	SYM
ap-9037	79	4	−ψ1(x	−ψ1(x	NOUN
ap-9037	79	5	)	)	PUNCT
ap-9037	79	6	.	.	PUNCT
ap-9037	80	1	returning	return	VERB
ap-9037	80	2	to	to	ADP
ap-9037	80	3	the	the	DET
ap-9037	80	4	general	general	ADJ
ap-9037	80	5	case	case	NOUN
ap-9037	80	6	,	,	PUNCT
ap-9037	80	7	if	if	SCONJ
ap-9037	80	8	solutions	solution	NOUN
ap-9037	80	9	ψ1	ψ1	VERB
ap-9037	80	10	,	,	PUNCT
ap-9037	80	11	.	.	PUNCT
ap-9037	80	12	.	.	PUNCT
ap-9037	81	1	.	.	PUNCT
ap-9037	82	1	,	,	PUNCT
ap-9037	82	2	ψn	ψn	X
ap-9037	82	3	of	of	ADP
ap-9037	82	4	the	the	DET
ap-9037	82	5	system	system	NOUN
ap-9037	82	6	(	(	PUNCT
ap-9037	82	7	8)	8)	NUM
ap-9037	82	8	,	,	PUNCT
ap-9037	82	9	(	(	PUNCT
ap-9037	82	10	9	9	X
ap-9037	82	11	)	)	PUNCT
ap-9037	82	12	are	be	AUX
ap-9037	82	13	known	know	VERB
ap-9037	82	14	,	,	PUNCT
ap-9037	82	15	then	then	ADV
ap-9037	82	16	the	the	DET
ap-9037	82	17	function	function	NOUN
ap-9037	82	18	:	:	PUNCT
ap-9037	82	19	ψ̂(x	ψ̂(x	NUM
ap-9037	82	20	)	)	PUNCT
ap-9037	82	21	=	=	PUNCT
ap-9037	82	22	wψ1,ψ2,	wψ1,ψ2,	NOUN
ap-9037	82	23	...	...	PUNCT
ap-9037	82	24	,ψn	,ψn	PUNCT
ap-9037	82	25	,	,	PUNCT
ap-9037	82	26	ψ(x	ψ(x	NOUN
ap-9037	82	27	)	)	PUNCT
ap-9037	82	28	wψ1,ψ2,	wψ1,ψ2,	NOUN
ap-9037	82	29	...	...	PUNCT
ap-9037	82	30	,ψn(x	,ψn(x	PUNCT
ap-9037	82	31	)	)	PUNCT
ap-9037	82	32	,	,	PUNCT
ap-9037	82	33	(	(	PUNCT
ap-9037	82	34	10	10	X
ap-9037	82	35	)	)	PUNCT
ap-9037	82	36	solves	solve	VERB
ap-9037	82	37	equation	equation	NOUN
ap-9037	82	38	(	(	PUNCT
ap-9037	82	39	5	5	NUM
ap-9037	82	40	)	)	PUNCT
ap-9037	82	41	,	,	PUNCT
ap-9037	82	42	as	as	ADV
ap-9037	82	43	long	long	ADV
ap-9037	82	44	as	as	SCONJ
ap-9037	82	45	the	the	DET
ap-9037	82	46	potential	potential	NOUN
ap-9037	82	47	û	û	PRON
ap-9037	82	48	is	be	AUX
ap-9037	82	49	interrelated	interrelate	VERB
ap-9037	82	50	to	to	ADP
ap-9037	82	51	its	its	PRON
ap-9037	82	52	counterpart	counterpart	NOUN
ap-9037	82	53	u	u	NOUN
ap-9037	82	54	by	by	ADP
ap-9037	82	55	means	mean	NOUN
ap-9037	82	56	of	of	ADP
ap-9037	82	57	:	:	PUNCT
ap-9037	82	58	û(x	û(x	X
ap-9037	82	59	)	)	PUNCT
ap-9037	82	60	=	=	SYM
ap-9037	82	61	u(x	u(x	PROPN
ap-9037	82	62	)	)	PUNCT
ap-9037	82	63	−	−	PROPN
ap-9037	82	64	2	2	NUM
ap-9037	82	65	d2	d2	PROPN
ap-9037	82	66	dx2	dx2	PROPN
ap-9037	82	67	log	log	NOUN
ap-9037	82	68	[	[	X
ap-9037	82	69	wψ1,ψ2,	wψ1,ψ2,	NOUN
ap-9037	82	70	...	...	PUNCT
ap-9037	82	71	,ψn(x	,ψn(x	PUNCT
ap-9037	82	72	)	)	PUNCT
ap-9037	82	73	]	]	PUNCT
ap-9037	82	74	.	.	PUNCT
ap-9037	83	1	(	(	PUNCT
ap-9037	83	2	11	11	X
ap-9037	83	3	)	)	PUNCT
ap-9037	83	4	note	note	NOUN
ap-9037	83	5	that	that	SCONJ
ap-9037	83	6	the	the	DET
ap-9037	83	7	forms	form	NOUN
ap-9037	83	8	(	(	PUNCT
ap-9037	83	9	10	10	NUM
ap-9037	83	10	)	)	PUNCT
ap-9037	83	11	and	and	CCONJ
ap-9037	83	12	(	(	PUNCT
ap-9037	83	13	11	11	NUM
ap-9037	83	14	)	)	PUNCT
ap-9037	83	15	are	be	AUX
ap-9037	83	16	the	the	DET
ap-9037	83	17	same	same	ADJ
ap-9037	83	18	as	as	ADP
ap-9037	83	19	(	(	PUNCT
ap-9037	83	20	6	6	NUM
ap-9037	83	21	)	)	PUNCT
ap-9037	83	22	and	and	CCONJ
ap-9037	83	23	(	(	PUNCT
ap-9037	83	24	7	7	NUM
ap-9037	83	25	)	)	PUNCT
ap-9037	83	26	,	,	PUNCT
ap-9037	83	27	respectively	respectively	ADV
ap-9037	83	28	,	,	PUNCT
ap-9037	83	29	so	so	SCONJ
ap-9037	83	30	that	that	SCONJ
ap-9037	83	31	the	the	DET
ap-9037	83	32	only	only	ADJ
ap-9037	83	33	difference	difference	NOUN
ap-9037	83	34	between	between	ADP
ap-9037	83	35	the	the	DET
ap-9037	83	36	two	two	NUM
ap-9037	83	37	algorithms	algorithm	NOUN
ap-9037	83	38	lies	lie	VERB
ap-9037	83	39	in	in	ADP
ap-9037	83	40	the	the	DET
ap-9037	83	41	equations	equation	NOUN
ap-9037	83	42	that	that	PRON
ap-9037	83	43	determine	determine	VERB
ap-9037	83	44	the	the	DET
ap-9037	83	45	transformation	transformation	NOUN
ap-9037	83	46	functions	function	NOUN
ap-9037	83	47	.	.	PUNCT
ap-9037	84	1	3	3	X
ap-9037	84	2	.	.	X
ap-9037	84	3	the	the	DET
ap-9037	84	4	dunkl	dunkl	NOUN
ap-9037	84	5	-	-	PUNCT
ap-9037	84	6	schrödinger	schrödinger	NOUN
ap-9037	84	7	system	system	NOUN
ap-9037	84	8	our	our	PRON
ap-9037	84	9	first	first	ADJ
ap-9037	84	10	goal	goal	NOUN
ap-9037	84	11	is	be	AUX
ap-9037	84	12	to	to	PART
ap-9037	84	13	derive	derive	VERB
ap-9037	84	14	the	the	DET
ap-9037	84	15	stationary	stationary	ADJ
ap-9037	84	16	schrödinger	schrödinger	ADJ
ap-9037	84	17	equation	equation	NOUN
ap-9037	84	18	in	in	ADP
ap-9037	84	19	three	three	NUM
ap-9037	84	20	dimensions	dimension	NOUN
ap-9037	84	21	within	within	ADP
ap-9037	84	22	the	the	DET
ap-9037	84	23	dunkl	dunkl	NOUN
ap-9037	84	24	formalism	formalism	NOUN
ap-9037	84	25	.	.	PUNCT
ap-9037	85	1	this	this	DET
ap-9037	85	2	procedure	procedure	NOUN
ap-9037	85	3	is	be	AUX
ap-9037	85	4	well	well	ADV
ap-9037	85	5	known	know	VERB
ap-9037	85	6	–	–	PUNCT
ap-9037	85	7	see	see	VERB
ap-9037	85	8	e.g.	e.g.	ADV
ap-9037	85	9	[	[	X
ap-9037	85	10	18	18	NUM
ap-9037	85	11	]	]	PUNCT
ap-9037	85	12	–	–	PUNCT
ap-9037	85	13	so	so	ADV
ap-9037	85	14	we	we	PRON
ap-9037	85	15	will	will	AUX
ap-9037	85	16	not	not	PART
ap-9037	85	17	give	give	VERB
ap-9037	85	18	the	the	DET
ap-9037	85	19	details	detail	NOUN
ap-9037	85	20	of	of	ADP
ap-9037	85	21	the	the	DET
ap-9037	85	22	derivation	derivation	NOUN
ap-9037	85	23	here	here	ADV
ap-9037	85	24	.	.	PUNCT
ap-9037	86	1	while	while	SCONJ
ap-9037	86	2	the	the	DET
ap-9037	86	3	hamiltonian	hamiltonian	ADJ
ap-9037	86	4	governing	govern	VERB
ap-9037	86	5	our	our	PRON
ap-9037	86	6	system	system	NOUN
ap-9037	86	7	is	be	AUX
ap-9037	86	8	written	write	VERB
ap-9037	86	9	in	in	ADP
ap-9037	86	10	the	the	DET
ap-9037	86	11	usual	usual	ADJ
ap-9037	86	12	form	form	NOUN
ap-9037	86	13	:	:	PUNCT
ap-9037	86	14	h	h	NOUN
ap-9037	86	15	=	=	NOUN
ap-9037	86	16	p2	p2	PROPN
ap-9037	86	17	1	1	NUM
ap-9037	87	1	+	+	CCONJ
ap-9037	87	2	p̂2	p̂2	NOUN
ap-9037	87	3	2	2	NUM
ap-9037	87	4	+	+	CCONJ
ap-9037	87	5	p̂2	p̂2	NOUN
ap-9037	87	6	3	3	NUM
ap-9037	87	7	+	+	SYM
ap-9037	87	8	v	v	PROPN
ap-9037	87	9	(	(	PUNCT
ap-9037	87	10	x1	x1	PROPN
ap-9037	87	11	,	,	PUNCT
ap-9037	87	12	x2	x2	PROPN
ap-9037	87	13	,	,	PUNCT
ap-9037	87	14	x3	x3	ADJ
ap-9037	87	15	)	)	PUNCT
ap-9037	87	16	,	,	PUNCT
ap-9037	87	17	(	(	PUNCT
ap-9037	87	18	12	12	NUM
ap-9037	87	19	)	)	PUNCT
ap-9037	87	20	where	where	SCONJ
ap-9037	87	21	v	v	NOUN
ap-9037	87	22	represents	represent	VERB
ap-9037	87	23	the	the	DET
ap-9037	87	24	potential	potential	NOUN
ap-9037	87	25	,	,	PUNCT
ap-9037	87	26	and	and	CCONJ
ap-9037	87	27	the	the	DET
ap-9037	87	28	momentum	momentum	NOUN
ap-9037	87	29	operators	operator	NOUN
ap-9037	87	30	p1	p1	PROPN
ap-9037	87	31	,	,	PUNCT
ap-9037	87	32	p2	p2	NOUN
ap-9037	87	33	,	,	PUNCT
ap-9037	87	34	p3	p3	PROPN
ap-9037	87	35	obey	obey	VERB
ap-9037	87	36	a	a	DET
ap-9037	87	37	nonstandard	nonstandard	ADJ
ap-9037	87	38	definition	definition	NOUN
ap-9037	87	39	.	.	PUNCT
ap-9037	88	1	here	here	ADV
ap-9037	88	2	,	,	PUNCT
ap-9037	88	3	they	they	PRON
ap-9037	88	4	are	be	AUX
ap-9037	88	5	given	give	VERB
ap-9037	88	6	as	as	ADP
ap-9037	88	7	:	:	PUNCT
ap-9037	88	8	pj	pj	PROPN
ap-9037	88	9	=	=	PUNCT
ap-9037	88	10	−idj	−idj	NOUN
ap-9037	88	11	=	=	SYM
ap-9037	88	12	−i	−i	NOUN
ap-9037	88	13	(	(	PUNCT
ap-9037	88	14	∂	∂	NUM
ap-9037	88	15	∂xj	∂xj	NOUN
ap-9037	88	16	+	+	CCONJ
ap-9037	88	17	νj	νj	PROPN
ap-9037	88	18	xj	xj	NOUN
ap-9037	88	19	−	−	PROPN
ap-9037	88	20	νj	νj	PROPN
ap-9037	88	21	xj	xj	PROPN
ap-9037	88	22	rj	rj	PROPN
ap-9037	88	23	)	)	PUNCT
ap-9037	88	24	,	,	PUNCT
ap-9037	88	25	j	j	PROPN
ap-9037	88	26	=	=	SYM
ap-9037	88	27	1	1	NUM
ap-9037	88	28	,	,	PUNCT
ap-9037	88	29	2	2	NUM
ap-9037	88	30	,	,	PUNCT
ap-9037	88	31	3	3	NUM
ap-9037	88	32	,	,	PUNCT
ap-9037	88	33	(	(	PUNCT
ap-9037	88	34	13	13	X
ap-9037	88	35	)	)	PUNCT
ap-9037	88	36	introducing	introduce	VERB
ap-9037	88	37	real	real	ADV
ap-9037	88	38	-	-	PUNCT
ap-9037	88	39	valued	value	VERB
ap-9037	88	40	constants	constant	NOUN
ap-9037	88	41	ν1	ν1	NOUN
ap-9037	88	42	,	,	PUNCT
ap-9037	88	43	ν2	ν2	NOUN
ap-9037	88	44	,	,	PUNCT
ap-9037	88	45	ν3	ν3	NOUN
ap-9037	88	46	,	,	PUNCT
ap-9037	88	47	and	and	CCONJ
ap-9037	88	48	dunkl	dunkl	VERB
ap-9037	88	49	operators	operator	NOUN
ap-9037	88	50	d1	d1	PROPN
ap-9037	88	51	,	,	PUNCT
ap-9037	88	52	d2	d2	PROPN
ap-9037	88	53	,	,	PUNCT
ap-9037	88	54	d3	d3	PROPN
ap-9037	88	55	,	,	PUNCT
ap-9037	88	56	that	that	PRON
ap-9037	88	57	generalise	generalise	VERB
ap-9037	88	58	the	the	DET
ap-9037	88	59	conventional	conventional	ADJ
ap-9037	88	60	partial	partial	ADJ
ap-9037	88	61	derivative	derivative	NOUN
ap-9037	88	62	.	.	PUNCT
ap-9037	89	1	furthermore	furthermore	ADV
ap-9037	89	2	,	,	PUNCT
ap-9037	89	3	the	the	DET
ap-9037	89	4	rj	rj	PROPN
ap-9037	89	5	stand	stand	VERB
ap-9037	89	6	for	for	ADP
ap-9037	89	7	parity	parity	NOUN
ap-9037	89	8	or	or	CCONJ
ap-9037	89	9	reflection	reflection	NOUN
ap-9037	89	10	operators	operator	NOUN
ap-9037	89	11	with	with	ADP
ap-9037	89	12	respect	respect	NOUN
ap-9037	89	13	to	to	ADP
ap-9037	89	14	the	the	DET
ap-9037	89	15	variable	variable	NOUN
ap-9037	89	16	xj	xj	PROPN
ap-9037	89	17	.	.	PUNCT
ap-9037	90	1	this	this	DET
ap-9037	90	2	operator	operator	NOUN
ap-9037	90	3	acts	act	VERB
ap-9037	90	4	on	on	ADP
ap-9037	90	5	the	the	DET
ap-9037	90	6	admissible	admissible	ADJ
ap-9037	90	7	functions	function	NOUN
ap-9037	90	8	ψ	ψ	NOUN
ap-9037	90	9	in	in	ADP
ap-9037	90	10	the	the	DET
ap-9037	90	11	following	following	ADJ
ap-9037	90	12	way	way	NOUN
ap-9037	90	13	:	:	PUNCT
ap-9037	90	14	r1	r1	PROPN
ap-9037	90	15	ψ(x1	ψ(x1	PROPN
ap-9037	90	16	,	,	PUNCT
ap-9037	90	17	x2	x2	PROPN
ap-9037	90	18	,	,	PUNCT
ap-9037	90	19	x3	x3	ADJ
ap-9037	90	20	)	)	PUNCT
ap-9037	90	21	=	=	SYM
ap-9037	91	1	ψ(−x1	ψ(−x1	PROPN
ap-9037	91	2	,	,	PUNCT
ap-9037	91	3	x2	x2	NOUN
ap-9037	91	4	,	,	PUNCT
ap-9037	91	5	x3	x3	ADJ
ap-9037	91	6	)	)	PUNCT
ap-9037	91	7	,	,	PUNCT
ap-9037	91	8	(	(	PUNCT
ap-9037	91	9	14	14	X
ap-9037	91	10	)	)	PUNCT
ap-9037	91	11	r2	r2	PROPN
ap-9037	91	12	ψ(x1	ψ(x1	NOUN
ap-9037	91	13	,	,	PUNCT
ap-9037	91	14	x2	x2	PROPN
ap-9037	91	15	,	,	PUNCT
ap-9037	91	16	x3	x3	ADJ
ap-9037	91	17	)	)	PUNCT
ap-9037	91	18	=	=	SYM
ap-9037	91	19	ψ(x1	ψ(x1	NOUN
ap-9037	91	20	,	,	PUNCT
ap-9037	91	21	−x2	−x2	PROPN
ap-9037	91	22	,	,	PUNCT
ap-9037	91	23	x3	x3	ADJ
ap-9037	91	24	)	)	PUNCT
ap-9037	91	25	,	,	PUNCT
ap-9037	91	26	(	(	PUNCT
ap-9037	91	27	15	15	X
ap-9037	91	28	)	)	PUNCT
ap-9037	91	29	r3	r3	PROPN
ap-9037	91	30	ψ(x1	ψ(x1	NOUN
ap-9037	91	31	,	,	PUNCT
ap-9037	91	32	x2	x2	PROPN
ap-9037	91	33	,	,	PUNCT
ap-9037	91	34	x3	x3	ADJ
ap-9037	91	35	)	)	PUNCT
ap-9037	91	36	=	=	SYM
ap-9037	91	37	ψ(x1	ψ(x1	NOUN
ap-9037	91	38	,	,	PUNCT
ap-9037	91	39	x2	x2	PROPN
ap-9037	91	40	,	,	PUNCT
ap-9037	91	41	−x3	−x3	PROPN
ap-9037	91	42	)	)	PUNCT
ap-9037	91	43	.	.	PUNCT
ap-9037	92	1	(	(	PUNCT
ap-9037	92	2	16	16	X
ap-9037	92	3	)	)	PUNCT
ap-9037	92	4	note	note	NOUN
ap-9037	92	5	that	that	SCONJ
ap-9037	92	6	setting	set	VERB
ap-9037	92	7	ν1	ν1	NOUN
ap-9037	92	8	=	=	SYM
ap-9037	92	9	ν2	ν2	NOUN
ap-9037	92	10	=	=	SYM
ap-9037	92	11	ν3	ν3	NOUN
ap-9037	92	12	=	=	SYM
ap-9037	92	13	0	0	NUM
ap-9037	92	14	converts	convert	VERB
ap-9037	92	15	the	the	DET
ap-9037	92	16	momentum	momentum	NOUN
ap-9037	92	17	operators	operator	NOUN
ap-9037	92	18	in	in	ADP
ap-9037	92	19	(	(	PUNCT
ap-9037	92	20	13	13	NUM
ap-9037	92	21	)	)	PUNCT
ap-9037	92	22	to	to	ADP
ap-9037	92	23	their	their	PRON
ap-9037	92	24	conventional	conventional	ADJ
ap-9037	92	25	form	form	NOUN
ap-9037	92	26	,	,	PUNCT
ap-9037	92	27	thus	thus	ADV
ap-9037	92	28	removing	remove	VERB
ap-9037	92	29	the	the	DET
ap-9037	92	30	reflection	reflection	NOUN
ap-9037	92	31	operators	operator	NOUN
ap-9037	92	32	.	.	PUNCT
ap-9037	93	1	now	now	ADV
ap-9037	93	2	,	,	PUNCT
ap-9037	93	3	our	our	PRON
ap-9037	93	4	hamiltonian	hamiltonian	NOUN
ap-9037	93	5	(	(	PUNCT
ap-9037	93	6	12	12	NUM
ap-9037	93	7	)	)	PUNCT
ap-9037	93	8	is	be	AUX
ap-9037	93	9	defined	define	VERB
ap-9037	93	10	on	on	ADP
ap-9037	93	11	a	a	DET
ap-9037	93	12	weighted	weight	VERB
ap-9037	93	13	hilbert	hilbert	NOUN
ap-9037	93	14	space	space	NOUN
ap-9037	93	15	l2	l2	NOUN
ap-9037	93	16	w	w	NOUN
ap-9037	93	17	with	with	ADP
ap-9037	93	18	weight	weight	NOUN
ap-9037	93	19	function	function	NOUN
ap-9037	93	20	:	:	PUNCT
ap-9037	93	21	w(x1	w(x1	NOUN
ap-9037	93	22	,	,	PUNCT
ap-9037	93	23	x2	x2	PROPN
ap-9037	93	24	,	,	PUNCT
ap-9037	93	25	x3	x3	ADJ
ap-9037	93	26	)	)	PUNCT
ap-9037	93	27	=	=	SYM
ap-9037	93	28	|x1|2ν1	|x1|2ν1	NUM
ap-9037	93	29	|x2|2ν2	|x2|2ν2	PUNCT
ap-9037	93	30	|x3|2ν3	|x3|2ν3	PUNCT
ap-9037	93	31	,	,	PUNCT
ap-9037	93	32	so	so	SCONJ
ap-9037	93	33	that	that	SCONJ
ap-9037	93	34	the	the	DET
ap-9037	93	35	norm	norm	NOUN
ap-9037	93	36	of	of	ADP
ap-9037	93	37	a	a	DET
ap-9037	93	38	function	function	NOUN
ap-9037	93	39	ψ	ψ	X
ap-9037	93	40	∈	∈	NOUN
ap-9037	93	41	l2	l2	NOUN
ap-9037	93	42	w	w	PROPN
ap-9037	93	43	(	(	PUNCT
ap-9037	93	44	r3	r3	PROPN
ap-9037	93	45	)	)	PUNCT
ap-9037	93	46	takes	take	VERB
ap-9037	93	47	the	the	DET
ap-9037	93	48	form	form	NOUN
ap-9037	93	49	:	:	PUNCT
ap-9037	93	50	∥ψ∥	∥ψ∥	X
ap-9037	93	51	=	=	SYM
ap-9037	93	52	√√√√∫	√√√√∫	PROPN
ap-9037	93	53	r3	r3	PROPN
ap-9037	93	54	ψ(x1	ψ(x1	NOUN
ap-9037	93	55	,	,	PUNCT
ap-9037	93	56	x2	x2	PROPN
ap-9037	93	57	,	,	PUNCT
ap-9037	93	58	x3)∗ψ(x1	x3)∗ψ(x1	PROPN
ap-9037	93	59	,	,	PUNCT
ap-9037	93	60	x2	x2	PROPN
ap-9037	93	61	,	,	PUNCT
ap-9037	93	62	x3)|x1|2ν1	x3)|x1|2ν1	PROPN
ap-9037	93	63	|x2|2ν2	|x2|2ν2	X
ap-9037	93	64	|x3|2ν3	|x3|2ν3	X
ap-9037	94	1	dx1	dx1	PROPN
ap-9037	94	2	dx2	dx2	PROPN
ap-9037	94	3	dx3	dx3	PROPN
ap-9037	94	4	,	,	PUNCT
ap-9037	94	5	(	(	PUNCT
ap-9037	94	6	17	17	NUM
ap-9037	94	7	)	)	PUNCT
ap-9037	94	8	where	where	SCONJ
ap-9037	94	9	the	the	DET
ap-9037	94	10	asterisk	asterisk	NOUN
ap-9037	94	11	denotes	denote	VERB
ap-9037	94	12	a	a	DET
ap-9037	94	13	complex	complex	ADJ
ap-9037	94	14	conjugation	conjugation	NOUN
ap-9037	94	15	.	.	PUNCT
ap-9037	95	1	it	it	PRON
ap-9037	95	2	is	be	AUX
ap-9037	95	3	understood	understand	VERB
ap-9037	95	4	that	that	SCONJ
ap-9037	95	5	the	the	DET
ap-9037	95	6	admissible	admissible	ADJ
ap-9037	95	7	values	value	NOUN
ap-9037	95	8	of	of	ADP
ap-9037	95	9	the	the	DET
ap-9037	95	10	parameters	parameter	NOUN
ap-9037	95	11	ν1	ν1	NOUN
ap-9037	95	12	,	,	PUNCT
ap-9037	95	13	ν2	ν2	NOUN
ap-9037	95	14	,	,	PUNCT
ap-9037	95	15	and	and	CCONJ
ap-9037	95	16	ν3	ν3	NOUN
ap-9037	95	17	must	must	AUX
ap-9037	95	18	be	be	AUX
ap-9037	95	19	suitably	suitably	ADV
ap-9037	95	20	restricted	restrict	VERB
ap-9037	95	21	so	so	SCONJ
ap-9037	95	22	that	that	SCONJ
ap-9037	95	23	they	they	PRON
ap-9037	95	24	do	do	AUX
ap-9037	95	25	not	not	PART
ap-9037	95	26	contribute	contribute	VERB
ap-9037	95	27	singularities	singularity	NOUN
ap-9037	95	28	in	in	ADP
ap-9037	95	29	the	the	DET
ap-9037	95	30	integrand	integrand	NOUN
ap-9037	95	31	.	.	PUNCT
ap-9037	96	1	in	in	ADP
ap-9037	96	2	the	the	DET
ap-9037	96	3	next	next	ADJ
ap-9037	96	4	step	step	NOUN
ap-9037	96	5	we	we	PRON
ap-9037	96	6	introduce	introduce	VERB
ap-9037	96	7	spherical	spherical	ADJ
ap-9037	96	8	coordinates	coordinate	NOUN
ap-9037	96	9	r	r	NOUN
ap-9037	96	10	≥	≥	NOUN
ap-9037	96	11	0	0	NUM
ap-9037	96	12	,	,	PUNCT
ap-9037	96	13	0	0	NUM
ap-9037	96	14	≤	≤	NUM
ap-9037	96	15	θ	θ	PROPN
ap-9037	96	16	≤	≤	NUM
ap-9037	96	17	π	π	PROPN
ap-9037	96	18	,	,	PUNCT
ap-9037	96	19	and	and	CCONJ
ap-9037	96	20	0	0	NUM
ap-9037	96	21	≤	≤	NOUN
ap-9037	96	22	ϕ	ϕ	X
ap-9037	96	23	≤	≤	ADJ
ap-9037	96	24	2π	2π	NOUN
ap-9037	96	25	by	by	ADP
ap-9037	96	26	means	mean	NOUN
ap-9037	96	27	of	of	ADP
ap-9037	96	28	the	the	DET
ap-9037	96	29	usual	usual	ADJ
ap-9037	96	30	relations	relation	NOUN
ap-9037	96	31	:	:	PUNCT
ap-9037	97	1	x1	x1	PROPN
ap-9037	97	2	=	=	PUNCT
ap-9037	97	3	r	r	NOUN
ap-9037	97	4	sin(θ	sin(θ	PROPN
ap-9037	97	5	)	)	PUNCT
ap-9037	97	6	cos(ϕ	cos(ϕ	PROPN
ap-9037	97	7	)	)	PUNCT
ap-9037	97	8	,	,	PUNCT
ap-9037	98	1	x2	x2	NOUN
ap-9037	98	2	=	=	PUNCT
ap-9037	98	3	r	r	NOUN
ap-9037	98	4	sin(θ	sin(θ	PROPN
ap-9037	98	5	)	)	PUNCT
ap-9037	98	6	sin(ϕ	sin(ϕ	NOUN
ap-9037	98	7	)	)	PUNCT
ap-9037	98	8	,	,	PUNCT
ap-9037	98	9	(	(	PUNCT
ap-9037	98	10	18	18	NUM
ap-9037	98	11	)	)	PUNCT
ap-9037	98	12	x3	x3	NOUN
ap-9037	99	1	=	=	PUNCT
ap-9037	99	2	r	r	NOUN
ap-9037	99	3	cos(θ	cos(θ	PROPN
ap-9037	99	4	)	)	PUNCT
ap-9037	99	5	.	.	PUNCT
ap-9037	100	1	275	275	NUM
ap-9037	100	2	axel	axel	PROPN
ap-9037	100	3	schulze	schulze	NOUN
ap-9037	100	4	-	-	PUNCT
ap-9037	100	5	halberg	halberg	PROPN
ap-9037	100	6	acta	acta	PROPN
ap-9037	100	7	polytechnica	polytechnica	PROPN
ap-9037	100	8	furthermore	furthermore	ADV
ap-9037	100	9	,	,	PUNCT
ap-9037	100	10	we	we	PRON
ap-9037	100	11	restrict	restrict	VERB
ap-9037	100	12	the	the	DET
ap-9037	100	13	class	class	NOUN
ap-9037	100	14	of	of	ADP
ap-9037	100	15	potentials	potential	NOUN
ap-9037	100	16	in	in	ADP
ap-9037	100	17	(	(	PUNCT
ap-9037	100	18	12	12	NUM
ap-9037	100	19	)	)	PUNCT
ap-9037	100	20	to	to	ADP
ap-9037	100	21	functions	function	NOUN
ap-9037	100	22	of	of	ADP
ap-9037	100	23	the	the	DET
ap-9037	100	24	following	follow	VERB
ap-9037	100	25	form	form	NOUN
ap-9037	100	26	:	:	PUNCT
ap-9037	100	27	v	v	NOUN
ap-9037	100	28	(	(	PUNCT
ap-9037	100	29	r	r	NOUN
ap-9037	100	30	,	,	PUNCT
ap-9037	100	31	ϕ	ϕ	NOUN
ap-9037	100	32	,	,	PUNCT
ap-9037	100	33	θ	θ	NOUN
ap-9037	100	34	)	)	PUNCT
ap-9037	100	35	=	=	SYM
ap-9037	100	36	vr(r	vr(r	NOUN
ap-9037	100	37	)	)	PUNCT
ap-9037	101	1	+	+	CCONJ
ap-9037	101	2	1	1	NUM
ap-9037	101	3	r2	r2	NOUN
ap-9037	101	4	vθ(θ	vθ(θ	NUM
ap-9037	101	5	)	)	PUNCT
ap-9037	101	6	+	+	CCONJ
ap-9037	101	7	1	1	NUM
ap-9037	101	8	r2	r2	NOUN
ap-9037	101	9	sin(θ)2	sin(θ)2	NOUN
ap-9037	101	10	vϕ(ϕ	vϕ(ϕ	NUM
ap-9037	101	11	)	)	PUNCT
ap-9037	101	12	,	,	PUNCT
ap-9037	101	13	(	(	PUNCT
ap-9037	101	14	19	19	X
ap-9037	101	15	)	)	PUNCT
ap-9037	101	16	introducing	introduce	VERB
ap-9037	101	17	the	the	DET
ap-9037	101	18	three	three	NUM
ap-9037	101	19	single	single	ADJ
ap-9037	101	20	-	-	PUNCT
ap-9037	101	21	variable	variable	ADJ
ap-9037	101	22	parameters	parameter	NOUN
ap-9037	101	23	vr	vr	VERB
ap-9037	101	24	,	,	PUNCT
ap-9037	101	25	vθ	vθ	NOUN
ap-9037	101	26	,	,	PUNCT
ap-9037	101	27	and	and	CCONJ
ap-9037	101	28	vϕ.	vϕ.	ADP
ap-9037	101	29	the	the	DET
ap-9037	101	30	purpose	purpose	NOUN
ap-9037	101	31	of	of	ADP
ap-9037	101	32	representing	represent	VERB
ap-9037	101	33	our	our	PRON
ap-9037	101	34	potential	potential	NOUN
ap-9037	101	35	in	in	ADP
ap-9037	101	36	the	the	DET
ap-9037	101	37	form	form	NOUN
ap-9037	101	38	(	(	PUNCT
ap-9037	101	39	19	19	NUM
ap-9037	101	40	)	)	PUNCT
ap-9037	101	41	is	be	AUX
ap-9037	101	42	to	to	PART
ap-9037	101	43	ensure	ensure	VERB
ap-9037	101	44	that	that	SCONJ
ap-9037	101	45	we	we	PRON
ap-9037	101	46	can	can	AUX
ap-9037	101	47	separate	separate	VERB
ap-9037	101	48	variables	variable	NOUN
ap-9037	101	49	in	in	ADP
ap-9037	101	50	the	the	DET
ap-9037	101	51	schrödinger	schrödinger	ADJ
ap-9037	101	52	equation	equation	NOUN
ap-9037	101	53	that	that	SCONJ
ap-9037	101	54	we	we	PRON
ap-9037	101	55	will	will	AUX
ap-9037	101	56	now	now	ADV
ap-9037	101	57	construct	construct	VERB
ap-9037	101	58	.	.	PUNCT
ap-9037	102	1	writing	write	VERB
ap-9037	102	2	our	our	PRON
ap-9037	102	3	hamiltonian	hamiltonian	NOUN
ap-9037	102	4	(	(	PUNCT
ap-9037	102	5	12	12	NUM
ap-9037	102	6	)	)	PUNCT
ap-9037	102	7	into	into	ADP
ap-9037	102	8	the	the	DET
ap-9037	102	9	form	form	NOUN
ap-9037	102	10	of	of	ADP
ap-9037	102	11	spherical	spherical	ADJ
ap-9037	102	12	coordinates	coordinate	NOUN
ap-9037	102	13	(	(	PUNCT
ap-9037	102	14	18	18	NUM
ap-9037	102	15	)	)	PUNCT
ap-9037	102	16	,	,	PUNCT
ap-9037	102	17	and	and	CCONJ
ap-9037	102	18	substituting	substitute	VERB
ap-9037	102	19	the	the	DET
ap-9037	102	20	potential	potential	ADJ
ap-9037	102	21	(	(	PUNCT
ap-9037	102	22	19	19	NUM
ap-9037	102	23	)	)	PUNCT
ap-9037	102	24	,	,	PUNCT
ap-9037	102	25	we	we	PRON
ap-9037	102	26	obtain	obtain	VERB
ap-9037	102	27	the	the	DET
ap-9037	102	28	schrödinger	schrödinger	ADJ
ap-9037	102	29	equation	equation	NOUN
ap-9037	102	30	hψ	hψ	NOUN
ap-9037	102	31	=	=	SYM
ap-9037	102	32	eψ	eψ	PROPN
ap-9037	102	33	in	in	ADP
ap-9037	102	34	the	the	DET
ap-9037	102	35	form	form	NOUN
ap-9037	102	36	:	:	PUNCT
ap-9037	102	37	0	0	PUNCT
ap-9037	102	38	=	=	SYM
ap-9037	102	39	{	{	PUNCT
ap-9037	102	40	∂2	∂2	X
ap-9037	102	41	∂r2	∂r2	NOUN
ap-9037	102	42	+	+	CCONJ
ap-9037	102	43	2	2	NUM
ap-9037	102	44	r	r	NOUN
ap-9037	102	45	(	(	PUNCT
ap-9037	102	46	1	1	NUM
ap-9037	102	47	+	+	NUM
ap-9037	102	48	ν1	ν1	NOUN
ap-9037	102	49	+	+	CCONJ
ap-9037	102	50	ν2	ν2	NOUN
ap-9037	102	51	+	+	CCONJ
ap-9037	102	52	ν3	ν3	NOUN
ap-9037	102	53	)	)	PUNCT
ap-9037	102	54	∂	∂	NOUN
ap-9037	103	1	∂r	∂r	NOUN
ap-9037	104	1	+	+	NUM
ap-9037	104	2	e	e	NOUN
ap-9037	104	3	−	−	PROPN
ap-9037	104	4	vr(r	vr(r	NOUN
ap-9037	104	5	)	)	PUNCT
ap-9037	105	1	+	+	CCONJ
ap-9037	105	2	1	1	NUM
ap-9037	105	3	r2	r2	NOUN
ap-9037	105	4	sin(θ)2	sin(θ)2	NOUN
ap-9037	105	5	{	{	PUNCT
ap-9037	105	6	∂2	∂2	NOUN
ap-9037	105	7	∂ϕ2	∂ϕ2	NOUN
ap-9037	105	8	−	−	PROPN
ap-9037	105	9	2	2	NUM
ap-9037	105	10	[	[	PUNCT
ap-9037	105	11	ν1	ν1	NOUN
ap-9037	105	12	tan(ϕ	tan(ϕ	NOUN
ap-9037	105	13	)	)	PUNCT
ap-9037	105	14	−	−	PROPN
ap-9037	105	15	ν2	ν2	PROPN
ap-9037	105	16	cot(ϕ	cot(ϕ	PROPN
ap-9037	105	17	)	)	PUNCT
ap-9037	105	18	]	]	PUNCT
ap-9037	105	19	∂	∂	NUM
ap-9037	106	1	∂ϕ	∂ϕ	NUM
ap-9037	106	2	−	−	PROPN
ap-9037	106	3	ν1(1	ν1(1	PROPN
ap-9037	106	4	−	−	PROPN
ap-9037	106	5	r1	r1	PROPN
ap-9037	106	6	)	)	PUNCT
ap-9037	106	7	cos(ϕ)2	cos(ϕ)2	NOUN
ap-9037	106	8	−	−	PROPN
ap-9037	106	9	ν2(1	ν2(1	NOUN
ap-9037	106	10	−	−	NOUN
ap-9037	106	11	r2	r2	NOUN
ap-9037	106	12	)	)	PUNCT
ap-9037	106	13	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	106	14	−	−	NOUN
ap-9037	106	15	vϕ(ϕ	vϕ(ϕ	NOUN
ap-9037	106	16	)	)	PUNCT
ap-9037	106	17	}	}	PUNCT
ap-9037	106	18	(	(	PUNCT
ap-9037	106	19	20	20	NUM
ap-9037	106	20	)	)	PUNCT
ap-9037	107	1	+	+	CCONJ
ap-9037	107	2	1	1	NUM
ap-9037	107	3	r2	r2	NOUN
ap-9037	107	4	{	{	PUNCT
ap-9037	107	5	∂2	∂2	NOUN
ap-9037	107	6	∂θ2	∂θ2	PRON
ap-9037	107	7	−	−	PROPN
ap-9037	107	8	2	2	NUM
ap-9037	107	9	[	[	PUNCT
ap-9037	107	10	ν3	ν3	NOUN
ap-9037	107	11	tan(θ	tan(θ	NOUN
ap-9037	107	12	)	)	PUNCT
ap-9037	107	13	−	−	PROPN
ap-9037	108	1	(	(	PUNCT
ap-9037	108	2	1	1	NUM
ap-9037	108	3	2	2	NUM
ap-9037	108	4	+	+	NUM
ap-9037	108	5	ν1	ν1	NOUN
ap-9037	108	6	+	+	CCONJ
ap-9037	108	7	ν2	ν2	NOUN
ap-9037	108	8	)	)	PUNCT
ap-9037	108	9	cot(θ	cot(θ	PROPN
ap-9037	108	10	)	)	PUNCT
ap-9037	108	11	]	]	PUNCT
ap-9037	108	12	∂	∂	NUM
ap-9037	109	1	∂θ	∂θ	CCONJ
ap-9037	109	2	−	−	PROPN
ap-9037	109	3	ν3(1	ν3(1	PROPN
ap-9037	109	4	−	−	PROPN
ap-9037	109	5	r3	r3	PROPN
ap-9037	109	6	)	)	PUNCT
ap-9037	109	7	cos(θ)2	cos(θ)2	NOUN
ap-9037	109	8	−	−	NOUN
ap-9037	109	9	vθ(θ	vθ(θ	ADP
ap-9037	109	10	)	)	PUNCT
ap-9037	109	11	}	}	PUNCT
ap-9037	109	12	}	}	PUNCT
ap-9037	109	13	ψ(r	ψ(r	PROPN
ap-9037	109	14	,	,	PUNCT
ap-9037	109	15	θ	θ	NOUN
ap-9037	109	16	,	,	PUNCT
ap-9037	109	17	ϕ	ϕ	NOUN
ap-9037	109	18	)	)	PUNCT
ap-9037	109	19	.	.	PUNCT
ap-9037	110	1	recall	recall	VERB
ap-9037	110	2	that	that	SCONJ
ap-9037	110	3	we	we	PRON
ap-9037	110	4	can	can	AUX
ap-9037	110	5	recover	recover	VERB
ap-9037	110	6	the	the	DET
ap-9037	110	7	known	know	VERB
ap-9037	110	8	,	,	PUNCT
ap-9037	110	9	conventional	conventional	ADJ
ap-9037	110	10	schrödinger	schrödinger	ADJ
ap-9037	110	11	equation	equation	NOUN
ap-9037	110	12	by	by	ADP
ap-9037	110	13	removing	remove	VERB
ap-9037	110	14	the	the	DET
ap-9037	110	15	dunkl	dunkl	NOUN
ap-9037	110	16	context	context	NOUN
ap-9037	110	17	through	through	ADP
ap-9037	110	18	the	the	DET
ap-9037	110	19	setting	set	VERB
ap-9037	110	20	ν1	ν1	NOUN
ap-9037	110	21	=	=	SYM
ap-9037	110	22	ν2	ν2	NOUN
ap-9037	110	23	=	=	SYM
ap-9037	110	24	ν3	ν3	NOUN
ap-9037	110	25	=	=	SYM
ap-9037	110	26	0	0	X
ap-9037	110	27	.	.	PUNCT
ap-9037	111	1	next	next	ADV
ap-9037	111	2	,	,	PUNCT
ap-9037	111	3	in	in	ADP
ap-9037	111	4	order	order	NOUN
ap-9037	111	5	to	to	PART
ap-9037	111	6	separate	separate	VERB
ap-9037	111	7	variables	variable	NOUN
ap-9037	111	8	in	in	ADP
ap-9037	111	9	(	(	PUNCT
ap-9037	111	10	20	20	NUM
ap-9037	111	11	)	)	PUNCT
ap-9037	111	12	,	,	PUNCT
ap-9037	111	13	we	we	PRON
ap-9037	111	14	first	first	ADV
ap-9037	111	15	need	need	VERB
ap-9037	111	16	to	to	PART
ap-9037	111	17	apply	apply	VERB
ap-9037	111	18	the	the	DET
ap-9037	111	19	reflection	reflection	NOUN
ap-9037	111	20	operators	operator	NOUN
ap-9037	111	21	to	to	ADP
ap-9037	111	22	the	the	DET
ap-9037	111	23	solution	solution	NOUN
ap-9037	111	24	.	.	PUNCT
ap-9037	112	1	by	by	ADP
ap-9037	112	2	rewriting	rewrite	VERB
ap-9037	112	3	(	(	PUNCT
ap-9037	112	4	14)–(16	14)–(16	NUM
ap-9037	112	5	)	)	PUNCT
ap-9037	112	6	to	to	ADP
ap-9037	112	7	the	the	DET
ap-9037	112	8	case	case	NOUN
ap-9037	112	9	of	of	ADP
ap-9037	112	10	spherical	spherical	ADJ
ap-9037	112	11	coordinates	coordinate	NOUN
ap-9037	112	12	,	,	PUNCT
ap-9037	112	13	we	we	PRON
ap-9037	112	14	obtain	obtain	VERB
ap-9037	112	15	:	:	PUNCT
ap-9037	112	16	r1	r1	PROPN
ap-9037	112	17	ψ(r	ψ(r	PROPN
ap-9037	112	18	,	,	PUNCT
ap-9037	112	19	θ	θ	PROPN
ap-9037	112	20	,	,	PUNCT
ap-9037	112	21	ϕ	ϕ	NOUN
ap-9037	112	22	)	)	PUNCT
ap-9037	112	23	=	=	SYM
ap-9037	113	1	ψ(r	ψ(r	PROPN
ap-9037	113	2	,	,	PUNCT
ap-9037	113	3	θ	θ	PROPN
ap-9037	113	4	,	,	PUNCT
ap-9037	113	5	π	π	PROPN
ap-9037	113	6	−	−	PROPN
ap-9037	113	7	ϕ	ϕ	PROPN
ap-9037	113	8	)	)	PUNCT
ap-9037	113	9	,	,	PUNCT
ap-9037	113	10	(	(	PUNCT
ap-9037	113	11	21	21	NUM
ap-9037	113	12	)	)	PUNCT
ap-9037	113	13	r2	r2	PROPN
ap-9037	113	14	ψ(r	ψ(r	PROPN
ap-9037	113	15	,	,	PUNCT
ap-9037	113	16	θ	θ	PROPN
ap-9037	113	17	,	,	PUNCT
ap-9037	113	18	ϕ	ϕ	NOUN
ap-9037	113	19	)	)	PUNCT
ap-9037	113	20	=	=	SYM
ap-9037	113	21	ψ(r	ψ(r	PROPN
ap-9037	113	22	,	,	PUNCT
ap-9037	113	23	θ	θ	NOUN
ap-9037	113	24	,	,	PUNCT
ap-9037	113	25	−ϕ	−ϕ	ADV
ap-9037	113	26	)	)	PUNCT
ap-9037	113	27	,	,	PUNCT
ap-9037	113	28	(	(	PUNCT
ap-9037	113	29	22	22	X
ap-9037	113	30	)	)	PUNCT
ap-9037	113	31	r3	r3	PROPN
ap-9037	113	32	ψ(r	ψ(r	PROPN
ap-9037	113	33	,	,	PUNCT
ap-9037	113	34	θ	θ	PROPN
ap-9037	113	35	,	,	PUNCT
ap-9037	113	36	ϕ	ϕ	NOUN
ap-9037	113	37	)	)	PUNCT
ap-9037	113	38	=	=	SYM
ap-9037	113	39	ψ(r	ψ(r	PROPN
ap-9037	113	40	,	,	PUNCT
ap-9037	113	41	π	π	PROPN
ap-9037	113	42	−	−	PROPN
ap-9037	113	43	θ	θ	PROPN
ap-9037	113	44	,	,	PUNCT
ap-9037	113	45	ϕ	ϕ	NOUN
ap-9037	113	46	)	)	PUNCT
ap-9037	113	47	.	.	PUNCT
ap-9037	114	1	(	(	PUNCT
ap-9037	114	2	23	23	NUM
ap-9037	114	3	)	)	PUNCT
ap-9037	114	4	without	without	ADP
ap-9037	114	5	further	further	ADJ
ap-9037	114	6	information	information	NOUN
ap-9037	114	7	on	on	ADP
ap-9037	114	8	the	the	DET
ap-9037	114	9	solution	solution	NOUN
ap-9037	114	10	ψ	ψ	NOUN
ap-9037	114	11	of	of	ADP
ap-9037	114	12	equation	equation	NOUN
ap-9037	114	13	(	(	PUNCT
ap-9037	114	14	20	20	NUM
ap-9037	114	15	)	)	PUNCT
ap-9037	114	16	,	,	PUNCT
ap-9037	114	17	application	application	NOUN
ap-9037	114	18	of	of	ADP
ap-9037	114	19	the	the	DET
ap-9037	114	20	actions	action	NOUN
ap-9037	114	21	(	(	PUNCT
ap-9037	114	22	21)–(23	21)–(23	NOUN
ap-9037	114	23	)	)	PUNCT
ap-9037	114	24	would	would	AUX
ap-9037	114	25	render	render	VERB
ap-9037	114	26	our	our	PRON
ap-9037	114	27	equation	equation	NOUN
ap-9037	114	28	in	in	ADP
ap-9037	114	29	a	a	DET
ap-9037	114	30	form	form	NOUN
ap-9037	114	31	that	that	PRON
ap-9037	114	32	does	do	AUX
ap-9037	114	33	not	not	PART
ap-9037	114	34	allow	allow	VERB
ap-9037	114	35	for	for	ADP
ap-9037	114	36	separation	separation	NOUN
ap-9037	114	37	of	of	ADP
ap-9037	114	38	variables	variable	NOUN
ap-9037	114	39	.	.	PUNCT
ap-9037	115	1	therefore	therefore	ADV
ap-9037	115	2	,	,	PUNCT
ap-9037	115	3	we	we	PRON
ap-9037	115	4	impose	impose	VERB
ap-9037	115	5	the	the	DET
ap-9037	115	6	following	follow	VERB
ap-9037	115	7	restrictions	restriction	NOUN
ap-9037	115	8	on	on	ADP
ap-9037	115	9	ψ	ψ	NOUN
ap-9037	115	10	:	:	PUNCT
ap-9037	115	11	rj	rj	PROPN
ap-9037	115	12	ψ(r	ψ(r	PROPN
ap-9037	115	13	,	,	PUNCT
ap-9037	115	14	θ	θ	PROPN
ap-9037	115	15	,	,	PUNCT
ap-9037	115	16	ϕ	ϕ	NOUN
ap-9037	115	17	)	)	PUNCT
ap-9037	115	18	=	=	SYM
ap-9037	115	19	rj	rj	PROPN
ap-9037	115	20	ψ(r	ψ(r	PROPN
ap-9037	115	21	,	,	PUNCT
ap-9037	115	22	θ	θ	PROPN
ap-9037	115	23	,	,	PUNCT
ap-9037	115	24	ϕ	ϕ	NOUN
ap-9037	115	25	)	)	PUNCT
ap-9037	115	26	,	,	PUNCT
ap-9037	115	27	j	j	PROPN
ap-9037	115	28	=	=	SYM
ap-9037	115	29	1	1	NUM
ap-9037	115	30	,	,	PUNCT
ap-9037	115	31	2	2	NUM
ap-9037	115	32	,	,	PUNCT
ap-9037	115	33	3	3	NUM
ap-9037	115	34	,	,	PUNCT
ap-9037	115	35	(	(	PUNCT
ap-9037	115	36	24	24	NUM
ap-9037	115	37	)	)	PUNCT
ap-9037	115	38	introducing	introduce	VERB
ap-9037	115	39	constants	constant	NOUN
ap-9037	115	40	r1	r1	NOUN
ap-9037	115	41	,	,	PUNCT
ap-9037	115	42	r2	r2	PROPN
ap-9037	115	43	,	,	PUNCT
ap-9037	115	44	r3	r3	PROPN
ap-9037	115	45	,	,	PUNCT
ap-9037	115	46	the	the	DET
ap-9037	115	47	existence	existence	NOUN
ap-9037	115	48	and	and	CCONJ
ap-9037	115	49	values	value	NOUN
ap-9037	115	50	of	of	ADP
ap-9037	115	51	which	which	PRON
ap-9037	115	52	must	must	AUX
ap-9037	115	53	be	be	AUX
ap-9037	115	54	determined	determine	VERB
ap-9037	115	55	separately	separately	ADV
ap-9037	115	56	for	for	ADP
ap-9037	115	57	each	each	DET
ap-9037	115	58	particular	particular	ADJ
ap-9037	115	59	potential	potential	NOUN
ap-9037	115	60	(	(	PUNCT
ap-9037	115	61	19	19	NUM
ap-9037	115	62	)	)	PUNCT
ap-9037	115	63	.	.	PUNCT
ap-9037	116	1	we	we	PRON
ap-9037	116	2	refer	refer	VERB
ap-9037	116	3	to	to	ADP
ap-9037	116	4	the	the	DET
ap-9037	116	5	r1	r1	NOUN
ap-9037	116	6	,	,	PUNCT
ap-9037	116	7	r2	r2	PROPN
ap-9037	116	8	,	,	PUNCT
ap-9037	116	9	r3	r3	PROPN
ap-9037	116	10	as	as	ADP
ap-9037	116	11	parity	parity	NOUN
ap-9037	116	12	constants	constant	NOUN
ap-9037	116	13	because	because	SCONJ
ap-9037	116	14	they	they	PRON
ap-9037	116	15	typically	typically	ADV
ap-9037	116	16	dictate	dictate	VERB
ap-9037	116	17	that	that	SCONJ
ap-9037	116	18	the	the	DET
ap-9037	116	19	function	function	NOUN
ap-9037	116	20	ψ	ψ	NOUN
ap-9037	116	21	must	must	AUX
ap-9037	116	22	exhibit	exhibit	VERB
ap-9037	116	23	a	a	DET
ap-9037	116	24	certain	certain	ADJ
ap-9037	116	25	type	type	NOUN
ap-9037	116	26	of	of	ADP
ap-9037	116	27	symmetry	symmetry	NOUN
ap-9037	116	28	or	or	CCONJ
ap-9037	116	29	parity	parity	NOUN
ap-9037	116	30	.	.	PUNCT
ap-9037	117	1	now	now	ADV
ap-9037	117	2	,	,	PUNCT
ap-9037	117	3	after	after	ADP
ap-9037	117	4	substituting	substitute	VERB
ap-9037	117	5	(	(	PUNCT
ap-9037	117	6	24	24	NUM
ap-9037	117	7	)	)	PUNCT
ap-9037	117	8	in	in	ADP
ap-9037	117	9	(	(	PUNCT
ap-9037	117	10	20	20	NUM
ap-9037	117	11	)	)	PUNCT
ap-9037	117	12	,	,	PUNCT
ap-9037	117	13	and	and	CCONJ
ap-9037	117	14	requiring	require	VERB
ap-9037	117	15	the	the	DET
ap-9037	117	16	usual	usual	ADJ
ap-9037	117	17	factored	factored	ADJ
ap-9037	117	18	form	form	NOUN
ap-9037	117	19	of	of	ADP
ap-9037	117	20	the	the	DET
ap-9037	117	21	solution	solution	NOUN
ap-9037	117	22	:	:	PUNCT
ap-9037	117	23	ψ(r	ψ(r	PROPN
ap-9037	117	24	,	,	PUNCT
ap-9037	117	25	θ	θ	PROPN
ap-9037	117	26	,	,	PUNCT
ap-9037	117	27	ϕ	ϕ	NOUN
ap-9037	117	28	)	)	PUNCT
ap-9037	117	29	=	=	SYM
ap-9037	117	30	r(r	r(r	NOUN
ap-9037	117	31	)	)	PUNCT
ap-9037	117	32	θ(θ	θ(θ	VERB
ap-9037	117	33	)	)	PUNCT
ap-9037	117	34	φ(ϕ	φ(ϕ	PROPN
ap-9037	117	35	)	)	PUNCT
ap-9037	117	36	,	,	PUNCT
ap-9037	117	37	we	we	PRON
ap-9037	117	38	can	can	AUX
ap-9037	117	39	separate	separate	VERB
ap-9037	117	40	variables	variable	NOUN
ap-9037	117	41	,	,	PUNCT
ap-9037	117	42	so	so	SCONJ
ap-9037	117	43	that	that	SCONJ
ap-9037	117	44	(	(	PUNCT
ap-9037	117	45	20	20	NUM
ap-9037	117	46	)	)	PUNCT
ap-9037	117	47	decomposes	decompose	NOUN
ap-9037	117	48	into	into	ADP
ap-9037	117	49	three	three	NUM
ap-9037	117	50	equations	equation	NOUN
ap-9037	117	51	.	.	PUNCT
ap-9037	118	1	the	the	DET
ap-9037	118	2	radial	radial	ADJ
ap-9037	118	3	equation	equation	NOUN
ap-9037	118	4	is	be	AUX
ap-9037	118	5	given	give	VERB
ap-9037	118	6	by	by	ADP
ap-9037	118	7	:	:	PUNCT
ap-9037	118	8	r′′(r	r′′(r	NOUN
ap-9037	118	9	)	)	PUNCT
ap-9037	119	1	+	+	CCONJ
ap-9037	119	2	2	2	NUM
ap-9037	119	3	r	r	NOUN
ap-9037	119	4	(	(	PUNCT
ap-9037	119	5	1	1	NUM
ap-9037	119	6	+	+	NUM
ap-9037	119	7	ν1	ν1	NOUN
ap-9037	119	8	+	+	CCONJ
ap-9037	119	9	ν2	ν2	NOUN
ap-9037	119	10	+	+	CCONJ
ap-9037	119	11	ν3	ν3	NOUN
ap-9037	119	12	)	)	PUNCT
ap-9037	119	13	r′(r	r′(r	PROPN
ap-9037	119	14	)	)	PUNCT
ap-9037	120	1	+	+	CCONJ
ap-9037	120	2	[	[	PUNCT
ap-9037	120	3	e	e	NOUN
ap-9037	120	4	−	−	PROPN
ap-9037	120	5	vr(r	vr(r	NOUN
ap-9037	120	6	)	)	PUNCT
ap-9037	120	7	−	−	PROPN
ap-9037	121	1	λ	λ	PROPN
ap-9037	121	2	r2	r2	PROPN
ap-9037	121	3	]	]	PUNCT
ap-9037	121	4	r(r	r(r	PROPN
ap-9037	121	5	)	)	PUNCT
ap-9037	121	6	=	=	SYM
ap-9037	121	7	0	0	NUM
ap-9037	121	8	,	,	PUNCT
ap-9037	121	9	(	(	PUNCT
ap-9037	121	10	25	25	NUM
ap-9037	121	11	)	)	PUNCT
ap-9037	121	12	where	where	SCONJ
ap-9037	121	13	λ	λ	PROPN
ap-9037	121	14	is	be	AUX
ap-9037	121	15	the	the	DET
ap-9037	121	16	separation	separation	NOUN
ap-9037	121	17	constant	constant	ADJ
ap-9037	121	18	.	.	PUNCT
ap-9037	122	1	let	let	VERB
ap-9037	122	2	us	we	PRON
ap-9037	122	3	simplify	simplify	VERB
ap-9037	122	4	the	the	DET
ap-9037	122	5	form	form	NOUN
ap-9037	122	6	of	of	ADP
ap-9037	122	7	this	this	DET
ap-9037	122	8	equation	equation	NOUN
ap-9037	122	9	by	by	ADP
ap-9037	122	10	removing	remove	VERB
ap-9037	122	11	the	the	DET
ap-9037	122	12	first	first	ADJ
ap-9037	122	13	-	-	PUNCT
ap-9037	122	14	order	order	NOUN
ap-9037	122	15	derivative	derivative	ADJ
ap-9037	122	16	term	term	NOUN
ap-9037	122	17	∼	∼	NOUN
ap-9037	122	18	ψ′.	ψ′.	NOUN
ap-9037	122	19	to	to	ADP
ap-9037	122	20	this	this	DET
ap-9037	122	21	end	end	NOUN
ap-9037	122	22	,	,	PUNCT
ap-9037	122	23	we	we	PRON
ap-9037	122	24	set	set	VERB
ap-9037	122	25	:	:	PUNCT
ap-9037	122	26	r(r	r(r	PROPN
ap-9037	122	27	)	)	PUNCT
ap-9037	122	28	=	=	SYM
ap-9037	123	1	1	1	NUM
ap-9037	123	2	rν1+ν2+ν3	rν1+ν2+ν3	NOUN
ap-9037	123	3	s(r	s(r	PROPN
ap-9037	123	4	)	)	PUNCT
ap-9037	123	5	,	,	PUNCT
ap-9037	123	6	which	which	PRON
ap-9037	123	7	,	,	PUNCT
ap-9037	123	8	after	after	ADP
ap-9037	123	9	substitution	substitution	NOUN
ap-9037	123	10	in	in	ADP
ap-9037	123	11	(	(	PUNCT
ap-9037	123	12	25	25	NUM
ap-9037	123	13	)	)	PUNCT
ap-9037	123	14	,	,	PUNCT
ap-9037	123	15	gives	give	VERB
ap-9037	123	16	our	our	PRON
ap-9037	123	17	radial	radial	ADJ
ap-9037	123	18	equation	equation	NOUN
ap-9037	123	19	in	in	ADP
ap-9037	123	20	the	the	DET
ap-9037	123	21	form	form	NOUN
ap-9037	123	22	:	:	PUNCT
ap-9037	123	23	s′′(r	s′′(r	NOUN
ap-9037	123	24	)	)	PUNCT
ap-9037	124	1	+	+	CCONJ
ap-9037	124	2	[	[	PUNCT
ap-9037	124	3	e	e	X
ap-9037	124	4	−	−	PROPN
ap-9037	124	5	λ	λ	PROPN
ap-9037	124	6	r2	r2	PROPN
ap-9037	124	7	+	+	CCONJ
ap-9037	124	8	(	(	PUNCT
ap-9037	124	9	ν1	ν1	NOUN
ap-9037	124	10	+	+	CCONJ
ap-9037	124	11	ν2	ν2	NOUN
ap-9037	124	12	+	+	NUM
ap-9037	124	13	ν3)(ν1	ν3)(ν1	NOUN
ap-9037	124	14	+	+	CCONJ
ap-9037	124	15	ν2	ν2	NOUN
ap-9037	124	16	+	+	CCONJ
ap-9037	124	17	ν3	ν3	NOUN
ap-9037	124	18	−	−	NOUN
ap-9037	124	19	1	1	NUM
ap-9037	124	20	)	)	PUNCT
ap-9037	124	21	r2	r2	NOUN
ap-9037	124	22	−	−	PROPN
ap-9037	124	23	vr(r	vr(r	NOUN
ap-9037	124	24	)	)	PUNCT
ap-9037	124	25	]	]	PUNCT
ap-9037	125	1	s(r	s(r	X
ap-9037	125	2	)	)	PUNCT
ap-9037	125	3	=	=	SYM
ap-9037	125	4	0	0	X
ap-9037	125	5	.	.	PUNCT
ap-9037	126	1	(	(	PUNCT
ap-9037	126	2	26	26	NUM
ap-9037	126	3	)	)	PUNCT
ap-9037	126	4	we	we	PRON
ap-9037	126	5	observe	observe	VERB
ap-9037	126	6	that	that	SCONJ
ap-9037	126	7	the	the	DET
ap-9037	126	8	terms	term	NOUN
ap-9037	126	9	containing	contain	VERB
ap-9037	126	10	the	the	DET
ap-9037	126	11	parameters	parameter	NOUN
ap-9037	126	12	ν1	ν1	NOUN
ap-9037	126	13	,	,	PUNCT
ap-9037	126	14	ν2	ν2	NOUN
ap-9037	126	15	,	,	PUNCT
ap-9037	126	16	and	and	CCONJ
ap-9037	126	17	ν3	ν3	NOUN
ap-9037	126	18	,	,	PUNCT
ap-9037	126	19	can	can	AUX
ap-9037	126	20	be	be	AUX
ap-9037	126	21	formally	formally	ADV
ap-9037	126	22	the	the	DET
ap-9037	126	23	separation	separation	NOUN
ap-9037	126	24	constant	constant	ADJ
ap-9037	126	25	λ	λ	PROPN
ap-9037	126	26	.	.	PUNCT
ap-9037	127	1	as	as	ADP
ap-9037	127	2	such	such	ADJ
ap-9037	127	3	,	,	PUNCT
ap-9037	127	4	we	we	PRON
ap-9037	127	5	can	can	AUX
ap-9037	127	6	conclude	conclude	VERB
ap-9037	127	7	that	that	SCONJ
ap-9037	127	8	the	the	DET
ap-9037	127	9	general	general	ADJ
ap-9037	127	10	form	form	NOUN
ap-9037	127	11	of	of	ADP
ap-9037	127	12	our	our	PRON
ap-9037	127	13	radial	radial	ADJ
ap-9037	127	14	equation	equation	NOUN
ap-9037	127	15	is	be	AUX
ap-9037	127	16	the	the	DET
ap-9037	127	17	same	same	ADJ
ap-9037	127	18	with	with	ADP
ap-9037	127	19	and	and	CCONJ
ap-9037	127	20	without	without	ADP
ap-9037	127	21	the	the	DET
ap-9037	127	22	dunkl	dunkl	NOUN
ap-9037	127	23	context	context	NOUN
ap-9037	127	24	.	.	PUNCT
ap-9037	128	1	for	for	ADP
ap-9037	128	2	this	this	DET
ap-9037	128	3	reason	reason	NOUN
ap-9037	128	4	,	,	PUNCT
ap-9037	128	5	we	we	PRON
ap-9037	128	6	will	will	AUX
ap-9037	128	7	not	not	PART
ap-9037	128	8	consider	consider	VERB
ap-9037	128	9	this	this	DET
ap-9037	128	10	equation	equation	NOUN
ap-9037	128	11	any	any	ADV
ap-9037	128	12	further	far	ADV
ap-9037	128	13	in	in	ADP
ap-9037	128	14	the	the	DET
ap-9037	128	15	present	present	ADJ
ap-9037	128	16	work	work	NOUN
ap-9037	128	17	,	,	PUNCT
ap-9037	128	18	except	except	SCONJ
ap-9037	128	19	when	when	SCONJ
ap-9037	128	20	stating	state	VERB
ap-9037	128	21	examples	example	NOUN
ap-9037	128	22	.	.	PUNCT
ap-9037	129	1	next	next	ADV
ap-9037	129	2	,	,	PUNCT
ap-9037	129	3	the	the	DET
ap-9037	129	4	polar	polar	ADJ
ap-9037	129	5	equation	equation	NOUN
ap-9037	129	6	resulting	result	VERB
ap-9037	129	7	from	from	ADP
ap-9037	129	8	the	the	DET
ap-9037	129	9	separation	separation	NOUN
ap-9037	129	10	of	of	ADP
ap-9037	129	11	(	(	PUNCT
ap-9037	129	12	20	20	NUM
ap-9037	129	13	)	)	PUNCT
ap-9037	129	14	takes	take	VERB
ap-9037	129	15	the	the	DET
ap-9037	129	16	form	form	NOUN
ap-9037	129	17	:	:	PUNCT
ap-9037	129	18	θ′′(θ	θ′′(θ	NUM
ap-9037	129	19	)	)	PUNCT
ap-9037	129	20	−	−	NOUN
ap-9037	129	21	2	2	NUM
ap-9037	129	22	[	[	PUNCT
ap-9037	129	23	ν3	ν3	NOUN
ap-9037	129	24	tan(θ	tan(θ	NOUN
ap-9037	129	25	)	)	PUNCT
ap-9037	129	26	−	−	PROPN
ap-9037	130	1	(	(	PUNCT
ap-9037	130	2	1	1	NUM
ap-9037	130	3	2	2	NUM
ap-9037	130	4	+	+	NUM
ap-9037	130	5	ν1	ν1	NOUN
ap-9037	130	6	+	+	CCONJ
ap-9037	130	7	ν2	ν2	NOUN
ap-9037	130	8	)	)	PUNCT
ap-9037	130	9	cot(θ	cot(θ	PROPN
ap-9037	130	10	)	)	PUNCT
ap-9037	130	11	]	]	PUNCT
ap-9037	131	1	θ′(θ	θ′(θ	ADP
ap-9037	131	2	)	)	PUNCT
ap-9037	131	3	+	+	CCONJ
ap-9037	131	4	[	[	PUNCT
ap-9037	131	5	λ	λ	X
ap-9037	131	6	−	−	NOUN
ap-9037	131	7	m2	m2	PROPN
ap-9037	131	8	sin(θ)2	sin(θ)2	NOUN
ap-9037	131	9	−	−	PROPN
ap-9037	131	10	ν3(1	ν3(1	ADJ
ap-9037	131	11	−	−	PROPN
ap-9037	131	12	r3	r3	PROPN
ap-9037	131	13	)	)	PUNCT
ap-9037	131	14	cos(θ)2	cos(θ)2	NOUN
ap-9037	131	15	−	−	NOUN
ap-9037	131	16	vθ(θ	vθ(θ	NUM
ap-9037	131	17	)	)	PUNCT
ap-9037	131	18	]	]	PUNCT
ap-9037	131	19	θ(θ	θ(θ	VERB
ap-9037	131	20	)	)	PUNCT
ap-9037	131	21	=	=	SYM
ap-9037	131	22	0	0	NUM
ap-9037	131	23	,	,	PUNCT
ap-9037	131	24	(	(	PUNCT
ap-9037	131	25	27	27	NUM
ap-9037	131	26	)	)	SYM
ap-9037	131	27	276	276	NUM
ap-9037	131	28	vol	vol	NOUN
ap-9037	131	29	.	.	PUNCT
ap-9037	132	1	63	63	NUM
ap-9037	132	2	no	no	INTJ
ap-9037	132	3	.	.	PUNCT
ap-9037	133	1	4/2023	4/2023	NOUN
ap-9037	133	2	construction	construction	NOUN
ap-9037	133	3	of	of	ADP
ap-9037	133	4	angular	angular	ADJ
ap-9037	133	5	-	-	PUNCT
ap-9037	133	6	dependent	dependent	ADJ
ap-9037	133	7	potentials	potential	NOUN
ap-9037	133	8	introducing	introduce	VERB
ap-9037	133	9	another	another	DET
ap-9037	133	10	separation	separation	NOUN
ap-9037	133	11	constant	constant	ADJ
ap-9037	133	12	m2	m2	PROPN
ap-9037	133	13	.	.	PUNCT
ap-9037	134	1	finally	finally	ADV
ap-9037	134	2	,	,	PUNCT
ap-9037	134	3	the	the	DET
ap-9037	134	4	azimuthal	azimuthal	ADJ
ap-9037	134	5	equation	equation	NOUN
ap-9037	134	6	can	can	AUX
ap-9037	134	7	be	be	AUX
ap-9037	134	8	written	write	VERB
ap-9037	134	9	as	as	ADP
ap-9037	134	10	:	:	PUNCT
ap-9037	134	11	φ′′(ϕ	φ′′(ϕ	NOUN
ap-9037	134	12	)	)	PUNCT
ap-9037	134	13	−	−	NUM
ap-9037	134	14	2	2	NUM
ap-9037	134	15	[	[	PUNCT
ap-9037	134	16	ν1	ν1	NOUN
ap-9037	134	17	tan(ϕ	tan(ϕ	NOUN
ap-9037	134	18	)	)	PUNCT
ap-9037	134	19	−	−	PROPN
ap-9037	134	20	ν2	ν2	PROPN
ap-9037	134	21	cot(ϕ	cot(ϕ	PROPN
ap-9037	134	22	)	)	PUNCT
ap-9037	134	23	]	]	PUNCT
ap-9037	135	1	φ′(ϕ	φ′(ϕ	PROPN
ap-9037	135	2	)	)	PUNCT
ap-9037	135	3	+	+	CCONJ
ap-9037	135	4	[	[	PUNCT
ap-9037	135	5	m2	m2	PROPN
ap-9037	135	6	−	−	PROPN
ap-9037	135	7	ν1(1	ν1(1	PROPN
ap-9037	135	8	−	−	PROPN
ap-9037	135	9	r1	r1	PROPN
ap-9037	135	10	)	)	PUNCT
ap-9037	135	11	cos(ϕ)2	cos(ϕ)2	NOUN
ap-9037	135	12	−	−	PROPN
ap-9037	135	13	ν2(1	ν2(1	NOUN
ap-9037	135	14	−	−	NOUN
ap-9037	135	15	r2	r2	NOUN
ap-9037	135	16	)	)	PUNCT
ap-9037	135	17	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	135	18	−	−	NOUN
ap-9037	135	19	vϕ(ϕ	vϕ(ϕ	NOUN
ap-9037	135	20	)	)	PUNCT
ap-9037	135	21	]	]	PUNCT
ap-9037	136	1	φ(ϕ	φ(ϕ	PROPN
ap-9037	136	2	)	)	PUNCT
ap-9037	136	3	=	=	SYM
ap-9037	136	4	0	0	PUNCT
ap-9037	136	5	.	.	PUNCT
