id	sid	tid	token	lemma	pos
ejpam-6146	1	1	european	european	PROPN
ejpam-6146	1	2	journal	journal	PROPN
ejpam-6146	1	3	of	of	ADP
ejpam-6146	1	4	pure	pure	ADJ
ejpam-6146	1	5	and	and	CCONJ
ejpam-6146	1	6	applied	applied	ADJ
ejpam-6146	1	7	mathematics	mathematic	NOUN
ejpam-6146	1	8	2025	2025	NUM
ejpam-6146	1	9	,	,	PUNCT
ejpam-6146	1	10	vol	vol	NOUN
ejpam-6146	1	11	.	.	PROPN
ejpam-6146	1	12	18	18	NUM
ejpam-6146	1	13	,	,	PUNCT
ejpam-6146	1	14	issue	issue	NOUN
ejpam-6146	1	15	4	4	NUM
ejpam-6146	1	16	,	,	PUNCT
ejpam-6146	1	17	article	article	NOUN
ejpam-6146	1	18	number	number	NOUN
ejpam-6146	1	19	6146	6146	NUM
ejpam-6146	1	20	issn	issn	VERB
ejpam-6146	1	21	1307	1307	NUM
ejpam-6146	1	22	-	-	SYM
ejpam-6146	1	23	5543	5543	NUM
ejpam-6146	1	24	–	–	PUNCT
ejpam-6146	1	25	ejpam.com	ejpam.com	X
ejpam-6146	1	26	published	publish	VERB
ejpam-6146	1	27	by	by	ADP
ejpam-6146	1	28	new	new	PROPN
ejpam-6146	1	29	york	york	PROPN
ejpam-6146	1	30	business	business	PROPN
ejpam-6146	1	31	global	global	ADJ
ejpam-6146	1	32	deducing	deduce	VERB
ejpam-6146	1	33	trigonometric	trigonometric	ADJ
ejpam-6146	1	34	functions	function	NOUN
ejpam-6146	1	35	from	from	ADP
ejpam-6146	1	36	differential	differential	ADJ
ejpam-6146	1	37	equations	equation	NOUN
ejpam-6146	1	38	:	:	PUNCT
ejpam-6146	1	39	an	an	DET
ejpam-6146	1	40	educational	educational	ADJ
ejpam-6146	1	41	perspective	perspective	NOUN
ejpam-6146	1	42	sameen	sameen	PROPN
ejpam-6146	1	43	ahmed	ahmed	PROPN
ejpam-6146	1	44	khan1	khan1	PROPN
ejpam-6146	1	45	,	,	PUNCT
ejpam-6146	1	46	manisha	manisha	PROPN
ejpam-6146	1	47	m.	m.	PROPN
ejpam-6146	1	48	kankarej2,∗	kankarej2,∗	PROPN
ejpam-6146	1	49	,	,	PUNCT
ejpam-6146	1	50	mohammad	mohammad	PROPN
ejpam-6146	1	51	nazrul	nazrul	PROPN
ejpam-6146	1	52	islam	islam	PROPN
ejpam-6146	1	53	khan3	khan3	PROPN
ejpam-6146	1	54	1	1	NUM
ejpam-6146	1	55	department	department	NOUN
ejpam-6146	1	56	of	of	ADP
ejpam-6146	1	57	mathematics	mathematic	NOUN
ejpam-6146	1	58	and	and	CCONJ
ejpam-6146	1	59	sciences	science	NOUN
ejpam-6146	1	60	,	,	PUNCT
ejpam-6146	1	61	college	college	NOUN
ejpam-6146	1	62	of	of	ADP
ejpam-6146	1	63	arts	art	NOUN
ejpam-6146	1	64	and	and	CCONJ
ejpam-6146	1	65	applied	apply	VERB
ejpam-6146	1	66	sciences	science	NOUN
ejpam-6146	1	67	,	,	PUNCT
ejpam-6146	1	68	dhofar	dhofar	ADJ
ejpam-6146	1	69	university	university	PROPN
ejpam-6146	1	70	,	,	PUNCT
ejpam-6146	1	71	salalah	salalah	PROPN
ejpam-6146	1	72	,	,	PUNCT
ejpam-6146	1	73	sultanate	sultanate	NOUN
ejpam-6146	1	74	of	of	ADP
ejpam-6146	1	75	oman	oman	PROPN
ejpam-6146	1	76	2	2	NUM
ejpam-6146	1	77	rochester	rochester	PROPN
ejpam-6146	1	78	institute	institute	PROPN
ejpam-6146	1	79	of	of	ADP
ejpam-6146	1	80	technology	technology	PROPN
ejpam-6146	1	81	,	,	PUNCT
ejpam-6146	1	82	dubai	dubai	PROPN
ejpam-6146	1	83	,	,	PUNCT
ejpam-6146	1	84	uae	uae	PROPN
ejpam-6146	1	85	3	3	NUM
ejpam-6146	1	86	department	department	NOUN
ejpam-6146	1	87	of	of	ADP
ejpam-6146	1	88	computer	computer	NOUN
ejpam-6146	1	89	engineering	engineering	NOUN
ejpam-6146	1	90	,	,	PUNCT
ejpam-6146	1	91	college	college	NOUN
ejpam-6146	1	92	of	of	ADP
ejpam-6146	1	93	computer	computer	NOUN
ejpam-6146	1	94	,	,	PUNCT
ejpam-6146	1	95	qassim	qassim	PROPN
ejpam-6146	1	96	university	university	PROPN
ejpam-6146	1	97	,	,	PUNCT
ejpam-6146	1	98	saudi	saudi	PROPN
ejpam-6146	1	99	arabia	arabia	PROPN
ejpam-6146	1	100	abstract	abstract	NOUN
ejpam-6146	1	101	.	.	PUNCT
ejpam-6146	2	1	we	we	PRON
ejpam-6146	2	2	derive	derive	VERB
ejpam-6146	2	3	the	the	DET
ejpam-6146	2	4	two	two	NUM
ejpam-6146	2	5	trigonometric	trigonometric	ADJ
ejpam-6146	2	6	functions	function	NOUN
ejpam-6146	2	7	,	,	PUNCT
ejpam-6146	2	8	sine	sine	NOUN
ejpam-6146	2	9	and	and	CCONJ
ejpam-6146	2	10	cosine	cosine	NOUN
ejpam-6146	2	11	using	use	VERB
ejpam-6146	2	12	the	the	DET
ejpam-6146	2	13	qualitative	qualitative	ADJ
ejpam-6146	2	14	properties	property	NOUN
ejpam-6146	2	15	of	of	ADP
ejpam-6146	2	16	differential	differential	ADJ
ejpam-6146	2	17	equations	equation	NOUN
ejpam-6146	2	18	and	and	CCONJ
ejpam-6146	2	19	further	far	ADV
ejpam-6146	2	20	conclude	conclude	VERB
ejpam-6146	2	21	that	that	SCONJ
ejpam-6146	2	22	the	the	DET
ejpam-6146	2	23	two	two	NUM
ejpam-6146	2	24	functions	function	NOUN
ejpam-6146	2	25	are	be	AUX
ejpam-6146	2	26	periodic	periodic	ADJ
ejpam-6146	2	27	.	.	PUNCT
ejpam-6146	3	1	some	some	DET
ejpam-6146	3	2	properties	property	NOUN
ejpam-6146	3	3	such	such	ADJ
ejpam-6146	3	4	as	as	ADP
ejpam-6146	3	5	the	the	DET
ejpam-6146	3	6	sine	sine	NOUN
ejpam-6146	3	7	/	/	SYM
ejpam-6146	3	8	cosine	cosine	NOUN
ejpam-6146	3	9	of	of	ADP
ejpam-6146	3	10	sums	sum	NOUN
ejpam-6146	3	11	and	and	CCONJ
ejpam-6146	3	12	differences	difference	NOUN
ejpam-6146	3	13	of	of	ADP
ejpam-6146	3	14	two	two	NUM
ejpam-6146	3	15	angles	angle	NOUN
ejpam-6146	3	16	are	be	AUX
ejpam-6146	3	17	also	also	ADV
ejpam-6146	3	18	derived	derive	VERB
ejpam-6146	3	19	from	from	ADP
ejpam-6146	3	20	the	the	DET
ejpam-6146	3	21	defining	define	VERB
ejpam-6146	3	22	differential	differential	ADJ
ejpam-6146	3	23	equations	equation	NOUN
ejpam-6146	3	24	.	.	PUNCT
ejpam-6146	4	1	the	the	DET
ejpam-6146	4	2	oscillations	oscillation	NOUN
ejpam-6146	4	3	of	of	ADP
ejpam-6146	4	4	the	the	DET
ejpam-6146	4	5	solutions	solution	NOUN
ejpam-6146	4	6	of	of	ADP
ejpam-6146	4	7	second	second	ADJ
ejpam-6146	4	8	-	-	PUNCT
ejpam-6146	4	9	order	order	NOUN
ejpam-6146	4	10	differential	differential	ADJ
ejpam-6146	4	11	equations	equation	NOUN
ejpam-6146	4	12	are	be	AUX
ejpam-6146	4	13	covered	cover	VERB
ejpam-6146	4	14	along	along	ADP
ejpam-6146	4	15	with	with	ADP
ejpam-6146	4	16	the	the	DET
ejpam-6146	4	17	related	relate	VERB
ejpam-6146	4	18	theorems	theorem	NOUN
ejpam-6146	4	19	and	and	CCONJ
ejpam-6146	4	20	techniques	technique	NOUN
ejpam-6146	4	21	.	.	PUNCT
ejpam-6146	5	1	we	we	PRON
ejpam-6146	5	2	also	also	ADV
ejpam-6146	5	3	derive	derive	VERB
ejpam-6146	5	4	the	the	DET
ejpam-6146	5	5	oscillatory	oscillatory	ADJ
ejpam-6146	5	6	behaviour	behaviour	NOUN
ejpam-6146	5	7	of	of	ADP
ejpam-6146	5	8	the	the	DET
ejpam-6146	5	9	bessel	bessel	NOUN
ejpam-6146	5	10	functions	function	NOUN
ejpam-6146	5	11	from	from	ADP
ejpam-6146	5	12	its	its	PRON
ejpam-6146	5	13	defining	define	VERB
ejpam-6146	5	14	differential	differential	NOUN
ejpam-6146	5	15	equation	equation	NOUN
ejpam-6146	5	16	.	.	PUNCT
ejpam-6146	6	1	2020	2020	NUM
ejpam-6146	6	2	mathematics	mathematic	NOUN
ejpam-6146	6	3	subject	subject	NOUN
ejpam-6146	6	4	classifications	classification	NOUN
ejpam-6146	6	5	:	:	PUNCT
ejpam-6146	6	6	53d10	53d10	NUM
ejpam-6146	6	7	,	,	PUNCT
ejpam-6146	6	8	53c25	53c25	NUM
ejpam-6146	6	9	,	,	PUNCT
ejpam-6146	6	10	53c15	53c15	NUM
ejpam-6146	6	11	,	,	PUNCT
ejpam-6146	6	12	58a30	58a30	NUM
ejpam-6146	6	13	key	key	ADJ
ejpam-6146	6	14	words	word	NOUN
ejpam-6146	6	15	and	and	CCONJ
ejpam-6146	6	16	phrases	phrase	NOUN
ejpam-6146	6	17	:	:	PUNCT
ejpam-6146	6	18	trigonometric	trigonometric	ADJ
ejpam-6146	6	19	functions	function	NOUN
ejpam-6146	6	20	,	,	PUNCT
ejpam-6146	6	21	differential	differential	ADJ
ejpam-6146	6	22	equations	equation	NOUN
ejpam-6146	6	23	,	,	PUNCT
ejpam-6146	6	24	qualitative	qualitative	ADJ
ejpam-6146	6	25	theory	theory	NOUN
ejpam-6146	6	26	of	of	ADP
ejpam-6146	6	27	differential	differential	ADJ
ejpam-6146	6	28	equations	equation	NOUN
ejpam-6146	6	29	,	,	PUNCT
ejpam-6146	6	30	oscillation	oscillation	NOUN
ejpam-6146	6	31	phenomenon	phenomenon	NOUN
ejpam-6146	6	32	,	,	PUNCT
ejpam-6146	6	33	sturm	sturm	PROPN
ejpam-6146	6	34	separation	separation	NOUN
ejpam-6146	6	35	theorem	theorem	PROPN
ejpam-6146	6	36	,	,	PUNCT
ejpam-6146	6	37	sturm	sturm	PROPN
ejpam-6146	6	38	comparison	comparison	NOUN
ejpam-6146	6	39	theorem	theorem	VERB
ejpam-6146	6	40	,	,	PUNCT
ejpam-6146	6	41	orthogonal	orthogonal	ADJ
ejpam-6146	6	42	polynomials	polynomial	NOUN
ejpam-6146	6	43	1	1	NUM
ejpam-6146	6	44	.	.	PUNCT
ejpam-6146	7	1	introduction	introduction	NOUN
ejpam-6146	7	2	trigonometry	trigonometry	NOUN
ejpam-6146	7	3	is	be	AUX
ejpam-6146	7	4	a	a	DET
ejpam-6146	7	5	branch	branch	NOUN
ejpam-6146	7	6	of	of	ADP
ejpam-6146	7	7	mathematics	mathematic	NOUN
ejpam-6146	7	8	that	that	PRON
ejpam-6146	7	9	studies	study	VERB
ejpam-6146	7	10	the	the	DET
ejpam-6146	7	11	relationships	relationship	NOUN
ejpam-6146	7	12	between	between	ADP
ejpam-6146	7	13	the	the	DET
ejpam-6146	7	14	angles	angle	NOUN
ejpam-6146	7	15	and	and	CCONJ
ejpam-6146	7	16	sides	side	NOUN
ejpam-6146	7	17	of	of	ADP
ejpam-6146	7	18	triangles	triangle	NOUN
ejpam-6146	7	19	.	.	PUNCT
ejpam-6146	8	1	it	it	PRON
ejpam-6146	8	2	is	be	AUX
ejpam-6146	8	3	widely	widely	ADV
ejpam-6146	8	4	used	use	VERB
ejpam-6146	8	5	in	in	ADP
ejpam-6146	8	6	geometry	geometry	NOUN
ejpam-6146	8	7	,	,	PUNCT
ejpam-6146	8	8	physics	physics	NOUN
ejpam-6146	8	9	,	,	PUNCT
ejpam-6146	8	10	engineering	engineering	NOUN
ejpam-6146	8	11	,	,	PUNCT
ejpam-6146	8	12	astronomy	astronomy	NOUN
ejpam-6146	8	13	,	,	PUNCT
ejpam-6146	8	14	and	and	CCONJ
ejpam-6146	8	15	many	many	ADJ
ejpam-6146	8	16	other	other	ADJ
ejpam-6146	8	17	fields	field	NOUN
ejpam-6146	8	18	.	.	PUNCT
ejpam-6146	9	1	the	the	DET
ejpam-6146	9	2	origins	origin	NOUN
ejpam-6146	9	3	of	of	ADP
ejpam-6146	9	4	trigonometry	trigonometry	NOUN
ejpam-6146	9	5	can	can	AUX
ejpam-6146	9	6	be	be	AUX
ejpam-6146	9	7	traced	trace	VERB
ejpam-6146	9	8	back	back	ADV
ejpam-6146	9	9	to	to	ADP
ejpam-6146	9	10	ancient	ancient	ADJ
ejpam-6146	9	11	civilizations	civilization	NOUN
ejpam-6146	9	12	such	such	ADJ
ejpam-6146	9	13	as	as	ADP
ejpam-6146	9	14	the	the	DET
ejpam-6146	9	15	babylonians	babylonians	PROPN
ejpam-6146	9	16	,	,	PUNCT
ejpam-6146	9	17	egyptians	egyptians	PROPN
ejpam-6146	9	18	and	and	CCONJ
ejpam-6146	9	19	indians	indians	PROPN
ejpam-6146	9	20	,	,	PUNCT
ejpam-6146	9	21	who	who	PRON
ejpam-6146	9	22	used	use	VERB
ejpam-6146	9	23	basic	basic	ADJ
ejpam-6146	9	24	trigonometric	trigonometric	ADJ
ejpam-6146	9	25	concepts	concept	NOUN
ejpam-6146	9	26	for	for	ADP
ejpam-6146	9	27	practical	practical	ADJ
ejpam-6146	9	28	purposes	purpose	NOUN
ejpam-6146	9	29	[	[	X
ejpam-6146	9	30	1	1	NUM
ejpam-6146	9	31	]	]	PUNCT
ejpam-6146	9	32	.	.	PUNCT
ejpam-6146	10	1	the	the	DET
ejpam-6146	10	2	trigonometric	trigonometric	ADJ
ejpam-6146	10	3	functions	function	NOUN
ejpam-6146	10	4	are	be	AUX
ejpam-6146	10	5	traditionally	traditionally	ADV
ejpam-6146	10	6	introduced	introduce	VERB
ejpam-6146	10	7	using	use	VERB
ejpam-6146	10	8	the	the	DET
ejpam-6146	10	9	geometric	geometric	ADJ
ejpam-6146	10	10	approach	approach	NOUN
ejpam-6146	10	11	of	of	ADP
ejpam-6146	10	12	right	right	ADJ
ejpam-6146	10	13	-	-	PUNCT
ejpam-6146	10	14	angled	angle	VERB
ejpam-6146	10	15	triangles	triangle	NOUN
ejpam-6146	10	16	.	.	PUNCT
ejpam-6146	11	1	a	a	DET
ejpam-6146	11	2	more	more	ADV
ejpam-6146	11	3	formal	formal	ADJ
ejpam-6146	11	4	approach	approach	NOUN
ejpam-6146	11	5	is	be	AUX
ejpam-6146	11	6	done	do	VERB
ejpam-6146	11	7	using	use	VERB
ejpam-6146	11	8	the	the	DET
ejpam-6146	11	9	unit	unit	NOUN
ejpam-6146	11	10	circle	circle	NOUN
ejpam-6146	12	1	[	[	X
ejpam-6146	12	2	2–5	2–5	X
ejpam-6146	12	3	]	]	PUNCT
ejpam-6146	12	4	.	.	PUNCT
ejpam-6146	13	1	we	we	PRON
ejpam-6146	13	2	know	know	VERB
ejpam-6146	13	3	that	that	SCONJ
ejpam-6146	13	4	the	the	DET
ejpam-6146	13	5	trigonometric	trigonometric	ADJ
ejpam-6146	13	6	functions	function	NOUN
ejpam-6146	13	7	are	be	AUX
ejpam-6146	13	8	solutions	solution	NOUN
ejpam-6146	13	9	of	of	ADP
ejpam-6146	13	10	differential	differential	ADJ
ejpam-6146	13	11	equations	equation	NOUN
ejpam-6146	14	1	[	[	X
ejpam-6146	14	2	6–9	6–9	NOUN
ejpam-6146	14	3	]	]	PUNCT
ejpam-6146	14	4	.	.	PUNCT
ejpam-6146	15	1	trigonometric	trigonometric	ADJ
ejpam-6146	15	2	arise	arise	NOUN
ejpam-6146	15	3	as	as	ADP
ejpam-6146	15	4	solutions	solution	NOUN
ejpam-6146	15	5	of	of	ADP
ejpam-6146	15	6	linear	linear	NOUN
ejpam-6146	15	7	as	as	ADP
ejpam-6146	15	8	well	well	ADV
ejpam-6146	15	9	partial	partial	ADJ
ejpam-6146	15	10	differential	differential	ADJ
ejpam-6146	15	11	equations	equation	NOUN
ejpam-6146	15	12	[	[	X
ejpam-6146	15	13	10–15	10–15	NUM
ejpam-6146	15	14	]	]	PUNCT
ejpam-6146	15	15	.	.	PUNCT
ejpam-6146	16	1	individual	individual	ADJ
ejpam-6146	16	2	trigonometric	trigonometric	ADJ
ejpam-6146	16	3	ratios	ratio	NOUN
ejpam-6146	16	4	can	can	AUX
ejpam-6146	16	5	be	be	AUX
ejpam-6146	16	6	calculated	calculate	VERB
ejpam-6146	16	7	using	use	VERB
ejpam-6146	16	8	geometric	geometric	ADJ
ejpam-6146	16	9	∗corresponding	∗corresponding	NOUN
ejpam-6146	16	10	author	author	NOUN
ejpam-6146	16	11	.	.	PUNCT
ejpam-6146	17	1	doi	doi	NOUN
ejpam-6146	17	2	:	:	PUNCT
ejpam-6146	17	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6146	https://doi.org/10.29020/nybg.ejpam.v18i4.6146	NOUN
ejpam-6146	17	4	email	email	NOUN
ejpam-6146	17	5	addresses	address	VERB
ejpam-6146	17	6	:	:	PUNCT
ejpam-6146	18	1	rohelakhan@yahoo.com	rohelakhan@yahoo.com	X
ejpam-6146	18	2	(	(	PUNCT
ejpam-6146	18	3	s.	s.	PROPN
ejpam-6146	18	4	a.	a.	PROPN
ejpam-6146	18	5	khan	khan	PROPN
ejpam-6146	18	6	)	)	PUNCT
ejpam-6146	18	7	,	,	PUNCT
ejpam-6146	18	8	manisha.kankarej@gmail.com	manisha.kankarej@gmail.com	X
ejpam-6146	18	9	(	(	PUNCT
ejpam-6146	18	10	m.	m.	NOUN
ejpam-6146	18	11	m.	m.	PROPN
ejpam-6146	18	12	kankarej	kankarej	PROPN
ejpam-6146	18	13	)	)	PUNCT
ejpam-6146	18	14	,	,	PUNCT
ejpam-6146	18	15	m.nazrul@qu.edu.sa	m.nazrul@qu.edu.sa	PROPN
ejpam-6146	18	16	(	(	PUNCT
ejpam-6146	18	17	m.	m.	PROPN
ejpam-6146	18	18	n.	n.	PROPN
ejpam-6146	18	19	i.	i.	PROPN
ejpam-6146	18	20	khan	khan	PROPN
ejpam-6146	18	21	)	)	PUNCT
ejpam-6146	18	22	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6146	19	1	1	1	NUM
ejpam-6146	19	2	copyright	copyright	NOUN
ejpam-6146	19	3	:	:	PUNCT
ejpam-6146	19	4	©	©	PROPN
ejpam-6146	19	5	2025	2025	NUM
ejpam-6146	19	6	the	the	DET
ejpam-6146	19	7	author(s	author(s	NOUN
ejpam-6146	19	8	)	)	PUNCT
ejpam-6146	19	9	.	.	PUNCT
ejpam-6146	20	1	(	(	PUNCT
ejpam-6146	20	2	cc	cc	NOUN
ejpam-6146	20	3	by	by	ADP
ejpam-6146	20	4	-	-	PUNCT
ejpam-6146	20	5	nc	nc	PROPN
ejpam-6146	20	6	4.0	4.0	NUM
ejpam-6146	20	7	)	)	PUNCT
ejpam-6146	20	8	s.	s.	PROPN
ejpam-6146	20	9	a.	a.	PROPN
ejpam-6146	20	10	khan	khan	PROPN
ejpam-6146	20	11	,	,	PUNCT
ejpam-6146	20	12	m.	m.	NOUN
ejpam-6146	20	13	m.	m.	PROPN
ejpam-6146	20	14	kankarej	kankarej	PROPN
ejpam-6146	20	15	,	,	PUNCT
ejpam-6146	20	16	m.	m.	PROPN
ejpam-6146	20	17	n.	n.	PROPN
ejpam-6146	20	18	i.	i.	PROPN
ejpam-6146	20	19	khan	khan	PROPN
ejpam-6146	20	20	/	/	SYM
ejpam-6146	20	21	eur	eur	PROPN
ejpam-6146	20	22	.	.	PUNCT
ejpam-6146	21	1	j.	j.	PROPN
ejpam-6146	21	2	pure	pure	PROPN
ejpam-6146	21	3	appl	appl	PROPN
ejpam-6146	21	4	.	.	PROPN
ejpam-6146	21	5	math	math	PROPN
ejpam-6146	21	6	,	,	PUNCT
ejpam-6146	21	7	18	18	NUM
ejpam-6146	21	8	(	(	PUNCT
ejpam-6146	21	9	4	4	NUM
ejpam-6146	21	10	)	)	PUNCT
ejpam-6146	21	11	(	(	PUNCT
ejpam-6146	21	12	2025	2025	NUM
ejpam-6146	21	13	)	)	PUNCT
ejpam-6146	21	14	,	,	PUNCT
ejpam-6146	21	15	6146	6146	NUM
ejpam-6146	21	16	2	2	NUM
ejpam-6146	21	17	of	of	ADP
ejpam-6146	21	18	14	14	NUM
ejpam-6146	21	19	techniques	technique	NOUN
ejpam-6146	21	20	[	[	X
ejpam-6146	21	21	16	16	NUM
ejpam-6146	21	22	]	]	PUNCT
ejpam-6146	21	23	and	and	CCONJ
ejpam-6146	21	24	algebraic	algebraic	ADJ
ejpam-6146	21	25	techniques	technique	NOUN
ejpam-6146	22	1	[	[	X
ejpam-6146	22	2	17	17	NUM
ejpam-6146	22	3	]	]	PUNCT
ejpam-6146	22	4	.	.	PUNCT
ejpam-6146	23	1	trigonometric	trigonometric	ADJ
ejpam-6146	23	2	ratios	ratio	NOUN
ejpam-6146	23	3	can	can	AUX
ejpam-6146	23	4	also	also	ADV
ejpam-6146	23	5	be	be	AUX
ejpam-6146	23	6	used	use	VERB
ejpam-6146	23	7	as	as	ADP
ejpam-6146	23	8	a	a	DET
ejpam-6146	23	9	tool	tool	NOUN
ejpam-6146	23	10	to	to	PART
ejpam-6146	23	11	teach	teach	VERB
ejpam-6146	23	12	irrational	irrational	ADJ
ejpam-6146	23	13	numbers	number	NOUN
ejpam-6146	23	14	[	[	X
ejpam-6146	23	15	18	18	NUM
ejpam-6146	23	16	]	]	PUNCT
ejpam-6146	23	17	.	.	PUNCT
ejpam-6146	24	1	as	as	SCONJ
ejpam-6146	24	2	trigonometric	trigonometric	ADJ
ejpam-6146	24	3	functions	function	NOUN
ejpam-6146	24	4	are	be	AUX
ejpam-6146	24	5	solutions	solution	NOUN
ejpam-6146	24	6	of	of	ADP
ejpam-6146	24	7	differential	differential	ADJ
ejpam-6146	24	8	equations	equation	NOUN
ejpam-6146	24	9	,	,	PUNCT
ejpam-6146	24	10	it	it	PRON
ejpam-6146	24	11	is	be	AUX
ejpam-6146	24	12	natural	natural	ADJ
ejpam-6146	24	13	to	to	PART
ejpam-6146	24	14	explore	explore	VERB
ejpam-6146	24	15	them	they	PRON
ejpam-6146	24	16	starting	start	VERB
ejpam-6146	24	17	with	with	ADP
ejpam-6146	24	18	a	a	DET
ejpam-6146	24	19	differential	differential	ADJ
ejpam-6146	24	20	equation	equation	NOUN
ejpam-6146	24	21	[	[	X
ejpam-6146	24	22	6	6	NUM
ejpam-6146	24	23	,	,	PUNCT
ejpam-6146	24	24	7	7	NUM
ejpam-6146	24	25	,	,	PUNCT
ejpam-6146	24	26	9	9	NUM
ejpam-6146	24	27	]	]	PUNCT
ejpam-6146	24	28	.	.	PUNCT
ejpam-6146	25	1	in	in	ADP
ejpam-6146	25	2	this	this	DET
ejpam-6146	25	3	article	article	NOUN
ejpam-6146	25	4	,	,	PUNCT
ejpam-6146	25	5	we	we	PRON
ejpam-6146	25	6	use	use	VERB
ejpam-6146	25	7	the	the	DET
ejpam-6146	25	8	qualitative	qualitative	ADJ
ejpam-6146	25	9	properties	property	NOUN
ejpam-6146	25	10	of	of	ADP
ejpam-6146	25	11	differential	differential	ADJ
ejpam-6146	25	12	equations	equation	NOUN
ejpam-6146	25	13	to	to	PART
ejpam-6146	25	14	deduce	deduce	VERB
ejpam-6146	25	15	the	the	DET
ejpam-6146	25	16	periodicity	periodicity	NOUN
ejpam-6146	25	17	and	and	CCONJ
ejpam-6146	25	18	other	other	ADJ
ejpam-6146	25	19	properties	property	NOUN
ejpam-6146	25	20	of	of	ADP
ejpam-6146	25	21	the	the	DET
ejpam-6146	25	22	two	two	NUM
ejpam-6146	25	23	trigonometric	trigonometric	ADJ
ejpam-6146	25	24	functions	function	NOUN
ejpam-6146	25	25	,	,	PUNCT
ejpam-6146	25	26	sine	sine	NOUN
ejpam-6146	25	27	and	and	CCONJ
ejpam-6146	25	28	cosine	cosine	NOUN
ejpam-6146	25	29	respectively	respectively	ADV
ejpam-6146	25	30	.	.	PUNCT
ejpam-6146	26	1	first	first	ADV
ejpam-6146	26	2	,	,	PUNCT
ejpam-6146	26	3	we	we	PRON
ejpam-6146	26	4	deduce	deduce	VERB
ejpam-6146	26	5	the	the	DET
ejpam-6146	26	6	sine	sine	ADJ
ejpam-6146	26	7	and	and	CCONJ
ejpam-6146	26	8	cosine	cosine	NOUN
ejpam-6146	26	9	functions	function	NOUN
ejpam-6146	26	10	along	along	ADP
ejpam-6146	26	11	with	with	ADP
ejpam-6146	26	12	their	their	PRON
ejpam-6146	26	13	basic	basic	ADJ
ejpam-6146	26	14	properties	property	NOUN
ejpam-6146	26	15	from	from	ADP
ejpam-6146	26	16	the	the	DET
ejpam-6146	26	17	differential	differential	ADJ
ejpam-6146	26	18	equation	equation	NOUN
ejpam-6146	26	19	,	,	PUNCT
ejpam-6146	26	20	y′′+y	y′′+y	NOUN
ejpam-6146	26	21	=	=	SYM
ejpam-6146	26	22	0	0	NUM
ejpam-6146	26	23	.	.	PUNCT
ejpam-6146	27	1	then	then	ADV
ejpam-6146	27	2	,	,	PUNCT
ejpam-6146	27	3	we	we	PRON
ejpam-6146	27	4	see	see	VERB
ejpam-6146	27	5	that	that	SCONJ
ejpam-6146	27	6	such	such	ADJ
ejpam-6146	27	7	deductions	deduction	NOUN
ejpam-6146	27	8	,	,	PUNCT
ejpam-6146	27	9	particularly	particularly	ADV
ejpam-6146	27	10	the	the	DET
ejpam-6146	27	11	oscillation	oscillation	NOUN
ejpam-6146	27	12	phenomenon	phenomenon	NOUN
ejpam-6146	27	13	can	can	AUX
ejpam-6146	27	14	be	be	AUX
ejpam-6146	27	15	extended	extend	VERB
ejpam-6146	27	16	to	to	ADP
ejpam-6146	27	17	the	the	DET
ejpam-6146	27	18	second	second	ADJ
ejpam-6146	27	19	-	-	PUNCT
ejpam-6146	27	20	order	order	NOUN
ejpam-6146	27	21	linear	linear	ADJ
ejpam-6146	27	22	differential	differential	NOUN
ejpam-6146	27	23	equations	equation	NOUN
ejpam-6146	27	24	under	under	ADP
ejpam-6146	27	25	certain	certain	ADJ
ejpam-6146	27	26	conditions	condition	NOUN
ejpam-6146	27	27	.	.	PUNCT
ejpam-6146	28	1	the	the	DET
ejpam-6146	28	2	related	related	ADJ
ejpam-6146	28	3	theorems	theorem	NOUN
ejpam-6146	28	4	and	and	CCONJ
ejpam-6146	28	5	techniques	technique	NOUN
ejpam-6146	28	6	are	be	AUX
ejpam-6146	28	7	done	do	VERB
ejpam-6146	28	8	in	in	ADP
ejpam-6146	28	9	detail	detail	NOUN
ejpam-6146	28	10	.	.	PUNCT
ejpam-6146	29	1	we	we	PRON
ejpam-6146	29	2	also	also	ADV
ejpam-6146	29	3	consider	consider	VERB
ejpam-6146	29	4	the	the	DET
ejpam-6146	29	5	classical	classical	ADJ
ejpam-6146	29	6	orthogonal	orthogonal	ADJ
ejpam-6146	29	7	polynomial	polynomial	NOUN
ejpam-6146	29	8	,	,	PUNCT
ejpam-6146	29	9	which	which	PRON
ejpam-6146	29	10	are	be	AUX
ejpam-6146	29	11	governed	govern	VERB
ejpam-6146	29	12	by	by	ADP
ejpam-6146	29	13	second	second	ADJ
ejpam-6146	29	14	-	-	PUNCT
ejpam-6146	29	15	order	order	NOUN
ejpam-6146	29	16	linear	linear	ADJ
ejpam-6146	29	17	differential	differential	NOUN
ejpam-6146	29	18	equations	equation	NOUN
ejpam-6146	29	19	.	.	PUNCT
ejpam-6146	30	1	teaching	teach	VERB
ejpam-6146	30	2	trigonometry	trigonometry	NOUN
ejpam-6146	30	3	from	from	ADP
ejpam-6146	30	4	its	its	PRON
ejpam-6146	30	5	defining	define	VERB
ejpam-6146	30	6	differential	differential	ADJ
ejpam-6146	30	7	equations	equation	NOUN
ejpam-6146	30	8	is	be	AUX
ejpam-6146	30	9	a	a	DET
ejpam-6146	30	10	valuable	valuable	ADJ
ejpam-6146	30	11	but	but	CCONJ