ap-9037	137	1	(	(	PUNCT
ap-9037	137	2	28	28	NUM
ap-9037	137	3	)	)	PUNCT
ap-9037	137	4	we	we	PRON
ap-9037	137	5	point	point	VERB
ap-9037	137	6	out	out	ADP
ap-9037	137	7	that	that	SCONJ
ap-9037	137	8	the	the	DET
ap-9037	137	9	three	three	NUM
ap-9037	137	10	equations	equation	NOUN
ap-9037	137	11	(	(	PUNCT
ap-9037	137	12	25	25	NUM
ap-9037	137	13	)	)	PUNCT
ap-9037	137	14	,	,	PUNCT
ap-9037	137	15	(	(	PUNCT
ap-9037	137	16	27	27	NUM
ap-9037	137	17	)	)	PUNCT
ap-9037	137	18	and	and	CCONJ
ap-9037	137	19	(	(	PUNCT
ap-9037	137	20	28	28	NUM
ap-9037	137	21	)	)	PUNCT
ap-9037	137	22	turn	turn	VERB
ap-9037	137	23	into	into	ADP
ap-9037	137	24	their	their	PRON
ap-9037	137	25	known	know	VERB
ap-9037	137	26	counterparts	counterpart	NOUN
ap-9037	137	27	if	if	SCONJ
ap-9037	137	28	we	we	PRON
ap-9037	137	29	remove	remove	VERB
ap-9037	137	30	the	the	DET
ap-9037	137	31	dunkl	dunkl	NOUN
ap-9037	137	32	context	context	NOUN
ap-9037	137	33	by	by	ADP
ap-9037	137	34	setting	set	VERB
ap-9037	137	35	ν1	ν1	NOUN
ap-9037	137	36	=	=	SYM
ap-9037	137	37	ν2	ν2	NOUN
ap-9037	137	38	=	=	SYM
ap-9037	137	39	ν3	ν3	NOUN
ap-9037	137	40	=	=	SYM
ap-9037	137	41	0	0	NUM
ap-9037	137	42	.	.	NOUN
ap-9037	138	1	4	4	NUM
ap-9037	138	2	.	.	X
ap-9037	139	1	the	the	DET
ap-9037	139	2	polar	polar	ADJ
ap-9037	139	3	equation	equation	NOUN
ap-9037	139	4	in	in	ADP
ap-9037	139	5	this	this	DET
ap-9037	139	6	section	section	NOUN
ap-9037	139	7	we	we	PRON
ap-9037	139	8	show	show	VERB
ap-9037	139	9	that	that	SCONJ
ap-9037	139	10	equation	equation	NOUN
ap-9037	139	11	(	(	PUNCT
ap-9037	139	12	27	27	NUM
ap-9037	139	13	)	)	PUNCT
ap-9037	139	14	is	be	AUX
ap-9037	139	15	closely	closely	ADV
ap-9037	139	16	related	relate	VERB
ap-9037	139	17	to	to	ADP
ap-9037	139	18	trigonometric	trigonometric	ADJ
ap-9037	139	19	pöschl	pöschl	NOUN
ap-9037	139	20	-	-	PUNCT
ap-9037	139	21	teller	teller	NOUN
ap-9037	139	22	systems	system	NOUN
ap-9037	139	23	.	.	PUNCT
ap-9037	140	1	this	this	DET
ap-9037	140	2	relation	relation	NOUN
ap-9037	140	3	can	can	AUX
ap-9037	140	4	be	be	AUX
ap-9037	140	5	used	use	VERB
ap-9037	140	6	directly	directly	ADV
ap-9037	140	7	to	to	PART
ap-9037	140	8	find	find	VERB
ap-9037	140	9	solutions	solution	NOUN
ap-9037	140	10	of	of	ADP
ap-9037	140	11	the	the	DET
ap-9037	140	12	polar	polar	ADJ
ap-9037	140	13	equation	equation	NOUN
ap-9037	140	14	for	for	ADP
ap-9037	140	15	different	different	ADJ
ap-9037	140	16	potentials	potential	NOUN
ap-9037	140	17	vθ	vθ	VERB
ap-9037	140	18	.	.	PUNCT
ap-9037	141	1	furthermore	furthermore	ADV
ap-9037	141	2	,	,	PUNCT
ap-9037	141	3	these	these	DET
ap-9037	141	4	solutions	solution	NOUN
ap-9037	141	5	can	can	AUX
ap-9037	141	6	be	be	AUX
ap-9037	141	7	generalised	generalise	VERB
ap-9037	141	8	by	by	ADP
ap-9037	141	9	the	the	DET
ap-9037	141	10	application	application	NOUN
ap-9037	141	11	of	of	ADP
ap-9037	141	12	the	the	DET
ap-9037	141	13	darboux	darboux	ADJ
ap-9037	141	14	-	-	PUNCT
ap-9037	141	15	crum	crum	NOUN
ap-9037	141	16	transformation	transformation	NOUN
ap-9037	141	17	.	.	PUNCT
ap-9037	142	1	4.1	4.1	NUM
ap-9037	142	2	.	.	PUNCT
ap-9037	142	3	extended	extend	VERB
ap-9037	142	4	trigonometric	trigonometric	ADJ
ap-9037	142	5	pöschl	pöschl	NOUN
ap-9037	142	6	-	-	PUNCT
ap-9037	142	7	teller	teller	NOUN
ap-9037	142	8	systems	system	NOUN
ap-9037	142	9	in	in	ADP
ap-9037	142	10	order	order	NOUN
ap-9037	142	11	to	to	PART
ap-9037	142	12	observe	observe	VERB
ap-9037	142	13	the	the	DET
ap-9037	142	14	similarity	similarity	NOUN
ap-9037	142	15	between	between	ADP
ap-9037	142	16	equation	equation	NOUN
ap-9037	142	17	(	(	PUNCT
ap-9037	142	18	27	27	NUM
ap-9037	142	19	)	)	PUNCT
ap-9037	142	20	and	and	CCONJ
ap-9037	142	21	trigonometric	trigonometric	ADJ
ap-9037	142	22	pöschl	pöschl	NOUN
ap-9037	142	23	-	-	PUNCT
ap-9037	142	24	teller	teller	NOUN
ap-9037	142	25	systems	system	NOUN
ap-9037	142	26	,	,	PUNCT
ap-9037	142	27	we	we	PRON
ap-9037	142	28	rewrite	rewrite	VERB
ap-9037	142	29	the	the	DET
ap-9037	142	30	solution	solution	NOUN
ap-9037	142	31	θ	θ	PROPN
ap-9037	142	32	in	in	ADP
ap-9037	142	33	terms	term	NOUN
ap-9037	142	34	of	of	ADP
ap-9037	142	35	a	a	DET
ap-9037	142	36	new	new	ADJ
ap-9037	142	37	function	function	NOUN
ap-9037	142	38	η	η	PROPN
ap-9037	142	39	as	as	SCONJ
ap-9037	142	40	follows	follow	VERB
ap-9037	142	41	:	:	PUNCT
ap-9037	142	42	θ(θ	θ(θ	VERB
ap-9037	142	43	)	)	PUNCT
ap-9037	143	1	=	=	SYM
ap-9037	143	2	cos(θ)−ν3	cos(θ)−ν3	PROPN
ap-9037	143	3	sin(θ)−ν1−ν2−	sin(θ)−ν1−ν2−	VERB
ap-9037	143	4	1	1	NUM
ap-9037	143	5	2	2	NUM
ap-9037	143	6	η(θ	η(θ	NOUN
ap-9037	143	7	)	)	PUNCT
ap-9037	143	8	.	.	PUNCT
ap-9037	144	1	(	(	PUNCT
ap-9037	144	2	29	29	NUM
ap-9037	144	3	)	)	PUNCT
ap-9037	144	4	substitution	substitution	NOUN
ap-9037	144	5	renders	render	VERB
ap-9037	144	6	the	the	DET
ap-9037	144	7	polar	polar	ADJ
ap-9037	144	8	equation	equation	NOUN
ap-9037	144	9	(	(	PUNCT
ap-9037	144	10	27	27	NUM
ap-9037	144	11	)	)	PUNCT
ap-9037	144	12	in	in	ADP
ap-9037	144	13	the	the	DET
ap-9037	144	14	form	form	NOUN
ap-9037	144	15	:	:	PUNCT
ap-9037	144	16	η′′(θ	η′′(θ	NOUN
ap-9037	144	17	)	)	PUNCT
ap-9037	144	18	+	+	PUNCT
ap-9037	145	1	[	[	PUNCT
ap-9037	145	2	λ	λ	X
ap-9037	145	3	+	+	CCONJ
ap-9037	145	4	(	(	PUNCT
ap-9037	145	5	1	1	NUM
ap-9037	145	6	+	+	NUM
ap-9037	145	7	2ν1	2ν1	NUM
ap-9037	145	8	+	+	SYM
ap-9037	145	9	2ν2	2ν2	NUM
ap-9037	145	10	+	+	CCONJ
ap-9037	145	11	2ν3)2	2ν3)2	PROPN
ap-9037	145	12	4	4	NUM
ap-9037	145	13	+	+	CCONJ
ap-9037	145	14	(	(	PUNCT
ap-9037	145	15	r3	r3	PROPN
ap-9037	145	16	−	−	PROPN
ap-9037	145	17	ν3)ν3	ν3)ν3	PART
ap-9037	145	18	cos(θ)2	cos(θ)2	NOUN
ap-9037	145	19	+	+	CCONJ
ap-9037	145	20	1	1	NUM
ap-9037	145	21	−	−	NOUN
ap-9037	145	22	4m2	4m2	NUM
ap-9037	145	23	−	−	NOUN
ap-9037	146	1	4(ν1	4(ν1	NOUN
ap-9037	147	1	+	+	CCONJ
ap-9037	147	2	ν2)2	ν2)2	ADP
ap-9037	147	3	4	4	NUM
ap-9037	147	4	sin(θ)2	sin(θ)2	NOUN
ap-9037	147	5	−	−	NOUN
ap-9037	147	6	vθ(θ	vθ(θ	NUM
ap-9037	147	7	)	)	PUNCT
ap-9037	147	8	]	]	PUNCT
ap-9037	148	1	η(θ	η(θ	X
ap-9037	148	2	)	)	PUNCT
ap-9037	148	3	=	=	SYM
ap-9037	148	4	0	0	PUNCT
ap-9037	148	5	.	.	PUNCT
ap-9037	149	1	(	(	PUNCT
ap-9037	149	2	30	30	X
ap-9037	149	3	)	)	PUNCT
ap-9037	149	4	comparison	comparison	NOUN
ap-9037	149	5	with	with	ADP
ap-9037	149	6	(	(	PUNCT
ap-9037	149	7	1	1	X
ap-9037	149	8	)	)	PUNCT
ap-9037	149	9	shows	show	VERB
ap-9037	149	10	that	that	SCONJ
ap-9037	149	11	this	this	DET
ap-9037	149	12	equation	equation	NOUN
ap-9037	149	13	has	have	VERB
ap-9037	149	14	schrödinger	schrödinger	ADJ
ap-9037	149	15	form	form	NOUN
ap-9037	149	16	for	for	ADP
ap-9037	149	17	a	a	DET
ap-9037	149	18	trigonometric	trigonometric	ADJ
ap-9037	149	19	pöschl	pöschl	NOUN
ap-9037	149	20	-	-	PUNCT
ap-9037	149	21	teller	teller	NOUN
ap-9037	149	22	potential	potential	NOUN
ap-9037	149	23	,	,	PUNCT
ap-9037	149	24	extended	extend	VERB
ap-9037	149	25	by	by	ADP
ap-9037	149	26	the	the	DET
ap-9037	149	27	function	function	NOUN
ap-9037	149	28	vθ	vθ	PROPN
ap-9037	149	29	.	.	PUNCT
ap-9037	150	1	as	as	ADP
ap-9037	150	2	such	such	ADJ
ap-9037	150	3	,	,	PUNCT
ap-9037	150	4	any	any	DET
ap-9037	150	5	extension	extension	NOUN
ap-9037	150	6	that	that	PRON
ap-9037	150	7	maintains	maintain	VERB
ap-9037	150	8	solvability	solvability	NOUN
ap-9037	150	9	of	of	ADP
ap-9037	150	10	(	(	PUNCT
ap-9037	150	11	30	30	NUM
ap-9037	150	12	)	)	PUNCT
ap-9037	150	13	,	,	PUNCT
ap-9037	150	14	leads	lead	VERB
ap-9037	150	15	to	to	ADP
ap-9037	150	16	a	a	DET
ap-9037	150	17	new	new	ADJ
ap-9037	150	18	polar	polar	ADJ
ap-9037	150	19	term	term	NOUN
ap-9037	150	20	in	in	ADP
ap-9037	150	21	the	the	DET
ap-9037	150	22	potential	potential	ADJ
ap-9037	150	23	(	(	PUNCT
ap-9037	150	24	19	19	NUM
ap-9037	150	25	)	)	PUNCT
ap-9037	150	26	,	,	PUNCT
ap-9037	150	27	along	along	ADP
ap-9037	150	28	with	with	ADP
ap-9037	150	29	a	a	DET
ap-9037	150	30	solution	solution	NOUN
ap-9037	150	31	of	of	ADP
ap-9037	150	32	the	the	DET
ap-9037	150	33	associated	associated	ADJ
ap-9037	150	34	schrödinger	schrödinger	ADJ
ap-9037	150	35	equation	equation	NOUN
ap-9037	150	36	(	(	PUNCT
ap-9037	150	37	20	20	NUM
ap-9037	150	38	)	)	PUNCT
ap-9037	150	39	.	.	PUNCT
ap-9037	151	1	straightforward	straightforward	ADJ
ap-9037	151	2	examples	example	NOUN
ap-9037	151	3	for	for	ADP
ap-9037	151	4	such	such	ADJ
ap-9037	151	5	extensions	extension	NOUN
ap-9037	151	6	vθ	vθ	VERB
ap-9037	151	7	are	be	AUX
ap-9037	151	8	given	give	VERB
ap-9037	151	9	by	by	ADP
ap-9037	151	10	functions	function	NOUN
ap-9037	151	11	that	that	PRON
ap-9037	151	12	resemble	resemble	VERB
ap-9037	151	13	the	the	DET
ap-9037	151	14	shape	shape	NOUN
ap-9037	151	15	of	of	ADP
ap-9037	151	16	the	the	DET
ap-9037	151	17	trigonometric	trigonometric	ADJ
ap-9037	151	18	pöschl	pöschl	NOUN
ap-9037	151	19	-	-	PUNCT
ap-9037	151	20	teller	teller	NOUN
ap-9037	151	21	potential	potential	NOUN
ap-9037	151	22	,	,	PUNCT
ap-9037	151	23	one	one	NUM
ap-9037	151	24	of	of	ADP
ap-9037	151	25	its	its	PRON
ap-9037	151	26	special	special	ADJ
ap-9037	151	27	cases	case	NOUN
ap-9037	151	28	(	(	PUNCT
ap-9037	151	29	as	as	SCONJ
ap-9037	151	30	will	will	AUX
ap-9037	151	31	be	be	AUX
ap-9037	151	32	shown	show	VERB
ap-9037	151	33	below	below	ADP
ap-9037	151	34	)	)	PUNCT
ap-9037	151	35	,	,	PUNCT
ap-9037	151	36	or	or	CCONJ
ap-9037	151	37	their	their	PRON
ap-9037	151	38	generalisations	generalisation	NOUN
ap-9037	151	39	[	[	X
ap-9037	151	40	23	23	NUM
ap-9037	151	41	]	]	PUNCT
ap-9037	151	42	.	.	PUNCT
ap-9037	152	1	while	while	SCONJ
ap-9037	152	2	the	the	DET
ap-9037	152	3	last	last	ADJ
ap-9037	152	4	case	case	NOUN
ap-9037	152	5	is	be	AUX
ap-9037	152	6	beyond	beyond	ADP
ap-9037	152	7	the	the	DET
ap-9037	152	8	scope	scope	NOUN
ap-9037	152	9	of	of	ADP
ap-9037	152	10	this	this	DET
ap-9037	152	11	article	article	NOUN
ap-9037	152	12	,	,	PUNCT
ap-9037	152	13	let	let	VERB
ap-9037	152	14	us	we	PRON
ap-9037	152	15	consider	consider	VERB
ap-9037	152	16	a	a	DET
ap-9037	152	17	function	function	NOUN
ap-9037	152	18	vθ	vθ	NOUN
ap-9037	152	19	that	that	PRON
ap-9037	152	20	has	have	VERB
ap-9037	152	21	the	the	DET
ap-9037	152	22	shape	shape	NOUN
ap-9037	152	23	of	of	ADP
ap-9037	152	24	a	a	DET
ap-9037	152	25	trigonometric	trigonometric	ADJ
ap-9037	152	26	pöschl	pöschl	NOUN
ap-9037	152	27	-	-	PUNCT
ap-9037	152	28	teller	teller	NOUN
ap-9037	152	29	potential	potential	NOUN
ap-9037	152	30	.	.	PUNCT
ap-9037	153	1	we	we	PRON
ap-9037	153	2	let	let	VERB
ap-9037	153	3	:	:	PUNCT
ap-9037	153	4	vθ(θ	vθ(θ	X
ap-9037	153	5	)	)	PUNCT
ap-9037	154	1	=	=	SYM
ap-9037	154	2	α	α	PROPN
ap-9037	155	1	+	+	X
ap-9037	155	2	β	β	NOUN
ap-9037	155	3	sin(θ)2	sin(θ)2	NOUN
ap-9037	155	4	+	+	CCONJ
ap-9037	155	5	γ	γ	NOUN
ap-9037	155	6	cos(θ)2	cos(θ)2	NOUN
ap-9037	155	7	,	,	PUNCT
ap-9037	155	8	(	(	PUNCT
ap-9037	155	9	31	31	NUM
ap-9037	155	10	)	)	PUNCT
ap-9037	155	11	where	where	SCONJ
ap-9037	155	12	α	α	X
ap-9037	155	13	,	,	PUNCT
ap-9037	155	14	β	β	X
ap-9037	155	15	and	and	CCONJ
ap-9037	155	16	γ	γ	PROPN
ap-9037	155	17	are	be	AUX
ap-9037	155	18	constants	constant	NOUN
ap-9037	155	19	.	.	PUNCT
ap-9037	156	1	observe	observe	VERB
ap-9037	156	2	that	that	SCONJ
ap-9037	156	3	there	there	PRON
ap-9037	156	4	are	be	VERB
ap-9037	156	5	equivalent	equivalent	ADJ
ap-9037	156	6	ways	way	NOUN
ap-9037	156	7	of	of	ADP
ap-9037	156	8	expressing	express	VERB
ap-9037	156	9	this	this	DET
ap-9037	156	10	function	function	NOUN
ap-9037	156	11	,	,	PUNCT
ap-9037	156	12	for	for	ADP
ap-9037	156	13	example	example	NOUN
ap-9037	156	14	,	,	PUNCT
ap-9037	156	15	through	through	ADP
ap-9037	156	16	tangent	tangent	NOUN
ap-9037	156	17	and	and	CCONJ
ap-9037	156	18	cotangent	cotangent	NOUN
ap-9037	156	19	functions	function	NOUN
ap-9037	156	20	.	.	PUNCT
ap-9037	157	1	we	we	PRON
ap-9037	157	2	have	have	VERB
ap-9037	157	3	,	,	PUNCT
ap-9037	157	4	for	for	ADP
ap-9037	157	5	example	example	NOUN
ap-9037	157	6	,	,	PUNCT
ap-9037	157	7	the	the	DET
ap-9037	157	8	two	two	NUM
ap-9037	157	9	following	follow	VERB
ap-9037	157	10	representations	representation	NOUN
ap-9037	157	11	:	:	PUNCT
ap-9037	157	12	vθ(θ	vθ(θ	NUM
ap-9037	157	13	)	)	PUNCT
ap-9037	158	1	=	=	SYM
ap-9037	158	2	α	α	PROPN
ap-9037	159	1	+	+	X
ap-9037	160	1	β	β	X
ap-9037	161	1	+	+	X
ap-9037	161	2	γ	γ	X
ap-9037	161	3	+	+	X
ap-9037	161	4	γ	γ	X
ap-9037	161	5	tan(θ)2	tan(θ)2	NOUN
ap-9037	161	6	+	+	NUM
ap-9037	161	7	β	β	X
ap-9037	161	8	tan(θ)2	tan(θ)2	NOUN
ap-9037	161	9	,	,	PUNCT
ap-9037	161	10	vθ(θ	vθ(θ	X
ap-9037	161	11	)	)	PUNCT
ap-9037	162	1	=	=	SYM
ap-9037	162	2	β	β	X
ap-9037	163	1	+	+	X
ap-9037	163	2	(	(	PUNCT
ap-9037	163	3	α	α	NOUN
ap-9037	163	4	−	−	NOUN
ap-9037	163	5	β	β	NOUN
ap-9037	163	6	+	+	CCONJ
ap-9037	163	7	γ	γ	X
ap-9037	163	8	)	)	PUNCT
ap-9037	163	9	sin(θ)2	sin(θ)2	NOUN
ap-9037	163	10	−	−	NOUN
ap-9037	163	11	α	α	DET
ap-9037	163	12	sin(θ)4	sin(θ)4	ADJ
ap-9037	163	13	sin(θ)2	sin(θ)2	NOUN
ap-9037	163	14	cos(θ)2	cos(θ)2	NOUN
ap-9037	163	15	,	,	PUNCT
ap-9037	163	16	both	both	PRON
ap-9037	163	17	of	of	ADP
ap-9037	163	18	which	which	PRON
ap-9037	163	19	coincide	coincide	NOUN
ap-9037	163	20	with	with	ADP
ap-9037	163	21	(	(	PUNCT
ap-9037	163	22	31	31	NUM
ap-9037	163	23	)	)	PUNCT
ap-9037	163	24	.	.	PUNCT
ap-9037	164	1	by	by	ADP
ap-9037	164	2	substituting	substitute	VERB
ap-9037	164	3	(	(	PUNCT
ap-9037	164	4	31	31	NUM
ap-9037	164	5	)	)	PUNCT
ap-9037	164	6	into	into	ADP
ap-9037	164	7	(	(	PUNCT
ap-9037	164	8	30	30	NUM
ap-9037	164	9	)	)	PUNCT
ap-9037	164	10	,	,	PUNCT
ap-9037	164	11	we	we	PRON
ap-9037	164	12	obtain	obtain	VERB
ap-9037	164	13	:	:	PUNCT
ap-9037	164	14	η′′(θ	η′′(θ	NOUN
ap-9037	164	15	)	)	PUNCT
ap-9037	164	16	+	+	PUNCT
ap-9037	165	1	[	[	PUNCT
ap-9037	165	2	λ	λ	X
ap-9037	165	3	+	+	NUM
ap-9037	165	4	α	α	NOUN
ap-9037	165	5	+	+	CCONJ
ap-9037	165	6	(	(	PUNCT
ap-9037	165	7	1	1	NUM
ap-9037	165	8	+	+	NUM
ap-9037	165	9	2ν1	2ν1	NUM
ap-9037	165	10	+	+	SYM
ap-9037	165	11	2ν2	2ν2	NUM
ap-9037	165	12	+	+	CCONJ
ap-9037	165	13	2ν3)2	2ν3)2	PROPN
ap-9037	165	14	4	4	NUM
ap-9037	165	15	+	+	CCONJ
ap-9037	165	16	(	(	PUNCT
ap-9037	165	17	r3	r3	PROPN
ap-9037	165	18	−	−	PROPN
ap-9037	165	19	ν3)ν3	ν3)ν3	PROPN
ap-9037	165	20	+	+	NUM
ap-9037	165	21	γ	γ	NOUN
ap-9037	165	22	cos(θ)2	cos(θ)2	NOUN
ap-9037	165	23	+	+	CCONJ
ap-9037	165	24	1	1	NUM
ap-9037	165	25	−	−	NOUN
ap-9037	165	26	4m2	4m2	NUM
ap-9037	165	27	−	−	NOUN
ap-9037	165	28	4(ν1	4(ν1	NOUN
ap-9037	166	1	+	+	CCONJ
ap-9037	166	2	ν2)2	ν2)2	PUNCT
ap-9037	166	3	+	+	NUM
ap-9037	166	4	β	β	X
ap-9037	166	5	4	4	NUM
ap-9037	166	6	sin(θ)2	sin(θ)2	NOUN
ap-9037	166	7	]	]	PUNCT
ap-9037	166	8	η(θ	η(θ	NOUN
ap-9037	166	9	)	)	PUNCT
ap-9037	166	10	=	=	SYM
ap-9037	166	11	0	0	PUNCT
ap-9037	166	12	.	.	PUNCT
ap-9037	167	1	(	(	PUNCT
ap-9037	167	2	32	32	NUM
ap-9037	167	3	)	)	PUNCT
ap-9037	167	4	this	this	DET
ap-9037	167	5	equation	equation	NOUN
ap-9037	167	6	is	be	AUX
ap-9037	167	7	exactly	exactly	ADV
ap-9037	167	8	-	-	PUNCT
ap-9037	167	9	solvable	solvable	ADJ
ap-9037	167	10	,	,	PUNCT
ap-9037	167	11	as	as	SCONJ
ap-9037	167	12	we	we	PRON
ap-9037	167	13	can	can	AUX
ap-9037	167	14	see	see	VERB
ap-9037	167	15	from	from	ADP
ap-9037	167	16	the	the	DET
ap-9037	167	17	comparison	comparison	NOUN
ap-9037	167	18	with	with	ADP
ap-9037	167	19	(	(	PUNCT
ap-9037	167	20	1	1	NUM
ap-9037	167	21	)	)	PUNCT
ap-9037	167	22	.	.	PUNCT
ap-9037	168	1	in	in	ADP
ap-9037	168	2	addition	addition	NOUN
ap-9037	168	3	,	,	PUNCT
ap-9037	168	4	renaming	rename	VERB
ap-9037	168	5	the	the	DET
ap-9037	168	6	solution	solution	NOUN
ap-9037	168	7	and	and	CCONJ
ap-9037	168	8	the	the	DET
ap-9037	168	9	variable	variable	NOUN
ap-9037	168	10	,	,	PUNCT
ap-9037	168	11	we	we	PRON
ap-9037	168	12	can	can	AUX
ap-9037	168	13	match	match	VERB
ap-9037	168	14	the	the	DET
ap-9037	168	15	two	two	NUM
ap-9037	168	16	equations	equation	NOUN
ap-9037	168	17	by	by	ADP
ap-9037	168	18	setting	set	VERB
ap-9037	168	19	:	:	PUNCT
ap-9037	168	20	a	a	DET
ap-9037	168	21	=	=	PUNCT
ap-9037	168	22	λ	λ	X
ap-9037	168	23	+	+	NUM
ap-9037	168	24	α	α	NOUN
ap-9037	168	25	+	+	CCONJ
ap-9037	168	26	(	(	PUNCT
ap-9037	168	27	1	1	NUM
ap-9037	168	28	+	+	NUM
ap-9037	168	29	2ν1	2ν1	NUM
ap-9037	168	30	+	+	SYM
ap-9037	168	31	2ν2	2ν2	NUM
ap-9037	168	32	+	+	CCONJ
ap-9037	168	33	2ν3)2	2ν3)2	PROPN
ap-9037	168	34	4	4	NUM
ap-9037	168	35	,	,	PUNCT
ap-9037	168	36	(	(	PUNCT
ap-9037	168	37	33	33	NUM
ap-9037	168	38	)	)	PUNCT
ap-9037	168	39	b	b	NOUN
ap-9037	168	40	=	=	SYM
ap-9037	168	41	1	1	NUM
ap-9037	168	42	4	4	NUM
ap-9037	168	43	−	−	NOUN
ap-9037	168	44	m2	m2	PROPN
ap-9037	168	45	−	−	PROPN
ap-9037	168	46	(	(	PUNCT
ap-9037	168	47	ν1	ν1	NOUN
ap-9037	168	48	+	+	CCONJ
ap-9037	168	49	ν2)2	ν2)2	PUNCT
ap-9037	168	50	+	+	CCONJ
ap-9037	168	51	β	β	X
ap-9037	168	52	4	4	NUM
ap-9037	168	53	,	,	PUNCT
ap-9037	168	54	(	(	PUNCT
ap-9037	168	55	34	34	NUM
ap-9037	168	56	)	)	PUNCT
ap-9037	168	57	c	c	NOUN
ap-9037	168	58	=	=	SYM
ap-9037	168	59	(	(	PUNCT
ap-9037	168	60	r3	r3	PROPN
ap-9037	168	61	−	−	PROPN
ap-9037	168	62	ν3)ν3	ν3)ν3	PROPN
ap-9037	168	63	−	−	PROPN
ap-9037	168	64	γ	γ	X
ap-9037	168	65	.	.	PUNCT
ap-9037	169	1	(	(	PUNCT
ap-9037	169	2	35	35	NUM
ap-9037	169	3	)	)	PUNCT
ap-9037	169	4	277	277	NUM
ap-9037	169	5	axel	axel	PROPN
ap-9037	169	6	schulze	schulze	NOUN
ap-9037	169	7	-	-	PUNCT
ap-9037	169	8	halberg	halberg	PROPN
ap-9037	169	9	acta	acta	PROPN
ap-9037	169	10	polytechnica	polytechnica	PROPN
ap-9037	169	11	hence	hence	ADV
ap-9037	169	12	,	,	PUNCT
ap-9037	169	13	implementing	implement	VERB
ap-9037	169	14	these	these	DET
ap-9037	169	15	settings	setting	NOUN
ap-9037	169	16	in	in	ADP
ap-9037	169	17	(	(	PUNCT
ap-9037	169	18	2	2	X
ap-9037	169	19	)	)	PUNCT
ap-9037	169	20	provides	provide	VERB
ap-9037	169	21	the	the	DET
ap-9037	169	22	general	general	ADJ
ap-9037	169	23	solution	solution	NOUN
ap-9037	169	24	of	of	ADP
ap-9037	169	25	(	(	PUNCT
ap-9037	169	26	32	32	NUM
ap-9037	169	27	)	)	PUNCT
ap-9037	169	28	.	.	PUNCT
ap-9037	170	1	let	let	VERB
ap-9037	170	2	us	we	PRON
ap-9037	170	3	now	now	ADV
ap-9037	170	4	continue	continue	VERB
ap-9037	170	5	by	by	ADP
ap-9037	170	6	presenting	present	VERB
ap-9037	170	7	an	an	DET
ap-9037	170	8	example	example	NOUN
ap-9037	170	9	that	that	PRON
ap-9037	170	10	focuses	focus	VERB
ap-9037	170	11	on	on	ADP
ap-9037	170	12	the	the	DET
ap-9037	170	13	special	special	ADJ
ap-9037	170	14	case	case	NOUN
ap-9037	170	15	of	of	ADP
ap-9037	170	16	vθ	vθ	NOUN
ap-9037	170	17	being	be	AUX
ap-9037	170	18	a	a	DET
ap-9037	170	19	trigonometric	trigonometric	ADJ
ap-9037	170	20	pöschl	pöschl	NOUN
ap-9037	170	21	-	-	PUNCT
ap-9037	170	22	teller	teller	NOUN
ap-9037	170	23	i	i	PRON
ap-9037	170	24	potential	potential	VERB
ap-9037	170	25	.	.	PUNCT
ap-9037	171	1	we	we	PRON
ap-9037	171	2	set	set	VERB
ap-9037	171	3	:	:	PUNCT
ap-9037	171	4	α	α	X
ap-9037	171	5	=	=	PUNCT
ap-9037	171	6	β	β	X
ap-9037	171	7	=	=	SYM
ap-9037	171	8	0	0	PROPN
ap-9037	171	9	.	.	PUNCT
ap-9037	172	1	(	(	PUNCT
ap-9037	172	2	36	36	NUM
ap-9037	172	3	)	)	PUNCT
ap-9037	172	4	in	in	ADP
ap-9037	172	5	addition	addition	NOUN
ap-9037	172	6	,	,	PUNCT
ap-9037	172	7	we	we	PRON
ap-9037	172	8	will	will	AUX
ap-9037	172	9	aim	aim	VERB
ap-9037	172	10	for	for	ADP
ap-9037	172	11	constructing	construct	VERB
ap-9037	172	12	a	a	DET
ap-9037	172	13	solution	solution	NOUN
ap-9037	172	14	to	to	ADP
ap-9037	172	15	our	our	PRON
ap-9037	172	16	polar	polar	ADJ
ap-9037	172	17	equation	equation	NOUN
ap-9037	172	18	(	(	PUNCT
ap-9037	172	19	27	27	NUM
ap-9037	172	20	)	)	PUNCT
ap-9037	172	21	that	that	PRON
ap-9037	172	22	forms	form	VERB
ap-9037	172	23	part	part	NOUN
ap-9037	172	24	of	of	ADP
ap-9037	172	25	a	a	DET
ap-9037	172	26	bound	bound	ADJ
ap-9037	172	27	state	state	NOUN
ap-9037	172	28	to	to	ADP
ap-9037	172	29	(	(	PUNCT
ap-9037	172	30	20	20	NUM
ap-9037	172	31	)	)	PUNCT
ap-9037	172	32	.	.	PUNCT
ap-9037	173	1	keeping	keep	VERB
ap-9037	173	2	this	this	PRON
ap-9037	173	3	in	in	ADP
ap-9037	173	4	mind	mind	NOUN
ap-9037	173	5	,	,	PUNCT
ap-9037	173	6	the	the	DET
ap-9037	173	7	inspection	inspection	NOUN
ap-9037	173	8	of	of	ADP
ap-9037	173	9	the	the	DET
ap-9037	173	10	general	general	ADJ
ap-9037	173	11	solution	solution	NOUN
ap-9037	173	12	(	(	PUNCT
ap-9037	173	13	2	2	X
ap-9037	173	14	)	)	PUNCT
ap-9037	173	15	shows	show	VERB
ap-9037	173	16	that	that	SCONJ
ap-9037	173	17	we	we	PRON
ap-9037	173	18	must	must	AUX
ap-9037	173	19	impose	impose	VERB
ap-9037	173	20	the	the	DET
ap-9037	173	21	condition	condition	NOUN
ap-9037	173	22	c1	c1	NOUN
ap-9037	173	23	=	=	PROPN
ap-9037	173	24	0	0	PROPN
ap-9037	174	1	in	in	ADP
ap-9037	174	2	order	order	NOUN
ap-9037	174	3	to	to	PART
ap-9037	174	4	avoid	avoid	VERB
ap-9037	174	5	singularities	singularity	NOUN
ap-9037	174	6	generated	generate	VERB
ap-9037	174	7	by	by	ADP
ap-9037	174	8	the	the	DET
ap-9037	174	9	coefficient	coefficient	NOUN
ap-9037	174	10	of	of	ADP
ap-9037	174	11	the	the	DET
ap-9037	174	12	hypergeometric	hypergeometric	ADJ
ap-9037	174	13	function	function	NOUN
ap-9037	174	14	.	.	PUNCT
ap-9037	175	1	furthermore	furthermore	ADV
ap-9037	175	2	,	,	PUNCT
ap-9037	175	3	the	the	DET
ap-9037	175	4	hypergeometric	hypergeometric	ADJ
ap-9037	175	5	function	function	NOUN
ap-9037	175	6	itself	itself	PRON
ap-9037	175	7	produces	produce	VERB
ap-9037	175	8	a	a	DET
ap-9037	175	9	singularity	singularity	NOUN
ap-9037	175	10	at	at	ADP
ap-9037	175	11	θ	θ	PROPN
ap-9037	175	12	=	=	SYM
ap-9037	175	13	π/2	π/2	NUM
ap-9037	175	14	,	,	PUNCT
ap-9037	175	15	unless	unless	SCONJ
ap-9037	175	16	it	it	PRON
ap-9037	175	17	degenerates	degenerate	VERB
ap-9037	175	18	to	to	ADP
ap-9037	175	19	a	a	DET
ap-9037	175	20	polynomial	polynomial	NOUN
ap-9037	175	21	.	.	PUNCT
ap-9037	176	1	it	it	PRON
ap-9037	176	2	is	be	AUX
ap-9037	176	3	known	know	VERB
ap-9037	176	4	that	that	SCONJ
ap-9037	176	5	this	this	DET
ap-9037	176	6	case	case	NOUN
ap-9037	176	7	occurs	occur	VERB
ap-9037	176	8	if	if	SCONJ
ap-9037	176	9	its	its	PRON
ap-9037	176	10	first	first	ADJ
ap-9037	176	11	argument	argument	NOUN
ap-9037	176	12	equals	equal	VERB
ap-9037	176	13	to	to	ADP
ap-9037	176	14	a	a	DET
ap-9037	176	15	nonpositive	nonpositive	ADJ
ap-9037	176	16	integer	integer	NOUN
ap-9037	176	17	.	.	PUNCT
ap-9037	177	1	from	from	ADP
ap-9037	177	2	(	(	PUNCT
ap-9037	177	3	2	2	NUM
ap-9037	177	4	)	)	PUNCT
ap-9037	177	5	and	and	CCONJ
ap-9037	177	6	(	(	PUNCT
ap-9037	177	7	3	3	NUM
ap-9037	177	8	)	)	PUNCT
ap-9037	177	9	,	,	PUNCT
ap-9037	177	10	we	we	PRON
ap-9037	177	11	have	have	VERB
ap-9037	177	12	the	the	DET
ap-9037	177	13	condition	condition	NOUN
ap-9037	177	14	:	:	PUNCT
ap-9037	177	15	1	1	NUM
ap-9037	177	16	2	2	NUM
ap-9037	177	17	−	−	NOUN
ap-9037	177	18	√	√	NUM
ap-9037	177	19	a	a	DET
ap-9037	177	20	2	2	NUM
ap-9037	177	21	+	+	CCONJ
ap-9037	177	22	√	√	NUM
ap-9037	177	23	1	1	NUM
ap-9037	177	24	−	−	PROPN
ap-9037	177	25	4b	4b	NOUN
ap-9037	177	26	+	+	CCONJ
ap-9037	178	1	√	√	NUM
ap-9037	178	2	1	1	NUM
ap-9037	178	3	−	−	PROPN
ap-9037	178	4	4c	4c	NOUN
ap-9037	178	5	4	4	NUM
ap-9037	178	6	=	=	SYM
ap-9037	178	7	−n	−n	NOUN
ap-9037	178	8	,	,	PUNCT
ap-9037	178	9	n	n	NOUN
ap-9037	178	10	=	=	SYM
ap-9037	178	11	0	0	NUM
ap-9037	178	12	,	,	PUNCT
ap-9037	178	13	1	1	NUM
ap-9037	178	14	,	,	PUNCT
ap-9037	178	15	2	2	NUM
ap-9037	178	16	,	,	PUNCT
ap-9037	178	17	.	.	PUNCT
ap-9037	178	18	.	.	PUNCT
ap-9037	178	19	.	.	PUNCT
ap-9037	178	20	.	.	PUNCT
ap-9037	179	1	(	(	PUNCT
ap-9037	179	2	37	37	NUM
ap-9037	179	3	)	)	PUNCT
ap-9037	179	4	by	by	ADP
ap-9037	179	5	substituting	substitute	VERB
ap-9037	179	6	(	(	PUNCT
ap-9037	179	7	33)–(35	33)–(35	NUM
ap-9037	179	8	)	)	PUNCT
ap-9037	179	9	,	,	PUNCT
ap-9037	179	10	and	and	CCONJ
ap-9037	179	11	solving	solve	VERB
ap-9037	179	12	for	for	ADP
ap-9037	179	13	the	the	DET
ap-9037	179	14	separation	separation	NOUN
ap-9037	179	15	constant	constant	ADJ
ap-9037	179	16	λ	λ	PROPN
ap-9037	179	17	,	,	PUNCT
ap-9037	179	18	we	we	PRON
ap-9037	179	19	obtain	obtain	VERB
ap-9037	179	20	:	:	PUNCT
ap-9037	179	21	λ	λ	X
ap-9037	179	22	=	=	SYM
ap-9037	180	1	−	−	PROPN
ap-9037	180	2	(	(	PUNCT
ap-9037	180	3	1	1	NUM
ap-9037	180	4	+	+	NUM
ap-9037	180	5	2ν1	2ν1	NUM
ap-9037	180	6	+	+	SYM
ap-9037	180	7	2ν2	2ν2	NUM
ap-9037	180	8	+	+	CCONJ
ap-9037	180	9	2ν3)2	2ν3)2	PROPN
ap-9037	180	10	4	4	NUM
ap-9037	180	11	+	+	NUM
ap-9037	180	12	1	1	NUM
ap-9037	180	13	4	4	NUM
ap-9037	180	14	[	[	PUNCT
ap-9037	180	15	4n	4n	NOUN
ap-9037	180	16	+	+	NOUN
ap-9037	180	17	2	2	NUM
ap-9037	180	18	+	+	SYM
ap-9037	180	19	2	2	NUM
ap-9037	180	20	√	√	NOUN
ap-9037	180	21	m2	m2	PROPN
ap-9037	180	22	+	+	CCONJ
ap-9037	180	23	(	(	PUNCT
ap-9037	180	24	ν1	ν1	NOUN
ap-9037	180	25	+	+	CCONJ
ap-9037	180	26	ν2)2	ν2)2	PUNCT
ap-9037	180	27	+	+	CCONJ
ap-9037	180	28	√	√	NUM
ap-9037	180	29	1	1	NUM
ap-9037	180	30	+	+	NUM
ap-9037	180	31	4γ	4γ	NOUN
ap-9037	180	32	+	+	CCONJ
ap-9037	180	33	4ν3(ν3	4ν3(ν3	NUM
ap-9037	180	34	−	−	PROPN
ap-9037	180	35	r3	r3	PROPN
ap-9037	180	36	)	)	PUNCT
ap-9037	180	37	]	]	PUNCT
ap-9037	180	38	2	2	X
ap-9037	180	39	.	.	PUNCT
ap-9037	181	1	(	(	PUNCT
ap-9037	181	2	38	38	NUM
ap-9037	181	3	)	)	PUNCT
ap-9037	181	4	now	now	ADV
ap-9037	181	5	,	,	PUNCT
ap-9037	181	6	taking	take	VERB
ap-9037	181	7	into	into	ADP
ap-9037	181	8	account	account	NOUN
ap-9037	181	9	the	the	DET
ap-9037	181	10	present	present	ADJ
ap-9037	181	11	settings	setting	NOUN
ap-9037	181	12	(	(	PUNCT
ap-9037	181	13	33)–(38	33)–(38	NUM
ap-9037	181	14	)	)	PUNCT
ap-9037	181	15	,	,	PUNCT
ap-9037	181	16	we	we	PRON
ap-9037	181	17	can	can	AUX
ap-9037	181	18	get	get	VERB
ap-9037	181	19	the	the	DET
ap-9037	181	20	solution	solution	NOUN
ap-9037	181	21	of	of	ADP
ap-9037	181	22	equation	equation	NOUN
ap-9037	181	23	(	(	PUNCT
ap-9037	181	24	32	32	NUM
ap-9037	181	25	)	)	PUNCT
ap-9037	181	26	from	from	ADP
ap-9037	181	27	(	(	PUNCT
ap-9037	181	28	2	2	NUM
ap-9037	181	29	)	)	PUNCT
ap-9037	181	30	and	and	CCONJ
ap-9037	181	31	(	(	PUNCT
ap-9037	181	32	3	3	X
ap-9037	181	33	)	)	PUNCT
ap-9037	181	34	in	in	ADP
ap-9037	181	35	the	the	DET
ap-9037	181	36	form	form	NOUN
ap-9037	181	37	:	:	PUNCT
ap-9037	181	38	η(θ	η(θ	NOUN
ap-9037	181	39	)	)	PUNCT
ap-9037	181	40	=	=	SYM
ap-9037	181	41	sin(θ	sin(θ	PROPN
ap-9037	181	42	)	)	PUNCT
ap-9037	181	43	1	1	NUM
ap-9037	181	44	2	2	NUM
ap-9037	181	45	+	+	CCONJ
ap-9037	181	46	1	1	NUM
ap-9037	181	47	2	2	NUM
ap-9037	181	48	√	√	NUM
ap-9037	181	49	1−4b	1−4b	NUM
ap-9037	181	50	cos(θ	cos(θ	NOUN
ap-9037	181	51	)	)	PUNCT
ap-9037	181	52	1	1	NUM
ap-9037	181	53	2	2	NUM
ap-9037	181	54	+	+	CCONJ
ap-9037	181	55	1	1	NUM
ap-9037	181	56	2	2	NUM
ap-9037	181	57	√	√	NUM
ap-9037	181	58	1−4c	1−4c	NUM
ap-9037	181	59	2f1	2f1	NUM
ap-9037	181	60	[	[	PUNCT
ap-9037	181	61	−	−	NUM
ap-9037	181	62	n	n	CCONJ
ap-9037	181	63	,	,	PUNCT
ap-9037	181	64	1	1	NUM
ap-9037	181	65	2	2	NUM
ap-9037	181	66	+	+	CCONJ
ap-9037	181	67	√	√	DET
ap-9037	181	68	a	a	DET
ap-9037	181	69	2	2	NUM
ap-9037	181	70	+	+	CCONJ
ap-9037	181	71	k+	k+	NOUN
ap-9037	181	72	,	,	PUNCT
ap-9037	181	73	1	1	NUM
ap-9037	181	74	+	+	CCONJ
ap-9037	181	75	√	√	NUM
ap-9037	181	76	1	1	NUM
ap-9037	181	77	−	−	PROPN
ap-9037	181	78	4b	4b	X
ap-9037	181	79	2	2	NUM
ap-9037	181	80	,	,	PUNCT
ap-9037	181	81	sin(θ)2	sin(θ)2	NOUN
ap-9037	181	82	]	]	PUNCT
ap-9037	181	83	,	,	PUNCT
ap-9037	181	84	(	(	PUNCT
ap-9037	181	85	39	39	NUM
ap-9037	181	86	)	)	PUNCT
ap-9037	181	87	where	where	SCONJ
ap-9037	181	88	we	we	PRON
ap-9037	181	89	have	have	AUX
ap-9037	181	90	not	not	PART
ap-9037	181	91	substituted	substitute	VERB
ap-9037	181	92	the	the	DET
ap-9037	181	93	explicit	explicit	ADJ
ap-9037	181	94	form	form	NOUN
ap-9037	181	95	of	of	ADP
ap-9037	181	96	our	our	PRON
ap-9037	181	97	parameters	parameter	NOUN
ap-9037	181	98	,	,	PUNCT
ap-9037	181	99	as	as	SCONJ
ap-9037	181	100	the	the	DET
ap-9037	181	101	resulting	result	VERB
ap-9037	181	102	expression	expression	NOUN
ap-9037	181	103	would	would	AUX
ap-9037	181	104	become	become	VERB
ap-9037	181	105	very	very	ADV
ap-9037	181	106	long	long	ADJ
ap-9037	181	107	.	.	PUNCT
ap-9037	182	1	in	in	ADP
ap-9037	182	2	the	the	DET
ap-9037	182	3	next	next	ADJ
ap-9037	182	4	step	step	NOUN
ap-9037	182	5	,	,	PUNCT
ap-9037	182	6	we	we	PRON
ap-9037	182	7	build	build	VERB
ap-9037	182	8	a	a	DET
ap-9037	182	9	solution	solution	NOUN
ap-9037	182	10	θ	θ	NOUN
ap-9037	182	11	of	of	ADP
ap-9037	182	12	our	our	PRON
ap-9037	182	13	polar	polar	ADJ
ap-9037	182	14	equation	equation	NOUN
ap-9037	182	15	(	(	PUNCT
ap-9037	182	16	28	28	NUM
ap-9037	182	17	)	)	PUNCT
ap-9037	182	18	for	for	ADP
ap-9037	182	19	the	the	DET
ap-9037	182	20	current	current	ADJ
ap-9037	182	21	settings	setting	NOUN
ap-9037	182	22	by	by	ADP
ap-9037	182	23	means	mean	NOUN
ap-9037	182	24	of	of	ADP
ap-9037	182	25	the	the	DET
ap-9037	182	26	point	point	NOUN
ap-9037	182	27	transformation	transformation	NOUN
ap-9037	182	28	(	(	PUNCT
ap-9037	182	29	29	29	NUM
ap-9037	182	30	)	)	PUNCT
ap-9037	182	31	.	.	PUNCT
ap-9037	183	1	this	this	PRON
ap-9037	183	2	gives	give	VERB
ap-9037	183	3	our	our	PRON
ap-9037	183	4	solution	solution	NOUN
ap-9037	183	5	(	(	PUNCT
ap-9037	183	6	39	39	NUM
ap-9037	183	7	)	)	PUNCT
ap-9037	183	8	as	as	ADP
ap-9037	183	9	:	:	PUNCT
ap-9037	183	10	θ(θ	θ(θ	VERB
ap-9037	183	11	)	)	PUNCT
ap-9037	183	12	=	=	SYM
ap-9037	183	13	sin(θ	sin(θ	PROPN
ap-9037	183	14	)	)	PUNCT
ap-9037	183	15	1	1	NUM
ap-9037	183	16	2	2	NUM
ap-9037	183	17	+	+	CCONJ
ap-9037	183	18	1	1	NUM
ap-9037	183	19	2	2	NUM
ap-9037	183	20	√	√	NUM
ap-9037	183	21	1−4b−ν1−ν2−	1−4b−ν1−ν2−	NUM
ap-9037	183	22	1	1	NUM
ap-9037	183	23	2	2	NUM
ap-9037	183	24	cos(θ	cos(θ	PROPN
ap-9037	183	25	)	)	PUNCT
ap-9037	183	26	1	1	NUM
ap-9037	183	27	2	2	NUM
ap-9037	183	28	+	+	CCONJ
ap-9037	183	29	1	1	NUM
ap-9037	183	30	2	2	NUM
ap-9037	183	31	√	√	NUM
ap-9037	183	32	1−4c−ν3	1−4c−ν3	NUM
ap-9037	183	33	×	×	NOUN
ap-9037	183	34	2f1	2f1	NUM
ap-9037	183	35	[	[	PUNCT
ap-9037	183	36	−	−	NUM
ap-9037	183	37	n	n	CCONJ
ap-9037	183	38	,	,	PUNCT
ap-9037	183	39	1	1	NUM
ap-9037	183	40	2	2	NUM
ap-9037	183	41	+	+	CCONJ
ap-9037	183	42	√	√	DET
ap-9037	183	43	a	a	DET
ap-9037	183	44	2	2	NUM
ap-9037	183	45	+	+	CCONJ
ap-9037	183	46	k+	k+	NOUN
ap-9037	183	47	,	,	PUNCT
ap-9037	183	48	1	1	NUM
ap-9037	183	49	+	+	CCONJ
ap-9037	183	50	√	√	NUM
ap-9037	183	51	1	1	NUM
ap-9037	183	52	−	−	PROPN
ap-9037	183	53	4b	4b	X
ap-9037	183	54	2	2	NUM
ap-9037	183	55	,	,	PUNCT
ap-9037	183	56	sin(θ)2	sin(θ)2	NOUN
ap-9037	183	57	]	]	PUNCT
ap-9037	183	58	.	.	PUNCT
ap-9037	184	1	(	(	PUNCT
ap-9037	184	2	40	40	NUM
ap-9037	184	3	)	)	PUNCT
ap-9037	184	4	the	the	DET
ap-9037	184	5	final	final	ADJ
ap-9037	184	6	task	task	NOUN
ap-9037	184	7	consists	consist	VERB
ap-9037	184	8	in	in	ADP
ap-9037	184	9	determining	determine	VERB
ap-9037	184	10	the	the	DET
ap-9037	184	11	constant	constant	ADJ
ap-9037	184	12	r3	r3	NOUN
ap-9037	184	13	that	that	PRON
ap-9037	184	14	is	be	AUX
ap-9037	184	15	inserted	insert	VERB
ap-9037	184	16	in	in	ADP
ap-9037	184	17	(	(	PUNCT
ap-9037	184	18	35	35	NUM
ap-9037	184	19	)	)	PUNCT
ap-9037	184	20	.	.	PUNCT
ap-9037	185	1	recall	recall	VERB
ap-9037	185	2	that	that	SCONJ
ap-9037	185	3	this	this	DET
ap-9037	185	4	constant	constant	ADJ
ap-9037	185	5	results	result	NOUN
ap-9037	185	6	from	from	ADP
ap-9037	185	7	applying	apply	VERB
ap-9037	185	8	the	the	DET
ap-9037	185	9	reflection	reflection	NOUN
ap-9037	185	10	operator	operator	NOUN
ap-9037	185	11	r3	r3	PROPN
ap-9037	185	12	,	,	PUNCT
ap-9037	185	13	as	as	SCONJ
ap-9037	185	14	shown	show	VERB
ap-9037	185	15	in	in	ADP
ap-9037	185	16	(	(	PUNCT
ap-9037	185	17	24	24	NUM
ap-9037	185	18	)	)	PUNCT
ap-9037	185	19	.	.	PUNCT
ap-9037	186	1	we	we	PRON
ap-9037	186	2	can	can	AUX
ap-9037	186	3	determine	determine	VERB
ap-9037	186	4	r3	r3	PROPN
ap-9037	186	5	by	by	ADP
ap-9037	186	6	evaluating	evaluate	VERB
ap-9037	186	7	(	(	PUNCT
ap-9037	186	8	23	23	NUM
ap-9037	186	9	)	)	PUNCT
ap-9037	186	10	,	,	PUNCT
ap-9037	186	11	taking	take	VERB
ap-9037	186	12	into	into	ADP
ap-9037	186	13	account	account	NOUN
ap-9037	186	14	that	that	SCONJ
ap-9037	186	15	it	it	PRON
ap-9037	186	16	only	only	ADV
ap-9037	186	17	acts	act	VERB
ap-9037	186	18	on	on	ADP
ap-9037	186	19	the	the	DET
ap-9037	186	20	function	function	NOUN
ap-9037	186	21	θ	θ	PROPN
ap-9037	186	22	.	.	PUNCT
ap-9037	187	1	since	since	SCONJ
ap-9037	187	2	we	we	PRON
ap-9037	187	3	know	know	VERB
ap-9037	187	4	that	that	PRON
ap-9037	187	5	:	:	PUNCT
ap-9037	187	6	r3	r3	PROPN
ap-9037	187	7	sin(θ	sin(θ	PROPN
ap-9037	187	8	)	)	PUNCT
ap-9037	188	1	=	=	PUNCT
ap-9037	188	2	sin(π	sin(π	PROPN
ap-9037	188	3	−	−	NUM
ap-9037	188	4	θ	θ	NOUN
ap-9037	188	5	)	)	PUNCT
ap-9037	188	6	=	=	SYM
ap-9037	188	7	sin(θ	sin(θ	PROPN
ap-9037	188	8	)	)	PUNCT
ap-9037	188	9	,	,	PUNCT
ap-9037	188	10	r3	r3	PROPN
ap-9037	188	11	cos(θ	cos(θ	PROPN
ap-9037	188	12	)	)	PUNCT
ap-9037	188	13	=	=	SYM
ap-9037	188	14	cos(π	cos(π	NOUN
ap-9037	188	15	−	−	PROPN
ap-9037	188	16	θ	θ	NOUN
ap-9037	188	17	)	)	PUNCT
ap-9037	188	18	=	=	SYM
ap-9037	189	1	−	−	PROPN
ap-9037	189	2	cos(θ	cos(θ	PROPN
ap-9037	189	3	)	)	PUNCT
ap-9037	189	4	,	,	PUNCT
ap-9037	189	5	from	from	ADP
ap-9037	189	6	(	(	PUNCT
ap-9037	189	7	40	40	NUM
ap-9037	189	8	)	)	PUNCT
ap-9037	189	9	and	and	CCONJ
ap-9037	189	10	(	(	PUNCT
ap-9037	189	11	35	35	NUM
ap-9037	189	12	)	)	PUNCT
ap-9037	189	13	we	we	PRON
ap-9037	189	14	obtain	obtain	VERB
ap-9037	190	1	that	that	PRON
ap-9037	190	2	:	:	PUNCT
ap-9037	190	3	r3θ(θ	r3θ(θ	VERB
ap-9037	190	4	)	)	PUNCT
ap-9037	190	5	=	=	PRON
ap-9037	190	6	(	(	PUNCT
ap-9037	190	7	−1	−1	NOUN
ap-9037	190	8	)	)	PUNCT
ap-9037	190	9	1	1	NUM
ap-9037	190	10	2	2	NUM
ap-9037	190	11	+	+	CCONJ
ap-9037	190	12	1	1	NUM
ap-9037	190	13	2	2	NUM
ap-9037	190	14	√	√	PROPN
ap-9037	190	15	1−4c−ν3	1−4c−ν3	NUM
ap-9037	190	16	θ(θ	θ(θ	VERB
ap-9037	190	17	)	)	PUNCT
ap-9037	190	18	=	=	PRON
ap-9037	190	19	(	(	PUNCT
ap-9037	190	20	−1	−1	NOUN
ap-9037	190	21	)	)	PUNCT
ap-9037	190	22	1	1	NUM
ap-9037	190	23	2	2	NUM
ap-9037	190	24	+	+	CCONJ
ap-9037	190	25	1	1	NUM
ap-9037	190	26	2	2	NUM
ap-9037	190	27	√	√	NUM
ap-9037	190	28	1−4(r3−ν3)ν3	1−4(r3−ν3)ν3	NUM
ap-9037	190	29	+	+	PROPN
ap-9037	190	30	4γ−ν3	4γ−ν3	NOUN
ap-9037	190	31	θ(θ	θ(θ	VERB
ap-9037	190	32	)	)	PUNCT
ap-9037	190	33	.	.	PUNCT
ap-9037	191	1	comparison	comparison	NOUN
ap-9037	191	2	of	of	ADP
ap-9037	191	3	this	this	DET
ap-9037	191	4	result	result	NOUN
ap-9037	191	5	with	with	ADP
ap-9037	191	6	(	(	PUNCT
ap-9037	191	7	24	24	NUM
ap-9037	191	8	)	)	PUNCT
ap-9037	191	9	gives	give	VERB
ap-9037	191	10	the	the	DET
ap-9037	191	11	following	follow	VERB
ap-9037	191	12	equation	equation	NOUN
ap-9037	191	13	for	for	ADP
ap-9037	191	14	the	the	DET
ap-9037	191	15	constant	constant	ADJ
ap-9037	191	16	r3	r3	PROPN
ap-9037	191	17	:	:	PUNCT
ap-9037	191	18	r3	r3	PROPN
ap-9037	191	19	=	=	SYM
ap-9037	191	20	(	(	PUNCT
ap-9037	191	21	−1	−1	NOUN
ap-9037	191	22	)	)	PUNCT
ap-9037	191	23	1	1	NUM
ap-9037	191	24	2	2	NUM
ap-9037	191	25	+	+	CCONJ
ap-9037	191	26	1	1	NUM
ap-9037	191	27	2	2	NUM
ap-9037	191	28	√	√	NUM
ap-9037	191	29	1−4(r3−ν3)ν3	1−4(r3−ν3)ν3	NUM
ap-9037	191	30	+	+	PROPN
ap-9037	191	31	4γ−ν3	4γ−ν3	NUM
ap-9037	191	32	.	.	PUNCT
ap-9037	192	1	(	(	PUNCT
ap-9037	192	2	41	41	NUM
ap-9037	192	3	)	)	PUNCT
ap-9037	192	4	since	since	SCONJ
ap-9037	192	5	we	we	PRON
ap-9037	192	6	assume	assume	VERB
ap-9037	192	7	that	that	SCONJ
ap-9037	192	8	r3	r3	PROPN
ap-9037	192	9	is	be	AUX
ap-9037	192	10	real	real	ADV
ap-9037	192	11	-	-	PUNCT
ap-9037	192	12	valued	value	VERB
ap-9037	192	13	,	,	PUNCT
ap-9037	192	14	the	the	DET
ap-9037	192	15	right	right	ADJ
ap-9037	192	16	side	side	NOUN
ap-9037	192	17	of	of	ADP
ap-9037	192	18	this	this	DET
ap-9037	192	19	condition	condition	NOUN
ap-9037	192	20	can	can	AUX
ap-9037	192	21	only	only	ADV
ap-9037	192	22	attain	attain	VERB
ap-9037	192	23	the	the	DET
ap-9037	192	24	two	two	NUM
ap-9037	192	25	values	value	NOUN
ap-9037	192	26	−1	−1	NOUN
ap-9037	192	27	and	and	CCONJ
ap-9037	192	28	1	1	NUM
ap-9037	192	29	,	,	PUNCT
ap-9037	192	30	thus	thus	ADV
ap-9037	192	31	making	make	VERB
ap-9037	192	32	it	it	PRON
ap-9037	192	33	the	the	DET
ap-9037	192	34	only	only	ADJ
ap-9037	192	35	two	two	NUM
ap-9037	192	36	possible	possible	ADJ
ap-9037	192	37	values	value	NOUN
ap-9037	192	38	for	for	ADP
ap-9037	192	39	r3	r3	PROPN
ap-9037	192	40	.	.	PUNCT
ap-9037	193	1	consequently	consequently	ADV
ap-9037	193	2	,	,	PUNCT
ap-9037	193	3	in	in	ADP
ap-9037	193	4	the	the	DET
ap-9037	193	5	case	case	NOUN
ap-9037	193	6	r3	r3	PROPN
ap-9037	193	7	=	=	SYM
ap-9037	193	8	−1	−1	NOUN
ap-9037	193	9	,	,	PUNCT
ap-9037	193	10	the	the	DET
ap-9037	193	11	exponent	exponent	NOUN
ap-9037	193	12	of	of	ADP
ap-9037	193	13	the	the	DET
ap-9037	193	14	right	right	ADJ
ap-9037	193	15	side	side	NOUN
ap-9037	193	16	of	of	ADP
ap-9037	193	17	(	(	PUNCT
ap-9037	193	18	41	41	NUM
ap-9037	193	19	)	)	PUNCT
ap-9037	193	20	must	must	AUX
ap-9037	193	21	be	be	AUX
ap-9037	193	22	an	an	DET
ap-9037	193	23	odd	odd	ADJ
ap-9037	193	24	number	number	NOUN
ap-9037	193	25	,	,	PUNCT
ap-9037	193	26	giving	give	VERB
ap-9037	193	27	the	the	DET
ap-9037	193	28	condition	condition	NOUN
ap-9037	193	29	:	:	PUNCT
ap-9037	193	30	1	1	NUM
ap-9037	193	31	2	2	NUM
ap-9037	193	32	+	+	CCONJ
ap-9037	193	33	1	1	NUM
ap-9037	193	34	2	2	NUM
ap-9037	193	35	√	√	NUM
ap-9037	193	36	1	1	NUM
ap-9037	193	37	+	+	NUM
ap-9037	193	38	4(1	4(1	NUM
ap-9037	193	39	+	+	CCONJ
ap-9037	193	40	ν3)ν3	ν3)ν3	PROPN
ap-9037	193	41	+	+	NUM
ap-9037	193	42	4	4	NUM
ap-9037	193	43	γ	γ	NOUN
ap-9037	193	44	−	−	NOUN
ap-9037	193	45	ν3	ν3	NOUN
ap-9037	193	46	=	=	SYM
ap-9037	193	47	2k	2k	NOUN
ap-9037	193	48	+	+	CCONJ
ap-9037	193	49	1	1	NUM
ap-9037	193	50	,	,	PUNCT
ap-9037	193	51	(	(	PUNCT
ap-9037	193	52	42	42	NUM
ap-9037	193	53	)	)	PUNCT
ap-9037	193	54	where	where	SCONJ
ap-9037	193	55	k	k	PROPN
ap-9037	193	56	denotes	denote	VERB
ap-9037	193	57	any	any	DET
ap-9037	193	58	integer	integer	NOUN
ap-9037	193	59	.	.	PUNCT
ap-9037	194	1	observe	observe	VERB
ap-9037	194	2	further	far	ADV
ap-9037	194	3	that	that	SCONJ
ap-9037	194	4	if	if	SCONJ
ap-9037	194	5	(	(	PUNCT
ap-9037	194	6	42	42	NUM
ap-9037	194	7	)	)	PUNCT
ap-9037	194	8	is	be	AUX
ap-9037	194	9	satisfied	satisfied	ADJ
ap-9037	194	10	,	,	PUNCT
ap-9037	194	11	then	then	ADV
ap-9037	194	12	the	the	DET
ap-9037	194	13	function	function	NOUN
ap-9037	194	14	θ	θ	PROPN
ap-9037	194	15	in	in	ADP
ap-9037	194	16	(	(	PUNCT
ap-9037	194	17	40	40	NUM
ap-9037	194	18	)	)	PUNCT
ap-9037	194	19	is	be	AUX
ap-9037	194	20	odd	odd	ADJ
ap-9037	194	21	with	with	ADP
ap-9037	194	22	respect	respect	NOUN
ap-9037	194	23	to	to	ADP
ap-9037	194	24	the	the	DET
ap-9037	194	25	point	point	NOUN
ap-9037	194	26	θ	θ	NOUN
ap-9037	194	27	=	=	PUNCT
ap-9037	194	28	π/2	π/2	NUM
ap-9037	194	29	.	.	PUNCT
ap-9037	195	1	in	in	ADP
ap-9037	195	2	the	the	DET
ap-9037	195	3	other	other	ADJ
ap-9037	195	4	case	case	NOUN
ap-9037	195	5	r3	r3	PROPN
ap-9037	195	6	=	=	SYM
ap-9037	195	7	1	1	NUM
ap-9037	195	8	,	,	PUNCT
ap-9037	195	9	the	the	DET
ap-9037	195	10	exponent	exponent	NOUN
ap-9037	195	11	must	must	AUX
ap-9037	195	12	be	be	AUX
ap-9037	195	13	an	an	DET
ap-9037	195	14	even	even	ADJ
ap-9037	195	15	number	number	NOUN
ap-9037	195	16	,	,	PUNCT
ap-9037	195	17	so	so	SCONJ
ap-9037	195	18	that	that	SCONJ
ap-9037	195	19	we	we	PRON
ap-9037	195	20	must	must	AUX
ap-9037	195	21	have	have	VERB
ap-9037	195	22	:	:	PUNCT
ap-9037	195	23	1	1	NUM
ap-9037	195	24	2	2	NUM
ap-9037	195	25	+	+	CCONJ
ap-9037	195	26	1	1	NUM
ap-9037	195	27	2	2	NUM
ap-9037	195	28	√	√	NUM
ap-9037	195	29	1	1	NUM
ap-9037	195	30	−	−	PROPN
ap-9037	195	31	4(1	4(1	NOUN
ap-9037	195	32	−	−	PROPN
ap-9037	195	33	ν3)ν3	ν3)ν3	PROPN
ap-9037	195	34	+	+	NUM
ap-9037	195	35	4γ	4γ	NOUN
ap-9037	195	36	−	−	NOUN
ap-9037	195	37	ν3	ν3	NOUN
ap-9037	195	38	=	=	SYM
ap-9037	195	39	2k	2k	NUM
ap-9037	195	40	,	,	PUNCT
ap-9037	195	41	(	(	PUNCT
ap-9037	195	42	43	43	NUM
ap-9037	195	43	)	)	PUNCT
ap-9037	195	44	278	278	NUM
ap-9037	195	45	vol	vol	NOUN
ap-9037	195	46	.	.	PUNCT
ap-9037	196	1	63	63	NUM
ap-9037	197	1	no	no	INTJ
ap-9037	197	2	.	.	PUNCT
ap-9037	198	1	4/2023	4/2023	NOUN
ap-9037	198	2	construction	construction	NOUN
ap-9037	198	3	of	of	ADP
ap-9037	198	4	angular	angular	ADJ
ap-9037	198	5	-	-	PUNCT
ap-9037	198	6	dependent	dependent	ADJ
ap-9037	198	7	potentials	potential	NOUN
ap-9037	198	8	for	for	ADP
ap-9037	198	9	any	any	DET
ap-9037	198	10	integer	integer	NOUN
ap-9037	198	11	k.	k.	PROPN
ap-9037	199	1	if	if	SCONJ
ap-9037	199	2	this	this	DET
ap-9037	199	3	condition	condition	NOUN
ap-9037	199	4	is	be	AUX
ap-9037	199	5	satisfied	satisfied	ADJ
ap-9037	199	6	,	,	PUNCT
ap-9037	199	7	then	then	ADV
ap-9037	199	8	the	the	DET
ap-9037	199	9	function	function	NOUN
ap-9037	199	10	θ	θ	PROPN
ap-9037	199	11	in	in	ADP
ap-9037	199	12	(	(	PUNCT
ap-9037	199	13	40	40	NUM
ap-9037	199	14	)	)	PUNCT
ap-9037	199	15	is	be	AUX
ap-9037	199	16	even	even	ADV
ap-9037	199	17	with	with	ADP
ap-9037	199	18	respect	respect	NOUN
ap-9037	199	19	to	to	ADP
ap-9037	199	20	the	the	DET
ap-9037	199	21	point	point	NOUN
ap-9037	199	22	θ	θ	NOUN
ap-9037	199	23	=	=	PUNCT
ap-9037	199	24	π/2	π/2	NUM
ap-9037	199	25	.	.	PUNCT
ap-9037	200	1	in	in	ADP
ap-9037	200	2	summary	summary	NOUN
ap-9037	200	3	,	,	PUNCT
ap-9037	200	4	the	the	DET
ap-9037	200	5	latter	latter	ADJ
ap-9037	200	6	function	function	NOUN