ejpam-6146	30	12	advanced	advanced	ADJ
ejpam-6146	30	13	approach	approach	NOUN
ejpam-6146	30	14	that	that	PRON
ejpam-6146	30	15	offers	offer	VERB
ejpam-6146	30	16	a	a	DET
ejpam-6146	30	17	deeper	deep	ADJ
ejpam-6146	30	18	,	,	PUNCT
ejpam-6146	30	19	more	more	ADV
ejpam-6146	30	20	rigorous	rigorous	ADJ
ejpam-6146	30	21	understanding	understanding	NOUN
ejpam-6146	30	22	of	of	ADP
ejpam-6146	30	23	the	the	DET
ejpam-6146	30	24	functions	function	NOUN
ejpam-6146	30	25	than	than	ADP
ejpam-6146	30	26	the	the	DET
ejpam-6146	30	27	traditional	traditional	ADJ
ejpam-6146	30	28	geometrical	geometrical	ADJ
ejpam-6146	30	29	methods	method	NOUN
ejpam-6146	30	30	.	.	PUNCT
ejpam-6146	31	1	it	it	PRON
ejpam-6146	31	2	connects	connect	VERB
ejpam-6146	31	3	trigonometry	trigonometry	NOUN
ejpam-6146	31	4	directly	directly	ADV
ejpam-6146	31	5	to	to	ADP
ejpam-6146	31	6	calculus	calculus	NOUN
ejpam-6146	31	7	.	.	PUNCT
ejpam-6146	32	1	2	2	X
ejpam-6146	32	2	.	.	X
ejpam-6146	32	3	the	the	DET
ejpam-6146	32	4	sine	sine	NOUN
ejpam-6146	32	5	and	and	CCONJ
ejpam-6146	32	6	cosine	cosine	NOUN
ejpam-6146	32	7	functions	function	NOUN
ejpam-6146	32	8	from	from	ADP
ejpam-6146	32	9	y′′	y′′	PROPN
ejpam-6146	32	10	+	+	CCONJ
ejpam-6146	32	11	y	y	PROPN
ejpam-6146	32	12	=	=	NOUN
ejpam-6146	32	13	0	0	NUM
ejpam-6146	32	14	in	in	ADP
ejpam-6146	32	15	this	this	DET
ejpam-6146	32	16	article	article	NOUN
ejpam-6146	32	17	,	,	PUNCT
ejpam-6146	32	18	we	we	PRON
ejpam-6146	32	19	will	will	AUX
ejpam-6146	32	20	be	be	AUX
ejpam-6146	32	21	dealing	deal	VERB
ejpam-6146	32	22	with	with	ADP
ejpam-6146	32	23	the	the	DET
ejpam-6146	32	24	second	second	ADJ
ejpam-6146	32	25	-	-	PUNCT
ejpam-6146	32	26	order	order	NOUN
ejpam-6146	32	27	linear	linear	PROPN
ejpam-6146	32	28	differential	differential	NOUN
ejpam-6146	32	29	equations	equation	NOUN
ejpam-6146	32	30	,	,	PUNCT
ejpam-6146	32	31	which	which	PRON
ejpam-6146	32	32	in	in	ADP
ejpam-6146	32	33	the	the	DET
ejpam-6146	32	34	nonhomogeneous	nonhomogeneous	ADJ
ejpam-6146	32	35	case	case	NOUN
ejpam-6146	32	36	can	can	AUX
ejpam-6146	32	37	be	be	AUX
ejpam-6146	32	38	written	write	VERB
ejpam-6146	32	39	as	as	ADP
ejpam-6146	32	40	y′′	y′′	PROPN
ejpam-6146	32	41	+	+	CCONJ
ejpam-6146	32	42	p	p	X
ejpam-6146	32	43	(	(	PUNCT
ejpam-6146	32	44	x)y′	x)y′	PROPN
ejpam-6146	33	1	+	+	NOUN
ejpam-6146	33	2	q(x)y	q(x)y	PROPN
ejpam-6146	33	3	=	=	SYM
ejpam-6146	33	4	r(x	r(x	PROPN
ejpam-6146	33	5	)	)	PUNCT
ejpam-6146	33	6	,	,	PUNCT
ejpam-6146	33	7	(	(	PUNCT
ejpam-6146	33	8	1	1	X
ejpam-6146	33	9	)	)	PUNCT
ejpam-6146	33	10	where	where	SCONJ
ejpam-6146	33	11	p	p	NOUN
ejpam-6146	33	12	(	(	PUNCT
ejpam-6146	33	13	x	x	NOUN
ejpam-6146	33	14	)	)	PUNCT
ejpam-6146	33	15	,	,	PUNCT
ejpam-6146	33	16	q(x	q(x	PROPN
ejpam-6146	33	17	)	)	PUNCT
ejpam-6146	33	18	and	and	CCONJ
ejpam-6146	33	19	r(x	r(x	PROPN
ejpam-6146	33	20	)	)	PUNCT
ejpam-6146	33	21	are	be	AUX
ejpam-6146	33	22	continuous	continuous	ADJ
ejpam-6146	33	23	.	.	PUNCT
ejpam-6146	34	1	if	if	SCONJ
ejpam-6146	34	2	r(x	r(x	PROPN
ejpam-6146	34	3	)	)	PUNCT
ejpam-6146	34	4	=	=	SYM
ejpam-6146	34	5	0	0	NUM
ejpam-6146	34	6	,	,	PUNCT
ejpam-6146	34	7	the	the	DET
ejpam-6146	34	8	equation	equation	NOUN
ejpam-6146	34	9	is	be	AUX
ejpam-6146	34	10	said	say	VERB
ejpam-6146	34	11	to	to	PART
ejpam-6146	34	12	be	be	AUX
ejpam-6146	34	13	homogeneous	homogeneous	ADJ
ejpam-6146	34	14	.	.	PUNCT
ejpam-6146	35	1	in	in	ADP
ejpam-6146	35	2	this	this	DET
ejpam-6146	35	3	article	article	NOUN
ejpam-6146	35	4	,	,	PUNCT
ejpam-6146	35	5	we	we	PRON
ejpam-6146	35	6	will	will	AUX
ejpam-6146	35	7	be	be	AUX
ejpam-6146	35	8	dealing	deal	VERB
ejpam-6146	35	9	only	only	ADV
ejpam-6146	35	10	with	with	ADP
ejpam-6146	35	11	the	the	DET
ejpam-6146	35	12	second	second	ADJ
ejpam-6146	35	13	-	-	PUNCT
ejpam-6146	35	14	order	order	NOUN
ejpam-6146	35	15	linear	linear	ADJ
ejpam-6146	35	16	homogeneous	homogeneous	ADJ
ejpam-6146	35	17	differential	differential	ADJ
ejpam-6146	35	18	equations	equation	NOUN
ejpam-6146	35	19	.	.	PUNCT
ejpam-6146	36	1	a	a	DET
ejpam-6146	36	2	second	second	ADJ
ejpam-6146	36	3	-	-	PUNCT
ejpam-6146	36	4	order	order	NOUN
ejpam-6146	36	5	linear	linear	ADJ
ejpam-6146	36	6	homogeneous	homogeneous	ADJ
ejpam-6146	36	7	differential	differential	NOUN
ejpam-6146	36	8	equation	equation	NOUN
ejpam-6146	36	9	has	have	VERB
ejpam-6146	36	10	two	two	NUM
ejpam-6146	36	11	linearly	linearly	ADV
ejpam-6146	36	12	independent	independent	ADJ
ejpam-6146	36	13	solutions	solution	NOUN
ejpam-6146	36	14	,	,	PUNCT
ejpam-6146	36	15	say	say	VERB
ejpam-6146	36	16	y1(x	y1(x	NOUN
ejpam-6146	36	17	)	)	PUNCT
ejpam-6146	36	18	and	and	CCONJ
ejpam-6146	36	19	y2(x	y2(x	NOUN
ejpam-6146	36	20	)	)	PUNCT
ejpam-6146	36	21	.	.	PUNCT
ejpam-6146	37	1	two	two	NUM
ejpam-6146	37	2	functions	function	NOUN
ejpam-6146	37	3	are	be	AUX
ejpam-6146	37	4	said	say	VERB
ejpam-6146	37	5	to	to	PART
ejpam-6146	37	6	be	be	AUX
ejpam-6146	37	7	linearly	linearly	ADV
ejpam-6146	37	8	independent	independent	ADJ
ejpam-6146	37	9	,	,	PUNCT
ejpam-6146	37	10	if	if	SCONJ
ejpam-6146	37	11	one	one	PRON
ejpam-6146	37	12	is	be	AUX
ejpam-6146	37	13	not	not	PART
ejpam-6146	37	14	a	a	DET
ejpam-6146	37	15	multiple	multiple	NOUN
ejpam-6146	37	16	of	of	ADP
ejpam-6146	37	17	the	the	DET
ejpam-6146	37	18	other	other	ADJ
ejpam-6146	37	19	.	.	PUNCT
ejpam-6146	38	1	the	the	DET
ejpam-6146	38	2	linear	linear	ADJ
ejpam-6146	38	3	independence	independence	NOUN
ejpam-6146	38	4	of	of	ADP
ejpam-6146	38	5	y1(x	y1(x	NOUN
ejpam-6146	38	6	)	)	PUNCT
ejpam-6146	38	7	and	and	CCONJ
ejpam-6146	38	8	y2(x	y2(x	NOUN
ejpam-6146	38	9	)	)	PUNCT
ejpam-6146	38	10	is	be	AUX
ejpam-6146	38	11	established	establish	VERB
ejpam-6146	38	12	from	from	ADP
ejpam-6146	38	13	their	their	PRON
ejpam-6146	38	14	wronskian	wronskian	NOUN
ejpam-6146	38	15	w	w	PROPN
ejpam-6146	38	16	(	(	PUNCT
ejpam-6146	38	17	x	x	NOUN
ejpam-6146	38	18	)	)	PUNCT
ejpam-6146	38	19	=	=	PUNCT
ejpam-6146	39	1	w	w	PROPN
ejpam-6146	40	1	[	[	X
ejpam-6146	40	2	y1(x	y1(x	NOUN
ejpam-6146	40	3	)	)	PUNCT
ejpam-6146	40	4	,	,	PUNCT
ejpam-6146	40	5	y2(x	y2(x	PROPN
ejpam-6146	40	6	)	)	PUNCT
ejpam-6146	40	7	]	]	PUNCT
ejpam-6146	41	1	=	=	PUNCT
ejpam-6146	41	2	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6146	41	3	y1(x	y1(x	NOUN
ejpam-6146	41	4	)	)	PUNCT
ejpam-6146	41	5	y2(x	y2(x	NOUN
ejpam-6146	41	6	)	)	PUNCT
ejpam-6146	41	7	y′1(x	y′1(x	NOUN
ejpam-6146	41	8	)	)	PUNCT
ejpam-6146	41	9	y′2(x	y′2(x	PROPN
ejpam-6146	41	10	)	)	PUNCT
ejpam-6146	41	11	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6146	41	12	=	=	PUNCT
ejpam-6146	42	1	y1(x)y	y1(x)y	NUM
ejpam-6146	42	2	′	′	NUM
ejpam-6146	42	3	2(x)−	2(x)−	NUM
ejpam-6146	43	1	y′1(x)y2(x	y′1(x)y2(x	NOUN
ejpam-6146	43	2	)	)	PUNCT
ejpam-6146	43	3	.	.	PUNCT
ejpam-6146	44	1	(	(	PUNCT
ejpam-6146	44	2	2	2	X
ejpam-6146	44	3	)	)	PUNCT
ejpam-6146	44	4	if	if	SCONJ
ejpam-6146	44	5	the	the	DET
ejpam-6146	44	6	wronskian	wronskian	NOUN
ejpam-6146	44	7	is	be	AUX
ejpam-6146	44	8	not	not	PART
ejpam-6146	44	9	zero	zero	NUM
ejpam-6146	44	10	,	,	PUNCT
ejpam-6146	44	11	the	the	DET
ejpam-6146	44	12	two	two	NUM
ejpam-6146	44	13	solutions	solution	NOUN
ejpam-6146	44	14	y1(x	y1(x	NOUN
ejpam-6146	44	15	)	)	PUNCT
ejpam-6146	44	16	and	and	CCONJ
ejpam-6146	44	17	y2(x	y2(x	NOUN
ejpam-6146	44	18	)	)	PUNCT
ejpam-6146	44	19	are	be	AUX
ejpam-6146	44	20	linearly	linearly	ADV
ejpam-6146	44	21	independent	independent	ADJ
ejpam-6146	44	22	.	.	PUNCT
ejpam-6146	45	1	the	the	DET
ejpam-6146	45	2	wronskian	wronskian	NOUN
ejpam-6146	45	3	is	be	AUX
ejpam-6146	45	4	named	name	VERB
ejpam-6146	45	5	in	in	ADP
ejpam-6146	45	6	honour	honour	NOUN
ejpam-6146	45	7	of	of	ADP
ejpam-6146	45	8	the	the	DET
ejpam-6146	45	9	polish	polish	ADJ
ejpam-6146	45	10	mathematician	mathematician	NOUN
ejpam-6146	45	11	,	,	PUNCT
ejpam-6146	45	12	józef	józef	PROPN
ejpam-6146	45	13	maria	maria	PROPN
ejpam-6146	45	14	hoënéwroński	hoënéwroński	PROPN
ejpam-6146	45	15	(	(	PUNCT
ejpam-6146	45	16	1776	1776	NUM
ejpam-6146	45	17	-	-	SYM
ejpam-6146	45	18	1853	1853	NUM
ejpam-6146	45	19	)	)	PUNCT
ejpam-6146	45	20	.	.	PUNCT
ejpam-6146	46	1	a	a	DET
ejpam-6146	46	2	general	general	ADJ
ejpam-6146	46	3	solution	solution	NOUN
ejpam-6146	46	4	is	be	AUX
ejpam-6146	46	5	obtained	obtain	VERB
ejpam-6146	46	6	by	by	ADP
ejpam-6146	46	7	taking	take	VERB
ejpam-6146	46	8	a	a	DET
ejpam-6146	46	9	linear	linear	ADJ
ejpam-6146	46	10	combination	combination	NOUN
ejpam-6146	46	11	,	,	PUNCT
ejpam-6146	46	12	y	y	PROPN
ejpam-6146	46	13	=	=	PUNCT
ejpam-6146	46	14	c1y1(x	c1y1(x	PROPN
ejpam-6146	46	15	)	)	PUNCT
ejpam-6146	47	1	+	+	CCONJ
ejpam-6146	47	2	c2y2(x	c2y2(x	X
ejpam-6146	47	3	)	)	PUNCT
ejpam-6146	47	4	,	,	PUNCT
ejpam-6146	47	5	where	where	SCONJ
ejpam-6146	47	6	c1	c1	PROPN
ejpam-6146	47	7	and	and	CCONJ
ejpam-6146	47	8	c2	c2	PROPN
ejpam-6146	47	9	are	be	AUX
ejpam-6146	47	10	arbitrary	arbitrary	ADJ
ejpam-6146	47	11	constants	constant	NOUN
ejpam-6146	47	12	.	.	PUNCT
ejpam-6146	48	1	the	the	DET
ejpam-6146	48	2	unique	unique	ADJ
ejpam-6146	48	3	solution	solution	NOUN
ejpam-6146	48	4	is	be	AUX
ejpam-6146	48	5	determined	determine	VERB
ejpam-6146	48	6	by	by	ADP
ejpam-6146	48	7	fixing	fix	VERB
ejpam-6146	48	8	the	the	DET
ejpam-6146	48	9	values	value	NOUN
ejpam-6146	48	10	of	of	ADP
ejpam-6146	48	11	the	the	DET
ejpam-6146	48	12	arbitrary	arbitrary	ADJ
ejpam-6146	48	13	constants	constant	NOUN
ejpam-6146	48	14	,	,	PUNCT
ejpam-6146	48	15	c1	c1	PROPN
ejpam-6146	48	16	and	and	CCONJ
ejpam-6146	48	17	c2	c2	PROPN
ejpam-6146	48	18	by	by	ADP
ejpam-6146	48	19	an	an	DET
ejpam-6146	48	20	additional	additional	ADJ
ejpam-6146	48	21	input	input	NOUN
ejpam-6146	48	22	,	,	PUNCT
ejpam-6146	48	23	which	which	PRON
ejpam-6146	48	24	is	be	AUX
ejpam-6146	48	25	the	the	DET
ejpam-6146	48	26	set	set	NOUN
ejpam-6146	48	27	of	of	ADP
ejpam-6146	48	28	initial	initial	ADJ
ejpam-6146	48	29	conditions	condition	NOUN
ejpam-6146	48	30	.	.	PUNCT
ejpam-6146	49	1	the	the	DET
ejpam-6146	49	2	initial	initial	ADJ
ejpam-6146	49	3	conditions	condition	NOUN
ejpam-6146	49	4	are	be	AUX
ejpam-6146	49	5	the	the	DET
ejpam-6146	49	6	values	value	NOUN
ejpam-6146	49	7	of	of	ADP
ejpam-6146	49	8	the	the	DET
ejpam-6146	49	9	solutions	solution	NOUN
ejpam-6146	49	10	,	,	PUNCT
ejpam-6146	49	11	y(x0	y(x0	NOUN
ejpam-6146	49	12	)	)	PUNCT
ejpam-6146	49	13	=	=	SYM
ejpam-6146	49	14	y0	y0	NOUN
ejpam-6146	49	15	and	and	CCONJ
ejpam-6146	49	16	their	their	PRON
ejpam-6146	49	17	derivatives	derivative	NOUN
ejpam-6146	49	18	,	,	PUNCT
ejpam-6146	49	19	y′0(x0	y′0(x0	NOUN
ejpam-6146	49	20	)	)	PUNCT
ejpam-6146	49	21	=	=	PUNCT
ejpam-6146	49	22	y′0	y′0	NOUN
ejpam-6146	49	23	respectively	respectively	ADV
ejpam-6146	49	24	.	.	PUNCT
ejpam-6146	50	1	equation	equation	NOUN
ejpam-6146	50	2	(	(	PUNCT
ejpam-6146	50	3	1	1	X
ejpam-6146	50	4	)	)	PUNCT
ejpam-6146	50	5	has	have	VERB
ejpam-6146	50	6	a	a	DET
ejpam-6146	50	7	unique	unique	ADJ
ejpam-6146	50	8	solution	solution	NOUN
ejpam-6146	50	9	,	,	PUNCT
ejpam-6146	50	10	which	which	PRON
ejpam-6146	50	11	is	be	AUX
ejpam-6146	50	12	summarised	summarise	VERB
ejpam-6146	50	13	in	in	ADP
ejpam-6146	50	14	the	the	DET
ejpam-6146	50	15	following	following	NOUN
ejpam-6146	50	16	theorem	theorem	NOUN
ejpam-6146	50	17	.	.	PUNCT
ejpam-6146	51	1	s.	s.	PROPN
ejpam-6146	51	2	a.	a.	PROPN
ejpam-6146	51	3	khan	khan	PROPN
ejpam-6146	51	4	,	,	PUNCT
ejpam-6146	51	5	m.	m.	NOUN
ejpam-6146	51	6	m.	m.	PROPN
ejpam-6146	51	7	kankarej	kankarej	PROPN
ejpam-6146	51	8	,	,	PUNCT
ejpam-6146	51	9	m.	m.	PROPN
ejpam-6146	51	10	n.	n.	PROPN
ejpam-6146	51	11	i.	i.	PROPN
ejpam-6146	51	12	khan	khan	PROPN
ejpam-6146	51	13	/	/	SYM
ejpam-6146	51	14	eur	eur	PROPN
ejpam-6146	51	15	.	.	PUNCT
ejpam-6146	52	1	j.	j.	PROPN
ejpam-6146	52	2	pure	pure	PROPN
ejpam-6146	52	3	appl	appl	PROPN
ejpam-6146	52	4	.	.	PROPN
ejpam-6146	52	5	math	math	PROPN
ejpam-6146	52	6	,	,	PUNCT
ejpam-6146	52	7	18	18	NUM
ejpam-6146	52	8	(	(	PUNCT
ejpam-6146	52	9	4	4	NUM
ejpam-6146	52	10	)	)	PUNCT
ejpam-6146	52	11	(	(	PUNCT
ejpam-6146	52	12	2025	2025	NUM
ejpam-6146	52	13	)	)	PUNCT
ejpam-6146	52	14	,	,	PUNCT
ejpam-6146	52	15	6146	6146	NUM
ejpam-6146	52	16	3	3	NUM
ejpam-6146	52	17	of	of	ADP
ejpam-6146	52	18	14	14	NUM
ejpam-6146	52	19	theorem	theorem	NOUN
ejpam-6146	52	20	1	1	NUM
ejpam-6146	52	21	.	.	NOUN
ejpam-6146	52	22	existence	existence	NOUN
ejpam-6146	52	23	and	and	CCONJ
ejpam-6146	52	24	uniqueness	uniqueness	NOUN
ejpam-6146	52	25	theorem	theorem	VERB
ejpam-6146	52	26	:	:	PUNCT
ejpam-6146	52	27	let	let	VERB
ejpam-6146	52	28	p	p	NOUN
ejpam-6146	52	29	(	(	PUNCT
ejpam-6146	52	30	x	x	NOUN
ejpam-6146	52	31	)	)	PUNCT
ejpam-6146	52	32	,	,	PUNCT
ejpam-6146	52	33	q(x	q(x	PROPN
ejpam-6146	52	34	)	)	PUNCT
ejpam-6146	52	35	and	and	CCONJ
ejpam-6146	52	36	r(x	r(x	PROPN
ejpam-6146	52	37	)	)	PUNCT
ejpam-6146	52	38	be	be	VERB
ejpam-6146	52	39	continuous	continuous	ADJ
ejpam-6146	52	40	functions	function	NOUN
ejpam-6146	52	41	on	on	ADP
ejpam-6146	52	42	a	a	DET
ejpam-6146	52	43	closed	closed	ADJ
ejpam-6146	52	44	interval	interval	NOUN
ejpam-6146	52	45	[	[	X
ejpam-6146	52	46	a	a	DET
ejpam-6146	52	47	,	,	PUNCT
ejpam-6146	52	48	b	b	NOUN
ejpam-6146	52	49	]	]	X
ejpam-6146	52	50	.	.	PUNCT
ejpam-6146	53	1	if	if	SCONJ
ejpam-6146	53	2	x0	x0	PROPN
ejpam-6146	53	3	is	be	AUX
ejpam-6146	53	4	any	any	DET
ejpam-6146	53	5	point	point	NOUN
ejpam-6146	53	6	in	in	ADP
ejpam-6146	53	7	[	[	X
ejpam-6146	53	8	a	a	DET
ejpam-6146	53	9	,	,	PUNCT
ejpam-6146	53	10	b	b	NOUN
ejpam-6146	53	11	]	]	X
ejpam-6146	53	12	,	,	PUNCT
ejpam-6146	53	13	and	and	CCONJ
ejpam-6146	53	14	if	if	SCONJ
ejpam-6146	53	15	y0	y0	PROPN
ejpam-6146	53	16	and	and	CCONJ
ejpam-6146	53	17	y′0	y′0	NOUN
ejpam-6146	53	18	are	be	AUX
ejpam-6146	53	19	any	any	DET
ejpam-6146	53	20	numbers	number	NOUN
ejpam-6146	53	21	whatever	whatever	PRON
ejpam-6146	53	22	,	,	PUNCT
ejpam-6146	53	23	then	then	ADV
ejpam-6146	53	24	y′′	y′′	PROPN
ejpam-6146	53	25	+	+	CCONJ
ejpam-6146	53	26	p	p	X
ejpam-6146	53	27	(	(	PUNCT
ejpam-6146	53	28	x)y′	x)y′	PROPN
ejpam-6146	53	29	+	+	NOUN
ejpam-6146	53	30	q(x)y	q(x)y	PROPN
ejpam-6146	53	31	=	=	SYM
ejpam-6146	53	32	r(x	r(x	PROPN
ejpam-6146	53	33	)	)	PUNCT
ejpam-6146	53	34	has	have	VERB
ejpam-6146	53	35	one	one	NUM
ejpam-6146	53	36	and	and	CCONJ
ejpam-6146	53	37	only	only	ADV
ejpam-6146	53	38	one	one	NUM
ejpam-6146	53	39	solution	solution	NOUN
ejpam-6146	53	40	y(x	y(x	NOUN
ejpam-6146	53	41	)	)	PUNCT
ejpam-6146	53	42	on	on	ADP
ejpam-6146	53	43	the	the	DET
ejpam-6146	53	44	entire	entire	ADJ
ejpam-6146	53	45	interval	interval	NOUN
ejpam-6146	53	46	such	such	ADJ
ejpam-6146	53	47	that	that	DET
ejpam-6146	53	48	y(x0	y(x0	NOUN
ejpam-6146	53	49	)	)	PUNCT
ejpam-6146	54	1	=	=	SYM
ejpam-6146	54	2	y0	y0	PROPN
ejpam-6146	54	3	and	and	CCONJ
ejpam-6146	54	4	y′0(x0	y′0(x0	NOUN
ejpam-6146	54	5	)	)	PUNCT
ejpam-6146	54	6	=	=	SYM
ejpam-6146	54	7	y′0	y′0	NOUN
ejpam-6146	54	8	.	.	PUNCT
ejpam-6146	55	1	this	this	PRON
ejpam-6146	55	2	theorem	theorem	VERB
ejpam-6146	55	3	along	along	ADP
ejpam-6146	55	4	with	with	ADP
ejpam-6146	55	5	the	the	DET
ejpam-6146	55	6	wronskian	wronskian	NOUN
ejpam-6146	55	7	is	be	AUX
ejpam-6146	55	8	sufficient	sufficient	ADJ
ejpam-6146	55	9	to	to	PART
ejpam-6146	55	10	deduce	deduce	VERB
ejpam-6146	55	11	the	the	DET
ejpam-6146	55	12	properties	property	NOUN
ejpam-6146	55	13	of	of	ADP
ejpam-6146	55	14	the	the	DET
ejpam-6146	55	15	two	two	NUM
ejpam-6146	55	16	trigonometric	trigonometric	ADJ
ejpam-6146	55	17	functions	function	NOUN
ejpam-6146	55	18	,	,	PUNCT
ejpam-6146	55	19	sine	sine	NOUN
ejpam-6146	55	20	and	and	CCONJ
ejpam-6146	55	21	cosine	cosine	NOUN
ejpam-6146	55	22	.	.	PUNCT
ejpam-6146	56	1	let	let	VERB
ejpam-6146	56	2	us	we	PRON
ejpam-6146	56	3	consider	consider	VERB
ejpam-6146	56	4	the	the	DET
ejpam-6146	56	5	second	second	ADJ
ejpam-6146	56	6	-	-	PUNCT
ejpam-6146	56	7	order	order	NOUN
ejpam-6146	56	8	linear	linear	ADJ
ejpam-6146	56	9	homogeneous	homogeneous	ADJ
ejpam-6146	56	10	differential	differential	NOUN
ejpam-6146	56	11	equation	equation	NOUN
ejpam-6146	56	12	y′′	y′′	PROPN
ejpam-6146	56	13	+	+	CCONJ
ejpam-6146	56	14	y	y	PROPN
ejpam-6146	56	15	=	=	SYM
ejpam-6146	56	16	0	0	NUM
ejpam-6146	56	17	,	,	PUNCT
ejpam-6146	56	18	(	(	PUNCT
ejpam-6146	56	19	3	3	X
ejpam-6146	56	20	)	)	PUNCT
ejpam-6146	56	21	with	with	ADP
ejpam-6146	56	22	the	the	DET
ejpam-6146	56	23	initial	initial	ADJ
ejpam-6146	56	24	conditions	condition	NOUN
ejpam-6146	56	25	y1(0	y1(0	PROPN
ejpam-6146	56	26	)	)	PUNCT
ejpam-6146	57	1	=	=	SYM
ejpam-6146	57	2	0	0	NUM
ejpam-6146	57	3	,	,	PUNCT
ejpam-6146	57	4	y′1(0	y′1(0	PROPN
ejpam-6146	57	5	)	)	PUNCT
ejpam-6146	57	6	=	=	SYM
ejpam-6146	57	7	1	1	NUM
ejpam-6146	57	8	and	and	CCONJ
ejpam-6146	57	9	y2(0	y2(0	PROPN
ejpam-6146	57	10	)	)	PUNCT
ejpam-6146	57	11	=	=	SYM
ejpam-6146	57	12	1	1	NUM
ejpam-6146	57	13	,	,	PUNCT
ejpam-6146	57	14	y′2(0	y′2(0	NUM
ejpam-6146	57	15	)	)	PUNCT
ejpam-6146	57	16	=	=	SYM
ejpam-6146	58	1	0	0	X
ejpam-6146	58	2	.	.	PUNCT
ejpam-6146	59	1	we	we	PRON
ejpam-6146	59	2	use	use	VERB
ejpam-6146	59	3	only	only	ADV
ejpam-6146	59	4	the	the	DET
ejpam-6146	59	5	qualitative	qualitative	ADJ
ejpam-6146	59	6	properties	property	NOUN
ejpam-6146	59	7	of	of	ADP
ejpam-6146	59	8	the	the	DET
ejpam-6146	59	9	differential	differential	ADJ
ejpam-6146	59	10	equations	equation	NOUN
ejpam-6146	59	11	to	to	PART
ejpam-6146	59	12	deduce	deduce	VERB
ejpam-6146	59	13	the	the	DET
ejpam-6146	59	14	trigonometric	trigonometric	ADJ
ejpam-6146	59	15	functions	function	NOUN
ejpam-6146	59	16	as	as	ADV
ejpam-6146	59	17	well	well	ADV
ejpam-6146	59	18	as	as	ADP
ejpam-6146	59	19	some	some	PRON
ejpam-6146	59	20	of	of	ADP
ejpam-6146	59	21	their	their	PRON
ejpam-6146	59	22	properties	property	NOUN
ejpam-6146	59	23	.	.	PUNCT
ejpam-6146	60	1	theorem	theorem	NOUN
ejpam-6146	60	2	(	(	PUNCT
ejpam-6146	60	3	1	1	X
ejpam-6146	60	4	)	)	PUNCT
ejpam-6146	60	5	ensures	ensure	VERB
ejpam-6146	60	6	that	that	SCONJ
ejpam-6146	60	7	,	,	PUNCT
ejpam-6146	60	8	this	this	DET
ejpam-6146	60	9	equation	equation	NOUN
ejpam-6146	60	10	has	have	VERB
ejpam-6146	60	11	a	a	DET
ejpam-6146	60	12	unique	unique	ADJ
ejpam-6146	60	13	solution	solution	NOUN
ejpam-6146	60	14	.	.	PUNCT
ejpam-6146	61	1	let	let	VERB
ejpam-6146	61	2	the	the	DET
ejpam-6146	61	3	two	two	NUM
ejpam-6146	61	4	linearly	linearly	ADV
ejpam-6146	61	5	independent	independent	ADJ
ejpam-6146	61	6	solutions	solution	NOUN
ejpam-6146	61	7	be	be	VERB
ejpam-6146	61	8	y1	y1	NOUN
ejpam-6146	61	9	=	=	SYM
ejpam-6146	61	10	s(x	s(x	PROPN
ejpam-6146	61	11	)	)	PUNCT
ejpam-6146	61	12	and	and	CCONJ
ejpam-6146	61	13	y2	y2	NOUN
ejpam-6146	61	14	=	=	SYM
ejpam-6146	61	15	c(x	c(x	NOUN
ejpam-6146	61	16	)	)	PUNCT
ejpam-6146	61	17	with	with	ADP
ejpam-6146	61	18	the	the	DET
ejpam-6146	61	19	initial	initial	ADJ
ejpam-6146	61	20	conditions	condition	NOUN
ejpam-6146	61	21	s(0	s(0	PROPN
ejpam-6146	61	22	)	)	PUNCT
ejpam-6146	61	23	=	=	SYM
ejpam-6146	61	24	0	0	NUM
ejpam-6146	61	25	,	,	PUNCT
ejpam-6146	61	26	s′(0	s′(0	NOUN
ejpam-6146	61	27	)	)	PUNCT
ejpam-6146	61	28	=	=	SYM
ejpam-6146	61	29	1	1	NUM
ejpam-6146	61	30	and	and	CCONJ
ejpam-6146	61	31	c(0	c(0	NOUN
ejpam-6146	61	32	)	)	PUNCT
ejpam-6146	61	33	=	=	SYM
ejpam-6146	62	1	1	1	NUM
ejpam-6146	62	2	,	,	PUNCT
ejpam-6146	62	3	c′(0	c′(0	NOUN
ejpam-6146	62	4	)	)	PUNCT
ejpam-6146	62	5	=	=	NOUN
ejpam-6146	63	1	0	0	X
ejpam-6146	63	2	.	.	PUNCT
ejpam-6146	64	1	from	from	ADP
ejpam-6146	64	2	the	the	DET
ejpam-6146	64	3	initial	initial	ADJ
ejpam-6146	64	4	conditions	condition	NOUN
ejpam-6146	64	5	,	,	PUNCT
ejpam-6146	64	6	the	the	DET
ejpam-6146	64	7	graph	graph	NOUN
ejpam-6146	64	8	of	of	ADP
ejpam-6146	64	9	s(x	s(x	PROPN
ejpam-6146	64	10	)	)	PUNCT
ejpam-6146	64	11	starts	start	VERB
ejpam-6146	64	12	at	at	ADP
ejpam-6146	64	13	origin	origin	NOUN
ejpam-6146	64	14	with	with	ADP
ejpam-6146	64	15	slope	slope	NOUN
ejpam-6146	64	16	of	of	ADP
ejpam-6146	64	17	unity	unity	NOUN
ejpam-6146	64	18	.	.	PUNCT
ejpam-6146	65	1	from	from	ADP
ejpam-6146	65	2	the	the	DET
ejpam-6146	65	3	defining	define	VERB
ejpam-6146	65	4	equation	equation	NOUN
ejpam-6146	65	5	eq	eq	ADJ
ejpam-6146	65	6	.	.	PUNCT
ejpam-6146	66	1	(	(	PUNCT
ejpam-6146	66	2	3	3	NUM
ejpam-6146	66	3	)	)	PUNCT
ejpam-6146	66	4	,	,	PUNCT
ejpam-6146	66	5	s′′(x	s′′(x	NOUN
ejpam-6146	66	6	)	)	PUNCT
ejpam-6146	66	7	=	=	SYM
ejpam-6146	66	8	−s(x	−s(x	NOUN
ejpam-6146	66	9	)	)	PUNCT
ejpam-6146	66	10	.	.	PUNCT
ejpam-6146	67	1	so	so	ADV
ejpam-6146	67	2	,	,	PUNCT
ejpam-6146	67	3	s′′(x	s′′(x	PROPN
ejpam-6146	67	4	)	)	PUNCT
ejpam-6146	67	5	is	be	AUX
ejpam-6146	67	6	a	a	DET
ejpam-6146	67	7	negative	negative	ADJ
ejpam-6146	67	8	number	number	NOUN
ejpam-6146	67	9	whenever	whenever	SCONJ
ejpam-6146	67	10	s(x	s(x	PROPN
ejpam-6146	67	11	)	)	PUNCT
ejpam-6146	67	12	is	be	AUX
ejpam-6146	67	13	above	above	ADP
ejpam-6146	67	14	the	the	DET
ejpam-6146	67	15	x	x	NOUN
ejpam-6146	67	16	-	-	NOUN
ejpam-6146	67	17	axis	axis	NOUN
ejpam-6146	67	18	.	.	PUNCT
ejpam-6146	68	1	furthermore	furthermore	ADV