ap-9037	200	7	is	be	AUX
ap-9037	200	8	a	a	DET
ap-9037	200	9	solution	solution	NOUN
ap-9037	200	10	to	to	ADP
ap-9037	200	11	our	our	PRON
ap-9037	200	12	polar	polar	ADJ
ap-9037	200	13	equation	equation	NOUN
ap-9037	200	14	(	(	PUNCT
ap-9037	200	15	27	27	NUM
ap-9037	200	16	)	)	PUNCT
ap-9037	200	17	for	for	ADP
ap-9037	200	18	the	the	DET
ap-9037	200	19	potential	potential	ADJ
ap-9037	200	20	(	(	PUNCT
ap-9037	200	21	31	31	NUM
ap-9037	200	22	)	)	PUNCT
ap-9037	200	23	with	with	ADP
ap-9037	200	24	the	the	DET
ap-9037	200	25	settings	setting	NOUN
ap-9037	200	26	(	(	PUNCT
ap-9037	200	27	3	3	NUM
ap-9037	200	28	)	)	PUNCT
ap-9037	200	29	and	and	CCONJ
ap-9037	200	30	(	(	PUNCT
ap-9037	200	31	33)–(38	33)–(38	NUM
ap-9037	200	32	)	)	PUNCT
ap-9037	200	33	,	,	PUNCT
ap-9037	200	34	provided	provide	VERB
ap-9037	200	35	the	the	DET
ap-9037	200	36	parameters	parameter	NOUN
ap-9037	200	37	are	be	AUX
ap-9037	200	38	chosen	choose	VERB
ap-9037	200	39	to	to	PART
ap-9037	200	40	comply	comply	VERB
ap-9037	200	41	with	with	ADP
ap-9037	200	42	(	(	PUNCT
ap-9037	200	43	42	42	NUM
ap-9037	200	44	)	)	PUNCT
ap-9037	200	45	or	or	CCONJ
ap-9037	200	46	(	(	PUNCT
ap-9037	200	47	43	43	NUM
ap-9037	200	48	)	)	PUNCT
ap-9037	200	49	.	.	PUNCT
ap-9037	201	1	let	let	VERB
ap-9037	201	2	us	we	PRON
ap-9037	201	3	now	now	ADV
ap-9037	201	4	present	present	ADJ
ap-9037	201	5	examples	example	NOUN
ap-9037	201	6	for	for	ADP
ap-9037	201	7	solutions	solution	NOUN
ap-9037	201	8	of	of	ADP
ap-9037	201	9	our	our	PRON
ap-9037	201	10	polar	polar	ADJ
ap-9037	201	11	equation	equation	NOUN
ap-9037	201	12	for	for	ADP
ap-9037	201	13	the	the	DET
ap-9037	201	14	overall	overall	ADJ
ap-9037	201	15	settings	setting	NOUN
ap-9037	201	16	:	:	PUNCT
ap-9037	201	17	ν1	ν1	NOUN
ap-9037	201	18	=	=	SYM
ap-9037	201	19	ν2	ν2	NOUN
ap-9037	201	20	=	=	SYM
ap-9037	201	21	ν3	ν3	NOUN
ap-9037	201	22	=	=	SYM
ap-9037	201	23	1	1	NUM
ap-9037	201	24	2	2	NUM
ap-9037	201	25	;	;	PUNCT
ap-9037	201	26	m	m	VERB
ap-9037	201	27	=	=	SYM
ap-9037	201	28	5	5	NUM
ap-9037	201	29	.	.	PUNCT
ap-9037	202	1	(	(	PUNCT
ap-9037	202	2	44	44	NUM
ap-9037	202	3	)	)	PUNCT
ap-9037	202	4	odd	odd	ADJ
ap-9037	202	5	-	-	PUNCT
ap-9037	202	6	parity	parity	NOUN
ap-9037	202	7	solutions	solution	NOUN
ap-9037	202	8	are	be	AUX
ap-9037	202	9	obtained	obtain	VERB
ap-9037	202	10	by	by	ADP
ap-9037	202	11	setting	set	VERB
ap-9037	202	12	r3	r3	NOUN
ap-9037	202	13	=	=	SYM
ap-9037	202	14	−1	−1	NOUN
ap-9037	202	15	and	and	CCONJ
ap-9037	202	16	γ	γ	X
ap-9037	202	17	=	=	SYM
ap-9037	202	18	8	8	NUM
ap-9037	202	19	,	,	PUNCT
ap-9037	202	20	note	note	VERB
ap-9037	202	21	that	that	SCONJ
ap-9037	202	22	these	these	DET
ap-9037	202	23	parameter	parameter	NOUN
ap-9037	202	24	values	value	NOUN
ap-9037	202	25	satisfy	satisfy	VERB
ap-9037	202	26	the	the	DET
ap-9037	202	27	constraint	constraint	NOUN
ap-9037	202	28	(	(	PUNCT
ap-9037	202	29	42	42	NUM
ap-9037	202	30	)	)	PUNCT
ap-9037	202	31	for	for	ADP
ap-9037	202	32	k	k	PROPN
ap-9037	202	33	=	=	SYM
ap-9037	202	34	1	1	X
ap-9037	202	35	.	.	PUNCT
ap-9037	203	1	substituting	substitute	VERB
ap-9037	203	2	the	the	DET
ap-9037	203	3	latter	latter	ADJ
ap-9037	203	4	values	value	NOUN
ap-9037	203	5	and	and	CCONJ
ap-9037	203	6	(	(	PUNCT
ap-9037	203	7	44	44	NUM
ap-9037	203	8	)	)	PUNCT
ap-9037	203	9	in	in	ADP
ap-9037	203	10	combination	combination	NOUN
ap-9037	203	11	with	with	ADP
ap-9037	203	12	(	(	PUNCT
ap-9037	203	13	33)–(36	33)–(36	NUM
ap-9037	203	14	)	)	PUNCT
ap-9037	203	15	into	into	ADP
ap-9037	203	16	(	(	PUNCT
ap-9037	203	17	40	40	NUM
ap-9037	203	18	)	)	PUNCT
ap-9037	203	19	gives	give	VERB
ap-9037	203	20	:	:	PUNCT
ap-9037	203	21	θ(θ	θ(θ	VERB
ap-9037	203	22	)	)	PUNCT
ap-9037	203	23	=	=	SYM
ap-9037	203	24	sin(θ)−1	sin(θ)−1	NOUN
ap-9037	203	25	+	+	CCONJ
ap-9037	203	26	√	√	PROPN
ap-9037	203	27	26	26	NUM
ap-9037	203	28	cos(θ)3	cos(θ)3	NOUN
ap-9037	203	29	2f1	2f1	NUM
ap-9037	203	30	[	[	PUNCT
ap-9037	203	31	−	−	NUM
ap-9037	203	32	n	n	CCONJ
ap-9037	203	33	,	,	PUNCT
ap-9037	203	34	n	n	PROPN
ap-9037	203	35	+	+	CCONJ
ap-9037	203	36	4	4	NUM
ap-9037	203	37	+	+	CCONJ
ap-9037	203	38	√	√	NUM
ap-9037	203	39	26	26	NUM
ap-9037	203	40	,	,	PUNCT
ap-9037	203	41	1	1	NUM
ap-9037	203	42	+	+	CCONJ
ap-9037	203	43	√	√	NUM
ap-9037	203	44	26	26	NUM
ap-9037	203	45	,	,	PUNCT
ap-9037	203	46	sin(θ)2	sin(θ)2	NOUN
ap-9037	203	47	]	]	PUNCT
ap-9037	203	48	.	.	PUNCT
ap-9037	204	1	(	(	PUNCT
ap-9037	204	2	45	45	NUM
ap-9037	204	3	)	)	PUNCT
ap-9037	204	4	graphs	graph	NOUN
ap-9037	204	5	of	of	ADP
ap-9037	204	6	this	this	DET
ap-9037	204	7	function	function	NOUN
ap-9037	204	8	for	for	ADP
ap-9037	204	9	different	different	ADJ
ap-9037	204	10	values	value	NOUN
ap-9037	204	11	of	of	ADP
ap-9037	204	12	n	n	PRON
ap-9037	204	13	can	can	AUX
ap-9037	204	14	be	be	AUX
ap-9037	204	15	found	find	VERB
ap-9037	204	16	in	in	ADP
ap-9037	204	17	figure	figure	NOUN
ap-9037	204	18	1	1	NUM
ap-9037	204	19	.	.	PUNCT
ap-9037	205	1	for	for	ADP
ap-9037	205	2	the	the	DET
ap-9037	205	3	even	even	ADJ
ap-9037	205	4	case	case	NOUN
ap-9037	205	5	,	,	PUNCT
ap-9037	205	6	we	we	PRON
ap-9037	205	7	set	set	VERB
ap-9037	205	8	r3	r3	PROPN
ap-9037	205	9	=	=	SYM
ap-9037	205	10	1	1	NUM
ap-9037	205	11	,	,	PUNCT
ap-9037	205	12	γ	γ	NOUN
ap-9037	205	13	=	=	SYM
ap-9037	205	14	4	4	NUM
ap-9037	205	15	,	,	PUNCT
ap-9037	205	16	and	and	CCONJ
ap-9037	205	17	keep	keep	VERB
ap-9037	205	18	the	the	DET
ap-9037	205	19	remaining	remain	VERB
ap-9037	205	20	parameters	parameter	NOUN
ap-9037	205	21	at	at	ADP
ap-9037	205	22	the	the	DET
ap-9037	205	23	same	same	ADJ
ap-9037	205	24	values	value	NOUN
ap-9037	205	25	they	they	PRON
ap-9037	205	26	were	be	AUX
ap-9037	205	27	assigned	assign	VERB
ap-9037	205	28	in	in	ADP
ap-9037	205	29	the	the	DET
ap-9037	205	30	previous	previous	ADJ
ap-9037	205	31	case	case	NOUN
ap-9037	205	32	.	.	PUNCT
ap-9037	206	1	from	from	ADP
ap-9037	206	2	(	(	PUNCT
ap-9037	206	3	40	40	NUM
ap-9037	206	4	)	)	PUNCT
ap-9037	206	5	,	,	PUNCT
ap-9037	206	6	we	we	PRON
ap-9037	206	7	obtain	obtain	VERB
ap-9037	206	8	that	that	DET
ap-9037	206	9	:	:	PUNCT
ap-9037	206	10	θ(θ	θ(θ	VERB
ap-9037	206	11	)	)	PUNCT
ap-9037	206	12	=	=	SYM
ap-9037	206	13	sin(θ)−1	sin(θ)−1	NOUN
ap-9037	206	14	+	+	CCONJ
ap-9037	206	15	√	√	PROPN
ap-9037	206	16	26	26	NUM
ap-9037	206	17	cos(θ)2	cos(θ)2	NOUN
ap-9037	206	18	2f1	2f1	NUM
ap-9037	206	19	[	[	PUNCT
ap-9037	206	20	−	−	NUM
ap-9037	206	21	n	n	CCONJ
ap-9037	206	22	,	,	PUNCT
ap-9037	206	23	n	n	PROPN
ap-9037	206	24	+	+	CCONJ
ap-9037	206	25	3	3	NUM
ap-9037	206	26	+	+	CCONJ
ap-9037	206	27	√	√	NUM
ap-9037	206	28	26	26	NUM
ap-9037	206	29	,	,	PUNCT
ap-9037	206	30	1	1	NUM
ap-9037	206	31	+	+	CCONJ
ap-9037	206	32	√	√	NUM
ap-9037	206	33	26	26	NUM
ap-9037	206	34	,	,	PUNCT
ap-9037	206	35	sin(θ)2	sin(θ)2	NOUN
ap-9037	206	36	]	]	PUNCT
ap-9037	206	37	.	.	PUNCT
ap-9037	207	1	(	(	PUNCT
ap-9037	207	2	46	46	NUM
ap-9037	207	3	)	)	PUNCT
ap-9037	207	4	0.5	0.5	NUM
ap-9037	207	5	1.0	1.0	NUM
ap-9037	207	6	1.5	1.5	NUM
ap-9037	207	7	2.0	2.0	NUM
ap-9037	207	8	2.5	2.5	NUM
ap-9037	207	9	3.0	3.0	NUM
ap-9037	207	10	θ	θ	NOUN
ap-9037	207	11	-0.5	-0.5	X
ap-9037	207	12	0.5	0.5	NUM
ap-9037	207	13	qhθl	qhθl	NOUN
ap-9037	207	14	n	n	NOUN
ap-9037	207	15	=	=	SYM
ap-9037	207	16	2	2	NUM
ap-9037	207	17	n	n	NOUN
ap-9037	207	18	=	=	SYM
ap-9037	207	19	1	1	NUM
ap-9037	207	20	n	n	NOUN
ap-9037	207	21	=	=	PRON
ap-9037	207	22	figure	figure	NOUN
ap-9037	207	23	1	1	NUM
ap-9037	207	24	.	.	PUNCT
ap-9037	208	1	graphs	graph	NOUN
ap-9037	208	2	of	of	ADP
ap-9037	208	3	odd	odd	ADJ
ap-9037	208	4	solutions	solution	NOUN
ap-9037	208	5	(	(	PUNCT
ap-9037	208	6	45	45	NUM
ap-9037	208	7	)	)	PUNCT
ap-9037	208	8	for	for	ADP
ap-9037	208	9	different	different	ADJ
ap-9037	208	10	values	value	NOUN
ap-9037	208	11	of	of	ADP
ap-9037	208	12	n	n	PROPN
ap-9037	208	13	.	.	PUNCT
ap-9037	209	1	figure	figure	NOUN
ap-9037	209	2	2	2	NUM
ap-9037	209	3	shows	show	VERB
ap-9037	209	4	graphs	graph	NOUN
ap-9037	209	5	of	of	ADP
ap-9037	209	6	this	this	DET
ap-9037	209	7	solution	solution	NOUN
ap-9037	209	8	for	for	ADP
ap-9037	209	9	several	several	ADJ
ap-9037	209	10	values	value	NOUN
ap-9037	209	11	of	of	ADP
ap-9037	209	12	the	the	DET
ap-9037	209	13	parameter	parameter	NOUN
ap-9037	209	14	n	n	PROPN
ap-9037	209	15	.	.	PUNCT
ap-9037	210	1	0.5	0.5	NUM
ap-9037	210	2	1.0	1.0	NUM
ap-9037	210	3	1.5	1.5	NUM
ap-9037	210	4	2.0	2.0	NUM
ap-9037	210	5	2.5	2.5	NUM
ap-9037	210	6	3.0	3.0	NUM
ap-9037	210	7	θ	θ	PROPN
ap-9037	210	8	-0.6	-0.6	X
ap-9037	210	9	-0.4	-0.4	X
ap-9037	210	10	-0.2	-0.2	PROPN
ap-9037	210	11	0.2	0.2	NUM
ap-9037	210	12	0.4	0.4	NUM
ap-9037	210	13	0.6	0.6	NUM
ap-9037	210	14	0.8	0.8	NUM
ap-9037	210	15	qhθl	qhθl	NOUN
ap-9037	210	16	n	n	NOUN
ap-9037	210	17	=	=	SYM
ap-9037	210	18	2	2	NUM
ap-9037	210	19	n	n	NOUN
ap-9037	210	20	=	=	SYM
ap-9037	210	21	1	1	NUM
ap-9037	210	22	n	n	NOUN
ap-9037	210	23	=	=	SYM
ap-9037	210	24	0	0	NUM
ap-9037	210	25	figure	figure	NOUN
ap-9037	210	26	2	2	NUM
ap-9037	210	27	.	.	PUNCT
ap-9037	210	28	graphs	graph	NOUN
ap-9037	210	29	of	of	ADP
ap-9037	210	30	even	even	ADV
ap-9037	210	31	solutions	solution	NOUN
ap-9037	210	32	(	(	PUNCT
ap-9037	210	33	46	46	NUM
ap-9037	210	34	)	)	PUNCT
ap-9037	210	35	for	for	ADP
ap-9037	210	36	different	different	ADJ
ap-9037	210	37	values	value	NOUN
ap-9037	210	38	of	of	ADP
ap-9037	210	39	n	n	NOUN
ap-9037	210	40	.	.	PUNCT
ap-9037	211	1	4.2	4.2	NUM
ap-9037	211	2	.	.	PUNCT
ap-9037	212	1	the	the	DET
ap-9037	212	2	darboux	darboux	VERB
ap-9037	212	3	-	-	PUNCT
ap-9037	212	4	crum	crum	NOUN
ap-9037	212	5	transformation	transformation	NOUN
ap-9037	212	6	in	in	ADP
ap-9037	212	7	addition	addition	NOUN
ap-9037	212	8	to	to	ADP
ap-9037	212	9	using	use	VERB
ap-9037	212	10	results	result	NOUN
ap-9037	212	11	on	on	ADP
ap-9037	212	12	extended	extended	ADJ
ap-9037	212	13	pöschl	pöschl	NOUN
ap-9037	212	14	-	-	PUNCT
ap-9037	212	15	teller	teller	NOUN
ap-9037	212	16	systems	system	NOUN
ap-9037	212	17	directly	directly	ADV
ap-9037	212	18	in	in	ADP
ap-9037	212	19	the	the	DET
ap-9037	212	20	polar	polar	ADJ
ap-9037	212	21	equation	equation	NOUN
ap-9037	212	22	(	(	PUNCT
ap-9037	212	23	27	27	NUM
ap-9037	212	24	)	)	PUNCT
ap-9037	212	25	,	,	PUNCT
ap-9037	212	26	we	we	PRON
ap-9037	212	27	can	can	AUX
ap-9037	212	28	also	also	ADV
ap-9037	212	29	generate	generate	VERB
ap-9037	212	30	solvable	solvable	ADJ
ap-9037	212	31	cases	case	NOUN
ap-9037	212	32	of	of	ADP
ap-9037	212	33	the	the	DET
ap-9037	212	34	latter	latter	ADJ
ap-9037	212	35	equation	equation	NOUN
ap-9037	212	36	by	by	ADP
ap-9037	212	37	means	mean	NOUN
ap-9037	212	38	of	of	ADP
ap-9037	212	39	the	the	DET
ap-9037	212	40	darboux	darboux	ADJ
ap-9037	212	41	-	-	PUNCT
ap-9037	212	42	crum	crum	NOUN
ap-9037	212	43	transformation	transformation	NOUN
ap-9037	212	44	,	,	PUNCT
ap-9037	212	45	if	if	SCONJ
ap-9037	212	46	we	we	PRON
ap-9037	212	47	consider	consider	VERB
ap-9037	212	48	our	our	PRON
ap-9037	212	49	polar	polar	ADJ
ap-9037	212	50	equation	equation	NOUN
ap-9037	212	51	in	in	ADP
ap-9037	212	52	the	the	DET
ap-9037	212	53	form	form	NOUN
ap-9037	212	54	(	(	PUNCT
ap-9037	212	55	30	30	NUM
ap-9037	212	56	)	)	PUNCT
ap-9037	212	57	.	.	PUNCT
ap-9037	213	1	as	as	SCONJ
ap-9037	213	2	will	will	AUX
ap-9037	213	3	be	be	AUX
ap-9037	213	4	discussed	discuss	VERB
ap-9037	213	5	below	below	ADV
ap-9037	213	6	,	,	PUNCT
ap-9037	213	7	this	this	DET
ap-9037	213	8	equation	equation	NOUN
ap-9037	213	9	matches	match	VERB
ap-9037	213	10	the	the	DET
ap-9037	213	11	form	form	NOUN
ap-9037	213	12	of	of	ADP
ap-9037	213	13	(	(	PUNCT
ap-9037	213	14	4	4	NUM
ap-9037	213	15	)	)	PUNCT
ap-9037	213	16	,	,	PUNCT
ap-9037	213	17	so	so	SCONJ
ap-9037	213	18	that	that	SCONJ
ap-9037	213	19	the	the	DET
ap-9037	213	20	darboux	darboux	VERB
ap-9037	213	21	-	-	PUNCT
ap-9037	213	22	crum	crum	NOUN
ap-9037	213	23	transformation	transformation	NOUN
ap-9037	213	24	is	be	AUX
ap-9037	213	25	applicable	applicable	ADJ
ap-9037	213	26	.	.	PUNCT
ap-9037	214	1	it	it	PRON
ap-9037	214	2	is	be	AUX
ap-9037	214	3	important	important	ADJ
ap-9037	214	4	to	to	PART
ap-9037	214	5	stress	stress	VERB
ap-9037	214	6	that	that	SCONJ
ap-9037	214	7	the	the	DET
ap-9037	214	8	transformation	transformation	NOUN
ap-9037	214	9	parameter	parameter	NOUN
ap-9037	214	10	,	,	PUNCT
ap-9037	214	11	in	in	ADP
ap-9037	214	12	our	our	PRON
ap-9037	214	13	case	case	NOUN
ap-9037	214	14	,	,	PUNCT
ap-9037	214	15	is	be	AUX
ap-9037	214	16	not	not	PART
ap-9037	214	17	the	the	DET
ap-9037	214	18	stationary	stationary	ADJ
ap-9037	214	19	energy	energy	NOUN
ap-9037	214	20	e	e	NOUN
ap-9037	214	21	of	of	ADP
ap-9037	214	22	the	the	DET
ap-9037	214	23	system	system	NOUN
ap-9037	214	24	,	,	PUNCT
ap-9037	214	25	as	as	SCONJ
ap-9037	214	26	this	this	DET
ap-9037	214	27	energy	energy	NOUN
ap-9037	214	28	does	do	AUX
ap-9037	214	29	not	not	PART
ap-9037	214	30	enter	enter	VERB
ap-9037	214	31	in	in	ADP
ap-9037	214	32	the	the	DET
ap-9037	214	33	polar	polar	ADJ
ap-9037	214	34	equation	equation	NOUN
ap-9037	214	35	.	.	PUNCT
ap-9037	215	1	279	279	NUM
ap-9037	215	2	axel	axel	PROPN
ap-9037	215	3	schulze	schulze	NOUN
ap-9037	215	4	-	-	PUNCT
ap-9037	215	5	halberg	halberg	PROPN
ap-9037	215	6	acta	acta	PROPN
ap-9037	215	7	polytechnica	polytechnica	PROPN
ap-9037	215	8	instead	instead	ADV
ap-9037	215	9	,	,	PUNCT
ap-9037	215	10	we	we	PRON
ap-9037	215	11	will	will	AUX
ap-9037	215	12	use	use	VERB
ap-9037	215	13	the	the	DET
ap-9037	215	14	separation	separation	NOUN
ap-9037	215	15	constants	constant	NOUN
ap-9037	215	16	λ	λ	PROPN
ap-9037	215	17	and	and	CCONJ
ap-9037	215	18	m2	m2	PROPN
ap-9037	215	19	as	as	ADP
ap-9037	215	20	our	our	PRON
ap-9037	215	21	transformation	transformation	NOUN
ap-9037	215	22	parameters	parameter	NOUN
ap-9037	215	23	.	.	PUNCT
ap-9037	216	1	more	more	ADV
ap-9037	216	2	precisely	precisely	ADV
ap-9037	216	3	,	,	PUNCT
ap-9037	216	4	we	we	PRON
ap-9037	216	5	will	will	AUX
ap-9037	216	6	now	now	ADV
ap-9037	216	7	show	show	VERB
ap-9037	216	8	that	that	SCONJ
ap-9037	216	9	each	each	PRON
ap-9037	216	10	of	of	ADP
ap-9037	216	11	these	these	DET
ap-9037	216	12	two	two	NUM
ap-9037	216	13	separation	separation	NOUN
ap-9037	216	14	constants	constant	NOUN
ap-9037	216	15	can	can	AUX
ap-9037	216	16	be	be	AUX
ap-9037	216	17	used	use	VERB
ap-9037	216	18	as	as	ADP
ap-9037	216	19	a	a	DET
ap-9037	216	20	transformation	transformation	NOUN
ap-9037	216	21	parameter	parameter	NOUN
ap-9037	216	22	for	for	ADP
ap-9037	216	23	a	a	DET
ap-9037	216	24	darboux	darboux	ADJ
ap-9037	216	25	-	-	PUNCT
ap-9037	216	26	crum	crum	NOUN
ap-9037	216	27	transformation	transformation	NOUN
ap-9037	216	28	,	,	PUNCT
ap-9037	216	29	while	while	SCONJ
ap-9037	216	30	the	the	DET
ap-9037	216	31	remaining	remain	VERB
ap-9037	216	32	parameters	parameter	NOUN
ap-9037	216	33	must	must	AUX
ap-9037	216	34	be	be	AUX
ap-9037	216	35	kept	keep	VERB
ap-9037	216	36	at	at	ADP
ap-9037	216	37	fixed	fix	VERB
ap-9037	216	38	values	value	NOUN
ap-9037	216	39	.	.	PUNCT
ap-9037	217	1	this	this	PRON
ap-9037	217	2	is	be	AUX
ap-9037	217	3	a	a	DET
ap-9037	217	4	strong	strong	ADJ
ap-9037	217	5	restriction	restriction	NOUN
ap-9037	217	6	,	,	PUNCT
ap-9037	217	7	as	as	SCONJ
ap-9037	217	8	we	we	PRON
ap-9037	217	9	will	will	AUX
ap-9037	217	10	discuss	discuss	VERB
ap-9037	217	11	further	far	ADV
ap-9037	217	12	below	below	ADV
ap-9037	217	13	.	.	PUNCT
ap-9037	218	1	before	before	SCONJ
ap-9037	218	2	we	we	PRON
ap-9037	218	3	continue	continue	VERB
ap-9037	218	4	,	,	PUNCT
ap-9037	218	5	let	let	VERB
ap-9037	218	6	us	we	PRON
ap-9037	218	7	remark	remark	VERB
ap-9037	218	8	that	that	SCONJ
ap-9037	218	9	in	in	ADP
ap-9037	218	10	the	the	DET
ap-9037	218	11	vast	vast	ADJ
ap-9037	218	12	majority	majority	NOUN
ap-9037	218	13	of	of	ADP
ap-9037	218	14	cases	case	NOUN
ap-9037	218	15	,	,	PUNCT
ap-9037	218	16	where	where	SCONJ
ap-9037	218	17	the	the	DET
ap-9037	218	18	darboux	darboux	VERB
ap-9037	218	19	-	-	PUNCT
ap-9037	218	20	crum	crum	NOUN
ap-9037	218	21	transformation	transformation	NOUN
ap-9037	218	22	is	be	AUX
ap-9037	218	23	applied	apply	VERB
ap-9037	218	24	to	to	ADP
ap-9037	218	25	a	a	DET
ap-9037	218	26	schrödinger	schrödinger	ADJ
ap-9037	218	27	equation	equation	NOUN
ap-9037	218	28	,	,	PUNCT
ap-9037	218	29	the	the	DET
ap-9037	218	30	transformation	transformation	NOUN
ap-9037	218	31	parameter	parameter	NOUN
ap-9037	218	32	is	be	AUX
ap-9037	218	33	given	give	VERB
ap-9037	218	34	by	by	ADP
ap-9037	218	35	the	the	DET
ap-9037	218	36	stationary	stationary	ADJ
ap-9037	218	37	energy	energy	NOUN
ap-9037	218	38	.	.	PUNCT
ap-9037	219	1	this	this	PRON
ap-9037	219	2	allows	allow	VERB
ap-9037	219	3	to	to	PART
ap-9037	219	4	control	control	VERB
ap-9037	219	5	the	the	DET
ap-9037	219	6	discrete	discrete	ADJ
ap-9037	219	7	spectrum	spectrum	NOUN
ap-9037	219	8	of	of	ADP
ap-9037	219	9	the	the	DET
ap-9037	219	10	system	system	NOUN
ap-9037	219	11	,	,	PUNCT
ap-9037	219	12	for	for	ADP
ap-9037	219	13	instance	instance	NOUN
ap-9037	219	14	,	,	PUNCT
ap-9037	219	15	energy	energy	NOUN
ap-9037	219	16	levels	level	NOUN
ap-9037	219	17	can	can	AUX
ap-9037	219	18	be	be	AUX
ap-9037	219	19	added	add	VERB
ap-9037	219	20	or	or	CCONJ
ap-9037	219	21	deleted	delete	VERB
ap-9037	219	22	.	.	PUNCT
ap-9037	220	1	however	however	ADV
ap-9037	220	2	,	,	PUNCT
ap-9037	220	3	there	there	PRON
ap-9037	220	4	are	be	VERB
ap-9037	220	5	applications	application	NOUN
ap-9037	220	6	that	that	PRON
ap-9037	220	7	involve	involve	VERB
ap-9037	220	8	non	non	ADJ
ap-9037	220	9	-	-	ADJ
ap-9037	220	10	energy	energy	ADJ
ap-9037	220	11	transformation	transformation	NOUN
ap-9037	220	12	parameters	parameter	NOUN
ap-9037	220	13	.	.	PUNCT
ap-9037	221	1	an	an	DET
ap-9037	221	2	example	example	NOUN
ap-9037	221	3	for	for	ADP
ap-9037	221	4	such	such	DET
ap-9037	221	5	a	a	DET
ap-9037	221	6	case	case	NOUN
ap-9037	221	7	is	be	AUX
ap-9037	221	8	given	give	VERB
ap-9037	221	9	by	by	ADP
ap-9037	221	10	the	the	DET
ap-9037	221	11	two	two	NUM
ap-9037	221	12	-	-	PUNCT
ap-9037	221	13	dimensional	dimensional	ADJ
ap-9037	221	14	massless	massless	NOUN
ap-9037	221	15	dirac	dirac	NOUN
ap-9037	221	16	equation	equation	NOUN
ap-9037	221	17	at	at	ADP
ap-9037	221	18	zero	zero	NUM
ap-9037	221	19	energy	energy	NOUN
ap-9037	221	20	,	,	PUNCT
ap-9037	221	21	where	where	SCONJ
ap-9037	221	22	the	the	DET
ap-9037	221	23	transformation	transformation	NOUN
ap-9037	221	24	parameter	parameter	NOUN
ap-9037	221	25	for	for	ADP
ap-9037	221	26	the	the	DET
ap-9037	221	27	darboux	darboux	VERB
ap-9037	221	28	-	-	PUNCT
ap-9037	221	29	crum	crum	NOUN
ap-9037	221	30	transformations	transformation	NOUN
ap-9037	221	31	is	be	AUX
ap-9037	221	32	given	give	VERB
ap-9037	221	33	by	by	ADP
ap-9037	221	34	the	the	DET
ap-9037	221	35	wave	wave	NOUN
ap-9037	221	36	number	number	NOUN
ap-9037	221	37	[	[	X
ap-9037	221	38	24	24	NUM
ap-9037	221	39	]	]	PUNCT
ap-9037	221	40	.	.	PUNCT
ap-9037	222	1	in	in	ADP
ap-9037	222	2	another	another	DET
ap-9037	222	3	application	application	NOUN
ap-9037	222	4	to	to	ADP
ap-9037	222	5	the	the	DET
ap-9037	222	6	one	one	NUM
ap-9037	222	7	-	-	PUNCT
ap-9037	222	8	dimensional	dimensional	ADJ
ap-9037	222	9	dunkl	dunkl	NOUN
ap-9037	222	10	-	-	PUNCT
ap-9037	222	11	schrödinger	schrödinger	NOUN
ap-9037	222	12	equation	equation	NOUN
ap-9037	222	13	,	,	PUNCT
ap-9037	222	14	the	the	DET
ap-9037	222	15	transformation	transformation	NOUN
ap-9037	222	16	parameter	parameter	NOUN
ap-9037	222	17	consists	consist	VERB
ap-9037	222	18	of	of	ADP
ap-9037	222	19	a	a	DET
ap-9037	222	20	constant	constant	ADJ
ap-9037	222	21	involving	involve	VERB
ap-9037	222	22	the	the	DET
ap-9037	222	23	dunkl	dunkl	NOUN
ap-9037	222	24	constants	constant	NOUN
ap-9037	222	25	[	[	X
ap-9037	222	26	25	25	NUM
ap-9037	222	27	]	]	PUNCT
ap-9037	222	28	.	.	PUNCT
ap-9037	223	1	as	as	ADP
ap-9037	223	2	a	a	DET
ap-9037	223	3	result	result	NOUN
ap-9037	223	4	,	,	PUNCT
ap-9037	223	5	transformed	transform	VERB
ap-9037	223	6	potentials	potential	NOUN
ap-9037	223	7	in	in	ADP
ap-9037	223	8	the	the	DET
ap-9037	223	9	latter	latter	ADJ
ap-9037	223	10	reference	reference	NOUN
ap-9037	223	11	are	be	AUX
ap-9037	223	12	dependent	dependent	ADJ
ap-9037	223	13	on	on	ADP
ap-9037	223	14	the	the	DET
ap-9037	223	15	stationary	stationary	ADJ
ap-9037	223	16	energy	energy	NOUN
ap-9037	223	17	.	.	PUNCT
ap-9037	224	1	4.2.1	4.2.1	NOUN
ap-9037	224	2	.	.	PUNCT
ap-9037	225	1	transformation	transformation	NOUN
ap-9037	225	2	parameter	parameter	NOUN
ap-9037	225	3	λ	λ	PROPN
ap-9037	225	4	our	our	PRON
ap-9037	225	5	goal	goal	NOUN
ap-9037	225	6	in	in	ADP
ap-9037	225	7	this	this	DET
ap-9037	225	8	section	section	NOUN
ap-9037	225	9	is	be	AUX
ap-9037	225	10	to	to	PART
ap-9037	225	11	apply	apply	VERB
ap-9037	225	12	our	our	PRON
ap-9037	225	13	darboux	darboux	ADJ
ap-9037	225	14	-	-	PUNCT
ap-9037	225	15	crum	crum	NOUN
ap-9037	225	16	transformation	transformation	NOUN
ap-9037	225	17	to	to	ADP
ap-9037	225	18	the	the	DET
ap-9037	225	19	polar	polar	ADJ
ap-9037	225	20	equation	equation	NOUN
ap-9037	225	21	,	,	PUNCT
ap-9037	225	22	where	where	SCONJ
ap-9037	225	23	the	the	DET
ap-9037	225	24	separation	separation	NOUN
ap-9037	225	25	constant	constant	ADJ
ap-9037	225	26	λ	λ	PROPN
ap-9037	225	27	plays	play	VERB
ap-9037	225	28	the	the	DET
ap-9037	225	29	role	role	NOUN
ap-9037	225	30	of	of	ADP
ap-9037	225	31	the	the	DET
ap-9037	225	32	transformation	transformation	NOUN
ap-9037	225	33	parameter	parameter	NOUN
ap-9037	225	34	.	.	PUNCT
ap-9037	226	1	our	our	PRON
ap-9037	226	2	starting	starting	NOUN
ap-9037	226	3	point	point	NOUN
ap-9037	226	4	is	be	AUX
ap-9037	226	5	equation	equation	NOUN
ap-9037	226	6	(	(	PUNCT
ap-9037	226	7	30	30	NUM
ap-9037	226	8	)	)	PUNCT
ap-9037	226	9	,	,	PUNCT
ap-9037	226	10	where	where	SCONJ
ap-9037	226	11	we	we	PRON
ap-9037	226	12	observe	observe	VERB
ap-9037	226	13	that	that	SCONJ
ap-9037	226	14	after	after	ADP
ap-9037	226	15	renaming	rename	VERB
ap-9037	226	16	the	the	DET
ap-9037	226	17	solution	solution	NOUN
ap-9037	226	18	and	and	CCONJ
ap-9037	226	19	the	the	DET
ap-9037	226	20	variable	variable	NOUN
ap-9037	226	21	,	,	PUNCT
ap-9037	226	22	this	this	DET
ap-9037	226	23	equation	equation	NOUN
ap-9037	226	24	matches	match	NOUN
ap-9037	226	25	(	(	PUNCT
ap-9037	226	26	4	4	NUM
ap-9037	226	27	)	)	PUNCT
ap-9037	226	28	,	,	PUNCT
ap-9037	226	29	if	if	SCONJ
ap-9037	226	30	we	we	PRON
ap-9037	226	31	set	set	VERB
ap-9037	226	32	:	:	PUNCT
ap-9037	226	33	e	e	X
ap-9037	226	34	=	=	SYM
ap-9037	226	35	λ	λ	PROPN
ap-9037	226	36	,	,	PUNCT
ap-9037	226	37	u(θ	u(θ	NOUN
ap-9037	226	38	)	)	PUNCT
ap-9037	226	39	=	=	SYM
ap-9037	227	1	−	−	PROPN
ap-9037	227	2	(	(	PUNCT
ap-9037	227	3	1	1	NUM
ap-9037	227	4	+	+	NUM
ap-9037	227	5	2ν1	2ν1	NUM
ap-9037	227	6	+	+	SYM
ap-9037	227	7	2ν2	2ν2	NUM
ap-9037	227	8	+	+	CCONJ
ap-9037	227	9	2ν3)2	2ν3)2	PROPN
ap-9037	227	10	4	4	NUM
ap-9037	227	11	−	−	NOUN
ap-9037	227	12	(	(	PUNCT
ap-9037	227	13	r3	r3	PROPN
ap-9037	227	14	−	−	PROPN
ap-9037	227	15	ν3)ν3	ν3)ν3	PROPN
ap-9037	227	16	cos(θ)2	cos(θ)2	NOUN
ap-9037	227	17	−	−	PROPN
ap-9037	227	18	1	1	NUM
ap-9037	227	19	−	−	NOUN
ap-9037	227	20	4m2	4m2	NUM
ap-9037	227	21	−	−	NOUN
ap-9037	227	22	4(ν1	4(ν1	NOUN
ap-9037	228	1	+	+	CCONJ
ap-9037	228	2	ν2)2	ν2)2	ADP
ap-9037	228	3	4	4	NUM
ap-9037	228	4	sin(θ)2	sin(θ)2	NOUN
ap-9037	228	5	+	+	CCONJ
ap-9037	228	6	vθ(θ	vθ(θ	NUM
ap-9037	228	7	)	)	PUNCT
ap-9037	228	8	.	.	PUNCT
ap-9037	229	1	(	(	PUNCT
ap-9037	229	2	47	47	NUM
ap-9037	229	3	)	)	PUNCT
ap-9037	229	4	consequently	consequently	ADV
ap-9037	229	5	,	,	PUNCT
ap-9037	229	6	the	the	DET
ap-9037	229	7	darboux	darboux	VERB
ap-9037	229	8	-	-	PUNCT
ap-9037	229	9	crum	crum	NOUN
ap-9037	229	10	transformation	transformation	NOUN
ap-9037	229	11	can	can	AUX
ap-9037	229	12	be	be	AUX
ap-9037	229	13	applied	apply	VERB
ap-9037	229	14	to	to	ADP
ap-9037	229	15	equation	equation	NOUN
ap-9037	229	16	(	(	PUNCT
ap-9037	229	17	30	30	NUM
ap-9037	229	18	)	)	PUNCT
ap-9037	229	19	.	.	PUNCT
ap-9037	230	1	let	let	VERB
ap-9037	230	2	us	we	PRON
ap-9037	230	3	now	now	ADV
ap-9037	230	4	assume	assume	VERB
ap-9037	230	5	that	that	SCONJ
ap-9037	230	6	ν1	ν1	NOUN
ap-9037	230	7	,	,	PUNCT
ap-9037	230	8	ν2	ν2	NOUN
ap-9037	230	9	,	,	PUNCT
ap-9037	230	10	.	.	PUNCT
ap-9037	230	11	.	.	PUNCT
ap-9037	231	1	.	.	PUNCT
ap-9037	232	1	,	,	PUNCT
ap-9037	232	2	νn	νn	AUX
ap-9037	232	3	are	be	AUX
ap-9037	232	4	transformation	transformation	NOUN
ap-9037	232	5	functions	function	NOUN
ap-9037	232	6	that	that	PRON
ap-9037	232	7	solve	solve	VERB
ap-9037	232	8	the	the	DET
ap-9037	232	9	following	follow	VERB
ap-9037	232	10	version	version	NOUN
ap-9037	232	11	of	of	ADP
ap-9037	232	12	equation	equation	NOUN
ap-9037	232	13	(	(	PUNCT
ap-9037	232	14	30	30	NUM
ap-9037	232	15	):	):	PUNCT
ap-9037	232	16	η′′	η′′	PROPN
ap-9037	232	17	j	j	PROPN
ap-9037	232	18	(	(	PUNCT
ap-9037	232	19	θ	θ	PROPN
ap-9037	232	20	)	)	PUNCT
ap-9037	233	1	+	+	CCONJ
ap-9037	233	2	[	[	PUNCT
ap-9037	233	3	λj	λj	X
ap-9037	233	4	+	+	X
ap-9037	233	5	(	(	PUNCT
ap-9037	233	6	1	1	NUM
ap-9037	233	7	+	+	NUM
ap-9037	233	8	2ν1	2ν1	NUM
ap-9037	233	9	+	+	SYM
ap-9037	233	10	2ν2	2ν2	NUM
ap-9037	233	11	+	+	CCONJ
ap-9037	233	12	2ν3)2	2ν3)2	PROPN
ap-9037	233	13	4	4	NUM
ap-9037	233	14	+	+	CCONJ
ap-9037	233	15	(	(	PUNCT
ap-9037	233	16	r3	r3	PROPN
ap-9037	233	17	−	−	PROPN
ap-9037	233	18	ν3)ν3	ν3)ν3	PART
ap-9037	233	19	cos(θ)2	cos(θ)2	NOUN
ap-9037	233	20	+	+	CCONJ
ap-9037	233	21	1	1	NUM
ap-9037	233	22	−	−	NOUN
ap-9037	233	23	4m2	4m2	NUM
ap-9037	233	24	−	−	NOUN
ap-9037	233	25	4(ν1	4(ν1	NOUN
ap-9037	234	1	+	+	CCONJ
ap-9037	234	2	ν2)2	ν2)2	ADP
ap-9037	234	3	4	4	NUM
ap-9037	234	4	sin(θ)2	sin(θ)2	NOUN
ap-9037	234	5	−	−	NOUN
ap-9037	234	6	vθ(θ	vθ(θ	NUM
ap-9037	234	7	)	)	PUNCT
ap-9037	234	8	]	]	PUNCT
ap-9037	235	1	ηj(θ	ηj(θ	X
ap-9037	235	2	)	)	PUNCT
ap-9037	235	3	=	=	SYM
ap-9037	235	4	0	0	NUM
ap-9037	235	5	,	,	PUNCT
ap-9037	235	6	j	j	PROPN
ap-9037	235	7	=	=	SYM
ap-9037	235	8	1	1	NUM
ap-9037	235	9	,	,	PUNCT
ap-9037	235	10	.	.	PUNCT
ap-9037	235	11	.	.	PUNCT
ap-9037	235	12	.	.	PUNCT
ap-9037	235	13	,	,	PUNCT
ap-9037	235	14	n	n	X
ap-9037	235	15	,	,	PUNCT
ap-9037	235	16	(	(	PUNCT
ap-9037	235	17	48	48	NUM
ap-9037	235	18	)	)	PUNCT
ap-9037	235	19	where	where	SCONJ
ap-9037	235	20	λ1	λ1	ADJ
ap-9037	235	21	,	,	PUNCT
ap-9037	235	22	.	.	PUNCT
ap-9037	235	23	.	.	PUNCT
ap-9037	235	24	.	.	PUNCT
ap-9037	236	1	,	,	PUNCT
ap-9037	236	2	λn	λn	PROPN
ap-9037	236	3	are	be	AUX
ap-9037	236	4	transformation	transformation	NOUN
ap-9037	236	5	parameters	parameter	NOUN
ap-9037	236	6	,	,	PUNCT
ap-9037	236	7	and	and	CCONJ
ap-9037	236	8	the	the	DET
ap-9037	236	9	potential	potential	ADJ
ap-9037	236	10	vθ	vθ	NOUN
ap-9037	236	11	is	be	AUX
ap-9037	236	12	given	give	VERB
ap-9037	236	13	by	by	ADP
ap-9037	236	14	(	(	PUNCT
ap-9037	236	15	31	31	NUM
ap-9037	236	16	)	)	PUNCT
ap-9037	236	17	.	.	PUNCT
ap-9037	237	1	then	then	ADV
ap-9037	237	2	,	,	PUNCT
ap-9037	237	3	substituting	substitute	VERB
ap-9037	237	4	into	into	ADP
ap-9037	237	5	the	the	DET
ap-9037	237	6	expressions	expression	NOUN
ap-9037	237	7	(	(	PUNCT
ap-9037	237	8	6	6	NUM
ap-9037	237	9	)	)	PUNCT
ap-9037	237	10	and	and	CCONJ
ap-9037	237	11	(	(	PUNCT
ap-9037	237	12	7	7	X
ap-9037	237	13	)	)	PUNCT
ap-9037	237	14	gives	give	VERB
ap-9037	237	15	the	the	DET
ap-9037	237	16	results	result	NOUN
ap-9037	237	17	:	:	PUNCT
ap-9037	237	18	η̂(θ	η̂(θ	NUM
ap-9037	237	19	)	)	PUNCT
ap-9037	237	20	=	=	SYM
ap-9037	237	21	wη1,η2,	wη1,η2,	PROPN
ap-9037	237	22	...	...	PUNCT
ap-9037	237	23	,ηn	,ηn	PUNCT
ap-9037	237	24	,	,	PUNCT
ap-9037	237	25	η(θ	η(θ	NOUN
ap-9037	237	26	)	)	PUNCT
ap-9037	237	27	wη1,η2,	wη1,η2,	PROPN
ap-9037	237	28	...	...	PUNCT
ap-9037	237	29	,ηn	,ηn	PUNCT
ap-9037	237	30	(	(	PUNCT
ap-9037	237	31	θ	θ	NOUN
ap-9037	237	32	)	)	PUNCT
ap-9037	237	33	,	,	PUNCT
ap-9037	237	34	(	(	PUNCT
ap-9037	237	35	49	49	NUM
ap-9037	237	36	)	)	PUNCT
ap-9037	237	37	v̂θ(θ	v̂θ(θ	NUM
ap-9037	237	38	)	)	PUNCT
ap-9037	237	39	=	=	SYM
ap-9037	237	40	vθ(θ	vθ(θ	X
ap-9037	237	41	)	)	PUNCT
ap-9037	237	42	−	−	PROPN
ap-9037	237	43	2	2	NUM
ap-9037	237	44	d2	d2	NOUN
ap-9037	237	45	dθ2	dθ2	NOUN
ap-9037	237	46	log	log	NOUN
ap-9037	238	1	[	[	X
ap-9037	238	2	wη1,η2,	wη1,η2,	NOUN
ap-9037	238	3	...	...	PUNCT
ap-9037	238	4	,ηn(θ	,ηn(θ	PROPN
ap-9037	238	5	)	)	PUNCT
ap-9037	238	6	]	]	PUNCT
ap-9037	238	7	.	.	PUNCT
ap-9037	239	1	(	(	PUNCT
ap-9037	239	2	50	50	X
ap-9037	239	3	)	)	PUNCT
ap-9037	239	4	let	let	VERB
ap-9037	239	5	us	we	PRON
ap-9037	239	6	point	point	VERB
ap-9037	239	7	out	out	ADP
ap-9037	239	8	that	that	SCONJ
ap-9037	239	9	due	due	ADP
ap-9037	239	10	to	to	ADP
ap-9037	239	11	(	(	PUNCT
ap-9037	239	12	48	48	NUM
ap-9037	239	13	)	)	PUNCT
ap-9037	239	14	,	,	PUNCT
ap-9037	239	15	the	the	DET
ap-9037	239	16	function	function	NOUN
ap-9037	239	17	η̂	η̂	NUM
ap-9037	239	18	depends	depend	VERB
ap-9037	239	19	on	on	ADP
ap-9037	239	20	the	the	DET
ap-9037	239	21	transformation	transformation	NOUN
ap-9037	239	22	parameters	parameter	NOUN
ap-9037	239	23	λ1	λ1	ADJ
ap-9037	239	24	,	,	PUNCT
ap-9037	239	25	.	.	PUNCT
ap-9037	239	26	.	.	PUNCT
ap-9037	240	1	.	.	PUNCT
ap-9037	241	1	,	,	PUNCT
ap-9037	241	2	λn	λn	NOUN
ap-9037	241	3	,	,	PUNCT
ap-9037	241	4	and	and	CCONJ
ap-9037	241	5	also	also	ADV
ap-9037	241	6	on	on	ADP
ap-9037	241	7	the	the	DET
ap-9037	241	8	separation	separation	NOUN
ap-9037	241	9	constant	constant	ADJ
ap-9037	241	10	λ	λ	PROPN
ap-9037	241	11	from	from	ADP
ap-9037	241	12	equation	equation	NOUN
ap-9037	241	13	(	(	PUNCT
ap-9037	241	14	30	30	NUM
ap-9037	241	15	)	)	PUNCT
ap-9037	241	16	.	.	PUNCT
ap-9037	242	1	furthermore	furthermore	ADV
ap-9037	242	2	,	,	PUNCT
ap-9037	242	3	the	the	DET
ap-9037	242	4	associated	associated	ADJ
ap-9037	242	5	potential	potential	NOUN
ap-9037	242	6	v̂θ	v̂θ	X
ap-9037	242	7	also	also	ADV
ap-9037	242	8	depends	depend	VERB
ap-9037	242	9	on	on	ADP
ap-9037	242	10	the	the	DET
ap-9037	242	11	transformation	transformation	NOUN
ap-9037	242	12	parameters	parameter	NOUN
ap-9037	242	13	λ1	λ1	ADJ
ap-9037	242	14	,	,	PUNCT
ap-9037	242	15	.	.	PUNCT
ap-9037	242	16	.	.	PUNCT
ap-9037	243	1	.	.	PUNCT
ap-9037	244	1	,	,	PUNCT
ap-9037	244	2	λn	λn	NOUN
ap-9037	244	3	,	,	PUNCT
ap-9037	244	4	but	but	CCONJ
ap-9037	244	5	it	it	PRON
ap-9037	244	6	does	do	AUX
ap-9037	244	7	not	not	PART
ap-9037	244	8	depend	depend	VERB
ap-9037	244	9	on	on	ADP
ap-9037	244	10	λ	λ	NOUN
ap-9037	244	11	.	.	PUNCT
ap-9037	245	1	the	the	DET
ap-9037	245	2	function	function	NOUN
ap-9037	245	3	η̂	η̂	PUNCT
ap-9037	245	4	and	and	CCONJ
ap-9037	245	5	the	the	DET
ap-9037	245	6	associated	associated	ADJ
ap-9037	245	7	potential	potential	NOUN
ap-9037	245	8	v̂θ	v̂θ	X
ap-9037	245	9	are	be	AUX
ap-9037	245	10	inserted	insert	VERB
ap-9037	245	11	in	in	ADP
ap-9037	245	12	the	the	DET
ap-9037	245	13	transformed	transform	VERB
ap-9037	245	14	counterpart	counterpart	NOUN
ap-9037	245	15	of	of	ADP
ap-9037	245	16	(	(	PUNCT
ap-9037	245	17	30	30	NUM
ap-9037	245	18	)	)	PUNCT
ap-9037	245	19	that	that	PRON
ap-9037	245	20	reads	read	VERB
ap-9037	245	21	:	:	PUNCT
ap-9037	245	22	η̂′′(θ	η̂′′(θ	NOUN
ap-9037	245	23	)	)	PUNCT
ap-9037	246	1	+	+	CCONJ
ap-9037	246	2	[	[	PUNCT
ap-9037	246	3	λ	λ	X
ap-9037	246	4	+	+	CCONJ
ap-9037	246	5	(	(	PUNCT
ap-9037	246	6	1	1	NUM
ap-9037	246	7	+	+	NUM
ap-9037	246	8	2ν1	2ν1	NUM
ap-9037	246	9	+	+	SYM
ap-9037	246	10	2ν2	2ν2	NUM
ap-9037	246	11	+	+	CCONJ
ap-9037	246	12	2ν3)2	2ν3)2	PROPN
ap-9037	246	13	4	4	NUM
ap-9037	246	14	+	+	CCONJ
ap-9037	246	15	(	(	PUNCT
ap-9037	246	16	r3	r3	PROPN
ap-9037	246	17	−	−	PROPN
ap-9037	246	18	ν3)ν3	ν3)ν3	PART
ap-9037	246	19	cos(θ)2	cos(θ)2	NOUN
ap-9037	246	20	+	+	CCONJ
ap-9037	246	21	1	1	NUM
ap-9037	246	22	−	−	NOUN
ap-9037	246	23	4m2	4m2	NUM
ap-9037	246	24	−	−	NOUN
ap-9037	246	25	4(ν1	4(ν1	NOUN
ap-9037	247	1	+	+	CCONJ
ap-9037	247	2	ν2)2	ν2)2	ADP
ap-9037	247	3	4	4	NUM
ap-9037	247	4	sin(θ)2	sin(θ)2	NOUN
ap-9037	247	5	−	−	NOUN
ap-9037	247	6	v̂θ(θ	v̂θ(θ	NUM
ap-9037	247	7	)	)	PUNCT
ap-9037	247	8	]	]	PUNCT
ap-9037	248	1	η̂(θ	η̂(θ	NUM
ap-9037	248	2	)	)	PUNCT
ap-9037	248	3	=	=	SYM
ap-9037	248	4	0	0	X
ap-9037	248	5	.	.	PUNCT
ap-9037	249	1	in	in	ADP
ap-9037	249	2	the	the	DET
ap-9037	249	3	next	next	ADJ
ap-9037	249	4	step	step	NOUN
ap-9037	249	5	,	,	PUNCT
ap-9037	249	6	we	we	PRON
ap-9037	249	7	must	must	AUX
ap-9037	249	8	rewrite	rewrite	VERB
ap-9037	249	9	this	this	DET
ap-9037	249	10	equation	equation	NOUN
ap-9037	249	11	in	in	ADP
ap-9037	249	12	the	the	DET
ap-9037	249	13	form	form	NOUN
ap-9037	249	14	(	(	PUNCT
ap-9037	249	15	27	27	NUM
ap-9037	249	16	)	)	PUNCT
ap-9037	249	17	that	that	PRON
ap-9037	249	18	was	be	AUX
ap-9037	249	19	obtained	obtain	VERB
ap-9037	249	20	after	after	ADP
ap-9037	249	21	the	the	DET
ap-9037	249	22	separation	separation	NOUN
ap-9037	249	23	of	of	ADP
ap-9037	249	24	variables	variable	NOUN
ap-9037	249	25	.	.	PUNCT
ap-9037	250	1	this	this	DET
ap-9037	250	2	form	form	NOUN
ap-9037	250	3	is	be	AUX
ap-9037	250	4	given	give	VERB
ap-9037	250	5	by	by	ADP
ap-9037	250	6	:	:	PUNCT
ap-9037	250	7	θ̂′′(θ	θ̂′′(θ	NOUN
ap-9037	250	8	)	)	PUNCT
ap-9037	250	9	−	−	NOUN
ap-9037	250	10	2	2	NUM
ap-9037	250	11	[	[	PUNCT
ap-9037	250	12	ν3	ν3	NOUN
ap-9037	250	13	tan(θ	tan(θ	NOUN
ap-9037	250	14	)	)	PUNCT
ap-9037	251	1	−	−	PROPN
ap-9037	251	2	(	(	PUNCT
ap-9037	251	3	1	1	NUM
ap-9037	251	4	2	2	NUM
ap-9037	251	5	+	+	NUM
ap-9037	251	6	ν1	ν1	NOUN
ap-9037	251	7	+	+	CCONJ
ap-9037	251	8	ν2	ν2	NOUN
ap-9037	251	9	)	)	PUNCT
ap-9037	251	10	cot(θ	cot(θ	PROPN
ap-9037	251	11	)	)	PUNCT
ap-9037	251	12	]	]	PUNCT
ap-9037	251	13	θ̂′(θ	θ̂′(θ	NOUN
ap-9037	251	14	)	)	PUNCT
ap-9037	251	15	+	+	CCONJ
ap-9037	251	16	[	[	PUNCT
ap-9037	251	17	λ	λ	X
ap-9037	251	18	−	−	NOUN
ap-9037	251	19	m2	m2	PROPN
ap-9037	251	20	sin(θ)2	sin(θ)2	NOUN
ap-9037	251	21	−	−	PROPN
ap-9037	251	22	ν3(1	ν3(1	ADJ
ap-9037	251	23	−	−	PROPN
ap-9037	251	24	r3	r3	PROPN
ap-9037	251	25	)	)	PUNCT
ap-9037	251	26	cos(θ)2	cos(θ)2	NOUN
ap-9037	251	27	−	−	PROPN
ap-9037	251	28	v̂θ(θ	v̂θ(θ	NUM
ap-9037	251	29	)	)	PUNCT
ap-9037	251	30	]	]	PUNCT
ap-9037	252	1	θ̂(θ	θ̂(θ	VERB
ap-9037	252	2	)	)	PUNCT
ap-9037	252	3	=	=	SYM
ap-9037	252	4	0	0	NUM
ap-9037	252	5	,	,	PUNCT
ap-9037	252	6	(	(	PUNCT
ap-9037	252	7	51	51	NUM
ap-9037	252	8	)	)	PUNCT
ap-9037	252	9	where	where	SCONJ
ap-9037	252	10	the	the	DET
ap-9037	252	11	functions	function	NOUN
ap-9037	252	12	θ̂	θ̂	VERB
ap-9037	252	13	and	and	CCONJ
ap-9037	252	14	v̂θ	v̂θ	X
ap-9037	252	15	will	will	AUX
ap-9037	252	16	be	be	AUX
ap-9037	252	17	determined	determine	VERB
ap-9037	252	18	now	now	ADV
ap-9037	252	19	.	.	PUNCT
ap-9037	253	1	in	in	ADP
ap-9037	253	2	order	order	NOUN
ap-9037	253	3	to	to	PART
ap-9037	253	4	do	do	AUX
ap-9037	253	5	so	so	ADV
ap-9037	253	6	,	,	PUNCT
ap-9037	253	7	we	we	PRON
ap-9037	253	8	have	have	VERB
ap-9037	253	9	to	to	PART
ap-9037	253	10	rewrite	rewrite	VERB
ap-9037	253	11	both	both	DET
ap-9037	253	12	(	(	PUNCT
ap-9037	253	13	49	49	NUM
ap-9037	253	14	)	)	PUNCT
ap-9037	253	15	and	and	CCONJ
ap-9037	253	16	(	(	PUNCT
ap-9037	253	17	50	50	NUM
ap-9037	253	18	)	)	PUNCT
ap-9037	253	19	in	in	ADP
ap-9037	253	20	terms	term	NOUN
ap-9037	253	21	of	of	ADP
ap-9037	253	22	quantities	quantity	NOUN
ap-9037	253	23	pertaining	pertain	VERB
ap-9037	253	24	to	to	ADP
ap-9037	253	25	equation	equation	NOUN
ap-9037	253	26	(	(	PUNCT
ap-9037	253	27	27	27	NUM
ap-9037	253	28	)	)	PUNCT
ap-9037	253	29	.	.	PUNCT
ap-9037	254	1	the	the	DET
ap-9037	254	2	functions	function	NOUN
ap-9037	254	3	η1	η1	NOUN
ap-9037	254	4	,	,	PUNCT
ap-9037	254	5	η2	η2	NOUN
ap-9037	254	6	,	,	PUNCT
ap-9037	254	7	.	.	PUNCT
ap-9037	254	8	.	.	PUNCT
ap-9037	254	9	.	.	PUNCT
ap-9037	255	1	,	,	PUNCT
ap-9037	255	2	ηn	ηn	PROPN
ap-9037	255	3	can	can	AUX
ap-9037	255	4	expressed	express	VERB
ap-9037	255	5	through	through	ADP
ap-9037	255	6	solutions	solution	NOUN
ap-9037	255	7	θ1	θ1	NOUN
ap-9037	255	8	,	,	PUNCT
ap-9037	255	9	θ2	θ2	PROPN
ap-9037	255	10	,	,	PUNCT
ap-9037	255	11	.	.	PUNCT
ap-9037	255	12	.	.	PUNCT
ap-9037	255	13	.	.	PUNCT
ap-9037	256	1	,	,	PUNCT
ap-9037	256	2	θn	θn	NOUN
ap-9037	256	3	of	of	ADP
ap-9037	256	4	the	the	DET
ap-9037	256	5	latter	latter	ADJ
ap-9037	256	6	equation	equation	NOUN
ap-9037	256	7	by	by	ADP
ap-9037	256	8	means	mean	NOUN
ap-9037	256	9	of	of	ADP
ap-9037	256	10	the	the	DET
ap-9037	256	11	point	point	NOUN
ap-9037	256	12	transformation	transformation	NOUN
ap-9037	256	13	(	(	PUNCT
ap-9037	256	14	29	29	NUM
ap-9037	256	15	)	)	PUNCT
ap-9037	256	16	.	.	PUNCT
ap-9037	257	1	we	we	PRON
ap-9037	257	2	obtain	obtain	VERB
ap-9037	257	3	:	:	PUNCT
ap-9037	257	4	θj(θ	θj(θ	NUM
ap-9037	257	5	)	)	PUNCT
ap-9037	258	1	=	=	SYM
ap-9037	258	2	cos(θ)−ν3	cos(θ)−ν3	PROPN
ap-9037	258	3	sin(θ)−ν1−ν2−	sin(θ)−ν1−ν2−	VERB
ap-9037	258	4	1	1	NUM
ap-9037	258	5	2	2	NUM
ap-9037	258	6	ηj(θ	ηj(θ	NUM
ap-9037	258	7	)	)	PUNCT
ap-9037	258	8	,	,	PUNCT
ap-9037	258	9	j	j	PROPN
ap-9037	259	1	=	=	SYM
ap-9037	259	2	1	1	NUM
ap-9037	259	3	,	,	PUNCT
ap-9037	259	4	2	2	NUM
ap-9037	259	5	,	,	PUNCT
ap-9037	259	6	.	.	PUNCT
ap-9037	259	7	.	.	PUNCT
ap-9037	260	1	.	.	PUNCT
ap-9037	261	1	,	,	PUNCT
ap-9037	261	2	n	n	X
ap-9037	261	3	.	.	PUNCT
ap-9037	262	1	280	280	NUM
ap-9037	262	2	vol	vol	NOUN
ap-9037	262	3	.	.	PUNCT
ap-9037	262	4	63	63	NUM
ap-9037	262	5	no	no	NOUN
ap-9037	262	6	.	.	PUNCT
ap-9037	263	1	4/2023	4/2023	NOUN
ap-9037	263	2	construction	construction	NOUN
ap-9037	263	3	of	of	ADP
ap-9037	263	4	angular	angular	ADJ
ap-9037	263	5	-	-	PUNCT
ap-9037	263	6	dependent	dependent	ADJ
ap-9037	263	7	potentials	potential	NOUN
ap-9037	263	8	by	by	ADP
ap-9037	263	9	using	use	VERB
ap-9037	263	10	(	(	PUNCT
ap-9037	263	11	29	29	NUM
ap-9037	263	12	)	)	PUNCT
ap-9037	263	13	once	once	ADV
ap-9037	263	14	more	more	ADJ
ap-9037	263	15	for	for	ADP
ap-9037	263	16	relating	relate	VERB
ap-9037	263	17	η̂	η̂	PUNCT
ap-9037	263	18	to	to	ADP
ap-9037	263	19	a	a	DET
ap-9037	263	20	new	new	ADJ
ap-9037	263	21	function	function	NOUN
ap-9037	263	22	θ̂	θ̂	NUM
ap-9037	263	23	,	,	PUNCT
ap-9037	263	24	we	we	PRON
ap-9037	263	25	obtain	obtain	VERB
ap-9037	263	26	from	from	ADP
ap-9037	263	27	(	(	PUNCT
ap-9037	263	28	49	49	NUM
ap-9037	263	29	):	):	PUNCT
ap-9037	263	30	θ̂(θ	θ̂(θ	NUM
ap-9037	263	31	)	)	PUNCT
ap-9037	263	32	=	=	SYM
ap-9037	264	1	cos(θ)−ν3	cos(θ)−ν3	PROPN
ap-9037	264	2	sin(θ)−ν1−ν2−	sin(θ)−ν1−ν2−	VERB
ap-9037	264	3	1	1	NUM
ap-9037	264	4	2	2	NUM
ap-9037	264	5	η̂(θ	η̂(θ	NUM
ap-9037	264	6	)	)	PUNCT
ap-9037	264	7	=	=	SYM
ap-9037	264	8	cos(θ)−ν3	cos(θ)−ν3	PROPN
ap-9037	264	9	sin(θ)−ν1−ν2−	sin(θ)−ν1−ν2−	VERB
ap-9037	264	10	1	1	NUM
ap-9037	264	11	2	2	NUM
ap-9037	264	12	wη1,η2,	wη1,η2,	NOUN
ap-9037	264	13	...	...	PUNCT
ap-9037	264	14	,ηn	,ηn	PUNCT
ap-9037	264	15	,	,	PUNCT
ap-9037	264	16	η(θ	η(θ	NOUN
ap-9037	264	17	)	)	PUNCT
ap-9037	264	18	wη1,η2,	wη1,η2,	PROPN
ap-9037	264	19	...	...	PUNCT
ap-9037	264	20	,ηn	,ηn	PUNCT
ap-9037	264	21	(	(	PUNCT
ap-9037	264	22	θ	θ	NOUN
ap-9037	264	23	)	)	PUNCT
ap-9037	264	24	(	(	PUNCT
ap-9037	264	25	52	52	NUM
ap-9037	264	26	)	)	PUNCT
ap-9037	264	27	=	=	SYM
ap-9037	264	28	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	264	29	...	...	PUNCT
ap-9037	264	30	,θn	,θn	PUNCT
ap-9037	264	31	,	,	PUNCT
ap-9037	264	32	θ(θ	θ(θ	VERB
ap-9037	264	33	)	)	PUNCT
ap-9037	264	34	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	264	35	...	...	PUNCT
ap-9037	264	36	,θn(θ	,θn(θ	NUM
ap-9037	264	37	)	)	PUNCT
ap-9037	264	38	.	.	PUNCT
ap-9037	265	1	this	this	PRON
ap-9037	265	2	is	be	AUX
ap-9037	265	3	the	the	DET
ap-9037	265	4	explicit	explicit	ADJ
ap-9037	265	5	form	form	NOUN
ap-9037	265	6	of	of	ADP
ap-9037	265	7	the	the	DET
ap-9037	265	8	transformed	transform	VERB
ap-9037	265	9	solution	solution	NOUN
ap-9037	265	10	to	to	ADP
ap-9037	265	11	our	our	PRON
ap-9037	265	12	polar	polar	ADJ
ap-9037	265	13	equation	equation	NOUN
ap-9037	265	14	(	(	PUNCT
ap-9037	265	15	51	51	NUM
ap-9037	265	16	)	)	PUNCT
ap-9037	265	17	.	.	PUNCT
ap-9037	266	1	the	the	DET
ap-9037	266	2	associated	associated	ADJ
ap-9037	266	3	potential	potential	NOUN
ap-9037	266	4	can	can	AUX
ap-9037	266	5	be	be	AUX
ap-9037	266	6	obtained	obtain	VERB
ap-9037	266	7	from	from	ADP
ap-9037	266	8	(	(	PUNCT
ap-9037	266	9	50	50	NUM
ap-9037	266	10	)	)	PUNCT
ap-9037	266	11	as	as	SCONJ
ap-9037	266	12	follows	follow	VERB
ap-9037	266	13	:	:	PUNCT
ap-9037	266	14	v̂θ(θ	v̂θ(θ	NUM
ap-9037	266	15	)	)	PUNCT
ap-9037	266	16	=	=	SYM
ap-9037	266	17	vθ(θ	vθ(θ	X
ap-9037	266	18	)	)	PUNCT
ap-9037	267	1	−	−	PROPN
ap-9037	267	2	2	2	NUM
ap-9037	267	3	d2	d2	PROPN
ap-9037	267	4	dθ2	dθ2	NOUN
ap-9037	267	5	{	{	PUNCT
ap-9037	267	6	log	log	PROPN
ap-9037	267	7	[	[	PUNCT
ap-9037	267	8	cos(θ)nν3	cos(θ)nν3	PROPN
ap-9037	267	9	sin(θ)n(ν1+ν2	sin(θ)n(ν1+ν2	PROPN
ap-9037	267	10	+	+	CCONJ
ap-9037	267	11	1	1	NUM
ap-9037	267	12	2	2	NUM
ap-9037	267	13	)	)	PUNCT
ap-9037	267	14	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	267	15	...	...	PUNCT
ap-9037	267	16	,θn	,θn	PUNCT
ap-9037	267	17	(	(	PUNCT
ap-9037	267	18	θ	θ	NOUN
ap-9037	267	19	)	)	PUNCT
ap-9037	267	20	]	]	PUNCT
ap-9037	267	21	}	}	PUNCT
ap-9037	267	22	=	=	SYM
ap-9037	267	23	vθ(θ	vθ(θ	X
ap-9037	267	24	)	)	PUNCT
ap-9037	267	25	−	−	PROPN
ap-9037	267	26	2n	2n	NUM
ap-9037	267	27	(	(	PUNCT
ap-9037	267	28	ν1	ν1	NOUN
ap-9037	267	29	+	+	CCONJ
ap-9037	267	30	ν2	ν2	NOUN
ap-9037	267	31	+	+	CCONJ
ap-9037	267	32	1	1	NUM
ap-9037	267	33	2	2	NUM
ap-9037	267	34	)	)	PUNCT
ap-9037	267	35	cos(θ)2	cos(θ)2	NOUN
ap-9037	267	36	−	−	NOUN
ap-9037	267	37	2nν3	2nν3	NUM
ap-9037	267	38	sin(θ)2	sin(θ)2	NOUN
ap-9037	267	39	−	−	PROPN
ap-9037	267	40	2	2	NUM
ap-9037	267	41	d2	d2	NOUN
ap-9037	267	42	dθ2	dθ2	NOUN
ap-9037	267	43	log	log	NOUN
ap-9037	268	1	[	[	X
ap-9037	268	2	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	268	3	...	...	PUNCT
ap-9037	268	4	,θn	,θn	PUNCT
ap-9037	268	5	(	(	PUNCT
ap-9037	268	6	θ	θ	NOUN
ap-9037	268	7	)	)	PUNCT
ap-9037	268	8	]	]	PUNCT
ap-9037	268	9	.	.	PUNCT
ap-9037	269	1	(	(	PUNCT
ap-9037	269	2	53	53	NUM
ap-9037	269	3	)	)	PUNCT
ap-9037	269	4	we	we	PRON
ap-9037	269	5	observe	observe	VERB
ap-9037	269	6	that	that	SCONJ
ap-9037	269	7	this	this	PRON
ap-9037	269	8	transformed	transform	VERB
ap-9037	269	9	potential	potential	NOUN
ap-9037	269	10	contributes	contribute	VERB
ap-9037	269	11	terms	term	NOUN
ap-9037	269	12	of	of	ADP
ap-9037	269	13	the	the	DET
ap-9037	269	14	trigonometric	trigonometric	ADJ
ap-9037	269	15	pöschl	pöschl	NOUN
ap-9037	269	16	-	-	PUNCT
ap-9037	269	17	teller	teller	NOUN
ap-9037	269	18	form	form	NOUN
ap-9037	269	19	,	,	PUNCT
ap-9037	269	20	independent	independent	ADJ
ap-9037	269	21	of	of	ADP
ap-9037	269	22	the	the	DET
ap-9037	269	23	initial	initial	ADJ
ap-9037	269	24	potential	potential	ADJ
ap-9037	269	25	vθ	vθ	NOUN
ap-9037	269	26	.	.	PUNCT
ap-9037	270	1	in	in	ADP
ap-9037	270	2	summary	summary	NOUN
ap-9037	270	3	,	,	PUNCT
ap-9037	270	4	the	the	DET
ap-9037	270	5	solution	solution	NOUN
ap-9037	270	6	(	(	PUNCT
ap-9037	270	7	52	52	NUM
ap-9037	270	8	)	)	PUNCT
ap-9037	270	9	satisfies	satisfy	VERB
ap-9037	270	10	the	the	DET
ap-9037	270	11	transformed	transform	VERB
ap-9037	270	12	polar	polar	ADJ
ap-9037	270	13	equation	equation	NOUN
ap-9037	270	14	(	(	PUNCT
ap-9037	270	15	51	51	NUM
ap-9037	270	16	)	)	PUNCT
ap-9037	270	17	with	with	ADP
ap-9037	270	18	potential	potential	ADJ
ap-9037	270	19	(	(	PUNCT
ap-9037	270	20	53	53	NUM
ap-9037	270	21	)	)	PUNCT
ap-9037	270	22	.	.	PUNCT
ap-9037	271	1	as	as	SCONJ
ap-9037	271	2	indicated	indicate	VERB
ap-9037	271	3	above	above	ADV
ap-9037	271	4	,	,	PUNCT
ap-9037	271	5	the	the	DET
ap-9037	271	6	latter	latter	ADJ
ap-9037	271	7	potential	potential	NOUN
ap-9037	271	8	,	,	PUNCT
ap-9037	271	9	in	in	ADP
ap-9037	271	10	general	general	ADJ
ap-9037	271	11	,	,	PUNCT
ap-9037	271	12	depends	depend	VERB
ap-9037	271	13	on	on	ADP
ap-9037	271	14	all	all	DET
ap-9037	271	15	parameters	parameter	NOUN
ap-9037	271	16	except	except	SCONJ
ap-9037	271	17	on	on	ADP
ap-9037	271	18	the	the	DET
ap-9037	271	19	transformation	transformation	NOUN
ap-9037	271	20	parameter	parameter	PROPN
ap-9037	271	21	λ	λ	PROPN
ap-9037	271	22	.	.	PUNCT
ap-9037	272	1	this	this	PRON
ap-9037	272	2	includes	include	VERB
ap-9037	272	3	the	the	DET
ap-9037	272	4	separation	separation	NOUN
ap-9037	272	5	constant	constant	ADJ
ap-9037	272	6	m2	m2	PROPN
ap-9037	272	7	and	and	CCONJ
ap-9037	272	8	the	the	DET
ap-9037	272	9	parity	parity	NOUN
ap-9037	272	10	constant	constant	ADJ
ap-9037	272	11	r3	r3	PROPN
ap-9037	272	12	.	.	PUNCT
ap-9037	273	1	hence	hence	ADV
ap-9037	273	2	,	,	PUNCT
ap-9037	273	3	varying	vary	VERB
ap-9037	273	4	these	these	DET
ap-9037	273	5	parameters	parameter	NOUN
ap-9037	273	6	will	will	AUX
ap-9037	273	7	also	also	ADV
ap-9037	273	8	change	change	VERB
ap-9037	273	9	the	the	DET
ap-9037	273	10	transformed	transform	VERB
ap-9037	273	11	potential	potential	NOUN
ap-9037	273	12	.	.	PUNCT
ap-9037	274	1	since	since	SCONJ
ap-9037	274	2	this	this	PRON
ap-9037	274	3	is	be	AUX
ap-9037	274	4	not	not	PART
ap-9037	274	5	desirable	desirable	ADJ
ap-9037	274	6	,	,	PUNCT
ap-9037	274	7	the	the	DET
ap-9037	274	8	aforementioned	aforementioned	ADJ
ap-9037	274	9	parameters	parameter	NOUN
ap-9037	274	10	must	must	AUX
ap-9037	274	11	be	be	AUX
ap-9037	274	12	kept	keep	VERB
ap-9037	274	13	at	at	ADP
ap-9037	274	14	a	a	DET
ap-9037	274	15	fixed	fix	VERB
ap-9037	274	16	value	value	NOUN
ap-9037	274	17	.	.	PUNCT
ap-9037	275	1	recall	recall	VERB
ap-9037	275	2	that	that	SCONJ
ap-9037	275	3	this	this	PRON
ap-9037	275	4	does	do	AUX
ap-9037	275	5	not	not	PART
ap-9037	275	6	hold	hold	VERB
ap-9037	275	7	for	for	ADP
ap-9037	275	8	λ	λ	PROPN
ap-9037	275	9	.	.	PROPN
ap-9037	275	10	example	example	NOUN
ap-9037	275	11	:	:	PUNCT
ap-9037	275	12	second	second	ADJ
ap-9037	275	13	-	-	PUNCT
ap-9037	275	14	order	order	NOUN
ap-9037	275	15	darboux	darboux	VERB
ap-9037	275	16	-	-	PUNCT
ap-9037	275	17	crum	crum	NOUN
ap-9037	275	18	transformation	transformation	NOUN
ap-9037	275	19	.	.	PUNCT
ap-9037	276	1	let	let	VERB
ap-9037	276	2	us	we	PRON
ap-9037	276	3	now	now	ADV
ap-9037	276	4	give	give	VERB
ap-9037	276	5	an	an	DET
ap-9037	276	6	example	example	NOUN
ap-9037	276	7	of	of	ADP
ap-9037	276	8	a	a	DET
ap-9037	276	9	potential	potential	ADJ
ap-9037	276	10	vϕ	vϕ	NOUN
ap-9037	276	11	,	,	PUNCT
ap-9037	276	12	for	for	ADP
ap-9037	276	13	which	which	PRON
ap-9037	276	14	the	the	DET
ap-9037	276	15	polar	polar	ADJ
ap-9037	276	16	equation	equation	NOUN
ap-9037	276	17	(	(	PUNCT
ap-9037	276	18	27	27	NUM
ap-9037	276	19	)	)	PUNCT
ap-9037	276	20	is	be	AUX
ap-9037	276	21	in	in	ADP
ap-9037	276	22	a	a	DET
ap-9037	276	23	solvable	solvable	ADJ
ap-9037	276	24	form	form	NOUN
ap-9037	276	25	,	,	PUNCT
ap-9037	276	26	such	such	ADJ
ap-9037	276	27	that	that	SCONJ
ap-9037	276	28	a	a	DET
ap-9037	276	29	darboux	darboux	ADJ
ap-9037	276	30	-	-	PUNCT
ap-9037	276	31	crum	crum	NOUN
ap-9037	276	32	transformation	transformation	NOUN
ap-9037	276	33	can	can	AUX
ap-9037	276	34	be	be	AUX
ap-9037	276	35	applied	apply	VERB
ap-9037	276	36	.	.	PUNCT
ap-9037	277	1	we	we	PRON
ap-9037	277	2	take	take	VERB
ap-9037	277	3	the	the	DET
ap-9037	277	4	simplest	simple	ADJ
ap-9037	277	5	case	case	NOUN
ap-9037	277	6	vθ	vθ	VERB
ap-9037	277	7	=	=	SYM
ap-9037	277	8	0	0	NUM
ap-9037	277	9	,	,	PUNCT
ap-9037	277	10	that	that	ADV
ap-9037	277	11	is	is	ADV
ap-9037	277	12	,	,	PUNCT
ap-9037	277	13	we	we	PRON
ap-9037	277	14	have	have	VERB
ap-9037	277	15	our	our	PRON
ap-9037	277	16	initial	initial	ADJ
ap-9037	277	17	equation	equation	NOUN
ap-9037	277	18	in	in	ADP
ap-9037	277	19	the	the	DET
ap-9037	277	20	form	form	NOUN
ap-9037	277	21	:	:	PUNCT
ap-9037	277	22	θ′′(θ	θ′′(θ	NUM
ap-9037	277	23	)	)	PUNCT
ap-9037	277	24	−	−	NOUN
ap-9037	277	25	2	2	NUM
ap-9037	277	26	[	[	PUNCT
ap-9037	277	27	ν3	ν3	NOUN
ap-9037	277	28	tan(θ	tan(θ	NOUN
ap-9037	277	29	)	)	PUNCT
ap-9037	277	30	−	−	PROPN
ap-9037	278	1	(	(	PUNCT
ap-9037	278	2	1	1	NUM
ap-9037	278	3	2	2	NUM
ap-9037	278	4	+	+	NUM
ap-9037	278	5	ν1	ν1	NOUN
ap-9037	278	6	+	+	CCONJ
ap-9037	278	7	ν2	ν2	NOUN
ap-9037	278	8	)	)	PUNCT
ap-9037	278	9	cot(θ	cot(θ	PROPN
ap-9037	278	10	)	)	PUNCT
ap-9037	278	11	]	]	PUNCT
ap-9037	279	1	θ′(θ	θ′(θ	ADP
ap-9037	279	2	)	)	PUNCT
ap-9037	279	3	+	+	CCONJ
ap-9037	279	4	[	[	PUNCT
ap-9037	279	5	λ	λ	X
ap-9037	279	6	−	−	NOUN
ap-9037	279	7	m2	m2	PROPN
ap-9037	279	8	sin(θ)2	sin(θ)2	NOUN
ap-9037	279	9	−	−	PROPN
ap-9037	279	10	ν3(1	ν3(1	ADJ
ap-9037	279	11	−	−	PROPN
ap-9037	279	12	r3	r3	PROPN
ap-9037	279	13	)	)	PUNCT
ap-9037	279	14	cos(θ)2	cos(θ)2	NOUN
ap-9037	279	15	]	]	PUNCT
ap-9037	279	16	θ(θ	θ(θ	VERB
ap-9037	279	17	)	)	PUNCT
ap-9037	279	18	=	=	SYM
ap-9037	279	19	0	0	NUM
ap-9037	279	20	.	.	PUNCT
ap-9037	280	1	we	we	PRON
ap-9037	280	2	can	can	AUX
ap-9037	280	3	obtain	obtain	VERB
ap-9037	280	4	a	a	DET
ap-9037	280	5	particular	particular	ADJ
ap-9037	280	6	solution	solution	NOUN
ap-9037	280	7	of	of	ADP
ap-9037	280	8	this	this	DET
ap-9037	280	9	equation	equation	NOUN
ap-9037	280	10	from	from	ADP
ap-9037	280	11	our	our	PRON
ap-9037	280	12	prior	prior	ADJ
ap-9037	280	13	calculations	calculation	NOUN
ap-9037	280	14	.	.	PUNCT
ap-9037	281	1	more	more	ADV
ap-9037	281	2	precisely	precisely	ADV
ap-9037	281	3	,	,	PUNCT
ap-9037	281	4	this	this	DET
ap-9037	281	5	solution	solution	NOUN
ap-9037	281	6	is	be	AUX
ap-9037	281	7	given	give	VERB
ap-9037	281	8	by	by	ADP
ap-9037	281	9	(	(	PUNCT
ap-9037	281	10	40	40	NUM
ap-9037	281	11	)	)	PUNCT
ap-9037	281	12	for	for	ADP
ap-9037	281	13	the	the	DET
ap-9037	281	14	settings	setting	NOUN
ap-9037	281	15	:	:	PUNCT
ap-9037	281	16	λ	λ	X
ap-9037	281	17	=	=	SYM
ap-9037	281	18	−	−	PROPN
ap-9037	281	19	(	(	PUNCT
ap-9037	281	20	1	1	NUM
ap-9037	281	21	+	+	NUM
ap-9037	281	22	2ν1	2ν1	NUM
ap-9037	281	23	+	+	SYM
ap-9037	281	24	2ν2	2ν2	NUM
ap-9037	281	25	+	+	CCONJ
ap-9037	281	26	2ν3)2	2ν3)2	PROPN
ap-9037	281	27	4	4	NUM
ap-9037	281	28	+	+	CCONJ
ap-9037	281	29	[	[	PUNCT
ap-9037	281	30	2n	2n	NUM
ap-9037	282	1	+	+	CCONJ
ap-9037	282	2	1	1	NUM
ap-9037	282	3	+	+	CCONJ
ap-9037	282	4	√	√	PROPN
ap-9037	282	5	m2	m2	PROPN
ap-9037	282	6	+	+	CCONJ
ap-9037	282	7	(	(	PUNCT
ap-9037	282	8	ν1	ν1	NOUN
ap-9037	282	9	+	+	CCONJ
ap-9037	282	10	ν2)2	ν2)2	PUNCT
ap-9037	282	11	+	+	CCONJ
ap-9037	282	12	1	1	NUM
ap-9037	282	13	2	2	NUM
ap-9037	282	14	√	√	NUM
ap-9037	282	15	1	1	NUM
ap-9037	282	16	+	+	NUM
ap-9037	282	17	4ν3	4ν3	NUM
ap-9037	282	18	(	(	PUNCT
ap-9037	282	19	ν3	ν3	NOUN
ap-9037	282	20	−	−	PROPN
ap-9037	282	21	r3	r3	PROPN
ap-9037	282	22	)	)	PUNCT
ap-9037	282	23	]	]	PUNCT
ap-9037	282	24	2	2	NUM
ap-9037	282	25	,	,	PUNCT
ap-9037	282	26	(	(	PUNCT
ap-9037	282	27	54	54	NUM
ap-9037	282	28	)	)	PUNCT
ap-9037	282	29	a	a	PRON
ap-9037	282	30	=	=	X
ap-9037	282	31	[	[	PUNCT
ap-9037	282	32	2n	2n	NUM
ap-9037	283	1	+	+	CCONJ
ap-9037	283	2	1	1	NUM
ap-9037	283	3	+	+	CCONJ
ap-9037	283	4	√	√	PROPN
ap-9037	283	5	m2	m2	PROPN
ap-9037	283	6	+	+	CCONJ
ap-9037	283	7	(	(	PUNCT
ap-9037	283	8	ν1	ν1	NOUN
ap-9037	283	9	+	+	CCONJ
ap-9037	283	10	ν2)2	ν2)2	PUNCT
ap-9037	283	11	+	+	CCONJ
ap-9037	283	12	1	1	NUM
ap-9037	283	13	2	2	NUM
ap-9037	283	14	√	√	NUM
ap-9037	283	15	1	1	NUM
ap-9037	283	16	+	+	CCONJ
ap-9037	283	17	4ν3(ν3	4ν3(ν3	NUM
ap-9037	283	18	−	−	PROPN
ap-9037	283	19	r3	r3	PROPN
ap-9037	283	20	)	)	PUNCT
ap-9037	283	21	]	]	PUNCT
ap-9037	283	22	2	2	NUM
ap-9037	283	23	,	,	PUNCT
ap-9037	283	24	(	(	PUNCT
ap-9037	283	25	55	55	NUM
ap-9037	283	26	)	)	PUNCT
ap-9037	283	27	b	b	NOUN
ap-9037	284	1	=	=	SYM
ap-9037	284	2	1	1	NUM
ap-9037	284	3	4	4	NUM
ap-9037	284	4	−	−	NOUN
ap-9037	284	5	m2	m2	PROPN
ap-9037	284	6	−	−	PROPN
ap-9037	284	7	(	(	PUNCT
ap-9037	284	8	ν1	ν1	NOUN
ap-9037	284	9	+	+	CCONJ
ap-9037	284	10	ν2)2	ν2)2	CCONJ
ap-9037	284	11	,	,	PUNCT
ap-9037	284	12	(	(	PUNCT
ap-9037	284	13	56	56	NUM
ap-9037	284	14	)	)	PUNCT
ap-9037	284	15	c	c	NOUN
ap-9037	284	16	=	=	SYM
ap-9037	284	17	(	(	PUNCT
ap-9037	284	18	r3	r3	PROPN
ap-9037	284	19	−	−	PROPN
ap-9037	284	20	ν3)ν3	ν3)ν3	PROPN
ap-9037	284	21	.	.	PUNCT
ap-9037	285	1	(	(	PUNCT
ap-9037	285	2	57	57	NUM
ap-9037	285	3	)	)	PUNCT
ap-9037	285	4	now	now	ADV
ap-9037	285	5	,	,	PUNCT
ap-9037	285	6	in	in	ADP
ap-9037	285	7	order	order	NOUN
ap-9037	285	8	to	to	PART
ap-9037	285	9	apply	apply	VERB
ap-9037	285	10	a	a	DET
ap-9037	285	11	darboux	darboux	ADJ
ap-9037	285	12	-	-	PUNCT
ap-9037	285	13	crum	crum	NOUN
ap-9037	285	14	transformation	transformation	NOUN
ap-9037	285	15	of	of	ADP
ap-9037	285	16	second	second	ADJ
ap-9037	285	17	order	order	NOUN
ap-9037	285	18	,	,	PUNCT
ap-9037	285	19	we	we	PRON
ap-9037	285	20	must	must	AUX
ap-9037	285	21	provide	provide	VERB
ap-9037	285	22	two	two	NUM
ap-9037	285	23	transformation	transformation	NOUN
ap-9037	285	24	functions	function	NOUN
ap-9037	285	25	θ1	θ1	NOUN
ap-9037	285	26	and	and	CCONJ
ap-9037	285	27	θ2	θ2	NOUN
ap-9037	285	28	that	that	PRON
ap-9037	285	29	solve	solve	VERB
ap-9037	285	30	our	our	PRON
ap-9037	285	31	initial	initial	ADJ
ap-9037	285	32	equation	equation	NOUN
ap-9037	285	33	for	for	ADP
ap-9037	285	34	transformation	transformation	NOUN
ap-9037	285	35	parameters	parameter	NOUN
ap-9037	285	36	λ1	λ1	ADJ
ap-9037	285	37	and	and	CCONJ
ap-9037	285	38	λ2	λ2	PROPN
ap-9037	285	39	,	,	PUNCT
ap-9037	285	40	given	give	VERB
ap-9037	285	41	by	by	ADP
ap-9037	285	42	(	(	PUNCT
ap-9037	285	43	54	54	NUM
ap-9037	285	44	)	)	PUNCT
ap-9037	285	45	,	,	PUNCT
ap-9037	285	46	and	and	CCONJ
ap-9037	285	47	determined	determine	VERB
ap-9037	285	48	by	by	ADP
ap-9037	285	49	the	the	DET
ap-9037	285	50	values	value	NOUN
ap-9037	285	51	of	of	ADP
ap-9037	285	52	n1	n1	NOUN
ap-9037	285	53	and	and	CCONJ
ap-9037	285	54	n2	n2	ADJ
ap-9037	285	55	.	.	PUNCT
ap-9037	286	1	these	these	DET
ap-9037	286	2	values	value	NOUN
ap-9037	286	3	are	be	AUX
ap-9037	286	4	chosen	choose	VERB
ap-9037	286	5	as	as	ADP
ap-9037	286	6	:	:	PUNCT
ap-9037	286	7	n1	n1	NOUN
ap-9037	286	8	=	=	SYM
ap-9037	286	9	1	1	NUM
ap-9037	286	10	n2	n2	NOUN
ap-9037	286	11	=	=	SYM
ap-9037	286	12	2	2	NUM
ap-9037	286	13	.	.	PUNCT
ap-9037	287	1	by	by	ADP
ap-9037	287	2	substituting	substitute	VERB
ap-9037	287	3	them	they	PRON
ap-9037	287	4	into	into	ADP
ap-9037	287	5	(	(	PUNCT
ap-9037	287	6	40	40	NUM
ap-9037	287	7	)	)	PUNCT
ap-9037	287	8	along	along	ADP
ap-9037	287	9	with	with	ADP
ap-9037	287	10	the	the	DET
ap-9037	287	11	settings	setting	NOUN
ap-9037	287	12	(	(	PUNCT
ap-9037	287	13	54)–(57	54)–(57	NUM
ap-9037	287	14	)	)	PUNCT
ap-9037	287	15	,	,	PUNCT
ap-9037	287	16	we	we	PRON
ap-9037	287	17	obtain	obtain	VERB
ap-9037	287	18	our	our	PRON
ap-9037	287	19	transformation	transformation	NOUN
ap-9037	287	20	functions	function	NOUN
ap-9037	287	21	θ1	θ1	PROPN
ap-9037	287	22	and	and	CCONJ
ap-9037	287	23	θ2	θ2	PROPN
ap-9037	287	24	.	.	PUNCT
ap-9037	288	1	the	the	DET
ap-9037	288	2	first	first	ADJ
ap-9037	288	3	of	of	ADP
ap-9037	288	4	these	these	DET
ap-9037	288	5	functions	function	NOUN
ap-9037	288	6	reads	read	VERB
ap-9037	288	7	:	:	PUNCT
ap-9037	288	8	θ1(θ	θ1(θ	X
ap-9037	288	9	)	)	PUNCT
ap-9037	288	10	=	=	SYM
ap-9037	288	11	cos(θ	cos(θ	PROPN
ap-9037	288	12	)	)	PUNCT
ap-9037	288	13	1	1	NUM
ap-9037	288	14	2	2	NUM
ap-9037	288	15	−ν3	−ν3	NOUN
ap-9037	288	16	+	+	CCONJ
ap-9037	288	17	1	1	NUM
ap-9037	288	18	2	2	NUM
ap-9037	288	19	√	√	NUM
ap-9037	288	20	1	1	NUM
ap-9037	288	21	+	+	NOUN
ap-9037	288	22	4ν3(ν3−r3	4ν3(ν3−r3	NOUN
ap-9037	288	23	)	)	PUNCT
ap-9037	289	1	sin(θ)−ν1−ν2	sin(θ)−ν1−ν2	PROPN
ap-9037	289	2	+	+	NOUN
ap-9037	289	3	√	√	ADJ
ap-9037	289	4	m2+(ν1+ν2)2	m2+(ν1+ν2)2	NOUN
ap-9037	289	5	×	×	NOUN
ap-9037	289	6	{	{	PUNCT
ap-9037	289	7	2	2	NUM
ap-9037	289	8	+	+	NUM
ap-9037	289	9	2	2	NUM
ap-9037	289	10	√	√	NOUN
ap-9037	289	11	m2	m2	PROPN
ap-9037	289	12	+	+	CCONJ
ap-9037	289	13	(	(	PUNCT
ap-9037	289	14	ν1	ν1	NOUN
ap-9037	289	15	+	+	CCONJ
ap-9037	289	16	ν2)2	ν2)2	CCONJ
ap-9037	289	17	−	−	NOUN
ap-9037	289	18	sin(θ)2	sin(θ)2	NOUN
ap-9037	289	19	[	[	PUNCT
ap-9037	289	20	4	4	NUM
ap-9037	289	21	+	+	SYM
ap-9037	289	22	2	2	NUM
ap-9037	289	23	√	√	NOUN
ap-9037	289	24	m2	m2	PROPN
ap-9037	289	25	+	+	CCONJ
ap-9037	289	26	(	(	PUNCT
ap-9037	289	27	ν1	ν1	NOUN
ap-9037	289	28	+	+	CCONJ
ap-9037	289	29	ν2)2	ν2)2	PUNCT
ap-9037	289	30	+	+	CCONJ
ap-9037	289	31	√	√	NUM
ap-9037	289	32	1	1	NUM
ap-9037	290	1	+	+	NUM
ap-9037	290	2	4ν3(ν3	4ν3(ν3	NUM
ap-9037	290	3	−	−	PROPN
ap-9037	290	4	r3	r3	PROPN
ap-9037	290	5	)	)	PUNCT
ap-9037	290	6	]	]	PUNCT
ap-9037	290	7	}	}	PUNCT
ap-9037	290	8	,	,	PUNCT
ap-9037	290	9	(	(	PUNCT
ap-9037	290	10	58	58	X
ap-9037	290	11	)	)	PUNCT
ap-9037	290	12	we	we	PRON
ap-9037	290	13	omit	omit	VERB
ap-9037	290	14	the	the	DET
ap-9037	290	15	explicit	explicit	ADJ
ap-9037	290	16	form	form	NOUN
ap-9037	290	17	of	of	ADP
ap-9037	290	18	the	the	DET
ap-9037	290	19	remaining	remain	VERB
ap-9037	290	20	transformation	transformation	NOUN
ap-9037	290	21	function	function	NOUN
ap-9037	290	22	θ2	θ2	PROPN
ap-9037	290	23	due	due	ADP
ap-9037	290	24	to	to	ADP
ap-9037	290	25	its	its	PRON
ap-9037	290	26	length	length	NOUN
ap-9037	290	27	.	.	PUNCT
ap-9037	291	1	let	let	VERB
ap-9037	291	2	us	we	PRON
ap-9037	291	3	now	now	ADV
ap-9037	291	4	perform	perform	VERB
ap-9037	291	5	the	the	DET
ap-9037	291	6	darboux	darboux	ADJ
ap-9037	291	7	-	-	PUNCT
ap-9037	291	8	crum	crum	NOUN
ap-9037	291	9	transformation	transformation	NOUN
ap-9037	291	10	by	by	ADP
ap-9037	291	11	inserting	insert	VERB
ap-9037	291	12	our	our	PRON
ap-9037	291	13	transformation	transformation	NOUN
ap-9037	291	14	functions	function	NOUN
ap-9037	291	15	,	,	PUNCT
ap-9037	291	16	and	and	CCONJ
ap-9037	291	17	the	the	DET
ap-9037	291	18	solution	solution	NOUN
ap-9037	291	19	(	(	PUNCT
ap-9037	291	20	40	40	NUM
ap-9037	291	21	)	)	PUNCT
ap-9037	291	22	with	with	ADP
ap-9037	291	23	(	(	PUNCT
ap-9037	291	24	54)–(57	54)–(57	NUM
ap-9037	291	25	)	)	PUNCT
ap-9037	291	26	281	281	NUM
ap-9037	291	27	axel	axel	PROPN
ap-9037	291	28	schulze	schulze	NOUN
ap-9037	291	29	-	-	PUNCT
ap-9037	291	30	halberg	halberg	PROPN
ap-9037	291	31	acta	acta	PROPN
ap-9037	291	32	polytechnica	polytechnica	PROPN
ap-9037	291	33	into	into	ADP
ap-9037	291	34	(	(	PUNCT
ap-9037	291	35	52	52	NUM
ap-9037	291	36	)	)	PUNCT
ap-9037	291	37	and	and	CCONJ
ap-9037	291	38	(	(	PUNCT
ap-9037	291	39	53	53	NUM
ap-9037	291	40	)	)	PUNCT
ap-9037	291	41	.	.	PUNCT
ap-9037	292	1	since	since	SCONJ
ap-9037	292	2	the	the	DET
ap-9037	292	3	resulting	result	VERB
ap-9037	292	4	expressions	expression	NOUN
ap-9037	292	5	are	be	AUX
ap-9037	292	6	very	very	ADV
ap-9037	292	7	long	long	ADJ
ap-9037	292	8	,	,	PUNCT
ap-9037	292	9	we	we	PRON
ap-9037	292	10	restrict	restrict	VERB
ap-9037	292	11	ourselves	ourselves	PRON
ap-9037	292	12	here	here	ADV
ap-9037	292	13	to	to	PART
ap-9037	292	14	show	show	VERB
ap-9037	292	15	a	a	DET
ap-9037	292	16	particular	particular	ADJ
ap-9037	292	17	case	case	NOUN
ap-9037	292	18	of	of	ADP
ap-9037	292	19	the	the	DET
ap-9037	292	20	solution	solution	NOUN
ap-9037	292	21	and	and	CCONJ
ap-9037	292	22	the	the	DET
ap-9037	292	23	potential	potential	NOUN
ap-9037	292	24	that	that	PRON
ap-9037	292	25	enter	enter	VERB
ap-9037	292	26	in	in	ADP
ap-9037	292	27	the	the	DET
ap-9037	292	28	transformed	transform	VERB
ap-9037	292	29	polar	polar	ADJ
ap-9037	292	30	equation	equation	NOUN
ap-9037	292	31	(	(	PUNCT
ap-9037	292	32	51	51	NUM
ap-9037	292	33	)	)	PUNCT
ap-9037	292	34	.	.	PUNCT
ap-9037	293	1	we	we	PRON
ap-9037	293	2	choose	choose	VERB
ap-9037	293	3	our	our	PRON
ap-9037	293	4	parameters	parameter	NOUN
ap-9037	293	5	as	as	SCONJ
ap-9037	293	6	follows	follow	VERB
ap-9037	293	7	:	:	PUNCT
ap-9037	293	8	ν1	ν1	NOUN
ap-9037	293	9	=	=	SYM
ap-9037	293	10	1	1	NUM
ap-9037	293	11	ν2	ν2	NOUN
ap-9037	293	12	=	=	SYM
ap-9037	293	13	2	2	NUM
ap-9037	293	14	ν3	ν3	NOUN
ap-9037	293	15	=	=	SYM
ap-9037	293	16	3	3	NUM
ap-9037	293	17	m	m	NOUN
ap-9037	293	18	=	=	SYM
ap-9037	293	19	5	5	NUM
ap-9037	293	20	.	.	PUNCT
ap-9037	294	1	(	(	PUNCT
ap-9037	294	2	59	59	NUM
ap-9037	294	3	)	)	PUNCT
ap-9037	294	4	next	next	ADV
ap-9037	294	5	,	,	PUNCT
ap-9037	294	6	we	we	PRON
ap-9037	294	7	insert	insert	VERB
ap-9037	294	8	them	they	PRON
ap-9037	294	9	into	into	ADP
ap-9037	294	10	(	(	PUNCT
ap-9037	294	11	52	52	NUM
ap-9037	294	12	)	)	PUNCT
ap-9037	294	13	.	.	PUNCT
ap-9037	295	1	the	the	DET
ap-9037	295	2	evaluation	evaluation	NOUN
ap-9037	295	3	gives	give	VERB
ap-9037	295	4	the	the	DET
ap-9037	295	5	simplest	simple	ADJ
ap-9037	295	6	solutions	solution	NOUN
ap-9037	295	7	in	in	ADP
ap-9037	295	8	the	the	DET
ap-9037	295	9	form	form	NOUN
ap-9037	295	10	:	:	PUNCT
ap-9037	295	11	θ̂(θ)|n=0,r3=−1	θ̂(θ)|n=0,r3=−1	ADP
ap-9037	295	12	=	=	PUNCT
ap-9037	295	13	8(2483	8(2483	NUM
ap-9037	295	14	+	+	NOUN
ap-9037	295	15	426	426	NUM
ap-9037	295	16	√	√	NUM
ap-9037	295	17	34	34	NUM
ap-9037	295	18	)	)	PUNCT
ap-9037	295	19	cos(θ)3	cos(θ)3	NOUN
ap-9037	295	20	sin(θ)−1	sin(θ)−1	NOUN
ap-9037	295	21	+	+	CCONJ
ap-9037	295	22	√	√	NUM
ap-9037	295	23	34	34	NUM
ap-9037	295	24	992	992	NUM
ap-9037	295	25	+	+	NOUN
ap-9037	295	26	156	156	NUM
ap-9037	295	27	√	√	NUM
ap-9037	295	28	34−4(745	34−4(745	NUM
ap-9037	295	29	+	+	NOUN
ap-9037	295	30	129	129	NUM
ap-9037	295	31	√	√	NUM
ap-9037	295	32	34	34	NUM
ap-9037	295	33	)	)	PUNCT
ap-9037	295	34	sin(θ)2+(2483	sin(θ)2+(2483	NOUN
ap-9037	295	35	+	+	PROPN
ap-9037	295	36	426	426	NUM
ap-9037	295	37	√	√	NUM
ap-9037	295	38	34	34	NUM
ap-9037	295	39	)	)	PUNCT
ap-9037	295	40	sin(θ)4	sin(θ)4	NOUN
ap-9037	295	41	,	,	PUNCT
ap-9037	295	42	θ̂(θ)|n=0,r3=1	θ̂(θ)|n=0,r3=1	ADP
ap-9037	295	43	=	=	SYM
ap-9037	295	44	440(35	440(35	NUM
ap-9037	295	45	+	+	PROPN
ap-9037	295	46	6	6	NUM
ap-9037	295	47	√	√	NUM
ap-9037	295	48	34	34	NUM
ap-9037	295	49	)	)	PUNCT
ap-9037	295	50	cos(θ)3	cos(θ)3	NOUN
ap-9037	295	51	sin(θ)−1	sin(θ)−1	NOUN
ap-9037	295	52	+	+	CCONJ
ap-9037	295	53	√	√	PROPN
ap-9037	295	54	34	34	NUM
ap-9037	295	55	896	896	NUM
ap-9037	295	56	+	+	NOUN
ap-9037	295	57	148	148	NUM
ap-9037	295	58	√	√	NUM
ap-9037	295	59	34−4(637	34−4(637	NUM
ap-9037	296	1	+	+	NOUN
ap-9037	296	2	109	109	NUM
ap-9037	296	3	√	√	NUM
ap-9037	296	4	34	34	NUM
ap-9037	296	5	)	)	PUNCT
ap-9037	296	6	sin(θ)2	sin(θ)2	NOUN
ap-9037	296	7	+	+	NOUN
ap-9037	296	8	55(35	55(35	NOUN
ap-9037	296	9	+	+	NOUN
ap-9037	296	10	6	6	NUM
ap-9037	296	11	√	√	NUM
ap-9037	296	12	34	34	NUM
ap-9037	296	13	)	)	PUNCT
ap-9037	296	14	sin(θ)4	sin(θ)4	NOUN
ap-9037	296	15	.	.	PUNCT
ap-9037	297	1	graphs	graph	NOUN
ap-9037	297	2	of	of	ADP
ap-9037	297	3	these	these	DET
ap-9037	297	4	functions	function	NOUN
ap-9037	297	5	can	can	AUX
ap-9037	297	6	be	be	AUX
ap-9037	297	7	found	find	VERB
ap-9037	297	8	in	in	ADP
ap-9037	297	9	figures	figure	NOUN
ap-9037	297	10	3	3	NUM
ap-9037	297	11	and	and	CCONJ
ap-9037	297	12	4	4	NUM
ap-9037	297	13	.	.	X
ap-9037	297	14	note	note	VERB
ap-9037	297	15	that	that	SCONJ
ap-9037	297	16	the	the	DET
ap-9037	297	17	parameter	parameter	NOUN
ap-9037	297	18	values	value	NOUN
ap-9037	297	19	chosen	choose	VERB
ap-9037	297	20	in	in	ADP
ap-9037	297	21	the	the	DET
ap-9037	297	22	latter	latter	ADJ
ap-9037	297	23	functions	function	NOUN
ap-9037	297	24	comply	comply	VERB
ap-9037	297	25	with	with	ADP
ap-9037	297	26	the	the	DET
ap-9037	297	27	appropriate	appropriate	ADJ
ap-9037	297	28	conditions	condition	NOUN
ap-9037	297	29	(	(	PUNCT
ap-9037	297	30	42	42	NUM
ap-9037	297	31	)	)	PUNCT
ap-9037	297	32	and	and	CCONJ
ap-9037	297	33	(	(	PUNCT
ap-9037	297	34	43	43	NUM
ap-9037	297	35	)	)	PUNCT
ap-9037	297	36	,	,	PUNCT
ap-9037	297	37	respectively	respectively	ADV
ap-9037	297	38	.	.	PUNCT
ap-9037	298	1	0.5	0.5	NUM
ap-9037	298	2	1.0	1.0	NUM
ap-9037	298	3	1.5	1.5	NUM
ap-9037	298	4	2.0	2.0	NUM
ap-9037	298	5	2.5	2.5	NUM
ap-9037	298	6	3.0	3.0	NUM
ap-9037	298	7	θ	θ	NOUN
ap-9037	298	8	-0.5	-0.5	X
ap-9037	298	9	0.5	0.5	NUM
ap-9037	298	10	q	q	NOUN
ap-9037	298	11	`	`	PUNCT
ap-9037	298	12	hθl	hθl	PROPN
ap-9037	298	13	n	n	PROPN
ap-9037	298	14	=	=	SYM
ap-9037	298	15	4	4	NUM
ap-9037	298	16	n	n	NOUN
ap-9037	298	17	=	=	SYM
ap-9037	298	18	3	3	NUM
ap-9037	298	19	n	n	NOUN
ap-9037	298	20	=	=	SYM
ap-9037	298	21	0	0	NUM
ap-9037	298	22	figure	figure	NOUN
ap-9037	298	23	3	3	NUM
ap-9037	298	24	.	.	PUNCT
ap-9037	298	25	graphs	graph	NOUN
ap-9037	298	26	of	of	ADP
ap-9037	298	27	odd	odd	ADJ
ap-9037	298	28	solutions	solution	NOUN
ap-9037	298	29	(	(	PUNCT
ap-9037	298	30	52	52	NUM
ap-9037	298	31	)	)	PUNCT
ap-9037	298	32	for	for	ADP
ap-9037	298	33	the	the	DET
ap-9037	298	34	settings	setting	NOUN
ap-9037	298	35	(	(	PUNCT
ap-9037	298	36	59	59	NUM
ap-9037	298	37	)	)	PUNCT
ap-9037	298	38	,	,	PUNCT
ap-9037	298	39	r3	r3	PROPN
ap-9037	298	40	=	=	SYM
ap-9037	298	41	−1	−1	NOUN
ap-9037	298	42	,	,	PUNCT
ap-9037	298	43	and	and	CCONJ
ap-9037	298	44	different	different	ADJ
ap-9037	298	45	values	value	NOUN
ap-9037	298	46	of	of	ADP
ap-9037	298	47	n	n	PROPN
ap-9037	298	48	.	.	PUNCT
ap-9037	299	1	the	the	DET
ap-9037	299	2	transformation	transformation	NOUN
ap-9037	299	3	functions	function	NOUN
ap-9037	299	4	for	for	ADP
ap-9037	299	5	the	the	DET
ap-9037	299	6	darboux	darboux	VERB
ap-9037	299	7	-	-	PUNCT
ap-9037	299	8	crum	crum	NOUN
ap-9037	299	9	transformation	transformation	NOUN
ap-9037	299	10	are	be	AUX
ap-9037	299	11	obtained	obtain	VERB
ap-9037	299	12	from	from	ADP
ap-9037	299	13	(	(	PUNCT
ap-9037	299	14	40	40	NUM
ap-9037	299	15	)	)	PUNCT
ap-9037	299	16	with	with	ADP
ap-9037	299	17	the	the	DET
ap-9037	299	18	settings	setting	NOUN
ap-9037	299	19	(	(	PUNCT
ap-9037	299	20	54)–(57	54)–(57	NUM
ap-9037	299	21	)	)	PUNCT
ap-9037	299	22	,	,	PUNCT
ap-9037	299	23	and	and	CCONJ
ap-9037	299	24	n1	n1	NOUN
ap-9037	299	25	=	=	SYM
ap-9037	299	26	1	1	NUM
ap-9037	299	27	,	,	PUNCT
ap-9037	299	28	n2	n2	NOUN
ap-9037	299	29	=	=	SYM
ap-9037	299	30	2	2	NUM
ap-9037	299	31	.	.	NOUN
ap-9037	299	32	0.5	0.5	NUM
ap-9037	299	33	1.0	1.0	NUM
ap-9037	299	34	1.5	1.5	NUM
ap-9037	299	35	2.0	2.0	NUM
ap-9037	299	36	2.5	2.5	NUM
ap-9037	299	37	3.0	3.0	NUM
ap-9037	299	38	θ	θ	NOUN
ap-9037	299	39	-0.5	-0.5	X
ap-9037	299	40	0.5	0.5	NUM
ap-9037	299	41	1.0	1.0	NUM
ap-9037	299	42	q	q	NOUN
ap-9037	299	43	`	`	PUNCT
ap-9037	299	44	hθl	hθl	PROPN
ap-9037	299	45	n	n	PROPN
ap-9037	299	46	=	=	SYM
ap-9037	299	47	4	4	NUM
ap-9037	299	48	n	n	NOUN
ap-9037	299	49	=	=	SYM
ap-9037	299	50	3	3	NUM
ap-9037	299	51	n	n	NOUN
ap-9037	299	52	=	=	PRON
ap-9037	299	53	figure	figure	NOUN
ap-9037	299	54	4	4	NUM
ap-9037	299	55	.	.	PUNCT
ap-9037	300	1	graphs	graph	NOUN
ap-9037	300	2	of	of	ADP
ap-9037	300	3	even	even	ADV
ap-9037	300	4	solutions	solution	NOUN
ap-9037	300	5	(	(	PUNCT
ap-9037	300	6	52	52	NUM
ap-9037	300	7	)	)	PUNCT
ap-9037	300	8	for	for	ADP
ap-9037	300	9	the	the	DET
ap-9037	300	10	settings	setting	NOUN
ap-9037	300	11	(	(	PUNCT
ap-9037	300	12	59	59	NUM
ap-9037	300	13	)	)	PUNCT
ap-9037	300	14	,	,	PUNCT
ap-9037	300	15	r3	r3	PROPN
ap-9037	300	16	=	=	SYM
ap-9037	300	17	1	1	NUM
ap-9037	300	18	,	,	PUNCT
ap-9037	300	19	and	and	CCONJ
ap-9037	300	20	different	different	ADJ
ap-9037	300	21	values	value	NOUN
ap-9037	300	22	of	of	ADP
ap-9037	300	23	n	n	PROPN
ap-9037	300	24	.	.	PUNCT
ap-9037	301	1	the	the	DET
ap-9037	301	2	transformation	transformation	NOUN
ap-9037	301	3	functions	function	NOUN
ap-9037	301	4	for	for	ADP
ap-9037	301	5	the	the	DET
ap-9037	301	6	darboux	darboux	VERB
ap-9037	301	7	-	-	PUNCT
ap-9037	301	8	crum	crum	NOUN
ap-9037	301	9	transformation	transformation	NOUN
ap-9037	301	10	are	be	AUX
ap-9037	301	11	obtained	obtain	VERB
ap-9037	301	12	from	from	ADP
ap-9037	301	13	(	(	PUNCT
ap-9037	301	14	40	40	NUM
ap-9037	301	15	)	)	PUNCT
ap-9037	301	16	with	with	ADP
ap-9037	301	17	the	the	DET
ap-9037	301	18	settings	setting	NOUN
ap-9037	301	19	(	(	PUNCT
ap-9037	301	20	54)–(57	54)–(57	NUM
ap-9037	301	21	)	)	PUNCT
ap-9037	301	22	,	,	PUNCT
ap-9037	301	23	and	and	CCONJ
ap-9037	301	24	n1	n1	NOUN
ap-9037	301	25	=	=	SYM
ap-9037	301	26	1	1	NUM
ap-9037	301	27	,	,	PUNCT
ap-9037	301	28	n2	n2	NOUN
ap-9037	301	29	=	=	ADJ
ap-9037	301	30	2	2	NUM
ap-9037	301	31	.	.	NOUN
ap-9037	301	32	282	282	NUM
ap-9037	301	33	vol	vol	NOUN
ap-9037	301	34	.	.	PUNCT
ap-9037	302	1	63	63	NUM
ap-9037	303	1	no	no	INTJ
ap-9037	303	2	.	.	PUNCT
ap-9037	304	1	4/2023	4/2023	NOUN
ap-9037	304	2	construction	construction	NOUN
ap-9037	304	3	of	of	ADP
ap-9037	304	4	angular	angular	ADJ
ap-9037	304	5	-	-	PUNCT
ap-9037	304	6	dependent	dependent	ADJ
ap-9037	304	7	potentials	potential	NOUN
ap-9037	304	8	before	before	SCONJ
ap-9037	304	9	we	we	PRON
ap-9037	304	10	conclude	conclude	VERB
ap-9037	304	11	this	this	DET
ap-9037	304	12	example	example	NOUN
ap-9037	304	13	,	,	PUNCT
ap-9037	304	14	let	let	VERB
ap-9037	304	15	us	we	PRON
ap-9037	304	16	state	state	VERB
ap-9037	304	17	a	a	DET
ap-9037	304	18	particular	particular	ADJ
ap-9037	304	19	case	case	NOUN
ap-9037	304	20	of	of	ADP
ap-9037	304	21	the	the	DET
ap-9037	304	22	transformed	transform	VERB
ap-9037	304	23	potential	potential	NOUN
ap-9037	304	24	v̂θ	v̂θ	ADV
ap-9037	304	25	that	that	PRON
ap-9037	304	26	is	be	AUX
ap-9037	304	27	obtained	obtain	VERB
ap-9037	304	28	from	from	ADP
ap-9037	304	29	(	(	PUNCT
ap-9037	304	30	53	53	NUM
ap-9037	304	31	)	)	PUNCT
ap-9037	304	32	,	,	PUNCT
ap-9037	304	33	as	as	SCONJ
ap-9037	304	34	described	describe	VERB
ap-9037	304	35	above	above	ADV
ap-9037	304	36	.	.	PUNCT
ap-9037	305	1	upon	upon	SCONJ
ap-9037	305	2	substituting	substitute	VERB
ap-9037	305	3	the	the	DET
ap-9037	305	4	parameter	parameter	NOUN
ap-9037	305	5	settings	setting	NOUN
ap-9037	305	6	(	(	PUNCT
ap-9037	305	7	59	59	NUM
ap-9037	305	8	)	)	PUNCT
ap-9037	305	9	and	and	CCONJ
ap-9037	305	10	r3	r3	PROPN
ap-9037	305	11	=	=	SYM
ap-9037	305	12	−1	−1	NOUN
ap-9037	305	13	,	,	PUNCT
ap-9037	305	14	we	we	PRON
ap-9037	305	15	find	find	VERB
ap-9037	305	16	:	:	PUNCT
ap-9037	305	17	v̂θ(θ	v̂θ(θ	NUM
ap-9037	305	18	)	)	PUNCT
ap-9037	305	19	=	=	PRON
ap-9037	305	20	{	{	PUNCT
ap-9037	305	21	4	4	NUM
ap-9037	305	22	[	[	PUNCT
ap-9037	305	23	−	−	NUM
ap-9037	305	24	89	89	NUM
ap-9037	305	25	(	(	PUNCT
ap-9037	305	26	14609723	14609723	NUM
ap-9037	305	27	+	+	NUM
ap-9037	305	28	2505554	2505554	NUM
ap-9037	305	29	√	√	NUM
ap-9037	305	30	34	34	NUM
ap-9037	305	31	)	)	PUNCT
ap-9037	305	32	cos(2θ	cos(2θ	PROPN
ap-9037	305	33	)	)	PUNCT
ap-9037	306	1	−	−	PROPN
ap-9037	306	2	10(53136719	10(53136719	NUM
ap-9037	307	1	+	+	CCONJ
ap-9037	307	2	9112838	9112838	NUM
ap-9037	307	3	√	√	NUM
ap-9037	307	4	34	34	NUM
ap-9037	307	5	)	)	PUNCT
ap-9037	307	6	×	×	NOUN
ap-9037	307	7	cos(4θ	cos(4θ	NOUN
ap-9037	307	8	)	)	PUNCT
ap-9037	307	9	−	−	PROPN
ap-9037	307	10	29(1632977	29(1632977	PRON
ap-9037	308	1	+	+	CCONJ
ap-9037	308	2	280054	280054	NUM
ap-9037	308	3	√	√	NUM
ap-9037	308	4	34	34	NUM
ap-9037	308	5	)	)	PUNCT
ap-9037	308	6	cos(6θ	cos(6θ	NOUN
ap-9037	308	7	)	)	PUNCT
ap-9037	309	1	+	+	CCONJ
ap-9037	309	2	2048(385457	2048(385457	NUM
ap-9037	309	3	+	+	CCONJ
ap-9037	309	4	66281	66281	NUM
ap-9037	309	5	√	√	NUM
ap-9037	309	6	34	34	NUM
ap-9037	309	7	)	)	PUNCT
ap-9037	309	8	sin(θ)2	sin(θ)2	NOUN
ap-9037	309	9	+	+	NUM
ap-9037	309	10	313632(361	313632(361	NUM
ap-9037	309	11	+	+	CCONJ
ap-9037	309	12	60	60	NUM
ap-9037	309	13	√	√	NUM
ap-9037	309	14	34	34	NUM
ap-9037	309	15	)	)	PUNCT
ap-9037	309	16	cos(θ)2	cos(θ)2	NOUN
ap-9037	309	17	−	−	PROPN
ap-9037	309	18	398655716	398655716	NUM
ap-9037	309	19	√	√	PROPN
ap-9037	309	20	34	34	NUM
ap-9037	309	21	−	−	PROPN
ap-9037	309	22	2327701978	2327701978	NUM
ap-9037	309	23	]	]	PUNCT
ap-9037	309	24	}	}	PUNCT
ap-9037	309	25	(	(	PUNCT
ap-9037	309	26	60	60	NUM
ap-9037	309	27	)	)	PUNCT
ap-9037	309	28	×	×	NOUN
ap-9037	309	29	{	{	PUNCT
ap-9037	309	30	[	[	PUNCT
ap-9037	309	31	4(497	4(497	NUM
ap-9037	309	32	+	+	CCONJ
ap-9037	309	33	90	90	NUM
ap-9037	309	34	√	√	NUM
ap-9037	309	35	34	34	NUM
ap-9037	309	36	)	)	PUNCT
ap-9037	309	37	cos(2θ	cos(2θ	NOUN
ap-9037	309	38	)	)	PUNCT
ap-9037	310	1	+	+	CCONJ
ap-9037	310	2	(	(	PUNCT
ap-9037	310	3	2483	2483	NUM
ap-9037	310	4	+	+	CCONJ
ap-9037	310	5	426	426	NUM
ap-9037	310	6	√	√	NUM
ap-9037	310	7	34	34	NUM
ap-9037	310	8	)	)	PUNCT
ap-9037	310	9	cos(4θ	cos(4θ	NOUN
ap-9037	310	10	)	)	PUNCT
ap-9037	311	1	+	+	NUM
ap-9037	311	2	231(15	231(15	NUM
ap-9037	311	3	+	+	CCONJ
ap-9037	311	4	2	2	NUM
ap-9037	311	5	√	√	NUM
ap-9037	311	6	34))2	34))2	NUM
ap-9037	311	7	]	]	PUNCT
ap-9037	311	8	}	}	PUNCT
ap-9037	311	9	−1	−1	NOUN
ap-9037	311	10	.	.	PUNCT
ap-9037	312	1	the	the	DET
ap-9037	312	2	graph	graph	NOUN
ap-9037	312	3	of	of	ADP
ap-9037	312	4	this	this	DET
ap-9037	312	5	potential	potential	NOUN
ap-9037	312	6	,	,	PUNCT
ap-9037	312	7	along	along	ADP
ap-9037	312	8	with	with	ADP
ap-9037	312	9	its	its	PRON
ap-9037	312	10	counterpart	counterpart	NOUN
ap-9037	312	11	for	for	ADP
ap-9037	312	12	r3	r3	PROPN
ap-9037	312	13	=	=	SYM
ap-9037	312	14	1	1	NUM
ap-9037	312	15	,	,	PUNCT
ap-9037	312	16	can	can	AUX
ap-9037	312	17	be	be	AUX
ap-9037	312	18	seen	see	VERB
ap-9037	312	19	in	in	ADP
ap-9037	312	20	figure	figure	NOUN
ap-9037	312	21	5	5	NUM
ap-9037	312	22	.	.	NOUN
ap-9037	312	23	0.5	0.5	NUM
ap-9037	312	24	1.0	1.0	NUM
ap-9037	312	25	1.5	1.5	NUM
ap-9037	312	26	2.0	2.0	NUM
ap-9037	312	27	2.5	2.5	NUM
ap-9037	312	28	3.0	3.0	NUM
ap-9037	312	29	θ	θ	NOUN
ap-9037	313	1	-100	-100	NUM
ap-9037	313	2	100	100	NUM
ap-9037	313	3	200	200	NUM
ap-9037	313	4	300	300	NUM
ap-9037	313	5	400	400	NUM
ap-9037	313	6	500	500	NUM
ap-9037	313	7	v	v	PRON
ap-9037	313	8	`	`	PUNCT
ap-9037	313	9	θhθl	θhθl	PROPN
ap-9037	313	10	r3	r3	PROPN
ap-9037	313	11	=	=	SYM
ap-9037	313	12	1	1	NUM
ap-9037	313	13	r3	r3	NOUN
ap-9037	313	14	=	=	SYM
ap-9037	313	15	-1	-1	NOUN
ap-9037	313	16	figure	figure	NOUN
ap-9037	313	17	5	5	NUM
ap-9037	313	18	.	.	PUNCT
ap-9037	314	1	graphs	graph	NOUN
ap-9037	314	2	of	of	ADP
ap-9037	314	3	the	the	DET
ap-9037	314	4	potential	potential	ADJ
ap-9037	314	5	(	(	PUNCT
ap-9037	314	6	60	60	NUM
ap-9037	314	7	)	)	PUNCT
ap-9037	314	8	and	and	CCONJ
ap-9037	314	9	its	its	PRON
ap-9037	314	10	counterpart	counterpart	NOUN
ap-9037	314	11	for	for	ADP
ap-9037	314	12	the	the	DET
ap-9037	314	13	two	two	NUM
ap-9037	314	14	admissible	admissible	ADJ
ap-9037	314	15	values	value	NOUN
ap-9037	314	16	of	of	ADP
ap-9037	314	17	r3	r3	PROPN
ap-9037	314	18	.	.	PUNCT
ap-9037	315	1	4.2.2	4.2.2	X
ap-9037	315	2	.	.	PUNCT
ap-9037	316	1	transformation	transformation	NOUN
ap-9037	316	2	parameter	parameter	PROPN
ap-9037	316	3	m2	m2	PROPN
ap-9037	316	4	instead	instead	ADV
ap-9037	316	5	of	of	ADP
ap-9037	316	6	λ	λ	NOUN
ap-9037	316	7	,	,	PUNCT
ap-9037	316	8	we	we	PRON
ap-9037	316	9	can	can	AUX
ap-9037	316	10	also	also	ADV
ap-9037	316	11	use	use	VERB
ap-9037	316	12	the	the	DET
ap-9037	316	13	separation	separation	NOUN
ap-9037	316	14	constant	constant	ADJ
ap-9037	316	15	m2	m2	PROPN
ap-9037	316	16	as	as	ADP
ap-9037	316	17	a	a	DET
ap-9037	316	18	darboux	darboux	ADJ
ap-9037	316	19	-	-	PUNCT
ap-9037	316	20	crum	crum	NOUN
ap-9037	316	21	transformation	transformation	NOUN
ap-9037	316	22	parameter	parameter	NOUN
ap-9037	316	23	.	.	PUNCT
ap-9037	317	1	in	in	ADP
ap-9037	317	2	order	order	NOUN
ap-9037	317	3	to	to	PART
ap-9037	317	4	do	do	AUX
ap-9037	317	5	so	so	ADV
ap-9037	317	6	,	,	PUNCT
ap-9037	317	7	we	we	PRON
ap-9037	317	8	must	must	AUX
ap-9037	317	9	change	change	VERB
ap-9037	317	10	the	the	DET
ap-9037	317	11	form	form	NOUN
ap-9037	317	12	of	of	ADP
ap-9037	317	13	our	our	PRON
ap-9037	317	14	equation	equation	NOUN
ap-9037	317	15	,	,	PUNCT
ap-9037	317	16	so	so	SCONJ
ap-9037	317	17	that	that	SCONJ
ap-9037	317	18	it	it	PRON
ap-9037	317	19	matches	match	VERB
ap-9037	317	20	(	(	PUNCT
ap-9037	317	21	4	4	NUM
ap-9037	317	22	)	)	PUNCT
ap-9037	317	23	.	.	PUNCT
ap-9037	318	1	this	this	PRON
ap-9037	318	2	requires	require	VERB
ap-9037	318	3	a	a	DET
ap-9037	318	4	point	point	NOUN
ap-9037	318	5	transformation	transformation	NOUN
ap-9037	318	6	that	that	PRON
ap-9037	318	7	changes	change	VERB
ap-9037	318	8	both	both	CCONJ
ap-9037	318	9	the	the	DET
ap-9037	318	10	dependent	dependent	ADJ
ap-9037	318	11	and	and	CCONJ
ap-9037	318	12	the	the	DET
ap-9037	318	13	independent	independent	ADJ
ap-9037	318	14	variable	variable	NOUN
ap-9037	318	15	.	.	PUNCT
ap-9037	319	1	we	we	PRON
ap-9037	319	2	set	set	VERB
ap-9037	319	3	:	:	PUNCT
ap-9037	319	4	θ(y	θ(y	X
ap-9037	319	5	)	)	PUNCT
ap-9037	319	6	=	=	SYM
ap-9037	319	7	2arccot[exp(iy	2arccot[exp(iy	NOUN
ap-9037	319	8	)	)	PUNCT
ap-9037	319	9	]	]	PUNCT
ap-9037	319	10	,	,	PUNCT
ap-9037	319	11	(	(	PUNCT
ap-9037	319	12	61	61	NUM
ap-9037	319	13	)	)	PUNCT
ap-9037	319	14	η(y	η(y	NOUN
ap-9037	319	15	)	)	PUNCT
ap-9037	320	1	=	=	SYM
ap-9037	320	2	exp	exp	NOUN
ap-9037	321	1	[	[	X
ap-9037	321	2	i(ν1	i(ν1	NOUN
ap-9037	321	3	+	+	X
ap-9037	321	4	ν2)y	ν2)y	NOUN
ap-9037	321	5	]	]	X
ap-9037	322	1	[	[	X
ap-9037	322	2	exp(2iy	exp(2iy	ADP
ap-9037	322	3	)	)	PUNCT
ap-9037	322	4	−	−	NOUN
ap-9037	323	1	1]ν3	1]ν3	NUM
ap-9037	324	1	[	[	X
ap-9037	324	2	exp(2iy	exp(2iy	ADP
ap-9037	324	3	)	)	PUNCT
ap-9037	324	4	+	+	NUM
ap-9037	324	5	1]ν1+ν2+ν3	1]ν1+ν2+ν3	NUM
ap-9037	324	6	θ{2arccot[exp(iy	θ{2arccot[exp(iy	NOUN
ap-9037	324	7	)	)	PUNCT
ap-9037	324	8	]	]	PUNCT
ap-9037	324	9	}	}	PUNCT
ap-9037	324	10	,	,	PUNCT
ap-9037	324	11	(	(	PUNCT
ap-9037	324	12	62	62	X
ap-9037	324	13	)	)	PUNCT
ap-9037	324	14	introducing	introduce	VERB
ap-9037	324	15	a	a	DET
ap-9037	324	16	new	new	ADJ
ap-9037	324	17	function	function	NOUN
ap-9037	324	18	η	η	PROPN
ap-9037	324	19	.	.	PROPN
ap-9037	324	20	by	by	ADP
ap-9037	324	21	inserting	insert	VERB
ap-9037	324	22	this	this	DET
ap-9037	324	23	point	point	NOUN
ap-9037	324	24	transformation	transformation	NOUN
ap-9037	324	25	in	in	ADP
ap-9037	324	26	(	(	PUNCT
ap-9037	324	27	27	27	NUM
ap-9037	324	28	)	)	PUNCT
ap-9037	324	29	,	,	PUNCT
ap-9037	324	30	we	we	PRON
ap-9037	324	31	obtain	obtain	VERB
ap-9037	324	32	our	our	PRON
ap-9037	324	33	equation	equation	NOUN
ap-9037	324	34	as	as	ADP
ap-9037	324	35	:	:	PUNCT
ap-9037	324	36	η′′(y	η′′(y	NOUN
ap-9037	324	37	)	)	PUNCT
ap-9037	324	38	+	+	CCONJ
ap-9037	324	39	{	{	PUNCT
ap-9037	324	40	m2	m2	PROPN
ap-9037	324	41	+	+	CCONJ
ap-9037	324	42	(	(	PUNCT
ap-9037	324	43	ν1	ν1	NOUN
ap-9037	324	44	+	+	CCONJ
ap-9037	324	45	ν2)2	ν2)2	CCONJ
ap-9037	324	46	−	−	PROPN
ap-9037	324	47	λ	λ	X
ap-9037	324	48	+	+	CCONJ
ap-9037	324	49	(	(	PUNCT
ap-9037	324	50	ν1	ν1	NOUN
ap-9037	324	51	+	+	CCONJ
ap-9037	324	52	ν2	ν2	NOUN
ap-9037	324	53	+	+	CCONJ
ap-9037	324	54	ν3)(1	ν3)(1	NOUN
ap-9037	324	55	+	+	NUM
ap-9037	324	56	ν1	ν1	NOUN
ap-9037	324	57	+	+	CCONJ
ap-9037	324	58	ν2	ν2	NOUN
ap-9037	324	59	+	+	CCONJ
ap-9037	324	60	ν3	ν3	NOUN
ap-9037	324	61	)	)	PUNCT
ap-9037	324	62	cos(y)2	cos(y)2	NOUN
ap-9037	324	63	+	+	CCONJ
ap-9037	324	64	(	(	PUNCT
ap-9037	324	65	r3	r3	PROPN
ap-9037	324	66	−	−	PROPN
ap-9037	324	67	ν3)ν3	ν3)ν3	ADP
ap-9037	324	68	sin(y)2	sin(y)2	ADJ
ap-9037	324	69	+	+	CCONJ
ap-9037	324	70	1	1	NUM
ap-9037	324	71	cos(y)2	cos(y)2	NOUN
ap-9037	324	72	vθ{2arccot[exp(iy	vθ{2arccot[exp(iy	NOUN
ap-9037	324	73	)	)	PUNCT
ap-9037	324	74	]	]	PUNCT
ap-9037	324	75	}	}	PUNCT
ap-9037	324	76	}	}	PUNCT
ap-9037	324	77	η(y	η(y	NOUN
ap-9037	324	78	)	)	PUNCT
ap-9037	324	79	=	=	SYM
ap-9037	324	80	0	0	PUNCT
ap-9037	324	81	.	.	PUNCT
ap-9037	325	1	(	(	PUNCT
ap-9037	325	2	63	63	NUM
ap-9037	325	3	)	)	PUNCT
ap-9037	325	4	note	note	NOUN
ap-9037	325	5	that	that	SCONJ
ap-9037	325	6	the	the	DET
ap-9037	325	7	coordinate	coordinate	NOUN
ap-9037	325	8	change	change	NOUN
ap-9037	325	9	(	(	PUNCT
ap-9037	325	10	61	61	NUM
ap-9037	325	11	)	)	PUNCT
ap-9037	325	12	in	in	ADP
ap-9037	325	13	our	our	PRON
ap-9037	325	14	point	point	NOUN
ap-9037	325	15	transformation	transformation	NOUN
ap-9037	325	16	was	be	AUX
ap-9037	325	17	necessary	necessary	ADJ
ap-9037	325	18	because	because	SCONJ
ap-9037	325	19	in	in	SCONJ
ap-9037	325	20	(	(	PUNCT
ap-9037	325	21	27	27	NUM
ap-9037	325	22	)	)	PUNCT
ap-9037	325	23	the	the	DET
ap-9037	325	24	separation	separation	NOUN
ap-9037	325	25	constant	constant	ADJ
ap-9037	325	26	m2	m2	PROPN
ap-9037	325	27	has	have	VERB
ap-9037	325	28	a	a	DET
ap-9037	325	29	nonconstant	nonconstant	ADJ
ap-9037	325	30	coefficient	coefficient	NOUN
ap-9037	325	31	that	that	PRON
ap-9037	325	32	needs	need	VERB
ap-9037	325	33	to	to	PART
ap-9037	325	34	be	be	AUX
ap-9037	325	35	removed	remove	VERB
ap-9037	325	36	.	.	PUNCT
ap-9037	326	1	we	we	PRON
ap-9037	326	2	observe	observe	VERB
ap-9037	326	3	that	that	SCONJ
ap-9037	326	4	equations	equation	NOUN
ap-9037	326	5	(	(	PUNCT
ap-9037	326	6	63	63	NUM
ap-9037	326	7	)	)	PUNCT
ap-9037	326	8	and	and	CCONJ
ap-9037	326	9	(	(	PUNCT
ap-9037	326	10	4	4	X
ap-9037	326	11	)	)	PUNCT
ap-9037	326	12	can	can	AUX
ap-9037	326	13	now	now	ADV
ap-9037	326	14	be	be	AUX
ap-9037	326	15	matched	match	VERB
ap-9037	326	16	,	,	PUNCT
ap-9037	326	17	if	if	SCONJ
ap-9037	326	18	we	we	PRON
ap-9037	326	19	choose	choose	VERB
ap-9037	326	20	:	:	PUNCT
ap-9037	326	21	e	e	PROPN
ap-9037	326	22	=	=	PROPN
ap-9037	326	23	m2	m2	PROPN
ap-9037	326	24	,	,	PUNCT
ap-9037	326	25	u(y	u(y	PROPN
ap-9037	326	26	)	)	PUNCT
ap-9037	326	27	=	=	NOUN
ap-9037	326	28	−(ν1	−(ν1	ADP
ap-9037	326	29	+	+	X
ap-9037	326	30	ν2)2	ν2)2	PUNCT
ap-9037	326	31	+	+	NUM
ap-9037	326	32	λ	λ	X
ap-9037	326	33	+	+	CCONJ
ap-9037	326	34	(	(	PUNCT
ap-9037	326	35	ν1	ν1	NOUN
ap-9037	326	36	+	+	CCONJ
ap-9037	326	37	ν2	ν2	NOUN
ap-9037	326	38	+	+	CCONJ
ap-9037	326	39	ν3)(1	ν3)(1	NOUN
ap-9037	326	40	+	+	NUM
ap-9037	326	41	ν1	ν1	NOUN
ap-9037	326	42	+	+	CCONJ
ap-9037	326	43	ν2	ν2	NOUN
ap-9037	326	44	+	+	CCONJ
ap-9037	326	45	ν3	ν3	NOUN
ap-9037	326	46	)	)	PUNCT
ap-9037	326	47	cos(y)2	cos(y)2	NOUN
ap-9037	326	48	−	−	PROPN
ap-9037	327	1	(	(	PUNCT
ap-9037	327	2	r3	r3	PROPN
ap-9037	327	3	−	−	PROPN
ap-9037	327	4	ν3)ν3	ν3)ν3	ADV
ap-9037	327	5	sin(y)2	sin(y)2	ADJ
ap-9037	327	6	−	−	NUM
ap-9037	327	7	1	1	NUM
ap-9037	327	8	cos(y)2	cos(y)2	NOUN
ap-9037	327	9	vθ{2arccot[exp(iy	vθ{2arccot[exp(iy	NOUN
ap-9037	327	10	)	)	PUNCT
ap-9037	327	11	]	]	PUNCT
ap-9037	327	12	}	}	PUNCT
ap-9037	327	13	.	.	PUNCT
ap-9037	328	1	283	283	NUM
ap-9037	328	2	axel	axel	PROPN
ap-9037	328	3	schulze	schulze	NOUN
ap-9037	328	4	-	-	PUNCT
ap-9037	328	5	halberg	halberg	PROPN
ap-9037	328	6	acta	acta	PROPN
ap-9037	328	7	polytechnica	polytechnica	PROPN
ap-9037	328	8	consequently	consequently	ADV
ap-9037	328	9	,	,	PUNCT
ap-9037	328	10	the	the	DET
ap-9037	328	11	darboux	darboux	VERB
ap-9037	328	12	-	-	PUNCT
ap-9037	328	13	crum	crum	NOUN
ap-9037	328	14	transformation	transformation	NOUN