ejpam-6146	68	2	,	,	PUNCT
ejpam-6146	68	3	s′′(x	s′′(x	NOUN
ejpam-6146	68	4	)	)	PUNCT
ejpam-6146	68	5	is	be	AUX
ejpam-6146	68	6	a	a	DET
ejpam-6146	68	7	negative	negative	ADJ
ejpam-6146	68	8	number	number	NOUN
ejpam-6146	68	9	,	,	PUNCT
ejpam-6146	68	10	which	which	PRON
ejpam-6146	68	11	increases	increase	VERB
ejpam-6146	68	12	in	in	ADP
ejpam-6146	68	13	magnitude	magnitude	NOUN
ejpam-6146	68	14	as	as	ADP
ejpam-6146	68	15	the	the	DET
ejpam-6146	68	16	curve	curve	NOUN
ejpam-6146	68	17	s(x	s(x	PROPN
ejpam-6146	68	18	)	)	PUNCT
ejpam-6146	68	19	rises	rise	NOUN
ejpam-6146	68	20	.	.	PUNCT
ejpam-6146	69	1	note	note	NOUN
ejpam-6146	69	2	,	,	PUNCT
ejpam-6146	69	3	that	that	DET
ejpam-6146	69	4	s′′(x	s′′(x	NOUN
ejpam-6146	69	5	)	)	PUNCT
ejpam-6146	69	6	is	be	AUX
ejpam-6146	69	7	the	the	DET
ejpam-6146	69	8	rate	rate	NOUN
ejpam-6146	69	9	of	of	ADP
ejpam-6146	69	10	change	change	NOUN
ejpam-6146	69	11	of	of	ADP
ejpam-6146	69	12	the	the	DET
ejpam-6146	69	13	slope	slope	NOUN
ejpam-6146	69	14	s′(x	s′(x	NOUN
ejpam-6146	69	15	)	)	PUNCT
ejpam-6146	69	16	.	.	PUNCT
ejpam-6146	70	1	this	this	DET
ejpam-6146	70	2	slope	slope	NOUN
ejpam-6146	70	3	decreases	decrease	VERB
ejpam-6146	70	4	at	at	ADP
ejpam-6146	70	5	an	an	DET
ejpam-6146	70	6	increasing	increase	VERB
ejpam-6146	70	7	rate	rate	NOUN
ejpam-6146	70	8	and	and	CCONJ
ejpam-6146	70	9	it	it	PRON
ejpam-6146	70	10	must	must	AUX
ejpam-6146	70	11	reach	reach	VERB
ejpam-6146	70	12	zero	zero	NUM
ejpam-6146	70	13	at	at	ADP
ejpam-6146	70	14	some	some	DET
ejpam-6146	70	15	point	point	NOUN
ejpam-6146	70	16	x	x	X
ejpam-6146	70	17	=	=	SYM
ejpam-6146	70	18	x0	x0	PROPN
ejpam-6146	70	19	.	.	PUNCT
ejpam-6146	71	1	with	with	ADP
ejpam-6146	71	2	further	further	ADJ
ejpam-6146	71	3	increase	increase	NOUN
ejpam-6146	71	4	in	in	ADP
ejpam-6146	71	5	x	x	PRON
ejpam-6146	71	6	,	,	PUNCT
ejpam-6146	71	7	the	the	DET
ejpam-6146	71	8	curve	curve	NOUN
ejpam-6146	71	9	falls	fall	VERB
ejpam-6146	71	10	towards	towards	ADP
ejpam-6146	71	11	the	the	DET
ejpam-6146	71	12	x	x	NOUN
ejpam-6146	71	13	-	-	NOUN
ejpam-6146	71	14	axis	axis	ADJ
ejpam-6146	71	15	,	,	PUNCT
ejpam-6146	71	16	s′(x	s′(x	PUNCT
ejpam-6146	71	17	)	)	PUNCT
ejpam-6146	71	18	decreases	decrease	NOUN
ejpam-6146	71	19	at	at	ADP
ejpam-6146	71	20	a	a	DET
ejpam-6146	71	21	decreasing	decrease	VERB
ejpam-6146	71	22	rate	rate	NOUN
ejpam-6146	71	23	.	.	PUNCT
ejpam-6146	72	1	the	the	DET
ejpam-6146	72	2	curve	curve	NOUN
ejpam-6146	72	3	crosses	cross	VERB
ejpam-6146	72	4	the	the	DET
ejpam-6146	72	5	x	x	NOUN
ejpam-6146	72	6	-	-	NOUN
ejpam-6146	72	7	axis	axis	NOUN
ejpam-6146	72	8	at	at	ADP
ejpam-6146	72	9	the	the	DET
ejpam-6146	72	10	point	point	NOUN
ejpam-6146	72	11	x	x	PUNCT
ejpam-6146	73	1	=	=	SYM
ejpam-6146	74	1	p.	p.	NOUN
ejpam-6146	74	2	the	the	DET
ejpam-6146	74	3	value	value	NOUN
ejpam-6146	74	4	of	of	ADP
ejpam-6146	74	5	p	p	NOUN
ejpam-6146	74	6	will	will	AUX
ejpam-6146	74	7	be	be	AUX
ejpam-6146	74	8	later	later	ADV
ejpam-6146	74	9	determined	determined	ADJ
ejpam-6146	74	10	to	to	PART
ejpam-6146	74	11	be	be	AUX
ejpam-6146	74	12	π	π	X
ejpam-6146	74	13	.	.	PUNCT
ejpam-6146	75	1	as	as	ADP
ejpam-6146	75	2	s′′(x	s′′(x	NOUN
ejpam-6146	75	3	)	)	PUNCT
ejpam-6146	75	4	depends	depend	VERB
ejpam-6146	75	5	only	only	ADV
ejpam-6146	75	6	on	on	ADP
ejpam-6146	75	7	s(x	s(x	NOUN
ejpam-6146	75	8	)	)	PUNCT
ejpam-6146	75	9	,	,	PUNCT
ejpam-6146	75	10	the	the	DET
ejpam-6146	75	11	graph	graph	NOUN
ejpam-6146	75	12	between	between	ADP
ejpam-6146	75	13	x	x	PROPN
ejpam-6146	75	14	=	=	SYM
ejpam-6146	75	15	0	0	NUM
ejpam-6146	75	16	and	and	CCONJ
ejpam-6146	75	17	x	x	X
ejpam-6146	75	18	=	=	PRON
ejpam-6146	75	19	p	p	NOUN
ejpam-6146	75	20	is	be	AUX
ejpam-6146	75	21	symmetric	symmetric	ADJ
ejpam-6146	75	22	about	about	ADP
ejpam-6146	75	23	the	the	DET
ejpam-6146	75	24	line	line	NOUN
ejpam-6146	75	25	x	x	X
ejpam-6146	76	1	=	=	SYM
ejpam-6146	76	2	x0	x0	PROPN
ejpam-6146	76	3	.	.	PUNCT
ejpam-6146	77	1	so	so	ADV
ejpam-6146	77	2	,	,	PUNCT
ejpam-6146	77	3	x0	x0	PROPN
ejpam-6146	77	4	=	=	X
ejpam-6146	77	5	p/2	p/2	NOUN
ejpam-6146	77	6	and	and	CCONJ
ejpam-6146	77	7	s′(p	s′(p	ADJ
ejpam-6146	77	8	)	)	PUNCT
ejpam-6146	77	9	=	=	SYM
ejpam-6146	77	10	−1	−1	NOUN
ejpam-6146	77	11	.	.	PUNCT
ejpam-6146	78	1	the	the	DET
ejpam-6146	78	2	maximum	maximum	NOUN
ejpam-6146	78	3	of	of	ADP
ejpam-6146	78	4	s(x	s(x	PROPN
ejpam-6146	78	5	)	)	PUNCT
ejpam-6146	78	6	is	be	AUX
ejpam-6146	78	7	at	at	ADP
ejpam-6146	78	8	x	x	X
ejpam-6146	78	9	=	=	PUNCT
ejpam-6146	78	10	p/2	p/2	NUM
ejpam-6146	78	11	.	.	PUNCT
ejpam-6146	79	1	a	a	DET
ejpam-6146	79	2	similar	similar	ADJ
ejpam-6146	79	3	line	line	NOUN
ejpam-6146	79	4	of	of	ADP
ejpam-6146	79	5	arguments	argument	NOUN
ejpam-6146	79	6	leads	lead	VERB
ejpam-6146	79	7	to	to	ADP
ejpam-6146	79	8	the	the	DET
ejpam-6146	79	9	conclusion	conclusion	NOUN
ejpam-6146	79	10	that	that	SCONJ
ejpam-6146	79	11	the	the	DET
ejpam-6146	79	12	portion	portion	NOUN
ejpam-6146	79	13	of	of	ADP
ejpam-6146	79	14	the	the	DET
ejpam-6146	79	15	curve	curve	NOUN
ejpam-6146	79	16	x	x	X
ejpam-6146	79	17	∈	∈	PROPN
ejpam-6146	79	18	(	(	PUNCT
ejpam-6146	79	19	p	p	NOUN
ejpam-6146	79	20	,	,	PUNCT
ejpam-6146	79	21	2p	2p	NUM
ejpam-6146	79	22	)	)	PUNCT
ejpam-6146	79	23	is	be	AUX
ejpam-6146	79	24	an	an	DET
ejpam-6146	79	25	inverted	inverted	ADJ
ejpam-6146	79	26	replica	replica	NOUN
ejpam-6146	79	27	of	of	ADP
ejpam-6146	79	28	the	the	DET
ejpam-6146	79	29	first	first	ADJ
ejpam-6146	79	30	arch	arch	NOUN
ejpam-6146	79	31	,	,	PUNCT
ejpam-6146	79	32	x	x	SYM
ejpam-6146	79	33	∈	∈	PROPN
ejpam-6146	79	34	(	(	PUNCT
ejpam-6146	79	35	0	0	NUM
ejpam-6146	79	36	,	,	PUNCT
ejpam-6146	79	37	p	p	NOUN
ejpam-6146	79	38	)	)	PUNCT
ejpam-6146	79	39	.	.	PUNCT
ejpam-6146	80	1	the	the	DET
ejpam-6146	80	2	pair	pair	NOUN
ejpam-6146	80	3	of	of	ADP
ejpam-6146	80	4	arches	arch	NOUN
ejpam-6146	80	5	(	(	PUNCT
ejpam-6146	80	6	one	one	NUM
ejpam-6146	80	7	above	above	ADV
ejpam-6146	80	8	and	and	CCONJ
ejpam-6146	80	9	one	one	NUM
ejpam-6146	80	10	below	below	ADP
ejpam-6146	80	11	the	the	DET
ejpam-6146	80	12	x	x	NOUN
ejpam-6146	80	13	-	-	NOUN
ejpam-6146	80	14	axis	axis	ADJ
ejpam-6146	80	15	)	)	PUNCT
ejpam-6146	80	16	are	be	AUX
ejpam-6146	80	17	found	find	VERB
ejpam-6146	80	18	to	to	PART
ejpam-6146	80	19	repeat	repeat	VERB
ejpam-6146	80	20	indefinitely	indefinitely	ADV
ejpam-6146	80	21	after	after	ADP
ejpam-6146	80	22	every	every	DET
ejpam-6146	80	23	2p	2p	NOUN
ejpam-6146	80	24	.	.	PUNCT
ejpam-6146	81	1	thus	thus	ADV
ejpam-6146	81	2	,	,	PUNCT
ejpam-6146	81	3	s(x	s(x	PROPN
ejpam-6146	81	4	)	)	PUNCT
ejpam-6146	81	5	is	be	AUX
ejpam-6146	81	6	periodic	periodic	ADJ
ejpam-6146	81	7	,	,	PUNCT
ejpam-6146	81	8	s(x	s(x	NOUN
ejpam-6146	81	9	)	)	PUNCT
ejpam-6146	81	10	=	=	PUNCT
ejpam-6146	81	11	s(x+	s(x+	NOUN
ejpam-6146	81	12	2p	2p	NUM
ejpam-6146	81	13	)	)	PUNCT
ejpam-6146	81	14	.	.	PUNCT
ejpam-6146	82	1	with	with	ADP
ejpam-6146	82	2	the	the	DET
ejpam-6146	82	3	initial	initial	ADJ
ejpam-6146	82	4	conditions	condition	NOUN
ejpam-6146	82	5	,	,	PUNCT
ejpam-6146	82	6	c(0	c(0	NOUN
ejpam-6146	82	7	)	)	PUNCT
ejpam-6146	82	8	=	=	SYM
ejpam-6146	82	9	1	1	NUM
ejpam-6146	82	10	,	,	PUNCT
ejpam-6146	82	11	c′(0	c′(0	NOUN
ejpam-6146	82	12	)	)	PUNCT
ejpam-6146	82	13	=	=	SYM
ejpam-6146	82	14	0	0	NUM
ejpam-6146	82	15	,	,	PUNCT
ejpam-6146	82	16	we	we	PRON
ejpam-6146	82	17	note	note	VERB
ejpam-6146	82	18	that	that	SCONJ
ejpam-6146	82	19	the	the	DET
ejpam-6146	82	20	graph	graph	NOUN
ejpam-6146	82	21	of	of	ADP
ejpam-6146	82	22	c(x	c(x	NOUN
ejpam-6146	82	23	)	)	PUNCT
ejpam-6146	82	24	starts	start	VERB
ejpam-6146	82	25	at	at	ADP
ejpam-6146	82	26	the	the	DET
ejpam-6146	82	27	point	point	NOUN
ejpam-6146	82	28	(	(	PUNCT
ejpam-6146	82	29	0	0	NUM
ejpam-6146	82	30	,	,	PUNCT
ejpam-6146	82	31	1	1	NUM
ejpam-6146	82	32	)	)	PUNCT
ejpam-6146	82	33	with	with	ADP
ejpam-6146	82	34	slope	slope	NOUN
ejpam-6146	82	35	zero	zero	NUM
ejpam-6146	82	36	.	.	PUNCT
ejpam-6146	83	1	using	use	VERB
ejpam-6146	83	2	the	the	DET
ejpam-6146	83	3	same	same	ADJ
ejpam-6146	83	4	reasoning	reasoning	NOUN
ejpam-6146	83	5	,	,	PUNCT
ejpam-6146	83	6	we	we	PRON
ejpam-6146	83	7	conclude	conclude	VERB
ejpam-6146	83	8	that	that	SCONJ
ejpam-6146	83	9	the	the	DET
ejpam-6146	83	10	graph	graph	NOUN
ejpam-6146	83	11	of	of	ADP
ejpam-6146	83	12	c(x	c(x	NOUN
ejpam-6146	83	13	)	)	PUNCT
ejpam-6146	83	14	also	also	ADV
ejpam-6146	83	15	oscillates	oscillate	VERB
ejpam-6146	83	16	about	about	ADP
ejpam-6146	83	17	the	the	DET
ejpam-6146	83	18	x	x	NOUN
ejpam-6146	83	19	-	-	NOUN
ejpam-6146	83	20	axis	axis	NOUN
ejpam-6146	83	21	with	with	ADP
ejpam-6146	83	22	a	a	DET
ejpam-6146	83	23	periodicity	periodicity	NOUN
ejpam-6146	83	24	2p̃.	2p̃.	NUM
ejpam-6146	83	25	later	later	ADV
ejpam-6146	83	26	,	,	PUNCT
ejpam-6146	83	27	we	we	PRON
ejpam-6146	83	28	shall	shall	AUX
ejpam-6146	83	29	see	see	VERB
ejpam-6146	83	30	that	that	DET
ejpam-6146	83	31	p̃	p̃	PROPN
ejpam-6146	83	32	=	=	SYM
ejpam-6146	83	33	p	p	X
ejpam-6146	83	34	=	=	SYM
ejpam-6146	83	35	π	π	PROPN
ejpam-6146	83	36	,	,	PUNCT
ejpam-6146	83	37	the	the	DET
ejpam-6146	83	38	transcendental	transcendental	ADJ
ejpam-6146	83	39	number	number	NOUN
ejpam-6146	83	40	from	from	ADP
ejpam-6146	83	41	the	the	DET
ejpam-6146	83	42	circle	circle	NOUN
ejpam-6146	83	43	.	.	PUNCT
ejpam-6146	84	1	in	in	ADP
ejpam-6146	84	2	order	order	NOUN
ejpam-6146	84	3	to	to	PART
ejpam-6146	84	4	interconnect	interconnect	VERB
ejpam-6146	84	5	s(x	s(x	PROPN
ejpam-6146	84	6	)	)	PUNCT
ejpam-6146	84	7	and	and	CCONJ
ejpam-6146	84	8	c(x	c(x	NOUN
ejpam-6146	84	9	)	)	PUNCT
ejpam-6146	84	10	(	(	PUNCT
ejpam-6146	84	11	so	so	ADV
ejpam-6146	84	12	also	also	ADV
ejpam-6146	84	13	p	p	NOUN
ejpam-6146	84	14	and	and	CCONJ
ejpam-6146	84	15	p̃	p̃	PROPN
ejpam-6146	84	16	)	)	PUNCT
ejpam-6146	84	17	,	,	PUNCT
ejpam-6146	84	18	we	we	PRON
ejpam-6146	84	19	differentiate	differentiate	VERB
ejpam-6146	84	20	eq	eq	ADP
ejpam-6146	84	21	.	.	PUNCT
ejpam-6146	85	1	(	(	PUNCT
ejpam-6146	85	2	3	3	NUM
ejpam-6146	85	3	)	)	PUNCT
ejpam-6146	85	4	and	and	CCONJ
ejpam-6146	85	5	obtain	obtain	VERB
ejpam-6146	85	6	y′′′	y′′′	PROPN
ejpam-6146	85	7	+	+	PUNCT
ejpam-6146	85	8	y′	y′	NOUN
ejpam-6146	85	9	=	=	SYM
ejpam-6146	85	10	0	0	NUM
ejpam-6146	85	11	or	or	CCONJ
ejpam-6146	85	12	(	(	PUNCT
ejpam-6146	85	13	y′)′′	y′)′′	NOUN
ejpam-6146	86	1	+	+	CCONJ
ejpam-6146	86	2	(	(	PUNCT
ejpam-6146	86	3	y′	y′	NUM
ejpam-6146	86	4	)	)	PUNCT
ejpam-6146	86	5	=	=	SYM
ejpam-6146	86	6	0	0	X
ejpam-6146	86	7	.	.	PUNCT
ejpam-6146	87	1	consequently	consequently	ADV
ejpam-6146	87	2	,	,	PUNCT
ejpam-6146	87	3	the	the	DET
ejpam-6146	87	4	derivative	derivative	NOUN
ejpam-6146	87	5	of	of	ADP
ejpam-6146	87	6	any	any	DET
ejpam-6146	87	7	solution	solution	NOUN
ejpam-6146	87	8	of	of	ADP
ejpam-6146	87	9	eq	eq	PROPN
ejpam-6146	87	10	.	.	PUNCT
ejpam-6146	88	1	(	(	PUNCT
ejpam-6146	88	2	3	3	X
ejpam-6146	88	3	)	)	PUNCT
ejpam-6146	88	4	is	be	AUX
ejpam-6146	88	5	also	also	ADV
ejpam-6146	88	6	a	a	DET
ejpam-6146	88	7	solution	solution	NOUN
ejpam-6146	88	8	.	.	PUNCT
ejpam-6146	89	1	thus	thus	ADV
ejpam-6146	89	2	s′(x	s′(x	NOUN
ejpam-6146	89	3	)	)	PUNCT
ejpam-6146	89	4	and	and	CCONJ
ejpam-6146	89	5	c(x	c(x	NOUN
ejpam-6146	89	6	)	)	PUNCT
ejpam-6146	89	7	are	be	AUX
ejpam-6146	89	8	both	both	DET
ejpam-6146	89	9	solutions	solution	NOUN
ejpam-6146	89	10	of	of	ADP
ejpam-6146	89	11	eq	eq	PROPN
ejpam-6146	89	12	.	.	PUNCT
ejpam-6146	90	1	(	(	PUNCT
ejpam-6146	90	2	3	3	NUM
ejpam-6146	90	3	)	)	PUNCT
ejpam-6146	90	4	.	.	PUNCT
ejpam-6146	91	1	so	so	ADV
ejpam-6146	91	2	also	also	ADV
ejpam-6146	91	3	c′(x	c′(x	PROPN
ejpam-6146	91	4	)	)	PUNCT
ejpam-6146	91	5	and	and	CCONJ
ejpam-6146	91	6	s(x	s(x	NOUN
ejpam-6146	91	7	)	)	PUNCT
ejpam-6146	91	8	are	be	AUX
ejpam-6146	91	9	both	both	DET
ejpam-6146	91	10	solutions	solution	NOUN
ejpam-6146	91	11	of	of	ADP
ejpam-6146	91	12	eq	eq	PROPN
ejpam-6146	91	13	.	.	PUNCT
ejpam-6146	92	1	(	(	PUNCT
ejpam-6146	92	2	3	3	NUM
ejpam-6146	92	3	)	)	PUNCT
ejpam-6146	92	4	.	.	PUNCT
ejpam-6146	93	1	from	from	ADP
ejpam-6146	93	2	the	the	DET
ejpam-6146	93	3	initial	initial	ADJ
ejpam-6146	93	4	conditions	condition	NOUN
ejpam-6146	93	5	,	,	PUNCT
ejpam-6146	93	6	s(0	s(0	PROPN
ejpam-6146	93	7	)	)	PUNCT
ejpam-6146	93	8	=	=	SYM
ejpam-6146	93	9	0	0	NUM
ejpam-6146	93	10	,	,	PUNCT
ejpam-6146	93	11	s′(0	s′(0	NOUN
ejpam-6146	93	12	)	)	PUNCT
ejpam-6146	93	13	=	=	SYM
ejpam-6146	93	14	1	1	NUM
ejpam-6146	93	15	and	and	CCONJ
ejpam-6146	93	16	c(0	c(0	NOUN
ejpam-6146	93	17	)	)	PUNCT
ejpam-6146	93	18	=	=	SYM
ejpam-6146	93	19	1	1	NUM
ejpam-6146	93	20	,	,	PUNCT
ejpam-6146	93	21	c′(0	c′(0	NOUN
ejpam-6146	93	22	)	)	PUNCT
ejpam-6146	93	23	=	=	SYM
ejpam-6146	93	24	0	0	NUM
ejpam-6146	93	25	,	,	PUNCT
ejpam-6146	93	26	along	along	ADP
ejpam-6146	93	27	with	with	ADP
ejpam-6146	93	28	theorem	theorem	ADJ
ejpam-6146	93	29	(	(	PUNCT
ejpam-6146	93	30	1	1	NUM
ejpam-6146	93	31	)	)	PUNCT
ejpam-6146	93	32	,	,	PUNCT
ejpam-6146	93	33	we	we	PRON
ejpam-6146	93	34	conclude	conclude	VERB
ejpam-6146	93	35	s′(x	s′(x	VERB
ejpam-6146	93	36	)	)	PUNCT
ejpam-6146	93	37	=	=	SYM
ejpam-6146	93	38	c(x	c(x	NOUN
ejpam-6146	93	39	)	)	PUNCT
ejpam-6146	93	40	c′(x	c′(x	NOUN
ejpam-6146	93	41	)	)	PUNCT
ejpam-6146	93	42	=	=	SYM
ejpam-6146	93	43	−s(x	−s(x	NOUN
ejpam-6146	93	44	)	)	PUNCT
ejpam-6146	93	45	.	.	PUNCT
ejpam-6146	94	1	(	(	PUNCT
ejpam-6146	94	2	4	4	X
ejpam-6146	94	3	)	)	PUNCT
ejpam-6146	94	4	s.	s.	PROPN
ejpam-6146	94	5	a.	a.	PROPN
ejpam-6146	94	6	khan	khan	PROPN
ejpam-6146	94	7	,	,	PUNCT
ejpam-6146	94	8	m.	m.	NOUN
ejpam-6146	94	9	m.	m.	PROPN
ejpam-6146	94	10	kankarej	kankarej	PROPN
ejpam-6146	94	11	,	,	PUNCT
ejpam-6146	94	12	m.	m.	PROPN
ejpam-6146	94	13	n.	n.	PROPN
ejpam-6146	94	14	i.	i.	PROPN
ejpam-6146	94	15	khan	khan	PROPN
ejpam-6146	94	16	/	/	SYM
ejpam-6146	94	17	eur	eur	PROPN
ejpam-6146	94	18	.	.	PUNCT
ejpam-6146	95	1	j.	j.	PROPN
ejpam-6146	95	2	pure	pure	PROPN
ejpam-6146	95	3	appl	appl	PROPN
ejpam-6146	95	4	.	.	PROPN
ejpam-6146	95	5	math	math	PROPN
ejpam-6146	95	6	,	,	PUNCT
ejpam-6146	95	7	18	18	NUM
ejpam-6146	95	8	(	(	PUNCT
ejpam-6146	95	9	4	4	NUM
ejpam-6146	95	10	)	)	PUNCT
ejpam-6146	95	11	(	(	PUNCT
ejpam-6146	95	12	2025	2025	NUM
ejpam-6146	95	13	)	)	PUNCT
ejpam-6146	95	14	,	,	PUNCT
ejpam-6146	95	15	6146	6146	NUM
ejpam-6146	95	16	4	4	NUM
ejpam-6146	95	17	of	of	ADP
ejpam-6146	95	18	14	14	NUM
ejpam-6146	95	19	if	if	SCONJ
ejpam-6146	95	20	f(x	f(x	PROPN
ejpam-6146	95	21	)	)	PUNCT
ejpam-6146	95	22	is	be	AUX
ejpam-6146	95	23	a	a	DET
ejpam-6146	95	24	solution	solution	NOUN
ejpam-6146	95	25	of	of	ADP
ejpam-6146	95	26	y′′	y′′	PROPN
ejpam-6146	95	27	+	+	CCONJ
ejpam-6146	95	28	y	y	PROPN
ejpam-6146	95	29	=	=	SYM
ejpam-6146	95	30	0	0	NUM
ejpam-6146	95	31	,	,	PUNCT
ejpam-6146	95	32	then	then	ADV
ejpam-6146	95	33	f(−x	f(−x	PUNCT
ejpam-6146	95	34	)	)	PUNCT
ejpam-6146	95	35	is	be	AUX
ejpam-6146	95	36	also	also	ADV
ejpam-6146	95	37	a	a	DET
ejpam-6146	95	38	solution	solution	NOUN
ejpam-6146	95	39	.	.	PUNCT
ejpam-6146	96	1	from	from	ADP
ejpam-6146	96	2	this	this	PRON
ejpam-6146	96	3	,	,	PUNCT
ejpam-6146	96	4	we	we	PRON
ejpam-6146	96	5	conclude	conclude	VERB
ejpam-6146	96	6	that	that	PRON
ejpam-6146	96	7	s(−x	s(−x	X
ejpam-6146	96	8	)	)	PUNCT
ejpam-6146	96	9	=	=	SYM
ejpam-6146	96	10	−s(x	−s(x	NOUN
ejpam-6146	96	11	)	)	PUNCT
ejpam-6146	96	12	c(−x	c(−x	PUNCT
ejpam-6146	96	13	)	)	PUNCT
ejpam-6146	97	1	=	=	SYM
ejpam-6146	97	2	c(x	c(x	NOUN
ejpam-6146	97	3	)	)	PUNCT
ejpam-6146	97	4	.	.	PUNCT
ejpam-6146	98	1	(	(	PUNCT
ejpam-6146	98	2	5	5	NUM
ejpam-6146	98	3	)	)	PUNCT
ejpam-6146	98	4	thus	thus	ADV
ejpam-6146	98	5	,	,	PUNCT
ejpam-6146	98	6	we	we	PRON
ejpam-6146	98	7	have	have	AUX
ejpam-6146	98	8	established	establish	VERB
ejpam-6146	98	9	that	that	SCONJ
ejpam-6146	98	10	s(x	s(x	NOUN
ejpam-6146	98	11	)	)	PUNCT
ejpam-6146	98	12	is	be	AUX
ejpam-6146	98	13	an	an	DET
ejpam-6146	98	14	odd	odd	ADJ
ejpam-6146	98	15	function	function	NOUN
ejpam-6146	98	16	and	and	CCONJ
ejpam-6146	98	17	c(x	c(x	NOUN
ejpam-6146	98	18	)	)	PUNCT
ejpam-6146	98	19	is	be	AUX
ejpam-6146	98	20	an	an	DET
ejpam-6146	98	21	even	even	ADJ
ejpam-6146	98	22	function	function	NOUN
ejpam-6146	98	23	.	.	PUNCT
ejpam-6146	99	1	the	the	DET
ejpam-6146	99	2	taylor	taylor	PROPN
ejpam-6146	99	3	series	series	NOUN
ejpam-6146	99	4	or	or	CCONJ
ejpam-6146	99	5	expansion	expansion	NOUN
ejpam-6146	99	6	of	of	ADP
ejpam-6146	99	7	a	a	DET
ejpam-6146	99	8	real	real	ADJ
ejpam-6146	99	9	function	function	NOUN
ejpam-6146	99	10	f(x	f(x	PROPN
ejpam-6146	99	11	)	)	PUNCT
ejpam-6146	99	12	that	that	PRON
ejpam-6146	99	13	is	be	AUX
ejpam-6146	99	14	infinitely	infinitely	ADV
ejpam-6146	99	15	differentiable	differentiable	ADJ
ejpam-6146	99	16	about	about	ADP
ejpam-6146	99	17	a	a	DET
ejpam-6146	99	18	point	point	NOUN
ejpam-6146	99	19	x	x	PUNCT
ejpam-6146	99	20	=	=	NOUN
ejpam-6146	99	21	a	a	PRON
ejpam-6146	99	22	is	be	AUX
ejpam-6146	99	23	given	give	VERB
ejpam-6146	99	24	by	by	ADP
ejpam-6146	99	25	f(x	f(x	PROPN
ejpam-6146	99	26	)	)	PUNCT
ejpam-6146	99	27	=	=	SYM
ejpam-6146	99	28	f(a	f(a	NOUN
ejpam-6146	99	29	)	)	PUNCT
ejpam-6146	100	1	+	+	CCONJ
ejpam-6146	100	2	1	1	NUM
ejpam-6146	100	3	1	1	NUM
ejpam-6146	100	4	!	!	PUNCT
ejpam-6146	100	5	f	f	PROPN
ejpam-6146	100	6	(	(	PUNCT
ejpam-6146	100	7	1)(a)(x−	1)(a)(x−	NUM
ejpam-6146	100	8	a	a	NOUN
ejpam-6146	100	9	)	)	PUNCT
ejpam-6146	101	1	+	+	CCONJ
ejpam-6146	101	2	1	1	NUM
ejpam-6146	101	3	2	2	NUM
ejpam-6146	101	4	!	!	PUNCT
ejpam-6146	101	5	f	f	X
ejpam-6146	101	6	(	(	PUNCT
ejpam-6146	101	7	2)(a)(x−	2)(a)(x−	NUM
ejpam-6146	101	8	a)2	a)2	NOUN
ejpam-6146	101	9	+	+	CCONJ
ejpam-6146	101	10	1	1	NUM
ejpam-6146	101	11	3	3	NUM
ejpam-6146	101	12	!	!	PUNCT
ejpam-6146	102	1	f	f	PROPN
ejpam-6146	102	2	(	(	PUNCT
ejpam-6146	102	3	3)(a)(x−	3)(a)(x−	NUM
ejpam-6146	102	4	a)3	a)3	ADJ
ejpam-6146	102	5	+	+	NUM
ejpam-6146	102	6	·	·	PUNCT
ejpam-6146	102	7	·	·	PUNCT
ejpam-6146	102	8	·	·	PUNCT
ejpam-6146	102	9	+	+	CCONJ
ejpam-6146	102	10	1	1	NUM
ejpam-6146	102	11	n	n	X
ejpam-6146	102	12	!	!	PUNCT
ejpam-6146	103	1	f	f	PROPN
ejpam-6146	103	2	(	(	PUNCT
ejpam-6146	103	3	n)(a)(x−	n)(a)(x−	NOUN
ejpam-6146	103	4	a)n	a)n	NOUN
ejpam-6146	103	5	+	+	CCONJ
ejpam-6146	103	6	·	·	PUNCT
ejpam-6146	103	7	·	·	PUNCT
ejpam-6146	103	8	·	·	PUNCT
ejpam-6146	104	1	=	=	PUNCT
ejpam-6146	104	2	∞∑	∞∑	NUM
ejpam-6146	104	3	k=0	k=0	PROPN
ejpam-6146	104	4	1	1	NUM
ejpam-6146	104	5	k	k	NOUN
ejpam-6146	104	6	!	!	PUNCT
ejpam-6146	105	1	f	f	PROPN
ejpam-6146	105	2	(	(	PUNCT
ejpam-6146	105	3	k)(a)(x−	k)(a)(x−	PROPN
ejpam-6146	105	4	a)k	a)k	PROPN
ejpam-6146	105	5	,	,	PUNCT
ejpam-6146	105	6	(	(	PUNCT
ejpam-6146	105	7	6	6	NUM
ejpam-6146	105	8	)	)	PUNCT
ejpam-6146	106	1	where	where	SCONJ
ejpam-6146	106	2	f	f	PROPN
ejpam-6146	106	3	(	(	PUNCT
ejpam-6146	106	4	n)(a	n)(a	NUM
ejpam-6146	106	5	)	)	PUNCT
ejpam-6146	106	6	=	=	PRON
ejpam-6146	106	7	dn	dn	PROPN
ejpam-6146	106	8	dxn	dxn	VERB
ejpam-6146	106	9	f(x)|x	f(x)|x	PROPN
ejpam-6146	106	10	=	=	NOUN
ejpam-6146	106	11	a.	a.	NOUN
ejpam-6146	106	12	the	the	DET
ejpam-6146	106	13	name	name	NOUN
ejpam-6146	106	14	is	be	AUX
ejpam-6146	106	15	in	in	ADP
ejpam-6146	106	16	honour	honour	NOUN
ejpam-6146	106	17	of	of	ADP
ejpam-6146	106	18	brook	brook	PROPN
ejpam-6146	106	19	taylor	taylor	PROPN
ejpam-6146	106	20	(	(	PUNCT
ejpam-6146	106	21	1685	1685	NUM
ejpam-6146	106	22	-	-	SYM
ejpam-6146	106	23	1731	1731	NUM
ejpam-6146	106	24	)	)	PUNCT
ejpam-6146	106	25	.	.	PUNCT
ejpam-6146	107	1	colin	colin	PROPN
ejpam-6146	107	2	maclaurin	maclaurin	PROPN
ejpam-6146	107	3	(	(	PUNCT
ejpam-6146	107	4	1698	1698	NUM
ejpam-6146	107	5	-	-	SYM
ejpam-6146	107	6	1746	1746	NUM
ejpam-6146	107	7	)	)	PUNCT
ejpam-6146	107	8	also	also	ADV
ejpam-6146	107	9	made	make	VERB
ejpam-6146	107	10	significant	significant	ADJ
ejpam-6146	107	11	contributions	contribution	NOUN
ejpam-6146	107	12	to	to	ADP
ejpam-6146	107	13	the	the	DET
ejpam-6146	107	14	series	series	NOUN
ejpam-6146	107	15	expansion	expansion	NOUN
ejpam-6146	107	16	of	of	ADP
ejpam-6146	107	17	functions	function	NOUN
ejpam-6146	107	18	.	.	PUNCT
ejpam-6146	108	1	in	in	ADP
ejpam-6146	108	2	maclaurin	maclaurin	PROPN
ejpam-6146	108	3	series	series	NOUN
ejpam-6146	108	4	,	,	PUNCT
ejpam-6146	108	5	the	the	DET
ejpam-6146	108	6	derivatives	derivative	NOUN
ejpam-6146	108	7	are	be	AUX
ejpam-6146	108	8	evaluated	evaluate	VERB
ejpam-6146	108	9	at	at	ADP
ejpam-6146	108	10	zero	zero	NUM
ejpam-6146	108	11	(	(	PUNCT
ejpam-6146	108	12	a	a	DET
ejpam-6146	108	13	=	=	SYM
ejpam-6146	108	14	0	0	NUM
ejpam-6146	108	15	in	in	ADP
ejpam-6146	108	16	eq	eq	ADP
ejpam-6146	108	17	.	.	PUNCT
ejpam-6146	109	1	(	(	PUNCT
ejpam-6146	109	2	6	6	NUM
ejpam-6146	109	3	)	)	PUNCT
ejpam-6146	109	4	)	)	PUNCT
ejpam-6146	110	1	and	and	CCONJ
ejpam-6146	110	2	hence	hence	ADV
ejpam-6146	110	3	,	,	PUNCT
ejpam-6146	110	4	it	it	PRON
ejpam-6146	110	5	is	be	AUX
ejpam-6146	110	6	a	a	DET
ejpam-6146	110	7	special	special	ADJ
ejpam-6146	110	8	case	case	NOUN
ejpam-6146	110	9	of	of	ADP
ejpam-6146	110	10	the	the	DET
ejpam-6146	110	11	taylor	taylor	PROPN
ejpam-6146	110	12	series	series	PROPN
ejpam-6146	110	13	.	.	PUNCT
ejpam-6146	111	1	from	from	ADP
ejpam-6146	111	2	eq	eq	ADP
ejpam-6146	111	3	.	.	PUNCT
ejpam-6146	112	1	(	(	PUNCT
ejpam-6146	112	2	4	4	NUM
ejpam-6146	112	3	)	)	PUNCT
ejpam-6146	112	4	,	,	PUNCT
ejpam-6146	112	5	it	it	PRON
ejpam-6146	112	6	is	be	AUX
ejpam-6146	112	7	seen	see	VERB
ejpam-6146	112	8	that	that	SCONJ
ejpam-6146	112	9	the	the	DET
ejpam-6146	112	10	higher	high	ADJ
ejpam-6146	112	11	derivatives	derivative	NOUN
ejpam-6146	112	12	of	of	ADP
ejpam-6146	112	13	s(x	s(x	PROPN
ejpam-6146	112	14	)	)	PUNCT
ejpam-6146	112	15	and	and	CCONJ
ejpam-6146	112	16	c(x	c(x	NOUN
ejpam-6146	112	17	)	)	PUNCT
ejpam-6146	112	18	have	have	VERB
ejpam-6146	112	19	a	a	DET
ejpam-6146	112	20	simple	simple	ADJ
ejpam-6146	112	21	pattern	pattern	NOUN
ejpam-6146	112	22	.	.	PUNCT
ejpam-6146	113	1	we	we	PRON
ejpam-6146	113	2	note	note	VERB
ejpam-6146	113	3	,	,	PUNCT
ejpam-6146	113	4	that	that	SCONJ
ejpam-6146	113	5	s(2n)(x	s(2n)(x	NOUN
ejpam-6146	113	6	)	)	PUNCT
ejpam-6146	113	7	=	=	PRON
ejpam-6146	113	8	(	(	PUNCT
ejpam-6146	113	9	−1)ns(x	−1)ns(x	X
ejpam-6146	113	10	)	)	PUNCT
ejpam-6146	113	11	and	and	CCONJ
ejpam-6146	113	12	s(2n+1)(x	s(2n+1)(x	PROPN
ejpam-6146	113	13	)	)	PUNCT
ejpam-6146	113	14	=	=	PUNCT
ejpam-6146	113	15	(	(	PUNCT
ejpam-6146	113	16	−1)nc(x	−1)nc(x	NOUN
ejpam-6146	113	17	)	)	PUNCT
ejpam-6146	113	18	,	,	PUNCT
ejpam-6146	113	19	and	and	CCONJ
ejpam-6146	113	20	c(2n)(x	c(2n)(x	PROPN
ejpam-6146	113	21	)	)	PUNCT
ejpam-6146	114	1	=	=	PRON
ejpam-6146	114	2	(	(	PUNCT
ejpam-6146	114	3	−1)nc(x	−1)nc(x	NOUN
ejpam-6146	114	4	)	)	PUNCT
ejpam-6146	114	5	and	and	CCONJ
ejpam-6146	114	6	c(2n+1)(x	c(2n+1)(x	PROPN
ejpam-6146	114	7	)	)	PUNCT
ejpam-6146	114	8	=	=	SYM
ejpam-6146	114	9	−(−1)ns(x	−(−1)ns(x	PROPN
ejpam-6146	114	10	)	)	PUNCT
ejpam-6146	114	11	,	,	PUNCT
ejpam-6146	114	12	for	for	ADP
ejpam-6146	114	13	n	n	NOUN
ejpam-6146	114	14	=	=	SYM
ejpam-6146	114	15	0	0	NUM
ejpam-6146	114	16	,	,	PUNCT
ejpam-6146	114	17	1	1	NUM
ejpam-6146	114	18	,	,	PUNCT
ejpam-6146	114	19	2	2	NUM
ejpam-6146	114	20	,	,	PUNCT
ejpam-6146	114	21	3	3	NUM
ejpam-6146	114	22	,	,	PUNCT
ejpam-6146	114	23	.	.	PUNCT
ejpam-6146	114	24	.	.	PUNCT
ejpam-6146	114	25	.	.	PUNCT
ejpam-6146	114	26	.	.	PUNCT
ejpam-6146	115	1	using	use	VERB
ejpam-6146	115	2	the	the	DET
ejpam-6146	115	3	initial	initial	ADJ
ejpam-6146	115	4	conditions	condition	NOUN
ejpam-6146	115	5	,	,	PUNCT
ejpam-6146	115	6	we	we	PRON
ejpam-6146	115	7	note	note	VERB
ejpam-6146	115	8	that	that	SCONJ
ejpam-6146	115	9	s(2n)(0	s(2n)(0	NOUN
ejpam-6146	115	10	)	)	PUNCT
ejpam-6146	115	11	=	=	SYM
ejpam-6146	115	12	0	0	NUM
ejpam-6146	115	13	and	and	CCONJ
ejpam-6146	115	14	s(2n+1)(0	s(2n+1)(0	ADJ
ejpam-6146	115	15	)	)	PUNCT
ejpam-6146	116	1	=	=	SYM
ejpam-6146	116	2	(	(	PUNCT
ejpam-6146	116	3	−1)n	−1)n	NOUN
ejpam-6146	116	4	,	,	PUNCT
ejpam-6146	116	5	c(2n)(0	c(2n)(0	NUM
ejpam-6146	116	6	)	)	PUNCT
ejpam-6146	116	7	=	=	SYM
ejpam-6146	116	8	(	(	PUNCT
ejpam-6146	117	1	−1)n	−1)n	X
ejpam-6146	117	2	and	and	CCONJ
ejpam-6146	117	3	c(2n+1)(0	c(2n+1)(0	NUM
ejpam-6146	117	4	)	)	PUNCT
ejpam-6146	117	5	=	=	SYM
ejpam-6146	118	1	0	0	X
ejpam-6146	118	2	.	.	PUNCT
ejpam-6146	119	1	the	the	DET
ejpam-6146	119	2	taylor	taylor	PROPN
ejpam-6146	119	3	series	series	PROPN
ejpam-6146	119	4	for	for	ADP
ejpam-6146	119	5	s(x	s(x	PROPN
ejpam-6146	119	6	)	)	PUNCT
ejpam-6146	119	7	and	and	CCONJ
ejpam-6146	119	8	c(x	c(x	NOUN
ejpam-6146	119	9	)	)	PUNCT
ejpam-6146	119	10	are	be	AUX
ejpam-6146	119	11	s(x	s(x	VERB
ejpam-6146	119	12	)	)	PUNCT
ejpam-6146	120	1	=	=	SYM
ejpam-6146	120	2	x−	x−	PROPN
ejpam-6146	120	3	1	1	NUM
ejpam-6146	120	4	3	3	X
ejpam-6146	120	5	!	!	X
ejpam-6146	121	1	x3	x3	ADJ
ejpam-6146	122	1	+	+	CCONJ
ejpam-6146	122	2	1	1	NUM
ejpam-6146	122	3	5	5	NUM
ejpam-6146	122	4	!	!	PUNCT
ejpam-6146	122	5	x5	x5	NOUN
ejpam-6146	122	6	−	−	PROPN
ejpam-6146	122	7	·	·	PUNCT
ejpam-6146	122	8	·	·	PUNCT
ejpam-6146	122	9	·	·	PUNCT
ejpam-6146	123	1	=	=	PUNCT
ejpam-6146	124	1	∞∑	∞∑	NUM
ejpam-6146	124	2	n=0	n=0	NUM
ejpam-6146	124	3	(	(	PUNCT
ejpam-6146	124	4	−1)n	−1)n	X
ejpam-6146	124	5	(	(	PUNCT
ejpam-6146	124	6	2n+	2n+	NUM
ejpam-6146	124	7	1	1	NUM
ejpam-6146	124	8	)	)	PUNCT
ejpam-6146	124	9	!	!	PUNCT
ejpam-6146	125	1	x2n+1	x2n+1	PROPN
ejpam-6146	125	2	,	,	PUNCT
ejpam-6146	125	3	∀x	∀x	X
ejpam-6146	125	4	,	,	PUNCT
ejpam-6146	125	5	(	(	PUNCT
ejpam-6146	125	6	7	7	NUM
ejpam-6146	125	7	)	)	PUNCT
ejpam-6146	125	8	c(x	c(x	NOUN
ejpam-6146	125	9	)	)	PUNCT
ejpam-6146	125	10	=	=	SYM
ejpam-6146	126	1	1−	1−	NUM
ejpam-6146	126	2	1	1	NUM
ejpam-6146	126	3	2	2	NUM
ejpam-6146	126	4	!	!	PUNCT
ejpam-6146	126	5	x2	x2	PROPN
ejpam-6146	127	1	+	+	CCONJ
ejpam-6146	127	2	1	1	NUM
ejpam-6146	127	3	4	4	NUM
ejpam-6146	127	4	!	!	PUNCT
ejpam-6146	127	5	x4	x4	PROPN
ejpam-6146	127	6	−	−	PROPN
ejpam-6146	127	7	·	·	PUNCT
ejpam-6146	127	8	·	·	PUNCT
ejpam-6146	127	9	·	·	PUNCT
ejpam-6146	128	1	=	=	PUNCT
ejpam-6146	128	2	∞∑	∞∑	NUM
ejpam-6146	128	3	n=0	n=0	NUM
ejpam-6146	128	4	(	(	PUNCT
ejpam-6146	128	5	−1)n	−1)n	X
ejpam-6146	128	6	(	(	PUNCT
ejpam-6146	128	7	2n	2n	NUM
ejpam-6146	128	8	)	)	PUNCT
ejpam-6146	128	9	!	!	PUNCT
ejpam-6146	129	1	x2n	x2n	PROPN
ejpam-6146	129	2	,	,	PUNCT
ejpam-6146	129	3	∀x	∀x	X
ejpam-6146	129	4	.	.	PUNCT
ejpam-6146	130	1	(	(	PUNCT
ejpam-6146	130	2	8)	8)	NUM
ejpam-6146	130	3	the	the	DET
ejpam-6146	130	4	odd	odd	ADJ
ejpam-6146	130	5	and	and	CCONJ
ejpam-6146	130	6	even	even	ADV
ejpam-6146	130	7	properties	property	NOUN
ejpam-6146	130	8	of	of	ADP
ejpam-6146	130	9	s(x	s(x	PROPN
ejpam-6146	130	10	)	)	PUNCT
ejpam-6146	130	11	and	and	CCONJ
ejpam-6146	130	12	c(x	c(x	NOUN
ejpam-6146	130	13	)	)	PUNCT
ejpam-6146	130	14	respectively	respectively	ADV
ejpam-6146	130	15	can	can	AUX
ejpam-6146	130	16	also	also	ADV
ejpam-6146	130	17	be	be	AUX
ejpam-6146	130	18	seen	see	VERB
ejpam-6146	130	19	from	from	ADP
ejpam-6146	130	20	their	their	PRON
ejpam-6146	130	21	taylor	taylor	PROPN
ejpam-6146	130	22	series	series	NOUN
ejpam-6146	130	23	.	.	PUNCT
ejpam-6146	131	1	the	the	DET
ejpam-6146	131	2	series	series	PROPN
ejpam-6146	131	3	expansion	expansion	NOUN
ejpam-6146	131	4	of	of	ADP
ejpam-6146	131	5	s(x	s(x	PROPN
ejpam-6146	131	6	)	)	PUNCT
ejpam-6146	131	7	and	and	CCONJ
ejpam-6146	131	8	c(x	c(x	NOUN
ejpam-6146	131	9	)	)	PUNCT
ejpam-6146	131	10	can	can	AUX
ejpam-6146	131	11	also	also	ADV
ejpam-6146	131	12	be	be	AUX
ejpam-6146	131	13	derived	derive	VERB
ejpam-6146	131	14	from	from	ADP
ejpam-6146	131	15	a	a	DET
ejpam-6146	131	16	matrix	matrix	NOUN
ejpam-6146	131	17	form	form	NOUN
ejpam-6146	131	18	of	of	ADP
ejpam-6146	131	19	the	the	DET
ejpam-6146	131	20	differential	differential	ADJ
ejpam-6146	131	21	equation	equation	NOUN
ejpam-6146	131	22	.	.	PUNCT
ejpam-6146	132	1	equation	equation	NOUN
ejpam-6146	132	2	(	(	PUNCT
ejpam-6146	132	3	4	4	NUM
ejpam-6146	132	4	)	)	PUNCT
ejpam-6146	132	5	,	,	PUNCT
ejpam-6146	132	6	s′(x	s′(x	PROPN
ejpam-6146	132	7	)	)	PUNCT
ejpam-6146	132	8	=	=	SYM
ejpam-6146	132	9	c(x	c(x	NOUN
ejpam-6146	132	10	)	)	PUNCT
ejpam-6146	132	11	and	and	CCONJ
ejpam-6146	132	12	c′(x	c′(x	NOUN
ejpam-6146	132	13	)	)	PUNCT
ejpam-6146	132	14	=	=	SYM
ejpam-6146	132	15	−s(x	−s(x	NOUN
ejpam-6146	132	16	)	)	PUNCT
ejpam-6146	132	17	has	have	VERB
ejpam-6146	132	18	the	the	DET
ejpam-6146	132	19	following	follow	VERB
ejpam-6146	132	20	matrix	matrix	NOUN
ejpam-6146	132	21	form	form	NOUN
ejpam-6146	132	22	d	d	X
ejpam-6146	132	23	dx	dx	PROPN
ejpam-6146	132	24	[	[	PUNCT
ejpam-6146	132	25	s(x	s(x	PROPN
ejpam-6146	132	26	)	)	PUNCT
ejpam-6146	132	27	c(x	c(x	NOUN
ejpam-6146	132	28	)	)	PUNCT
ejpam-6146	132	29	]	]	PUNCT
ejpam-6146	133	1	=	=	PUNCT
ejpam-6146	133	2	m	m	VERB
ejpam-6146	133	3	[	[	PUNCT
ejpam-6146	133	4	s(x	s(x	ADJ
ejpam-6146	133	5	)	)	PUNCT
ejpam-6146	133	6	c(x	c(x	NOUN
ejpam-6146	133	7	)	)	PUNCT
ejpam-6146	133	8	]	]	PUNCT
ejpam-6146	134	1	s.	s.	PROPN
ejpam-6146	134	2	a.	a.	PROPN
ejpam-6146	134	3	khan	khan	PROPN
ejpam-6146	134	4	,	,	PUNCT
ejpam-6146	134	5	m.	m.	NOUN
ejpam-6146	134	6	m.	m.	PROPN
ejpam-6146	134	7	kankarej	kankarej	PROPN
ejpam-6146	134	8	,	,	PUNCT
ejpam-6146	134	9	m.	m.	PROPN
ejpam-6146	134	10	n.	n.	PROPN
ejpam-6146	134	11	i.	i.	PROPN
ejpam-6146	134	12	khan	khan	PROPN
ejpam-6146	134	13	/	/	SYM
ejpam-6146	134	14	eur	eur	PROPN
ejpam-6146	134	15	.	.	PUNCT
ejpam-6146	135	1	j.	j.	PROPN
ejpam-6146	135	2	pure	pure	PROPN
ejpam-6146	135	3	appl	appl	PROPN
ejpam-6146	135	4	.	.	PROPN
ejpam-6146	135	5	math	math	PROPN
ejpam-6146	135	6	,	,	PUNCT
ejpam-6146	135	7	18	18	NUM
ejpam-6146	135	8	(	(	PUNCT
ejpam-6146	135	9	4	4	NUM
ejpam-6146	135	10	)	)	PUNCT
ejpam-6146	135	11	(	(	PUNCT
ejpam-6146	135	12	2025	2025	NUM
ejpam-6146	135	13	)	)	PUNCT
ejpam-6146	135	14	,	,	PUNCT
ejpam-6146	135	15	6146	6146	NUM
ejpam-6146	135	16	5	5	NUM
ejpam-6146	135	17	of	of	ADP
ejpam-6146	135	18	14	14	NUM
ejpam-6146	135	19	m	m	NOUN
ejpam-6146	135	20	=	=	PUNCT
ejpam-6146	136	1	[	[	PUNCT
ejpam-6146	136	2	0	0	NUM
ejpam-6146	136	3	1	1	NUM
ejpam-6146	136	4	−1	−1	NOUN
ejpam-6146	136	5	0	0	NUM
ejpam-6146	136	6	]	]	PUNCT
ejpam-6146	136	7	.	.	PUNCT
ejpam-6146	137	1	(	(	PUNCT
ejpam-6146	137	2	9	9	X
ejpam-6146	137	3	)	)	PUNCT
ejpam-6146	137	4	note	note	NOUN
ejpam-6146	137	5	,	,	PUNCT
ejpam-6146	137	6	that	that	SCONJ
ejpam-6146	137	7	m2	m2	PROPN
ejpam-6146	137	8	=	=	PROPN
ejpam-6146	137	9	−i	−i	PROPN
ejpam-6146	137	10	,	,	PUNCT
ejpam-6146	137	11	where	where	SCONJ
ejpam-6146	137	12	i	i	PRON
ejpam-6146	137	13	=	=	X
ejpam-6146	137	14	[	[	PUNCT
ejpam-6146	137	15	1	1	NUM
ejpam-6146	137	16	0	0	NUM
ejpam-6146	137	17	0	0	NUM
ejpam-6146	137	18	1	1	NUM
ejpam-6146	137	19	]	]	PUNCT
ejpam-6146	137	20	is	be	AUX
ejpam-6146	137	21	the	the	DET
ejpam-6146	137	22	2	2	NUM
ejpam-6146	137	23	×	×	NOUN
ejpam-6146	137	24	2	2	NUM
ejpam-6146	137	25	unit	unit	NOUN
ejpam-6146	137	26	matrix	matrix	NOUN
ejpam-6146	137	27	.	.	PUNCT
ejpam-6146	138	1	the	the	DET
ejpam-6146	138	2	formal	formal	ADJ
ejpam-6146	138	3	solution	solution	NOUN
ejpam-6146	138	4	incorporating	incorporate	VERB
ejpam-6146	138	5	the	the	DET
ejpam-6146	138	6	initial	initial	ADJ
ejpam-6146	138	7	conditions	condition	NOUN
ejpam-6146	138	8	is	be	AUX
ejpam-6146	138	9	[	[	PUNCT
ejpam-6146	138	10	s(x	s(x	NOUN
ejpam-6146	138	11	)	)	PUNCT
ejpam-6146	138	12	c(x	c(x	NOUN
ejpam-6146	138	13	)	)	PUNCT
ejpam-6146	138	14	]	]	PUNCT
ejpam-6146	139	1	=	=	PUNCT
ejpam-6146	139	2	emx	emx	PROPN
ejpam-6146	139	3	[	[	PUNCT
ejpam-6146	139	4	s(0	s(0	PROPN
ejpam-6146	139	5	)	)	PUNCT
ejpam-6146	139	6	c(0	c(0	PROPN
ejpam-6146	139	7	)	)	PUNCT
ejpam-6146	139	8	]	]	PUNCT
ejpam-6146	139	9	.	.	PUNCT
ejpam-6146	140	1	(	(	PUNCT
ejpam-6146	140	2	10	10	NUM
ejpam-6146	140	3	)	)	PUNCT
ejpam-6146	140	4	on	on	ADP
ejpam-6146	140	5	expanding	expand	VERB
ejpam-6146	140	6	the	the	DET
ejpam-6146	140	7	exponential	exponential	NOUN
ejpam-6146	140	8	[	[	PUNCT
ejpam-6146	140	9	s(x	s(x	NOUN
ejpam-6146	140	10	)	)	PUNCT
ejpam-6146	140	11	c(x	c(x	NOUN
ejpam-6146	140	12	)	)	PUNCT
ejpam-6146	140	13	]	]	PUNCT
ejpam-6146	141	1	=	=	PUNCT
ejpam-6146	141	2	{	{	PUNCT
ejpam-6146	141	3	i	i	PRON
ejpam-6146	141	4	+	+	PROPN
ejpam-6146	141	5	mx+	mx+	NOUN
ejpam-6146	141	6	1	1	NUM
ejpam-6146	141	7	2	2	NUM
ejpam-6146	141	8	!	!	PUNCT
ejpam-6146	141	9	m2x2	m2x2	NOUN
ejpam-6146	142	1	+	+	NOUN
ejpam-6146	142	2	1	1	NUM
ejpam-6146	142	3	3	3	NUM
ejpam-6146	142	4	!	!	PUNCT
ejpam-6146	143	1	m3x3	m3x3	X
ejpam-6146	143	2	+	+	X
ejpam-6146	143	3	·	·	PUNCT
ejpam-6146	143	4	·	·	PUNCT
ejpam-6146	143	5	·	·	PUNCT
ejpam-6146	143	6	}	}	PUNCT
ejpam-6146	144	1	=	=	SYM
ejpam-6146	144	2	{	{	PUNCT
ejpam-6146	144	3	(	(	PUNCT
ejpam-6146	144	4	x−	x−	PROPN
ejpam-6146	144	5	1	1	NUM
ejpam-6146	144	6	3	3	NUM
ejpam-6146	144	7	!	!	X
ejpam-6146	144	8	x3	x3	ADJ
ejpam-6146	145	1	+	+	CCONJ
ejpam-6146	145	2	1	1	NUM
ejpam-6146	145	3	5	5	NUM
ejpam-6146	145	4	!	!	PUNCT
ejpam-6146	145	5	x5	x5	NOUN
ejpam-6146	145	6	−	−	PROPN
ejpam-6146	145	7	·	·	PUNCT
ejpam-6146	145	8	·	·	PUNCT
ejpam-6146	145	9	·	·	PUNCT
ejpam-6146	145	10	)	)	PUNCT
ejpam-6146	146	1	m	m	VERB
ejpam-6146	146	2	+	+	PUNCT
ejpam-6146	146	3	(	(	PUNCT
ejpam-6146	146	4	1−	1−	NUM
ejpam-6146	146	5	1	1	NUM
ejpam-6146	146	6	2	2	NUM
ejpam-6146	146	7	!	!	PUNCT
ejpam-6146	146	8	x2	x2	PROPN
ejpam-6146	147	1	+	+	CCONJ
ejpam-6146	147	2	1	1	NUM
ejpam-6146	147	3	4	4	NUM
ejpam-6146	147	4	!	!	PUNCT
ejpam-6146	147	5	x4	x4	PROPN
ejpam-6146	148	1	−	−	PROPN
ejpam-6146	148	2	·	·	PUNCT
ejpam-6146	148	3	·	·	PUNCT
ejpam-6146	148	4	·	·	PUNCT
ejpam-6146	148	5	)	)	PUNCT
ejpam-6146	148	6	i	i	PRON
ejpam-6146	148	7	}	}	PUNCT
ejpam-6146	148	8	[	[	PUNCT
ejpam-6146	148	9	0	0	NUM
ejpam-6146	148	10	1	1	NUM
ejpam-6146	148	11	]	]	PUNCT
ejpam-6146	148	12	=	=	SYM
ejpam-6146	148	13	(	(	PUNCT
ejpam-6146	148	14	x−	x−	PROPN
ejpam-6146	148	15	1	1	NUM
ejpam-6146	148	16	3	3	NUM
ejpam-6146	148	17	!	!	X
ejpam-6146	149	1	x3	x3	ADJ
ejpam-6146	150	1	+	+	CCONJ
ejpam-6146	150	2	1	1	NUM
ejpam-6146	150	3	5	5	NUM
ejpam-6146	150	4	!	!	PUNCT
ejpam-6146	150	5	x5	x5	NOUN
ejpam-6146	150	6	−	−	PROPN
ejpam-6146	150	7	·	·	PUNCT
ejpam-6146	150	8	·	·	PUNCT
ejpam-6146	150	9	·	·	PUNCT
ejpam-6146	150	10	)	)	PUNCT
ejpam-6146	151	1	[	[	PUNCT
ejpam-6146	151	2	1	1	NUM
ejpam-6146	151	3	0	0	NUM
ejpam-6146	151	4	]	]	PUNCT
ejpam-6146	152	1	+	+	CCONJ
ejpam-6146	152	2	(	(	PUNCT
ejpam-6146	152	3	1−	1−	NUM
ejpam-6146	152	4	1	1	NUM
ejpam-6146	152	5	2	2	NUM
ejpam-6146	152	6	!	!	PUNCT
ejpam-6146	152	7	x2	x2	PROPN
ejpam-6146	153	1	+	+	CCONJ
ejpam-6146	153	2	1	1	NUM
ejpam-6146	153	3	4	4	NUM
ejpam-6146	153	4	!	!	PUNCT
ejpam-6146	153	5	x4	x4	PROPN
ejpam-6146	154	1	−	−	PROPN
ejpam-6146	154	2	·	·	PUNCT
ejpam-6146	154	3	·	·	PUNCT
ejpam-6146	154	4	·	·	PUNCT
ejpam-6146	154	5	)	)	PUNCT
ejpam-6146	155	1	[	[	PUNCT
ejpam-6146	155	2	0	0	NUM
ejpam-6146	155	3	1	1	NUM
ejpam-6146	155	4	]	]	PUNCT
ejpam-6146	155	5	.	.	PUNCT
ejpam-6146	156	1	(	(	PUNCT
ejpam-6146	156	2	11	11	NUM
ejpam-6146	156	3	)	)	PUNCT
ejpam-6146	156	4	this	this	PRON
ejpam-6146	156	5	leads	lead	VERB
ejpam-6146	156	6	to	to	ADP
ejpam-6146	156	7	the	the	DET
ejpam-6146	156	8	series	series	PROPN
ejpam-6146	156	9	expansion	expansion	NOUN
ejpam-6146	156	10	s(x	s(x	PROPN
ejpam-6146	156	11	)	)	PUNCT
ejpam-6146	157	1	=	=	SYM
ejpam-6146	157	2	x−	x−	PROPN
ejpam-6146	157	3	1	1	NUM
ejpam-6146	157	4	3	3	X
ejpam-6146	157	5	!	!	X
ejpam-6146	158	1	x3	x3	ADJ
ejpam-6146	159	1	+	+	CCONJ
ejpam-6146	159	2	1	1	NUM
ejpam-6146	159	3	5	5	NUM
ejpam-6146	159	4	!	!	PUNCT
ejpam-6146	159	5	x5	x5	NOUN
ejpam-6146	159	6	−	−	PROPN
ejpam-6146	159	7	·	·	PUNCT
ejpam-6146	159	8	·	·	PUNCT
ejpam-6146	159	9	·	·	PUNCT
ejpam-6146	159	10	c(x	c(x	NOUN
ejpam-6146	159	11	)	)	PUNCT
ejpam-6146	159	12	=	=	SYM
ejpam-6146	160	1	1−	1−	NUM
ejpam-6146	160	2	1	1	NUM
ejpam-6146	160	3	2	2	NUM
ejpam-6146	160	4	!	!	PUNCT
ejpam-6146	160	5	x2	x2	PROPN
ejpam-6146	161	1	+	+	CCONJ
ejpam-6146	161	2	1	1	NUM
ejpam-6146	161	3	4	4	NUM
ejpam-6146	161	4	!	!	PUNCT
ejpam-6146	161	5	x4	x4	PROPN
ejpam-6146	162	1	−	−	PROPN
ejpam-6146	162	2	·	·	PUNCT
ejpam-6146	162	3	·	·	PUNCT
ejpam-6146	162	4	·	·	PUNCT
ejpam-6146	162	5	.	.	PUNCT
ejpam-6146	163	1	(	(	PUNCT
ejpam-6146	163	2	12	12	NUM
ejpam-6146	163	3	)	)	PUNCT
ejpam-6146	163	4	to	to	PART
ejpam-6146	163	5	establish	establish	VERB
ejpam-6146	163	6	more	more	ADJ
ejpam-6146	163	7	connections	connection	NOUN
ejpam-6146	163	8	between	between	ADP
ejpam-6146	163	9	s(x	s(x	PROPN
ejpam-6146	163	10	)	)	PUNCT
ejpam-6146	163	11	and	and	CCONJ
ejpam-6146	163	12	c(x	c(x	NOUN
ejpam-6146	163	13	)	)	PUNCT
ejpam-6146	163	14	,	,	PUNCT
ejpam-6146	163	15	we	we	PRON
ejpam-6146	163	16	multiply	multiply	VERB
ejpam-6146	163	17	eq	eq	ADP
ejpam-6146	163	18	.	.	PUNCT
ejpam-6146	164	1	(	(	PUNCT
ejpam-6146	164	2	3	3	X
ejpam-6146	164	3	)	)	PUNCT
ejpam-6146	164	4	by	by	ADP
ejpam-6146	164	5	y′.	y′.	PROPN
ejpam-6146	164	6	then	then	ADV
ejpam-6146	164	7	,	,	PUNCT
ejpam-6146	164	8	we	we	PRON
ejpam-6146	164	9	obtain	obtain	VERB
ejpam-6146	164	10	y′y′′	y′y′′	NOUN
ejpam-6146	165	1	+	+	CCONJ
ejpam-6146	165	2	yy′	yy′	X
ejpam-6146	165	3	=	=	SYM
ejpam-6146	165	4	0	0	X
ejpam-6146	165	5	.	.	X
ejpam-6146	166	1	rewriting	rewrite	VERB
ejpam-6146	166	2	it	it	PRON
ejpam-6146	166	3	,	,	PUNCT
ejpam-6146	166	4	we	we	PRON
ejpam-6146	166	5	have	have	VERB
ejpam-6146	166	6	1	1	NUM
ejpam-6146	166	7	2	2	NUM
ejpam-6146	166	8	[	[	PUNCT
ejpam-6146	166	9	(	(	PUNCT
ejpam-6146	166	10	y′)2	y′)2	PROPN
ejpam-6146	166	11	+	+	PROPN
ejpam-6146	166	12	(	(	PUNCT
ejpam-6146	166	13	y2	y2	PROPN
ejpam-6146	166	14	)	)	PUNCT
ejpam-6146	166	15	]	]	PUNCT
ejpam-6146	167	1	′	′	NUM
ejpam-6146	167	2	=	=	SYM
ejpam-6146	167	3	0	0	X
ejpam-6146	167	4	.	.	PUNCT
ejpam-6146	168	1	so	so	ADV
ejpam-6146	168	2	,	,	PUNCT
ejpam-6146	168	3	(	(	PUNCT
ejpam-6146	168	4	y′)2	y′)2	PROPN
ejpam-6146	168	5	+	+	PROPN
ejpam-6146	168	6	(	(	PUNCT
ejpam-6146	168	7	y2	y2	NOUN
ejpam-6146	168	8	)	)	PUNCT
ejpam-6146	168	9	=	=	SYM
ejpam-6146	168	10	constant	constant	ADJ
ejpam-6146	168	11	=	=	PUNCT
ejpam-6146	168	12	c.	c.	NOUN
ejpam-6146	168	13	if	if	SCONJ
ejpam-6146	168	14	y	y	PROPN
ejpam-6146	168	15	=	=	SYM
ejpam-6146	168	16	s(x	s(x	PROPN
ejpam-6146	168	17	)	)	PUNCT
ejpam-6146	168	18	,	,	PUNCT
ejpam-6146	168	19	then	then	ADV
ejpam-6146	168	20	y′	y′	ADV
ejpam-6146	168	21	=	=	SYM
ejpam-6146	168	22	c(x	c(x	NOUN
ejpam-6146	168	23	)	)	PUNCT
ejpam-6146	168	24	and	and	CCONJ
ejpam-6146	168	25	c2(x	c2(x	NOUN
ejpam-6146	168	26	)	)	PUNCT
ejpam-6146	168	27	+	+	NUM
ejpam-6146	168	28	s2(x	s2(x	X
ejpam-6146	168	29	)	)	PUNCT
ejpam-6146	168	30	=	=	SYM
ejpam-6146	168	31	c.	c.	NOUN
ejpam-6146	168	32	using	use	VERB
ejpam-6146	168	33	the	the	DET
ejpam-6146	168	34	initial	initial	ADJ
ejpam-6146	168	35	conditions	condition	NOUN
ejpam-6146	168	36	,	,	PUNCT
ejpam-6146	168	37	s(0	s(0	PROPN
ejpam-6146	168	38	)	)	PUNCT
ejpam-6146	168	39	=	=	SYM
ejpam-6146	168	40	0	0	NUM
ejpam-6146	168	41	and	and	CCONJ
ejpam-6146	168	42	c(0	c(0	PROPN
ejpam-6146	168	43	)	)	PUNCT
ejpam-6146	168	44	=	=	SYM
ejpam-6146	169	1	1	1	NUM
ejpam-6146	169	2	,	,	PUNCT
ejpam-6146	169	3	we	we	PRON
ejpam-6146	169	4	have	have	VERB
ejpam-6146	169	5	c	c	NOUN
ejpam-6146	169	6	=	=	SYM
ejpam-6146	169	7	1	1	X
ejpam-6146	169	8	.	.	PUNCT
ejpam-6146	170	1	this	this	PRON
ejpam-6146	170	2	leads	lead	VERB
ejpam-6146	170	3	to	to	ADP
ejpam-6146	170	4	the	the	DET
ejpam-6146	170	5	identity	identity	NOUN
ejpam-6146	170	6	s2(x	s2(x	PROPN
ejpam-6146	170	7	)	)	PUNCT
ejpam-6146	170	8	+	+	PUNCT
ejpam-6146	170	9	c2(x	c2(x	NOUN
ejpam-6146	170	10	)	)	PUNCT
ejpam-6146	170	11	=	=	SYM
ejpam-6146	170	12	1	1	X
ejpam-6146	170	13	.	.	PUNCT
ejpam-6146	171	1	(	(	PUNCT
ejpam-6146	171	2	13	13	NUM
ejpam-6146	171	3	)	)	PUNCT
ejpam-6146	171	4	we	we	PRON
ejpam-6146	171	5	will	will	AUX
ejpam-6146	171	6	be	be	AUX
ejpam-6146	171	7	using	use	VERB
ejpam-6146	171	8	this	this	DET
ejpam-6146	171	9	identity	identity	NOUN
ejpam-6146	171	10	to	to	PART
ejpam-6146	171	11	connect	connect	VERB
ejpam-6146	171	12	the	the	DET
ejpam-6146	171	13	two	two	NUM
ejpam-6146	171	14	functions	function	NOUN
ejpam-6146	171	15	to	to	ADP
ejpam-6146	171	16	the	the	DET
ejpam-6146	171	17	circle	circle	NOUN
ejpam-6146	171	18	geometry	geometry	NOUN
ejpam-6146	171	19	,	,	PUNCT
ejpam-6146	171	20	but	but	CCONJ
ejpam-6146	171	21	after	after	ADP
ejpam-6146	171	22	deriving	derive	VERB
ejpam-6146	171	23	some	some	DET
ejpam-6146	171	24	more	more	ADJ
ejpam-6146	171	25	properties	property	NOUN
ejpam-6146	171	26	.	.	PUNCT
ejpam-6146	172	1	for	for	ADP
ejpam-6146	172	2	further	further	ADJ
ejpam-6146	172	3	discussion	discussion	NOUN
ejpam-6146	172	4	,	,	PUNCT
ejpam-6146	172	5	it	it	PRON
ejpam-6146	172	6	is	be	AUX
ejpam-6146	172	7	interesting	interesting	ADJ
ejpam-6146	172	8	to	to	PART
ejpam-6146	172	9	note	note	VERB
ejpam-6146	172	10	that	that	SCONJ
ejpam-6146	172	11	this	this	DET
ejpam-6146	172	12	identity	identity	NOUN
ejpam-6146	172	13	connecting	connect	VERB
ejpam-6146	172	14	s(x	s(x	PROPN
ejpam-6146	172	15	)	)	PUNCT
ejpam-6146	172	16	and	and	CCONJ
ejpam-6146	172	17	c(x	c(x	NOUN
ejpam-6146	172	18	)	)	PUNCT
ejpam-6146	172	19	is	be	AUX
ejpam-6146	172	20	free	free	ADJ
ejpam-6146	172	21	of	of	ADP
ejpam-6146	172	22	their	their	PRON
ejpam-6146	172	23	derivatives	derivative	NOUN
ejpam-6146	172	24	.	.	PUNCT
ejpam-6146	173	1	moreover	moreover	ADV
ejpam-6146	173	2	,	,	PUNCT
ejpam-6146	173	3	the	the	DET
ejpam-6146	173	4	two	two	NUM
ejpam-6146	173	5	functions	function	NOUN
ejpam-6146	173	6	s(x	s(x	NOUN
ejpam-6146	173	7	)	)	PUNCT
ejpam-6146	173	8	and	and	CCONJ
ejpam-6146	173	9	c(x	c(x	NOUN
ejpam-6146	173	10	)	)	PUNCT
ejpam-6146	173	11	occur	occur	VERB
ejpam-6146	173	12	on	on	ADP
ejpam-6146	173	13	an	an	DET
ejpam-6146	173	14	equal	equal	ADJ
ejpam-6146	173	15	footing	footing	NOUN
ejpam-6146	173	16	in	in	ADP
ejpam-6146	173	17	the	the	DET
ejpam-6146	173	18	identity	identity	NOUN
ejpam-6146	173	19	.	.	PUNCT
ejpam-6146	174	1	from	from	ADP
ejpam-6146	174	2	eq	eq	ADP
ejpam-6146	174	3	.	.	PUNCT
ejpam-6146	175	1	(	(	PUNCT