ap-9037	328	15	can	can	AUX
ap-9037	328	16	be	be	AUX
ap-9037	328	17	applied	apply	VERB
ap-9037	328	18	to	to	ADP
ap-9037	328	19	equation	equation	NOUN
ap-9037	328	20	(	(	PUNCT
ap-9037	328	21	63	63	NUM
ap-9037	328	22	)	)	PUNCT
ap-9037	328	23	.	.	PUNCT
ap-9037	329	1	in	in	ADP
ap-9037	329	2	order	order	NOUN
ap-9037	329	3	to	to	PART
ap-9037	329	4	apply	apply	VERB
ap-9037	329	5	this	this	DET
ap-9037	329	6	transformation	transformation	NOUN
ap-9037	329	7	,	,	PUNCT
ap-9037	329	8	we	we	PRON
ap-9037	329	9	need	need	VERB
ap-9037	329	10	:	:	PUNCT
ap-9037	329	11	η′′	η′′	PROPN
ap-9037	329	12	j	j	PROPN
ap-9037	329	13	(	(	PUNCT
ap-9037	329	14	y	y	PROPN
ap-9037	329	15	)	)	PUNCT
ap-9037	329	16	+	+	CCONJ
ap-9037	329	17	{	{	PUNCT
ap-9037	329	18	m2	m2	PROPN
ap-9037	329	19	j	j	PROPN
ap-9037	329	20	+	+	CCONJ
ap-9037	329	21	(	(	PUNCT
ap-9037	329	22	ν1	ν1	NOUN
ap-9037	329	23	+	+	CCONJ
ap-9037	329	24	ν2)2	ν2)2	CCONJ
ap-9037	329	25	−	−	PROPN
ap-9037	329	26	λ	λ	X
ap-9037	329	27	+	+	CCONJ
ap-9037	329	28	(	(	PUNCT
ap-9037	329	29	ν1	ν1	NOUN
ap-9037	329	30	+	+	CCONJ
ap-9037	329	31	ν2	ν2	NOUN
ap-9037	329	32	+	+	CCONJ
ap-9037	329	33	ν3)(1	ν3)(1	NOUN
ap-9037	329	34	+	+	NUM
ap-9037	329	35	ν1	ν1	NOUN
ap-9037	329	36	+	+	CCONJ
ap-9037	329	37	ν2	ν2	NOUN
ap-9037	329	38	+	+	CCONJ
ap-9037	329	39	ν3	ν3	NOUN
ap-9037	329	40	)	)	PUNCT
ap-9037	329	41	cos(y)2	cos(y)2	NOUN
ap-9037	329	42	+	+	CCONJ
ap-9037	329	43	(	(	PUNCT
ap-9037	329	44	r3	r3	PROPN
ap-9037	329	45	−	−	PROPN
ap-9037	329	46	ν3)ν3	ν3)ν3	ADP
ap-9037	329	47	sin(y)2	sin(y)2	ADJ
ap-9037	329	48	+	+	CCONJ
ap-9037	329	49	1	1	NUM
ap-9037	329	50	cos(y)2	cos(y)2	NOUN
ap-9037	329	51	vθ{2arccot[exp(iy	vθ{2arccot[exp(iy	NOUN
ap-9037	329	52	)	)	PUNCT
ap-9037	329	53	]	]	PUNCT
ap-9037	329	54	}	}	PUNCT
ap-9037	329	55	}	}	PUNCT
ap-9037	329	56	ηj(y	ηj(y	PUNCT
ap-9037	329	57	)	)	PUNCT
ap-9037	329	58	=	=	SYM
ap-9037	329	59	0	0	NUM
ap-9037	329	60	,	,	PUNCT
ap-9037	329	61	(	(	PUNCT
ap-9037	329	62	64	64	NUM
ap-9037	329	63	)	)	PUNCT
ap-9037	329	64	where	where	SCONJ
ap-9037	329	65	j	j	PROPN
ap-9037	329	66	=	=	NOUN
ap-9037	329	67	1	1	NUM
ap-9037	329	68	,	,	PUNCT
ap-9037	329	69	.	.	PUNCT
ap-9037	329	70	.	.	PUNCT
ap-9037	329	71	.	.	PUNCT
ap-9037	330	1	,	,	PUNCT
ap-9037	330	2	n	n	CCONJ
ap-9037	330	3	,	,	PUNCT
ap-9037	330	4	the	the	DET
ap-9037	330	5	transformation	transformation	NOUN
ap-9037	330	6	parameters	parameter	NOUN
ap-9037	330	7	are	be	AUX
ap-9037	330	8	given	give	VERB
ap-9037	330	9	by	by	ADP
ap-9037	330	10	m2	m2	PROPN
ap-9037	330	11	1	1	NUM
ap-9037	330	12	,	,	PUNCT
ap-9037	330	13	.	.	PUNCT
ap-9037	330	14	.	.	PUNCT
ap-9037	331	1	.	.	PUNCT
ap-9037	332	1	,	,	PUNCT
ap-9037	332	2	m2	m2	PROPN
ap-9037	332	3	n	n	CCONJ
ap-9037	332	4	,	,	PUNCT
ap-9037	332	5	and	and	CCONJ
ap-9037	332	6	the	the	DET
ap-9037	332	7	potential	potential	ADJ
ap-9037	332	8	vθ	vθ	NOUN
ap-9037	332	9	is	be	AUX
ap-9037	332	10	provided	provide	VERB
ap-9037	332	11	by	by	ADP
ap-9037	332	12	(	(	PUNCT
ap-9037	332	13	31	31	NUM
ap-9037	332	14	)	)	PUNCT
ap-9037	332	15	.	.	PUNCT
ap-9037	333	1	now	now	ADV
ap-9037	333	2	,	,	PUNCT
ap-9037	333	3	evaluation	evaluation	NOUN
ap-9037	333	4	of	of	ADP
ap-9037	333	5	(	(	PUNCT
ap-9037	333	6	6	6	NUM
ap-9037	333	7	)	)	PUNCT
ap-9037	333	8	and	and	CCONJ
ap-9037	333	9	(	(	PUNCT
ap-9037	333	10	7	7	X
ap-9037	333	11	)	)	PUNCT
ap-9037	333	12	gives	give	VERB
ap-9037	333	13	:	:	PUNCT
ap-9037	333	14	η̂(y	η̂(y	ADJ
ap-9037	333	15	)	)	PUNCT
ap-9037	333	16	=	=	PUNCT
ap-9037	333	17	wη1,η2,	wη1,η2,	PROPN
ap-9037	333	18	...	...	PUNCT
ap-9037	333	19	,ηn	,ηn	PROPN
ap-9037	333	20	,	,	PUNCT
ap-9037	333	21	η(y	η(y	PROPN
ap-9037	333	22	)	)	PUNCT
ap-9037	333	23	wη1,η2,	wη1,η2,	PROPN
ap-9037	333	24	...	...	PUNCT
ap-9037	333	25	,ηn	,ηn	PUNCT
ap-9037	333	26	(	(	PUNCT
ap-9037	333	27	y	y	NOUN
ap-9037	333	28	)	)	PUNCT
ap-9037	333	29	,	,	PUNCT
ap-9037	333	30	(	(	PUNCT
ap-9037	333	31	65	65	X
ap-9037	333	32	)	)	PUNCT
ap-9037	333	33	v̂θ{2arccot[exp(iy	v̂θ{2arccot[exp(iy	PROPN
ap-9037	333	34	)	)	PUNCT
ap-9037	333	35	]	]	PUNCT
ap-9037	333	36	}	}	PUNCT
ap-9037	333	37	=	=	SYM
ap-9037	333	38	vθ{2arccot[exp(iy	vθ{2arccot[exp(iy	NOUN
ap-9037	333	39	)	)	PUNCT
ap-9037	333	40	]	]	PUNCT
ap-9037	333	41	}	}	PUNCT
ap-9037	333	42	+	+	CCONJ
ap-9037	333	43	2	2	NUM
ap-9037	333	44	cos(y)2	cos(y)2	NOUN
ap-9037	333	45	d2	d2	PROPN
ap-9037	333	46	dy2	dy2	PROPN
ap-9037	333	47	log	log	NOUN
ap-9037	334	1	[	[	X
ap-9037	334	2	wη1,η2,	wη1,η2,	NOUN
ap-9037	334	3	...	...	X
ap-9037	334	4	,ηn(y	,ηn(y	PROPN
ap-9037	334	5	)	)	PUNCT
ap-9037	334	6	]	]	PUNCT
ap-9037	334	7	.	.	PUNCT
ap-9037	335	1	(	(	PUNCT
ap-9037	335	2	66	66	NUM
ap-9037	335	3	)	)	PUNCT
ap-9037	335	4	here	here	ADV
ap-9037	335	5	,	,	PUNCT
ap-9037	335	6	the	the	DET
ap-9037	335	7	function	function	NOUN
ap-9037	335	8	η̂	η̂	NUM
ap-9037	335	9	depends	depend	VERB
ap-9037	335	10	on	on	ADP
ap-9037	335	11	the	the	DET
ap-9037	335	12	transformation	transformation	NOUN
ap-9037	335	13	parameters	parameter	NOUN
ap-9037	335	14	m2	m2	PROPN
ap-9037	335	15	1	1	NUM
ap-9037	335	16	,	,	PUNCT
ap-9037	335	17	.	.	PUNCT
ap-9037	335	18	.	.	PUNCT
ap-9037	336	1	.	.	PUNCT
ap-9037	337	1	,	,	PUNCT
ap-9037	337	2	m2	m2	PROPN
ap-9037	337	3	n	n	CCONJ
ap-9037	337	4	,	,	PUNCT
ap-9037	337	5	and	and	CCONJ
ap-9037	337	6	also	also	ADV
ap-9037	337	7	on	on	ADP
ap-9037	337	8	the	the	DET
ap-9037	337	9	separation	separation	NOUN
ap-9037	337	10	constant	constant	ADJ
ap-9037	337	11	m2	m2	PROPN
ap-9037	337	12	from	from	ADP
ap-9037	337	13	equation	equation	NOUN
ap-9037	337	14	(	(	PUNCT
ap-9037	337	15	63	63	NUM
ap-9037	337	16	)	)	PUNCT
ap-9037	337	17	.	.	PUNCT
ap-9037	338	1	furthermore	furthermore	ADV
ap-9037	338	2	,	,	PUNCT
ap-9037	338	3	the	the	DET
ap-9037	338	4	associated	associated	ADJ
ap-9037	338	5	potential	potential	NOUN
ap-9037	338	6	v̂θ	v̂θ	X
ap-9037	338	7	also	also	ADV
ap-9037	338	8	depends	depend	VERB
ap-9037	338	9	on	on	ADP
ap-9037	338	10	the	the	DET
ap-9037	338	11	transformation	transformation	NOUN
ap-9037	338	12	parameters	parameter	NOUN
ap-9037	338	13	m2	m2	PROPN
ap-9037	338	14	1	1	NUM
ap-9037	338	15	,	,	PUNCT
ap-9037	338	16	.	.	PUNCT
ap-9037	338	17	.	.	PUNCT
ap-9037	339	1	.	.	PUNCT
ap-9037	340	1	,	,	PUNCT
ap-9037	340	2	m2	m2	PROPN
ap-9037	340	3	n	n	CCONJ
ap-9037	340	4	,	,	PUNCT
ap-9037	340	5	but	but	CCONJ
ap-9037	340	6	it	it	PRON
ap-9037	340	7	does	do	AUX
ap-9037	340	8	not	not	PART
ap-9037	340	9	depend	depend	VERB
ap-9037	340	10	on	on	ADP
ap-9037	340	11	m2	m2	PROPN
ap-9037	340	12	.	.	PUNCT
ap-9037	341	1	the	the	DET
ap-9037	341	2	function	function	NOUN
ap-9037	341	3	η̂	η̂	PUNCT
ap-9037	341	4	and	and	CCONJ
ap-9037	341	5	the	the	DET
ap-9037	341	6	potential	potential	NOUN
ap-9037	341	7	v̂θ	v̂θ	PRON
ap-9037	341	8	are	be	AUX
ap-9037	341	9	inserted	insert	VERB
ap-9037	341	10	in	in	ADP
ap-9037	341	11	the	the	DET
ap-9037	341	12	transformed	transform	VERB
ap-9037	341	13	version	version	NOUN
ap-9037	341	14	of	of	ADP
ap-9037	341	15	(	(	PUNCT
ap-9037	341	16	63	63	NUM
ap-9037	341	17	)	)	PUNCT
ap-9037	341	18	,	,	PUNCT
ap-9037	341	19	which	which	PRON
ap-9037	341	20	reads	read	VERB
ap-9037	341	21	:	:	PUNCT
ap-9037	341	22	η̂′′(y	η̂′′(y	NUM
ap-9037	341	23	)	)	PUNCT
ap-9037	342	1	+	+	CCONJ
ap-9037	342	2	{	{	PUNCT
ap-9037	342	3	m2	m2	PROPN
ap-9037	342	4	+	+	CCONJ
ap-9037	342	5	(	(	PUNCT
ap-9037	342	6	ν1	ν1	NOUN
ap-9037	342	7	+	+	CCONJ
ap-9037	342	8	ν2)2	ν2)2	CCONJ
ap-9037	342	9	−	−	PROPN
ap-9037	342	10	λ	λ	X
ap-9037	342	11	+	+	CCONJ
ap-9037	342	12	(	(	PUNCT
ap-9037	342	13	ν1	ν1	NOUN
ap-9037	342	14	+	+	CCONJ
ap-9037	342	15	ν2	ν2	NOUN
ap-9037	342	16	+	+	CCONJ
ap-9037	342	17	ν3)(1	ν3)(1	NOUN
ap-9037	342	18	+	+	NUM
ap-9037	342	19	ν1	ν1	NOUN
ap-9037	342	20	+	+	CCONJ
ap-9037	342	21	ν2	ν2	NOUN
ap-9037	342	22	+	+	CCONJ
ap-9037	342	23	ν3	ν3	NOUN
ap-9037	342	24	)	)	PUNCT
ap-9037	342	25	cos(y)2	cos(y)2	NOUN
ap-9037	342	26	+	+	CCONJ
ap-9037	342	27	(	(	PUNCT
ap-9037	342	28	r3	r3	PROPN
ap-9037	342	29	−	−	PROPN
ap-9037	342	30	ν3)ν3	ν3)ν3	ADP
ap-9037	342	31	sin(y)2	sin(y)2	ADJ
ap-9037	342	32	+	+	NOUN
ap-9037	342	33	1	1	NUM
ap-9037	342	34	cos(y)2	cos(y)2	NOUN
ap-9037	342	35	v̂θ{2arccot[exp(iy	v̂θ{2arccot[exp(iy	PROPN
ap-9037	342	36	)	)	PUNCT
ap-9037	342	37	]	]	PUNCT
ap-9037	342	38	}	}	PUNCT
ap-9037	342	39	}	}	PUNCT
ap-9037	342	40	η̂(y	η̂(y	X
ap-9037	342	41	)	)	PUNCT
ap-9037	342	42	=	=	SYM
ap-9037	342	43	0	0	X
ap-9037	342	44	.	.	PUNCT
ap-9037	343	1	now	now	ADV
ap-9037	343	2	we	we	PRON
ap-9037	343	3	need	need	VERB
ap-9037	343	4	to	to	PART
ap-9037	343	5	revert	revert	VERB
ap-9037	343	6	our	our	PRON
ap-9037	343	7	point	point	NOUN
ap-9037	343	8	transformation	transformation	NOUN
ap-9037	343	9	(	(	PUNCT
ap-9037	343	10	61	61	NUM
ap-9037	343	11	)	)	PUNCT
ap-9037	343	12	,	,	PUNCT
ap-9037	343	13	(	(	PUNCT
ap-9037	343	14	62	62	NUM
ap-9037	343	15	)	)	PUNCT
ap-9037	343	16	,	,	PUNCT
ap-9037	343	17	so	so	SCONJ
ap-9037	343	18	that	that	SCONJ
ap-9037	343	19	this	this	DET
ap-9037	343	20	equation	equation	NOUN
ap-9037	343	21	is	be	AUX
ap-9037	343	22	converted	convert	VERB
ap-9037	343	23	back	back	ADV
ap-9037	343	24	to	to	ADP
ap-9037	343	25	the	the	DET
ap-9037	343	26	form	form	NOUN
ap-9037	343	27	(	(	PUNCT
ap-9037	343	28	51	51	NUM
ap-9037	343	29	)	)	PUNCT
ap-9037	343	30	.	.	PUNCT
ap-9037	344	1	a	a	DET
ap-9037	344	2	solution	solution	NOUN
ap-9037	344	3	θ̂	θ̂	VERB
ap-9037	344	4	can	can	AUX
ap-9037	344	5	be	be	AUX
ap-9037	344	6	found	find	VERB
ap-9037	344	7	in	in	ADP
ap-9037	344	8	two	two	NUM
ap-9037	344	9	steps	step	NOUN
ap-9037	344	10	,	,	PUNCT
ap-9037	344	11	the	the	DET
ap-9037	344	12	first	first	ADJ
ap-9037	344	13	of	of	ADP
ap-9037	344	14	which	which	PRON
ap-9037	344	15	is	be	AUX
ap-9037	344	16	the	the	DET
ap-9037	344	17	implementation	implementation	NOUN
ap-9037	344	18	of	of	ADP
ap-9037	344	19	(	(	PUNCT
ap-9037	344	20	62	62	NUM
ap-9037	344	21	)	)	PUNCT
ap-9037	344	22	.	.	PUNCT
ap-9037	345	1	we	we	PRON
ap-9037	345	2	obtain	obtain	VERB
ap-9037	345	3	:	:	PUNCT
ap-9037	345	4	θ̂{2arccot[exp(iy	θ̂{2arccot[exp(iy	NOUN
ap-9037	345	5	)	)	PUNCT
ap-9037	345	6	]	]	PUNCT
ap-9037	345	7	}	}	PUNCT
ap-9037	345	8	=	=	PUNCT
ap-9037	346	1	[	[	X
ap-9037	346	2	exp(2iy	exp(2iy	ADP
ap-9037	346	3	)	)	PUNCT
ap-9037	346	4	+	+	NUM
ap-9037	347	1	1]ν1+ν2+ν3	1]ν1+ν2+ν3	NUM
ap-9037	347	2	exp	exp	NOUN
ap-9037	347	3	[	[	X
ap-9037	347	4	i(ν1	i(ν1	NOUN
ap-9037	347	5	+	+	X
ap-9037	347	6	ν2)y	ν2)y	NOUN
ap-9037	347	7	]	]	X
ap-9037	348	1	[	[	X
ap-9037	348	2	exp(2iy	exp(2iy	ADP
ap-9037	348	3	)	)	PUNCT
ap-9037	348	4	−	−	PROPN
ap-9037	348	5	1]ν3	1]ν3	NUM
ap-9037	348	6	η̂(y	η̂(y	NOUN
ap-9037	348	7	)	)	PUNCT
ap-9037	348	8	=	=	PUNCT
ap-9037	349	1	[	[	X
ap-9037	349	2	exp(2iy	exp(2iy	ADP
ap-9037	349	3	)	)	PUNCT
ap-9037	349	4	+	+	NUM
ap-9037	350	1	1]ν1+ν2+ν3	1]ν1+ν2+ν3	NUM
ap-9037	350	2	exp	exp	NOUN
ap-9037	350	3	[	[	X
ap-9037	350	4	i(ν1	i(ν1	NOUN
ap-9037	350	5	+	+	X
ap-9037	350	6	ν2)y	ν2)y	NOUN
ap-9037	350	7	]	]	X
ap-9037	351	1	[	[	X
ap-9037	351	2	exp(2iy	exp(2iy	ADP
ap-9037	351	3	)	)	PUNCT
ap-9037	351	4	−	−	PROPN
ap-9037	351	5	1]ν3	1]ν3	NUM
ap-9037	351	6	[	[	PUNCT
ap-9037	351	7	wη1,η2,	wη1,η2,	PROPN
ap-9037	351	8	...	...	X
ap-9037	351	9	,ηn	,ηn	PROPN
ap-9037	351	10	,	,	PUNCT
ap-9037	351	11	η(y	η(y	PROPN
ap-9037	351	12	)	)	PUNCT
ap-9037	351	13	wη1,η2,	wη1,η2,	PROPN
ap-9037	351	14	...	...	PUNCT
ap-9037	351	15	,ηn(y	,ηn(y	PROPN
ap-9037	351	16	)	)	PUNCT
ap-9037	351	17	]	]	PUNCT
ap-9037	351	18	.	.	PUNCT
ap-9037	352	1	(	(	PUNCT
ap-9037	352	2	67	67	NUM
ap-9037	352	3	)	)	PUNCT
ap-9037	352	4	in	in	ADP
ap-9037	352	5	the	the	DET
ap-9037	352	6	second	second	ADJ
ap-9037	352	7	step	step	NOUN
ap-9037	352	8	,	,	PUNCT
ap-9037	352	9	we	we	PRON
ap-9037	352	10	invert	invert	VERB
ap-9037	352	11	the	the	DET
ap-9037	352	12	coordinate	coordinate	NOUN
ap-9037	352	13	change	change	NOUN
ap-9037	352	14	(	(	PUNCT
ap-9037	352	15	61	61	NUM
ap-9037	352	16	)	)	PUNCT
ap-9037	352	17	.	.	PUNCT
ap-9037	353	1	taking	take	VERB
ap-9037	353	2	into	into	ADP
ap-9037	353	3	account	account	NOUN
ap-9037	353	4	that	that	SCONJ
ap-9037	353	5	:	:	PUNCT
ap-9037	353	6	y(θ	y(θ	X
ap-9037	353	7	)	)	PUNCT
ap-9037	354	1	=	=	SYM
ap-9037	354	2	−i	−i	ADJ
ap-9037	354	3	log	log	NOUN
ap-9037	354	4	[	[	PUNCT
ap-9037	354	5	cot	cot	NOUN
ap-9037	354	6	(	(	PUNCT
ap-9037	354	7	θ	θ	PROPN
ap-9037	354	8	2	2	NUM
ap-9037	354	9	)	)	PUNCT
ap-9037	354	10	]	]	PUNCT
ap-9037	354	11	,	,	PUNCT
ap-9037	354	12	and	and	CCONJ
ap-9037	354	13	defining	define	VERB
ap-9037	354	14	functions	function	NOUN
ap-9037	354	15	θ1	θ1	NOUN
ap-9037	354	16	,	,	PUNCT
ap-9037	354	17	θ2	θ2	PROPN
ap-9037	354	18	,	,	PUNCT
ap-9037	354	19	.	.	PUNCT
ap-9037	354	20	.	.	PUNCT
ap-9037	354	21	.	.	PUNCT
ap-9037	355	1	,	,	PUNCT
ap-9037	355	2	θn	θn	VERB
ap-9037	355	3	by	by	ADP
ap-9037	355	4	reverting	revert	VERB
ap-9037	355	5	(	(	PUNCT
ap-9037	355	6	61	61	NUM
ap-9037	355	7	)	)	PUNCT
ap-9037	355	8	,	,	PUNCT
ap-9037	355	9	(	(	PUNCT
ap-9037	355	10	62	62	NUM
ap-9037	355	11	)	)	PUNCT
ap-9037	355	12	as	as	ADP
ap-9037	355	13	:	:	PUNCT
ap-9037	355	14	θj(θ	θj(θ	NUM
ap-9037	355	15	)	)	PUNCT
ap-9037	356	1	=	=	PRON
ap-9037	356	2	tan	tan	PROPN
ap-9037	356	3	(	(	PUNCT
ap-9037	356	4	θ	θ	NOUN
ap-9037	356	5	2	2	NUM
ap-9037	356	6	)	)	PUNCT
ap-9037	356	7	ν1+ν2	ν1+ν2	NOUN
ap-9037	356	8	cos(θ)ν3	cos(θ)ν3	CCONJ
ap-9037	356	9	sin	sin	NOUN
ap-9037	356	10	(	(	PUNCT
ap-9037	356	11	θ	θ	NOUN
ap-9037	356	12	2	2	NUM
ap-9037	356	13	)	)	PUNCT
ap-9037	356	14	2ν1	2ν1	NUM
ap-9037	357	1	+	+	NOUN
ap-9037	357	2	2ν2	2ν2	NUM
ap-9037	357	3	ηj	ηj	ADP
ap-9037	357	4	{	{	PUNCT
ap-9037	357	5	−i	−i	ADV
ap-9037	357	6	log	log	NOUN
ap-9037	357	7	[	[	PUNCT
ap-9037	357	8	cot	cot	NOUN
ap-9037	357	9	(	(	PUNCT
ap-9037	357	10	θ	θ	PROPN
ap-9037	357	11	2	2	NUM
ap-9037	357	12	)	)	PUNCT
ap-9037	357	13	]	]	PUNCT
ap-9037	357	14	}	}	PUNCT
ap-9037	357	15	,	,	PUNCT
ap-9037	357	16	j	j	PROPN
ap-9037	357	17	=	=	SYM
ap-9037	357	18	1	1	NUM
ap-9037	357	19	,	,	PUNCT
ap-9037	357	20	2	2	NUM
ap-9037	357	21	,	,	PUNCT
ap-9037	357	22	.	.	PUNCT
ap-9037	357	23	.	.	PUNCT
ap-9037	358	1	.	.	PUNCT
ap-9037	359	1	,	,	PUNCT
ap-9037	359	2	n	n	X
ap-9037	359	3	,	,	PUNCT
ap-9037	359	4	we	we	PRON
ap-9037	359	5	find	find	VERB
ap-9037	359	6	,	,	PUNCT
ap-9037	359	7	after	after	ADP
ap-9037	359	8	substitution	substitution	NOUN
ap-9037	359	9	into	into	ADP
ap-9037	359	10	(	(	PUNCT
ap-9037	359	11	67	67	NUM
ap-9037	359	12	)	)	PUNCT
ap-9037	359	13	and	and	CCONJ
ap-9037	359	14	simplification	simplification	NOUN
ap-9037	359	15	of	of	ADP
ap-9037	359	16	the	the	DET
ap-9037	359	17	wronskians	wronskian	NOUN
ap-9037	359	18	,	,	PUNCT
ap-9037	359	19	that	that	PRON
ap-9037	359	20	:	:	PUNCT
ap-9037	359	21	θ̂(θ	θ̂(θ	NUM
ap-9037	359	22	)	)	PUNCT
ap-9037	359	23	=	=	NOUN
ap-9037	359	24	sin(θ)n	sin(θ)n	NOUN
ap-9037	359	25	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	359	26	...	...	PUNCT
ap-9037	359	27	,θn	,θn	PUNCT
ap-9037	359	28	,	,	PUNCT
ap-9037	359	29	θ(θ	θ(θ	VERB
ap-9037	359	30	)	)	PUNCT
ap-9037	359	31	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	359	32	...	...	PUNCT
ap-9037	359	33	,θn	,θn	PUNCT
ap-9037	359	34	(	(	PUNCT
ap-9037	359	35	θ	θ	NOUN
ap-9037	359	36	)	)	PUNCT
ap-9037	359	37	.	.	PUNCT
ap-9037	360	1	(	(	PUNCT
ap-9037	360	2	68	68	NUM
ap-9037	360	3	)	)	PUNCT
ap-9037	360	4	the	the	DET
ap-9037	360	5	remaining	remain	VERB
ap-9037	360	6	task	task	NOUN
ap-9037	360	7	is	be	AUX
ap-9037	360	8	to	to	PART
ap-9037	360	9	rewrite	rewrite	VERB
ap-9037	360	10	the	the	DET
ap-9037	360	11	transformed	transform	VERB
ap-9037	360	12	potential	potential	NOUN
ap-9037	360	13	(	(	PUNCT
ap-9037	360	14	66	66	NUM
ap-9037	360	15	)	)	PUNCT
ap-9037	360	16	in	in	ADP
ap-9037	360	17	terms	term	NOUN
ap-9037	360	18	of	of	ADP
ap-9037	360	19	the	the	DET
ap-9037	360	20	functions	function	NOUN
ap-9037	360	21	θ1	θ1	NOUN
ap-9037	360	22	,	,	PUNCT
ap-9037	360	23	θ2	θ2	PROPN
ap-9037	360	24	,	,	PUNCT
ap-9037	360	25	.	.	PUNCT
ap-9037	360	26	.	.	PUNCT
ap-9037	361	1	.	.	PUNCT
ap-9037	362	1	,	,	PUNCT
ap-9037	362	2	θn	θn	ADP
ap-9037	362	3	,	,	PUNCT
ap-9037	362	4	and	and	CCONJ
ap-9037	362	5	the	the	DET
ap-9037	362	6	coordinate	coordinate	NOUN
ap-9037	362	7	θ	θ	PROPN
ap-9037	362	8	.	.	PUNCT
ap-9037	362	9	by	by	ADP
ap-9037	362	10	implementing	implement	VERB
ap-9037	362	11	(	(	PUNCT
ap-9037	362	12	61	61	NUM
ap-9037	362	13	)	)	PUNCT
ap-9037	362	14	and	and	CCONJ
ap-9037	362	15	(	(	PUNCT
ap-9037	362	16	62	62	NUM
ap-9037	362	17	)	)	PUNCT
ap-9037	362	18	in	in	ADP
ap-9037	362	19	a	a	DET
ap-9037	362	20	first	first	ADJ
ap-9037	362	21	step	step	NOUN
ap-9037	362	22	:	:	PUNCT
ap-9037	362	23	v̂θ(θ	v̂θ(θ	X
ap-9037	362	24	)	)	PUNCT
ap-9037	362	25	=	=	PRON
ap-9037	362	26	vθ(θ	vθ(θ	X
ap-9037	362	27	)	)	PUNCT
ap-9037	363	1	+	+	CCONJ
ap-9037	363	2	2	2	NUM
ap-9037	363	3	sin(θ)2	sin(θ)2	NOUN
ap-9037	363	4	(	(	PUNCT
ap-9037	363	5	d2	d2	PROPN
ap-9037	363	6	dy2	dy2	PROPN
ap-9037	363	7	log	log	PROPN
ap-9037	363	8	{	{	PUNCT
ap-9037	363	9	exp	exp	NOUN
ap-9037	363	10	[	[	X
ap-9037	363	11	i(ν1	i(ν1	NOUN
ap-9037	363	12	+	+	X
ap-9037	363	13	ν2)y	ν2)y	NOUN
ap-9037	363	14	]	]	X
ap-9037	364	1	[	[	X
ap-9037	364	2	exp(2iy	exp(2iy	ADP
ap-9037	364	3	)	)	PUNCT
ap-9037	364	4	−	−	NOUN
ap-9037	365	1	1]ν3	1]ν3	NUM
ap-9037	366	1	[	[	X
ap-9037	366	2	exp(2iy	exp(2iy	ADP
ap-9037	366	3	)	)	PUNCT
ap-9037	366	4	+	+	NUM
ap-9037	366	5	1]ν1+ν2+ν3	1]ν1+ν2+ν3	NUM
ap-9037	366	6	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	366	7	...	...	PUNCT
ap-9037	366	8	,θn	,θn	PUNCT
ap-9037	366	9	(	(	PUNCT
ap-9037	366	10	y	y	NOUN
ap-9037	366	11	)	)	PUNCT
ap-9037	366	12	}	}	PUNCT
ap-9037	366	13	)	)	PUNCT
ap-9037	366	14	|y	|y	NOUN
ap-9037	366	15	=	=	SYM
ap-9037	366	16	y(θ	y(θ	PROPN
ap-9037	366	17	)	)	PUNCT
ap-9037	366	18	.	.	PUNCT
ap-9037	367	1	(	(	PUNCT
ap-9037	367	2	69	69	NUM
ap-9037	367	3	)	)	SYM
ap-9037	367	4	284	284	NUM
ap-9037	367	5	vol	vol	NOUN
ap-9037	367	6	.	.	PUNCT
ap-9037	368	1	63	63	NUM
ap-9037	368	2	no	no	INTJ
ap-9037	368	3	.	.	PUNCT
ap-9037	369	1	4/2023	4/2023	NOUN
ap-9037	369	2	construction	construction	NOUN
ap-9037	369	3	of	of	ADP
ap-9037	369	4	angular	angular	ADJ
ap-9037	369	5	-	-	PUNCT
ap-9037	369	6	dependent	dependent	ADJ
ap-9037	369	7	potentials	potential	NOUN
ap-9037	369	8	in	in	ADP
ap-9037	369	9	the	the	DET
ap-9037	369	10	next	next	ADJ
ap-9037	369	11	step	step	NOUN
ap-9037	369	12	,	,	PUNCT
ap-9037	369	13	we	we	PRON
ap-9037	369	14	focus	focus	VERB
ap-9037	369	15	on	on	ADP
ap-9037	369	16	simplifying	simplify	VERB
ap-9037	369	17	the	the	DET
ap-9037	369	18	second	second	ADJ
ap-9037	369	19	derivative	derivative	ADJ
ap-9037	369	20	term	term	NOUN
ap-9037	369	21	that	that	PRON
ap-9037	369	22	we	we	PRON
ap-9037	369	23	abbreviate	abbreviate	VERB
ap-9037	369	24	as	as	ADP
ap-9037	369	25	l.	l.	NOUN
ap-9037	369	26	by	by	ADP
ap-9037	369	27	expanding	expand	VERB
ap-9037	369	28	the	the	DET
ap-9037	369	29	logarithm	logarithm	NOUN
ap-9037	369	30	,	,	PUNCT
ap-9037	369	31	we	we	PRON
ap-9037	369	32	find	find	VERB
ap-9037	369	33	:	:	PUNCT
ap-9037	369	34	l(θ	l(θ	NOUN
ap-9037	369	35	)	)	PUNCT
ap-9037	369	36	=	=	PRON
ap-9037	369	37	(	(	PUNCT
ap-9037	369	38	d2	d2	PROPN
ap-9037	369	39	dy2	dy2	PROPN
ap-9037	369	40	log	log	PROPN
ap-9037	369	41	{	{	PUNCT
ap-9037	369	42	exp	exp	NOUN
ap-9037	369	43	[	[	X
ap-9037	369	44	i(ν1	i(ν1	NOUN
ap-9037	369	45	+	+	X
ap-9037	369	46	ν2)y	ν2)y	NOUN
ap-9037	369	47	]	]	X
ap-9037	370	1	[	[	X
ap-9037	370	2	exp(2iy	exp(2iy	ADP
ap-9037	370	3	)	)	PUNCT
ap-9037	370	4	−	−	NOUN
ap-9037	371	1	1]ν3	1]ν3	NUM
ap-9037	372	1	[	[	X
ap-9037	372	2	exp(2iy	exp(2iy	ADP
ap-9037	372	3	)	)	PUNCT
ap-9037	372	4	+	+	NUM
ap-9037	372	5	1]ν1+ν2+ν3	1]ν1+ν2+ν3	NUM
ap-9037	372	6	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	372	7	...	...	PUNCT
ap-9037	372	8	,θn	,θn	PUNCT
ap-9037	372	9	(	(	PUNCT
ap-9037	372	10	y	y	NOUN
ap-9037	372	11	)	)	PUNCT
ap-9037	372	12	}	}	PUNCT
ap-9037	372	13	)	)	PUNCT
ap-9037	372	14	|y	|y	NOUN
ap-9037	372	15	=	=	SYM
ap-9037	372	16	y(θ	y(θ	PROPN
ap-9037	372	17	)	)	PUNCT
ap-9037	372	18	=	=	PRON
ap-9037	372	19	(	(	PUNCT
ap-9037	372	20	d2	d2	PROPN
ap-9037	372	21	dy2	dy2	PROPN
ap-9037	372	22	log	log	PROPN
ap-9037	372	23	{	{	PUNCT
ap-9037	372	24	exp	exp	NOUN
ap-9037	372	25	[	[	X
ap-9037	372	26	i(ν1	i(ν1	NOUN
ap-9037	372	27	+	+	X
ap-9037	372	28	ν2)y	ν2)y	NOUN
ap-9037	372	29	]	]	X
ap-9037	373	1	[	[	X
ap-9037	373	2	exp(2iy	exp(2iy	ADP
ap-9037	373	3	)	)	PUNCT
ap-9037	373	4	−	−	NOUN
ap-9037	374	1	1]ν3	1]ν3	NUM
ap-9037	375	1	[	[	X
ap-9037	375	2	exp(2iy	exp(2iy	ADP
ap-9037	375	3	)	)	PUNCT
ap-9037	375	4	+	+	NUM
ap-9037	375	5	1]ν1+ν2+ν3	1]ν1+ν2+ν3	NUM
ap-9037	375	6	}	}	PUNCT
ap-9037	375	7	+	+	CCONJ
ap-9037	375	8	d2	d2	PROPN
ap-9037	375	9	dy2	dy2	PROPN
ap-9037	375	10	log	log	NOUN
ap-9037	376	1	[	[	X
ap-9037	376	2	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	376	3	...	...	PUNCT
ap-9037	376	4	,θn	,θn	PUNCT
ap-9037	376	5	(	(	PUNCT
ap-9037	376	6	y	y	NOUN
ap-9037	376	7	)	)	PUNCT
ap-9037	376	8	]	]	PUNCT
ap-9037	376	9	)	)	PUNCT
ap-9037	376	10	|y	|y	NOUN
ap-9037	376	11	=	=	SYM
ap-9037	376	12	y(θ	y(θ	PROPN
ap-9037	376	13	)	)	PUNCT
ap-9037	376	14	.	.	PUNCT
ap-9037	377	1	(	(	PUNCT
ap-9037	377	2	70	70	X
ap-9037	377	3	)	)	PUNCT
ap-9037	377	4	now	now	ADV
ap-9037	377	5	we	we	PRON
ap-9037	377	6	can	can	AUX
ap-9037	377	7	evaluate	evaluate	VERB
ap-9037	377	8	the	the	DET
ap-9037	377	9	second	second	ADJ
ap-9037	377	10	derivative	derivative	NOUN
ap-9037	377	11	in	in	ADP
ap-9037	377	12	the	the	DET
ap-9037	377	13	first	first	ADJ
ap-9037	377	14	term	term	NOUN
ap-9037	377	15	,	,	PUNCT
ap-9037	377	16	and	and	CCONJ
ap-9037	377	17	change	change	VERB
ap-9037	377	18	coordinates	coordinate	NOUN
ap-9037	377	19	to	to	ADP
ap-9037	377	20	θ	θ	PROPN
ap-9037	377	21	.	.	PUNCT
ap-9037	378	1	we	we	PRON
ap-9037	378	2	obtain	obtain	VERB
ap-9037	378	3	:	:	PUNCT
ap-9037	378	4	l(θ	l(θ	NOUN
ap-9037	378	5	)	)	PUNCT
ap-9037	379	1	=	=	PRON
ap-9037	379	2	[	[	PUNCT
ap-9037	379	3	ν1	ν1	NOUN
ap-9037	379	4	+	+	CCONJ
ap-9037	379	5	ν2	ν2	NOUN
ap-9037	379	6	+	+	CCONJ
ap-9037	379	7	ν3	ν3	NOUN
ap-9037	379	8	cos(y)2	cos(y)2	NOUN
ap-9037	379	9	−	−	NOUN
ap-9037	379	10	ν3	ν3	NOUN
ap-9037	379	11	sin(y)2	sin(y)2	X
ap-9037	379	12	]	]	PUNCT
ap-9037	379	13	|y	|y	NOUN
ap-9037	379	14	=	=	SYM
ap-9037	379	15	y(θ	y(θ	PROPN
ap-9037	379	16	)	)	PUNCT
ap-9037	380	1	+	+	CCONJ
ap-9037	380	2	{	{	PUNCT
ap-9037	380	3	d2	d2	PROPN
ap-9037	380	4	dy2	dy2	PROPN
ap-9037	380	5	log	log	NOUN
ap-9037	381	1	[	[	X
ap-9037	381	2	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	381	3	...	...	PUNCT
ap-9037	381	4	,θn	,θn	PUNCT
ap-9037	381	5	(	(	PUNCT
ap-9037	381	6	y	y	NOUN
ap-9037	381	7	)	)	PUNCT
ap-9037	381	8	]	]	PUNCT
ap-9037	381	9	}	}	PUNCT
ap-9037	381	10	|y	|y	NOUN
ap-9037	381	11	=	=	SYM
ap-9037	381	12	y(θ	y(θ	PROPN
ap-9037	381	13	)	)	PUNCT
ap-9037	382	1	=	=	SYM
ap-9037	382	2	sin(θ)2	sin(θ)2	NOUN
ap-9037	382	3	[	[	PUNCT
ap-9037	382	4	ν1	ν1	NOUN
ap-9037	382	5	+	+	CCONJ
ap-9037	382	6	ν2	ν2	NOUN
ap-9037	382	7	+	+	CCONJ
ap-9037	382	8	ν3	ν3	NOUN
ap-9037	382	9	cos(θ)2	cos(θ)2	NOUN
ap-9037	382	10	]	]	PUNCT
ap-9037	382	11	+	+	CCONJ
ap-9037	382	12	{	{	PUNCT
ap-9037	382	13	d2	d2	PROPN
ap-9037	382	14	dy2	dy2	PROPN
ap-9037	382	15	log	log	NOUN
ap-9037	383	1	[	[	X
ap-9037	383	2	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	383	3	...	...	PUNCT
ap-9037	383	4	,θn	,θn	PUNCT
ap-9037	383	5	(	(	PUNCT
ap-9037	383	6	y	y	NOUN
ap-9037	383	7	)	)	PUNCT
ap-9037	383	8	]	]	PUNCT
ap-9037	383	9	}	}	PUNCT
ap-9037	383	10	|y	|y	NOUN
ap-9037	383	11	=	=	SYM
ap-9037	383	12	y(θ	y(θ	NOUN
ap-9037	383	13	)	)	PUNCT
ap-9037	383	14	.	.	PUNCT
ap-9037	384	1	we	we	PRON
ap-9037	384	2	proceed	proceed	VERB
ap-9037	384	3	by	by	ADP
ap-9037	384	4	changing	change	VERB
ap-9037	384	5	the	the	DET
ap-9037	384	6	coordinate	coordinate	NOUN
ap-9037	384	7	in	in	ADP
ap-9037	384	8	our	our	PRON
ap-9037	384	9	remaining	remain	VERB
ap-9037	384	10	logarithmic	logarithmic	ADJ
ap-9037	384	11	term	term	NOUN
ap-9037	384	12	.	.	PUNCT
ap-9037	385	1	after	after	ADP
ap-9037	385	2	rewriting	rewrite	VERB
ap-9037	385	3	the	the	DET
ap-9037	385	4	second	second	ADJ
ap-9037	385	5	derivative	derivative	NOUN
ap-9037	385	6	in	in	ADP
ap-9037	385	7	terms	term	NOUN
ap-9037	385	8	of	of	ADP
ap-9037	385	9	the	the	DET
ap-9037	385	10	variable	variable	ADJ
ap-9037	385	11	θ	θ	PROPN
ap-9037	385	12	,	,	PUNCT
ap-9037	385	13	and	and	CCONJ
ap-9037	385	14	applying	apply	VERB
ap-9037	385	15	this	this	DET
ap-9037	385	16	derivative	derivative	NOUN
ap-9037	385	17	,	,	PUNCT
ap-9037	385	18	we	we	PRON
ap-9037	385	19	arrive	arrive	VERB
ap-9037	385	20	at	at	ADP
ap-9037	385	21	:	:	PUNCT
ap-9037	385	22	l(θ	l(θ	NOUN
ap-9037	385	23	)	)	PUNCT
ap-9037	386	1	=	=	PUNCT
ap-9037	386	2	sin(θ)2	sin(θ)2	NOUN
ap-9037	386	3	[	[	PUNCT
ap-9037	386	4	ν1	ν1	NOUN
ap-9037	386	5	+	+	CCONJ
ap-9037	386	6	ν2	ν2	NOUN
ap-9037	386	7	+	+	CCONJ
ap-9037	386	8	ν3	ν3	NOUN
ap-9037	386	9	cos(θ)2	cos(θ)2	NOUN
ap-9037	386	10	]	]	PUNCT
ap-9037	386	11	−	−	PROPN
ap-9037	386	12	[	[	PUNCT
ap-9037	386	13	sin(θ)2	sin(θ)2	NOUN
ap-9037	386	14	d2	d2	NOUN
ap-9037	386	15	dθ2	dθ2	NOUN
ap-9037	386	16	+	+	CCONJ
ap-9037	386	17	1	1	NUM
ap-9037	386	18	2	2	NUM
ap-9037	386	19	sin(2	sin(2	ADJ
ap-9037	386	20	θ	θ	NOUN
ap-9037	386	21	)	)	PUNCT
ap-9037	386	22	d	d	PROPN
ap-9037	386	23	dθ	dθ	PROPN
ap-9037	386	24	]	]	X
ap-9037	386	25	log	log	NOUN
ap-9037	386	26	{	{	PUNCT
ap-9037	387	1	[	[	X
ap-9037	387	2	−i	−i	NOUN
ap-9037	387	3	sin(θ	sin(θ	PROPN
ap-9037	387	4	)	)	PUNCT
ap-9037	387	5	]	]	PUNCT
ap-9037	387	6	n(n−1	n(n−1	X
ap-9037	387	7	)	)	PUNCT
ap-9037	387	8	2	2	NUM
ap-9037	387	9	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	387	10	...	...	PUNCT
ap-9037	387	11	,θn	,θn	PUNCT
ap-9037	387	12	(	(	PUNCT
ap-9037	387	13	θ	θ	NOUN
ap-9037	387	14	)	)	PUNCT
ap-9037	387	15	}	}	PUNCT
ap-9037	387	16	=	=	PUNCT
ap-9037	387	17	sin(θ)2	sin(θ)2	NOUN
ap-9037	387	18	[	[	PUNCT
ap-9037	387	19	ν1	ν1	NOUN
ap-9037	387	20	+	+	CCONJ
ap-9037	387	21	ν2	ν2	NOUN
ap-9037	387	22	+	+	CCONJ
ap-9037	387	23	ν3	ν3	NOUN
ap-9037	387	24	cos(θ)2	cos(θ)2	NOUN
ap-9037	387	25	]	]	PUNCT
ap-9037	388	1	+	+	CCONJ
ap-9037	388	2	n(n	n(n	NOUN
ap-9037	388	3	−	−	NOUN
ap-9037	388	4	1	1	NUM
ap-9037	388	5	)	)	PUNCT
ap-9037	388	6	2	2	NUM
ap-9037	388	7	+	+	CCONJ
ap-9037	388	8	sin(θ)2	sin(θ)2	NOUN
ap-9037	388	9	×	×	PROPN
ap-9037	388	10	d2	d2	NOUN
ap-9037	388	11	dθ2	dθ2	NOUN
ap-9037	388	12	log	log	NOUN
ap-9037	388	13	[	[	X
ap-9037	388	14	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	388	15	...	...	PUNCT
ap-9037	388	16	,θn	,θn	PUNCT
ap-9037	388	17	(	(	PUNCT
ap-9037	388	18	θ	θ	NOUN
ap-9037	388	19	)	)	PUNCT
ap-9037	388	20	]	]	PUNCT
ap-9037	388	21	+	+	CCONJ
ap-9037	388	22	n(1	n(1	NOUN
ap-9037	388	23	−	−	NOUN
ap-9037	388	24	n	n	CCONJ
ap-9037	388	25	)	)	PUNCT
ap-9037	388	26	2	2	NUM
ap-9037	388	27	cos(θ)2	cos(θ)2	NOUN
ap-9037	388	28	d	d	PROPN
ap-9037	388	29	dθ	dθ	PROPN
ap-9037	388	30	log	log	NOUN
ap-9037	388	31	[	[	X
ap-9037	388	32	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	388	33	...	...	PUNCT
ap-9037	388	34	,θn	,θn	PUNCT
ap-9037	388	35	(	(	PUNCT
ap-9037	388	36	θ	θ	NOUN
ap-9037	388	37	)	)	PUNCT
ap-9037	388	38	]	]	PUNCT
ap-9037	388	39	.	.	PUNCT
ap-9037	389	1	by	by	ADP
ap-9037	389	2	combining	combine	VERB
ap-9037	389	3	this	this	DET
ap-9037	389	4	result	result	NOUN
ap-9037	389	5	with	with	ADP
ap-9037	389	6	(	(	PUNCT
ap-9037	389	7	69	69	NUM
ap-9037	389	8	)	)	PUNCT
ap-9037	389	9	,	,	PUNCT
ap-9037	389	10	we	we	PRON
ap-9037	389	11	obtain	obtain	VERB
ap-9037	389	12	our	our	PRON
ap-9037	389	13	transformed	transform	VERB
ap-9037	389	14	potential	potential	NOUN
ap-9037	389	15	in	in	ADP
ap-9037	389	16	the	the	DET
ap-9037	389	17	form	form	NOUN
ap-9037	389	18	:	:	PUNCT
ap-9037	389	19	v̂θ(θ	v̂θ(θ	NUM
ap-9037	389	20	)	)	PUNCT
ap-9037	389	21	=	=	PRON
ap-9037	389	22	vθ(θ	vθ(θ	X
ap-9037	389	23	)	)	PUNCT
ap-9037	390	1	+	+	CCONJ
ap-9037	390	2	2	2	NUM
ap-9037	390	3	[	[	PUNCT
ap-9037	390	4	ν1	ν1	NOUN
ap-9037	390	5	+	+	CCONJ
ap-9037	390	6	ν2	ν2	NOUN
ap-9037	390	7	+	+	CCONJ
ap-9037	390	8	ν3	ν3	NOUN
ap-9037	390	9	cos(θ)2	cos(θ)2	NOUN
ap-9037	390	10	]	]	PUNCT
ap-9037	391	1	+	+	CCONJ
ap-9037	391	2	n(n	n(n	NOUN
ap-9037	391	3	−	−	NOUN
ap-9037	391	4	1	1	NUM
ap-9037	391	5	)	)	PUNCT
ap-9037	391	6	sin(θ)2	sin(θ)2	NOUN
ap-9037	391	7	+	+	SYM
ap-9037	391	8	2	2	NUM
ap-9037	391	9	d2	d2	NOUN
ap-9037	391	10	dθ2	dθ2	NOUN
ap-9037	391	11	log	log	NOUN
ap-9037	391	12	[	[	X
ap-9037	391	13	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	391	14	...	...	PUNCT
ap-9037	391	15	,θn	,θn	PUNCT
ap-9037	391	16	(	(	PUNCT
ap-9037	391	17	θ	θ	NOUN
ap-9037	391	18	)	)	PUNCT
ap-9037	391	19	]	]	PUNCT
ap-9037	391	20	+	+	CCONJ
ap-9037	391	21	n(1	n(1	NOUN
ap-9037	391	22	−	−	NOUN
ap-9037	391	23	n)cot(θ)2	n)cot(θ)2	NOUN
ap-9037	391	24	d	d	PROPN
ap-9037	391	25	dθ	dθ	PROPN
ap-9037	391	26	log	log	NOUN
ap-9037	391	27	[	[	X
ap-9037	391	28	wθ1,θ2,	wθ1,θ2,	NOUN
ap-9037	391	29	...	...	PUNCT
ap-9037	391	30	,θn	,θn	PUNCT
ap-9037	391	31	(	(	PUNCT
ap-9037	391	32	θ	θ	NOUN
ap-9037	391	33	)	)	PUNCT
ap-9037	391	34	]	]	PUNCT
ap-9037	391	35	.	.	PUNCT
ap-9037	392	1	(	(	PUNCT
ap-9037	392	2	71	71	NUM
ap-9037	392	3	)	)	PUNCT
ap-9037	392	4	in	in	ADP
ap-9037	392	5	conclusion	conclusion	NOUN
ap-9037	392	6	,	,	PUNCT
ap-9037	392	7	the	the	DET
ap-9037	392	8	function	function	NOUN
ap-9037	392	9	(	(	PUNCT
ap-9037	392	10	68	68	NUM
ap-9037	392	11	)	)	PUNCT
ap-9037	392	12	solves	solve	VERB
ap-9037	392	13	our	our	PRON
ap-9037	392	14	transformed	transform	VERB
ap-9037	392	15	polar	polar	ADJ
ap-9037	392	16	equation	equation	NOUN
ap-9037	392	17	(	(	PUNCT
ap-9037	392	18	51	51	NUM
ap-9037	392	19	)	)	PUNCT
ap-9037	392	20	for	for	ADP
ap-9037	392	21	the	the	DET
ap-9037	392	22	potential	potential	ADJ
ap-9037	392	23	(	(	PUNCT
ap-9037	392	24	71	71	NUM
ap-9037	392	25	)	)	PUNCT
ap-9037	392	26	.	.	PUNCT
ap-9037	393	1	recall	recall	VERB
ap-9037	393	2	that	that	SCONJ
ap-9037	393	3	the	the	DET
ap-9037	393	4	latter	latter	ADJ
ap-9037	393	5	potential	potential	NOUN
ap-9037	393	6	generally	generally	ADV
ap-9037	393	7	depends	depend	VERB
ap-9037	393	8	on	on	ADP
ap-9037	393	9	all	all	DET
ap-9037	393	10	parameters	parameter	NOUN
ap-9037	393	11	except	except	SCONJ
ap-9037	393	12	on	on	ADP
ap-9037	393	13	m2	m2	PROPN
ap-9037	393	14	.	.	PUNCT
ap-9037	394	1	it	it	PRON
ap-9037	394	2	is	be	AUX
ap-9037	394	3	therefore	therefore	ADV
ap-9037	394	4	,	,	PUNCT
ap-9037	394	5	desirable	desirable	ADJ
ap-9037	394	6	to	to	PART
ap-9037	394	7	keep	keep	VERB
ap-9037	394	8	the	the	DET
ap-9037	394	9	latter	latter	ADJ
ap-9037	394	10	parameters	parameter	NOUN
ap-9037	394	11	at	at	ADP
ap-9037	394	12	fixed	fix	VERB
ap-9037	394	13	values	value	NOUN
ap-9037	394	14	.	.	PUNCT
ap-9037	395	1	as	as	ADP
ap-9037	395	2	a	a	DET
ap-9037	395	3	further	further	ADJ
ap-9037	395	4	comment	comment	NOUN
ap-9037	395	5	,	,	PUNCT
ap-9037	395	6	let	let	VERB
ap-9037	395	7	us	we	PRON
ap-9037	395	8	point	point	VERB
ap-9037	395	9	out	out	ADP
ap-9037	395	10	that	that	SCONJ
ap-9037	395	11	the	the	DET
ap-9037	395	12	transformed	transform	VERB
ap-9037	395	13	potentials	potential	NOUN
ap-9037	395	14	,	,	PUNCT
ap-9037	395	15	for	for	ADP
ap-9037	395	16	the	the	DET
ap-9037	395	17	two	two	NUM
ap-9037	395	18	cases	case	NOUN
ap-9037	395	19	of	of	ADP
ap-9037	395	20	λ	λ	NOUN
ap-9037	395	21	and	and	CCONJ
ap-9037	395	22	m2	m2	PROPN
ap-9037	395	23	being	be	AUX
ap-9037	395	24	the	the	DET
ap-9037	395	25	transformation	transformation	NOUN
ap-9037	395	26	parameters	parameter	NOUN
ap-9037	395	27	,	,	PUNCT
ap-9037	395	28	are	be	AUX
ap-9037	395	29	entirely	entirely	ADV
ap-9037	395	30	different	different	ADJ
ap-9037	395	31	.	.	PUNCT
ap-9037	396	1	5	5	X
ap-9037	396	2	.	.	X
ap-9037	396	3	the	the	DET
ap-9037	396	4	azimuthal	azimuthal	ADJ
ap-9037	396	5	equation	equation	NOUN
ap-9037	396	6	similar	similar	ADJ
ap-9037	396	7	to	to	ADP
ap-9037	396	8	the	the	DET
ap-9037	396	9	previous	previous	ADJ
ap-9037	396	10	case	case	NOUN
ap-9037	396	11	,	,	PUNCT
ap-9037	396	12	the	the	DET
ap-9037	396	13	azimuthal	azimuthal	ADJ
ap-9037	396	14	equation	equation	NOUN
ap-9037	396	15	can	can	AUX
ap-9037	396	16	be	be	AUX
ap-9037	396	17	linked	link	VERB
ap-9037	396	18	to	to	ADP
ap-9037	396	19	trigonometric	trigonometric	ADJ
ap-9037	396	20	pöschl	pöschl	NOUN
ap-9037	396	21	-	-	PUNCT
ap-9037	396	22	teller	teller	NOUN
ap-9037	396	23	systems	system	NOUN
ap-9037	396	24	.	.	PUNCT
ap-9037	397	1	in	in	ADP
ap-9037	397	2	addition	addition	NOUN
ap-9037	397	3	to	to	ADP
ap-9037	397	4	using	use	VERB
ap-9037	397	5	results	result	NOUN
ap-9037	397	6	on	on	ADP
ap-9037	397	7	those	those	DET
ap-9037	397	8	systems	system	NOUN
ap-9037	397	9	to	to	PART
ap-9037	397	10	construct	construct	VERB
ap-9037	397	11	solutions	solution	NOUN
ap-9037	397	12	,	,	PUNCT
ap-9037	397	13	we	we	PRON
ap-9037	397	14	will	will	AUX
ap-9037	397	15	show	show	VERB
ap-9037	397	16	darboux	darboux	VERB
ap-9037	397	17	-	-	PUNCT
ap-9037	397	18	crum	crum	NOUN
ap-9037	397	19	transformations	transformation	NOUN
ap-9037	397	20	to	to	PART
ap-9037	397	21	be	be	AUX
ap-9037	397	22	applicable	applicable	ADJ
ap-9037	397	23	.	.	PUNCT
ap-9037	398	1	in	in	ADP
ap-9037	398	2	contrast	contrast	NOUN
ap-9037	398	3	to	to	ADP
ap-9037	398	4	the	the	DET
ap-9037	398	5	polar	polar	ADJ
ap-9037	398	6	equation	equation	NOUN
ap-9037	398	7	,	,	PUNCT
ap-9037	398	8	this	this	DET
ap-9037	398	9	time	time	NOUN
ap-9037	398	10	there	there	PRON
ap-9037	398	11	is	be	VERB
ap-9037	398	12	only	only	ADV
ap-9037	398	13	one	one	NUM
ap-9037	398	14	transformation	transformation	NOUN
ap-9037	398	15	parameter	parameter	NOUN
ap-9037	398	16	.	.	PUNCT
ap-9037	399	1	5.1	5.1	NUM
ap-9037	399	2	.	.	PUNCT
ap-9037	399	3	extended	extend	VERB
ap-9037	399	4	trigonometric	trigonometric	ADJ
ap-9037	399	5	pöschl	pöschl	NOUN
ap-9037	399	6	-	-	PUNCT
ap-9037	399	7	teller	teller	NOUN
ap-9037	399	8	systems	system	NOUN
ap-9037	399	9	we	we	PRON
ap-9037	399	10	will	will	AUX
ap-9037	399	11	approach	approach	VERB
ap-9037	399	12	the	the	DET
ap-9037	399	13	azimuthal	azimuthal	ADJ
ap-9037	399	14	equation	equation	NOUN
ap-9037	399	15	(	(	PUNCT
ap-9037	399	16	28	28	NUM
ap-9037	399	17	)	)	PUNCT
ap-9037	399	18	by	by	ADP
ap-9037	399	19	first	first	ADV
ap-9037	399	20	converting	convert	VERB
ap-9037	399	21	it	it	PRON
ap-9037	399	22	in	in	ADP
ap-9037	399	23	into	into	ADP
ap-9037	399	24	a	a	DET
ap-9037	399	25	form	form	NOUN
ap-9037	399	26	similar	similar	ADJ
ap-9037	399	27	to	to	ADP
ap-9037	399	28	the	the	DET
ap-9037	399	29	conventional	conventional	ADJ
ap-9037	399	30	schrödinger	schrödinger	ADJ
ap-9037	399	31	form	form	NOUN
ap-9037	399	32	.	.	PUNCT
ap-9037	400	1	to	to	PART
ap-9037	400	2	do	do	VERB
ap-9037	400	3	this	this	PRON
ap-9037	400	4	,	,	PUNCT
ap-9037	400	5	we	we	PRON
ap-9037	400	6	use	use	VERB
ap-9037	400	7	the	the	DET
ap-9037	400	8	point	point	NOUN
ap-9037	400	9	transformation	transformation	NOUN
ap-9037	400	10	:	:	PUNCT
ap-9037	400	11	φ(ϕ	φ(ϕ	NOUN
ap-9037	400	12	)	)	PUNCT
ap-9037	400	13	=	=	SYM
ap-9037	400	14	cos(ϕ)−ν1	cos(ϕ)−ν1	NOUN
ap-9037	400	15	sin(ϕ)−ν2ξ(ϕ	sin(ϕ)−ν2ξ(ϕ	PROPN
ap-9037	400	16	)	)	PUNCT
ap-9037	400	17	,	,	PUNCT
ap-9037	400	18	(	(	PUNCT
ap-9037	400	19	72	72	NUM
ap-9037	400	20	)	)	PUNCT
ap-9037	400	21	which	which	PRON
ap-9037	400	22	transforms	transform	VERB
ap-9037	400	23	(	(	PUNCT
ap-9037	400	24	28	28	NUM
ap-9037	400	25	)	)	PUNCT
ap-9037	400	26	into	into	ADP
ap-9037	400	27	:	:	PUNCT
ap-9037	400	28	ξ′′(ϕ	ξ′′(ϕ	NOUN
ap-9037	400	29	)	)	PUNCT
ap-9037	400	30	+	+	CCONJ
ap-9037	400	31	[	[	PUNCT
ap-9037	400	32	m2	m2	PROPN
ap-9037	400	33	+	+	CCONJ
ap-9037	400	34	(	(	PUNCT
ap-9037	400	35	ν1	ν1	NOUN
ap-9037	400	36	+	+	CCONJ
ap-9037	400	37	ν2)2	ν2)2	PUNCT
ap-9037	400	38	+	+	CCONJ
ap-9037	400	39	(	(	PUNCT
ap-9037	400	40	r1	r1	NOUN
ap-9037	400	41	−	−	PROPN
ap-9037	400	42	ν1)ν1	ν1)ν1	NOUN
ap-9037	400	43	cos(ϕ)2	cos(ϕ)2	NOUN
ap-9037	400	44	+	+	CCONJ
ap-9037	400	45	(	(	PUNCT
ap-9037	400	46	r2	r2	PROPN
ap-9037	400	47	−	−	PROPN
ap-9037	400	48	ν2)ν2	ν2)ν2	NOUN
ap-9037	400	49	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	400	50	−	−	PROPN
ap-9037	400	51	vϕ(ϕ	vϕ(ϕ	NOUN
ap-9037	400	52	)	)	PUNCT
ap-9037	400	53	]	]	PUNCT
ap-9037	401	1	ξ(ϕ	ξ(ϕ	X
ap-9037	401	2	)	)	PUNCT
ap-9037	401	3	=	=	SYM
ap-9037	401	4	0	0	X
ap-9037	401	5	.	.	PUNCT
ap-9037	402	1	(	(	PUNCT
ap-9037	402	2	73	73	NUM
ap-9037	402	3	)	)	PUNCT
ap-9037	402	4	similar	similar	ADJ
ap-9037	402	5	to	to	ADP
ap-9037	402	6	the	the	DET
ap-9037	402	7	polar	polar	ADJ
ap-9037	402	8	equation	equation	NOUN
ap-9037	402	9	(	(	PUNCT
ap-9037	402	10	51	51	NUM
ap-9037	402	11	)	)	PUNCT
ap-9037	402	12	,	,	PUNCT
ap-9037	402	13	this	this	PRON
ap-9037	402	14	can	can	AUX
ap-9037	402	15	be	be	AUX
ap-9037	402	16	interpreted	interpret	VERB
ap-9037	402	17	as	as	ADP
ap-9037	402	18	a	a	DET
ap-9037	402	19	schrödinger	schrödinger	ADJ
ap-9037	402	20	equation	equation	NOUN
ap-9037	402	21	for	for	ADP
ap-9037	402	22	a	a	DET
ap-9037	402	23	generalisation	generalisation	NOUN
ap-9037	402	24	of	of	ADP
ap-9037	402	25	the	the	DET
ap-9037	402	26	trigonometric	trigonometric	ADJ
ap-9037	402	27	pöschl	pöschl	NOUN
ap-9037	402	28	-	-	PUNCT
ap-9037	402	29	teller	teller	NOUN
ap-9037	402	30	potential	potential	NOUN
ap-9037	402	31	.	.	PUNCT
ap-9037	403	1	while	while	SCONJ
ap-9037	403	2	the	the	DET
ap-9037	403	3	two	two	NUM
ap-9037	403	4	equations	equation	NOUN
ap-9037	403	5	are	be	AUX
ap-9037	403	6	very	very	ADV
ap-9037	403	7	similar	similar	ADJ
ap-9037	403	8	in	in	ADP
ap-9037	403	9	this	this	DET
ap-9037	403	10	sense	sense	NOUN
ap-9037	403	11	,	,	PUNCT
ap-9037	403	12	the	the	DET
ap-9037	403	13	main	main	ADJ
ap-9037	403	14	285	285	NUM
ap-9037	403	15	axel	axel	NOUN
ap-9037	403	16	schulze	schulze	NOUN
ap-9037	403	17	-	-	PUNCT
ap-9037	403	18	halberg	halberg	PROPN
ap-9037	403	19	acta	acta	PROPN
ap-9037	403	20	polytechnica	polytechnica	PROPN
ap-9037	403	21	differences	difference	NOUN
ap-9037	403	22	are	be	AUX
ap-9037	403	23	in	in	ADP
ap-9037	403	24	the	the	DET
ap-9037	403	25	domains	domain	NOUN
ap-9037	403	26	and	and	CCONJ
ap-9037	403	27	in	in	ADP
ap-9037	403	28	the	the	DET
ap-9037	403	29	presence	presence	NOUN
ap-9037	403	30	of	of	ADP
ap-9037	403	31	two	two	NUM
ap-9037	403	32	parity	parity	NOUN
ap-9037	403	33	parameters	parameter	NOUN
ap-9037	403	34	in	in	ADP
ap-9037	403	35	(	(	PUNCT
ap-9037	403	36	73	73	NUM
ap-9037	403	37	)	)	PUNCT
ap-9037	403	38	.	.	PUNCT
ap-9037	404	1	to	to	PART
ap-9037	404	2	give	give	VERB
ap-9037	404	3	a	a	DET
ap-9037	404	4	simple	simple	ADJ
ap-9037	404	5	example	example	NOUN
ap-9037	404	6	of	of	ADP
ap-9037	404	7	a	a	DET
ap-9037	404	8	solution	solution	NOUN
ap-9037	404	9	to	to	ADP
ap-9037	404	10	(	(	PUNCT
ap-9037	404	11	73	73	NUM
ap-9037	404	12	)	)	PUNCT
ap-9037	404	13	,	,	PUNCT
ap-9037	404	14	we	we	PRON
ap-9037	404	15	choose	choose	VERB
ap-9037	404	16	the	the	DET
ap-9037	404	17	function	function	NOUN
ap-9037	404	18	vϕ	vϕ	NOUN
ap-9037	404	19	as	as	ADP
ap-9037	404	20	:	:	PUNCT
ap-9037	404	21	vϕ(ϕ	vϕ(ϕ	NUM
ap-9037	404	22	)	)	PUNCT
ap-9037	405	1	=	=	SYM
ap-9037	405	2	γ	γ	PROPN
ap-9037	405	3	cos(ϕ)2	cos(ϕ)2	NOUN
ap-9037	405	4	,	,	PUNCT
ap-9037	405	5	(	(	PUNCT
ap-9037	405	6	74	74	X
ap-9037	405	7	)	)	PUNCT
ap-9037	405	8	introducing	introduce	VERB
ap-9037	405	9	a	a	DET
ap-9037	405	10	real	real	ADV
ap-9037	405	11	-	-	PUNCT
ap-9037	405	12	valued	value	VERB
ap-9037	405	13	constant	constant	ADJ
ap-9037	405	14	γ	γ	X
ap-9037	405	15	.	.	PUNCT
ap-9037	405	16	by	by	ADP
ap-9037	405	17	substituting	substitute	VERB
ap-9037	405	18	vϕ	vϕ	NOUN
ap-9037	405	19	into	into	ADP
ap-9037	405	20	(	(	PUNCT
ap-9037	405	21	73	73	NUM
ap-9037	405	22	)	)	PUNCT
ap-9037	405	23	,	,	PUNCT
ap-9037	405	24	we	we	PRON
ap-9037	405	25	obtain	obtain	VERB
ap-9037	405	26	(	(	PUNCT
ap-9037	405	27	73	73	NUM
ap-9037	405	28	)	)	PUNCT
ap-9037	405	29	in	in	ADP
ap-9037	405	30	a	a	DET
ap-9037	405	31	form	form	NOUN
ap-9037	405	32	that	that	PRON
ap-9037	405	33	matches	match	VERB
ap-9037	405	34	(	(	PUNCT
ap-9037	405	35	4	4	NUM
ap-9037	405	36	)	)	PUNCT
ap-9037	405	37	,	,	PUNCT
ap-9037	405	38	provided	provide	VERB
ap-9037	405	39	we	we	PRON
ap-9037	405	40	choose	choose	VERB
ap-9037	405	41	the	the	DET
ap-9037	405	42	parameters	parameter	NOUN
ap-9037	405	43	a	a	DET
ap-9037	405	44	,	,	PUNCT
ap-9037	405	45	b	b	NOUN
ap-9037	405	46	,	,	PUNCT
ap-9037	405	47	and	and	CCONJ
ap-9037	405	48	c	c	PROPN
ap-9037	405	49	as	as	SCONJ
ap-9037	405	50	follows	follow	VERB
ap-9037	405	51	:	:	PUNCT
ap-9037	405	52	a	a	DET
ap-9037	405	53	=	=	PROPN
ap-9037	405	54	m2	m2	PROPN
ap-9037	405	55	+	+	CCONJ
ap-9037	405	56	(	(	PUNCT
ap-9037	405	57	ν1	ν1	NOUN
ap-9037	405	58	+	+	CCONJ
ap-9037	405	59	ν2)2	ν2)2	CCONJ
ap-9037	405	60	,	,	PUNCT
ap-9037	405	61	(	(	PUNCT
ap-9037	405	62	75	75	NUM
ap-9037	405	63	)	)	PUNCT
ap-9037	405	64	b	b	NOUN
ap-9037	405	65	=	=	SYM
ap-9037	405	66	(	(	PUNCT
ap-9037	405	67	r2	r2	PROPN
ap-9037	405	68	−	−	PROPN
ap-9037	405	69	ν2	ν2	NOUN
ap-9037	405	70	)	)	PUNCT
ap-9037	405	71	ν2	ν2	NOUN
ap-9037	405	72	,	,	PUNCT
ap-9037	405	73	(	(	PUNCT
ap-9037	405	74	76	76	NUM
ap-9037	405	75	)	)	PUNCT
ap-9037	405	76	c	c	NOUN