ejpam-6146	175	2	13	13	NUM
ejpam-6146	175	3	)	)	PUNCT
ejpam-6146	175	4	,	,	PUNCT
ejpam-6146	175	5	it	it	PRON
ejpam-6146	175	6	follows	follow	VERB
ejpam-6146	175	7	that	that	SCONJ
ejpam-6146	175	8	the	the	DET
ejpam-6146	175	9	height	height	NOUN
ejpam-6146	175	10	of	of	ADP
ejpam-6146	175	11	the	the	DET
ejpam-6146	175	12	first	first	ADJ
ejpam-6146	175	13	arch	arch	NOUN
ejpam-6146	175	14	of	of	ADP
ejpam-6146	175	15	s(x	s(x	PROPN
ejpam-6146	175	16	)	)	PUNCT
ejpam-6146	175	17	is	be	AUX
ejpam-6146	175	18	1	1	NUM
ejpam-6146	175	19	at	at	ADP
ejpam-6146	175	20	x	x	X
ejpam-6146	175	21	=	=	PUNCT
ejpam-6146	175	22	p/2	p/2	NOUN
ejpam-6146	175	23	and	and	CCONJ
ejpam-6146	175	24	the	the	DET
ejpam-6146	175	25	first	first	ADJ
ejpam-6146	175	26	zero	zero	NUM
ejpam-6146	175	27	of	of	ADP
ejpam-6146	175	28	c(x	c(x	NOUN
ejpam-6146	175	29	)	)	PUNCT
ejpam-6146	175	30	is	be	AUX
ejpam-6146	175	31	at	at	ADP
ejpam-6146	175	32	x	x	X
ejpam-6146	175	33	=	=	PUNCT
ejpam-6146	175	34	p/2	p/2	NOUN
ejpam-6146	175	35	.	.	PUNCT
ejpam-6146	176	1	so	so	ADV
ejpam-6146	176	2	,	,	PUNCT
ejpam-6146	176	3	we	we	PRON
ejpam-6146	176	4	conclude	conclude	VERB
ejpam-6146	176	5	that	that	PRON
ejpam-6146	177	1	p̃	p̃	PROPN
ejpam-6146	177	2	=	=	PUNCT
ejpam-6146	177	3	p.	p.	NOUN
ejpam-6146	177	4	from	from	ADP
ejpam-6146	177	5	this	this	PRON
ejpam-6146	177	6	we	we	PRON
ejpam-6146	177	7	conclude	conclude	VERB
ejpam-6146	177	8	that	that	SCONJ
ejpam-6146	177	9	the	the	DET
ejpam-6146	177	10	successive	successive	ADJ
ejpam-6146	177	11	zeros	zero	NOUN
ejpam-6146	177	12	of	of	ADP
ejpam-6146	177	13	s(x	s(x	PROPN
ejpam-6146	177	14	)	)	PUNCT
ejpam-6146	177	15	are	be	AUX
ejpam-6146	177	16	at	at	ADP
ejpam-6146	177	17	p	p	PRON
ejpam-6146	177	18	,	,	PUNCT
ejpam-6146	177	19	2p	2p	NUM
ejpam-6146	177	20	,	,	PUNCT
ejpam-6146	177	21	3p	3p	NUM
ejpam-6146	177	22	.	.	PUNCT
ejpam-6146	177	23	.	.	PUNCT
ejpam-6146	178	1	.	.	PUNCT
ejpam-6146	179	1	and	and	CCONJ
ejpam-6146	179	2	the	the	DET
ejpam-6146	179	3	successive	successive	ADJ
ejpam-6146	179	4	zeros	zero	NOUN
ejpam-6146	179	5	of	of	ADP
ejpam-6146	179	6	c(x	c(x	NOUN
ejpam-6146	179	7	)	)	PUNCT
ejpam-6146	179	8	are	be	AUX
ejpam-6146	179	9	at	at	ADP
ejpam-6146	179	10	p/2	p/2	NUM
ejpam-6146	179	11	,	,	PUNCT
ejpam-6146	179	12	3p/2	3p/2	NUM
ejpam-6146	179	13	,	,	PUNCT
ejpam-6146	179	14	5p/2	5p/2	NUM
ejpam-6146	179	15	,	,	PUNCT
ejpam-6146	179	16	.	.	PUNCT
ejpam-6146	179	17	.	.	PUNCT
ejpam-6146	179	18	.	.	PUNCT
ejpam-6146	180	1	.	.	PUNCT
ejpam-6146	181	1	it	it	PRON
ejpam-6146	181	2	is	be	AUX
ejpam-6146	181	3	worth	worth	ADJ
ejpam-6146	181	4	s.	s.	PROPN
ejpam-6146	181	5	a.	a.	PROPN
ejpam-6146	181	6	khan	khan	PROPN
ejpam-6146	181	7	,	,	PUNCT
ejpam-6146	181	8	m.	m.	NOUN
ejpam-6146	181	9	m.	m.	PROPN
ejpam-6146	181	10	kankarej	kankarej	PROPN
ejpam-6146	181	11	,	,	PUNCT
ejpam-6146	181	12	m.	m.	PROPN
ejpam-6146	181	13	n.	n.	PROPN
ejpam-6146	181	14	i.	i.	PROPN
ejpam-6146	181	15	khan	khan	PROPN
ejpam-6146	181	16	/	/	SYM
ejpam-6146	181	17	eur	eur	PROPN
ejpam-6146	181	18	.	.	PUNCT
ejpam-6146	182	1	j.	j.	PROPN
ejpam-6146	182	2	pure	pure	PROPN
ejpam-6146	182	3	appl	appl	PROPN
ejpam-6146	182	4	.	.	PROPN
ejpam-6146	182	5	math	math	PROPN
ejpam-6146	182	6	,	,	PUNCT
ejpam-6146	182	7	18	18	NUM
ejpam-6146	182	8	(	(	PUNCT
ejpam-6146	182	9	4	4	NUM
ejpam-6146	182	10	)	)	PUNCT
ejpam-6146	182	11	(	(	PUNCT
ejpam-6146	182	12	2025	2025	NUM
ejpam-6146	182	13	)	)	PUNCT
ejpam-6146	182	14	,	,	PUNCT
ejpam-6146	182	15	6146	6146	NUM
ejpam-6146	182	16	6	6	NUM
ejpam-6146	182	17	of	of	ADP
ejpam-6146	182	18	14	14	NUM
ejpam-6146	182	19	noting	note	VERB
ejpam-6146	182	20	,	,	PUNCT
ejpam-6146	182	21	that	that	SCONJ
ejpam-6146	182	22	between	between	ADP
ejpam-6146	182	23	any	any	DET
ejpam-6146	182	24	two	two	NUM
ejpam-6146	182	25	successive	successive	ADJ
ejpam-6146	182	26	zeros	zero	NOUN
ejpam-6146	182	27	of	of	ADP
ejpam-6146	182	28	s(x	s(x	PROPN
ejpam-6146	182	29	)	)	PUNCT
ejpam-6146	182	30	there	there	PRON
ejpam-6146	182	31	is	be	VERB
ejpam-6146	182	32	exactly	exactly	ADV
ejpam-6146	182	33	one	one	NUM
ejpam-6146	182	34	zero	zero	NUM
ejpam-6146	182	35	of	of	ADP
ejpam-6146	182	36	c(x	c(x	NOUN
ejpam-6146	182	37	)	)	PUNCT
ejpam-6146	182	38	and	and	CCONJ
ejpam-6146	182	39	between	between	ADP
ejpam-6146	182	40	any	any	DET
ejpam-6146	182	41	two	two	NUM
ejpam-6146	182	42	successive	successive	ADJ
ejpam-6146	182	43	zeros	zero	NOUN
ejpam-6146	182	44	of	of	ADP
ejpam-6146	182	45	c(x	c(x	NOUN
ejpam-6146	182	46	)	)	PUNCT
ejpam-6146	182	47	there	there	PRON
ejpam-6146	182	48	is	be	VERB
ejpam-6146	182	49	exactly	exactly	ADV
ejpam-6146	182	50	one	one	NUM
ejpam-6146	182	51	zero	zero	NUM
ejpam-6146	182	52	of	of	ADP
ejpam-6146	182	53	s(x	s(x	PROPN
ejpam-6146	182	54	)	)	PUNCT
ejpam-6146	182	55	.	.	PUNCT
ejpam-6146	183	1	to	to	PART
ejpam-6146	183	2	establish	establish	VERB
ejpam-6146	183	3	the	the	DET
ejpam-6146	183	4	linear	linear	ADJ
ejpam-6146	183	5	independence	independence	NOUN
ejpam-6146	183	6	of	of	ADP
ejpam-6146	183	7	s(x	s(x	PROPN
ejpam-6146	183	8	)	)	PUNCT
ejpam-6146	183	9	and	and	CCONJ
ejpam-6146	183	10	c(x	c(x	NOUN
ejpam-6146	183	11	)	)	PUNCT
ejpam-6146	183	12	,	,	PUNCT
ejpam-6146	183	13	we	we	PRON
ejpam-6146	183	14	calculate	calculate	VERB
ejpam-6146	183	15	their	their	PRON
ejpam-6146	183	16	wronskian	wronskian	NOUN
ejpam-6146	183	17	w	w	PROPN
ejpam-6146	183	18	(	(	PUNCT
ejpam-6146	183	19	x	x	NOUN
ejpam-6146	183	20	)	)	PUNCT
ejpam-6146	183	21	=	=	SYM
ejpam-6146	184	1	w	w	PROPN
ejpam-6146	185	1	[	[	X
ejpam-6146	185	2	s(x	s(x	X
ejpam-6146	185	3	)	)	PUNCT
ejpam-6146	185	4	,	,	PUNCT
ejpam-6146	185	5	c(x	c(x	NOUN
ejpam-6146	185	6	)	)	PUNCT
ejpam-6146	185	7	]	]	PUNCT
ejpam-6146	186	1	=	=	PUNCT
ejpam-6146	186	2	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6146	186	3	s(x	s(x	PROPN
ejpam-6146	186	4	)	)	PUNCT
ejpam-6146	186	5	c(x	c(x	NOUN
ejpam-6146	186	6	)	)	PUNCT
ejpam-6146	186	7	s′(x	s′(x	NOUN
ejpam-6146	186	8	)	)	PUNCT
ejpam-6146	186	9	c′(x	c′(x	NOUN
ejpam-6146	186	10	)	)	PUNCT
ejpam-6146	186	11	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6146	186	12	=	=	SYM
ejpam-6146	186	13	s(x)c′(x)−	s(x)c′(x)−	PROPN
ejpam-6146	186	14	s′(x)c(x	s′(x)c(x	NOUN
ejpam-6146	186	15	)	)	PUNCT
ejpam-6146	186	16	=	=	SYM
ejpam-6146	186	17	−s2(x)−	−s2(x)−	NOUN
ejpam-6146	186	18	c2(x	c2(x	PROPN
ejpam-6146	186	19	)	)	PUNCT
ejpam-6146	186	20	=	=	SYM
ejpam-6146	186	21	−1	−1	NOUN
ejpam-6146	186	22	.	.	PUNCT
ejpam-6146	187	1	(	(	PUNCT
ejpam-6146	187	2	14	14	NUM
ejpam-6146	187	3	)	)	PUNCT
ejpam-6146	187	4	as	as	SCONJ
ejpam-6146	187	5	the	the	DET
ejpam-6146	187	6	wronskian	wronskian	NOUN
ejpam-6146	187	7	is	be	AUX
ejpam-6146	187	8	not	not	PART
ejpam-6146	187	9	zero	zero	NUM
ejpam-6146	187	10	,	,	PUNCT
ejpam-6146	187	11	the	the	DET
ejpam-6146	187	12	two	two	NUM
ejpam-6146	187	13	solutions	solution	NOUN
ejpam-6146	187	14	s(x	s(x	NOUN
ejpam-6146	187	15	)	)	PUNCT
ejpam-6146	187	16	and	and	CCONJ
ejpam-6146	187	17	c(x	c(x	NOUN
ejpam-6146	187	18	)	)	PUNCT
ejpam-6146	187	19	are	be	AUX
ejpam-6146	187	20	linearly	linearly	ADV
ejpam-6146	187	21	independent	independent	ADJ
ejpam-6146	187	22	.	.	PUNCT
ejpam-6146	188	1	the	the	DET
ejpam-6146	188	2	addition	addition	NOUN
ejpam-6146	188	3	/	/	SYM
ejpam-6146	188	4	subtraction	subtraction	NOUN
ejpam-6146	188	5	formulae	formulae	NOUN
ejpam-6146	188	6	are	be	AUX
ejpam-6146	188	7	obtained	obtain	VERB
ejpam-6146	188	8	as	as	ADP
ejpam-6146	188	9	follows	follow	VERB
ejpam-6146	188	10	.	.	PUNCT
ejpam-6146	189	1	let	let	VERB
ejpam-6146	189	2	us	we	PRON
ejpam-6146	189	3	write	write	VERB
ejpam-6146	189	4	the	the	DET
ejpam-6146	189	5	general	general	ADJ
ejpam-6146	189	6	solutions	solution	NOUN
ejpam-6146	189	7	of	of	ADP
ejpam-6146	189	8	eq	eq	PROPN
ejpam-6146	189	9	.	.	PUNCT
ejpam-6146	190	1	(	(	PUNCT
ejpam-6146	190	2	3	3	NUM
ejpam-6146	190	3	)	)	PUNCT
ejpam-6146	190	4	as	as	ADP
ejpam-6146	190	5	c1s(x	c1s(x	PROPN
ejpam-6146	190	6	)	)	PUNCT
ejpam-6146	191	1	+	+	CCONJ
ejpam-6146	191	2	c2c(x	c2c(x	NOUN
ejpam-6146	191	3	)	)	PUNCT
ejpam-6146	191	4	=	=	PUNCT
ejpam-6146	192	1	rs(x+	rs(x+	ADV
ejpam-6146	192	2	a	a	X
ejpam-6146	192	3	)	)	PUNCT
ejpam-6146	192	4	respectively	respectively	ADV
ejpam-6146	192	5	,	,	PUNCT
ejpam-6146	192	6	where	where	SCONJ
ejpam-6146	192	7	c1	c1	PROPN
ejpam-6146	192	8	,	,	PUNCT
ejpam-6146	192	9	c2	c2	PROPN
ejpam-6146	192	10	,	,	PUNCT
ejpam-6146	192	11	r	r	NOUN
ejpam-6146	192	12	and	and	CCONJ
ejpam-6146	192	13	a	a	PRON
ejpam-6146	192	14	are	be	AUX
ejpam-6146	192	15	arbitrary	arbitrary	ADJ
ejpam-6146	192	16	constants	constant	NOUN
ejpam-6146	192	17	.	.	PUNCT
ejpam-6146	193	1	on	on	ADP
ejpam-6146	193	2	differentiating	differentiate	VERB
ejpam-6146	193	3	,	,	PUNCT
ejpam-6146	193	4	c1c(x)−c2s(x	c1c(x)−c2s(x	PROPN
ejpam-6146	193	5	)	)	PUNCT
ejpam-6146	193	6	=	=	SYM
ejpam-6146	193	7	rc(x+a	rc(x+a	NOUN
ejpam-6146	193	8	)	)	PUNCT
ejpam-6146	193	9	.	.	PUNCT
ejpam-6146	194	1	using	use	VERB
ejpam-6146	194	2	the	the	DET
ejpam-6146	194	3	initial	initial	ADJ
ejpam-6146	194	4	condition	condition	NOUN
ejpam-6146	194	5	at	at	ADP
ejpam-6146	194	6	x	x	X
ejpam-6146	194	7	=	=	SYM
ejpam-6146	194	8	0	0	NUM
ejpam-6146	194	9	,	,	PUNCT
ejpam-6146	194	10	we	we	PRON
ejpam-6146	194	11	obtain	obtain	VERB
ejpam-6146	194	12	c1	c1	NOUN
ejpam-6146	194	13	=	=	PUNCT
ejpam-6146	194	14	rc(a	rc(a	X
ejpam-6146	194	15	)	)	PUNCT
ejpam-6146	194	16	and	and	CCONJ
ejpam-6146	194	17	c2	c2	PROPN
ejpam-6146	194	18	=	=	PUNCT
ejpam-6146	194	19	rs(a	rs(a	PRON
ejpam-6146	194	20	)	)	PUNCT
ejpam-6146	194	21	.	.	PUNCT
ejpam-6146	195	1	substituting	substitute	VERB
ejpam-6146	195	2	back	back	ADP
ejpam-6146	195	3	the	the	DET
ejpam-6146	195	4	values	value	NOUN
ejpam-6146	195	5	of	of	ADP
ejpam-6146	195	6	c1	c1	PROPN
ejpam-6146	195	7	and	and	CCONJ
ejpam-6146	195	8	c2	c2	PROPN
ejpam-6146	195	9	and	and	CCONJ
ejpam-6146	195	10	cancelling	cancel	VERB
ejpam-6146	195	11	the	the	DET
ejpam-6146	195	12	constant	constant	ADJ
ejpam-6146	195	13	r	r	NOUN
ejpam-6146	195	14	,	,	PUNCT
ejpam-6146	195	15	we	we	PRON
ejpam-6146	195	16	obtain	obtain	VERB
ejpam-6146	195	17	s(x+	s(x+	NOUN
ejpam-6146	195	18	a	a	NOUN
ejpam-6146	195	19	)	)	PUNCT
ejpam-6146	195	20	=	=	SYM
ejpam-6146	195	21	s(x)c(a	s(x)c(a	NUM
ejpam-6146	195	22	)	)	PUNCT
ejpam-6146	195	23	+	+	NUM
ejpam-6146	195	24	c(x)s(a	c(x)s(a	NOUN
ejpam-6146	195	25	)	)	PUNCT
ejpam-6146	195	26	c(x+	c(x+	NOUN
ejpam-6146	195	27	a	a	PRON
ejpam-6146	195	28	)	)	PUNCT
ejpam-6146	195	29	=	=	PUNCT
ejpam-6146	195	30	c(x)c(a)−	c(x)c(a)−	NOUN
ejpam-6146	195	31	s(x)s(a	s(x)s(a	NOUN
ejpam-6146	195	32	)	)	PUNCT
ejpam-6146	195	33	.	.	PUNCT
ejpam-6146	196	1	(	(	PUNCT
ejpam-6146	196	2	15	15	NUM
ejpam-6146	196	3	)	)	PUNCT
ejpam-6146	196	4	in	in	ADP
ejpam-6146	196	5	eq	eq	ADP
ejpam-6146	196	6	.	.	PUNCT
ejpam-6146	197	1	(	(	PUNCT
ejpam-6146	197	2	15	15	NUM
ejpam-6146	197	3	)	)	PUNCT
ejpam-6146	197	4	,	,	PUNCT
ejpam-6146	197	5	with	with	ADP
ejpam-6146	197	6	a	a	DET
ejpam-6146	197	7	=	=	ADJ
ejpam-6146	197	8	x	x	NOUN
ejpam-6146	197	9	,	,	PUNCT
ejpam-6146	197	10	we	we	PRON
ejpam-6146	197	11	obtain	obtain	VERB
ejpam-6146	197	12	the	the	DET
ejpam-6146	197	13	double	double	ADJ
ejpam-6146	197	14	angle	angle	NOUN
ejpam-6146	197	15	identities	identity	NOUN
ejpam-6146	197	16	s(2x	s(2x	VERB
ejpam-6146	197	17	)	)	PUNCT
ejpam-6146	198	1	=	=	SYM
ejpam-6146	198	2	2s(x)c(x	2s(x)c(x	NOUN
ejpam-6146	198	3	)	)	PUNCT
ejpam-6146	198	4	c(2x	c(2x	VERB
ejpam-6146	198	5	)	)	PUNCT
ejpam-6146	199	1	=	=	PUNCT
ejpam-6146	199	2	c2(x)−	c2(x)−	PROPN
ejpam-6146	199	3	s2(x	s2(x	NOUN
ejpam-6146	199	4	)	)	PUNCT
ejpam-6146	199	5	.	.	PUNCT
ejpam-6146	200	1	(	(	PUNCT
ejpam-6146	200	2	16	16	NUM
ejpam-6146	200	3	)	)	PUNCT
ejpam-6146	200	4	equation	equation	NOUN
ejpam-6146	200	5	(	(	PUNCT
ejpam-6146	200	6	13	13	NUM
ejpam-6146	200	7	)	)	PUNCT
ejpam-6146	200	8	,	,	PUNCT
ejpam-6146	200	9	c2(x)+s2(x	c2(x)+s2(x	PROPN
ejpam-6146	200	10	)	)	PUNCT
ejpam-6146	200	11	=	=	SYM
ejpam-6146	200	12	1	1	NUM
ejpam-6146	200	13	is	be	AUX
ejpam-6146	200	14	recognized	recognize	VERB
ejpam-6146	200	15	as	as	ADP
ejpam-6146	200	16	the	the	DET
ejpam-6146	200	17	equation	equation	NOUN
ejpam-6146	200	18	of	of	ADP
ejpam-6146	200	19	the	the	DET
ejpam-6146	200	20	unit	unit	NOUN
ejpam-6146	200	21	circle	circle	NOUN
ejpam-6146	200	22	with	with	ADP
ejpam-6146	200	23	centre	centre	NOUN
ejpam-6146	200	24	at	at	ADP
ejpam-6146	200	25	the	the	DET
ejpam-6146	200	26	origin	origin	NOUN
ejpam-6146	200	27	and	and	CCONJ
ejpam-6146	200	28	the	the	DET
ejpam-6146	200	29	point	point	NOUN
ejpam-6146	200	30	(	(	PUNCT
ejpam-6146	200	31	c(x	c(x	NOUN
ejpam-6146	200	32	)	)	PUNCT
ejpam-6146	200	33	,	,	PUNCT
ejpam-6146	200	34	s(x	s(x	NOUN
ejpam-6146	200	35	)	)	PUNCT
ejpam-6146	200	36	)	)	PUNCT
ejpam-6146	200	37	on	on	ADP
ejpam-6146	200	38	it	it	PRON
ejpam-6146	200	39	.	.	PUNCT
ejpam-6146	201	1	this	this	PRON
ejpam-6146	201	2	connects	connect	VERB
ejpam-6146	201	3	the	the	DET
ejpam-6146	201	4	two	two	NUM
ejpam-6146	201	5	functions	function	NOUN
ejpam-6146	201	6	s(x	s(x	NOUN
ejpam-6146	201	7	)	)	PUNCT
ejpam-6146	201	8	and	and	CCONJ
ejpam-6146	201	9	c(x	c(x	NOUN
ejpam-6146	201	10	)	)	PUNCT
ejpam-6146	201	11	to	to	ADP
ejpam-6146	201	12	the	the	DET
ejpam-6146	201	13	circle	circle	NOUN
ejpam-6146	201	14	geometry	geometry	NOUN
ejpam-6146	201	15	,	,	PUNCT
ejpam-6146	201	16	establishing	establish	VERB
ejpam-6146	201	17	their	their	PRON
ejpam-6146	201	18	geometric	geometric	ADJ
ejpam-6146	201	19	origins	origin	NOUN
ejpam-6146	201	20	.	.	PUNCT
ejpam-6146	202	1	from	from	ADP
ejpam-6146	202	2	this	this	PRON
ejpam-6146	202	3	,	,	PUNCT
ejpam-6146	202	4	we	we	PRON
ejpam-6146	202	5	conclude	conclude	VERB
ejpam-6146	202	6	that	that	SCONJ
ejpam-6146	202	7	p	p	PROPN
ejpam-6146	202	8	=	=	X
ejpam-6146	202	9	π	π	PROPN
ejpam-6146	202	10	.	.	PUNCT
ejpam-6146	203	1	having	having	AUX
ejpam-6146	203	2	derived	derive	VERB
ejpam-6146	203	3	the	the	DET
ejpam-6146	203	4	various	various	ADJ
ejpam-6146	203	5	significant	significant	ADJ
ejpam-6146	203	6	properties	property	NOUN
ejpam-6146	203	7	of	of	ADP
ejpam-6146	203	8	the	the	DET
ejpam-6146	203	9	two	two	NUM
ejpam-6146	203	10	solutions	solution	NOUN
ejpam-6146	203	11	of	of	ADP
ejpam-6146	203	12	eq	eq	PROPN
ejpam-6146	203	13	.	.	PUNCT
ejpam-6146	204	1	(	(	PUNCT
ejpam-6146	204	2	3	3	NUM
ejpam-6146	204	3	)	)	PUNCT
ejpam-6146	204	4	,	,	PUNCT
ejpam-6146	204	5	we	we	PRON
ejpam-6146	204	6	can	can	AUX
ejpam-6146	204	7	now	now	ADV
ejpam-6146	204	8	emphatically	emphatically	ADV
ejpam-6146	204	9	state	state	VERB
ejpam-6146	204	10	that	that	SCONJ
ejpam-6146	204	11	s(x	s(x	NOUN
ejpam-6146	204	12	)	)	PUNCT
ejpam-6146	204	13	≡	≡	PROPN
ejpam-6146	204	14	sinx	sinx	PROPN
ejpam-6146	204	15	c(x	c(x	PROPN
ejpam-6146	204	16	)	)	PUNCT
ejpam-6146	204	17	≡	≡	PROPN
ejpam-6146	204	18	cosx	cosx	PROPN
ejpam-6146	204	19	.	.	PUNCT
ejpam-6146	205	1	(	(	PUNCT
ejpam-6146	205	2	17	17	NUM
ejpam-6146	205	3	)	)	PUNCT
ejpam-6146	205	4	from	from	ADP
ejpam-6146	205	5	the	the	DET
ejpam-6146	205	6	graphs	graph	NOUN
ejpam-6146	205	7	in	in	ADP
ejpam-6146	205	8	figure-1	figure-1	PROPN
ejpam-6146	205	9	we	we	PRON
ejpam-6146	205	10	can	can	AUX
ejpam-6146	205	11	see	see	VERB
ejpam-6146	205	12	the	the	DET
ejpam-6146	205	13	oscillations	oscillation	NOUN
ejpam-6146	205	14	of	of	ADP
ejpam-6146	205	15	the	the	DET
ejpam-6146	205	16	two	two	NUM
ejpam-6146	205	17	solutions	solution	NOUN
ejpam-6146	205	18	namely	namely	ADV
ejpam-6146	205	19	sine	sine	ADJ
ejpam-6146	205	20	and	and	CCONJ
ejpam-6146	205	21	cosine	cosine	NOUN
ejpam-6146	205	22	functions	function	NOUN
ejpam-6146	205	23	respectively	respectively	ADV
ejpam-6146	205	24	.	.	PUNCT
ejpam-6146	206	1	significantly	significantly	ADV
ejpam-6146	206	2	,	,	PUNCT
ejpam-6146	206	3	we	we	PRON
ejpam-6146	206	4	see	see	VERB
ejpam-6146	206	5	that	that	SCONJ
ejpam-6146	206	6	the	the	DET
ejpam-6146	206	7	roots	root	NOUN
ejpam-6146	206	8	of	of	ADP
ejpam-6146	206	9	the	the	DET
ejpam-6146	206	10	two	two	NUM
ejpam-6146	206	11	solutions	solution	NOUN
ejpam-6146	206	12	are	be	AUX
ejpam-6146	206	13	interlaced	interlaced	ADJ
ejpam-6146	206	14	.	.	PUNCT
ejpam-6146	207	1	in	in	ADP
ejpam-6146	207	2	this	this	DET
ejpam-6146	207	3	section	section	NOUN
ejpam-6146	207	4	,	,	PUNCT
ejpam-6146	207	5	we	we	PRON
ejpam-6146	207	6	deduced	deduce	VERB
ejpam-6146	207	7	the	the	DET
ejpam-6146	207	8	basic	basic	ADJ
ejpam-6146	207	9	properties	property	NOUN
ejpam-6146	207	10	of	of	ADP
ejpam-6146	207	11	the	the	DET
ejpam-6146	207	12	trigonometric	trigonometric	ADJ
ejpam-6146	207	13	functions	function	NOUN
ejpam-6146	207	14	,	,	PUNCT
ejpam-6146	207	15	sinx	sinx	NOUN
ejpam-6146	207	16	and	and	CCONJ
ejpam-6146	207	17	cosx	cosx	NOUN
ejpam-6146	207	18	starting	start	VERB
ejpam-6146	207	19	with	with	ADP
ejpam-6146	207	20	the	the	DET
ejpam-6146	207	21	second	second	ADJ
ejpam-6146	207	22	-	-	PUNCT
ejpam-6146	207	23	order	order	NOUN
ejpam-6146	207	24	linear	linear	NOUN
ejpam-6146	207	25	differential	differential	NOUN
ejpam-6146	207	26	equation	equation	NOUN
ejpam-6146	207	27	y′′	y′′	PROPN
ejpam-6146	207	28	+	+	CCONJ
ejpam-6146	207	29	y	y	PROPN
ejpam-6146	207	30	=	=	SYM
ejpam-6146	207	31	0	0	PROPN
ejpam-6146	207	32	,	,	PUNCT
ejpam-6146	207	33	using	use	VERB
ejpam-6146	207	34	the	the	DET
ejpam-6146	207	35	qualitative	qualitative	ADJ
ejpam-6146	207	36	properties	property	NOUN
ejpam-6146	207	37	of	of	ADP
ejpam-6146	207	38	differential	differential	ADJ
ejpam-6146	207	39	equations	equation	NOUN
ejpam-6146	207	40	(	(	PUNCT
ejpam-6146	207	41	essentially	essentially	ADV
ejpam-6146	207	42	the	the	DET
ejpam-6146	207	43	uniqueness	uniqueness	NOUN
ejpam-6146	207	44	of	of	ADP
ejpam-6146	207	45	the	the	DET
ejpam-6146	207	46	solutions	solution	NOUN
ejpam-6146	207	47	)	)	PUNCT
ejpam-6146	207	48	.	.	PUNCT
ejpam-6146	208	1	the	the	DET
ejpam-6146	208	2	derivations	derivation	NOUN
ejpam-6146	208	3	were	be	AUX
ejpam-6146	208	4	straightforward	straightforward	ADJ
ejpam-6146	208	5	based	base	VERB
ejpam-6146	208	6	on	on	ADP
ejpam-6146	208	7	the	the	DET
ejpam-6146	208	8	convexity	convexity	NOUN
ejpam-6146	208	9	arguments	argument	NOUN
ejpam-6146	208	10	(	(	PUNCT
ejpam-6146	208	11	signs	sign	NOUN
ejpam-6146	208	12	of	of	ADP
ejpam-6146	208	13	derivatives	derivative	NOUN
ejpam-6146	208	14	and	and	CCONJ
ejpam-6146	208	15	increasing	increase	VERB
ejpam-6146	208	16	/	/	SYM
ejpam-6146	208	17	decreasing	decreasing	NOUN
ejpam-6146	208	18	)	)	PUNCT
ejpam-6146	208	19	usually	usually	ADV
ejpam-6146	208	20	done	do	VERB
ejpam-6146	208	21	in	in	ADP
ejpam-6146	208	22	the	the	DET
ejpam-6146	208	23	introductory	introductory	ADJ
ejpam-6146	208	24	courses	course	NOUN
ejpam-6146	208	25	of	of	ADP
ejpam-6146	208	26	calculus	calculus	NOUN
ejpam-6146	208	27	.	.	PUNCT
ejpam-6146	209	1	it	it	PRON
ejpam-6146	209	2	s.	s.	PROPN
ejpam-6146	209	3	a.	a.	PROPN
ejpam-6146	209	4	khan	khan	PROPN
ejpam-6146	209	5	,	,	PUNCT
ejpam-6146	209	6	m.	m.	NOUN
ejpam-6146	209	7	m.	m.	PROPN
ejpam-6146	209	8	kankarej	kankarej	PROPN
ejpam-6146	209	9	,	,	PUNCT
ejpam-6146	209	10	m.	m.	PROPN
ejpam-6146	209	11	n.	n.	PROPN
ejpam-6146	209	12	i.	i.	PROPN
ejpam-6146	209	13	khan	khan	PROPN
ejpam-6146	209	14	/	/	SYM
ejpam-6146	209	15	eur	eur	PROPN
ejpam-6146	209	16	.	.	PUNCT
ejpam-6146	210	1	j.	j.	PROPN
ejpam-6146	210	2	pure	pure	PROPN
ejpam-6146	210	3	appl	appl	PROPN
ejpam-6146	210	4	.	.	PROPN
ejpam-6146	210	5	math	math	PROPN
ejpam-6146	210	6	,	,	PUNCT
ejpam-6146	210	7	18	18	NUM
ejpam-6146	210	8	(	(	PUNCT
ejpam-6146	210	9	4	4	NUM
ejpam-6146	210	10	)	)	PUNCT
ejpam-6146	210	11	(	(	PUNCT
ejpam-6146	210	12	2025	2025	NUM
ejpam-6146	210	13	)	)	PUNCT
ejpam-6146	210	14	,	,	PUNCT
ejpam-6146	210	15	6146	6146	NUM
ejpam-6146	210	16	7	7	NUM
ejpam-6146	210	17	of	of	ADP
ejpam-6146	210	18	14	14	NUM
ejpam-6146	210	19	-2π	-2π	NOUN
ejpam-6146	210	20	-π	-π	PUNCT
ejpam-6146	211	1	π	π	X
ejpam-6146	211	2	2π	2π	NUM
ejpam-6146	211	3	−1	−1	NOUN
ejpam-6146	211	4	1	1	NUM
ejpam-6146	211	5	x	x	NOUN
ejpam-6146	211	6	amplitude	amplitude	NOUN
ejpam-6146	211	7	sin(x	sin(x	PROPN
ejpam-6146	211	8	)	)	PUNCT
ejpam-6146	211	9	cos(x	cos(x	PROPN
ejpam-6146	211	10	)	)	PUNCT
ejpam-6146	211	11	figure	figure	NOUN
ejpam-6146	211	12	1	1	NUM
ejpam-6146	211	13	:	:	PUNCT
ejpam-6146	211	14	graphs	graph	NOUN
ejpam-6146	211	15	of	of	ADP
ejpam-6146	211	16	y	y	PROPN
ejpam-6146	211	17	=	=	SYM
ejpam-6146	211	18	sin(x	sin(x	PROPN
ejpam-6146	211	19	)	)	PUNCT
ejpam-6146	211	20	and	and	CCONJ
ejpam-6146	211	21	y	y	PROPN
ejpam-6146	211	22	=	=	SYM
ejpam-6146	211	23	cos(x	cos(x	PROPN
ejpam-6146	211	24	)	)	PUNCT
ejpam-6146	211	25	.	.	PUNCT
ejpam-6146	212	1	is	be	AUX
ejpam-6146	212	2	to	to	PART
ejpam-6146	212	3	be	be	AUX
ejpam-6146	212	4	noted	note	VERB
ejpam-6146	212	5	that	that	SCONJ
ejpam-6146	212	6	most	most	ADJ
ejpam-6146	212	7	of	of	ADP
ejpam-6146	212	8	the	the	DET
ejpam-6146	212	9	properties	property	NOUN
ejpam-6146	212	10	we	we	PRON
ejpam-6146	212	11	derived	derive	VERB
ejpam-6146	212	12	,	,	PUNCT
ejpam-6146	212	13	are	be	AUX
ejpam-6146	212	14	unique	unique	ADJ
ejpam-6146	212	15	to	to	ADP
ejpam-6146	212	16	the	the	DET
ejpam-6146	212	17	sine	sine	NOUN
ejpam-6146	212	18	and	and	CCONJ
ejpam-6146	212	19	cosine	cosine	NOUN
ejpam-6146	212	20	functions	function	NOUN
ejpam-6146	212	21	.	.	PUNCT
ejpam-6146	213	1	the	the	DET
ejpam-6146	213	2	two	two	NUM
ejpam-6146	213	3	solutions	solution	NOUN
ejpam-6146	213	4	were	be	AUX
ejpam-6146	213	5	found	find	VERB
ejpam-6146	213	6	to	to	PART
ejpam-6146	213	7	oscillate	oscillate	VERB
ejpam-6146	213	8	about	about	ADP
ejpam-6146	213	9	the	the	DET
ejpam-6146	213	10	x	x	NOUN
ejpam-6146	213	11	-	-	NOUN
ejpam-6146	213	12	axis	axis	ADJ
ejpam-6146	213	13	.	.	PUNCT
ejpam-6146	214	1	these	these	DET
ejpam-6146	214	2	oscillations	oscillation	NOUN
ejpam-6146	214	3	are	be	AUX
ejpam-6146	214	4	interrelated	interrelate	VERB
ejpam-6146	214	5	in	in	ADP
ejpam-6146	214	6	such	such	DET
ejpam-6146	214	7	a	a	DET
ejpam-6146	214	8	manner	manner	NOUN
ejpam-6146	214	9	that	that	SCONJ
ejpam-6146	214	10	the	the	DET
ejpam-6146	214	11	zeros	zero	NOUN
ejpam-6146	214	12	of	of	ADP
ejpam-6146	214	13	the	the	DET
ejpam-6146	214	14	two	two	NUM
ejpam-6146	214	15	solutions	solution	NOUN
ejpam-6146	214	16	are	be	AUX
ejpam-6146	214	17	distinct	distinct	ADJ
ejpam-6146	214	18	and	and	CCONJ
ejpam-6146	214	19	occur	occur	VERB
ejpam-6146	214	20	alternately	alternately	ADV
ejpam-6146	214	21	.	.	PUNCT
ejpam-6146	215	1	in	in	ADP
ejpam-6146	215	2	other	other	ADJ
ejpam-6146	215	3	words	word	NOUN
ejpam-6146	215	4	,	,	PUNCT
ejpam-6146	215	5	between	between	ADP
ejpam-6146	215	6	any	any	DET
ejpam-6146	215	7	two	two	NUM
ejpam-6146	215	8	successive	successive	ADJ
ejpam-6146	215	9	zeros	zero	NOUN
ejpam-6146	215	10	of	of	ADP
ejpam-6146	215	11	one	one	NUM
ejpam-6146	215	12	solution	solution	NOUN
ejpam-6146	215	13	there	there	PRON
ejpam-6146	215	14	is	be	VERB
ejpam-6146	215	15	exactly	exactly	ADV
ejpam-6146	215	16	one	one	NUM
ejpam-6146	215	17	zero	zero	NUM
ejpam-6146	215	18	of	of	ADP
ejpam-6146	215	19	the	the	DET
ejpam-6146	215	20	other	other	ADJ
ejpam-6146	215	21	solution	solution	NOUN
ejpam-6146	215	22	.	.	PUNCT
ejpam-6146	216	1	in	in	ADP
ejpam-6146	216	2	section-3	section-3	PROPN
ejpam-6146	216	3	,	,	PUNCT
ejpam-6146	216	4	we	we	PRON
ejpam-6146	216	5	shall	shall	AUX
ejpam-6146	216	6	see	see	VERB
ejpam-6146	216	7	that	that	SCONJ
ejpam-6146	216	8	the	the	DET
ejpam-6146	216	9	oscillation	oscillation	NOUN
ejpam-6146	216	10	and	and	CCONJ
ejpam-6146	216	11	this	this	DET
ejpam-6146	216	12	special	special	ADJ
ejpam-6146	216	13	pattern	pattern	NOUN
ejpam-6146	216	14	of	of	ADP
ejpam-6146	216	15	the	the	DET
ejpam-6146	216	16	zeros	zero	NOUN
ejpam-6146	216	17	is	be	AUX
ejpam-6146	216	18	a	a	DET
ejpam-6146	216	19	common	common	ADJ
ejpam-6146	216	20	feature	feature	NOUN
ejpam-6146	216	21	of	of	ADP
ejpam-6146	216	22	many	many	ADJ
ejpam-6146	216	23	second	second	ADJ
ejpam-6146	216	24	-	-	PUNCT
ejpam-6146	216	25	order	order	NOUN
ejpam-6146	216	26	linear	linear	ADJ
ejpam-6146	216	27	differential	differential	NOUN
ejpam-6146	216	28	equations	equation	NOUN
ejpam-6146	216	29	satisfying	satisfy	VERB
ejpam-6146	216	30	certain	certain	ADJ
ejpam-6146	216	31	conditions	condition	NOUN
ejpam-6146	216	32	.	.	PUNCT
ejpam-6146	217	1	in	in	ADP
ejpam-6146	217	2	section-4	section-4	NUM
ejpam-6146	217	3	,	,	PUNCT
ejpam-6146	217	4	we	we	PRON
ejpam-6146	217	5	shall	shall	AUX
ejpam-6146	217	6	discuss	discuss	VERB
ejpam-6146	217	7	the	the	DET
ejpam-6146	217	8	non	non	NOUN
ejpam-6146	217	9	-	-	NOUN
ejpam-6146	217	10	oscillation	oscillation	NOUN
ejpam-6146	217	11	of	of	ADP
ejpam-6146	217	12	the	the	DET
ejpam-6146	217	13	hyperbolic	hyperbolic	ADJ
ejpam-6146	217	14	functions	function	NOUN
ejpam-6146	217	15	from	from	ADP
ejpam-6146	217	16	the	the	DET
ejpam-6146	217	17	equation	equation	NOUN
ejpam-6146	217	18	y′′	y′′	NOUN
ejpam-6146	217	19	−	−	PROPN
ejpam-6146	217	20	y	y	PROPN
ejpam-6146	217	21	=	=	NOUN
ejpam-6146	217	22	0	0	PROPN
ejpam-6146	217	23	.	.	PUNCT
ejpam-6146	218	1	in	in	ADP
ejpam-6146	218	2	section-5	section-5	PROPN
ejpam-6146	218	3	,	,	PUNCT
ejpam-6146	218	4	we	we	PRON
ejpam-6146	218	5	shall	shall	AUX
ejpam-6146	218	6	discuss	discuss	VERB
ejpam-6146	218	7	the	the	DET
ejpam-6146	218	8	oscillation	oscillation	NOUN
ejpam-6146	218	9	of	of	ADP
ejpam-6146	218	10	the	the	DET
ejpam-6146	218	11	bessel	bessel	NOUN
ejpam-6146	218	12	functions	function	NOUN
ejpam-6146	218	13	from	from	ADP
ejpam-6146	218	14	the	the	DET
ejpam-6146	218	15	equation	equation	NOUN
ejpam-6146	218	16	x2y′′	x2y′′	PROPN
ejpam-6146	219	1	+	+	CCONJ
ejpam-6146	219	2	xy′	xy′	PROPN
ejpam-6146	220	1	+	+	CCONJ
ejpam-6146	220	2	(	(	PUNCT
ejpam-6146	220	3	x2	x2	INTJ
ejpam-6146	220	4	−	−	PROPN
ejpam-6146	220	5	p2	p2	PROPN
ejpam-6146	220	6	)	)	PUNCT
ejpam-6146	221	1	y	y	PROPN
ejpam-6146	221	2	=	=	PUNCT
ejpam-6146	221	3	0	0	NUM
ejpam-6146	221	4	.	.	NOUN
ejpam-6146	222	1	3	3	X
ejpam-6146	222	2	.	.	PUNCT
ejpam-6146	222	3	qualitative	qualitative	ADJ
ejpam-6146	222	4	theory	theory	NOUN
ejpam-6146	222	5	of	of	ADP
ejpam-6146	222	6	differential	differential	ADJ
ejpam-6146	222	7	equations	equation	NOUN
ejpam-6146	222	8	differential	differential	NOUN
ejpam-6146	222	9	equations	equation	NOUN
ejpam-6146	222	10	are	be	AUX
ejpam-6146	222	11	a	a	DET
ejpam-6146	222	12	fundamental	fundamental	ADJ
ejpam-6146	222	13	tool	tool	NOUN
ejpam-6146	222	14	to	to	PART
ejpam-6146	222	15	understand	understand	VERB
ejpam-6146	222	16	the	the	DET
ejpam-6146	222	17	physical	physical	ADJ
ejpam-6146	222	18	phenomena	phenomenon	NOUN
ejpam-6146	222	19	.	.	PUNCT
ejpam-6146	223	1	they	they	PRON
ejpam-6146	223	2	provide	provide	VERB
ejpam-6146	223	3	mathematical	mathematical	ADJ
ejpam-6146	223	4	models	model	NOUN
ejpam-6146	223	5	.	.	PUNCT
ejpam-6146	224	1	the	the	DET
ejpam-6146	224	2	area	area	NOUN
ejpam-6146	224	3	of	of	ADP
ejpam-6146	224	4	differential	differential	ADJ
ejpam-6146	224	5	equations	equation	NOUN
ejpam-6146	224	6	began	begin	VERB
ejpam-6146	224	7	with	with	ADP
ejpam-6146	224	8	different	different	ADJ
ejpam-6146	224	9	perspectives	perspective	NOUN
ejpam-6146	224	10	in	in	ADP
ejpam-6146	224	11	the	the	DET
ejpam-6146	224	12	works	work	NOUN
ejpam-6146	224	13	of	of	ADP
ejpam-6146	224	14	isaac	isaac	PROPN
ejpam-6146	224	15	newton	newton	PROPN
ejpam-6146	224	16	(	(	PUNCT
ejpam-6146	224	17	1642	1642	NUM
ejpam-6146	224	18	-	-	SYM
ejpam-6146	224	19	1727	1727	NUM
ejpam-6146	224	20	)	)	PUNCT
ejpam-6146	224	21	and	and	CCONJ
ejpam-6146	224	22	gottfried	gottfried	PROPN
ejpam-6146	224	23	wilhelm	wilhelm	PROPN
ejpam-6146	224	24	von	von	PROPN
ejpam-6146	224	25	leibniz	leibniz	PROPN
ejpam-6146	224	26	(	(	PUNCT
ejpam-6146	224	27	1646	1646	NUM
ejpam-6146	224	28	-	-	SYM
ejpam-6146	224	29	1716	1716	NUM
ejpam-6146	224	30	)	)	PUNCT
ejpam-6146	224	31	.	.	PUNCT
ejpam-6146	225	1	it	it	PRON
ejpam-6146	225	2	was	be	AUX
ejpam-6146	225	3	soon	soon	ADV
ejpam-6146	225	4	realized	realize	VERB
ejpam-6146	225	5	that	that	SCONJ
ejpam-6146	225	6	many	many	ADJ
ejpam-6146	225	7	of	of	ADP
ejpam-6146	225	8	the	the	DET
ejpam-6146	225	9	differential	differential	ADJ
ejpam-6146	225	10	equations	equation	NOUN
ejpam-6146	225	11	can	can	AUX
ejpam-6146	225	12	not	not	PART
ejpam-6146	225	13	be	be	AUX
ejpam-6146	225	14	solved	solve	VERB
ejpam-6146	225	15	exactly	exactly	ADV
ejpam-6146	225	16	.	.	PUNCT
ejpam-6146	226	1	in	in	ADP
ejpam-6146	226	2	fact	fact	NOUN
ejpam-6146	226	3	,	,	PUNCT
ejpam-6146	226	4	joseph	joseph	PROPN
ejpam-6146	226	5	liouville	liouville	PROPN
ejpam-6146	226	6	(	(	PUNCT
ejpam-6146	226	7	1809	1809	NUM
ejpam-6146	226	8	-	-	SYM
ejpam-6146	226	9	1882	1882	NUM
ejpam-6146	226	10	)	)	PUNCT
ejpam-6146	226	11	showed	show	VERB
ejpam-6146	226	12	that	that	SCONJ
ejpam-6146	226	13	it	it	PRON
ejpam-6146	226	14	is	be	AUX
ejpam-6146	226	15	impossible	impossible	ADJ
ejpam-6146	226	16	to	to	PART
ejpam-6146	226	17	obtain	obtain	VERB
ejpam-6146	226	18	solutions	solution	NOUN
ejpam-6146	226	19	of	of	ADP
ejpam-6146	226	20	a	a	DET
ejpam-6146	226	21	certain	certain	ADJ
ejpam-6146	226	22	class	class	NOUN
ejpam-6146	226	23	of	of	ADP
ejpam-6146	226	24	differential	differential	ADJ
ejpam-6146	226	25	equations	equation	NOUN
ejpam-6146	226	26	by	by	ADP
ejpam-6146	226	27	a	a	DET
ejpam-6146	226	28	finite	finite	ADJ
ejpam-6146	226	29	combination	combination	NOUN
ejpam-6146	226	30	of	of	ADP
ejpam-6146	226	31	elementary	elementary	ADJ
ejpam-6146	226	32	functions	function	NOUN
ejpam-6146	226	33	.	.	PUNCT
ejpam-6146	227	1	examples	example	NOUN
ejpam-6146	227	2	of	of	ADP
ejpam-6146	227	3	elementary	elementary	ADJ
ejpam-6146	227	4	functions	function	NOUN
ejpam-6146	227	5	are	be	AUX
ejpam-6146	227	6	the	the	DET
ejpam-6146	227	7	power	power	NOUN
ejpam-6146	227	8	function	function	NOUN
ejpam-6146	227	9	(	(	PUNCT
ejpam-6146	227	10	including	include	VERB
ejpam-6146	227	11	polynomials	polynomial	NOUN
ejpam-6146	227	12	)	)	PUNCT
ejpam-6146	227	13	,	,	PUNCT
ejpam-6146	227	14	exponential	exponential	ADJ
ejpam-6146	227	15	function	function	NOUN
ejpam-6146	227	16	,	,	PUNCT
ejpam-6146	227	17	(	(	PUNCT
ejpam-6146	227	18	including	include	VERB
ejpam-6146	227	19	the	the	DET
ejpam-6146	227	20	six	six	NUM
ejpam-6146	227	21	hyperbolic	hyperbolic	ADJ
ejpam-6146	227	22	functions	function	NOUN
ejpam-6146	227	23	)	)	PUNCT
ejpam-6146	227	24	,	,	PUNCT
ejpam-6146	227	25	logarithm	logarithm	NOUN
ejpam-6146	227	26	(	(	PUNCT
ejpam-6146	227	27	including	include	VERB
ejpam-6146	227	28	the	the	DET
ejpam-6146	227	29	six	six	NUM
ejpam-6146	227	30	inverse	inverse	ADJ
ejpam-6146	227	31	hyperbolic	hyperbolic	ADJ
ejpam-6146	227	32	functions	function	NOUN
ejpam-6146	227	33	)	)	PUNCT
ejpam-6146	227	34	,	,	PUNCT
ejpam-6146	227	35	and	and	CCONJ
ejpam-6146	227	36	the	the	DET
ejpam-6146	227	37	six	six	NUM
ejpam-6146	227	38	trigonometric	trigonometric	ADJ
ejpam-6146	227	39	functions	function	NOUN
ejpam-6146	227	40	along	along	ADP
ejpam-6146	227	41	with	with	ADP
ejpam-6146	227	42	their	their	PRON
ejpam-6146	227	43	inverses	inverse	NOUN
ejpam-6146	227	44	.	.	PUNCT
ejpam-6146	228	1	any	any	DET
ejpam-6146	228	2	function	function	NOUN
ejpam-6146	228	3	built	build	VERB
ejpam-6146	228	4	up	up	ADP
ejpam-6146	228	5	by	by	ADP
ejpam-6146	228	6	taking	take	VERB
ejpam-6146	228	7	sums	sum	NOUN
ejpam-6146	228	8	,	,	PUNCT
ejpam-6146	228	9	products	product	NOUN
ejpam-6146	228	10	,	,	PUNCT
ejpam-6146	228	11	and	and	CCONJ
ejpam-6146	228	12	compositions	composition	NOUN
ejpam-6146	228	13	of	of	ADP
ejpam-6146	228	14	finitely	finitely	ADV
ejpam-6146	228	15	many	many	ADJ
ejpam-6146	228	16	aforementioned	aforementione	VERB
ejpam-6146	228	17	functions	function	NOUN
ejpam-6146	228	18	also	also	ADV
ejpam-6146	228	19	results	result	VERB
ejpam-6146	228	20	in	in	ADP
ejpam-6146	228	21	an	an	DET
ejpam-6146	228	22	elementary	elementary	ADJ
ejpam-6146	228	23	function	function	NOUN
ejpam-6146	228	24	.	.	PUNCT
ejpam-6146	229	1	hence	hence	ADV
ejpam-6146	229	2	,	,	PUNCT
ejpam-6146	229	3	new	new	ADJ
ejpam-6146	229	4	approaches	approach	NOUN
ejpam-6146	229	5	had	have	VERB
ejpam-6146	229	6	to	to	PART
ejpam-6146	229	7	be	be	AUX
ejpam-6146	229	8	developed	develop	VERB
ejpam-6146	229	9	to	to	PART
ejpam-6146	229	10	solve	solve	VERB
ejpam-6146	229	11	differential	differential	ADJ
ejpam-6146	229	12	equations	equation	NOUN
ejpam-6146	229	13	such	such	ADJ
ejpam-6146	229	14	as	as	ADP
ejpam-6146	229	15	the	the	DET
ejpam-6146	229	16	approximate	approximate	ADJ
ejpam-6146	229	17	analytical	analytical	ADJ
ejpam-6146	229	18	solutions	solution	NOUN
ejpam-6146	229	19	and	and	CCONJ
ejpam-6146	229	20	numerical	numerical	ADJ
ejpam-6146	229	21	methods	method	NOUN
ejpam-6146	229	22	(	(	PUNCT
ejpam-6146	229	23	to	to	ADP
ejpam-6146	229	24	a	a	DET
ejpam-6146	229	25	desired	desire	VERB
ejpam-6146	229	26	degree	degree	NOUN
ejpam-6146	229	27	of	of	ADP
ejpam-6146	229	28	accuracy	accuracy	NOUN
ejpam-6146	229	29	)	)	PUNCT
ejpam-6146	229	30	.	.	PUNCT
ejpam-6146	230	1	here	here	ADV
ejpam-6146	230	2	,	,	PUNCT
ejpam-6146	230	3	we	we	PRON
ejpam-6146	230	4	shall	shall	AUX
ejpam-6146	230	5	focus	focus	VERB
ejpam-6146	230	6	on	on	ADP
ejpam-6146	230	7	another	another	DET
ejpam-6146	230	8	method	method	NOUN
ejpam-6146	230	9	used	use	VERB
ejpam-6146	230	10	for	for	ADP
ejpam-6146	230	11	deducing	deduce	VERB
ejpam-6146	230	12	the	the	DET
ejpam-6146	230	13	properties	property	NOUN
ejpam-6146	230	14	of	of	ADP
ejpam-6146	230	15	solutions	solution	NOUN
ejpam-6146	230	16	of	of	ADP
ejpam-6146	230	17	differential	differential	ADJ
ejpam-6146	230	18	equations	equation	NOUN
ejpam-6146	230	19	without	without	ADP
ejpam-6146	230	20	actually	actually	ADV
ejpam-6146	230	21	finding	find	VERB
ejpam-6146	230	22	their	their	PRON
ejpam-6146	230	23	solutions	solution	NOUN
ejpam-6146	230	24	[	[	X
ejpam-6146	230	25	6–9	6–9	NOUN
ejpam-6146	230	26	]	]	PUNCT
ejpam-6146	230	27	.	.	PUNCT
ejpam-6146	231	1	the	the	DET
ejpam-6146	231	2	work	work	NOUN
ejpam-6146	231	3	in	in	ADP
ejpam-6146	231	4	this	this	DET
ejpam-6146	231	5	direction	direction	NOUN
ejpam-6146	231	6	can	can	AUX
ejpam-6146	231	7	be	be	AUX
ejpam-6146	231	8	traced	trace	VERB
ejpam-6146	231	9	back	back	ADV
ejpam-6146	231	10	to	to	ADP
ejpam-6146	231	11	early	early	ADJ
ejpam-6146	231	12	19th	19th	ADJ
ejpam-6146	231	13	century	century	NOUN
ejpam-6146	231	14	,	,	PUNCT
ejpam-6146	231	15	but	but	CCONJ
ejpam-6146	231	16	a	a	DET
ejpam-6146	231	17	significant	significant	ADJ
ejpam-6146	231	18	growth	growth	NOUN
ejpam-6146	231	19	took	take	VERB
ejpam-6146	231	20	place	place	NOUN
ejpam-6146	231	21	in	in	ADP
ejpam-6146	231	22	the	the	DET
ejpam-6146	231	23	works	work	NOUN
ejpam-6146	231	24	of	of	ADP
ejpam-6146	231	25	jules	jules	PROPN
ejpam-6146	231	26	henri	henri	PROPN
ejpam-6146	231	27	poincaré	poincaré	ADJ
ejpam-6146	231	28	(	(	PUNCT
ejpam-6146	231	29	1854	1854	NUM
ejpam-6146	231	30	-	-	SYM
ejpam-6146	231	31	1912	1912	NUM
ejpam-6146	231	32	)	)	PUNCT
ejpam-6146	231	33	and	and	CCONJ
ejpam-6146	231	34	aleksandr	aleksandr	PROPN
ejpam-6146	231	35	mikhailovich	mikhailovich	X
ejpam-6146	231	36	lyapunov	lyapunov	PROPN
ejpam-6146	231	37	(	(	PUNCT
ejpam-6146	231	38	1857	1857	NUM
ejpam-6146	231	39	-	-	SYM
ejpam-6146	231	40	1918	1918	NUM
ejpam-6146	231	41	)	)	PUNCT
ejpam-6146	231	42	.	.	PUNCT
ejpam-6146	232	1	poincaré	poincaré	ADJ
ejpam-6146	232	2	was	be	AUX
ejpam-6146	232	3	then	then	ADV
ejpam-6146	232	4	analyzing	analyze	VERB
ejpam-6146	232	5	the	the	DET
ejpam-6146	232	6	stability	stability	NOUN
ejpam-6146	232	7	of	of	ADP
ejpam-6146	232	8	the	the	DET
ejpam-6146	232	9	solar	solar	ADJ
ejpam-6146	232	10	system	system	NOUN
ejpam-6146	232	11	[	[	X
ejpam-6146	232	12	19	19	NUM
ejpam-6146	232	13	]	]	PUNCT
ejpam-6146	232	14	.	.	PUNCT
ejpam-6146	233	1	this	this	DET
ejpam-6146	233	2	approach	approach	NOUN
ejpam-6146	233	3	of	of	ADP
ejpam-6146	233	4	deducing	deduce	VERB
ejpam-6146	233	5	properties	property	NOUN
ejpam-6146	233	6	and	and	CCONJ
ejpam-6146	233	7	behaviour	behaviour	NOUN
ejpam-6146	233	8	of	of	ADP
ejpam-6146	233	9	the	the	DET
ejpam-6146	233	10	solutions	solution	NOUN
ejpam-6146	233	11	is	be	AUX
ejpam-6146	233	12	known	know	VERB
ejpam-6146	233	13	as	as	ADP
ejpam-6146	233	14	qualitative	qualitative	ADJ
ejpam-6146	233	15	theory	theory	NOUN
ejpam-6146	233	16	of	of	ADP
ejpam-6146	233	17	differential	differential	ADJ
ejpam-6146	233	18	equations	equation	NOUN
ejpam-6146	233	19	.	.	PUNCT
ejpam-6146	234	1	we	we	PRON
ejpam-6146	234	2	have	have	AUX
ejpam-6146	234	3	used	use	VERB
ejpam-6146	234	4	this	this	DET
ejpam-6146	234	5	approach	approach	NOUN
ejpam-6146	234	6	to	to	PART
ejpam-6146	234	7	deduce	deduce	VERB
ejpam-6146	234	8	the	the	DET
ejpam-6146	234	9	trigonometric	trigonometric	PROPN
ejpam-6146	234	10	s.	s.	PROPN
ejpam-6146	234	11	a.	a.	PROPN
ejpam-6146	234	12	khan	khan	PROPN
ejpam-6146	234	13	,	,	PUNCT
ejpam-6146	234	14	m.	m.	NOUN
ejpam-6146	234	15	m.	m.	PROPN
ejpam-6146	234	16	kankarej	kankarej	PROPN
ejpam-6146	234	17	,	,	PUNCT
ejpam-6146	234	18	m.	m.	PROPN
ejpam-6146	234	19	n.	n.	PROPN
ejpam-6146	234	20	i.	i.	PROPN
ejpam-6146	234	21	khan	khan	PROPN
ejpam-6146	234	22	/	/	SYM
ejpam-6146	234	23	eur	eur	PROPN
ejpam-6146	234	24	.	.	PUNCT
ejpam-6146	235	1	j.	j.	PROPN
ejpam-6146	235	2	pure	pure	PROPN
ejpam-6146	235	3	appl	appl	PROPN
ejpam-6146	235	4	.	.	PROPN
ejpam-6146	235	5	math	math	PROPN
ejpam-6146	235	6	,	,	PUNCT
ejpam-6146	235	7	18	18	NUM
ejpam-6146	235	8	(	(	PUNCT
ejpam-6146	235	9	4	4	NUM
ejpam-6146	235	10	)	)	PUNCT
ejpam-6146	235	11	(	(	PUNCT
ejpam-6146	235	12	2025	2025	NUM
ejpam-6146	235	13	)	)	PUNCT
ejpam-6146	235	14	,	,	PUNCT
ejpam-6146	235	15	6146	6146	NUM
ejpam-6146	235	16	8	8	NUM
ejpam-6146	235	17	of	of	ADP
ejpam-6146	235	18	14	14	NUM
ejpam-6146	235	19	functions	function	NOUN
ejpam-6146	235	20	.	.	PUNCT
ejpam-6146	236	1	let	let	VERB
ejpam-6146	236	2	us	we	PRON
ejpam-6146	236	3	consider	consider	VERB
ejpam-6146	236	4	the	the	DET
ejpam-6146	236	5	second	second	ADJ
ejpam-6146	236	6	-	-	PUNCT
ejpam-6146	236	7	order	order	NOUN
ejpam-6146	236	8	linear	linear	ADJ
ejpam-6146	236	9	homogenous	homogenous	ADJ
ejpam-6146	236	10	differential	differential	NOUN
ejpam-6146	236	11	equation	equation	NOUN
ejpam-6146	236	12	y′′	y′′	NOUN
ejpam-6146	236	13	+	+	CCONJ
ejpam-6146	236	14	p	p	X
ejpam-6146	236	15	(	(	PUNCT
ejpam-6146	236	16	x)y′	x)y′	PROPN
ejpam-6146	237	1	+	+	NOUN
ejpam-6146	237	2	q(x)y	q(x)y	X
ejpam-6146	237	3	=	=	SYM
ejpam-6146	237	4	0	0	NUM
ejpam-6146	237	5	,	,	PUNCT
ejpam-6146	237	6	(	(	PUNCT
ejpam-6146	237	7	18	18	NUM
ejpam-6146	237	8	)	)	PUNCT
ejpam-6146	237	9	where	where	SCONJ
ejpam-6146	237	10	p	p	NOUN
ejpam-6146	237	11	(	(	PUNCT
ejpam-6146	237	12	x	x	NOUN
ejpam-6146	237	13	)	)	PUNCT
ejpam-6146	237	14	and	and	CCONJ
ejpam-6146	237	15	q(x	q(x	PROPN
ejpam-6146	237	16	)	)	PUNCT
ejpam-6146	237	17	are	be	AUX
ejpam-6146	237	18	continuous	continuous	ADJ
ejpam-6146	237	19	.	.	PUNCT
ejpam-6146	238	1	the	the	DET
ejpam-6146	238	2	behaviour	behaviour	NOUN
ejpam-6146	238	3	of	of	ADP
ejpam-6146	238	4	the	the	DET
ejpam-6146	238	5	zeros	zero	NOUN
ejpam-6146	238	6	(	(	PUNCT
ejpam-6146	238	7	roots	root	NOUN
ejpam-6146	238	8	)	)	PUNCT
ejpam-6146	238	9	of	of	ADP
ejpam-6146	238	10	the	the	DET
ejpam-6146	238	11	two	two	NUM
ejpam-6146	238	12	linearly	linearly	ADV
ejpam-6146	238	13	independent	independent	ADJ
ejpam-6146	238	14	solutions	solution	NOUN
ejpam-6146	238	15	is	be	AUX
ejpam-6146	238	16	summarised	summarise	VERB
ejpam-6146	238	17	in	in	ADP
ejpam-6146	238	18	the	the	DET
ejpam-6146	238	19	following	following	NOUN
ejpam-6146	238	20	theorem	theorem	NOUN
ejpam-6146	238	21	due	due	ADP
ejpam-6146	238	22	to	to	ADP
ejpam-6146	238	23	jacques	jacques	PROPN
ejpam-6146	238	24	charles	charles	PROPN
ejpam-6146	238	25	françois	françois	PROPN
ejpam-6146	238	26	sturm	sturm	X
ejpam-6146	238	27	(	(	PUNCT
ejpam-6146	238	28	1803	1803	NUM
ejpam-6146	238	29	-	-	SYM
ejpam-6146	238	30	1855	1855	NUM
ejpam-6146	238	31	)	)	PUNCT
ejpam-6146	238	32	.	.	PUNCT
ejpam-6146	239	1	theorem	theorem	NOUN
ejpam-6146	239	2	2	2	NUM
ejpam-6146	239	3	.	.	PUNCT
ejpam-6146	239	4	sturm	sturm	PROPN
ejpam-6146	239	5	separation	separation	NOUN
ejpam-6146	239	6	theorem	theorem	VERB
ejpam-6146	239	7	:	:	PUNCT
ejpam-6146	239	8	if	if	SCONJ
ejpam-6146	239	9	y1(x	y1(x	NOUN
ejpam-6146	239	10	)	)	PUNCT
ejpam-6146	239	11	and	and	CCONJ
ejpam-6146	239	12	y2(x	y2(x	NOUN
ejpam-6146	239	13	)	)	PUNCT
ejpam-6146	239	14	are	be	AUX
ejpam-6146	239	15	two	two	NUM
ejpam-6146	239	16	linearly	linearly	ADV
ejpam-6146	239	17	independent	independent	ADJ
ejpam-6146	239	18	solutions	solution	NOUN
ejpam-6146	239	19	of	of	ADP
ejpam-6146	239	20	y′′	y′′	PROPN
ejpam-6146	240	1	+	+	CCONJ
ejpam-6146	241	1	p	p	X
ejpam-6146	241	2	(	(	PUNCT
ejpam-6146	241	3	x)y′	x)y′	PROPN
ejpam-6146	241	4	+	+	NOUN
ejpam-6146	241	5	q(x)y	q(x)y	X
ejpam-6146	241	6	=	=	SYM
ejpam-6146	241	7	0	0	NUM
ejpam-6146	241	8	,	,	PUNCT
ejpam-6146	241	9	then	then	ADV
ejpam-6146	241	10	,	,	PUNCT
ejpam-6146	241	11	the	the	DET
ejpam-6146	241	12	zeros	zero	NOUN
ejpam-6146	241	13	of	of	ADP
ejpam-6146	241	14	these	these	DET
ejpam-6146	241	15	functions	function	NOUN
ejpam-6146	241	16	are	be	AUX
ejpam-6146	241	17	distinct	distinct	ADJ
ejpam-6146	241	18	and	and	CCONJ
ejpam-6146	241	19	occur	occur	VERB
ejpam-6146	241	20	alternately	alternately	ADV
ejpam-6146	241	21	—	—	PUNCT
ejpam-6146	241	22	in	in	ADP
ejpam-6146	241	23	the	the	DET
ejpam-6146	241	24	sense	sense	NOUN
ejpam-6146	241	25	that	that	PRON
ejpam-6146	241	26	y1(x	y1(x	NOUN
ejpam-6146	241	27	)	)	PUNCT
ejpam-6146	241	28	vanishes	vanish	VERB
ejpam-6146	241	29	exactly	exactly	ADV
ejpam-6146	241	30	once	once	ADV
ejpam-6146	241	31	between	between	ADP
ejpam-6146	241	32	any	any	DET
ejpam-6146	241	33	two	two	NUM
ejpam-6146	241	34	successive	successive	ADJ
ejpam-6146	241	35	zeros	zero	NOUN
ejpam-6146	241	36	of	of	ADP
ejpam-6146	241	37	y2(x	y2(x	PROPN
ejpam-6146	241	38	)	)	PUNCT