ap-9037	406	1	=	=	SYM
ap-9037	406	2	(	(	PUNCT
ap-9037	406	3	r1	r1	PROPN
ap-9037	406	4	−	−	PROPN
ap-9037	406	5	ν1	ν1	NOUN
ap-9037	406	6	)	)	PUNCT
ap-9037	406	7	ν1	ν1	NOUN
ap-9037	406	8	−	−	PROPN
ap-9037	406	9	γ	γ	X
ap-9037	406	10	.	.	PUNCT
ap-9037	407	1	(	(	PUNCT
ap-9037	407	2	77	77	NUM
ap-9037	407	3	)	)	PUNCT
ap-9037	407	4	furthermore	furthermore	ADV
ap-9037	407	5	,	,	PUNCT
ap-9037	407	6	we	we	PRON
ap-9037	407	7	will	will	AUX
ap-9037	407	8	employ	employ	VERB
ap-9037	407	9	the	the	DET
ap-9037	407	10	condition	condition	NOUN
ap-9037	407	11	(	(	PUNCT
ap-9037	407	12	37	37	NUM
ap-9037	407	13	)	)	PUNCT
ap-9037	407	14	here	here	ADV
ap-9037	407	15	,	,	PUNCT
ap-9037	407	16	which	which	PRON
ap-9037	407	17	requires	require	VERB
ap-9037	407	18	that	that	SCONJ
ap-9037	407	19	the	the	DET
ap-9037	407	20	separation	separation	NOUN
ap-9037	407	21	constant	constant	ADJ
ap-9037	407	22	m2	m2	PROPN
ap-9037	407	23	is	be	AUX
ap-9037	407	24	given	give	VERB
ap-9037	407	25	by	by	ADP
ap-9037	407	26	:	:	PUNCT
ap-9037	407	27	m2	m2	PROPN
ap-9037	407	28	=	=	PROPN
ap-9037	407	29	−(ν1	−(ν1	ADP
ap-9037	407	30	+	+	X
ap-9037	407	31	ν2)2	ν2)2	PUNCT
ap-9037	407	32	+	+	CCONJ
ap-9037	407	33	[	[	PUNCT
ap-9037	407	34	1	1	NUM
ap-9037	407	35	+	+	NUM
ap-9037	407	36	2n	2n	NUM
ap-9037	408	1	+	+	CCONJ
ap-9037	408	2	1	1	NUM
ap-9037	408	3	2	2	NUM
ap-9037	408	4	√	√	NUM
ap-9037	408	5	1	1	NUM
ap-9037	408	6	+	+	NUM
ap-9037	408	7	4γ	4γ	NOUN
ap-9037	408	8	+	+	CCONJ
ap-9037	408	9	4ν1(ν1	4ν1(ν1	NUM
ap-9037	408	10	−	−	PROPN
ap-9037	408	11	r1	r1	PROPN
ap-9037	408	12	)	)	PUNCT
ap-9037	408	13	+	+	CCONJ
ap-9037	408	14	1	1	NUM
ap-9037	408	15	2	2	NUM
ap-9037	408	16	√	√	NUM
ap-9037	408	17	1	1	NUM
ap-9037	408	18	+	+	CCONJ
ap-9037	408	19	4ν2(ν2	4ν2(ν2	NUM
ap-9037	408	20	−	−	PROPN
ap-9037	408	21	r2	r2	PROPN
ap-9037	408	22	)	)	PUNCT
ap-9037	408	23	]	]	PUNCT
ap-9037	408	24	2	2	X
ap-9037	408	25	.	.	PUNCT
ap-9037	409	1	(	(	PUNCT
ap-9037	409	2	78	78	NUM
ap-9037	409	3	)	)	PUNCT
ap-9037	409	4	now	now	ADV
ap-9037	409	5	,	,	PUNCT
ap-9037	409	6	according	accord	VERB
ap-9037	409	7	to	to	ADP
ap-9037	409	8	(	(	PUNCT
ap-9037	409	9	2	2	NUM
ap-9037	409	10	)	)	PUNCT
ap-9037	409	11	and	and	CCONJ
ap-9037	409	12	(	(	PUNCT
ap-9037	409	13	3	3	NUM
ap-9037	409	14	)	)	PUNCT
ap-9037	409	15	,	,	PUNCT
ap-9037	409	16	a	a	DET
ap-9037	409	17	particular	particular	ADJ
ap-9037	409	18	solution	solution	NOUN
ap-9037	409	19	of	of	ADP
ap-9037	409	20	(	(	PUNCT
ap-9037	409	21	73	73	NUM
ap-9037	409	22	)	)	PUNCT
ap-9037	409	23	can	can	AUX
ap-9037	409	24	be	be	AUX
ap-9037	409	25	written	write	VERB
ap-9037	409	26	in	in	ADP
ap-9037	409	27	the	the	DET
ap-9037	409	28	form	form	NOUN
ap-9037	409	29	:	:	PUNCT
ap-9037	409	30	ξ(ϕ	ξ(ϕ	X
ap-9037	409	31	)	)	PUNCT
ap-9037	409	32	=	=	SYM
ap-9037	409	33	sin(ϕ	sin(ϕ	PROPN
ap-9037	409	34	)	)	PUNCT
ap-9037	409	35	1	1	NUM
ap-9037	409	36	2	2	NUM
ap-9037	409	37	+	+	CCONJ
ap-9037	409	38	1	1	NUM
ap-9037	409	39	2	2	NUM
ap-9037	409	40	√	√	NUM
ap-9037	409	41	1−4b	1−4b	NUM
ap-9037	409	42	cos(ϕ	cos(ϕ	NOUN
ap-9037	409	43	)	)	PUNCT
ap-9037	409	44	1	1	NUM
ap-9037	409	45	2	2	NUM
ap-9037	409	46	+	+	CCONJ
ap-9037	409	47	1	1	NUM
ap-9037	409	48	2	2	NUM
ap-9037	409	49	√	√	NUM
ap-9037	409	50	1−4c	1−4c	NUM
ap-9037	409	51	2f1	2f1	NUM
ap-9037	409	52	[	[	PUNCT
ap-9037	409	53	−	−	NUM
ap-9037	409	54	n	n	CCONJ
ap-9037	409	55	,	,	PUNCT
ap-9037	409	56	1	1	NUM
ap-9037	409	57	2	2	NUM
ap-9037	409	58	+	+	CCONJ
ap-9037	409	59	√	√	DET
ap-9037	409	60	a	a	DET
ap-9037	409	61	2	2	NUM
ap-9037	409	62	+	+	CCONJ
ap-9037	409	63	k+	k+	NOUN
ap-9037	409	64	,	,	PUNCT
ap-9037	409	65	1	1	NUM
ap-9037	409	66	+	+	CCONJ
ap-9037	409	67	√	√	NUM
ap-9037	409	68	1	1	NUM
ap-9037	409	69	−	−	PROPN
ap-9037	409	70	4b	4b	X
ap-9037	409	71	2	2	NUM
ap-9037	409	72	,	,	PUNCT
ap-9037	409	73	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	409	74	]	]	PUNCT
ap-9037	409	75	,	,	PUNCT
ap-9037	409	76	(	(	PUNCT
ap-9037	409	77	79	79	X
ap-9037	409	78	)	)	PUNCT
ap-9037	409	79	recall	recall	VERB
ap-9037	409	80	that	that	SCONJ
ap-9037	409	81	the	the	DET
ap-9037	409	82	parameters	parameter	NOUN
ap-9037	409	83	a	a	DET
ap-9037	409	84	,	,	PUNCT
ap-9037	409	85	b	b	NOUN
ap-9037	409	86	,	,	PUNCT
ap-9037	409	87	c	c	PROPN
ap-9037	409	88	are	be	AUX
ap-9037	409	89	defined	define	VERB
ap-9037	409	90	in	in	ADP
ap-9037	409	91	(	(	PUNCT
ap-9037	409	92	75)–(77	75)–(77	NUM
ap-9037	409	93	)	)	PUNCT
ap-9037	409	94	,	,	PUNCT
ap-9037	409	95	where	where	SCONJ
ap-9037	409	96	a	a	PRON
ap-9037	409	97	depends	depend	VERB
ap-9037	409	98	on	on	ADP
ap-9037	409	99	m2	m2	PROPN
ap-9037	409	100	from	from	ADP
ap-9037	409	101	(	(	PUNCT
ap-9037	409	102	78	78	NUM
ap-9037	409	103	)	)	PUNCT
ap-9037	409	104	.	.	PUNCT
ap-9037	410	1	in	in	ADP
ap-9037	410	2	the	the	DET
ap-9037	410	3	next	next	ADJ
ap-9037	410	4	step	step	NOUN
ap-9037	410	5	,	,	PUNCT
ap-9037	410	6	we	we	PRON
ap-9037	410	7	use	use	VERB
ap-9037	410	8	our	our	PRON
ap-9037	410	9	point	point	NOUN
ap-9037	410	10	transformation	transformation	NOUN
ap-9037	410	11	(	(	PUNCT
ap-9037	410	12	72	72	NUM
ap-9037	410	13	)	)	PUNCT
ap-9037	410	14	to	to	PART
ap-9037	410	15	build	build	VERB
ap-9037	410	16	the	the	DET
ap-9037	410	17	solution	solution	NOUN
ap-9037	410	18	for	for	ADP
ap-9037	410	19	our	our	PRON
ap-9037	410	20	azimuthal	azimuthal	ADJ
ap-9037	410	21	equation	equation	NOUN
ap-9037	410	22	(	(	PUNCT
ap-9037	410	23	28	28	NUM
ap-9037	410	24	)	)	PUNCT
ap-9037	410	25	with	with	ADP
ap-9037	410	26	(	(	PUNCT
ap-9037	410	27	74	74	NUM
ap-9037	410	28	)	)	PUNCT
ap-9037	410	29	.	.	PUNCT
ap-9037	411	1	the	the	DET
ap-9037	411	2	result	result	NOUN
ap-9037	411	3	reads	read	VERB
ap-9037	411	4	:	:	PUNCT
ap-9037	411	5	φ(ϕ	φ(ϕ	NUM
ap-9037	411	6	)	)	PUNCT
ap-9037	411	7	=	=	PUNCT
ap-9037	412	1	sin(ϕ)−ν2	sin(ϕ)−ν2	VERB
ap-9037	412	2	+	+	CCONJ
ap-9037	412	3	1	1	NUM
ap-9037	412	4	2	2	NUM
ap-9037	412	5	+	+	CCONJ
ap-9037	412	6	1	1	NUM
ap-9037	412	7	2	2	NUM
ap-9037	412	8	√	√	NUM
ap-9037	412	9	1−4b	1−4b	NUM
ap-9037	412	10	cos(ϕ)−ν1	cos(ϕ)−ν1	NOUN
ap-9037	412	11	+	+	CCONJ
ap-9037	412	12	1	1	NUM
ap-9037	412	13	2	2	NUM
ap-9037	412	14	+	+	CCONJ
ap-9037	412	15	1	1	NUM
ap-9037	412	16	2	2	NUM
ap-9037	412	17	√	√	NUM
ap-9037	412	18	1−4c	1−4c	NUM
ap-9037	412	19	2f1	2f1	NUM
ap-9037	412	20	[	[	PUNCT
ap-9037	412	21	−	−	NUM
ap-9037	412	22	n	n	CCONJ
ap-9037	412	23	,	,	PUNCT
ap-9037	412	24	1	1	NUM
ap-9037	412	25	2	2	NUM
ap-9037	412	26	+	+	CCONJ
ap-9037	412	27	√	√	DET
ap-9037	412	28	a	a	DET
ap-9037	412	29	2	2	NUM
ap-9037	412	30	+	+	CCONJ
ap-9037	412	31	k+	k+	NOUN
ap-9037	412	32	,	,	PUNCT
ap-9037	412	33	1	1	NUM
ap-9037	412	34	+	+	CCONJ
ap-9037	412	35	√	√	NUM
ap-9037	412	36	1	1	NUM
ap-9037	412	37	−	−	PROPN
ap-9037	412	38	4b	4b	X
ap-9037	412	39	2	2	NUM
ap-9037	412	40	,	,	PUNCT
ap-9037	412	41	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	412	42	]	]	PUNCT
ap-9037	412	43	.	.	PUNCT
ap-9037	413	1	(	(	PUNCT
ap-9037	413	2	80	80	NUM
ap-9037	413	3	)	)	PUNCT
ap-9037	413	4	we	we	PRON
ap-9037	413	5	now	now	ADV
ap-9037	413	6	need	need	VERB
ap-9037	413	7	to	to	PART
ap-9037	413	8	find	find	VERB
ap-9037	413	9	the	the	DET
ap-9037	413	10	values	value	NOUN
ap-9037	413	11	of	of	ADP
ap-9037	413	12	our	our	PRON
ap-9037	413	13	parity	parity	NOUN
ap-9037	413	14	parameters	parameter	NOUN
ap-9037	413	15	r1	r1	PROPN
ap-9037	413	16	and	and	CCONJ
ap-9037	413	17	r2	r2	PROPN
ap-9037	413	18	using	use	VERB
ap-9037	413	19	the	the	DET
ap-9037	413	20	relations	relation	NOUN
ap-9037	413	21	(	(	PUNCT
ap-9037	413	22	21	21	NUM
ap-9037	413	23	)	)	PUNCT
ap-9037	413	24	,	,	PUNCT
ap-9037	413	25	(	(	PUNCT
ap-9037	413	26	22	22	NUM
ap-9037	413	27	)	)	PUNCT
ap-9037	413	28	,	,	PUNCT
ap-9037	413	29	and	and	CCONJ
ap-9037	413	30	(	(	PUNCT
ap-9037	413	31	24	24	NUM
ap-9037	413	32	)	)	PUNCT
ap-9037	413	33	.	.	PUNCT
ap-9037	414	1	before	before	SCONJ
ap-9037	414	2	we	we	PRON
ap-9037	414	3	do	do	VERB
ap-9037	414	4	so	so	ADV
ap-9037	414	5	,	,	PUNCT
ap-9037	414	6	it	it	PRON
ap-9037	414	7	is	be	AUX
ap-9037	414	8	convenient	convenient	ADJ
ap-9037	414	9	to	to	PART
ap-9037	414	10	find	find	VERB
ap-9037	414	11	the	the	DET
ap-9037	414	12	actions	action	NOUN
ap-9037	414	13	of	of	ADP
ap-9037	414	14	the	the	DET
ap-9037	414	15	reflection	reflection	NOUN
ap-9037	414	16	operators	operator	NOUN
ap-9037	414	17	on	on	ADP
ap-9037	414	18	the	the	DET
ap-9037	414	19	sine	sine	NOUN
ap-9037	414	20	and	and	CCONJ
ap-9037	414	21	cosine	cosine	NOUN
ap-9037	414	22	functions	function	NOUN
ap-9037	414	23	.	.	PUNCT
ap-9037	415	1	we	we	PRON
ap-9037	415	2	have	have	VERB
ap-9037	415	3	:	:	PUNCT
ap-9037	415	4	r1	r1	PROPN
ap-9037	415	5	sin(ϕ	sin(ϕ	PROPN
ap-9037	415	6	)	)	PUNCT
ap-9037	415	7	=	=	SYM
ap-9037	415	8	sin(ϕ	sin(ϕ	PROPN
ap-9037	415	9	)	)	PUNCT
ap-9037	415	10	,	,	PUNCT
ap-9037	415	11	r1	r1	PROPN
ap-9037	415	12	cos(ϕ	cos(ϕ	PROPN
ap-9037	415	13	)	)	PUNCT
ap-9037	415	14	=	=	SYM
ap-9037	416	1	−	−	PROPN
ap-9037	416	2	cos(ϕ	cos(ϕ	PROPN
ap-9037	416	3	)	)	PUNCT
ap-9037	416	4	,	,	PUNCT
ap-9037	416	5	r2	r2	PROPN
ap-9037	416	6	sin(ϕ	sin(ϕ	PROPN
ap-9037	416	7	)	)	PUNCT
ap-9037	416	8	=	=	SYM
ap-9037	416	9	−	−	NOUN
ap-9037	416	10	sin(ϕ	sin(ϕ	NOUN
ap-9037	416	11	)	)	PUNCT
ap-9037	416	12	,	,	PUNCT
ap-9037	416	13	r2	r2	PROPN
ap-9037	416	14	cos(ϕ	cos(ϕ	PROPN
ap-9037	416	15	)	)	PUNCT
ap-9037	416	16	=	=	SYM
ap-9037	416	17	cos(ϕ	cos(ϕ	PROPN
ap-9037	416	18	)	)	PUNCT
ap-9037	416	19	.	.	PUNCT
ap-9037	417	1	thus	thus	ADV
ap-9037	417	2	,	,	PUNCT
ap-9037	417	3	applying	apply	VERB
ap-9037	417	4	these	these	DET
ap-9037	417	5	operators	operator	NOUN
ap-9037	417	6	to	to	ADP
ap-9037	417	7	the	the	DET
ap-9037	417	8	solution	solution	NOUN
ap-9037	417	9	(	(	PUNCT
ap-9037	417	10	80	80	NUM
ap-9037	417	11	)	)	PUNCT
ap-9037	417	12	results	result	NOUN
ap-9037	417	13	in	in	ADP
ap-9037	417	14	:	:	PUNCT
ap-9037	417	15	r1	r1	PROPN
ap-9037	417	16	φ(ϕ	φ(ϕ	PROPN
ap-9037	417	17	)	)	PUNCT
ap-9037	417	18	=	=	PUNCT
ap-9037	417	19	(	(	PUNCT
ap-9037	417	20	−1)−ν1	−1)−ν1	NUM
ap-9037	417	21	+	+	SYM
ap-9037	417	22	1	1	NUM
ap-9037	417	23	2	2	NUM
ap-9037	417	24	+	+	CCONJ
ap-9037	417	25	1	1	NUM
ap-9037	417	26	2	2	NUM
ap-9037	417	27	√	√	NUM
ap-9037	417	28	1−4cφ(ϕ	1−4cφ(ϕ	NUM
ap-9037	417	29	)	)	PUNCT
ap-9037	417	30	,	,	PUNCT
ap-9037	417	31	r2	r2	PROPN
ap-9037	417	32	φ(ϕ	φ(ϕ	PROPN
ap-9037	417	33	)	)	PUNCT
ap-9037	417	34	=	=	SYM
ap-9037	417	35	(	(	PUNCT
ap-9037	417	36	−1)−ν2	−1)−ν2	NOUN
ap-9037	417	37	+	+	NOUN
ap-9037	417	38	1	1	NUM
ap-9037	417	39	2	2	NUM
ap-9037	417	40	+	+	CCONJ
ap-9037	417	41	1	1	NUM
ap-9037	417	42	2	2	NUM
ap-9037	417	43	√	√	NUM
ap-9037	417	44	1−4bφ(ϕ	1−4bφ(ϕ	NUM
ap-9037	417	45	)	)	PUNCT
ap-9037	417	46	.	.	PUNCT
ap-9037	418	1	now	now	ADV
ap-9037	418	2	,	,	PUNCT
ap-9037	418	3	if	if	SCONJ
ap-9037	418	4	we	we	PRON
ap-9037	418	5	combine	combine	VERB
ap-9037	418	6	this	this	PRON
ap-9037	418	7	with	with	ADP
ap-9037	418	8	(	(	PUNCT
ap-9037	418	9	21	21	NUM
ap-9037	418	10	)	)	PUNCT
ap-9037	418	11	and	and	CCONJ
ap-9037	418	12	(	(	PUNCT
ap-9037	418	13	22	22	NUM
ap-9037	418	14	)	)	PUNCT
ap-9037	418	15	,	,	PUNCT
ap-9037	418	16	we	we	PRON
ap-9037	418	17	obtain	obtain	VERB
ap-9037	418	18	conditions	condition	NOUN
ap-9037	418	19	for	for	ADP
ap-9037	418	20	the	the	DET
ap-9037	418	21	parity	parity	NOUN
ap-9037	418	22	parameters	parameter	NOUN
ap-9037	418	23	r1	r1	NOUN
ap-9037	418	24	and	and	CCONJ
ap-9037	418	25	r2	r2	PROPN
ap-9037	418	26	.	.	PUNCT
ap-9037	419	1	these	these	PRON
ap-9037	419	2	read	read	NOUN
ap-9037	419	3	:	:	PUNCT
ap-9037	419	4	r1	r1	PROPN
ap-9037	419	5	=	=	SYM
ap-9037	419	6	(	(	PUNCT
ap-9037	419	7	−1)−ν1	−1)−ν1	NUM
ap-9037	419	8	+	+	SYM
ap-9037	420	1	1	1	NUM
ap-9037	420	2	2	2	NUM
ap-9037	420	3	+	+	CCONJ
ap-9037	420	4	1	1	NUM
ap-9037	420	5	2	2	NUM
ap-9037	420	6	√	√	NUM
ap-9037	420	7	1−4c	1−4c	NUM
ap-9037	420	8	,	,	PUNCT
ap-9037	420	9	(	(	PUNCT
ap-9037	420	10	81	81	NUM
ap-9037	420	11	)	)	PUNCT
ap-9037	420	12	r2	r2	NOUN
ap-9037	420	13	=	=	SYM
ap-9037	420	14	(	(	PUNCT
ap-9037	420	15	−1)−ν2	−1)−ν2	NOUN
ap-9037	420	16	+	+	NOUN
ap-9037	420	17	1	1	NUM
ap-9037	420	18	2	2	NUM
ap-9037	420	19	+	+	CCONJ
ap-9037	420	20	1	1	NUM
ap-9037	420	21	2	2	NUM
ap-9037	420	22	√	√	NUM
ap-9037	420	23	1−4b	1−4b	NUM
ap-9037	420	24	,	,	PUNCT
ap-9037	420	25	(	(	PUNCT
ap-9037	420	26	82	82	X
ap-9037	420	27	)	)	PUNCT
ap-9037	420	28	note	note	NOUN
ap-9037	420	29	that	that	SCONJ
ap-9037	420	30	the	the	DET
ap-9037	420	31	constants	constant	NOUN
ap-9037	420	32	b	b	PROPN
ap-9037	420	33	and	and	CCONJ
ap-9037	420	34	c	c	PROPN
ap-9037	420	35	are	be	AUX
ap-9037	420	36	defined	define	VERB
ap-9037	420	37	in	in	ADP
ap-9037	420	38	(	(	PUNCT
ap-9037	420	39	76	76	NUM
ap-9037	420	40	)	)	PUNCT
ap-9037	420	41	and	and	CCONJ
ap-9037	420	42	(	(	PUNCT
ap-9037	420	43	77	77	NUM
ap-9037	420	44	)	)	PUNCT
ap-9037	420	45	,	,	PUNCT
ap-9037	420	46	respectively	respectively	ADV
ap-9037	420	47	.	.	PUNCT
ap-9037	421	1	the	the	DET
ap-9037	421	2	first	first	ADJ
ap-9037	421	3	of	of	ADP
ap-9037	421	4	these	these	DET
ap-9037	421	5	conditions	condition	NOUN
ap-9037	421	6	is	be	AUX
ap-9037	421	7	satisfied	satisfied	ADJ
ap-9037	421	8	if	if	SCONJ
ap-9037	421	9	the	the	DET
ap-9037	421	10	exponent	exponent	NOUN
ap-9037	421	11	equals	equal	VERB
ap-9037	421	12	an	an	DET
ap-9037	421	13	even	even	ADJ
ap-9037	421	14	or	or	CCONJ
ap-9037	421	15	an	an	DET
ap-9037	421	16	odd	odd	ADJ
ap-9037	421	17	number	number	NOUN
ap-9037	421	18	,	,	PUNCT
ap-9037	421	19	depending	depend	VERB
ap-9037	421	20	on	on	ADP
ap-9037	421	21	the	the	DET
ap-9037	421	22	value	value	NOUN
ap-9037	421	23	of	of	ADP
ap-9037	421	24	r1	r1	PROPN
ap-9037	421	25	.	.	PUNCT
ap-9037	422	1	we	we	PRON
ap-9037	422	2	have	have	VERB
ap-9037	422	3	the	the	DET
ap-9037	422	4	two	two	NUM
ap-9037	422	5	conditions	condition	NOUN
ap-9037	422	6	:	:	PUNCT
ap-9037	423	1	−ν1	−ν1	PROPN
ap-9037	423	2	+	+	CCONJ
ap-9037	423	3	1	1	NUM
ap-9037	423	4	2	2	NUM
ap-9037	423	5	+	+	CCONJ
ap-9037	423	6	1	1	NUM
ap-9037	423	7	2	2	NUM
ap-9037	423	8	√	√	NUM
ap-9037	423	9	1	1	NUM
ap-9037	423	10	+	+	NUM
ap-9037	423	11	4γ	4γ	NOUN
ap-9037	423	12	+	+	CCONJ
ap-9037	423	13	4ν1(ν1	4ν1(ν1	NUM
ap-9037	423	14	−	−	PROPN
ap-9037	423	15	r1	r1	PROPN
ap-9037	423	16	)	)	PUNCT
ap-9037	423	17	=	=	SYM
ap-9037	423	18	2k1	2k1	NUM
ap-9037	424	1	+	+	NUM
ap-9037	424	2	1	1	NUM
ap-9037	424	3	,	,	PUNCT
ap-9037	424	4	(	(	PUNCT
ap-9037	424	5	83	83	NUM
ap-9037	424	6	)	)	PUNCT
ap-9037	425	1	−ν1	−ν1	PROPN
ap-9037	425	2	+	+	CCONJ
ap-9037	425	3	1	1	NUM
ap-9037	425	4	2	2	NUM
ap-9037	425	5	+	+	CCONJ
ap-9037	425	6	1	1	NUM
ap-9037	425	7	2	2	NUM
ap-9037	425	8	√	√	NUM
ap-9037	425	9	1	1	NUM
ap-9037	425	10	+	+	NUM
ap-9037	425	11	4γ	4γ	NOUN
ap-9037	425	12	+	+	CCONJ
ap-9037	425	13	4ν1(ν1	4ν1(ν1	NUM
ap-9037	425	14	−	−	PROPN
ap-9037	425	15	r1	r1	PROPN
ap-9037	425	16	)	)	PUNCT
ap-9037	425	17	=	=	SYM
ap-9037	425	18	2k1	2k1	NUM
ap-9037	425	19	,	,	PUNCT
ap-9037	425	20	(	(	PUNCT
ap-9037	425	21	84	84	NUM
ap-9037	425	22	)	)	PUNCT
ap-9037	425	23	286	286	NUM
ap-9037	425	24	vol	vol	NOUN
ap-9037	425	25	.	.	PUNCT
ap-9037	426	1	63	63	NUM
ap-9037	426	2	no	no	INTJ
ap-9037	426	3	.	.	PUNCT
ap-9037	427	1	4/2023	4/2023	NOUN
ap-9037	427	2	construction	construction	NOUN
ap-9037	427	3	of	of	ADP
ap-9037	427	4	angular	angular	ADJ
ap-9037	427	5	-	-	PUNCT
ap-9037	427	6	dependent	dependent	ADJ
ap-9037	427	7	potentials	potential	NOUN
ap-9037	427	8	where	where	SCONJ
ap-9037	427	9	k1	k1	NOUN
ap-9037	427	10	is	be	AUX
ap-9037	427	11	an	an	DET
ap-9037	427	12	integer	integer	NOUN
ap-9037	427	13	.	.	PUNCT
ap-9037	428	1	if	if	SCONJ
ap-9037	428	2	(	(	PUNCT
ap-9037	428	3	83	83	NUM
ap-9037	428	4	)	)	PUNCT
ap-9037	428	5	is	be	AUX
ap-9037	428	6	satisfied	satisfied	ADJ
ap-9037	428	7	,	,	PUNCT
ap-9037	428	8	then	then	ADV
ap-9037	428	9	r1	r1	PROPN
ap-9037	428	10	=	=	SYM
ap-9037	428	11	−1	−1	NOUN
ap-9037	428	12	,	,	PUNCT
ap-9037	428	13	meaning	mean	VERB
ap-9037	428	14	that	that	SCONJ
ap-9037	428	15	the	the	DET
ap-9037	428	16	solution	solution	NOUN
ap-9037	428	17	(	(	PUNCT
ap-9037	428	18	80	80	NUM
ap-9037	428	19	)	)	PUNCT
ap-9037	428	20	has	have	VERB
ap-9037	428	21	odd	odd	ADJ
ap-9037	428	22	parity	parity	NOUN
ap-9037	428	23	with	with	ADP
ap-9037	428	24	respect	respect	NOUN
ap-9037	428	25	to	to	ADP
ap-9037	428	26	the	the	DET
ap-9037	428	27	points	point	NOUN
ap-9037	428	28	ϕ	ϕ	X
ap-9037	428	29	=	=	X
ap-9037	428	30	π/2	π/2	NUM
ap-9037	428	31	and	and	CCONJ
ap-9037	428	32	ϕ	ϕ	X
ap-9037	428	33	=	=	SYM
ap-9037	428	34	3π/2	3π/2	NUM
ap-9037	428	35	.	.	PUNCT
ap-9037	429	1	in	in	ADP
ap-9037	429	2	the	the	DET
ap-9037	429	3	other	other	ADJ
ap-9037	429	4	case	case	NOUN
ap-9037	429	5	that	that	SCONJ
ap-9037	429	6	the	the	DET
ap-9037	429	7	parameters	parameter	NOUN
ap-9037	429	8	comply	comply	VERB
ap-9037	429	9	with	with	ADP
ap-9037	429	10	(	(	PUNCT
ap-9037	429	11	84	84	NUM
ap-9037	429	12	)	)	PUNCT
ap-9037	429	13	,	,	PUNCT
ap-9037	429	14	then	then	ADV
ap-9037	429	15	r1	r1	PROPN
ap-9037	429	16	=	=	SYM
ap-9037	429	17	1	1	NUM
ap-9037	429	18	,	,	PUNCT
ap-9037	429	19	so	so	SCONJ
ap-9037	429	20	that	that	SCONJ
ap-9037	429	21	(	(	PUNCT
ap-9037	429	22	80	80	NUM
ap-9037	429	23	)	)	PUNCT
ap-9037	429	24	has	have	AUX
ap-9037	429	25	even	even	ADV
ap-9037	429	26	parity	parity	VERB
ap-9037	429	27	with	with	ADP
ap-9037	429	28	respect	respect	NOUN
ap-9037	429	29	to	to	ADP
ap-9037	429	30	the	the	DET
ap-9037	429	31	points	point	NOUN
ap-9037	429	32	ϕ	ϕ	X
ap-9037	429	33	=	=	X
ap-9037	429	34	π/2	π/2	NUM
ap-9037	429	35	and	and	CCONJ
ap-9037	429	36	ϕ	ϕ	X
ap-9037	429	37	=	=	SYM
ap-9037	429	38	3π/2	3π/2	NUM
ap-9037	429	39	.	.	PUNCT
ap-9037	430	1	a	a	DET
ap-9037	430	2	similar	similar	ADJ
ap-9037	430	3	way	way	NOUN
ap-9037	430	4	of	of	ADP
ap-9037	430	5	reasoning	reasoning	NOUN
ap-9037	430	6	can	can	AUX
ap-9037	430	7	be	be	AUX
ap-9037	430	8	applied	apply	VERB
ap-9037	430	9	to	to	ADP
ap-9037	430	10	the	the	DET
ap-9037	430	11	second	second	ADJ
ap-9037	430	12	condition	condition	NOUN
ap-9037	430	13	(	(	PUNCT
ap-9037	430	14	82	82	NUM
ap-9037	430	15	)	)	PUNCT
ap-9037	430	16	.	.	PUNCT
ap-9037	431	1	we	we	PRON
ap-9037	431	2	have	have	VERB
ap-9037	431	3	:	:	PUNCT
ap-9037	431	4	−ν2	−ν2	NOUN
ap-9037	431	5	+	+	CCONJ
ap-9037	431	6	1	1	NUM
ap-9037	431	7	2	2	NUM
ap-9037	431	8	+	+	CCONJ
ap-9037	431	9	1	1	NUM
ap-9037	431	10	2	2	NUM
ap-9037	431	11	√	√	NUM
ap-9037	431	12	1	1	NUM
ap-9037	432	1	+	+	CCONJ
ap-9037	432	2	4ν2(ν2	4ν2(ν2	NUM
ap-9037	432	3	−	−	NOUN
ap-9037	432	4	r2	r2	NOUN
ap-9037	432	5	)	)	PUNCT
ap-9037	432	6	=	=	PUNCT
ap-9037	433	1	2k2	2k2	NUM
ap-9037	434	1	+	+	NUM
ap-9037	434	2	1	1	NUM
ap-9037	434	3	,	,	PUNCT
ap-9037	434	4	(	(	PUNCT
ap-9037	434	5	85	85	NUM
ap-9037	434	6	)	)	PUNCT
ap-9037	434	7	−ν2	−ν2	NOUN
ap-9037	434	8	+	+	CCONJ
ap-9037	434	9	1	1	NUM
ap-9037	434	10	2	2	NUM
ap-9037	434	11	+	+	CCONJ
ap-9037	434	12	1	1	NUM
ap-9037	434	13	2	2	NUM
ap-9037	434	14	√	√	NUM
ap-9037	434	15	1	1	NUM
ap-9037	435	1	+	+	CCONJ
ap-9037	435	2	4ν2(ν2	4ν2(ν2	NUM
ap-9037	435	3	−	−	NOUN
ap-9037	435	4	r2	r2	NOUN
ap-9037	435	5	)	)	PUNCT
ap-9037	436	1	=	=	PUNCT
ap-9037	436	2	2k2	2k2	NUM
ap-9037	436	3	,	,	PUNCT
ap-9037	436	4	(	(	PUNCT
ap-9037	436	5	86	86	NUM
ap-9037	436	6	)	)	PUNCT
ap-9037	436	7	for	for	ADP
ap-9037	436	8	an	an	DET
ap-9037	436	9	integer	integer	NOUN
ap-9037	436	10	k2	k2	NOUN
ap-9037	436	11	.	.	PUNCT
ap-9037	437	1	if	if	SCONJ
ap-9037	437	2	(	(	PUNCT
ap-9037	437	3	85	85	NUM
ap-9037	437	4	)	)	PUNCT
ap-9037	437	5	is	be	AUX
ap-9037	437	6	satisfied	satisfied	ADJ
ap-9037	437	7	,	,	PUNCT
ap-9037	437	8	then	then	ADV
ap-9037	437	9	r2	r2	PROPN
ap-9037	437	10	=	=	SYM
ap-9037	437	11	−1	−1	NOUN
ap-9037	437	12	,	,	PUNCT
ap-9037	437	13	so	so	SCONJ
ap-9037	437	14	that	that	SCONJ
ap-9037	437	15	(	(	PUNCT
ap-9037	437	16	80	80	NUM
ap-9037	437	17	)	)	PUNCT
ap-9037	437	18	is	be	AUX
ap-9037	437	19	odd	odd	ADJ
ap-9037	437	20	with	with	ADP
ap-9037	437	21	respect	respect	NOUN
ap-9037	437	22	to	to	ADP
ap-9037	437	23	the	the	DET
ap-9037	437	24	origin	origin	NOUN
ap-9037	437	25	.	.	PUNCT
ap-9037	438	1	otherwise	otherwise	ADV
ap-9037	438	2	,	,	PUNCT
ap-9037	438	3	if	if	SCONJ
ap-9037	438	4	(	(	PUNCT
ap-9037	438	5	86	86	NUM
ap-9037	438	6	)	)	PUNCT
ap-9037	438	7	is	be	AUX
ap-9037	438	8	true	true	ADJ
ap-9037	438	9	,	,	PUNCT
ap-9037	438	10	then	then	ADV
ap-9037	438	11	r2	r2	PROPN
ap-9037	438	12	=	=	SYM
ap-9037	438	13	1	1	NUM
ap-9037	438	14	,	,	PUNCT
ap-9037	438	15	meaning	mean	VERB
ap-9037	438	16	that	that	SCONJ
ap-9037	438	17	(	(	PUNCT
ap-9037	438	18	80	80	NUM
ap-9037	438	19	)	)	PUNCT
ap-9037	438	20	is	be	AUX
ap-9037	438	21	even	even	ADV
ap-9037	438	22	with	with	ADP
ap-9037	438	23	respect	respect	NOUN
ap-9037	438	24	to	to	ADP
ap-9037	438	25	the	the	DET
ap-9037	438	26	origin	origin	NOUN
ap-9037	438	27	.	.	PUNCT
ap-9037	439	1	let	let	VERB
ap-9037	439	2	us	we	PRON
ap-9037	439	3	show	show	VERB
ap-9037	439	4	examples	example	NOUN
ap-9037	439	5	of	of	ADP
ap-9037	439	6	our	our	PRON
ap-9037	439	7	solutions	solution	NOUN
ap-9037	439	8	(	(	PUNCT
ap-9037	439	9	80	80	NUM
ap-9037	439	10	)	)	PUNCT
ap-9037	439	11	for	for	ADP
ap-9037	439	12	the	the	DET
ap-9037	439	13	following	follow	VERB
ap-9037	439	14	parameter	parameter	NOUN
ap-9037	439	15	setting	setting	NOUN
ap-9037	439	16	:	:	PUNCT
ap-9037	439	17	ν1	ν1	NOUN
ap-9037	439	18	=	=	SYM
ap-9037	439	19	2	2	NUM
ap-9037	439	20	ν2	ν2	NOUN
ap-9037	439	21	=	=	SYM
ap-9037	439	22	−1	−1	NOUN
ap-9037	439	23	2	2	NUM
ap-9037	439	24	ν3	ν3	NOUN
ap-9037	439	25	=	=	SYM
ap-9037	439	26	2	2	NUM
ap-9037	439	27	γ	γ	X
ap-9037	439	28	=	=	SYM
ap-9037	439	29	14	14	NUM
ap-9037	439	30	.	.	PUNCT
ap-9037	440	1	(	(	PUNCT
ap-9037	440	2	87	87	NUM
ap-9037	440	3	)	)	PUNCT
ap-9037	440	4	figures	figure	NOUN
ap-9037	440	5	6	6	NUM
ap-9037	440	6	and	and	CCONJ
ap-9037	440	7	7	7	NUM
ap-9037	440	8	show	show	NOUN
ap-9037	440	9	graphs	graph	NOUN
ap-9037	440	10	of	of	ADP
ap-9037	440	11	the	the	DET
ap-9037	440	12	functions	function	NOUN
ap-9037	440	13	(	(	PUNCT
ap-9037	440	14	80	80	NUM
ap-9037	440	15	)	)	PUNCT
ap-9037	440	16	for	for	ADP
ap-9037	440	17	the	the	DET
ap-9037	440	18	parameter	parameter	NOUN
ap-9037	440	19	setting	setting	NOUN
ap-9037	440	20	(	(	PUNCT
ap-9037	440	21	87	87	NUM
ap-9037	440	22	)	)	PUNCT
ap-9037	440	23	,	,	PUNCT
ap-9037	440	24	and	and	CCONJ
ap-9037	440	25	different	different	ADJ
ap-9037	440	26	choices	choice	NOUN
ap-9037	440	27	for	for	ADP
ap-9037	440	28	the	the	DET
ap-9037	440	29	parity	parity	NOUN
ap-9037	440	30	parameters	parameter	NOUN
ap-9037	440	31	r1	r1	NOUN
ap-9037	440	32	and	and	CCONJ
ap-9037	440	33	r2	r2	PROPN
ap-9037	440	34	.	.	PUNCT
ap-9037	441	1	1	1	NUM
ap-9037	441	2	2	2	NUM
ap-9037	441	3	3	3	NUM
ap-9037	441	4	4	4	NUM
ap-9037	441	5	5	5	NUM
ap-9037	441	6	6	6	NUM
ap-9037	441	7	φ	φ	PROPN
ap-9037	441	8	-0.4	-0.4	SYM
ap-9037	441	9	-0.2	-0.2	PROPN
ap-9037	441	10	0.2	0.2	NUM
ap-9037	441	11	0.4	0.4	NUM
ap-9037	441	12	fhφl	fhφl	ADJ
ap-9037	441	13	n	n	NOUN
ap-9037	441	14	=	=	SYM
ap-9037	441	15	2	2	NUM
ap-9037	441	16	n	n	NOUN
ap-9037	441	17	=	=	SYM
ap-9037	441	18	1	1	NUM
ap-9037	441	19	n	n	NOUN
ap-9037	441	20	=	=	SYM
ap-9037	441	21	0	0	NUM
ap-9037	441	22	figure	figure	NOUN
ap-9037	441	23	6	6	NUM
ap-9037	441	24	.	.	PUNCT
ap-9037	442	1	graphs	graph	NOUN
ap-9037	442	2	of	of	ADP
ap-9037	442	3	the	the	DET
ap-9037	442	4	solution	solution	NOUN
ap-9037	442	5	(	(	PUNCT
ap-9037	442	6	80	80	NUM
ap-9037	442	7	)	)	PUNCT
ap-9037	442	8	for	for	ADP
ap-9037	442	9	the	the	DET
ap-9037	442	10	parameter	parameter	NOUN
ap-9037	442	11	settings	setting	NOUN
ap-9037	442	12	(	(	PUNCT
ap-9037	442	13	87	87	NUM
ap-9037	442	14	)	)	PUNCT
ap-9037	442	15	and	and	CCONJ
ap-9037	442	16	r1	r1	PROPN
ap-9037	442	17	=	=	SYM
ap-9037	442	18	−1	−1	NOUN
ap-9037	442	19	,	,	PUNCT
ap-9037	442	20	r2	r2	PROPN
ap-9037	442	21	=	=	NOUN
ap-9037	442	22	1	1	NUM
ap-9037	442	23	for	for	ADP
ap-9037	442	24	different	different	ADJ
ap-9037	442	25	values	value	NOUN
ap-9037	442	26	of	of	ADP
ap-9037	442	27	n	n	PROPN
ap-9037	442	28	.	.	PUNCT
ap-9037	443	1	1	1	NUM
ap-9037	443	2	2	2	NUM
ap-9037	443	3	3	3	NUM
ap-9037	443	4	4	4	NUM
ap-9037	443	5	5	5	NUM
ap-9037	443	6	6	6	NUM
ap-9037	443	7	φ	φ	PROPN
ap-9037	443	8	-0.4	-0.4	SYM
ap-9037	443	9	-0.2	-0.2	PROPN
ap-9037	443	10	0.2	0.2	NUM
ap-9037	443	11	0.4	0.4	NUM
ap-9037	443	12	fhφl	fhφl	ADJ
ap-9037	443	13	n	n	NOUN
ap-9037	443	14	=	=	SYM
ap-9037	443	15	2	2	NUM
ap-9037	443	16	n	n	NOUN
ap-9037	443	17	=	=	SYM
ap-9037	443	18	1	1	NUM
ap-9037	443	19	n	n	NOUN
ap-9037	443	20	=	=	SYM
ap-9037	443	21	0	0	NUM
ap-9037	443	22	figure	figure	NOUN
ap-9037	443	23	7	7	NUM
ap-9037	443	24	.	.	PUNCT
ap-9037	443	25	graphs	graph	NOUN
ap-9037	443	26	of	of	ADP
ap-9037	443	27	the	the	DET
ap-9037	443	28	solution	solution	NOUN
ap-9037	443	29	(	(	PUNCT
ap-9037	443	30	80	80	NUM
ap-9037	443	31	)	)	PUNCT
ap-9037	443	32	for	for	ADP
ap-9037	443	33	the	the	DET
ap-9037	443	34	parameter	parameter	NOUN
ap-9037	443	35	settings	setting	NOUN
ap-9037	443	36	(	(	PUNCT
ap-9037	443	37	87	87	NUM
ap-9037	443	38	)	)	PUNCT
ap-9037	443	39	and	and	CCONJ
ap-9037	443	40	r1	r1	PROPN
ap-9037	443	41	=	=	SYM
ap-9037	443	42	r2	r2	PROPN
ap-9037	443	43	=	=	SYM
ap-9037	443	44	1	1	NUM
ap-9037	443	45	for	for	ADP
ap-9037	443	46	different	different	ADJ
ap-9037	443	47	values	value	NOUN
ap-9037	443	48	of	of	ADP
ap-9037	443	49	n	n	PROPN
ap-9037	443	50	.	.	PUNCT
ap-9037	444	1	it	it	PRON
ap-9037	444	2	is	be	AUX
ap-9037	444	3	important	important	ADJ
ap-9037	444	4	to	to	PART
ap-9037	444	5	point	point	VERB
ap-9037	444	6	out	out	ADP
ap-9037	444	7	that	that	SCONJ
ap-9037	444	8	the	the	DET
ap-9037	444	9	parameter	parameter	NOUN
ap-9037	444	10	settings	setting	NOUN
ap-9037	444	11	used	use	VERB
ap-9037	444	12	in	in	ADP
ap-9037	444	13	figures	figure	NOUN
ap-9037	444	14	6	6	NUM
ap-9037	444	15	and	and	CCONJ
ap-9037	444	16	7	7	NUM
ap-9037	444	17	,	,	PUNCT
ap-9037	444	18	are	be	AUX
ap-9037	444	19	compatible	compatible	ADJ
ap-9037	444	20	with	with	ADP
ap-9037	444	21	the	the	DET
ap-9037	444	22	conditions	condition	NOUN
ap-9037	444	23	(	(	PUNCT
ap-9037	444	24	81	81	NUM
ap-9037	444	25	)	)	PUNCT
ap-9037	444	26	and	and	CCONJ
ap-9037	444	27	(	(	PUNCT
ap-9037	444	28	82	82	NUM
ap-9037	444	29	)	)	PUNCT
ap-9037	444	30	.	.	PUNCT
ap-9037	445	1	5.2	5.2	NUM
ap-9037	445	2	.	.	PUNCT
ap-9037	446	1	the	the	DET
ap-9037	446	2	darboux	darboux	VERB
ap-9037	446	3	-	-	PUNCT
ap-9037	446	4	crum	crum	NOUN
ap-9037	446	5	transformation	transformation	NOUN
ap-9037	446	6	we	we	PRON
ap-9037	446	7	will	will	AUX
ap-9037	446	8	follow	follow	VERB
ap-9037	446	9	an	an	DET
ap-9037	446	10	analogous	analogous	ADJ
ap-9037	446	11	approach	approach	NOUN
ap-9037	446	12	as	as	ADP
ap-9037	446	13	in	in	ADP
ap-9037	446	14	section	section	NOUN
ap-9037	446	15	4.1	4.1	NUM
ap-9037	446	16	.	.	PUNCT
ap-9037	447	1	there	there	PRON
ap-9037	447	2	is	be	VERB
ap-9037	447	3	only	only	ADV
ap-9037	447	4	one	one	NUM
ap-9037	447	5	separation	separation	NOUN
ap-9037	447	6	constant	constant	ADJ
ap-9037	447	7	m2	m2	PROPN
ap-9037	447	8	that	that	PRON
ap-9037	447	9	appears	appear	VERB
ap-9037	447	10	in	in	ADP
ap-9037	447	11	our	our	PRON
ap-9037	447	12	equation	equation	NOUN
ap-9037	447	13	.	.	PUNCT
ap-9037	448	1	this	this	DET
ap-9037	448	2	separation	separation	NOUN
ap-9037	448	3	constant	constant	ADJ
ap-9037	448	4	will	will	AUX
ap-9037	448	5	act	act	VERB
ap-9037	448	6	as	as	ADP
ap-9037	448	7	transformation	transformation	NOUN
ap-9037	448	8	parameter	parameter	NOUN
ap-9037	448	9	in	in	ADP
ap-9037	448	10	the	the	DET
ap-9037	448	11	darboux	darboux	ADJ
ap-9037	448	12	-	-	PUNCT
ap-9037	448	13	crum	crum	NOUN
ap-9037	448	14	transformation	transformation	NOUN
ap-9037	448	15	.	.	PUNCT
ap-9037	449	1	as	as	ADP
ap-9037	449	2	a	a	DET
ap-9037	449	3	first	first	ADJ
ap-9037	449	4	step	step	NOUN
ap-9037	449	5	we	we	PRON
ap-9037	449	6	see	see	VERB
ap-9037	449	7	that	that	SCONJ
ap-9037	449	8	(	(	PUNCT
ap-9037	449	9	73	73	NUM
ap-9037	449	10	)	)	PUNCT
ap-9037	449	11	matches	match	NOUN
ap-9037	449	12	(	(	PUNCT
ap-9037	449	13	4	4	NUM
ap-9037	449	14	)	)	PUNCT
ap-9037	449	15	if	if	SCONJ
ap-9037	449	16	the	the	DET
ap-9037	449	17	following	follow	VERB
ap-9037	449	18	settings	setting	NOUN
ap-9037	449	19	are	be	AUX
ap-9037	449	20	made	make	VERB
ap-9037	449	21	:	:	PUNCT
ap-9037	449	22	e	e	X
ap-9037	449	23	=	=	PROPN
ap-9037	449	24	m2	m2	PROPN
ap-9037	449	25	,	,	PUNCT
ap-9037	449	26	u(ϕ	u(ϕ	PROPN
ap-9037	449	27	)	)	PUNCT
ap-9037	449	28	=	=	NOUN
ap-9037	449	29	−(ν1	−(ν1	ADP
ap-9037	449	30	+	+	X
ap-9037	449	31	ν2)2	ν2)2	PUNCT
ap-9037	449	32	−	−	PROPN
ap-9037	449	33	(	(	PUNCT
ap-9037	449	34	r1	r1	NOUN
ap-9037	449	35	−	−	NOUN
ap-9037	449	36	ν1)ν1	ν1)ν1	NOUN
ap-9037	449	37	cos(ϕ)2	cos(ϕ)2	NOUN
ap-9037	449	38	−	−	PROPN
ap-9037	449	39	(	(	PUNCT
ap-9037	449	40	r2	r2	PROPN
ap-9037	449	41	−	−	PROPN
ap-9037	449	42	ν2)ν2	ν2)ν2	NOUN
ap-9037	449	43	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	449	44	+	+	CCONJ
ap-9037	449	45	vϕ(ϕ	vϕ(ϕ	NUM
ap-9037	449	46	)	)	PUNCT
ap-9037	449	47	.	.	PUNCT
ap-9037	450	1	(	(	PUNCT
ap-9037	450	2	88	88	NUM
ap-9037	450	3	)	)	SYM
ap-9037	450	4	287	287	NUM
ap-9037	450	5	axel	axel	PROPN
ap-9037	450	6	schulze	schulze	NOUN
ap-9037	450	7	-	-	PUNCT
ap-9037	450	8	halberg	halberg	PROPN
ap-9037	450	9	acta	acta	PROPN
ap-9037	450	10	polytechnica	polytechnica	PROPN
ap-9037	450	11	since	since	SCONJ
ap-9037	450	12	these	these	DET
ap-9037	450	13	settings	setting	NOUN
ap-9037	450	14	make	make	VERB
ap-9037	450	15	our	our	PRON
ap-9037	450	16	equations	equation	NOUN
ap-9037	450	17	(	(	PUNCT
ap-9037	450	18	4	4	NUM
ap-9037	450	19	)	)	PUNCT
ap-9037	450	20	and	and	CCONJ
ap-9037	450	21	(	(	PUNCT
ap-9037	450	22	73	73	NUM
ap-9037	450	23	)	)	PUNCT
ap-9037	450	24	match	match	NOUN
ap-9037	450	25	,	,	PUNCT
ap-9037	450	26	our	our	PRON
ap-9037	450	27	darboux	darboux	VERB
ap-9037	450	28	-	-	PUNCT
ap-9037	450	29	crum	crum	NOUN
ap-9037	450	30	transformation	transformation	NOUN
ap-9037	450	31	becomes	become	VERB
ap-9037	450	32	applicable	applicable	ADJ
ap-9037	450	33	.	.	PUNCT
ap-9037	451	1	before	before	SCONJ
ap-9037	451	2	we	we	PRON
ap-9037	451	3	construct	construct	VERB
ap-9037	451	4	this	this	DET
ap-9037	451	5	transformation	transformation	NOUN
ap-9037	451	6	,	,	PUNCT
ap-9037	451	7	let	let	VERB
ap-9037	451	8	us	we	PRON
ap-9037	451	9	observe	observe	VERB
ap-9037	451	10	that	that	SCONJ
ap-9037	451	11	(	(	PUNCT
ap-9037	451	12	73	73	NUM
ap-9037	451	13	)	)	PUNCT
ap-9037	451	14	can	can	AUX
ap-9037	451	15	be	be	AUX
ap-9037	451	16	interpreted	interpret	VERB
ap-9037	451	17	as	as	ADP
ap-9037	451	18	a	a	DET
ap-9037	451	19	schrödinger	schrödinger	ADJ
ap-9037	451	20	equation	equation	NOUN
ap-9037	451	21	for	for	ADP
ap-9037	451	22	an	an	DET
ap-9037	451	23	extended	extended	ADJ
ap-9037	451	24	trigonometric	trigonometric	ADJ
ap-9037	451	25	pöschl	pöschl	NOUN
ap-9037	451	26	-	-	PUNCT
ap-9037	451	27	teller	teller	NOUN
ap-9037	451	28	potential	potential	NOUN
ap-9037	451	29	,	,	PUNCT
ap-9037	451	30	similar	similar	ADJ
ap-9037	451	31	to	to	ADP
ap-9037	451	32	the	the	DET
ap-9037	451	33	rewritten	rewrite	VERB
ap-9037	451	34	polar	polar	ADJ
ap-9037	451	35	equations	equation	NOUN
ap-9037	451	36	(	(	PUNCT
ap-9037	451	37	30	30	NUM
ap-9037	451	38	)	)	PUNCT
ap-9037	451	39	and	and	CCONJ
ap-9037	451	40	(	(	PUNCT
ap-9037	451	41	63	63	NUM
ap-9037	451	42	)	)	PUNCT
ap-9037	451	43	.	.	PUNCT
ap-9037	452	1	for	for	ADP
ap-9037	452	2	our	our	PRON
ap-9037	452	3	darboux	darboux	ADJ
ap-9037	452	4	-	-	PUNCT
ap-9037	452	5	crum	crum	NOUN
ap-9037	452	6	transformation	transformation	NOUN
ap-9037	452	7	,	,	PUNCT
ap-9037	452	8	we	we	PRON
ap-9037	452	9	require	require	VERB
ap-9037	452	10	transformation	transformation	NOUN
ap-9037	452	11	functions	function	NOUN
ap-9037	452	12	ξ1	ξ1	NOUN
ap-9037	452	13	,	,	PUNCT
ap-9037	452	14	ξ2	ξ2	NOUN
ap-9037	452	15	,	,	PUNCT
ap-9037	452	16	.	.	PUNCT
ap-9037	452	17	.	.	PUNCT
ap-9037	453	1	.	.	PUNCT
ap-9037	454	1	,	,	PUNCT
ap-9037	454	2	ξn	ξn	INTJ
ap-9037	454	3	that	that	PRON
ap-9037	454	4	are	be	AUX
ap-9037	454	5	solutions	solution	NOUN
ap-9037	454	6	to	to	ADP
ap-9037	454	7	the	the	DET
ap-9037	454	8	following	follow	VERB
ap-9037	454	9	version	version	NOUN
ap-9037	454	10	of	of	ADP
ap-9037	454	11	equation	equation	NOUN
ap-9037	454	12	(	(	PUNCT
ap-9037	454	13	73	73	NUM
ap-9037	454	14	):	):	PUNCT
ap-9037	454	15	ξ′′	ξ′′	PROPN
ap-9037	454	16	j	j	PROPN
ap-9037	454	17	(	(	PUNCT
ap-9037	454	18	ϕ	ϕ	NOUN
ap-9037	454	19	)	)	PUNCT
ap-9037	454	20	+	+	CCONJ
ap-9037	454	21	[	[	PUNCT
ap-9037	454	22	m2	m2	PROPN
ap-9037	454	23	j	j	PROPN
ap-9037	454	24	+	+	CCONJ
ap-9037	454	25	(	(	PUNCT
ap-9037	454	26	ν1	ν1	NOUN
ap-9037	454	27	+	+	CCONJ
ap-9037	454	28	ν2)2	ν2)2	PUNCT
ap-9037	454	29	+	+	CCONJ
ap-9037	454	30	(	(	PUNCT
ap-9037	454	31	r1	r1	NOUN
ap-9037	454	32	−	−	PROPN
ap-9037	454	33	ν1)ν1	ν1)ν1	NOUN
ap-9037	454	34	cos(ϕ)2	cos(ϕ)2	NOUN
ap-9037	454	35	+	+	CCONJ
ap-9037	454	36	(	(	PUNCT
ap-9037	454	37	r2	r2	PROPN
ap-9037	454	38	−	−	PROPN
ap-9037	454	39	ν2)ν2	ν2)ν2	NOUN
ap-9037	454	40	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	454	41	−	−	PROPN
ap-9037	454	42	vϕ(ϕ	vϕ(ϕ	NOUN
ap-9037	454	43	)	)	PUNCT
ap-9037	454	44	]	]	PUNCT
ap-9037	455	1	ξj(ϕ	ξj(ϕ	X
ap-9037	455	2	)	)	PUNCT
ap-9037	455	3	=	=	SYM
ap-9037	455	4	0	0	NUM
ap-9037	455	5	,	,	PUNCT
ap-9037	455	6	j	j	PROPN
ap-9037	455	7	=	=	SYM
ap-9037	455	8	1	1	NUM
ap-9037	455	9	,	,	PUNCT
ap-9037	455	10	.	.	PUNCT
ap-9037	455	11	.	.	PUNCT
ap-9037	456	1	.	.	PUNCT
ap-9037	457	1	,	,	PUNCT
ap-9037	457	2	n	n	X
ap-9037	457	3	.	.	PUNCT
ap-9037	458	1	(	(	PUNCT
ap-9037	458	2	89	89	NUM
ap-9037	458	3	)	)	PUNCT
ap-9037	458	4	here	here	ADV
ap-9037	458	5	,	,	PUNCT
ap-9037	458	6	the	the	DET
ap-9037	458	7	constants	constant	NOUN
ap-9037	458	8	m2	m2	PROPN
ap-9037	458	9	1	1	NUM
ap-9037	458	10	,	,	PUNCT
ap-9037	458	11	.	.	PUNCT
ap-9037	458	12	.	.	PUNCT
ap-9037	459	1	.	.	PUNCT
ap-9037	460	1	,	,	PUNCT
ap-9037	460	2	m2	m2	PROPN
ap-9037	460	3	n	n	PROPN
ap-9037	460	4	are	be	AUX
ap-9037	460	5	the	the	DET
ap-9037	460	6	transformation	transformation	NOUN
ap-9037	460	7	parameters	parameter	NOUN
ap-9037	460	8	,	,	PUNCT
ap-9037	460	9	and	and	CCONJ
ap-9037	460	10	the	the	DET
ap-9037	460	11	potential	potential	ADJ
ap-9037	460	12	vϕ	vϕ	NOUN
ap-9037	460	13	is	be	AUX
ap-9037	460	14	given	give	VERB
ap-9037	460	15	by	by	ADP
ap-9037	460	16	(	(	PUNCT
ap-9037	460	17	74	74	NUM
ap-9037	460	18	)	)	PUNCT
ap-9037	460	19	.	.	PUNCT
ap-9037	461	1	evaluation	evaluation	NOUN
ap-9037	461	2	of	of	ADP
ap-9037	461	3	(	(	PUNCT
ap-9037	461	4	6	6	NUM
ap-9037	461	5	)	)	PUNCT
ap-9037	461	6	and	and	CCONJ
ap-9037	461	7	(	(	PUNCT
ap-9037	461	8	7	7	X
ap-9037	461	9	)	)	PUNCT
ap-9037	461	10	then	then	ADV
ap-9037	461	11	gives	give	VERB
ap-9037	461	12	:	:	PUNCT
ap-9037	461	13	ξ̂(ϕ	ξ̂(ϕ	SYM
ap-9037	461	14	)	)	PUNCT
ap-9037	461	15	=	=	SYM
ap-9037	461	16	wξ1,ξ2,	wξ1,ξ2,	NOUN
ap-9037	461	17	...	...	PUNCT
ap-9037	461	18	,ξn	,ξn	ADJ
ap-9037	461	19	,	,	PUNCT
ap-9037	461	20	ξ(ϕ	ξ(ϕ	PROPN
ap-9037	461	21	)	)	PUNCT
ap-9037	461	22	wξ1,ξ2,	wξ1,ξ2,	NOUN
ap-9037	461	23	...	...	PUNCT
ap-9037	461	24	,ξn(ϕ	,ξn(ϕ	PUNCT
ap-9037	461	25	)	)	PUNCT
ap-9037	461	26	,	,	PUNCT
ap-9037	461	27	(	(	PUNCT
ap-9037	461	28	90	90	NUM
ap-9037	461	29	)	)	PUNCT
ap-9037	461	30	v̂ϕ(ϕ	v̂ϕ(ϕ	PROPN
ap-9037	461	31	)	)	PUNCT
ap-9037	461	32	=	=	SYM
ap-9037	461	33	vϕ(ϕ	vϕ(ϕ	X
ap-9037	461	34	)	)	PUNCT
ap-9037	461	35	−	−	PROPN
ap-9037	461	36	2	2	NUM
ap-9037	461	37	d2	d2	PROPN
ap-9037	461	38	dϕ2	dϕ2	PROPN
ap-9037	461	39	log	log	NOUN
ap-9037	462	1	[	[	X
ap-9037	462	2	wξ1,ξ2,	wξ1,ξ2,	NOUN
ap-9037	462	3	...	...	PUNCT
ap-9037	462	4	,ξn(ϕ	,ξn(ϕ	PUNCT
ap-9037	462	5	)	)	PUNCT
ap-9037	462	6	]	]	PUNCT
ap-9037	462	7	.	.	PUNCT
ap-9037	463	1	(	(	PUNCT
ap-9037	463	2	91	91	NUM
ap-9037	463	3	)	)	PUNCT
ap-9037	463	4	for	for	ADP
ap-9037	463	5	the	the	DET
ap-9037	463	6	sake	sake	NOUN
ap-9037	463	7	of	of	ADP
ap-9037	463	8	clarity	clarity	NOUN
ap-9037	463	9	,	,	PUNCT
ap-9037	463	10	let	let	VERB
ap-9037	463	11	us	we	PRON
ap-9037	463	12	point	point	VERB
ap-9037	463	13	out	out	ADP
ap-9037	463	14	that	that	SCONJ
ap-9037	463	15	the	the	DET
ap-9037	463	16	function	function	NOUN
ap-9037	463	17	ξ̂	ξ̂	NUM
ap-9037	463	18	is	be	AUX
ap-9037	463	19	dependent	dependent	ADJ
ap-9037	463	20	on	on	ADP
ap-9037	463	21	the	the	DET
ap-9037	463	22	transformation	transformation	NOUN
ap-9037	463	23	parameters	parameter	NOUN
ap-9037	463	24	m2	m2	PROPN
ap-9037	463	25	1	1	NUM
ap-9037	463	26	,	,	PUNCT
ap-9037	463	27	.	.	PUNCT
ap-9037	463	28	.	.	PUNCT
ap-9037	464	1	.	.	PUNCT
ap-9037	465	1	,	,	PUNCT
ap-9037	465	2	m2	m2	PROPN
ap-9037	465	3	n	n	CCONJ
ap-9037	465	4	,	,	PUNCT
ap-9037	465	5	as	as	ADV
ap-9037	465	6	well	well	ADV
ap-9037	465	7	as	as	ADP
ap-9037	465	8	on	on	ADP
ap-9037	465	9	the	the	DET
ap-9037	465	10	separation	separation	NOUN
ap-9037	465	11	constant	constant	ADJ
ap-9037	465	12	m2	m2	PROPN
ap-9037	465	13	from	from	ADP
ap-9037	465	14	(	(	PUNCT
ap-9037	465	15	73	73	NUM
ap-9037	465	16	)	)	PUNCT
ap-9037	465	17	.	.	PUNCT
ap-9037	466	1	the	the	DET
ap-9037	466	2	potential	potential	ADJ
ap-9037	466	3	v̂ϕ	v̂ϕ	NOUN
ap-9037	466	4	is	be	AUX
ap-9037	466	5	dependent	dependent	ADJ
ap-9037	466	6	on	on	ADP
ap-9037	466	7	the	the	DET
ap-9037	466	8	transformation	transformation	NOUN
ap-9037	466	9	parameters	parameter	NOUN
ap-9037	466	10	m2	m2	PROPN
ap-9037	466	11	1	1	NUM
ap-9037	466	12	,	,	PUNCT
ap-9037	466	13	.	.	PUNCT
ap-9037	466	14	.	.	PUNCT
ap-9037	467	1	.	.	PUNCT
ap-9037	468	1	,	,	PUNCT
ap-9037	468	2	m2	m2	PROPN
ap-9037	468	3	n	n	CCONJ
ap-9037	468	4	,	,	PUNCT
ap-9037	468	5	but	but	CCONJ
ap-9037	468	6	not	not	PART
ap-9037	468	7	on	on	ADP
ap-9037	468	8	m2	m2	PROPN
ap-9037	468	9	.	.	PUNCT
ap-9037	469	1	now	now	ADV
ap-9037	469	2	,	,	PUNCT
ap-9037	469	3	after	after	ADP
ap-9037	469	4	applying	apply	VERB
ap-9037	469	5	the	the	DET
ap-9037	469	6	darboux	darboux	ADJ
ap-9037	469	7	-	-	PUNCT
ap-9037	469	8	crum	crum	NOUN
ap-9037	469	9	transformation	transformation	NOUN
ap-9037	469	10	,	,	PUNCT
ap-9037	469	11	these	these	DET
ap-9037	469	12	two	two	NUM
ap-9037	469	13	functions	function	NOUN
ap-9037	469	14	are	be	AUX
ap-9037	469	15	inserted	insert	VERB
ap-9037	469	16	into	into	ADP
ap-9037	469	17	the	the	DET
ap-9037	469	18	resulting	result	VERB
ap-9037	469	19	equation	equation	NOUN
ap-9037	469	20	as	as	SCONJ
ap-9037	469	21	follows	follow	VERB
ap-9037	469	22	:	:	PUNCT
ap-9037	469	23	ξ̂′′(ϕ	ξ̂′′(ϕ	NOUN
ap-9037	469	24	)	)	PUNCT
ap-9037	470	1	+	+	CCONJ
ap-9037	470	2	[	[	PUNCT
ap-9037	470	3	m2	m2	PROPN
ap-9037	470	4	+	+	CCONJ
ap-9037	470	5	(	(	PUNCT
ap-9037	470	6	ν1	ν1	NOUN
ap-9037	470	7	+	+	CCONJ
ap-9037	470	8	ν2)2	ν2)2	PUNCT
ap-9037	470	9	+	+	CCONJ
ap-9037	470	10	(	(	PUNCT
ap-9037	470	11	r1	r1	NOUN
ap-9037	470	12	−	−	PROPN
ap-9037	470	13	ν1)ν1	ν1)ν1	NOUN
ap-9037	470	14	cos(ϕ)2	cos(ϕ)2	NOUN
ap-9037	470	15	+	+	CCONJ
ap-9037	470	16	(	(	PUNCT
ap-9037	470	17	r2	r2	PROPN
ap-9037	470	18	−	−	PROPN
ap-9037	470	19	ν2)ν2	ν2)ν2	NOUN
ap-9037	470	20	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	470	21	−	−	PROPN
ap-9037	470	22	v̂ϕ(ϕ	v̂ϕ(ϕ	PROPN
ap-9037	470	23	)	)	PUNCT
ap-9037	470	24	]	]	PUNCT
ap-9037	471	1	ξ̂(ϕ	ξ̂(ϕ	X
ap-9037	471	2	)	)	PUNCT
ap-9037	471	3	=	=	SYM
ap-9037	471	4	0	0	X
ap-9037	471	5	.	.	PUNCT
ap-9037	472	1	we	we	PRON
ap-9037	472	2	now	now	ADV
ap-9037	472	3	need	need	VERB
ap-9037	472	4	to	to	PART
ap-9037	472	5	convert	convert	VERB
ap-9037	472	6	this	this	DET
ap-9037	472	7	equation	equation	NOUN
ap-9037	472	8	into	into	ADP
ap-9037	472	9	a	a	DET
ap-9037	472	10	transformed	transform	VERB
ap-9037	472	11	version	version	NOUN
ap-9037	472	12	of	of	ADP
ap-9037	472	13	(	(	PUNCT
ap-9037	472	14	28	28	NUM
ap-9037	472	15	)	)	PUNCT
ap-9037	472	16	that	that	PRON
ap-9037	472	17	reads	read	VERB
ap-9037	472	18	:	:	PUNCT
ap-9037	472	19	φ̂′′(ϕ	φ̂′′(ϕ	VERB
ap-9037	472	20	)	)	PUNCT
ap-9037	472	21	−	−	PROPN
ap-9037	472	22	2	2	NUM
ap-9037	472	23	[	[	PUNCT
ap-9037	472	24	ν1	ν1	NOUN
ap-9037	472	25	tan(ϕ	tan(ϕ	NOUN
ap-9037	472	26	)	)	PUNCT
ap-9037	472	27	−	−	PROPN
ap-9037	472	28	ν2	ν2	PROPN
ap-9037	472	29	cot(ϕ	cot(ϕ	PROPN
ap-9037	472	30	)	)	PUNCT
ap-9037	472	31	]	]	PUNCT
ap-9037	473	1	φ′(ϕ	φ′(ϕ	PROPN
ap-9037	473	2	)	)	PUNCT
ap-9037	473	3	+	+	CCONJ
ap-9037	473	4	[	[	PUNCT
ap-9037	473	5	m2	m2	PROPN
ap-9037	473	6	−	−	PROPN
ap-9037	473	7	ν1(1	ν1(1	PROPN
ap-9037	473	8	−	−	PROPN
ap-9037	473	9	r1	r1	PROPN
ap-9037	473	10	)	)	PUNCT
ap-9037	473	11	cos(ϕ)2	cos(ϕ)2	NOUN
ap-9037	473	12	−	−	PROPN
ap-9037	473	13	ν2(1	ν2(1	NOUN
ap-9037	473	14	−	−	NOUN
ap-9037	473	15	r2	r2	NOUN
ap-9037	473	16	)	)	PUNCT
ap-9037	473	17	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	473	18	−	−	PROPN
ap-9037	473	19	v̂ϕ(ϕ	v̂ϕ(ϕ	PROPN
ap-9037	473	20	)	)	PUNCT
ap-9037	473	21	]	]	PUNCT
ap-9037	474	1	φ̂(ϕ	φ̂(ϕ	X
ap-9037	474	2	)	)	PUNCT
ap-9037	474	3	=	=	SYM
ap-9037	474	4	0	0	X
ap-9037	474	5	.	.	PUNCT
ap-9037	475	1	(	(	PUNCT
ap-9037	475	2	92	92	NUM
ap-9037	475	3	)	)	PUNCT
ap-9037	475	4	the	the	DET
ap-9037	475	5	solution	solution	NOUN
ap-9037	475	6	of	of	ADP
ap-9037	475	7	this	this	DET
ap-9037	475	8	equation	equation	NOUN
ap-9037	475	9	can	can	AUX
ap-9037	475	10	be	be	AUX
ap-9037	475	11	found	find	VERB
ap-9037	475	12	from	from	ADP
ap-9037	475	13	(	(	PUNCT
ap-9037	475	14	90	90	NUM
ap-9037	475	15	)	)	PUNCT
ap-9037	475	16	by	by	ADP
ap-9037	475	17	introducing	introduce	VERB
ap-9037	475	18	functions	function	NOUN
ap-9037	475	19	φ1	φ1	PROPN
ap-9037	475	20	,	,	PUNCT
ap-9037	475	21	φ2	φ2	PROPN
ap-9037	475	22	,	,	PUNCT
ap-9037	475	23	.	.	PUNCT
ap-9037	475	24	.	.	PUNCT
ap-9037	476	1	.	.	PUNCT
ap-9037	477	1	,	,	PUNCT
ap-9037	477	2	φn	φn	ADP
ap-9037	477	3	through	through	ADP
ap-9037	477	4	our	our	PRON
ap-9037	477	5	point	point	NOUN
ap-9037	477	6	transformation	transformation	NOUN
ap-9037	477	7	(	(	PUNCT
ap-9037	477	8	72	72	NUM
ap-9037	477	9	)	)	PUNCT
ap-9037	477	10	as	as	ADP
ap-9037	477	11	:	:	PUNCT
ap-9037	477	12	φj(ϕ	φj(ϕ	X
ap-9037	477	13	)	)	PUNCT
ap-9037	477	14	=	=	SYM
ap-9037	477	15	cos(ϕ)−ν1	cos(ϕ)−ν1	NOUN
ap-9037	477	16	sin(ϕ)−ν2ξj(ϕ	sin(ϕ)−ν2ξj(ϕ	NOUN
ap-9037	477	17	)	)	PUNCT
ap-9037	477	18	,	,	PUNCT
ap-9037	477	19	j	j	PROPN
ap-9037	477	20	=	=	SYM
ap-9037	477	21	1	1	NUM
ap-9037	477	22	,	,	PUNCT
ap-9037	477	23	2	2	NUM
ap-9037	477	24	,	,	PUNCT
ap-9037	477	25	.	.	PUNCT
ap-9037	477	26	.	.	PUNCT
ap-9037	478	1	.	.	PUNCT
ap-9037	479	1	,	,	PUNCT
ap-9037	479	2	n	n	X
ap-9037	479	3	.	.	PUNCT
ap-9037	480	1	(	(	PUNCT
ap-9037	480	2	93	93	NUM
ap-9037	480	3	)	)	PUNCT
ap-9037	480	4	substitution	substitution	NOUN
ap-9037	480	5	into	into	ADP
ap-9037	480	6	(	(	PUNCT
ap-9037	480	7	90	90	NUM
ap-9037	480	8	)	)	PUNCT
ap-9037	480	9	gives	give	VERB
ap-9037	480	10	,	,	PUNCT
ap-9037	480	11	after	after	ADP
ap-9037	480	12	simplification	simplification	NOUN
ap-9037	480	13	:	:	PUNCT
ap-9037	480	14	φ̂(ϕ	φ̂(ϕ	X
ap-9037	480	15	)	)	PUNCT
ap-9037	480	16	=	=	SYM
ap-9037	480	17	wφ1,φ2,	wφ1,φ2,	NOUN
ap-9037	480	18	...	...	PUNCT
ap-9037	480	19	,φn	,φn	NOUN
ap-9037	480	20	,	,	PUNCT
ap-9037	480	21	φ(ϕ	φ(ϕ	PROPN
ap-9037	480	22	)	)	PUNCT
ap-9037	480	23	wφ1,φ2,	wφ1,φ2,	NOUN
ap-9037	480	24	...	...	PUNCT
ap-9037	480	25	,φn	,φn	PUNCT
ap-9037	480	26	(	(	PUNCT
ap-9037	480	27	ϕ	ϕ	NOUN
ap-9037	480	28	)	)	PUNCT
ap-9037	480	29	.	.	PUNCT
ap-9037	481	1	(	(	PUNCT
ap-9037	481	2	94	94	X
ap-9037	481	3	)	)	PUNCT
ap-9037	481	4	the	the	DET
ap-9037	481	5	transformed	transform	VERB
ap-9037	481	6	potential	potential	NOUN
ap-9037	481	7	v̂	v̂	PRON
ap-9037	481	8	inserted	insert	VERB
ap-9037	481	9	into	into	ADP
ap-9037	481	10	equation	equation	NOUN
ap-9037	481	11	(	(	PUNCT
ap-9037	481	12	73	73	NUM
ap-9037	481	13	)	)	PUNCT
ap-9037	481	14	is	be	AUX
ap-9037	481	15	obtained	obtain	VERB
ap-9037	481	16	by	by	ADP
ap-9037	481	17	combining	combine	VERB
ap-9037	481	18	(	(	PUNCT
ap-9037	481	19	72	72	NUM
ap-9037	481	20	)	)	PUNCT
ap-9037	481	21	and	and	CCONJ
ap-9037	481	22	(	(	PUNCT
ap-9037	481	23	91	91	NUM
ap-9037	481	24	)	)	PUNCT
ap-9037	481	25	.	.	PUNCT
ap-9037	482	1	substitution	substitution	NOUN
ap-9037	482	2	yields	yield	NOUN
ap-9037	482	3	:	:	PUNCT
ap-9037	482	4	v̂ϕ(ϕ	v̂ϕ(ϕ	NUM
ap-9037	482	5	)	)	PUNCT
ap-9037	482	6	=	=	SYM
ap-9037	482	7	vϕ(ϕ	vϕ(ϕ	X
ap-9037	482	8	)	)	PUNCT
ap-9037	482	9	−	−	PROPN
ap-9037	482	10	2	2	NUM
ap-9037	482	11	d2	d2	PROPN
ap-9037	482	12	dϕ2	dϕ2	PROPN
ap-9037	482	13	log	log	NOUN
ap-9037	482	14	[	[	PUNCT
ap-9037	482	15	cos(ϕ)nν1	cos(ϕ)nν1	PROPN
ap-9037	482	16	sin(ϕ)nν2wφ1,φ2,	sin(ϕ)nν2wφ1,φ2,	PROPN
ap-9037	482	17	...	...	PUNCT
ap-9037	482	18	,φn	,φn	PUNCT
ap-9037	482	19	(	(	PUNCT
ap-9037	482	20	ϕ	ϕ	NOUN
ap-9037	482	21	)	)	PUNCT
ap-9037	482	22	]	]	PUNCT
ap-9037	482	23	=	=	SYM
ap-9037	482	24	vϕ(ϕ	vϕ(ϕ	X
ap-9037	482	25	)	)	PUNCT
ap-9037	483	1	+	+	NUM
ap-9037	483	2	2n	2n	NUM
ap-9037	483	3	[	[	PUNCT
ap-9037	483	4	ν1	ν1	NOUN
ap-9037	483	5	cos(ϕ)2	cos(ϕ)2	NOUN
ap-9037	483	6	+	+	CCONJ
ap-9037	483	7	ν2	ν2	NOUN
ap-9037	483	8	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	483	9	]	]	PUNCT
ap-9037	483	10	−	−	PROPN
ap-9037	483	11	2	2	NUM
ap-9037	483	12	d2	d2	PROPN
ap-9037	483	13	dϕ2	dϕ2	PROPN
ap-9037	483	14	log	log	PROPN
ap-9037	483	15	[	[	PUNCT
ap-9037	483	16	wφ1,φ2,	wφ1,φ2,	NOUN
ap-9037	483	17	...	...	PUNCT
ap-9037	483	18	,φn(ϕ	,φn(ϕ	PUNCT
ap-9037	483	19	)	)	PUNCT
ap-9037	483	20	]	]	PUNCT
ap-9037	483	21	.	.	PUNCT
ap-9037	484	1	(	(	PUNCT
ap-9037	484	2	95	95	NUM
ap-9037	484	3	)	)	PUNCT
ap-9037	484	4	we	we	PRON
ap-9037	484	5	observe	observe	VERB
ap-9037	484	6	that	that	SCONJ
ap-9037	484	7	this	this	DET
ap-9037	484	8	potential	potential	NOUN
ap-9037	484	9	contributes	contribute	VERB
ap-9037	484	10	a	a	DET
ap-9037	484	11	term	term	NOUN
ap-9037	484	12	of	of	ADP
ap-9037	484	13	trigonometric	trigonometric	ADJ
ap-9037	484	14	pöschl	pöschl	NOUN
ap-9037	484	15	-	-	PUNCT
ap-9037	484	16	teller	teller	NOUN
ap-9037	484	17	type	type	NOUN
ap-9037	484	18	,	,	PUNCT
ap-9037	484	19	independent	independent	ADJ
ap-9037	484	20	of	of	ADP
ap-9037	484	21	its	its	PRON
ap-9037	484	22	initial	initial	ADJ
ap-9037	484	23	counterpart	counterpart	NOUN
ap-9037	484	24	vϕ.	vϕ.	CCONJ
ap-9037	484	25	since	since	SCONJ
ap-9037	484	26	(	(	PUNCT
ap-9037	484	27	95	95	NUM
ap-9037	484	28	)	)	PUNCT
ap-9037	484	29	generally	generally	ADV
ap-9037	484	30	depends	depend	VERB
ap-9037	484	31	on	on	ADP
ap-9037	484	32	all	all	DET
ap-9037	484	33	parameters	parameter	NOUN
ap-9037	484	34	in	in	ADP
ap-9037	484	35	the	the	DET
ap-9037	484	36	function	function	NOUN
ap-9037	484	37	(	(	PUNCT
ap-9037	484	38	88	88	NUM
ap-9037	484	39	)	)	PUNCT
ap-9037	484	40	,	,	PUNCT
ap-9037	484	41	these	these	DET
ap-9037	484	42	parameters	parameter	NOUN
ap-9037	484	43	must	must	AUX
ap-9037	484	44	be	be	AUX
ap-9037	484	45	assigned	assign	VERB
ap-9037	484	46	fixed	fix	VERB
ap-9037	484	47	values	value	NOUN
ap-9037	484	48	in	in	ADP
ap-9037	484	49	order	order	NOUN
ap-9037	484	50	not	not	PART
ap-9037	484	51	to	to	PART
ap-9037	484	52	change	change	VERB
ap-9037	484	53	the	the	DET
ap-9037	484	54	potential	potential	NOUN
ap-9037	484	55	.	.	PUNCT
ap-9037	485	1	this	this	PRON
ap-9037	485	2	is	be	AUX
ap-9037	485	3	not	not	PART
ap-9037	485	4	valid	valid	ADJ
ap-9037	485	5	for	for	ADP
ap-9037	485	6	the	the	DET
ap-9037	485	7	transformation	transformation	NOUN
ap-9037	485	8	parameter	parameter	PROPN
ap-9037	485	9	m2	m2	PROPN
ap-9037	485	10	,	,	PUNCT
ap-9037	485	11	since	since	SCONJ
ap-9037	485	12	(	(	PUNCT
ap-9037	485	13	95	95	NUM
ap-9037	485	14	)	)	PUNCT
ap-9037	485	15	does	do	AUX
ap-9037	485	16	not	not	PART
ap-9037	485	17	depend	depend	VERB
ap-9037	485	18	on	on	ADP
ap-9037	485	19	it	it	PRON
ap-9037	485	20	.	.	PUNCT
ap-9037	486	1	in	in	ADP
ap-9037	486	2	summary	summary	NOUN
ap-9037	486	3	,	,	PUNCT
ap-9037	486	4	the	the	DET
ap-9037	486	5	transformed	transform	VERB
ap-9037	486	6	equation	equation	NOUN
ap-9037	486	7	(	(	PUNCT
ap-9037	486	8	92	92	NUM
ap-9037	486	9	)	)	PUNCT
ap-9037	486	10	for	for	ADP
ap-9037	486	11	this	this	DET
ap-9037	486	12	potential	potential	NOUN
ap-9037	486	13	is	be	AUX
ap-9037	486	14	solved	solve	VERB
ap-9037	486	15	by	by	ADP
ap-9037	486	16	the	the	DET
ap-9037	486	17	function	function	NOUN
ap-9037	486	18	(	(	PUNCT
ap-9037	486	19	94	94	NUM
ap-9037	486	20	)	)	PUNCT
ap-9037	486	21	.	.	PUNCT
ap-9037	487	1	example	example	NOUN
ap-9037	487	2	:	:	PUNCT
ap-9037	487	3	confluent	confluent	ADJ
ap-9037	487	4	second	second	ADJ
ap-9037	487	5	-	-	PUNCT
ap-9037	487	6	order	order	NOUN
ap-9037	487	7	darboux	darboux	VERB
ap-9037	487	8	-	-	PUNCT
ap-9037	487	9	crum	crum	NOUN
ap-9037	487	10	transformation	transformation	NOUN
ap-9037	487	11	.	.	PUNCT
ap-9037	488	1	in	in	ADP
ap-9037	488	2	this	this	DET
ap-9037	488	3	application	application	NOUN
ap-9037	488	4	,	,	PUNCT
ap-9037	488	5	we	we	PRON
ap-9037	488	6	will	will	AUX
ap-9037	488	7	generate	generate	VERB
ap-9037	488	8	a	a	DET
ap-9037	488	9	solvable	solvable	ADJ
ap-9037	488	10	case	case	NOUN
ap-9037	488	11	of	of	ADP
ap-9037	488	12	the	the	DET
ap-9037	488	13	azimuthal	azimuthal	ADJ
ap-9037	488	14	equation	equation	NOUN
ap-9037	488	15	by	by	ADP
ap-9037	488	16	means	mean	NOUN
ap-9037	488	17	of	of	ADP
ap-9037	488	18	a	a	DET
ap-9037	488	19	confluent	confluent	ADJ
ap-9037	488	20	darboux	darboux	NOUN
ap-9037	488	21	-	-	PUNCT
ap-9037	488	22	crum	crum	NOUN
ap-9037	488	23	transformation	transformation	NOUN
ap-9037	488	24	.	.	PUNCT
ap-9037	489	1	our	our	PRON
ap-9037	489	2	starting	starting	NOUN
ap-9037	489	3	point	point	NOUN
ap-9037	489	4	is	be	AUX
ap-9037	489	5	equation	equation	NOUN
ap-9037	489	6	(	(	PUNCT
ap-9037	489	7	28	28	NUM
ap-9037	489	8	)	)	PUNCT
ap-9037	489	9	for	for	ADP
ap-9037	489	10	the	the	DET
ap-9037	489	11	potential	potential	ADJ
ap-9037	489	12	vϕ	vϕ	NOUN
ap-9037	489	13	=	=	SYM
ap-9037	489	14	0	0	NUM
ap-9037	489	15	.	.	PUNCT
ap-9037	490	1	this	this	DET
ap-9037	490	2	results	result	VERB
ap-9037	490	3	in	in	ADP
ap-9037	490	4	equation	equation	NOUN
ap-9037	490	5	:	:	PUNCT
ap-9037	490	6	φ′′(ϕ	φ′′(ϕ	NOUN
ap-9037	490	7	)	)	PUNCT
ap-9037	490	8	−	−	NUM
ap-9037	490	9	2	2	NUM
ap-9037	491	1	[	[	X
ap-9037	491	2	ν1	ν1	NOUN
ap-9037	491	3	tan(ϕ	tan(ϕ	NOUN
ap-9037	491	4	)	)	PUNCT
ap-9037	491	5	−	−	PROPN
ap-9037	491	6	ν2	ν2	NOUN
ap-9037	491	7	cot(ϕ	cot(ϕ	PROPN
ap-9037	491	8	)	)	PUNCT
ap-9037	491	9	]	]	PUNCT
ap-9037	492	1	φ′(ϕ	φ′(ϕ	VERB
ap-9037	492	2	)	)	PUNCT
ap-9037	492	3	+	+	CCONJ
ap-9037	492	4	[	[	PUNCT
ap-9037	492	5	m2	m2	PROPN
ap-9037	492	6	−	−	PROPN
ap-9037	492	7	ν1(1	ν1(1	PROPN
ap-9037	492	8	−	−	PROPN
ap-9037	492	9	r1	r1	PROPN
ap-9037	492	10	)	)	PUNCT
ap-9037	492	11	cos(ϕ)2	cos(ϕ)2	NOUN
ap-9037	492	12	−	−	PROPN
ap-9037	492	13	ν2(1	ν2(1	NOUN
ap-9037	492	14	−	−	NOUN
ap-9037	492	15	r2	r2	NOUN
ap-9037	492	16	)	)	PUNCT
ap-9037	492	17	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	492	18	]	]	PUNCT
ap-9037	492	19	φ(ϕ	φ(ϕ	PROPN
ap-9037	492	20	)	)	PUNCT
ap-9037	492	21	=	=	SYM
ap-9037	492	22	0	0	X
ap-9037	492	23	.	.	PUNCT
ap-9037	493	1	(	(	PUNCT
ap-9037	493	2	96	96	NUM
ap-9037	493	3	)	)	PUNCT
ap-9037	493	4	288	288	NUM
ap-9037	493	5	vol	vol	NOUN
ap-9037	493	6	.	.	PUNCT
ap-9037	494	1	63	63	NUM
ap-9037	494	2	no	no	INTJ
ap-9037	494	3	.	.	PUNCT
ap-9037	495	1	4/2023	4/2023	NOUN
ap-9037	495	2	construction	construction	NOUN
ap-9037	495	3	of	of	ADP
ap-9037	495	4	angular	angular	ADJ
ap-9037	495	5	-	-	PUNCT
ap-9037	495	6	dependent	dependent	ADJ
ap-9037	495	7	potentials	potential	NOUN
ap-9037	495	8	in	in	ADP
ap-9037	495	9	order	order	NOUN
ap-9037	495	10	to	to	PART
ap-9037	495	11	perform	perform	VERB
ap-9037	495	12	a	a	DET
ap-9037	495	13	confluent	confluent	ADJ
ap-9037	495	14	darboux	darboux	NOUN
ap-9037	495	15	-	-	PUNCT
ap-9037	495	16	crum	crum	NOUN
ap-9037	495	17	transformation	transformation	NOUN
ap-9037	495	18	of	of	ADP
ap-9037	495	19	second	second	ADJ
ap-9037	495	20	order	order	NOUN
ap-9037	495	21	,	,	PUNCT
ap-9037	495	22	we	we	PRON
ap-9037	495	23	resort	resort	VERB
ap-9037	495	24	to	to	ADP
ap-9037	495	25	the	the	DET
ap-9037	495	26	version	version	NOUN
ap-9037	495	27	of	of	ADP
ap-9037	495	28	our	our	PRON
ap-9037	495	29	equation	equation	NOUN
ap-9037	495	30	that	that	PRON
ap-9037	495	31	is	be	AUX
ap-9037	495	32	obtained	obtain	VERB
ap-9037	495	33	after	after	ADP
ap-9037	495	34	applying	apply	VERB
ap-9037	495	35	the	the	DET
ap-9037	495	36	point	point	NOUN
ap-9037	495	37	transformation	transformation	NOUN
ap-9037	495	38	(	(	PUNCT
ap-9037	495	39	72	72	NUM
ap-9037	495	40	)	)	PUNCT
ap-9037	495	41	.	.	PUNCT
ap-9037	496	1	this	this	DET
ap-9037	496	2	equation	equation	NOUN
ap-9037	496	3	is	be	AUX
ap-9037	496	4	given	give	VERB
ap-9037	496	5	by	by	ADP
ap-9037	496	6	(	(	PUNCT
ap-9037	496	7	73	73	NUM
ap-9037	496	8	)	)	PUNCT
ap-9037	496	9	for	for	ADP
ap-9037	496	10	vϕ	vϕ	NOUN
ap-9037	496	11	=	=	SYM
ap-9037	496	12	0	0	NUM
ap-9037	496	13	,	,	PUNCT
ap-9037	496	14	that	that	ADV
ap-9037	496	15	is	is	ADV
ap-9037	496	16	,	,	PUNCT
ap-9037	496	17	we	we	PRON
ap-9037	496	18	have	have	AUX
ap-9037	496	19	:	:	PUNCT
ap-9037	496	20	ξ′′(ϕ	ξ′′(ϕ	NOUN
ap-9037	496	21	)	)	PUNCT
ap-9037	497	1	+	+	CCONJ
ap-9037	497	2	[	[	PUNCT
ap-9037	497	3	m2	m2	PROPN
ap-9037	497	4	+	+	CCONJ
ap-9037	497	5	(	(	PUNCT
ap-9037	497	6	ν1	ν1	NOUN
ap-9037	497	7	+	+	CCONJ
ap-9037	497	8	ν2)2	ν2)2	PUNCT
ap-9037	497	9	+	+	CCONJ
ap-9037	497	10	(	(	PUNCT
ap-9037	497	11	r1	r1	NOUN
ap-9037	497	12	−	−	PROPN
ap-9037	497	13	ν1)ν1	ν1)ν1	NOUN
ap-9037	497	14	cos(ϕ)2	cos(ϕ)2	NOUN
ap-9037	497	15	+	+	CCONJ
ap-9037	497	16	(	(	PUNCT
ap-9037	497	17	r2	r2	PROPN
ap-9037	497	18	−	−	PROPN
ap-9037	497	19	ν2)ν2	ν2)ν2	NOUN
ap-9037	497	20	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	497	21	]	]	PUNCT
ap-9037	497	22	ξ(ϕ	ξ(ϕ	X
ap-9037	497	23	)	)	PUNCT
ap-9037	497	24	=	=	SYM
ap-9037	497	25	0	0	NUM
ap-9037	497	26	.	.	PUNCT
ap-9037	498	1	a	a	DET
ap-9037	498	2	particular	particular	ADJ
ap-9037	498	3	solution	solution	NOUN
ap-9037	498	4	is	be	AUX
ap-9037	498	5	found	find	VERB
ap-9037	498	6	from	from	ADP
ap-9037	498	7	(	(	PUNCT
ap-9037	498	8	79	79	NUM
ap-9037	498	9	)	)	PUNCT
ap-9037	498	10	,	,	PUNCT
ap-9037	498	11	where	where	SCONJ
ap-9037	498	12	the	the	DET
ap-9037	498	13	parameters	parameter	NOUN
ap-9037	498	14	must	must	AUX
ap-9037	498	15	be	be	AUX
ap-9037	498	16	chosen	choose	VERB
ap-9037	498	17	according	accord	VERB
ap-9037	498	18	to	to	ADP
ap-9037	498	19	(	(	PUNCT
ap-9037	498	20	75)–(78	75)–(78	NOUN
ap-9037	498	21	)	)	PUNCT
ap-9037	498	22	for	for	ADP
ap-9037	498	23	γ	γ	X
ap-9037	498	24	=	=	SYM
ap-9037	498	25	0	0	PROPN
ap-9037	498	26	.	.	PUNCT
ap-9037	499	1	this	this	PRON
ap-9037	499	2	gives	give	VERB
ap-9037	499	3	:	:	PUNCT
ap-9037	499	4	a	a	DET
ap-9037	499	5	=	=	X
ap-9037	499	6	[	[	PUNCT
ap-9037	499	7	1	1	NUM
ap-9037	499	8	+	+	NUM
ap-9037	499	9	2n	2n	NUM
ap-9037	499	10	+	+	CCONJ
ap-9037	499	11	1	1	NUM
ap-9037	499	12	2	2	NUM
ap-9037	499	13	√	√	NUM
ap-9037	499	14	1	1	NUM
ap-9037	499	15	+	+	CCONJ
ap-9037	499	16	4ν1(ν1	4ν1(ν1	NUM
ap-9037	499	17	−	−	PROPN
ap-9037	499	18	r1	r1	PROPN
ap-9037	499	19	)	)	PUNCT
ap-9037	500	1	+	+	CCONJ
ap-9037	500	2	1	1	NUM
ap-9037	500	3	2	2	NUM
ap-9037	500	4	√	√	NUM
ap-9037	500	5	1	1	NUM
ap-9037	501	1	+	+	CCONJ
ap-9037	501	2	4ν2(ν2	4ν2(ν2	NUM
ap-9037	501	3	−	−	PROPN
ap-9037	501	4	r2	r2	PROPN
ap-9037	501	5	)	)	PUNCT
ap-9037	502	1	]	]	PUNCT
ap-9037	502	2	2	2	NUM
ap-9037	502	3	,	,	PUNCT
ap-9037	502	4	b	b	NOUN
ap-9037	502	5	=	=	SYM
ap-9037	502	6	(	(	PUNCT
ap-9037	502	7	r2	r2	PROPN
ap-9037	502	8	−	−	PROPN
ap-9037	502	9	ν2)ν2	ν2)ν2	NOUN
ap-9037	502	10	,	,	PUNCT
ap-9037	502	11	c	c	X
ap-9037	502	12	=	=	SYM
ap-9037	502	13	(	(	PUNCT
ap-9037	502	14	r1	r1	NOUN
ap-9037	502	15	−	−	PROPN
ap-9037	502	16	ν1)ν1	ν1)ν1	NOUN
ap-9037	502	17	,	,	PUNCT
ap-9037	502	18	(	(	PUNCT
ap-9037	502	19	97	97	NUM
ap-9037	502	20	)	)	PUNCT
ap-9037	502	21	m2	m2	PROPN
ap-9037	502	22	=	=	PUNCT
ap-9037	502	23	−(ν1	−(ν1	ADP
ap-9037	502	24	+	+	X
ap-9037	502	25	ν2)2	ν2)2	PUNCT
ap-9037	502	26	+	+	CCONJ
ap-9037	502	27	[	[	PUNCT
ap-9037	502	28	1	1	NUM
ap-9037	502	29	+	+	NUM
ap-9037	502	30	2n	2n	NUM
ap-9037	502	31	+	+	CCONJ
ap-9037	502	32	1	1	NUM
ap-9037	502	33	2	2	NUM
ap-9037	502	34	√	√	NUM
ap-9037	502	35	1	1	NUM
ap-9037	502	36	+	+	CCONJ
ap-9037	502	37	4ν1(ν1	4ν1(ν1	NUM
ap-9037	502	38	−	−	PROPN
ap-9037	502	39	r1	r1	PROPN
ap-9037	502	40	)	)	PUNCT
ap-9037	502	41	+	+	CCONJ
ap-9037	502	42	1	1	NUM
ap-9037	502	43	2	2	NUM
ap-9037	502	44	√	√	NUM
ap-9037	502	45	1	1	NUM
ap-9037	502	46	+	+	CCONJ
ap-9037	502	47	4ν2(ν2	4ν2(ν2	NUM
ap-9037	502	48	−	−	PROPN
ap-9037	502	49	r2	r2	PROPN
ap-9037	502	50	)	)	PUNCT
ap-9037	502	51	]	]	PUNCT
ap-9037	502	52	2	2	X
ap-9037	502	53	.	.	PUNCT
ap-9037	503	1	in	in	ADP
ap-9037	503	2	the	the	DET
ap-9037	503	3	next	next	ADJ
ap-9037	503	4	step	step	NOUN
ap-9037	503	5	we	we	PRON
ap-9037	503	6	must	must	AUX
ap-9037	503	7	provide	provide	VERB
ap-9037	503	8	two	two	NUM
ap-9037	503	9	transformation	transformation	NOUN
ap-9037	503	10	functions	function	NOUN
ap-9037	503	11	ξ1	ξ1	NOUN
ap-9037	503	12	and	and	CCONJ
ap-9037	503	13	ξ2	ξ2	NOUN
ap-9037	503	14	,	,	PUNCT
ap-9037	503	15	together	together	ADV
ap-9037	503	16	with	with	ADP
ap-9037	503	17	a	a	DET
ap-9037	503	18	single	single	ADJ
ap-9037	503	19	transformation	transformation	NOUN
ap-9037	503	20	parameter	parameter	NOUN
ap-9037	503	21	m2	m2	PROPN
ap-9037	503	22	1	1	NUM
ap-9037	503	23	.	.	PUNCT
ap-9037	504	1	this	this	DET
ap-9037	504	2	parameter	parameter	NOUN
ap-9037	504	3	is	be	AUX
ap-9037	504	4	determined	determine	VERB
ap-9037	504	5	by	by	ADP
ap-9037	504	6	choosing	choose	VERB
ap-9037	504	7	a	a	DET
ap-9037	504	8	value	value	NOUN
ap-9037	504	9	for	for	ADP
ap-9037	504	10	the	the	DET
ap-9037	504	11	constant	constant	ADJ
ap-9037	504	12	n	n	NOUN
ap-9037	504	13	in	in	ADP
ap-9037	504	14	(	(	PUNCT
ap-9037	504	15	97	97	NUM
ap-9037	504	16	)	)	PUNCT
ap-9037	504	17	.	.	PUNCT
ap-9037	505	1	in	in	ADP
ap-9037	505	2	the	the	DET
ap-9037	505	3	present	present	ADJ
ap-9037	505	4	case	case	NOUN
ap-9037	505	5	,	,	PUNCT
ap-9037	505	6	we	we	PRON
ap-9037	505	7	choose	choose	VERB
ap-9037	505	8	n	n	NOUN
ap-9037	505	9	=	=	SYM
ap-9037	505	10	n1	n1	PROPN
ap-9037	505	11	=	=	SYM
ap-9037	505	12	0	0	NUM
ap-9037	505	13	,	,	PUNCT
ap-9037	505	14	so	so	SCONJ
ap-9037	505	15	we	we	PRON
ap-9037	505	16	can	can	AUX
ap-9037	505	17	extract	extract	VERB
ap-9037	505	18	the	the	DET
ap-9037	505	19	first	first	ADJ
ap-9037	505	20	transformation	transformation	NOUN
ap-9037	505	21	function	function	NOUN
ap-9037	505	22	from	from	ADP
ap-9037	505	23	(	(	PUNCT
ap-9037	505	24	79	79	NUM
ap-9037	505	25	)	)	PUNCT
ap-9037	505	26	.	.	PUNCT
ap-9037	506	1	this	this	PRON
ap-9037	506	2	gives	give	VERB
ap-9037	506	3	:	:	PUNCT
ap-9037	506	4	ξ1(ϕ	ξ1(ϕ	NUM
ap-9037	506	5	)	)	PUNCT
ap-9037	506	6	=	=	SYM
ap-9037	506	7	ξ(ϕ)|n1=0	ξ(ϕ)|n1=0	NOUN
ap-9037	506	8	=	=	SYM
ap-9037	506	9	cos(ϕ	cos(ϕ	PROPN
ap-9037	506	10	)	)	PUNCT
ap-9037	506	11	1	1	NUM
ap-9037	506	12	2	2	NUM
ap-9037	506	13	+	+	CCONJ
ap-9037	506	14	1	1	NUM
ap-9037	506	15	2	2	NUM
ap-9037	506	16	√	√	NUM
ap-9037	506	17	1	1	NUM
ap-9037	506	18	+	+	NOUN
ap-9037	506	19	4ν1(ν1−r1	4ν1(ν1−r1	NOUN
ap-9037	506	20	)	)	PUNCT
ap-9037	506	21	sin(ϕ	sin(ϕ	NOUN
ap-9037	506	22	)	)	PUNCT
ap-9037	506	23	1	1	NUM
ap-9037	506	24	2	2	NUM
ap-9037	506	25	+	+	CCONJ
ap-9037	506	26	1	1	NUM
ap-9037	506	27	2	2	NUM
ap-9037	506	28	√	√	NUM
ap-9037	506	29	1	1	NUM
ap-9037	506	30	+	+	NOUN
ap-9037	506	31	4ν2(ν2−r2	4ν2(ν2−r2	NUM
ap-9037	506	32	)	)	PUNCT
ap-9037	506	33	.	.	PUNCT
ap-9037	507	1	(	(	PUNCT
ap-9037	507	2	98	98	NUM
ap-9037	507	3	)	)	PUNCT
ap-9037	507	4	the	the	DET
ap-9037	507	5	remaining	remain	VERB
ap-9037	507	6	transformation	transformation	NOUN
ap-9037	507	7	function	function	NOUN
ap-9037	507	8	is	be	AUX
ap-9037	507	9	a	a	DET
ap-9037	507	10	solution	solution	NOUN
ap-9037	507	11	of	of	ADP
ap-9037	507	12	equation	equation	NOUN
ap-9037	507	13	(	(	PUNCT
ap-9037	507	14	9	9	NUM
ap-9037	507	15	)	)	PUNCT
ap-9037	507	16	for	for	ADP
ap-9037	507	17	j	j	PROPN
ap-9037	507	18	=	=	SYM
ap-9037	507	19	2	2	NUM
ap-9037	507	20	and	and	CCONJ
ap-9037	507	21	the	the	DET
ap-9037	507	22	present	present	ADJ
ap-9037	507	23	settings	setting	NOUN
ap-9037	507	24	.	.	PUNCT
ap-9037	508	1	this	this	DET
ap-9037	508	2	equation	equation	NOUN
ap-9037	508	3	reads	read	VERB
ap-9037	508	4	:	:	PUNCT
ap-9037	508	5	ξ′′	ξ′′	NOUN
ap-9037	508	6	2	2	NUM
ap-9037	508	7	(	(	PUNCT
ap-9037	508	8	ϕ	ϕ	NOUN
ap-9037	508	9	)	)	PUNCT
ap-9037	508	10	+	+	CCONJ
ap-9037	508	11	[	[	PUNCT
ap-9037	508	12	m2	m2	PROPN
ap-9037	508	13	+	+	CCONJ
ap-9037	508	14	(	(	PUNCT
ap-9037	508	15	ν1	ν1	NOUN
ap-9037	508	16	+	+	CCONJ
ap-9037	508	17	ν2)2	ν2)2	PUNCT
ap-9037	508	18	+	+	CCONJ
ap-9037	508	19	(	(	PUNCT
ap-9037	508	20	r1	r1	NOUN
ap-9037	508	21	−	−	PROPN
ap-9037	508	22	ν1)ν1	ν1)ν1	NOUN
ap-9037	508	23	cos(ϕ)2	cos(ϕ)2	NOUN
ap-9037	508	24	+	+	CCONJ
ap-9037	508	25	(	(	PUNCT
ap-9037	508	26	r2	r2	PROPN
ap-9037	508	27	−	−	PROPN
ap-9037	508	28	ν2)ν2	ν2)ν2	NOUN
ap-9037	508	29	sin(ϕ)2	sin(ϕ)2	VERB
ap-9037	508	30	]	]	PUNCT
ap-9037	508	31	ξ2(ϕ	ξ2(ϕ	X
ap-9037	508	32	)	)	PUNCT
ap-9037	508	33	=	=	SYM
ap-9037	508	34	−ξ1(ϕ	−ξ1(ϕ	PROPN
ap-9037	508	35	)	)	PUNCT
ap-9037	508	36	,	,	PUNCT
ap-9037	508	37	(	(	PUNCT
ap-9037	508	38	99	99	NUM
ap-9037	508	39	)	)	PUNCT
ap-9037	508	40	recall	recall	NOUN
ap-9037	508	41	that	that	PRON
ap-9037	508	42	m2	m2	PROPN
ap-9037	508	43	and	and	CCONJ
ap-9037	508	44	ξ1	ξ1	PROPN
ap-9037	508	45	are	be	AUX
ap-9037	508	46	given	give	VERB
ap-9037	508	47	in	in	ADP
ap-9037	508	48	(	(	PUNCT
ap-9037	508	49	97	97	NUM
ap-9037	508	50	)	)	PUNCT
ap-9037	508	51	and	and	CCONJ
ap-9037	508	52	(	(	PUNCT
ap-9037	508	53	98	98	NUM
ap-9037	508	54	)	)	PUNCT
ap-9037	508	55	,	,	PUNCT
ap-9037	508	56	respectively	respectively	ADV
ap-9037	508	57	.	.	PUNCT
ap-9037	509	1	a	a	DET
ap-9037	509	2	solution	solution	NOUN
ap-9037	509	3	of	of	ADP
ap-9037	509	4	(	(	PUNCT
ap-9037	509	5	99	99	NUM
ap-9037	509	6	)	)	PUNCT
ap-9037	509	7	can	can	AUX
ap-9037	509	8	be	be	AUX
ap-9037	509	9	found	find	VERB
ap-9037	509	10	by	by	ADP
ap-9037	509	11	taking	take	VERB
ap-9037	509	12	the	the	DET
ap-9037	509	13	derivative	derivative	NOUN
ap-9037	509	14	of	of	ADP
ap-9037	509	15	ξ	ξ	PROPN
ap-9037	509	16	with	with	ADP
ap-9037	509	17	respect	respect	NOUN
ap-9037	509	18	to	to	ADP
ap-9037	509	19	the	the	DET
ap-9037	509	20	transformation	transformation	NOUN
ap-9037	509	21	parameter	parameter	NOUN
ap-9037	510	1	[	[	X
ap-9037	510	2	26	26	NUM
ap-9037	510	3	]	]	PUNCT
ap-9037	510	4	.	.	PUNCT
ap-9037	511	1	applying	apply	VERB
ap-9037	511	2	the	the	DET
ap-9037	511	3	chain	chain	NOUN
ap-9037	511	4	rule	rule	NOUN
ap-9037	511	5	gives	give	VERB
ap-9037	511	6	:	:	PUNCT
ap-9037	511	7	ξ2(ϕ	ξ2(ϕ	X
ap-9037	511	8	)	)	PUNCT
ap-9037	511	9	=	=	PUNCT
ap-9037	512	1	[	[	PUNCT
ap-9037	512	2	d	d	X
ap-9037	512	3	d(m2	d(m2	NOUN
ap-9037	512	4	)	)	PUNCT
ap-9037	512	5	ξ(ϕ	ξ(ϕ	PROPN
ap-9037	512	6	)	)	PUNCT
ap-9037	512	7	]	]	PUNCT
ap-9037	513	1	|n	|n	X
ap-9037	513	2	=	=	SYM
ap-9037	513	3	n1=0	n1=0	PROPN
ap-9037	513	4	=	=	PRON
ap-9037	513	5	{	{	PUNCT
ap-9037	513	6	[	[	PUNCT
ap-9037	513	7	4	4	NUM
ap-9037	513	8	+	+	SYM
ap-9037	513	9	8n	8n	NOUN
ap-9037	513	10	+	+	CCONJ
ap-9037	513	11	2	2	NUM
ap-9037	513	12	√	√	NUM
ap-9037	513	13	1	1	NUM
ap-9037	513	14	+	+	CCONJ
ap-9037	513	15	4ν1(ν1	4ν1(ν1	NUM
ap-9037	513	16	−	−	PROPN
ap-9037	513	17	r1	r1	PROPN
ap-9037	513	18	)	)	PUNCT
ap-9037	513	19	+	+	CCONJ
ap-9037	513	20	2	2	NUM
ap-9037	513	21	√	√	NUM
ap-9037	513	22	1	1	NUM
ap-9037	513	23	+	+	CCONJ
ap-9037	513	24	4ν2(ν2	4ν2(ν2	NUM
ap-9037	513	25	−	−	PROPN
ap-9037	513	26	r2	r2	PROPN
ap-9037	513	27	)	)	PUNCT
ap-9037	513	28	]	]	PUNCT
ap-9037	513	29	−1	−1	NOUN
ap-9037	514	1	d	d	X
ap-9037	514	2	dn	dn	PROPN
ap-9037	514	3	ξ(ϕ	ξ(ϕ	PROPN
ap-9037	514	4	)	)	PUNCT
ap-9037	514	5	}	}	PUNCT
ap-9037	514	6	|n	|n	NOUN
ap-9037	514	7	=	=	SYM
ap-9037	514	8	n1=0	n1=0	NOUN
ap-9037	514	9	=	=	SYM
ap-9037	515	1	1	1	NUM
ap-9037	515	2	4	4	NUM
ap-9037	515	3	+	+	CCONJ
ap-9037	515	4	2	2	NUM
ap-9037	515	5	√	√	NUM
ap-9037	515	6	1	1	NUM
ap-9037	515	7	+	+	CCONJ
ap-9037	515	8	4ν1(ν1	4ν1(ν1	NUM
ap-9037	515	9	−	−	PROPN
ap-9037	515	10	r1	r1	PROPN
ap-9037	515	11	)	)	PUNCT
ap-9037	515	12	+	+	CCONJ
ap-9037	515	13	2	2	NUM
ap-9037	515	14	√	√	NUM
ap-9037	515	15	1	1	NUM
ap-9037	516	1	+	+	CCONJ
ap-9037	517	1	4ν2(ν2	4ν2(ν2	NUM
ap-9037	517	2	−	−	NOUN
ap-9037	517	3	r2	r2	PROPN
ap-9037	517	4	)	)	PUNCT
ap-9037	517	5	×	×	PROPN
ap-9037	517	6	cos(ϕ	cos(ϕ	PROPN
ap-9037	517	7	)	)	PUNCT
ap-9037	517	8	1	1	NUM
ap-9037	517	9	2	2	NUM
ap-9037	517	10	+	+	CCONJ
ap-9037	517	11	1	1	NUM
ap-9037	517	12	2	2	NUM
ap-9037	517	13	√	√	NUM
ap-9037	517	14	1	1	NUM
ap-9037	517	15	+	+	NOUN
ap-9037	517	16	4ν1(ν1−r1	4ν1(ν1−r1	NOUN
ap-9037	517	17	)	)	PUNCT
ap-9037	517	18	sin(ϕ	sin(ϕ	NOUN
ap-9037	517	19	)	)	PUNCT
ap-9037	517	20	1	1	NUM
ap-9037	517	21	2	2	NUM
ap-9037	517	22	+	+	CCONJ
ap-9037	517	23	1	1	NUM
ap-9037	517	24	2	2	NUM
ap-9037	517	25	√	√	NUM
ap-9037	517	26	1	1	NUM
ap-9037	517	27	+	+	NOUN
ap-9037	517	28	4ν2(ν2−r2	4ν2(ν2−r2	NUM
ap-9037	517	29	)	)	PUNCT
ap-9037	517	30	(	(	PUNCT
ap-9037	517	31	100	100	NUM
ap-9037	517	32	)	)	PUNCT
ap-9037	517	33	×	×	NOUN
ap-9037	517	34	{	{	PUNCT
ap-9037	517	35	d	d	NOUN
ap-9037	517	36	dn	dn	NOUN
ap-9037	517	37	2f1	2f1	NUM
ap-9037	517	38	[	[	PUNCT
ap-9037	517	39	−	−	NUM
ap-9037	517	40	n	n	CCONJ
ap-9037	517	41	,	,	PUNCT
ap-9037	517	42	1	1	NUM
ap-9037	517	43	+	+	SYM
ap-9037	517	44	n	n	CCONJ
ap-9037	517	45	+	+	CCONJ
ap-9037	517	46	1	1	NUM
ap-9037	517	47	2	2	NUM
ap-9037	517	48	√	√	NUM
ap-9037	517	49	1	1	NUM
ap-9037	517	50	+	+	CCONJ
ap-9037	517	51	4ν1(ν1	4ν1(ν1	NUM
ap-9037	517	52	−	−	PROPN
ap-9037	517	53	r1	r1	PROPN
ap-9037	517	54	)	)	PUNCT
ap-9037	517	55	+	+	CCONJ
ap-9037	517	56	1	1	NUM
ap-9037	517	57	2	2	NUM
ap-9037	517	58	√	√	NUM
ap-9037	517	59	1	1	NUM
ap-9037	517	60	+	+	CCONJ
ap-9037	517	61	4ν2(ν2	4ν2(ν2	NUM
ap-9037	517	62	−	−	NOUN
ap-9037	517	63	r2	r2	NOUN
ap-9037	517	64	)	)	PUNCT
ap-9037	517	65	,	,	PUNCT
ap-9037	517	66	1	1	NUM
ap-9037	517	67	+	+	SYM
ap-9037	517	68	1	1	NUM
ap-9037	517	69	2	2	NUM
ap-9037	517	70	√	√	NUM
ap-9037	517	71	1	1	NUM
ap-9037	517	72	+	+	CCONJ
ap-9037	517	73	4ν2(ν2	4ν2(ν2	NUM
ap-9037	517	74	−	−	NOUN
ap-9037	517	75	r2	r2	NOUN
ap-9037	517	76	)	)	PUNCT
ap-9037	517	77	,	,	PUNCT
ap-9037	517	78	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	517	79	]	]	PUNCT
ap-9037	517	80	}	}	PUNCT
ap-9037	517	81	|n	|n	NOUN
ap-9037	517	82	=	=	SYM
ap-9037	517	83	n1=0	n1=0	PROPN
ap-9037	517	84	.	.	PUNCT
ap-9037	518	1	we	we	PRON
ap-9037	518	2	substitute	substitute	VERB
ap-9037	518	3	the	the	DET
ap-9037	518	4	transformation	transformation	NOUN
ap-9037	518	5	functions	function	NOUN
ap-9037	518	6	(	(	PUNCT
ap-9037	518	7	98	98	NUM
ap-9037	518	8	)	)	PUNCT
ap-9037	518	9	and	and	CCONJ
ap-9037	518	10	(	(	PUNCT
ap-9037	518	11	100	100	NUM
ap-9037	518	12	)	)	PUNCT
ap-9037	518	13	into	into	ADP
ap-9037	518	14	the	the	DET
ap-9037	518	15	expressions	expression	NOUN
ap-9037	518	16	for	for	ADP
ap-9037	518	17	the	the	DET
ap-9037	518	18	solution	solution	NOUN
ap-9037	518	19	(	(	PUNCT
ap-9037	518	20	90	90	NUM
ap-9037	518	21	)	)	PUNCT
ap-9037	518	22	,	,	PUNCT
ap-9037	518	23	and	and	CCONJ
ap-9037	518	24	the	the	DET
ap-9037	518	25	potential	potential	ADJ
ap-9037	518	26	(	(	PUNCT
ap-9037	518	27	91	91	NUM
ap-9037	518	28	)	)	PUNCT
ap-9037	518	29	,	,	PUNCT
ap-9037	518	30	taking	take	VERB
ap-9037	518	31	into	into	ADP
ap-9037	518	32	account	account	NOUN
ap-9037	518	33	that	that	SCONJ
ap-9037	518	34	n	n	NOUN
ap-9037	518	35	=	=	SYM
ap-9037	518	36	2	2	X
ap-9037	518	37	.	.	PUNCT
ap-9037	519	1	the	the	DET
ap-9037	519	2	results	result	NOUN
ap-9037	519	3	are	be	AUX
ap-9037	519	4	long	long	ADJ
ap-9037	519	5	and	and	CCONJ
ap-9037	519	6	involved	involve	VERB
ap-9037	519	7	,	,	PUNCT
ap-9037	519	8	so	so	ADV
ap-9037	519	9	we	we	PRON
ap-9037	519	10	refrain	refrain	VERB
ap-9037	519	11	from	from	ADP
ap-9037	519	12	presenting	present	VERB
ap-9037	519	13	them	they	PRON
ap-9037	519	14	in	in	ADP
ap-9037	519	15	their	their	PRON
ap-9037	519	16	general	general	ADJ
ap-9037	519	17	form	form	NOUN
ap-9037	519	18	here	here	ADV
ap-9037	519	19	.	.	PUNCT
ap-9037	520	1	instead	instead	ADV
ap-9037	520	2	,	,	PUNCT
ap-9037	520	3	we	we	PRON
ap-9037	520	4	show	show	VERB
ap-9037	520	5	specific	specific	ADJ
ap-9037	520	6	cases	case	NOUN
ap-9037	520	7	of	of	ADP
ap-9037	520	8	the	the	DET
ap-9037	520	9	solution	solution	NOUN
ap-9037	520	10	φ̂	φ̂	VERB
ap-9037	520	11	to	to	ADP
ap-9037	520	12	our	our	PRON
ap-9037	520	13	transformed	transform	VERB
ap-9037	520	14	azimuthal	azimuthal	ADJ
ap-9037	520	15	equation	equation	NOUN
ap-9037	520	16	(	(	PUNCT
ap-9037	520	17	92	92	NUM
ap-9037	520	18	)	)	PUNCT
ap-9037	520	19	,	,	PUNCT
ap-9037	520	20	that	that	SCONJ
ap-9037	520	21	we	we	PRON
ap-9037	520	22	compute	compute	VERB
ap-9037	520	23	by	by	ADP
ap-9037	520	24	means	mean	NOUN
ap-9037	520	25	of	of	ADP
ap-9037	520	26	(	(	PUNCT
ap-9037	520	27	93	93	NUM
ap-9037	520	28	)	)	PUNCT
ap-9037	520	29	and	and	CCONJ
ap-9037	520	30	(	(	PUNCT
ap-9037	520	31	94	94	NUM
ap-9037	520	32	)	)	PUNCT
ap-9037	520	33	.	.	PUNCT
ap-9037	521	1	let	let	VERB
ap-9037	521	2	us	we	PRON
ap-9037	521	3	make	make	VERB
ap-9037	521	4	the	the	DET
ap-9037	521	5	overall	overall	ADJ
ap-9037	521	6	settings	setting	NOUN
ap-9037	521	7	:	:	PUNCT
ap-9037	521	8	ν1	ν1	NOUN
ap-9037	521	9	=	=	SYM
ap-9037	521	10	1	1	NUM
ap-9037	521	11	ν2	ν2	NOUN
ap-9037	521	12	=	=	SYM
ap-9037	521	13	2	2	NUM
ap-9037	521	14	n	n	NOUN
ap-9037	521	15	=	=	SYM
ap-9037	521	16	n1	n1	NOUN
ap-9037	521	17	=	=	SYM
ap-9037	521	18	0	0	NUM
ap-9037	521	19	.	.	PUNCT
ap-9037	522	1	(	(	PUNCT
ap-9037	522	2	101	101	NUM
ap-9037	522	3	)	)	PUNCT
ap-9037	522	4	289	289	NUM
ap-9037	522	5	axel	axel	PROPN
ap-9037	522	6	schulze	schulze	NOUN
ap-9037	522	7	-	-	PUNCT
ap-9037	522	8	halberg	halberg	PROPN
ap-9037	522	9	acta	acta	PROPN
ap-9037	522	10	polytechnica	polytechnica	PROPN
ap-9037	522	11	we	we	PRON
ap-9037	522	12	can	can	AUX
ap-9037	522	13	now	now	ADV
ap-9037	522	14	demonstrate	demonstrate	VERB
ap-9037	522	15	specific	specific	ADJ
ap-9037	522	16	cases	case	NOUN
ap-9037	522	17	of	of	ADP
ap-9037	522	18	our	our	PRON
ap-9037	522	19	solution	solution	NOUN
ap-9037	522	20	(	(	PUNCT
ap-9037	522	21	94	94	NUM
ap-9037	522	22	)	)	PUNCT
ap-9037	522	23	.	.	PUNCT
ap-9037	523	1	the	the	DET
ap-9037	523	2	two	two	NUM
ap-9037	523	3	solutions	solution	NOUN
ap-9037	523	4	exhibiting	exhibit	VERB
ap-9037	523	5	the	the	DET
ap-9037	523	6	simplest	simple	ADJ
ap-9037	523	7	form	form	NOUN
ap-9037	523	8	are	be	AUX
ap-9037	523	9	given	give	VERB
ap-9037	523	10	by	by	ADP
ap-9037	523	11	:	:	PUNCT
ap-9037	523	12	φ̂(ϕ)r1	φ̂(ϕ)r1	PROPN
ap-9037	523	13	=	=	SYM
ap-9037	523	14	r2=1	r2=1	NOUN
ap-9037	523	15	=	=	NOUN
ap-9037	523	16	cos(ϕ)5	cos(ϕ)5	NOUN
ap-9037	523	17	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	523	18	2f1	2f1	NUM
ap-9037	523	19	[	[	PUNCT
ap-9037	523	20	2	2	NUM
ap-9037	523	21	,	,	PUNCT
ap-9037	523	22	5	5	NUM
ap-9037	523	23	,	,	PUNCT
ap-9037	523	24	9	9	NUM
ap-9037	523	25	2	2	NUM
ap-9037	523	26	,	,	PUNCT
ap-9037	523	27	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	523	28	]	]	PUNCT
ap-9037	523	29	−3	−3	PROPN
ap-9037	523	30	arcsin[sin(ϕ	arcsin[sin(ϕ	PROPN
ap-9037	523	31	)	)	PUNCT
ap-9037	523	32	]	]	PUNCT
ap-9037	524	1	sin(ϕ)5	sin(ϕ)5	PROPN
ap-9037	524	2	+	+	CCONJ
ap-9037	524	3	cos(ϕ	cos(ϕ	NOUN
ap-9037	524	4	)	)	PUNCT
ap-9037	524	5	[	[	PUNCT
ap-9037	524	6	−8	−8	X
ap-9037	524	7	+	+	NOUN
ap-9037	524	8	2	2	NUM
ap-9037	524	9	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	524	10	+	+	CCONJ
ap-9037	524	11	3	3	NUM
ap-9037	524	12	sin(ϕ)4	sin(ϕ)4	ADV
ap-9037	524	13	]	]	PUNCT
ap-9037	524	14	φ̂(ϕ)r1	φ̂(ϕ)r1	NUM
ap-9037	524	15	=	=	SYM
ap-9037	524	16	r2=−1	r2=−1	NOUN
ap-9037	524	17	=	=	NOUN
ap-9037	524	18	cos(ϕ)8	cos(ϕ)8	VERB
ap-9037	524	19	sin(ϕ)3	sin(ϕ)3	VERB
ap-9037	524	20	2f1	2f1	NUM
ap-9037	524	21	[	[	PUNCT
ap-9037	524	22	2	2	NUM
ap-9037	524	23	,	,	PUNCT
ap-9037	524	24	7	7	NUM
ap-9037	524	25	,	,	PUNCT
ap-9037	524	26	11	11	NUM
ap-9037	524	27	2	2	NUM
ap-9037	524	28	,	,	PUNCT
ap-9037	524	29	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	524	30	]	]	PUNCT
ap-9037	524	31	−15	−15	PRON
ap-9037	524	32	arcsin[sin(ϕ	arcsin[sin(ϕ	PROPN
ap-9037	524	33	)	)	PUNCT
ap-9037	524	34	]	]	PUNCT
ap-9037	525	1	sin(ϕ)7	sin(ϕ)7	PROPN
ap-9037	525	2	+	+	NUM
ap-9037	525	3	cos(ϕ	cos(ϕ	NOUN
ap-9037	525	4	)	)	PUNCT
ap-9037	525	5	[	[	PUNCT
ap-9037	525	6	−112	−112	NOUN
ap-9037	525	7	−	−	PROPN
ap-9037	525	8	64	64	NUM
ap-9037	525	9	cos(2ϕ	cos(2ϕ	NOUN
ap-9037	525	10	)	)	PUNCT
ap-9037	526	1	+	+	CCONJ
ap-9037	526	2	8	8	NUM
ap-9037	526	3	sin(ϕ)2	sin(ϕ)2	NOUN
ap-9037	526	4	+	+	CCONJ
ap-9037	526	5	10	10	NUM
ap-9037	526	6	sin(ϕ)4	sin(ϕ)4	ADJ
ap-9037	526	7	+	+	SYM
ap-9037	526	8	15	15	NUM
ap-9037	526	9	sin(ϕ)6	sin(ϕ)6	NOUN
ap-9037	526	10	]	]	PUNCT
ap-9037	526	11	.	.	PUNCT
ap-9037	527	1	graphs	graph	NOUN
ap-9037	527	2	of	of	ADP
ap-9037	527	3	these	these	DET
ap-9037	527	4	solutions	solution	NOUN
ap-9037	527	5	and	and	CCONJ
ap-9037	527	6	of	of	ADP
ap-9037	527	7	more	more	ADJ
ap-9037	527	8	general	general	ADJ
ap-9037	527	9	cases	case	NOUN
ap-9037	527	10	are	be	AUX
ap-9037	527	11	shown	show	VERB
ap-9037	527	12	in	in	ADP
ap-9037	527	13	the	the	DET
ap-9037	527	14	figures	figure	NOUN
ap-9037	527	15	8	8	NUM
ap-9037	527	16	and	and	CCONJ
ap-9037	527	17	9	9	NUM
ap-9037	527	18	.	.	X
ap-9037	527	19	note	note	VERB
ap-9037	527	20	that	that	SCONJ
ap-9037	527	21	for	for	ADP
ap-9037	527	22	the	the	DET
ap-9037	527	23	sake	sake	NOUN
ap-9037	527	24	of	of	ADP
ap-9037	527	25	brevity	brevity	NOUN
ap-9037	527	26	,	,	PUNCT
ap-9037	527	27	we	we	PRON
ap-9037	527	28	will	will	AUX
ap-9037	527	29	not	not	PART
ap-9037	527	30	present	present	VERB
ap-9037	527	31	the	the	DET
ap-9037	527	32	explicit	explicit	ADJ
ap-9037	527	33	conditions	condition	NOUN
ap-9037	527	34	for	for	ADP
ap-9037	527	35	the	the	DET
ap-9037	527	36	parity	parity	NOUN
ap-9037	527	37	constants	constant	NOUN
ap-9037	527	38	r1	r1	NOUN
ap-9037	527	39	and	and	CCONJ
ap-9037	527	40	r2	r2	NOUN
ap-9037	527	41	.	.	PUNCT
ap-9037	528	1	1	1	NUM
ap-9037	528	2	2	2	NUM
ap-9037	528	3	3	3	NUM
ap-9037	528	4	4	4	NUM
ap-9037	528	5	5	5	NUM
ap-9037	528	6	6	6	NUM
ap-9037	528	7	φ	φ	PROPN
ap-9037	528	8	-0.4	-0.4	SYM
ap-9037	528	9	-0.2	-0.2	PROPN
ap-9037	528	10	0.2	0.2	NUM
ap-9037	528	11	0.4	0.4	NUM
ap-9037	528	12	f	f	NOUN
ap-9037	528	13	`	`	PUNCT
ap-9037	528	14	hφl	hφl	PROPN
ap-9037	528	15	n	n	NOUN
ap-9037	528	16	=	=	SYM
ap-9037	528	17	3	3	NUM
ap-9037	528	18	n	n	NOUN
ap-9037	528	19	=	=	SYM
ap-9037	528	20	2	2	NUM
ap-9037	528	21	n	n	NOUN
ap-9037	528	22	=	=	SYM
ap-9037	528	23	1	1	NUM
ap-9037	528	24	figure	figure	NOUN
ap-9037	528	25	8	8	NUM
ap-9037	528	26	.	.	PUNCT
ap-9037	529	1	graphs	graph	NOUN
ap-9037	529	2	of	of	ADP
ap-9037	529	3	the	the	DET
ap-9037	529	4	solution	solution	NOUN
ap-9037	529	5	(	(	PUNCT
ap-9037	529	6	80	80	NUM
ap-9037	529	7	)	)	PUNCT
ap-9037	529	8	for	for	ADP
ap-9037	529	9	the	the	DET
ap-9037	529	10	parameter	parameter	NOUN
ap-9037	529	11	settings	setting	NOUN
ap-9037	529	12	(	(	PUNCT
ap-9037	529	13	101	101	NUM
ap-9037	529	14	)	)	PUNCT
ap-9037	529	15	and	and	CCONJ
ap-9037	529	16	r1	r1	PROPN
ap-9037	529	17	=	=	SYM
ap-9037	529	18	r2	r2	PROPN
ap-9037	529	19	=	=	SYM
ap-9037	529	20	1	1	NUM
ap-9037	529	21	for	for	ADP
ap-9037	529	22	different	different	ADJ
ap-9037	529	23	values	value	NOUN
ap-9037	529	24	of	of	ADP
ap-9037	529	25	n	n	PROPN
ap-9037	529	26	.	.	PUNCT
ap-9037	530	1	1	1	NUM
ap-9037	530	2	2	2	NUM
ap-9037	530	3	3	3	NUM
ap-9037	530	4	4	4	NUM
ap-9037	530	5	5	5	NUM
ap-9037	530	6	6	6	NUM
ap-9037	530	7	φ	φ	PROPN
ap-9037	530	8	-0.3	-0.3	PROPN
ap-9037	530	9	-0.2	-0.2	PROPN
ap-9037	530	10	-0.1	-0.1	PROPN
ap-9037	530	11	0.1	0.1	NUM
ap-9037	530	12	0.2	0.2	NUM
ap-9037	530	13	0.3	0.3	NUM
ap-9037	530	14	f	f	NOUN
ap-9037	530	15	`	`	PUNCT
ap-9037	530	16	hφl	hφl	PROPN
ap-9037	530	17	n	n	NOUN
ap-9037	530	18	=	=	SYM
ap-9037	530	19	3	3	NUM
ap-9037	530	20	n	n	NOUN
ap-9037	530	21	=	=	SYM
ap-9037	530	22	2	2	NUM
ap-9037	530	23	n	n	NOUN
ap-9037	530	24	=	=	SYM
ap-9037	530	25	1	1	NUM
ap-9037	530	26	figure	figure	NOUN
ap-9037	530	27	9	9	NUM
ap-9037	530	28	.	.	PUNCT
ap-9037	530	29	graphs	graph	NOUN
ap-9037	530	30	of	of	ADP
ap-9037	530	31	the	the	DET
ap-9037	530	32	solution	solution	NOUN
ap-9037	530	33	(	(	PUNCT
ap-9037	530	34	80	80	NUM
ap-9037	530	35	)	)	PUNCT
ap-9037	530	36	for	for	ADP
ap-9037	530	37	the	the	DET
ap-9037	530	38	parameter	parameter	NOUN
ap-9037	530	39	settings	setting	NOUN
ap-9037	530	40	(	(	PUNCT
ap-9037	530	41	101	101	NUM
ap-9037	530	42	)	)	PUNCT
ap-9037	530	43	and	and	CCONJ
ap-9037	530	44	r1	r1	PROPN
ap-9037	530	45	=	=	SYM
ap-9037	530	46	r2	r2	PROPN
ap-9037	530	47	=	=	PUNCT
ap-9037	530	48	−1	−1	NOUN
ap-9037	530	49	for	for	ADP
ap-9037	530	50	different	different	ADJ
ap-9037	530	51	values	value	NOUN
ap-9037	530	52	of	of	ADP
ap-9037	530	53	n	n	PROPN
ap-9037	530	54	.	.	PUNCT
ap-9037	531	1	before	before	SCONJ
ap-9037	531	2	we	we	PRON
ap-9037	531	3	conclude	conclude	VERB
ap-9037	531	4	this	this	DET
ap-9037	531	5	application	application	NOUN
ap-9037	531	6	,	,	PUNCT
ap-9037	531	7	let	let	VERB
ap-9037	531	8	us	we	PRON
ap-9037	531	9	present	present	ADJ
ap-9037	531	10	examples	example	NOUN
ap-9037	531	11	of	of	ADP
ap-9037	531	12	the	the	DET
ap-9037	531	13	potential	potential	ADJ
ap-9037	531	14	(	(	PUNCT
ap-9037	531	15	95	95	NUM
ap-9037	531	16	)	)	PUNCT
ap-9037	531	17	that	that	PRON
ap-9037	531	18	is	be	AUX
ap-9037	531	19	inserted	insert	VERB
ap-9037	531	20	in	in	ADP
ap-9037	531	21	the	the	DET
ap-9037	531	22	transformed	transform	VERB
ap-9037	531	23	azimuthal	azimuthal	NOUN
ap-9037	531	24	equation	equation	NOUN
ap-9037	531	25	(	(	PUNCT
ap-9037	531	26	92	92	NUM
ap-9037	531	27	)	)	PUNCT
ap-9037	531	28	.	.	PUNCT
ap-9037	532	1	since	since	SCONJ
ap-9037	532	2	the	the	DET
ap-9037	532	3	transformation	transformation	NOUN
ap-9037	532	4	function	function	NOUN
ap-9037	532	5	(	(	PUNCT
ap-9037	532	6	100	100	NUM
ap-9037	532	7	)	)	PUNCT
ap-9037	532	8	is	be	AUX
ap-9037	532	9	not	not	PART
ap-9037	532	10	elementary	elementary	ADJ
ap-9037	532	11	,	,	PUNCT
ap-9037	532	12	the	the	DET
ap-9037	532	13	potential	potential	NOUN
ap-9037	532	14	can	can	AUX
ap-9037	532	15	not	not	PART
ap-9037	532	16	be	be	AUX
ap-9037	532	17	expressed	express	VERB
ap-9037	532	18	through	through	ADP
ap-9037	532	19	elementary	elementary	ADJ
ap-9037	532	20	functions	function	NOUN
ap-9037	532	21	either	either	ADV
ap-9037	532	22	.	.	PUNCT
ap-9037	533	1	since	since	SCONJ
ap-9037	533	2	its	its	PRON
ap-9037	533	3	general	general	ADJ
ap-9037	533	4	form	form	NOUN
ap-9037	533	5	is	be	AUX
ap-9037	533	6	very	very	ADV
ap-9037	533	7	long	long	ADJ
ap-9037	533	8	,	,	PUNCT
ap-9037	533	9	we	we	PRON
ap-9037	533	10	will	will	AUX
ap-9037	533	11	not	not	PART
ap-9037	533	12	show	show	VERB
ap-9037	533	13	it	it	PRON
ap-9037	533	14	here	here	ADV
ap-9037	533	15	.	.	PUNCT
ap-9037	534	1	instead	instead	ADV
ap-9037	534	2	,	,	PUNCT
ap-9037	534	3	we	we	PRON
ap-9037	534	4	show	show	VERB
ap-9037	534	5	examples	example	NOUN
ap-9037	534	6	in	in	ADP
ap-9037	534	7	figure	figure	NOUN
ap-9037	534	8	10	10	NUM
ap-9037	534	9	,	,	PUNCT
ap-9037	534	10	where	where	SCONJ
ap-9037	534	11	we	we	PRON
ap-9037	534	12	observe	observe	VERB
ap-9037	534	13	that	that	SCONJ
ap-9037	534	14	these	these	DET
ap-9037	534	15	potentials	potential	NOUN
ap-9037	534	16	can	can	AUX
ap-9037	534	17	be	be	AUX
ap-9037	534	18	interpreted	interpret	VERB
ap-9037	534	19	as	as	ADP
ap-9037	534	20	deformed	deform	VERB
ap-9037	534	21	trigonometric	trigonometric	ADJ
ap-9037	534	22	pöschl	pöschl	NOUN
ap-9037	534	23	-	-	PUNCT
ap-9037	534	24	teller	teller	NOUN
ap-9037	534	25	interactions	interaction	NOUN
ap-9037	534	26	.	.	PUNCT
ap-9037	535	1	note	note	VERB
ap-9037	535	2	that	that	SCONJ
ap-9037	535	3	this	this	DET
ap-9037	535	4	behaviour	behaviour	NOUN
ap-9037	535	5	is	be	AUX
ap-9037	535	6	expected	expect	VERB
ap-9037	535	7	due	due	ADP
ap-9037	535	8	to	to	ADP
ap-9037	535	9	the	the	DET
ap-9037	535	10	form	form	NOUN
ap-9037	535	11	of	of	ADP
ap-9037	535	12	(	(	PUNCT
ap-9037	535	13	95	95	NUM
ap-9037	535	14	)	)	PUNCT
ap-9037	535	15	.	.	PUNCT
ap-9037	536	1	290	290	NUM
ap-9037	536	2	vol	vol	NOUN
ap-9037	536	3	.	.	PUNCT
ap-9037	537	1	63	63	NUM
ap-9037	537	2	no	no	NOUN
ap-9037	537	3	.	.	PUNCT
ap-9037	538	1	4/2023	4/2023	NOUN
ap-9037	538	2	construction	construction	NOUN
ap-9037	538	3	of	of	ADP
ap-9037	538	4	angular	angular	ADJ
ap-9037	538	5	-	-	PUNCT
ap-9037	538	6	dependent	dependent	ADJ
ap-9037	538	7	potentials	potential	VERB
ap-9037	538	8	0	0	NUM
ap-9037	539	1	1	1	NUM
ap-9037	539	2	2	2	NUM
ap-9037	539	3	3	3	NUM
ap-9037	539	4	4	4	NUM
ap-9037	539	5	5	5	NUM
ap-9037	539	6	6	6	NUM
ap-9037	539	7	φ	φ	NUM
ap-9037	539	8	10	10	NUM
ap-9037	539	9	20	20	NUM
ap-9037	539	10	30	30	NUM
ap-9037	539	11	40	40	NUM
ap-9037	539	12	50	50	NUM
ap-9037	539	13	v	v	ADP
ap-9037	539	14	`	`	PUNCT
ap-9037	539	15	φhφl	φhφl	ADJ
ap-9037	539	16	r1	r1	NOUN
ap-9037	539	17	=	=	SYM
ap-9037	539	18	r2	r2	PROPN
ap-9037	539	19	=	=	SYM
ap-9037	539	20	1	1	NUM
ap-9037	539	21	r1	r1	NOUN
ap-9037	539	22	=	=	SYM
ap-9037	539	23	r2	r2	PROPN
ap-9037	539	24	=	=	SYM
ap-9037	539	25	-1	-1	PUNCT
ap-9037	539	26	figure	figure	NOUN
ap-9037	539	27	10	10	NUM
ap-9037	539	28	.	.	PUNCT
ap-9037	540	1	graphs	graph	NOUN
ap-9037	540	2	of	of	ADP
ap-9037	540	3	the	the	DET
ap-9037	540	4	potential	potential	ADJ
ap-9037	540	5	(	(	PUNCT
ap-9037	540	6	80	80	NUM
ap-9037	540	7	)	)	PUNCT
ap-9037	540	8	for	for	ADP
ap-9037	540	9	the	the	DET
ap-9037	540	10	parameter	parameter	NOUN
ap-9037	540	11	settings	setting	NOUN
ap-9037	540	12	(	(	PUNCT
ap-9037	540	13	101	101	NUM
ap-9037	540	14	)	)	PUNCT
ap-9037	540	15	and	and	CCONJ
ap-9037	540	16	different	different	ADJ
ap-9037	540	17	values	value	NOUN
ap-9037	540	18	of	of	ADP
ap-9037	540	19	r1	r1	NOUN
ap-9037	540	20	and	and	CCONJ
ap-9037	540	21	r2	r2	NOUN
ap-9037	540	22	.	.	PUNCT
ap-9037	541	1	6	6	X
ap-9037	541	2	.	.	X
ap-9037	541	3	concluding	conclude	VERB
ap-9037	541	4	remarks	remark	VERB
ap-9037	541	5	the	the	DET
ap-9037	541	6	main	main	ADJ
ap-9037	541	7	observation	observation	NOUN
ap-9037	541	8	in	in	ADP
ap-9037	541	9	this	this	DET