ejpam-6146	241	39	,	,	PUNCT
ejpam-6146	241	40	and	and	CCONJ
ejpam-6146	241	41	conversely	conversely	ADV
ejpam-6146	241	42	.	.	PUNCT
ejpam-6146	242	1	proof	proof	NOUN
ejpam-6146	242	2	.	.	PUNCT
ejpam-6146	243	1	let	let	VERB
ejpam-6146	243	2	x1	x1	PROPN
ejpam-6146	243	3	and	and	CCONJ
ejpam-6146	243	4	x2	x2	PROPN
ejpam-6146	243	5	be	be	VERB
ejpam-6146	243	6	the	the	DET
ejpam-6146	243	7	two	two	NUM
ejpam-6146	243	8	successive	successive	ADJ
ejpam-6146	243	9	zeros	zero	NOUN
ejpam-6146	243	10	of	of	ADP
ejpam-6146	243	11	y2(x	y2(x	PROPN
ejpam-6146	243	12	)	)	PUNCT
ejpam-6146	243	13	.	.	PUNCT
ejpam-6146	244	1	that	that	PRON
ejpam-6146	244	2	is	be	AUX
ejpam-6146	244	3	,	,	PUNCT
ejpam-6146	244	4	y2(x1	y2(x1	NUM
ejpam-6146	244	5	)	)	PUNCT
ejpam-6146	244	6	=	=	SYM
ejpam-6146	244	7	y2(x2	y2(x2	NOUN
ejpam-6146	244	8	)	)	PUNCT
ejpam-6146	244	9	=	=	SYM
ejpam-6146	244	10	0	0	NUM
ejpam-6146	244	11	and	and	CCONJ
ejpam-6146	244	12	y2(x	y2(x	PROPN
ejpam-6146	244	13	)	)	PUNCT
ejpam-6146	244	14	̸=	̸=	NOUN
ejpam-6146	244	15	0	0	NUM
ejpam-6146	245	1	on	on	ADP
ejpam-6146	245	2	(	(	PUNCT
ejpam-6146	245	3	x1	x1	INTJ
ejpam-6146	245	4	,	,	PUNCT
ejpam-6146	245	5	x2	x2	PROPN
ejpam-6146	245	6	)	)	PUNCT
ejpam-6146	245	7	.	.	PUNCT
ejpam-6146	246	1	since	since	SCONJ
ejpam-6146	246	2	,	,	PUNCT
ejpam-6146	246	3	y1(x	y1(x	NOUN
ejpam-6146	246	4	)	)	PUNCT
ejpam-6146	246	5	and	and	CCONJ
ejpam-6146	246	6	y2(x	y2(x	NOUN
ejpam-6146	246	7	)	)	PUNCT
ejpam-6146	246	8	are	be	AUX
ejpam-6146	246	9	linearly	linearly	ADV
ejpam-6146	246	10	independent	independent	ADJ
ejpam-6146	246	11	,	,	PUNCT
ejpam-6146	246	12	their	their	PRON
ejpam-6146	246	13	wronskian	wronskian	NOUN
ejpam-6146	246	14	w	w	PROPN
ejpam-6146	246	15	(	(	PUNCT
ejpam-6146	246	16	x	x	NOUN
ejpam-6146	246	17	)	)	PUNCT
ejpam-6146	246	18	=	=	PUNCT
ejpam-6146	247	1	w	w	PROPN
ejpam-6146	248	1	[	[	X
ejpam-6146	248	2	y1(x	y1(x	NOUN
ejpam-6146	248	3	)	)	PUNCT
ejpam-6146	248	4	,	,	PUNCT
ejpam-6146	248	5	y2(x	y2(x	PROPN
ejpam-6146	248	6	)	)	PUNCT
ejpam-6146	248	7	]	]	PUNCT
ejpam-6146	249	1	=	=	PUNCT
ejpam-6146	249	2	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6146	249	3	y1(x	y1(x	NOUN
ejpam-6146	249	4	)	)	PUNCT
ejpam-6146	249	5	y2(x	y2(x	NOUN
ejpam-6146	249	6	)	)	PUNCT
ejpam-6146	249	7	y′1(x	y′1(x	NOUN
ejpam-6146	249	8	)	)	PUNCT
ejpam-6146	249	9	y′2(x	y′2(x	PROPN
ejpam-6146	249	10	)	)	PUNCT
ejpam-6146	249	11	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6146	249	12	=	=	PUNCT
ejpam-6146	250	1	y1(x)y	y1(x)y	NUM
ejpam-6146	250	2	′	′	NUM
ejpam-6146	250	3	2(x)−	2(x)−	NUM
ejpam-6146	251	1	y′1(x)y2(x	y′1(x)y2(x	NOUN
ejpam-6146	251	2	)	)	PUNCT
ejpam-6146	251	3	(	(	PUNCT
ejpam-6146	251	4	19	19	NUM
ejpam-6146	251	5	)	)	PUNCT
ejpam-6146	251	6	is	be	AUX
ejpam-6146	251	7	never	never	ADV
ejpam-6146	251	8	zero	zero	NUM
ejpam-6146	251	9	.	.	PUNCT
ejpam-6146	252	1	let	let	VERB
ejpam-6146	252	2	us	we	PRON
ejpam-6146	252	3	assume	assume	VERB
ejpam-6146	252	4	that	that	SCONJ
ejpam-6146	252	5	w	w	PROPN
ejpam-6146	252	6	(	(	PUNCT
ejpam-6146	252	7	x	x	X
ejpam-6146	252	8	)	)	PUNCT
ejpam-6146	252	9	>	>	X
ejpam-6146	252	10	0	0	PUNCT
ejpam-6146	253	1	for	for	ADP
ejpam-6146	253	2	x	x	PROPN
ejpam-6146	253	3	∈	∈	PROPN
ejpam-6146	253	4	(	(	PUNCT
ejpam-6146	253	5	x1	x1	PROPN
ejpam-6146	253	6	,	,	PUNCT
ejpam-6146	253	7	x2	x2	PROPN
ejpam-6146	253	8	)	)	PUNCT
ejpam-6146	253	9	.	.	PUNCT
ejpam-6146	254	1	then	then	ADV
ejpam-6146	254	2	w	w	PROPN
ejpam-6146	254	3	(	(	PUNCT
ejpam-6146	254	4	xj	xj	PROPN
ejpam-6146	254	5	)	)	PUNCT
ejpam-6146	254	6	=	=	SYM
ejpam-6146	254	7	y1(xj)y	y1(xj)y	PROPN
ejpam-6146	254	8	′	′	NUM
ejpam-6146	254	9	2(xj	2(xj	ADV
ejpam-6146	254	10	)	)	PUNCT
ejpam-6146	254	11	>	>	X
ejpam-6146	254	12	0	0	PUNCT
ejpam-6146	254	13	,	,	PUNCT
ejpam-6146	254	14	j	j	PROPN
ejpam-6146	254	15	=	=	SYM
ejpam-6146	254	16	1	1	NUM
ejpam-6146	254	17	,	,	PUNCT
ejpam-6146	254	18	2	2	NUM
ejpam-6146	254	19	.	.	PUNCT
ejpam-6146	255	1	(	(	PUNCT
ejpam-6146	255	2	20	20	NUM
ejpam-6146	255	3	)	)	PUNCT
ejpam-6146	255	4	since	since	SCONJ
ejpam-6146	255	5	,	,	PUNCT
ejpam-6146	255	6	x1	x1	PROPN
ejpam-6146	255	7	and	and	CCONJ
ejpam-6146	255	8	x2	x2	PROPN
ejpam-6146	255	9	are	be	AUX
ejpam-6146	255	10	successive	successive	ADJ
ejpam-6146	255	11	zeros	zero	NOUN
ejpam-6146	255	12	of	of	ADP
ejpam-6146	255	13	y2(x	y2(x	PROPN
ejpam-6146	255	14	)	)	PUNCT
ejpam-6146	255	15	,	,	PUNCT
ejpam-6146	255	16	it	it	PRON
ejpam-6146	255	17	must	must	AUX
ejpam-6146	255	18	hold	hold	VERB
ejpam-6146	255	19	that	that	DET
ejpam-6146	255	20	y′2(x1	y′2(x1	NOUN
ejpam-6146	255	21	)	)	PUNCT
ejpam-6146	255	22	and	and	CCONJ
ejpam-6146	255	23	y′2(x2	y′2(x2	NOUN
ejpam-6146	255	24	)	)	PUNCT
ejpam-6146	255	25	have	have	VERB
ejpam-6146	255	26	opposite	opposite	ADJ
ejpam-6146	255	27	signs	sign	NOUN
ejpam-6146	255	28	.	.	PUNCT
ejpam-6146	256	1	if	if	SCONJ
ejpam-6146	256	2	y2(x	y2(x	NOUN
ejpam-6146	256	3	)	)	PUNCT
ejpam-6146	256	4	increases	increase	VERB
ejpam-6146	256	5	at	at	ADP
ejpam-6146	256	6	x1	x1	NOUN
ejpam-6146	256	7	then	then	ADV
ejpam-6146	256	8	it	it	PRON
ejpam-6146	256	9	must	must	AUX
ejpam-6146	256	10	decrease	decrease	VERB
ejpam-6146	256	11	at	at	ADP
ejpam-6146	256	12	x2	x2	PROPN
ejpam-6146	256	13	and	and	CCONJ
ejpam-6146	256	14	vice	vice	ADV
ejpam-6146	256	15	versa	versa	ADV
ejpam-6146	256	16	.	.	PUNCT
ejpam-6146	257	1	this	this	PRON
ejpam-6146	257	2	implies	imply	VERB
ejpam-6146	257	3	that	that	SCONJ
ejpam-6146	257	4	y1(x1	y1(x1	NUM
ejpam-6146	257	5	)	)	PUNCT
ejpam-6146	257	6	and	and	CCONJ
ejpam-6146	257	7	y1(x2	y1(x2	NOUN
ejpam-6146	257	8	)	)	PUNCT
ejpam-6146	257	9	must	must	AUX
ejpam-6146	257	10	have	have	VERB
ejpam-6146	257	11	opposite	opposite	ADJ
ejpam-6146	257	12	signs	sign	NOUN
ejpam-6146	257	13	.	.	PUNCT
ejpam-6146	258	1	from	from	ADP
ejpam-6146	258	2	the	the	DET
ejpam-6146	258	3	intermediate	intermediate	ADJ
ejpam-6146	258	4	value	value	NOUN
ejpam-6146	258	5	theorem	theorem	NOUN
ejpam-6146	258	6	,	,	PUNCT
ejpam-6146	258	7	y1(x	y1(x	NOUN
ejpam-6146	258	8	)	)	PUNCT
ejpam-6146	258	9	must	must	AUX
ejpam-6146	258	10	vanish	vanish	VERB
ejpam-6146	258	11	between	between	ADP
ejpam-6146	258	12	x1	x1	PROPN
ejpam-6146	258	13	and	and	CCONJ
ejpam-6146	258	14	x2	x2	PROPN
ejpam-6146	258	15	.	.	PUNCT
ejpam-6146	259	1	same	same	ADJ
ejpam-6146	259	2	arguments	argument	NOUN
ejpam-6146	259	3	work	work	VERB
ejpam-6146	259	4	for	for	ADP
ejpam-6146	259	5	the	the	DET
ejpam-6146	259	6	case	case	NOUN
ejpam-6146	259	7	,	,	PUNCT
ejpam-6146	259	8	w	w	PROPN
ejpam-6146	259	9	(	(	PUNCT
ejpam-6146	259	10	x	x	X
ejpam-6146	259	11	)	)	PUNCT
ejpam-6146	259	12	<	<	X
ejpam-6146	259	13	0	0	PUNCT
ejpam-6146	259	14	for	for	ADP
ejpam-6146	259	15	x	x	PROPN
ejpam-6146	259	16	∈	∈	PROPN
ejpam-6146	259	17	(	(	PUNCT
ejpam-6146	259	18	x1	x1	PROPN
ejpam-6146	259	19	,	,	PUNCT
ejpam-6146	259	20	x2	x2	PROPN
ejpam-6146	259	21	)	)	PUNCT
ejpam-6146	259	22	.	.	PUNCT
ejpam-6146	260	1	this	this	PRON
ejpam-6146	260	2	completes	complete	VERB
ejpam-6146	260	3	the	the	DET
ejpam-6146	260	4	proof	proof	NOUN
ejpam-6146	260	5	.	.	PUNCT
ejpam-6146	261	1	the	the	DET
ejpam-6146	261	2	roles	role	NOUN
ejpam-6146	261	3	of	of	ADP
ejpam-6146	261	4	y1(x	y1(x	NOUN
ejpam-6146	261	5	)	)	PUNCT
ejpam-6146	261	6	and	and	CCONJ
ejpam-6146	261	7	y2(x	y2(x	NOUN
ejpam-6146	261	8	)	)	PUNCT
ejpam-6146	261	9	can	can	AUX
ejpam-6146	261	10	be	be	AUX
ejpam-6146	261	11	interchanged	interchange	VERB
ejpam-6146	261	12	.	.	PUNCT
ejpam-6146	262	1	between	between	ADP
ejpam-6146	262	2	any	any	DET
ejpam-6146	262	3	two	two	NUM
ejpam-6146	262	4	consecutive	consecutive	ADJ
ejpam-6146	262	5	zeros	zero	NOUN
ejpam-6146	262	6	of	of	ADP
ejpam-6146	262	7	y1(x	y1(x	NOUN
ejpam-6146	262	8	)	)	PUNCT
ejpam-6146	262	9	there	there	PRON
ejpam-6146	262	10	is	be	VERB
ejpam-6146	262	11	only	only	ADV
ejpam-6146	262	12	one	one	NUM
ejpam-6146	262	13	zero	zero	NUM
ejpam-6146	262	14	of	of	ADP
ejpam-6146	262	15	y2(x	y2(x	PROPN
ejpam-6146	262	16	)	)	PUNCT
ejpam-6146	262	17	and	and	CCONJ
ejpam-6146	262	18	vice	vice	ADV
ejpam-6146	262	19	versa	versa	ADV
ejpam-6146	262	20	.	.	PUNCT
ejpam-6146	263	1	analogous	analogous	ADJ
ejpam-6146	263	2	theorems	theorem	NOUN
ejpam-6146	263	3	exist	exist	VERB
ejpam-6146	263	4	for	for	ADP
ejpam-6146	263	5	the	the	DET
ejpam-6146	263	6	eigenvalues	eigenvalue	NOUN
ejpam-6146	263	7	of	of	ADP
ejpam-6146	263	8	matrices	matrix	NOUN
ejpam-6146	263	9	,	,	PUNCT
ejpam-6146	263	10	which	which	PRON
ejpam-6146	263	11	we	we	PRON
ejpam-6146	263	12	illustrate	illustrate	VERB
ejpam-6146	263	13	below	below	ADV
ejpam-6146	263	14	.	.	PUNCT
ejpam-6146	264	1	so	so	ADV
ejpam-6146	264	2	,	,	PUNCT
ejpam-6146	264	3	also	also	ADV
ejpam-6146	264	4	for	for	ADP
ejpam-6146	264	5	the	the	DET
ejpam-6146	264	6	orthogonal	orthogonal	ADJ
ejpam-6146	264	7	functions	function	NOUN
ejpam-6146	264	8	,	,	PUNCT
ejpam-6146	264	9	which	which	PRON
ejpam-6146	264	10	we	we	PRON
ejpam-6146	264	11	shall	shall	AUX
ejpam-6146	264	12	see	see	VERB
ejpam-6146	264	13	in	in	ADP
ejpam-6146	264	14	some	some	DET
ejpam-6146	264	15	detail	detail	NOUN
ejpam-6146	264	16	in	in	ADP
ejpam-6146	264	17	section-6	section-6	PROPN
ejpam-6146	264	18	.	.	PUNCT
ejpam-6146	265	1	a	a	DET
ejpam-6146	265	2	symmetric	symmetric	ADJ
ejpam-6146	265	3	matrix	matrix	NOUN
ejpam-6146	265	4	is	be	AUX
ejpam-6146	265	5	a	a	DET
ejpam-6146	265	6	square	square	ADJ
ejpam-6146	265	7	matrix	matrix	NOUN
ejpam-6146	265	8	having	have	VERB
ejpam-6146	265	9	the	the	DET
ejpam-6146	265	10	property	property	NOUN
ejpam-6146	265	11	,	,	PUNCT
ejpam-6146	265	12	at	at	ADP
ejpam-6146	265	13	=	=	VERB
ejpam-6146	265	14	a.	a.	NOUN
ejpam-6146	265	15	in	in	ADP
ejpam-6146	265	16	other	other	ADJ
ejpam-6146	265	17	words	word	NOUN
ejpam-6146	265	18	the	the	DET
ejpam-6146	265	19	entries	entry	NOUN
ejpam-6146	265	20	satisfy	satisfy	VERB
ejpam-6146	265	21	the	the	DET
ejpam-6146	265	22	relation	relation	NOUN
ejpam-6146	265	23	aij	aij	PROPN
ejpam-6146	265	24	=	=	PROPN
ejpam-6146	265	25	aji	aji	PROPN
ejpam-6146	265	26	.	.	PUNCT
ejpam-6146	266	1	let	let	VERB
ejpam-6146	266	2	ar	ar	PROPN
ejpam-6146	266	3	=	=	PUNCT
ejpam-6146	266	4	aij	aij	PROPN
ejpam-6146	266	5	be	be	VERB
ejpam-6146	266	6	a	a	DET
ejpam-6146	266	7	sequence	sequence	NOUN
ejpam-6146	266	8	of	of	ADP
ejpam-6146	266	9	n	n	CCONJ
ejpam-6146	266	10	symmetric	symmetric	ADJ
ejpam-6146	266	11	matrices	matrix	NOUN
ejpam-6146	266	12	of	of	ADP
ejpam-6146	266	13	increasing	increase	VERB
ejpam-6146	266	14	order	order	NOUN
ejpam-6146	266	15	with	with	ADP
ejpam-6146	266	16	i	i	PRON
ejpam-6146	266	17	,	,	PUNCT
ejpam-6146	266	18	j	j	PROPN
ejpam-6146	266	19	=	=	SYM
ejpam-6146	266	20	1	1	NUM
ejpam-6146	266	21	,	,	PUNCT
ejpam-6146	266	22	2	2	NUM
ejpam-6146	266	23	,	,	PUNCT
ejpam-6146	266	24	3	3	NUM
ejpam-6146	266	25	,	,	PUNCT
ejpam-6146	266	26	.	.	PUNCT
ejpam-6146	266	27	.	.	PUNCT
ejpam-6146	267	1	.	.	PUNCT
ejpam-6146	268	1	,	,	PUNCT
ejpam-6146	268	2	r	r	NOUN
ejpam-6146	268	3	and	and	CCONJ
ejpam-6146	268	4	r	r	NOUN
ejpam-6146	268	5	=	=	SYM
ejpam-6146	268	6	1	1	NUM
ejpam-6146	268	7	,	,	PUNCT
ejpam-6146	268	8	2	2	NUM
ejpam-6146	268	9	,	,	PUNCT
ejpam-6146	268	10	3	3	NUM
ejpam-6146	268	11	,	,	PUNCT
ejpam-6146	268	12	.	.	PUNCT
ejpam-6146	268	13	.	.	PUNCT
ejpam-6146	268	14	.	.	PUNCT
ejpam-6146	269	1	n	n	X
ejpam-6146	269	2	.	.	PUNCT
ejpam-6146	270	1	let	let	VERB
ejpam-6146	270	2	λk(ar	λk(ar	NOUN
ejpam-6146	270	3	)	)	PUNCT
ejpam-6146	270	4	be	be	VERB
ejpam-6146	270	5	the	the	DET
ejpam-6146	270	6	k	k	NOUN
ejpam-6146	270	7	-	-	PUNCT
ejpam-6146	270	8	th	th	VERB
ejpam-6146	270	9	eigenvalue	eigenvalue	NOUN
ejpam-6146	270	10	of	of	ADP
ejpam-6146	270	11	ar	ar	PROPN
ejpam-6146	270	12	for	for	ADP
ejpam-6146	270	13	k	k	PROPN
ejpam-6146	270	14	=	=	SYM
ejpam-6146	270	15	1	1	NUM
ejpam-6146	270	16	,	,	PUNCT
ejpam-6146	270	17	2	2	NUM
ejpam-6146	270	18	,	,	PUNCT
ejpam-6146	270	19	3	3	NUM
ejpam-6146	270	20	,	,	PUNCT
ejpam-6146	270	21	.	.	PUNCT
ejpam-6146	270	22	.	.	PUNCT
ejpam-6146	271	1	.	.	PUNCT
ejpam-6146	272	1	,	,	PUNCT
ejpam-6146	272	2	r	r	NOUN
ejpam-6146	272	3	with	with	ADP
ejpam-6146	272	4	the	the	DET
ejpam-6146	272	5	ordering	order	VERB
ejpam-6146	272	6	λ1(ar	λ1(ar	PROPN
ejpam-6146	272	7	)	)	PUNCT
ejpam-6146	272	8	≥	≥	NOUN
ejpam-6146	272	9	λ2(ar	λ2(ar	NOUN
ejpam-6146	272	10	)	)	PUNCT
ejpam-6146	272	11	≥	≥	NOUN
ejpam-6146	272	12	λ3(ar	λ3(ar	NOUN
ejpam-6146	272	13	)	)	PUNCT
ejpam-6146	272	14	≥	≥	NOUN
ejpam-6146	272	15	.	.	PUNCT
ejpam-6146	272	16	.	.	PUNCT
ejpam-6146	272	17	.	.	PUNCT
ejpam-6146	273	1	≥	≥	PROPN
ejpam-6146	273	2	λr(ar	λr(ar	PROPN
ejpam-6146	273	3	)	)	PUNCT
ejpam-6146	273	4	.	.	PUNCT
ejpam-6146	274	1	then	then	ADV
ejpam-6146	274	2	λk+1(ai+1	λk+1(ai+1	NOUN
ejpam-6146	274	3	)	)	PUNCT
ejpam-6146	274	4	≤	≤	NUM
ejpam-6146	274	5	λk(ai	λk(ai	PROPN
ejpam-6146	274	6	)	)	PUNCT
ejpam-6146	274	7	≤	≤	NUM
ejpam-6146	274	8	λk(ai+1	λk(ai+1	NOUN
ejpam-6146	274	9	)	)	PUNCT
ejpam-6146	274	10	.	.	PUNCT
ejpam-6146	275	1	s.	s.	PROPN
ejpam-6146	275	2	a.	a.	PROPN
ejpam-6146	275	3	khan	khan	PROPN
ejpam-6146	275	4	,	,	PUNCT
ejpam-6146	275	5	m.	m.	NOUN
ejpam-6146	275	6	m.	m.	PROPN
ejpam-6146	275	7	kankarej	kankarej	PROPN
ejpam-6146	275	8	,	,	PUNCT
ejpam-6146	275	9	m.	m.	PROPN
ejpam-6146	275	10	n.	n.	PROPN
ejpam-6146	275	11	i.	i.	PROPN
ejpam-6146	275	12	khan	khan	PROPN
ejpam-6146	275	13	/	/	SYM
ejpam-6146	275	14	eur	eur	PROPN
ejpam-6146	275	15	.	.	PUNCT
ejpam-6146	276	1	j.	j.	PROPN
ejpam-6146	276	2	pure	pure	PROPN
ejpam-6146	276	3	appl	appl	PROPN
ejpam-6146	276	4	.	.	PROPN
ejpam-6146	276	5	math	math	PROPN
ejpam-6146	276	6	,	,	PUNCT
ejpam-6146	276	7	18	18	NUM
ejpam-6146	276	8	(	(	PUNCT
ejpam-6146	276	9	4	4	NUM
ejpam-6146	276	10	)	)	PUNCT
ejpam-6146	276	11	(	(	PUNCT
ejpam-6146	276	12	2025	2025	NUM
ejpam-6146	276	13	)	)	PUNCT
ejpam-6146	276	14	,	,	PUNCT
ejpam-6146	276	15	6146	6146	NUM
ejpam-6146	276	16	9	9	NUM
ejpam-6146	276	17	of	of	ADP
ejpam-6146	276	18	14	14	NUM
ejpam-6146	276	19	let	let	VERB
ejpam-6146	276	20	us	we	PRON
ejpam-6146	276	21	examine	examine	VERB
ejpam-6146	276	22	the	the	DET
ejpam-6146	276	23	equation	equation	NOUN
ejpam-6146	276	24	y′′	y′′	PROPN
ejpam-6146	276	25	+	+	PROPN
ejpam-6146	276	26	ay	ay	PROPN
ejpam-6146	276	27	=	=	SYM
ejpam-6146	276	28	0	0	NUM
ejpam-6146	276	29	,	,	PUNCT
ejpam-6146	276	30	where	where	SCONJ
ejpam-6146	276	31	a	a	PRON
ejpam-6146	276	32	is	be	AUX
ejpam-6146	276	33	any	any	DET
ejpam-6146	276	34	real	real	ADJ
ejpam-6146	276	35	number	number	NOUN
ejpam-6146	276	36	.	.	PUNCT
ejpam-6146	277	1	for	for	ADP
ejpam-6146	277	2	a	a	DET
ejpam-6146	277	3	>	>	X
ejpam-6146	277	4	0	0	NUM
ejpam-6146	277	5	,	,	PUNCT
ejpam-6146	277	6	the	the	DET
ejpam-6146	277	7	two	two	NUM
ejpam-6146	277	8	solutions	solution	NOUN
ejpam-6146	277	9	,	,	PUNCT
ejpam-6146	277	10	y1	y1	NOUN
ejpam-6146	277	11	=	=	PUNCT
ejpam-6146	277	12	sin	sin	NOUN
ejpam-6146	277	13	(	(	PUNCT
ejpam-6146	277	14	√	√	NUM
ejpam-6146	277	15	ax	ax	NOUN
ejpam-6146	277	16	)	)	PUNCT
ejpam-6146	277	17	and	and	CCONJ
ejpam-6146	277	18	y2	y2	PROPN
ejpam-6146	277	19	cos	cos	PROPN
ejpam-6146	277	20	(	(	PUNCT
ejpam-6146	277	21	√	√	NUM
ejpam-6146	277	22	ax	ax	NOUN
ejpam-6146	277	23	)	)	PUNCT
ejpam-6146	277	24	are	be	AUX
ejpam-6146	277	25	oscillatory	oscillatory	ADJ
ejpam-6146	277	26	and	and	CCONJ
ejpam-6146	277	27	we	we	PRON
ejpam-6146	277	28	have	have	AUX
ejpam-6146	277	29	seen	see	VERB
ejpam-6146	277	30	them	they	PRON
ejpam-6146	277	31	in	in	ADP
ejpam-6146	277	32	detail	detail	NOUN
ejpam-6146	277	33	with	with	ADP
ejpam-6146	277	34	a	a	DET
ejpam-6146	277	35	=	=	SYM
ejpam-6146	277	36	1	1	NUM
ejpam-6146	277	37	,	,	PUNCT
ejpam-6146	277	38	in	in	ADP
ejpam-6146	277	39	section-2	section-2	NUM
ejpam-6146	277	40	.	.	PUNCT
ejpam-6146	278	1	we	we	PRON
ejpam-6146	278	2	note	note	VERB
ejpam-6146	278	3	,	,	PUNCT
ejpam-6146	278	4	that	that	SCONJ
ejpam-6146	278	5	both	both	CCONJ
ejpam-6146	278	6	the	the	DET
ejpam-6146	278	7	solutions	solution	NOUN
ejpam-6146	278	8	have	have	AUX
ejpam-6146	278	9	infinite	infinite	ADJ
ejpam-6146	278	10	zeros	zero	NOUN
ejpam-6146	278	11	.	.	PUNCT
ejpam-6146	279	1	for	for	ADP
ejpam-6146	279	2	a	a	DET
ejpam-6146	279	3	<	<	X
ejpam-6146	279	4	0	0	NUM
ejpam-6146	279	5	,	,	PUNCT
ejpam-6146	279	6	the	the	DET
ejpam-6146	279	7	two	two	NUM
ejpam-6146	279	8	solutions	solution	NOUN
ejpam-6146	279	9	,	,	PUNCT
ejpam-6146	279	10	y1	y1	NOUN
ejpam-6146	279	11	=	=	PUNCT
ejpam-6146	279	12	e	e	NOUN
ejpam-6146	279	13	√	√	NOUN
ejpam-6146	279	14	ax	ax	NOUN
ejpam-6146	279	15	and	and	CCONJ
ejpam-6146	279	16	y2	y2	NOUN
ejpam-6146	279	17	=	=	SYM
ejpam-6146	279	18	e−	e−	ADJ
ejpam-6146	279	19	√	√	ADP
ejpam-6146	279	20	ax	ax	NOUN
ejpam-6146	279	21	or	or	CCONJ
ejpam-6146	279	22	equivalently	equivalently	ADV
ejpam-6146	279	23	y1	y1	NOUN
ejpam-6146	279	24	=	=	SYM
ejpam-6146	279	25	sinh	sinh	NOUN
ejpam-6146	279	26	(	(	PUNCT
ejpam-6146	279	27	√	√	NOUN
ejpam-6146	279	28	ax	ax	NOUN
ejpam-6146	279	29	)	)	PUNCT
ejpam-6146	279	30	and	and	CCONJ
ejpam-6146	279	31	y2	y2	NOUN
ejpam-6146	279	32	=	=	SYM
ejpam-6146	279	33	cosh	cosh	PROPN
ejpam-6146	279	34	(	(	PUNCT
ejpam-6146	279	35	√	√	NUM
ejpam-6146	279	36	ax	ax	NOUN
ejpam-6146	279	37	)	)	PUNCT
ejpam-6146	279	38	are	be	AUX
ejpam-6146	279	39	non	non	ADJ
ejpam-6146	279	40	-	-	ADJ
ejpam-6146	279	41	oscillatory	oscillatory	ADJ
ejpam-6146	279	42	.	.	PUNCT
ejpam-6146	280	1	the	the	DET
ejpam-6146	280	2	solution	solution	NOUN
ejpam-6146	280	3	y1	y1	NOUN
ejpam-6146	280	4	=	=	SYM
ejpam-6146	280	5	sinh	sinh	PROPN
ejpam-6146	280	6	(	(	PUNCT
ejpam-6146	280	7	√	√	NUM
ejpam-6146	280	8	ax	ax	NOUN
ejpam-6146	280	9	)	)	PUNCT
ejpam-6146	280	10	has	have	VERB
ejpam-6146	280	11	one	one	NUM
ejpam-6146	280	12	zero	zero	NUM
ejpam-6146	280	13	at	at	ADP
ejpam-6146	280	14	x	x	X
ejpam-6146	280	15	=	=	SYM
ejpam-6146	280	16	0	0	NUM
ejpam-6146	280	17	and	and	CCONJ
ejpam-6146	280	18	the	the	DET
ejpam-6146	280	19	other	other	ADJ
ejpam-6146	280	20	solution	solution	NOUN
ejpam-6146	280	21	has	have	VERB
ejpam-6146	280	22	no	no	DET
ejpam-6146	280	23	zeros	zero	NOUN
ejpam-6146	280	24	.	.	PUNCT
ejpam-6146	281	1	for	for	ADP
ejpam-6146	281	2	a	a	DET
ejpam-6146	281	3	=	=	SYM
ejpam-6146	281	4	0	0	NUM
ejpam-6146	281	5	,	,	PUNCT
ejpam-6146	281	6	the	the	DET
ejpam-6146	281	7	two	two	NUM
ejpam-6146	281	8	solutions	solution	NOUN
ejpam-6146	281	9	are	be	AUX
ejpam-6146	281	10	y1	y1	ADJ
ejpam-6146	281	11	=	=	PUNCT
ejpam-6146	281	12	x	x	X
ejpam-6146	281	13	and	and	CCONJ
ejpam-6146	281	14	y2	y2	PROPN
ejpam-6146	281	15	=	=	SYM
ejpam-6146	282	1	c	c	PROPN
ejpam-6146	282	2	and	and	CCONJ
ejpam-6146	282	3	the	the	DET
ejpam-6146	282	4	general	general	ADJ
ejpam-6146	282	5	solution	solution	NOUN
ejpam-6146	282	6	is	be	AUX
ejpam-6146	282	7	a	a	DET
ejpam-6146	282	8	straight	straight	ADJ
ejpam-6146	282	9	line	line	NOUN
ejpam-6146	282	10	,	,	PUNCT
ejpam-6146	282	11	y	y	PROPN
ejpam-6146	282	12	=	=	PUNCT
ejpam-6146	282	13	mx+	mx+	PROPN
ejpam-6146	282	14	c	c	NOUN
ejpam-6146	283	1	and	and	CCONJ
ejpam-6146	283	2	it	it	PRON
ejpam-6146	283	3	is	be	AUX
ejpam-6146	283	4	called	call	VERB
ejpam-6146	283	5	as	as	ADP
ejpam-6146	283	6	the	the	DET
ejpam-6146	283	7	trivial	trivial	ADJ
ejpam-6146	283	8	solution	solution	NOUN
ejpam-6146	283	9	.	.	PUNCT
ejpam-6146	284	1	the	the	DET
ejpam-6146	284	2	trivial	trivial	ADJ
ejpam-6146	284	3	solution	solution	NOUN
ejpam-6146	284	4	has	have	VERB
ejpam-6146	284	5	one	one	NUM
ejpam-6146	284	6	zero	zero	NUM
ejpam-6146	284	7	at	at	ADP
ejpam-6146	284	8	x	x	X
ejpam-6146	284	9	=	=	SYM
ejpam-6146	284	10	−c	−c	NOUN
ejpam-6146	284	11	/	/	SYM
ejpam-6146	284	12	m.	m.	NOUN
ejpam-6146	284	13	the	the	DET
ejpam-6146	284	14	other	other	ADJ
ejpam-6146	284	15	two	two	NUM
ejpam-6146	284	16	solutions	solution	NOUN
ejpam-6146	284	17	are	be	AUX
ejpam-6146	284	18	called	call	VERB
ejpam-6146	284	19	as	as	ADP
ejpam-6146	284	20	nontrivial	nontrivial	ADJ
ejpam-6146	284	21	solutions	solution	NOUN
ejpam-6146	284	22	.	.	PUNCT
ejpam-6146	285	1	in	in	ADP
ejpam-6146	285	2	the	the	DET
ejpam-6146	285	3	rest	rest	NOUN
ejpam-6146	285	4	of	of	ADP
ejpam-6146	285	5	this	this	DET
ejpam-6146	285	6	section	section	NOUN
ejpam-6146	285	7	,	,	PUNCT
ejpam-6146	285	8	we	we	PRON
ejpam-6146	285	9	shall	shall	AUX
ejpam-6146	285	10	study	study	VERB
ejpam-6146	285	11	the	the	DET
ejpam-6146	285	12	case	case	NOUN
ejpam-6146	285	13	,	,	PUNCT
ejpam-6146	285	14	when	when	SCONJ
ejpam-6146	285	15	the	the	DET
ejpam-6146	285	16	real	real	ADJ
ejpam-6146	285	17	number	number	NOUN
ejpam-6146	285	18	a	a	PRON
ejpam-6146	285	19	is	be	AUX
ejpam-6146	285	20	replaced	replace	VERB
ejpam-6146	285	21	by	by	ADP
ejpam-6146	285	22	a	a	DET
ejpam-6146	285	23	real	real	ADJ
ejpam-6146	285	24	and	and	CCONJ
ejpam-6146	285	25	continuous	continuous	ADJ
ejpam-6146	285	26	function	function	NOUN
ejpam-6146	285	27	q(x	q(x	NOUN
ejpam-6146	285	28	)	)	PUNCT
ejpam-6146	285	29	.	.	PUNCT
ejpam-6146	286	1	for	for	ADP
ejpam-6146	286	2	q(x	q(x	NOUN
ejpam-6146	286	3	)	)	PUNCT
ejpam-6146	286	4	<	<	X