ap-9037	541	10	work	work	NOUN
ap-9037	541	11	concerns	concern	VERB
ap-9037	541	12	the	the	DET
ap-9037	541	13	close	close	ADJ
ap-9037	541	14	interrelationship	interrelationship	NOUN
ap-9037	541	15	between	between	ADP
ap-9037	541	16	the	the	DET
ap-9037	541	17	angular	angular	ADJ
ap-9037	541	18	equations	equation	NOUN
ap-9037	541	19	associated	associate	VERB
ap-9037	541	20	with	with	ADP
ap-9037	541	21	three	three	NUM
ap-9037	541	22	-	-	PUNCT
ap-9037	541	23	dimensional	dimensional	ADJ
ap-9037	541	24	,	,	PUNCT
ap-9037	541	25	non	non	ADJ
ap-9037	541	26	-	-	ADJ
ap-9037	541	27	relativistic	relativistic	ADJ
ap-9037	541	28	dunkl	dunkl	NOUN
ap-9037	541	29	-	-	PUNCT
ap-9037	541	30	schrödinger	schrödinger	ADJ
ap-9037	541	31	equations	equation	NOUN
ap-9037	541	32	in	in	ADP
ap-9037	541	33	spherical	spherical	ADJ
ap-9037	541	34	coordinates	coordinate	NOUN
ap-9037	541	35	that	that	PRON
ap-9037	541	36	is	be	AUX
ap-9037	541	37	derived	derive	VERB
ap-9037	541	38	in	in	ADP
ap-9037	541	39	section	section	NOUN
ap-9037	541	40	3	3	NUM
ap-9037	541	41	,	,	PUNCT
ap-9037	541	42	and	and	CCONJ
ap-9037	541	43	one	one	NUM
ap-9037	541	44	-	-	PUNCT
ap-9037	541	45	dimensional	dimensional	ADJ
ap-9037	541	46	trigonometric	trigonometric	ADJ
ap-9037	541	47	pöschl	pöschl	NOUN
ap-9037	541	48	-	-	PUNCT
ap-9037	541	49	teller	teller	NOUN
ap-9037	541	50	systems	system	NOUN
ap-9037	541	51	.	.	PUNCT
ap-9037	542	1	this	this	DET
ap-9037	542	2	interrelationship	interrelationship	NOUN
ap-9037	542	3	can	can	AUX
ap-9037	542	4	be	be	AUX
ap-9037	542	5	seen	see	VERB
ap-9037	542	6	after	after	ADP
ap-9037	542	7	separating	separate	VERB
ap-9037	542	8	the	the	DET
ap-9037	542	9	variables	variable	NOUN
ap-9037	542	10	in	in	ADP
ap-9037	542	11	the	the	DET
ap-9037	542	12	three	three	NUM
ap-9037	542	13	-	-	PUNCT
ap-9037	542	14	dimensional	dimensional	ADJ
ap-9037	542	15	equation	equation	NOUN
ap-9037	542	16	.	.	PUNCT
ap-9037	543	1	as	as	SCONJ
ap-9037	543	2	shown	show	VERB
ap-9037	543	3	in	in	ADP
ap-9037	543	4	sections	section	NOUN
ap-9037	543	5	4.1	4.1	NUM
ap-9037	543	6	and	and	CCONJ
ap-9037	543	7	5.1	5.1	NUM
ap-9037	543	8	,	,	PUNCT
ap-9037	543	9	a	a	DET
ap-9037	543	10	simple	simple	ADJ
ap-9037	543	11	point	point	NOUN
ap-9037	543	12	transformation	transformation	NOUN
ap-9037	543	13	takes	take	VERB
ap-9037	543	14	each	each	PRON
ap-9037	543	15	of	of	ADP
ap-9037	543	16	the	the	DET
ap-9037	543	17	two	two	NUM
ap-9037	543	18	angular	angular	ADJ
ap-9037	543	19	equations	equation	NOUN
ap-9037	543	20	resulting	result	VERB
ap-9037	543	21	from	from	ADP
ap-9037	543	22	the	the	DET
ap-9037	543	23	separation	separation	NOUN
ap-9037	543	24	,	,	PUNCT
ap-9037	543	25	into	into	ADP
ap-9037	543	26	the	the	DET
ap-9037	543	27	form	form	NOUN
ap-9037	543	28	of	of	ADP
ap-9037	543	29	a	a	DET
ap-9037	543	30	schrödinger	schrödinger	ADJ
ap-9037	543	31	equation	equation	NOUN
ap-9037	543	32	for	for	ADP
ap-9037	543	33	a	a	DET
ap-9037	543	34	trigonometric	trigonometric	ADJ
ap-9037	543	35	pöschl	pöschl	NOUN
ap-9037	543	36	-	-	PUNCT
ap-9037	543	37	teller	teller	NOUN
ap-9037	543	38	potential	potential	NOUN
ap-9037	543	39	.	.	PUNCT
ap-9037	544	1	it	it	PRON
ap-9037	544	2	is	be	AUX
ap-9037	544	3	interesting	interesting	ADJ
ap-9037	544	4	to	to	PART
ap-9037	544	5	note	note	VERB
ap-9037	544	6	here	here	ADV
ap-9037	544	7	that	that	SCONJ
ap-9037	544	8	in	in	ADP
ap-9037	544	9	the	the	DET
ap-9037	544	10	standard	standard	ADJ
ap-9037	544	11	case	case	NOUN
ap-9037	544	12	without	without	ADP
ap-9037	544	13	the	the	DET
ap-9037	544	14	presence	presence	NOUN
ap-9037	544	15	of	of	ADP
ap-9037	544	16	dunkl	dunkl	PROPN
ap-9037	544	17	operators	operator	NOUN
ap-9037	544	18	ν1	ν1	NOUN
ap-9037	544	19	=	=	SYM
ap-9037	544	20	ν2	ν2	NOUN
ap-9037	544	21	=	=	SYM
ap-9037	544	22	ν3	ν3	NOUN
ap-9037	544	23	=	=	SYM
ap-9037	544	24	0	0	NUM
ap-9037	544	25	,	,	PUNCT
ap-9037	544	26	the	the	DET
ap-9037	544	27	two	two	NUM
ap-9037	544	28	angular	angular	ADJ
ap-9037	544	29	equations	equation	NOUN
ap-9037	544	30	(	(	PUNCT
ap-9037	544	31	27	27	NUM
ap-9037	544	32	)	)	PUNCT
ap-9037	544	33	and	and	CCONJ
ap-9037	544	34	(	(	PUNCT
ap-9037	544	35	28	28	NUM
ap-9037	544	36	)	)	PUNCT
ap-9037	544	37	do	do	AUX
ap-9037	544	38	not	not	PART
ap-9037	544	39	contain	contain	VERB
ap-9037	544	40	a	a	DET
ap-9037	544	41	trigonometric	trigonometric	ADJ
ap-9037	544	42	pöschl	pöschl	NOUN
ap-9037	544	43	-	-	PUNCT
ap-9037	544	44	teller	teller	NOUN
ap-9037	544	45	potential	potential	NOUN
ap-9037	544	46	.	.	PUNCT
ap-9037	545	1	more	more	ADV
ap-9037	545	2	precisely	precisely	ADV
ap-9037	545	3	,	,	PUNCT
ap-9037	545	4	the	the	DET
ap-9037	545	5	polar	polar	ADJ
ap-9037	545	6	equation	equation	NOUN
ap-9037	545	7	(	(	PUNCT
ap-9037	545	8	27	27	NUM
ap-9037	545	9	)	)	PUNCT
ap-9037	545	10	maintains	maintain	VERB
ap-9037	545	11	a	a	DET
ap-9037	545	12	term	term	NOUN
ap-9037	545	13	proportional	proportional	ADJ
ap-9037	545	14	to	to	ADP
ap-9037	545	15	an	an	DET
ap-9037	545	16	inverse	inverse	ADJ
ap-9037	545	17	sine	sine	NOUN
ap-9037	545	18	function	function	NOUN
ap-9037	545	19	,	,	PUNCT
ap-9037	545	20	while	while	SCONJ
ap-9037	545	21	there	there	PRON
ap-9037	545	22	is	be	VERB
ap-9037	545	23	no	no	DET
ap-9037	545	24	trigonometric	trigonometric	ADJ
ap-9037	545	25	term	term	NOUN
ap-9037	545	26	in	in	ADP
ap-9037	545	27	the	the	DET
ap-9037	545	28	azimuthal	azimuthal	ADJ
ap-9037	545	29	equation	equation	NOUN
ap-9037	545	30	(	(	PUNCT
ap-9037	545	31	28	28	NUM
ap-9037	545	32	)	)	PUNCT
ap-9037	545	33	.	.	PUNCT
ap-9037	546	1	returning	return	VERB
ap-9037	546	2	to	to	ADP
ap-9037	546	3	the	the	DET
ap-9037	546	4	dunkl	dunkl	PROPN
ap-9037	546	5	scenario	scenario	NOUN
ap-9037	546	6	,	,	PUNCT
ap-9037	546	7	we	we	PRON
ap-9037	546	8	can	can	AUX
ap-9037	546	9	find	find	VERB
ap-9037	546	10	solutions	solution	NOUN
ap-9037	546	11	of	of	ADP
ap-9037	546	12	our	our	PRON
ap-9037	546	13	angular	angular	ADJ
ap-9037	546	14	equations	equation	NOUN
ap-9037	546	15	by	by	ADP
ap-9037	546	16	using	use	VERB
ap-9037	546	17	known	know	VERB
ap-9037	546	18	results	result	NOUN
ap-9037	546	19	on	on	ADP
ap-9037	546	20	trigonometric	trigonometric	ADJ
ap-9037	546	21	pöschl	pöschl	NOUN
ap-9037	546	22	-	-	PUNCT
ap-9037	546	23	teller	teller	NOUN
ap-9037	546	24	systems	system	NOUN
ap-9037	546	25	,	,	PUNCT
ap-9037	546	26	as	as	SCONJ
ap-9037	546	27	demonstrated	demonstrate	VERB
ap-9037	546	28	in	in	ADP
ap-9037	546	29	sections	section	NOUN
ap-9037	546	30	4.1	4.1	NUM
ap-9037	546	31	and	and	CCONJ
ap-9037	546	32	5.1	5.1	NUM
ap-9037	546	33	,	,	PUNCT
ap-9037	546	34	respectively	respectively	ADV
ap-9037	546	35	.	.	PUNCT
ap-9037	547	1	since	since	SCONJ
ap-9037	547	2	the	the	DET
ap-9037	547	3	latter	latter	ADJ
ap-9037	547	4	angular	angular	ADJ
ap-9037	547	5	equations	equation	NOUN
ap-9037	547	6	in	in	ADP
ap-9037	547	7	their	their	PRON
ap-9037	547	8	versions	version	NOUN
ap-9037	547	9	(	(	PUNCT
ap-9037	547	10	30	30	NUM
ap-9037	547	11	)	)	PUNCT
ap-9037	547	12	and	and	CCONJ
ap-9037	547	13	(	(	PUNCT
ap-9037	547	14	73	73	NUM
ap-9037	547	15	)	)	PUNCT
ap-9037	547	16	match	match	VERB
ap-9037	547	17	the	the	DET
ap-9037	547	18	schrödinger	schrödinger	ADJ
ap-9037	547	19	form	form	NOUN
ap-9037	547	20	,	,	PUNCT
ap-9037	547	21	the	the	DET
ap-9037	547	22	darboux	darboux	VERB
ap-9037	547	23	-	-	PUNCT
ap-9037	547	24	crum	crum	NOUN
ap-9037	547	25	transformation	transformation	NOUN
ap-9037	547	26	becomes	become	VERB
ap-9037	547	27	applicable	applicable	ADJ
ap-9037	547	28	to	to	ADP
ap-9037	547	29	them	they	PRON
ap-9037	547	30	.	.	PUNCT
ap-9037	548	1	as	as	SCONJ
ap-9037	548	2	the	the	DET
ap-9037	548	3	examples	example	NOUN
ap-9037	548	4	in	in	ADP
ap-9037	548	5	sections	section	NOUN
ap-9037	548	6	4.2	4.2	NUM
ap-9037	548	7	and	and	CCONJ
ap-9037	548	8	5.2	5.2	NUM
ap-9037	548	9	illustrate	illustrate	NOUN
ap-9037	548	10	,	,	PUNCT
ap-9037	548	11	the	the	DET
ap-9037	548	12	application	application	NOUN
ap-9037	548	13	of	of	ADP
ap-9037	548	14	the	the	DET
ap-9037	548	15	darboux	darboux	ADJ
ap-9037	548	16	-	-	PUNCT
ap-9037	548	17	crum	crum	NOUN
ap-9037	548	18	transformation	transformation	NOUN
ap-9037	548	19	allows	allow	VERB
ap-9037	548	20	to	to	PART
ap-9037	548	21	construct	construct	VERB
ap-9037	548	22	solvable	solvable	ADJ
ap-9037	548	23	angular	angular	ADJ
ap-9037	548	24	equations	equation	NOUN
ap-9037	548	25	within	within	ADP
ap-9037	548	26	the	the	DET
ap-9037	548	27	dunkl	dunkl	NOUN
ap-9037	548	28	formalism	formalism	NOUN
ap-9037	548	29	by	by	ADP
ap-9037	548	30	using	use	VERB
ap-9037	548	31	results	result	NOUN
ap-9037	548	32	on	on	ADP
ap-9037	548	33	trigonometric	trigonometric	ADJ
ap-9037	548	34	pöschl	pöschl	NOUN
ap-9037	548	35	-	-	PUNCT
ap-9037	548	36	teller	teller	NOUN
ap-9037	548	37	systems	system	NOUN
ap-9037	548	38	and	and	CCONJ
ap-9037	548	39	their	their	PRON
ap-9037	548	40	generalisations	generalisation	NOUN
ap-9037	548	41	.	.	PUNCT
ap-9037	549	1	as	as	SCONJ
ap-9037	549	2	mentioned	mention	VERB
ap-9037	549	3	above	above	ADV
ap-9037	549	4	,	,	PUNCT
ap-9037	549	5	the	the	DET
ap-9037	549	6	darboux	darboux	VERB
ap-9037	549	7	-	-	PUNCT
ap-9037	549	8	crum	crum	NOUN
ap-9037	549	9	transformation	transformation	NOUN
ap-9037	549	10	generates	generate	VERB
ap-9037	549	11	potentials	potential	NOUN
ap-9037	549	12	that	that	PRON
ap-9037	549	13	will	will	AUX
ap-9037	549	14	generally	generally	ADV
ap-9037	549	15	depend	depend	VERB
ap-9037	549	16	on	on	ADP
ap-9037	549	17	all	all	DET
ap-9037	549	18	parameters	parameter	NOUN
ap-9037	549	19	in	in	ADP
ap-9037	549	20	the	the	DET
ap-9037	549	21	respective	respective	ADJ
ap-9037	549	22	angular	angular	ADJ
ap-9037	549	23	equation	equation	NOUN
ap-9037	549	24	except	except	SCONJ
ap-9037	549	25	for	for	ADP
ap-9037	549	26	the	the	DET
ap-9037	549	27	transformation	transformation	NOUN
ap-9037	549	28	parameter	parameter	NOUN
ap-9037	549	29	.	.	PUNCT
ap-9037	550	1	since	since	SCONJ
ap-9037	550	2	this	this	DET
ap-9037	550	3	behaviour	behaviour	NOUN
ap-9037	550	4	is	be	AUX
ap-9037	550	5	undesirable	undesirable	ADJ
ap-9037	550	6	,	,	PUNCT
ap-9037	550	7	it	it	PRON
ap-9037	550	8	is	be	AUX
ap-9037	550	9	necessary	necessary	ADJ
ap-9037	550	10	to	to	PART
ap-9037	550	11	assign	assign	VERB
ap-9037	550	12	fixed	fix	VERB
ap-9037	550	13	values	value	NOUN
ap-9037	550	14	to	to	ADP
ap-9037	550	15	all	all	DET
ap-9037	550	16	these	these	DET
ap-9037	550	17	parameters	parameter	NOUN
ap-9037	550	18	before	before	ADP
ap-9037	550	19	performing	perform	VERB
ap-9037	550	20	the	the	DET
ap-9037	550	21	darboux	darboux	ADJ
ap-9037	550	22	-	-	PUNCT
ap-9037	550	23	crum	crum	NOUN
ap-9037	550	24	transformation	transformation	NOUN
ap-9037	550	25	,	,	PUNCT
ap-9037	550	26	which	which	PRON
ap-9037	550	27	imposes	impose	VERB
ap-9037	550	28	a	a	DET
ap-9037	550	29	strong	strong	ADJ
ap-9037	550	30	constraint	constraint	NOUN
ap-9037	550	31	on	on	ADP
ap-9037	550	32	the	the	DET
ap-9037	550	33	transformed	transform	VERB
ap-9037	550	34	system	system	NOUN
ap-9037	550	35	.	.	PUNCT
ap-9037	551	1	references	reference	NOUN
ap-9037	551	2	[	[	X
ap-9037	551	3	1	1	NUM
ap-9037	551	4	]	]	PUNCT
ap-9037	551	5	c.	c.	PROPN
ap-9037	551	6	f.	f.	PROPN
ap-9037	551	7	dunkl	dunkl	PROPN
ap-9037	551	8	.	.	PUNCT
ap-9037	552	1	differential	differential	ADJ
ap-9037	552	2	-	-	PUNCT
ap-9037	552	3	difference	difference	NOUN
ap-9037	552	4	operators	operator	NOUN
ap-9037	552	5	associated	associate	VERB
ap-9037	552	6	to	to	ADP
ap-9037	552	7	reflection	reflection	NOUN
ap-9037	552	8	groups	group	NOUN
ap-9037	552	9	.	.	PUNCT
ap-9037	553	1	transactions	transaction	NOUN
ap-9037	553	2	of	of	ADP
ap-9037	553	3	the	the	DET
ap-9037	553	4	american	american	PROPN
ap-9037	553	5	mathematical	mathematical	PROPN
ap-9037	553	6	society	society	PROPN
ap-9037	553	7	311(1):167–183	311(1):167–183	NUM
ap-9037	553	8	,	,	PUNCT
ap-9037	553	9	1989	1989	NUM
ap-9037	553	10	.	.	PUNCT
ap-9037	554	1	https://doi.org/10.2307/2001022	https://doi.org/10.2307/2001022	PRON
ap-9037	555	1	[	[	X
ap-9037	555	2	2	2	NUM
ap-9037	555	3	]	]	PUNCT
ap-9037	555	4	m.	m.	NOUN
ap-9037	555	5	rösler	rösler	NOUN
ap-9037	555	6	.	.	PUNCT
ap-9037	556	1	orthogonal	orthogonal	ADJ
ap-9037	556	2	polynomials	polynomial	NOUN
ap-9037	556	3	and	and	CCONJ
ap-9037	556	4	special	special	ADJ
ap-9037	556	5	functions	function	NOUN
ap-9037	556	6	.	.	PUNCT
ap-9037	557	1	in	in	ADP
ap-9037	557	2	dunkl	dunkl	PROPN
ap-9037	557	3	operators	operator	NOUN
ap-9037	557	4	:	:	PUNCT
ap-9037	557	5	theory	theory	NOUN
ap-9037	557	6	and	and	CCONJ
ap-9037	557	7	applications	application	NOUN
ap-9037	557	8	,	,	PUNCT
ap-9037	557	9	vol	vol	NOUN
ap-9037	557	10	.	.	PROPN
ap-9037	557	11	1817	1817	NUM
ap-9037	557	12	,	,	PUNCT
ap-9037	557	13	pp	pp	ADP
ap-9037	557	14	.	.	PUNCT
ap-9037	558	1	93–135	93–135	NOUN
ap-9037	558	2	.	.	PUNCT
ap-9037	558	3	springer	springer	NOUN
ap-9037	558	4	,	,	PUNCT
ap-9037	558	5	berlin	berlin	PROPN
ap-9037	558	6	,	,	PUNCT
ap-9037	558	7	d	d	PROPN
ap-9037	558	8	,	,	PUNCT
ap-9037	558	9	2003	2003	NUM
ap-9037	558	10	.	.	PUNCT
ap-9037	559	1	https://doi.org/10.1007/3-540-44945-0_3	https://doi.org/10.1007/3-540-44945-0_3	VERB
ap-9037	560	1	[	[	X
ap-9037	560	2	3	3	X
ap-9037	560	3	]	]	PUNCT
ap-9037	560	4	j.	j.	PROPN
ap-9037	560	5	f.	f.	PROPN
ap-9037	560	6	diejen	diejen	PROPN
ap-9037	560	7	,	,	PUNCT
ap-9037	560	8	l.	l.	PROPN
ap-9037	560	9	vinet	vinet	PROPN
ap-9037	560	10	.	.	PUNCT
ap-9037	561	1	calogero	calogero	PROPN
ap-9037	561	2	-	-	PUNCT
ap-9037	561	3	moser	moser	PROPN
ap-9037	561	4	-	-	PUNCT
ap-9037	561	5	sutherland	sutherland	PROPN
ap-9037	561	6	models	model	NOUN
ap-9037	561	7	.	.	PUNCT
ap-9037	562	1	springer	springer	NOUN
ap-9037	562	2	,	,	PUNCT
ap-9037	562	3	new	new	PROPN
ap-9037	562	4	york	york	PROPN
ap-9037	562	5	,	,	PUNCT
ap-9037	562	6	ny	ny	PROPN
ap-9037	562	7	,	,	PUNCT
ap-9037	562	8	1st	1st	ADJ
ap-9037	562	9	edn	edn	PROPN
ap-9037	562	10	.	.	PUNCT
ap-9037	562	11	,	,	PUNCT
ap-9037	562	12	2000	2000	NUM
ap-9037	562	13	.	.	PUNCT
ap-9037	563	1	https://doi.org/10.1007/978-1-4612-1206-5	https://doi.org/10.1007/978-1-4612-1206-5	PROPN
ap-9037	564	1	[	[	X
ap-9037	564	2	4	4	X
ap-9037	564	3	]	]	X
ap-9037	564	4	p.	p.	NOUN
ap-9037	564	5	etingof	etingof	PROPN
ap-9037	564	6	.	.	PUNCT
ap-9037	565	1	calogero	calogero	PROPN
ap-9037	565	2	-	-	PUNCT
ap-9037	565	3	moser	moser	PROPN
ap-9037	565	4	systems	systems	PROPN
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ap-9037	565	7	theory	theory	NOUN
ap-9037	565	8	.	.	PUNCT
ap-9037	566	1	european	european	PROPN
ap-9037	566	2	mathematical	mathematical	PROPN
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ap-9037	566	4	,	,	PUNCT
ap-9037	566	5	zurich	zurich	PROPN
ap-9037	566	6	,	,	PUNCT
ap-9037	566	7	ch	ch	NOUN
ap-9037	566	8	,	,	PUNCT
ap-9037	566	9	2007	2007	NUM
ap-9037	566	10	.	.	PUNCT
ap-9037	567	1	https://doi.org/10.4171/034	https://doi.org/10.4171/034	NOUN
ap-9037	567	2	[	[	X
ap-9037	567	3	5	5	NUM
ap-9037	567	4	]	]	PUNCT
ap-9037	567	5	m.	m.	NOUN
ap-9037	567	6	feigin	feigin	NOUN
ap-9037	567	7	,	,	PUNCT
ap-9037	567	8	t.	t.	PROPN
ap-9037	567	9	hakobyan	hakobyan	PROPN
ap-9037	567	10	.	.	PUNCT
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ap-9037	568	2	dunkl	dunkl	PROPN
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ap-9037	568	4	momenta	momenta	NOUN
ap-9037	568	5	algebra	algebra	PROPN
ap-9037	568	6	.	.	PUNCT
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ap-9037	569	2	of	of	ADP
ap-9037	569	3	high	high	ADJ
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ap-9037	569	5	physics	physics	NOUN
ap-9037	569	6	2015:107	2015:107	PROPN
ap-9037	569	7	,	,	PUNCT
ap-9037	569	8	2015	2015	NUM
ap-9037	569	9	.	.	PUNCT
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ap-9037	571	2	6	6	NUM
ap-9037	571	3	]	]	PUNCT
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ap-9037	571	6	,	,	PUNCT
ap-9037	571	7	m.	m.	NOUN
ap-9037	571	8	vrabec	vrabec	PROPN
ap-9037	571	9	.	.	PUNCT
ap-9037	572	1	intertwining	intertwine	VERB
ap-9037	572	2	operator	operator	NOUN
ap-9037	572	3	for	for	ADP
ap-9037	572	4	ag2	ag2	PROPN
ap-9037	572	5	calogero	calogero	PROPN
ap-9037	572	6	-	-	PUNCT
ap-9037	572	7	moser	moser	PROPN
ap-9037	572	8	-	-	PUNCT
ap-9037	572	9	sutherland	sutherland	PROPN
ap-9037	572	10	system	system	NOUN
ap-9037	572	11	.	.	PUNCT
ap-9037	573	1	journal	journal	PROPN
ap-9037	573	2	of	of	ADP
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ap-9037	573	4	physics	physics	NOUN
ap-9037	573	5	60:073503	60:073503	PROPN
ap-9037	573	6	,	,	PUNCT
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ap-9037	573	8	.	.	PUNCT
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ap-9037	574	3	7	7	X
ap-9037	574	4	]	]	PUNCT
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ap-9037	574	8	.	.	PUNCT
ap-9037	575	1	an	an	DET
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ap-9037	575	3	to	to	PART
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ap-9037	575	10	.	.	PUNCT
ap-9037	576	1	in	in	ADP
ap-9037	576	2	analytic	analytic	ADJ
ap-9037	576	3	,	,	PUNCT
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ap-9037	576	5	and	and	CCONJ
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ap-9037	576	7	aspects	aspect	NOUN
ap-9037	576	8	of	of	ADP
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ap-9037	576	11	.	.	PUNCT
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ap-9037	577	2	in	in	ADP
ap-9037	577	3	mathematics	mathematic	NOUN
ap-9037	577	4	,	,	PUNCT
ap-9037	577	5	pp	pp	ADP
ap-9037	577	6	.	.	PUNCT
ap-9037	578	1	3–58	3–58	PROPN
ap-9037	578	2	.	.	PUNCT
ap-9037	579	1	birkhäuser	birkhäuser	PROPN
ap-9037	579	2	,	,	PUNCT
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ap-9037	579	4	,	,	PUNCT
ap-9037	579	5	ch	ch	NOUN
ap-9037	579	6	,	,	PUNCT
ap-9037	579	7	2017	2017	NUM
ap-9037	579	8	.	.	PUNCT
ap-9037	580	1	https://doi.org/10.1007/978-3-319-52842-7_1	https://doi.org/10.1007/978-3-319-52842-7_1	X
ap-9037	581	1	[	[	X
ap-9037	581	2	8	8	NUM
ap-9037	581	3	]	]	X
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ap-9037	582	4	-	-	PUNCT
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ap-9037	582	9	.	.	PUNCT
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ap-9037	583	5	and	and	CCONJ
ap-9037	583	6	applications	application	NOUN
ap-9037	583	7	375(1):118–138	375(1):118–138	NUM
ap-9037	583	8	,	,	PUNCT
ap-9037	583	9	2011	2011	NUM
ap-9037	583	10	.	.	PUNCT
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ap-9037	584	3	9	9	NUM
ap-9037	584	4	]	]	PUNCT
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ap-9037	584	6	f.	f.	PROPN
ap-9037	584	7	dunkl	dunkl	PROPN
ap-9037	584	8	,	,	PUNCT
ap-9037	584	9	y.	y.	PROPN
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ap-9037	584	11	.	.	PUNCT
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ap-9037	585	6	.	.	PUNCT
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ap-9037	586	4	,	,	PUNCT
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ap-9037	586	14	.	.	PUNCT
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ap-9037	588	8	h.	h.	PROPN
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ap-9037	588	10	.	.	PUNCT
ap-9037	589	1	one	one	NUM
ap-9037	589	2	-	-	PUNCT
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ap-9037	589	9	.	.	PUNCT
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ap-9037	590	5	34(24):1950190	34(24):1950190	NUM
ap-9037	590	6	,	,	PUNCT
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ap-9037	590	8	.	.	PUNCT
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ap-9037	591	12	https://doi.org/10.1142/s0217732319501906	https://doi.org/10.1142/s0217732319501906	NUM
ap-9037	591	13	axel	axel	ADJ
ap-9037	591	14	schulze	schulze	NOUN
ap-9037	591	15	-	-	PUNCT
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ap-9037	591	17	acta	acta	PROPN
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ap-9037	592	1	[	[	X
ap-9037	592	2	11	11	NUM
ap-9037	592	3	]	]	X
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ap-9037	592	16	a.	a.	PROPN
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ap-9037	593	7	ii	ii	NOUN
ap-9037	593	8	:	:	PUNCT
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ap-9037	593	13	.	.	PUNCT
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ap-9037	594	6	,	,	PUNCT
ap-9037	594	7	2014	2014	NUM
ap-9037	594	8	.	.	PUNCT
ap-9037	595	1	https://doi.org/10.1007/s00220-014-1915-2	https://doi.org/10.1007/s00220-014-1915-2	NUM
ap-9037	596	1	[	[	X
ap-9037	596	2	12	12	NUM
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ap-9037	596	5	x.	x.	PROPN
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ap-9037	596	8	m.	m.	PROPN
ap-9037	596	9	e.	e.	PROPN
ap-9037	596	10	h.	h.	PROPN
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ap-9037	597	7	i	i	PRON
ap-9037	597	8	:	:	PUNCT
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ap-9037	597	10	,	,	PUNCT
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ap-9037	597	16	.	.	PUNCT
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ap-9037	598	5	:	:	PUNCT
ap-9037	598	6	mathematical	mathematical	ADJ
ap-9037	598	7	and	and	CCONJ
ap-9037	598	8	theoretical	theoretical	ADJ
ap-9037	598	9	46(14):145201	46(14):145201	NUM
ap-9037	598	10	,	,	PUNCT
ap-9037	598	11	2013	2013	NUM
ap-9037	598	12	.	.	PUNCT
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ap-9037	600	13	.	.	PUNCT
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ap-9037	601	3	-	-	PUNCT
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ap-9037	602	5	,	,	PUNCT
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ap-9037	602	7	.	.	PUNCT
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ap-9037	603	3	14	14	NUM
ap-9037	603	4	]	]	PUNCT
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ap-9037	603	8	,	,	PUNCT
ap-9037	603	9	d.	d.	PROPN
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ap-9037	603	11	-	-	PUNCT
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ap-9037	603	13	,	,	PUNCT
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ap-9037	604	7	-	-	PUNCT
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ap-9037	604	9	-	-	PUNCT
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ap-9037	604	19	-	-	PUNCT
ap-9037	604	20	gordon	gordon	PROPN
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ap-9037	604	22	.	.	PUNCT
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ap-9037	605	8	.	.	PUNCT
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ap-9037	606	3	15	15	NUM
ap-9037	606	4	]	]	X
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ap-9037	606	7	.	.	PUNCT
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ap-9037	607	2	-	-	PUNCT
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ap-9037	609	1	oh	oh	INTJ
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ap-9037	609	3	a	a	PRON
ap-9037	609	4	:	:	PUNCT
ap-9037	609	5	mathematical	mathematical	ADJ
ap-9037	609	6	and	and	CCONJ
ap-9037	609	7	theoretical	theoretical	ADJ
ap-9037	609	8	56(26):265203	56(26):265203	NUM
ap-9037	609	9	,	,	PUNCT
ap-9037	609	10	2023	2023	NUM
ap-9037	609	11	.	.	PUNCT
ap-9037	610	1	https://doi.org/10.1088/1751-8121/acd736	https://doi.org/10.1088/1751-8121/acd736	PROPN
ap-9037	611	1	[	[	X
ap-9037	611	2	16	16	NUM
ap-9037	611	3	]	]	X
ap-9037	611	4	d.	d.	PROPN
ap-9037	611	5	ojeda	ojeda	PROPN
ap-9037	611	6	-	-	PUNCT
ap-9037	611	7	guillén	guillén	PROPN
ap-9037	611	8	,	,	PUNCT
ap-9037	611	9	r.	r.	PROPN
ap-9037	611	10	d.	d.	PROPN
ap-9037	611	11	mota	mota	PROPN
ap-9037	611	12	,	,	PUNCT
ap-9037	611	13	m.	m.	NOUN
ap-9037	611	14	salazar	salazar	NOUN
ap-9037	611	15	-	-	PUNCT
ap-9037	611	16	ramírez	ramírez	NOUN
ap-9037	611	17	,	,	PUNCT
ap-9037	611	18	v.	v.	PROPN
ap-9037	611	19	d.	d.	PROPN
ap-9037	611	20	granados	granados	PROPN
ap-9037	611	21	.	.	PUNCT
ap-9037	612	1	algebraic	algebraic	ADJ
ap-9037	612	2	approach	approach	NOUN
ap-9037	612	3	for	for	ADP
ap-9037	612	4	the	the	DET
ap-9037	612	5	one	one	NUM
ap-9037	612	6	-	-	PUNCT
ap-9037	612	7	dimensional	dimensional	ADJ
ap-9037	612	8	dirac	dirac	NOUN
ap-9037	612	9	-	-	PUNCT
ap-9037	612	10	dunkl	dunkl	NOUN
ap-9037	612	11	oscillator	oscillator	NOUN
ap-9037	612	12	.	.	PUNCT
ap-9037	613	1	modern	modern	ADJ
ap-9037	613	2	physics	physics	NOUN
ap-9037	613	3	letters	letter	NOUN
ap-9037	613	4	a	a	DET
ap-9037	613	5	35(31):2050255	35(31):2050255	NUM
ap-9037	613	6	,	,	PUNCT
ap-9037	613	7	2020	2020	NUM
ap-9037	613	8	.	.	PUNCT
ap-9037	614	1	https://doi.org/10.1142/s0217732320502557	https://doi.org/10.1142/s0217732320502557	NUM
ap-9037	614	2	[	[	X
ap-9037	614	3	17	17	NUM
ap-9037	614	4	]	]	PUNCT
ap-9037	614	5	s.	s.	PROPN
ap-9037	614	6	sargolzaeipor	sargolzaeipor	PROPN
ap-9037	614	7	,	,	PUNCT
ap-9037	614	8	h.	h.	PROPN
ap-9037	614	9	hassanabadi	hassanabadi	PROPN
ap-9037	614	10	,	,	PUNCT
ap-9037	614	11	w.	w.	PROPN
ap-9037	614	12	s.	s.	PROPN
ap-9037	614	13	chung	chung	PROPN
ap-9037	614	14	.	.	PUNCT
ap-9037	615	1	effect	effect	NOUN
ap-9037	615	2	of	of	ADP
ap-9037	615	3	the	the	DET
ap-9037	615	4	wigner	wigner	NOUN
ap-9037	615	5	-	-	PUNCT
ap-9037	615	6	dunkl	dunkl	NOUN
ap-9037	615	7	algebra	algebra	NOUN
ap-9037	615	8	on	on	ADP
ap-9037	615	9	the	the	DET
ap-9037	615	10	dirac	dirac	NOUN
ap-9037	615	11	equation	equation	NOUN
ap-9037	615	12	and	and	CCONJ
ap-9037	615	13	dirac	dirac	NOUN
ap-9037	615	14	harmonic	harmonic	ADJ
ap-9037	615	15	oscillator	oscillator	NOUN
ap-9037	615	16	.	.	PUNCT
ap-9037	616	1	modern	modern	ADJ
ap-9037	616	2	physics	physics	NOUN
ap-9037	616	3	letters	letter	NOUN
ap-9037	616	4	a	a	DET
ap-9037	616	5	33(25):1850146	33(25):1850146	NUM
ap-9037	616	6	,	,	PUNCT
ap-9037	616	7	2018	2018	NUM
ap-9037	616	8	.	.	PUNCT
ap-9037	617	1	https://doi.org/10.1142/s0217732318501468	https://doi.org/10.1142/s0217732318501468	NUM
ap-9037	617	2	[	[	X
ap-9037	617	3	18	18	NUM
ap-9037	617	4	]	]	PUNCT
ap-9037	617	5	r.	r.	PROPN
ap-9037	617	6	d.	d.	PROPN
ap-9037	617	7	mota	mota	PROPN
ap-9037	617	8	,	,	PUNCT
ap-9037	617	9	d.	d.	PROPN
ap-9037	617	10	ojeda	ojeda	PROPN
ap-9037	617	11	-	-	PUNCT
ap-9037	617	12	guillén	guillén	PROPN
ap-9037	617	13	.	.	PUNCT
ap-9037	618	1	exact	exact	ADJ
ap-9037	618	2	solutions	solution	NOUN
ap-9037	618	3	of	of	ADP
ap-9037	618	4	the	the	DET
ap-9037	618	5	schrödinger	schrödinger	ADJ
ap-9037	618	6	equation	equation	NOUN
ap-9037	618	7	with	with	ADP
ap-9037	618	8	dunkl	dunkl	PROPN
ap-9037	618	9	derivative	derivative	NOUN
ap-9037	618	10	for	for	ADP
ap-9037	618	11	the	the	DET
ap-9037	618	12	free	free	ADJ
ap-9037	618	13	-	-	PUNCT
ap-9037	618	14	particle	particle	NOUN
ap-9037	618	15	spherical	spherical	ADJ
ap-9037	618	16	waves	wave	NOUN
ap-9037	618	17	,	,	PUNCT
ap-9037	618	18	the	the	DET
ap-9037	618	19	pseudo	pseudo	NOUN
ap-9037	618	20	-	-	ADJ
ap-9037	618	21	harmonic	harmonic	ADJ
ap-9037	618	22	oscillator	oscillator	NOUN
ap-9037	618	23	and	and	CCONJ
ap-9037	618	24	the	the	DET
ap-9037	618	25	mie	mie	NOUN
ap-9037	618	26	-	-	PUNCT
ap-9037	618	27	type	type	NOUN
ap-9037	618	28	potential	potential	NOUN
ap-9037	618	29	.	.	PUNCT
ap-9037	619	1	modern	modern	ADJ
ap-9037	619	2	physics	physics	NOUN
ap-9037	619	3	letters	letter	NOUN
ap-9037	619	4	a	a	DET
ap-9037	619	5	37(1):2250006	37(1):2250006	NUM
ap-9037	619	6	,	,	PUNCT
ap-9037	619	7	2022	2022	NUM
ap-9037	619	8	.	.	PUNCT
ap-9037	620	1	https://doi.org/10.1142/s0217732322500067	https://doi.org/10.1142/s0217732322500067	NUM
ap-9037	620	2	[	[	SYM
ap-9037	620	3	19	19	NUM
ap-9037	620	4	]	]	PUNCT
ap-9037	620	5	a.	a.	PROPN
ap-9037	620	6	contreras	contreras	PROPN
ap-9037	620	7	-	-	PUNCT
ap-9037	620	8	astorga	astorga	PROPN
ap-9037	620	9	,	,	PUNCT
ap-9037	620	10	d.	d.	PROPN
ap-9037	620	11	j.	j.	PROPN
ap-9037	620	12	fernández	fernández	PROPN
ap-9037	620	13	.	.	PUNCT
ap-9037	621	1	supersymmetric	supersymmetric	ADJ
ap-9037	621	2	partners	partner	NOUN
ap-9037	621	3	of	of	ADP
ap-9037	621	4	the	the	DET
ap-9037	621	5	trigonometric	trigonometric	ADJ
ap-9037	621	6	pöschl	pöschl	NOUN
ap-9037	621	7	-	-	PUNCT
ap-9037	621	8	teller	teller	NOUN
ap-9037	621	9	potentials	potential	NOUN
ap-9037	621	10	.	.	PUNCT
ap-9037	622	1	journal	journal	PROPN
ap-9037	622	2	of	of	ADP
ap-9037	622	3	physics	physics	PROPN
ap-9037	622	4	a	a	PRON
ap-9037	622	5	:	:	PUNCT
ap-9037	622	6	mathematical	mathematical	ADJ
ap-9037	622	7	and	and	CCONJ
ap-9037	622	8	theoretical	theoretical	ADJ
ap-9037	622	9	41(47):475303	41(47):475303	NUM
ap-9037	622	10	,	,	PUNCT
ap-9037	622	11	2008	2008	NUM
ap-9037	622	12	.	.	PUNCT
ap-9037	623	1	https://doi.org/10.1088/1751-8113/41/47/475303	https://doi.org/10.1088/1751-8113/41/47/475303	VERB
ap-9037	624	1	[	[	X
ap-9037	624	2	20	20	NUM
ap-9037	624	3	]	]	PUNCT
ap-9037	624	4	a.	a.	NOUN
ap-9037	624	5	arabasghari	arabasghari	PROPN
ap-9037	624	6	,	,	PUNCT
ap-9037	624	7	h.	h.	PROPN
ap-9037	624	8	hassanabadi	hassanabadi	PROPN
ap-9037	624	9	,	,	PUNCT
ap-9037	624	10	a.	a.	PROPN
ap-9037	624	11	schulze	schulze	PROPN
ap-9037	624	12	-	-	PUNCT
ap-9037	624	13	halberg	halberg	PROPN
ap-9037	624	14	,	,	PUNCT
ap-9037	624	15	w.	w.	PROPN
ap-9037	624	16	s.	s.	PROPN
ap-9037	624	17	chung	chung	PROPN
ap-9037	624	18	.	.	PUNCT
ap-9037	625	1	bound	bind	VERB
ap-9037	625	2	state	state	NOUN
ap-9037	625	3	solutions	solution	NOUN
ap-9037	625	4	of	of	ADP
ap-9037	625	5	the	the	DET
ap-9037	625	6	dunkl	dunkl	NOUN
ap-9037	625	7	-	-	PUNCT
ap-9037	625	8	schrödinger	schrödinger	NOUN
ap-9037	625	9	equation	equation	NOUN
ap-9037	625	10	for	for	ADP
ap-9037	625	11	a	a	DET
ap-9037	625	12	class	class	NOUN
ap-9037	625	13	of	of	ADP
ap-9037	625	14	angular	angular	ADJ
ap-9037	625	15	-	-	PUNCT
ap-9037	625	16	dependent	dependent	ADJ
ap-9037	625	17	potentials	potential	NOUN
ap-9037	625	18	,	,	PUNCT
ap-9037	625	19	2023	2023	NUM
ap-9037	625	20	.	.	PUNCT
ap-9037	626	1	preprint	preprint	NOUN
ap-9037	626	2	.	.	PUNCT
ap-9037	627	1	[	[	X
ap-9037	627	2	21	21	NUM
ap-9037	627	3	]	]	X
ap-9037	627	4	d.	d.	PROPN
ap-9037	627	5	j.	j.	PROPN
ap-9037	627	6	fernández	fernández	PROPN
ap-9037	627	7	.	.	PUNCT
ap-9037	628	1	trends	trend	NOUN
ap-9037	628	2	in	in	ADP
ap-9037	628	3	supersymmetric	supersymmetric	ADJ
ap-9037	628	4	quantum	quantum	NOUN
ap-9037	628	5	mechanics	mechanic	NOUN
ap-9037	628	6	.	.	PUNCT
ap-9037	629	1	in	in	ADP
ap-9037	629	2	integrability	integrability	NOUN
ap-9037	629	3	,	,	PUNCT
ap-9037	629	4	supersymmetry	supersymmetry	NOUN
ap-9037	629	5	and	and	CCONJ
ap-9037	629	6	coherent	coherent	ADJ
ap-9037	629	7	states	state	NOUN
ap-9037	629	8	,	,	PUNCT
ap-9037	629	9	pp	pp	ADJ
ap-9037	629	10	.	.	PUNCT
ap-9037	630	1	37–68	37–68	NUM
ap-9037	630	2	.	.	PUNCT
ap-9037	631	1	springer	springer	NOUN
ap-9037	631	2	,	,	PUNCT
ap-9037	631	3	new	new	PROPN
ap-9037	631	4	york	york	PROPN
ap-9037	631	5	,	,	PUNCT
ap-9037	631	6	ny	ny	PROPN
ap-9037	631	7	,	,	PUNCT
ap-9037	631	8	2019	2019	NUM
ap-9037	631	9	.	.	PUNCT
ap-9037	632	1	https://doi.org/10.1007/978-3-030-20087-9_2	https://doi.org/10.1007/978-3-030-20087-9_2	PUNCT
ap-9037	633	1	[	[	X
ap-9037	633	2	22	22	NUM
ap-9037	633	3	]	]	PUNCT
ap-9037	633	4	m.	m.	NOUN
ap-9037	633	5	abramowitz	abramowitz	PROPN
ap-9037	633	6	,	,	PUNCT
ap-9037	633	7	i.	i.	PROPN
ap-9037	633	8	a.	a.	PROPN
ap-9037	633	9	stegun	stegun	PROPN
ap-9037	633	10	.	.	PUNCT
ap-9037	634	1	handbook	handbook	NOUN
ap-9037	634	2	of	of	ADP
ap-9037	634	3	mathematical	mathematical	ADJ
ap-9037	634	4	functions	function	NOUN
ap-9037	634	5	:	:	PUNCT
ap-9037	634	6	with	with	ADP
ap-9037	634	7	formulas	formula	NOUN
ap-9037	634	8	,	,	PUNCT
ap-9037	634	9	graphs	graph	NOUN
ap-9037	634	10	,	,	PUNCT
ap-9037	634	11	and	and	CCONJ
ap-9037	634	12	mathematical	mathematical	ADJ
ap-9037	634	13	tables	table	NOUN
ap-9037	634	14	.	.	PUNCT
ap-9037	635	1	dover	dover	PROPN
ap-9037	635	2	publications	publication	NOUN
ap-9037	635	3	,	,	PUNCT
ap-9037	635	4	new	new	PROPN
ap-9037	635	5	york	york	PROPN
ap-9037	635	6	,	,	PUNCT
ap-9037	635	7	ny	ny	PROPN
ap-9037	635	8	,	,	PUNCT
ap-9037	635	9	9th	9th	ADJ
ap-9037	635	10	edn	edn	NOUN
ap-9037	635	11	.	.	PUNCT
ap-9037	635	12	,	,	PUNCT
ap-9037	635	13	1964	1964	NUM
ap-9037	635	14	.	.	PUNCT
ap-9037	636	1	[	[	X
ap-9037	636	2	23	23	NUM
ap-9037	636	3	]	]	PUNCT
ap-9037	636	4	a.	a.	PROPN
ap-9037	636	5	d.	d.	PROPN
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ap-9037	636	7	,	,	PUNCT
ap-9037	636	8	i.	i.	PROPN
ap-9037	636	9	a.	a.	PROPN
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ap-9037	636	11	,	,	PUNCT
ap-9037	636	12	a.	a.	NOUN
ap-9037	636	13	mebirouk	mebirouk	NOUN
ap-9037	636	14	.	.	PUNCT
ap-9037	637	1	bound	bind	VERB
ap-9037	637	2	-	-	PUNCT
ap-9037	637	3	states	state	NOUN
ap-9037	637	4	for	for	ADP
ap-9037	637	5	generalized	generalized	ADJ
ap-9037	637	6	trigonometric	trigonometric	ADJ
ap-9037	637	7	and	and	CCONJ
ap-9037	637	8	hyperbolic	hyperbolic	ADJ
ap-9037	637	9	pöschl	pöschl	NOUN
ap-9037	637	10	-	-	PUNCT
ap-9037	637	11	teller	teller	NOUN
ap-9037	637	12	potentials	potential	NOUN
ap-9037	637	13	.	.	PUNCT
ap-9037	638	1	international	international	ADJ
ap-9037	638	2	journal	journal	NOUN
ap-9037	638	3	of	of	ADP
ap-9037	638	4	modern	modern	ADJ
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ap-9037	638	6	a	a	DET
ap-9037	638	7	37(3):2250012	37(3):2250012	NOUN
ap-9037	638	8	,	,	PUNCT
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ap-9037	639	1	https://doi.org/10.1142/s0217751x22500129	https://doi.org/10.1142/s0217751x22500129	X
ap-9037	640	1	[	[	X
ap-9037	640	2	24	24	NUM
ap-9037	640	3	]	]	PUNCT
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ap-9037	640	5	schulze	schulze	PROPN
ap-9037	640	6	-	-	PUNCT
ap-9037	640	7	halberg	halberg	PROPN
ap-9037	640	8	,	,	PUNCT
ap-9037	640	9	p.	p.	PROPN
ap-9037	640	10	roy	roy	PROPN
ap-9037	640	11	.	.	PROPN
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ap-9037	641	2	of	of	ADP
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ap-9037	641	4	-	-	PUNCT
ap-9037	641	5	energy	energy	NOUN
ap-9037	641	6	states	state	NOUN
ap-9037	641	7	in	in	ADP
ap-9037	641	8	graphene	graphene	NOUN
ap-9037	641	9	through	through	ADP
ap-9037	641	10	the	the	DET
ap-9037	641	11	supersymmetry	supersymmetry	NOUN
ap-9037	641	12	formalism	formalism	NOUN
ap-9037	641	13	.	.	PUNCT
ap-9037	642	1	journal	journal	PROPN
ap-9037	642	2	of	of	ADP
ap-9037	642	3	physics	physics	PROPN
ap-9037	642	4	a	a	PRON
ap-9037	642	5	:	:	PUNCT
ap-9037	642	6	mathematical	mathematical	ADJ
ap-9037	642	7	and	and	CCONJ
ap-9037	642	8	theoretical	theoretical	ADJ
ap-9037	642	9	50(36):365205	50(36):365205	NUM
ap-9037	642	10	,	,	PUNCT
ap-9037	642	11	2017	2017	NUM
ap-9037	642	12	.	.	PUNCT
ap-9037	643	1	https://doi.org/10.1088/1751-8121/aa8249	https://doi.org/10.1088/1751-8121/aa8249	PUNCT
ap-9037	643	2	[	[	X
ap-9037	643	3	25	25	NUM
ap-9037	643	4	]	]	PUNCT
ap-9037	643	5	a.	a.	NOUN
ap-9037	643	6	schulze	schulze	PROPN
ap-9037	643	7	-	-	PUNCT
ap-9037	643	8	halberg	halberg	PROPN
ap-9037	643	9	,	,	PUNCT
ap-9037	643	10	p.	p.	PROPN
ap-9037	643	11	roy	roy	PROPN
ap-9037	643	12	.	.	PROPN
ap-9037	644	1	darboux	darboux	VERB
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ap-9037	644	3	for	for	ADP
ap-9037	644	4	dunkl	dunkl	NOUN
ap-9037	644	5	-	-	PUNCT
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ap-9037	644	7	equations	equation	NOUN
ap-9037	644	8	with	with	ADP
ap-9037	644	9	energy	energy	NOUN
ap-9037	644	10	-	-	PUNCT
ap-9037	644	11	dependent	dependent	ADJ
ap-9037	644	12	potential	potential	NOUN
ap-9037	644	13	and	and	CCONJ
ap-9037	644	14	position	position	NOUN
ap-9037	644	15	-	-	PUNCT
ap-9037	644	16	dependent	dependent	ADJ
ap-9037	644	17	mass	mass	NOUN
ap-9037	644	18	.	.	PUNCT
ap-9037	645	1	[	[	X
ap-9037	645	2	2023	2023	NUM
ap-9037	645	3	-	-	SYM
ap-9037	645	4	01	01	NUM
ap-9037	645	5	-	-	SYM
ap-9037	645	6	27	27	NUM
ap-9037	645	7	]	]	PUNCT
ap-9037	645	8	.	.	PUNCT
ap-9037	645	9	arxiv:2301.11622	arxiv:2301.11622	VERB
ap-9037	645	10	[	[	X
ap-9037	645	11	26	26	NUM
ap-9037	645	12	]	]	PUNCT
ap-9037	645	13	d.	d.	PROPN
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ap-9037	645	15	,	,	PUNCT
ap-9037	645	16	d.	d.	PROPN
ap-9037	645	17	j.	j.	PROPN
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ap-9037	645	19	,	,	PUNCT
ap-9037	645	20	n.	n.	PROPN
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ap-9037	645	22	-	-	PUNCT
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ap-9037	645	24	.	.	PUNCT
ap-9037	646	1	wronskian	wronskian	ADJ
ap-9037	646	2	differential	differential	NOUN
ap-9037	646	3	formula	formula	NOUN
ap-9037	646	4	for	for	ADP
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ap-9037	646	6	supersymmetric	supersymmetric	ADJ
ap-9037	646	7	quantum	quantum	NOUN
ap-9037	646	8	mechanics	mechanic	NOUN
ap-9037	646	9	.	.	PUNCT
ap-9037	647	1	physics	physics	NOUN
ap-9037	647	2	letters	letter	NOUN
ap-9037	647	3	a	a	DET
ap-9037	647	4	376(5):692–696	376(5):692–696	NUM
ap-9037	647	5	,	,	PUNCT
ap-9037	647	6	2012	2012	NUM
ap-9037	647	7	.	.	PUNCT
ap-9037	648	1	https://doi.org/10.1016/j.physleta.2011.12.020	https://doi.org/10.1016/j.physleta.2011.12.020	NOUN
ap-9037	648	2	292	292	NUM
ap-9037	648	3	https://doi.org/10.1007/s00220-014-1915-2	https://doi.org/10.1007/s00220-014-1915-2	NUM
ap-9037	648	4	https://doi.org/10.1088/1751-8113/46/14/145201	https://doi.org/10.1088/1751-8113/46/14/145201	NOUN
ap-9037	648	5	https://doi.org/10.1016/j.physleta.2015.01.023	https://doi.org/10.1016/j.physleta.2015.01.023	PROPN
ap-9037	648	6	https://doi.org/10.1142/s0217732321501716	https://doi.org/10.1142/s0217732321501716	PROPN
ap-9037	648	7	https://doi.org/10.1088/1751-8121/acd736	https://doi.org/10.1088/1751-8121/acd736	PROPN
ap-9037	648	8	https://doi.org/10.1142/s0217732320502557	https://doi.org/10.1142/s0217732320502557	PROPN
ap-9037	648	9	https://doi.org/10.1142/s0217732318501468	https://doi.org/10.1142/s0217732318501468	NUM
ap-9037	648	10	https://doi.org/10.1142/s0217732322500067	https://doi.org/10.1142/s0217732322500067	NUM
ap-9037	648	11	https://doi.org/10.1088/1751-8113/41/47/475303	https://doi.org/10.1088/1751-8113/41/47/475303	NOUN
ap-9037	648	12	https://doi.org/10.1007/978-3-030-20087-9_2	https://doi.org/10.1007/978-3-030-20087-9_2	VERB
ap-9037	648	13	https://doi.org/10.1142/s0217751x22500129	https://doi.org/10.1142/s0217751x22500129	NUM
ap-9037	648	14	https://doi.org/10.1088/1751-8121/aa8249	https://doi.org/10.1088/1751-8121/aa8249	PROPN
ap-9037	648	15	https://arxiv.org/pdf/2301.11622.pdf	https://arxiv.org/pdf/2301.11622.pdf	PROPN
ap-9037	648	16	https://doi.org/10.1016/j.physleta.2011.12.020	https://doi.org/10.1016/j.physleta.2011.12.020	PROPN
ap-9037	648	17	acta	acta	PROPN
ap-9037	648	18	polytechnica	polytechnica	PROPN
ap-9037	648	19	63(4):1–20	63(4):1–20	PROPN
ap-9037	648	20	,	,	PUNCT
ap-9037	648	21	2023	2023	NUM
ap-9037	648	22	1	1	NUM
ap-9037	648	23	introduction	introduction	NOUN
ap-9037	648	24	2	2	NUM
ap-9037	648	25	preliminaries	preliminary	NOUN
ap-9037	648	26	2.1	2.1	NUM
ap-9037	648	27	the	the	DET
ap-9037	648	28	trigonometric	trigonometric	ADJ
ap-9037	648	29	pöschl	pöschl	NOUN
ap-9037	648	30	-	-	PUNCT
ap-9037	648	31	teller	teller	NOUN
ap-9037	648	32	system	system	NOUN
ap-9037	648	33	2.2	2.2	NUM
ap-9037	648	34	the	the	DET
ap-9037	648	35	darboux	darboux	ADJ
ap-9037	648	36	-	-	PUNCT
ap-9037	648	37	crum	crum	NOUN
ap-9037	648	38	transformation	transformation	NOUN
ap-9037	648	39	3	3	NUM
ap-9037	648	40	the	the	DET
ap-9037	648	41	dunkl	dunkl	NOUN
ap-9037	648	42	-	-	PUNCT
ap-9037	648	43	schrödinger	schrödinger	ADJ
ap-9037	648	44	system	system	NOUN
ap-9037	648	45	4	4	NUM
ap-9037	648	46	the	the	DET
ap-9037	648	47	polar	polar	ADJ
ap-9037	648	48	equation	equation	NOUN
ap-9037	648	49	4.1	4.1	NUM
ap-9037	648	50	extended	extend	VERB
ap-9037	648	51	trigonometric	trigonometric	ADJ
ap-9037	648	52	pöschl	pöschl	NOUN
ap-9037	648	53	-	-	PUNCT
ap-9037	648	54	teller	teller	NOUN
ap-9037	648	55	systems	system	NOUN
ap-9037	648	56	4.2	4.2	NUM
ap-9037	648	57	the	the	DET
ap-9037	648	58	darboux	darboux	ADJ
ap-9037	648	59	-	-	PUNCT
ap-9037	648	60	crum	crum	NOUN
ap-9037	648	61	transformation	transformation	NOUN
ap-9037	648	62	4.2.1	4.2.1	NUM
ap-9037	648	63	transformation	transformation	NOUN
ap-9037	648	64	parameter	parameter	NOUN
ap-9037	648	65	4.2.2	4.2.2	NUM
ap-9037	648	66	transformation	transformation	NOUN
ap-9037	648	67	parameter	parameter	NOUN
ap-9037	648	68	m2	m2	PROPN
ap-9037	648	69	5	5	NUM
ap-9037	648	70	the	the	DET
ap-9037	648	71	azimuthal	azimuthal	ADJ
ap-9037	648	72	equation	equation	NOUN
ap-9037	648	73	5.1	5.1	NUM
ap-9037	648	74	extended	extend	VERB
ap-9037	648	75	trigonometric	trigonometric	ADJ
ap-9037	648	76	pöschl	pöschl	NOUN
ap-9037	648	77	-	-	PUNCT
ap-9037	648	78	teller	teller	NOUN
ap-9037	648	79	systems	system	NOUN
ap-9037	648	80	5.2	5.2	NUM
ap-9037	648	81	the	the	DET
ap-9037	648	82	darboux	darboux	ADJ
ap-9037	648	83	-	-	PUNCT
ap-9037	648	84	crum	crum	NOUN
ap-9037	648	85	transformation	transformation	NOUN
ap-9037	648	86	6	6	NUM
ap-9037	648	87	concluding	conclude	VERB
ap-9037	648	88	remarks	remark	NOUN
ap-9037	648	89	references	reference	NOUN