ejpam-6146	286	5	0	0	PROPN
ejpam-6146	286	6	,	,	PUNCT
ejpam-6146	286	7	the	the	DET
ejpam-6146	286	8	solutions	solution	NOUN
ejpam-6146	286	9	do	do	AUX
ejpam-6146	286	10	not	not	PART
ejpam-6146	286	11	oscillate	oscillate	VERB
ejpam-6146	286	12	at	at	ADV
ejpam-6146	286	13	all	all	ADV
ejpam-6146	286	14	.	.	PUNCT
ejpam-6146	287	1	for	for	ADP
ejpam-6146	287	2	q(x	q(x	NOUN
ejpam-6146	287	3	)	)	PUNCT
ejpam-6146	287	4	>	>	X
ejpam-6146	287	5	0	0	NUM
ejpam-6146	287	6	,	,	PUNCT
ejpam-6146	287	7	the	the	DET
ejpam-6146	287	8	solutions	solution	NOUN
ejpam-6146	287	9	do	do	AUX
ejpam-6146	287	10	oscillate	oscillate	VERB
ejpam-6146	287	11	under	under	ADP
ejpam-6146	287	12	additional	additional	ADJ
ejpam-6146	287	13	conditions	condition	NOUN
ejpam-6146	287	14	,	,	PUNCT
ejpam-6146	287	15	which	which	PRON
ejpam-6146	287	16	we	we	PRON
ejpam-6146	287	17	shall	shall	AUX
ejpam-6146	287	18	cover	cover	VERB
ejpam-6146	287	19	in	in	ADP
ejpam-6146	287	20	theorem	theorem	NOUN
ejpam-6146	287	21	(	(	PUNCT
ejpam-6146	287	22	4	4	NUM
ejpam-6146	287	23	)	)	PUNCT
ejpam-6146	287	24	.	.	PUNCT
ejpam-6146	288	1	equation	equation	NOUN
ejpam-6146	288	2	(	(	PUNCT
ejpam-6146	288	3	18	18	NUM
ejpam-6146	288	4	)	)	PUNCT
ejpam-6146	288	5	is	be	AUX
ejpam-6146	288	6	in	in	ADP
ejpam-6146	288	7	the	the	DET
ejpam-6146	288	8	standard	standard	ADJ
ejpam-6146	288	9	form	form	NOUN
ejpam-6146	288	10	.	.	PUNCT
ejpam-6146	289	1	for	for	ADP
ejpam-6146	289	2	the	the	DET
ejpam-6146	289	3	purpose	purpose	NOUN
ejpam-6146	289	4	of	of	ADP
ejpam-6146	289	5	using	use	VERB
ejpam-6146	289	6	the	the	DET
ejpam-6146	289	7	convexity	convexity	NOUN
ejpam-6146	289	8	arguments	argument	NOUN
ejpam-6146	289	9	,	,	PUNCT
ejpam-6146	289	10	we	we	PRON
ejpam-6146	289	11	shall	shall	AUX
ejpam-6146	289	12	transform	transform	VERB
ejpam-6146	289	13	it	it	PRON
ejpam-6146	289	14	to	to	ADP
ejpam-6146	289	15	the	the	DET
ejpam-6146	289	16	following	follow	VERB
ejpam-6146	289	17	form	form	NOUN
ejpam-6146	289	18	known	know	VERB
ejpam-6146	289	19	as	as	ADP
ejpam-6146	289	20	the	the	DET
ejpam-6146	289	21	normal	normal	ADJ
ejpam-6146	289	22	form	form	NOUN
ejpam-6146	289	23	u′′	u′′	PROPN
ejpam-6146	289	24	+	+	CCONJ
ejpam-6146	289	25	q(x)u	q(x)u	VERB
ejpam-6146	289	26	=	=	SYM
ejpam-6146	289	27	0	0	NUM
ejpam-6146	289	28	q(x	q(x	PROPN
ejpam-6146	289	29	)	)	PUNCT
ejpam-6146	290	1	=	=	SYM
ejpam-6146	290	2	q(x)−	q(x)−	NOUN
ejpam-6146	290	3	1	1	NUM
ejpam-6146	290	4	4	4	NUM
ejpam-6146	290	5	p	p	NOUN
ejpam-6146	290	6	2(x)−	2(x)−	NUM
ejpam-6146	290	7	1	1	NUM
ejpam-6146	290	8	2	2	NUM
ejpam-6146	290	9	p	p	NOUN
ejpam-6146	290	10	′(x	′(x	NOUN
ejpam-6146	290	11	)	)	PUNCT
ejpam-6146	290	12	.	.	PUNCT
ejpam-6146	291	1	(	(	PUNCT
ejpam-6146	291	2	21	21	NUM
ejpam-6146	291	3	)	)	PUNCT
ejpam-6146	291	4	this	this	PRON
ejpam-6146	291	5	enables	enable	VERB
ejpam-6146	291	6	us	we	PRON
ejpam-6146	291	7	to	to	PART
ejpam-6146	291	8	relate	relate	VERB
ejpam-6146	291	9	the	the	DET
ejpam-6146	291	10	general	general	ADJ
ejpam-6146	291	11	equation	equation	NOUN
ejpam-6146	291	12	in	in	ADP
ejpam-6146	291	13	(	(	PUNCT
ejpam-6146	291	14	18	18	NUM
ejpam-6146	291	15	)	)	PUNCT
ejpam-6146	291	16	to	to	ADP
ejpam-6146	291	17	the	the	DET
ejpam-6146	291	18	familiar	familiar	ADJ
ejpam-6146	291	19	equation	equation	NOUN
ejpam-6146	291	20	,	,	PUNCT
ejpam-6146	291	21	y′′	y′′	PROPN
ejpam-6146	291	22	+	+	PROPN
ejpam-6146	291	23	ay	ay	PROPN
ejpam-6146	291	24	=	=	NOUN
ejpam-6146	291	25	0	0	PROPN
ejpam-6146	291	26	.	.	PUNCT
ejpam-6146	292	1	the	the	DET
ejpam-6146	292	2	change	change	NOUN
ejpam-6146	292	3	of	of	ADP
ejpam-6146	292	4	the	the	DET
ejpam-6146	292	5	dependent	dependent	ADJ
ejpam-6146	292	6	variable	variable	NOUN
ejpam-6146	292	7	is	be	AUX
ejpam-6146	292	8	done	do	VERB
ejpam-6146	292	9	by	by	ADP
ejpam-6146	292	10	choosing	choose	VERB
ejpam-6146	292	11	y(x	y(x	NOUN
ejpam-6146	292	12	)	)	PUNCT
ejpam-6146	292	13	=	=	SYM
ejpam-6146	292	14	u(x)v(x	u(x)v(x	NUM
ejpam-6146	292	15	)	)	PUNCT
ejpam-6146	292	16	.	.	PUNCT
ejpam-6146	293	1	this	this	DET
ejpam-6146	293	2	substitution	substitution	NOUN
ejpam-6146	293	3	transforms	transform	VERB
ejpam-6146	293	4	eq	eq	NOUN
ejpam-6146	293	5	.	.	PUNCT
ejpam-6146	294	1	(	(	PUNCT
ejpam-6146	294	2	18	18	NUM
ejpam-6146	294	3	)	)	PUNCT
ejpam-6146	294	4	to	to	ADP
ejpam-6146	294	5	vu′′	vu′′	PROPN
ejpam-6146	294	6	+	+	CCONJ
ejpam-6146	294	7	(	(	PUNCT
ejpam-6146	294	8	2v′	2v′	NUM
ejpam-6146	294	9	+	+	NOUN
ejpam-6146	294	10	pv)u′	pv)u′	NOUN
ejpam-6146	294	11	+	+	CCONJ
ejpam-6146	294	12	(	(	PUNCT
ejpam-6146	294	13	v′′	v′′	VERB
ejpam-6146	294	14	+	+	NOUN
ejpam-6146	294	15	pv′	pv′	NOUN
ejpam-6146	294	16	+	+	NOUN
ejpam-6146	294	17	qv)u	qv)u	NOUN
ejpam-6146	294	18	=	=	SYM
ejpam-6146	294	19	0	0	X
ejpam-6146	294	20	.	.	PUNCT
ejpam-6146	295	1	choosing	choose	VERB
ejpam-6146	295	2	the	the	DET
ejpam-6146	295	3	coefficient	coefficient	NOUN
ejpam-6146	295	4	of	of	ADP
ejpam-6146	295	5	the	the	DET
ejpam-6146	295	6	u′	u′	PROPN
ejpam-6146	295	7	to	to	PART
ejpam-6146	295	8	be	be	AUX
ejpam-6146	295	9	zero	zero	NUM
ejpam-6146	295	10	,	,	PUNCT
ejpam-6146	295	11	we	we	PRON
ejpam-6146	295	12	obtain	obtain	VERB
ejpam-6146	295	13	the	the	DET
ejpam-6146	295	14	equation	equation	NOUN
ejpam-6146	295	15	,	,	PUNCT
ejpam-6146	295	16	2v′	2v′	NUM
ejpam-6146	296	1	+	+	CCONJ
ejpam-6146	296	2	pv	pv	ADJ
ejpam-6146	296	3	=	=	NOUN
ejpam-6146	296	4	0	0	X
ejpam-6146	296	5	.	.	PUNCT
ejpam-6146	297	1	the	the	DET
ejpam-6146	297	2	solution	solution	NOUN
ejpam-6146	297	3	of	of	ADP
ejpam-6146	297	4	this	this	DET
ejpam-6146	297	5	equation	equation	NOUN
ejpam-6146	297	6	is	be	AUX
ejpam-6146	297	7	v	v	ADP
ejpam-6146	297	8	=	=	SYM
ejpam-6146	297	9	e−	e−	X
ejpam-6146	297	10	1	1	NUM
ejpam-6146	297	11	2	2	NUM
ejpam-6146	297	12	∫	∫	NOUN
ejpam-6146	297	13	p	p	PROPN
ejpam-6146	297	14	dx	dx	PROPN
ejpam-6146	297	15	.	.	PUNCT
ejpam-6146	298	1	using	use	VERB
ejpam-6146	298	2	this	this	DET
ejpam-6146	298	3	solution	solution	NOUN
ejpam-6146	298	4	,	,	PUNCT
ejpam-6146	298	5	we	we	PRON
ejpam-6146	298	6	obtain	obtain	VERB
ejpam-6146	298	7	the	the	DET
ejpam-6146	298	8	normal	normal	ADJ
ejpam-6146	298	9	form	form	NOUN
ejpam-6146	298	10	of	of	ADP
ejpam-6146	298	11	eq	eq	PROPN
ejpam-6146	298	12	.	.	PUNCT
ejpam-6146	299	1	(	(	PUNCT
ejpam-6146	299	2	18	18	NUM
ejpam-6146	299	3	)	)	PUNCT
ejpam-6146	299	4	in	in	ADP
ejpam-6146	299	5	eq	eq	ADP
ejpam-6146	299	6	.	.	PUNCT
ejpam-6146	300	1	(	(	PUNCT
ejpam-6146	300	2	21	21	NUM
ejpam-6146	300	3	)	)	PUNCT
ejpam-6146	300	4	.	.	PUNCT
ejpam-6146	301	1	as	as	ADP
ejpam-6146	301	2	the	the	DET
ejpam-6146	301	3	v(x	v(x	PROPN
ejpam-6146	301	4	)	)	PUNCT
ejpam-6146	301	5	in	in	ADP
ejpam-6146	301	6	this	this	DET
ejpam-6146	301	7	form	form	NOUN
ejpam-6146	301	8	is	be	AUX
ejpam-6146	301	9	never	never	ADV
ejpam-6146	301	10	zero	zero	NUM
ejpam-6146	301	11	,	,	PUNCT
ejpam-6146	301	12	the	the	DET
ejpam-6146	301	13	change	change	NOUN
ejpam-6146	301	14	of	of	ADP
ejpam-6146	301	15	dependent	dependent	ADJ
ejpam-6146	301	16	variable	variable	NOUN
ejpam-6146	301	17	from	from	ADP
ejpam-6146	301	18	y(x	y(x	PROPN
ejpam-6146	301	19	)	)	PUNCT
ejpam-6146	301	20	to	to	ADP
ejpam-6146	301	21	u(x	u(x	NOUN
ejpam-6146	301	22	)	)	PUNCT
ejpam-6146	301	23	does	do	AUX
ejpam-6146	301	24	not	not	PART
ejpam-6146	301	25	have	have	VERB
ejpam-6146	301	26	any	any	DET
ejpam-6146	301	27	effect	effect	NOUN
ejpam-6146	301	28	on	on	ADP
ejpam-6146	301	29	the	the	DET
ejpam-6146	301	30	zeros	zero	NOUN
ejpam-6146	301	31	of	of	ADP
ejpam-6146	301	32	eq	eq	PROPN
ejpam-6146	301	33	.	.	PUNCT
ejpam-6146	302	1	(	(	PUNCT
ejpam-6146	302	2	18	18	NUM
ejpam-6146	302	3	)	)	PUNCT
ejpam-6146	302	4	.	.	PUNCT
ejpam-6146	303	1	so	so	ADV
ejpam-6146	303	2	,	,	PUNCT
ejpam-6146	303	3	the	the	DET
ejpam-6146	303	4	oscillation	oscillation	NOUN
ejpam-6146	303	5	(	(	PUNCT
ejpam-6146	303	6	or	or	CCONJ
ejpam-6146	303	7	non	non	ADJ
ejpam-6146	303	8	-	-	NOUN
ejpam-6146	303	9	oscillation	oscillation	NOUN
ejpam-6146	303	10	)	)	PUNCT
ejpam-6146	303	11	of	of	ADP
ejpam-6146	303	12	the	the	DET
ejpam-6146	303	13	solutions	solution	NOUN
ejpam-6146	303	14	is	be	AUX
ejpam-6146	303	15	preserved	preserve	VERB
ejpam-6146	303	16	.	.	PUNCT
ejpam-6146	304	1	the	the	DET
ejpam-6146	304	2	following	follow	VERB
ejpam-6146	304	3	four	four	NUM
ejpam-6146	304	4	theorems	theorem	NOUN
ejpam-6146	304	5	summarise	summarise	VERB
ejpam-6146	304	6	the	the	DET
ejpam-6146	304	7	results	result	NOUN
ejpam-6146	304	8	required	require	VERB
ejpam-6146	304	9	to	to	PART
ejpam-6146	304	10	study	study	VERB
ejpam-6146	304	11	the	the	DET
ejpam-6146	304	12	oscillation	oscillation	NOUN
ejpam-6146	304	13	phenomenon	phenomenon	NOUN
ejpam-6146	304	14	in	in	ADP
ejpam-6146	304	15	the	the	DET
ejpam-6146	304	16	second	second	ADJ
ejpam-6146	304	17	-	-	PUNCT
ejpam-6146	304	18	order	order	NOUN
ejpam-6146	304	19	linear	linear	ADJ
ejpam-6146	304	20	homogeneous	homogeneous	ADJ
ejpam-6146	304	21	differential	differential	ADJ
ejpam-6146	304	22	equations	equation	NOUN
ejpam-6146	304	23	.	.	PUNCT
ejpam-6146	305	1	the	the	DET
ejpam-6146	305	2	proofs	proof	NOUN
ejpam-6146	305	3	are	be	AUX
ejpam-6146	305	4	straightforward	straightforward	ADJ
ejpam-6146	305	5	based	base	VERB
ejpam-6146	305	6	on	on	ADP
ejpam-6146	305	7	the	the	DET
ejpam-6146	305	8	convexity	convexity	NOUN
ejpam-6146	305	9	arguments	argument	NOUN
ejpam-6146	305	10	.	.	PUNCT
ejpam-6146	306	1	the	the	DET
ejpam-6146	306	2	following	follow	VERB
ejpam-6146	306	3	theorem	theorem	NOUN
ejpam-6146	306	4	ensures	ensure	VERB
ejpam-6146	306	5	the	the	DET
ejpam-6146	306	6	non	non	NOUN
ejpam-6146	306	7	-	-	NOUN
ejpam-6146	306	8	oscillation	oscillation	NOUN
ejpam-6146	306	9	of	of	ADP
ejpam-6146	306	10	the	the	DET
ejpam-6146	306	11	solutions	solution	NOUN
ejpam-6146	306	12	.	.	PUNCT
ejpam-6146	307	1	theorem	theorem	NOUN
ejpam-6146	307	2	3	3	NUM
ejpam-6146	307	3	.	.	PUNCT
ejpam-6146	307	4	non	non	ADJ
ejpam-6146	307	5	-	-	ADJ
ejpam-6146	307	6	oscillation	oscillation	NOUN
ejpam-6146	307	7	theorem	theorem	NOUN
ejpam-6146	307	8	:	:	PUNCT
ejpam-6146	307	9	if	if	SCONJ
ejpam-6146	307	10	q(x	q(x	NOUN
ejpam-6146	307	11	)	)	PUNCT
ejpam-6146	307	12	<	<	X
ejpam-6146	307	13	0	0	NUM
ejpam-6146	307	14	,	,	PUNCT
ejpam-6146	307	15	and	and	CCONJ
ejpam-6146	307	16	if	if	SCONJ
ejpam-6146	307	17	u(x	u(x	NOUN
ejpam-6146	307	18	)	)	PUNCT
ejpam-6146	307	19	is	be	AUX
ejpam-6146	307	20	a	a	DET
ejpam-6146	307	21	nontrivial	nontrivial	ADJ
ejpam-6146	307	22	solution	solution	NOUN
ejpam-6146	307	23	of	of	ADP
ejpam-6146	307	24	u′′	u′′	PROPN
ejpam-6146	307	25	+	+	CCONJ
ejpam-6146	307	26	q(x)u	q(x)u	X
ejpam-6146	307	27	=	=	SYM
ejpam-6146	307	28	0	0	NUM
ejpam-6146	307	29	,	,	PUNCT
ejpam-6146	307	30	then	then	ADV
ejpam-6146	307	31	u(x	u(x	VERB
ejpam-6146	307	32	)	)	PUNCT
ejpam-6146	307	33	has	have	VERB
ejpam-6146	307	34	at	at	ADP
ejpam-6146	307	35	most	most	ADJ
ejpam-6146	307	36	one	one	NUM
ejpam-6146	307	37	zero	zero	NUM
ejpam-6146	307	38	.	.	PUNCT
ejpam-6146	308	1	the	the	DET
ejpam-6146	308	2	proofs	proof	NOUN
ejpam-6146	308	3	of	of	ADP
ejpam-6146	308	4	theorem	theorem	NOUN
ejpam-6146	308	5	(	(	PUNCT
ejpam-6146	308	6	3	3	NUM
ejpam-6146	308	7	)	)	PUNCT
ejpam-6146	308	8	is	be	AUX
ejpam-6146	308	9	straightforward	straightforward	ADJ
ejpam-6146	308	10	based	base	VERB
ejpam-6146	308	11	on	on	ADP
ejpam-6146	308	12	the	the	DET
ejpam-6146	308	13	convexity	convexity	NOUN
ejpam-6146	308	14	arguments	argument	NOUN
ejpam-6146	308	15	[	[	X
ejpam-6146	308	16	6–9	6–9	NOUN
ejpam-6146	308	17	]	]	PUNCT
ejpam-6146	308	18	.	.	PUNCT
ejpam-6146	309	1	for	for	SCONJ
ejpam-6146	309	2	the	the	DET
ejpam-6146	309	3	oscillations	oscillation	NOUN
ejpam-6146	309	4	to	to	PART
ejpam-6146	309	5	take	take	VERB
ejpam-6146	309	6	place	place	NOUN
ejpam-6146	309	7	,	,	PUNCT
ejpam-6146	309	8	there	there	PRON
ejpam-6146	309	9	needs	need	VERB
ejpam-6146	309	10	to	to	PART
ejpam-6146	309	11	be	be	AUX
ejpam-6146	309	12	additional	additional	ADJ
ejpam-6146	309	13	conditions	condition	NOUN
ejpam-6146	309	14	as	as	SCONJ
ejpam-6146	309	15	given	give	VERB
ejpam-6146	309	16	in	in	ADP
ejpam-6146	309	17	the	the	DET
ejpam-6146	309	18	following	follow	VERB
ejpam-6146	309	19	theorem	theorem	NOUN
ejpam-6146	309	20	.	.	PUNCT
ejpam-6146	310	1	theorem	theorem	ADJ
ejpam-6146	310	2	4	4	NUM
ejpam-6146	310	3	.	.	PUNCT
ejpam-6146	311	1	oscillation	oscillation	NOUN
ejpam-6146	311	2	theorem	theorem	NOUN
ejpam-6146	311	3	:	:	PUNCT
ejpam-6146	311	4	let	let	VERB
ejpam-6146	311	5	u(x	u(x	NOUN
ejpam-6146	311	6	)	)	PUNCT
ejpam-6146	311	7	be	be	AUX
ejpam-6146	311	8	any	any	DET
ejpam-6146	311	9	nontrivial	nontrivial	ADJ
ejpam-6146	311	10	solution	solution	NOUN
ejpam-6146	311	11	of	of	ADP
ejpam-6146	311	12	u′′+	u′′+	X
ejpam-6146	311	13	q(x)u	q(x)u	X
ejpam-6146	311	14	=	=	SYM
ejpam-6146	311	15	0	0	NUM
ejpam-6146	311	16	,	,	PUNCT
ejpam-6146	311	17	where	where	SCONJ
ejpam-6146	311	18	q(x	q(x	NOUN
ejpam-6146	311	19	)	)	PUNCT
ejpam-6146	311	20	>	>	X
ejpam-6146	311	21	0	0	PUNCT
ejpam-6146	312	1	for	for	ADP
ejpam-6146	312	2	all	all	PRON
ejpam-6146	312	3	x	x	SYM
ejpam-6146	312	4	>	>	X
ejpam-6146	312	5	0	0	X
ejpam-6146	312	6	.	.	PUNCT
ejpam-6146	313	1	if	if	SCONJ
ejpam-6146	313	2	∫	∫	PROPN
ejpam-6146	313	3	∞	∞	PROPN
ejpam-6146	313	4	1	1	NUM
ejpam-6146	313	5	q(x	q(x	PROPN
ejpam-6146	313	6	)	)	PUNCT
ejpam-6146	313	7	dx	dx	PROPN
ejpam-6146	314	1	=	=	SYM
ejpam-6146	314	2	∞	∞	PROPN
ejpam-6146	314	3	,	,	PUNCT
ejpam-6146	314	4	then	then	ADV
ejpam-6146	314	5	u(x	u(x	VERB
ejpam-6146	314	6	)	)	PUNCT
ejpam-6146	314	7	has	have	VERB
ejpam-6146	314	8	infinitely	infinitely	ADV
ejpam-6146	314	9	many	many	ADJ
ejpam-6146	314	10	zeros	zero	NOUN
ejpam-6146	314	11	on	on	ADP
ejpam-6146	314	12	the	the	DET
ejpam-6146	314	13	positive	positive	ADJ
ejpam-6146	314	14	x	x	NOUN
ejpam-6146	314	15	-	-	NOUN
ejpam-6146	314	16	axis	axis	NOUN
ejpam-6146	314	17	.	.	PUNCT
ejpam-6146	315	1	s.	s.	PROPN
ejpam-6146	315	2	a.	a.	PROPN
ejpam-6146	315	3	khan	khan	PROPN
ejpam-6146	315	4	,	,	PUNCT
ejpam-6146	315	5	m.	m.	NOUN
ejpam-6146	315	6	m.	m.	PROPN
ejpam-6146	315	7	kankarej	kankarej	PROPN
ejpam-6146	315	8	,	,	PUNCT
ejpam-6146	315	9	m.	m.	PROPN
ejpam-6146	315	10	n.	n.	PROPN
ejpam-6146	315	11	i.	i.	PROPN
ejpam-6146	315	12	khan	khan	PROPN
ejpam-6146	315	13	/	/	SYM
ejpam-6146	315	14	eur	eur	PROPN
ejpam-6146	315	15	.	.	PUNCT
ejpam-6146	316	1	j.	j.	PROPN
ejpam-6146	316	2	pure	pure	PROPN
ejpam-6146	316	3	appl	appl	PROPN
ejpam-6146	316	4	.	.	PROPN
ejpam-6146	316	5	math	math	PROPN
ejpam-6146	316	6	,	,	PUNCT
ejpam-6146	316	7	18	18	NUM
ejpam-6146	316	8	(	(	PUNCT
ejpam-6146	316	9	4	4	NUM
ejpam-6146	316	10	)	)	PUNCT
ejpam-6146	316	11	(	(	PUNCT
ejpam-6146	316	12	2025	2025	NUM
ejpam-6146	316	13	)	)	PUNCT
ejpam-6146	316	14	,	,	PUNCT
ejpam-6146	316	15	6146	6146	NUM
ejpam-6146	316	16	10	10	NUM
ejpam-6146	316	17	of	of	ADP
ejpam-6146	316	18	14	14	NUM
ejpam-6146	316	19	the	the	DET
ejpam-6146	316	20	proofs	proof	NOUN
ejpam-6146	316	21	of	of	ADP
ejpam-6146	316	22	theorem	theorem	NOUN
ejpam-6146	316	23	(	(	PUNCT
ejpam-6146	316	24	4	4	NUM
ejpam-6146	316	25	)	)	PUNCT
ejpam-6146	316	26	is	be	AUX
ejpam-6146	316	27	straightforward	straightforward	ADJ
ejpam-6146	316	28	based	base	VERB
ejpam-6146	316	29	on	on	ADP
ejpam-6146	316	30	the	the	DET
ejpam-6146	316	31	convexity	convexity	NOUN
ejpam-6146	316	32	arguments	argument	NOUN
ejpam-6146	316	33	[	[	X
ejpam-6146	316	34	6–9	6–9	NOUN
ejpam-6146	316	35	]	]	PUNCT
ejpam-6146	316	36	.	.	PUNCT
ejpam-6146	317	1	in	in	ADP
ejpam-6146	317	2	many	many	ADJ
ejpam-6146	317	3	problems	problem	NOUN
ejpam-6146	317	4	,	,	PUNCT
ejpam-6146	317	5	we	we	PRON
ejpam-6146	317	6	are	be	AUX
ejpam-6146	317	7	dealing	deal	VERB
ejpam-6146	317	8	with	with	ADP
ejpam-6146	317	9	finite	finite	ADJ
ejpam-6146	317	10	intervals	interval	NOUN
ejpam-6146	317	11	.	.	PUNCT
ejpam-6146	318	1	the	the	DET
ejpam-6146	318	2	following	follow	VERB
ejpam-6146	318	3	theorem	theorem	ADJ
ejpam-6146	318	4	rules	rule	NOUN
ejpam-6146	318	5	out	out	ADP
ejpam-6146	318	6	the	the	DET
ejpam-6146	318	7	occurrence	occurrence	NOUN
ejpam-6146	318	8	of	of	ADP
ejpam-6146	318	9	infinitely	infinitely	ADV
ejpam-6146	318	10	many	many	ADJ
ejpam-6146	318	11	oscillations	oscillation	NOUN
ejpam-6146	318	12	on	on	ADP
ejpam-6146	318	13	closed	closed	ADJ
ejpam-6146	318	14	intervals	interval	NOUN
ejpam-6146	318	15	.	.	PUNCT
ejpam-6146	319	1	theorem	theorem	VERB
ejpam-6146	319	2	5	5	NUM
ejpam-6146	319	3	.	.	PUNCT
ejpam-6146	319	4	finite	finite	PROPN
ejpam-6146	319	5	zeros	zero	NOUN
ejpam-6146	319	6	in	in	ADP
ejpam-6146	319	7	a	a	DET
ejpam-6146	319	8	closed	closed	ADJ
ejpam-6146	319	9	interval	interval	NOUN
ejpam-6146	319	10	:	:	PUNCT
ejpam-6146	319	11	let	let	VERB
ejpam-6146	319	12	u(x	u(x	NOUN
ejpam-6146	319	13	)	)	PUNCT
ejpam-6146	319	14	be	be	AUX
ejpam-6146	319	15	a	a	DET
ejpam-6146	319	16	nontrivial	nontrivial	ADJ
ejpam-6146	319	17	solution	solution	NOUN
ejpam-6146	319	18	of	of	ADP
ejpam-6146	319	19	u′′	u′′	PROPN
ejpam-6146	319	20	+	+	CCONJ
ejpam-6146	319	21	q(x)u	q(x)u	X
ejpam-6146	319	22	=	=	SYM
ejpam-6146	319	23	0	0	NUM
ejpam-6146	319	24	on	on	ADP
ejpam-6146	319	25	a	a	DET
ejpam-6146	319	26	closed	closed	ADJ
ejpam-6146	319	27	interval	interval	NOUN
ejpam-6146	319	28	[	[	X
ejpam-6146	319	29	a	a	X
ejpam-6146	319	30	,	,	PUNCT
ejpam-6146	319	31	b	b	NOUN
ejpam-6146	319	32	]	]	X
ejpam-6146	319	33	.	.	PUNCT
ejpam-6146	320	1	then	then	ADV
ejpam-6146	320	2	u(x	u(x	VERB
ejpam-6146	320	3	)	)	PUNCT
ejpam-6146	320	4	has	have	VERB
ejpam-6146	320	5	at	at	ADP
ejpam-6146	320	6	most	most	ADJ
ejpam-6146	320	7	a	a	DET
ejpam-6146	320	8	finite	finite	ADJ
ejpam-6146	320	9	number	number	NOUN
ejpam-6146	320	10	of	of	ADP
ejpam-6146	320	11	zeros	zero	NOUN
ejpam-6146	320	12	in	in	ADP
ejpam-6146	320	13	this	this	DET
ejpam-6146	320	14	interval	interval	NOUN
ejpam-6146	320	15	.	.	PUNCT
ejpam-6146	321	1	the	the	DET
ejpam-6146	321	2	proofs	proof	NOUN
ejpam-6146	321	3	of	of	ADP
ejpam-6146	321	4	theorem	theorem	NOUN
ejpam-6146	321	5	(	(	PUNCT
ejpam-6146	321	6	5	5	NUM
ejpam-6146	321	7	)	)	PUNCT
ejpam-6146	321	8	is	be	AUX
ejpam-6146	321	9	straightforward	straightforward	ADJ
ejpam-6146	321	10	based	base	VERB
ejpam-6146	321	11	on	on	ADP
ejpam-6146	321	12	the	the	DET
ejpam-6146	321	13	convexity	convexity	NOUN
ejpam-6146	321	14	arguments	argument	NOUN
ejpam-6146	321	15	[	[	X
ejpam-6146	321	16	6–9	6–9	NOUN
ejpam-6146	321	17	]	]	PUNCT
ejpam-6146	321	18	.	.	PUNCT
ejpam-6146	322	1	from	from	ADP
ejpam-6146	322	2	the	the	DET
ejpam-6146	322	3	sturm	sturm	PROPN
ejpam-6146	322	4	separation	separation	NOUN
ejpam-6146	322	5	theorem	theorem	NOUN
ejpam-6146	322	6	(	(	PUNCT
ejpam-6146	322	7	2	2	NUM
ejpam-6146	322	8	)	)	PUNCT
ejpam-6146	322	9	,	,	PUNCT
ejpam-6146	322	10	we	we	PRON
ejpam-6146	322	11	know	know	VERB
ejpam-6146	322	12	that	that	SCONJ
ejpam-6146	322	13	the	the	DET
ejpam-6146	322	14	zeros	zero	NOUN
ejpam-6146	322	15	of	of	ADP
ejpam-6146	322	16	the	the	DET
ejpam-6146	322	17	two	two	NUM
ejpam-6146	322	18	solutions	solution	NOUN
ejpam-6146	322	19	of	of	ADP
ejpam-6146	322	20	eq	eq	PROPN
ejpam-6146	322	21	.	.	PUNCT
ejpam-6146	323	1	(	(	PUNCT
ejpam-6146	323	2	18	18	NUM
ejpam-6146	323	3	)	)	PUNCT
ejpam-6146	323	4	alternate	alternate	NOUN
ejpam-6146	323	5	.	.	PUNCT
ejpam-6146	324	1	so	so	ADV
ejpam-6146	324	2	,	,	PUNCT
ejpam-6146	324	3	the	the	DET
ejpam-6146	324	4	number	number	NOUN
ejpam-6146	324	5	of	of	ADP
ejpam-6146	324	6	zeros	zero	NOUN
ejpam-6146	324	7	of	of	ADP
ejpam-6146	324	8	the	the	DET
ejpam-6146	324	9	two	two	NUM
ejpam-6146	324	10	solutions	solution	NOUN
ejpam-6146	324	11	can	can	AUX
ejpam-6146	324	12	not	not	PART
ejpam-6146	324	13	differ	differ	VERB
ejpam-6146	324	14	by	by	ADP
ejpam-6146	324	15	more	more	ADJ
ejpam-6146	324	16	than	than	ADP
ejpam-6146	324	17	one	one	NUM
ejpam-6146	324	18	in	in	ADP
ejpam-6146	324	19	a	a	DET
ejpam-6146	324	20	given	give	VERB
ejpam-6146	324	21	interval	interval	NOUN
ejpam-6146	324	22	.	.	PUNCT
ejpam-6146	325	1	in	in	ADP
ejpam-6146	325	2	many	many	ADJ
ejpam-6146	325	3	problems	problem	NOUN
ejpam-6146	325	4	,	,	PUNCT
ejpam-6146	325	5	the	the	DET
ejpam-6146	325	6	decisive	decisive	ADJ
ejpam-6146	325	7	function	function	NOUN
ejpam-6146	325	8	,	,	PUNCT
ejpam-6146	325	9	q(x	q(x	PROPN
ejpam-6146	325	10	)	)	PUNCT
ejpam-6146	325	11	>	>	X
ejpam-6146	325	12	0	0	NUM
ejpam-6146	325	13	can	can	AUX
ejpam-6146	325	14	be	be	AUX
ejpam-6146	325	15	different	different	ADJ
ejpam-6146	325	16	.	.	PUNCT
ejpam-6146	326	1	this	this	PRON
ejpam-6146	326	2	leads	lead	VERB
ejpam-6146	326	3	to	to	ADP
ejpam-6146	326	4	the	the	DET
ejpam-6146	326	5	question	question	NOUN
ejpam-6146	326	6	of	of	ADP
ejpam-6146	326	7	the	the	DET
ejpam-6146	326	8	number	number	NOUN
ejpam-6146	326	9	of	of	ADP
ejpam-6146	326	10	zeros	zero	NOUN
ejpam-6146	326	11	for	for	ADP
ejpam-6146	326	12	different	different	ADJ
ejpam-6146	326	13	choices	choice	NOUN
ejpam-6146	326	14	of	of	ADP
ejpam-6146	326	15	q(x	q(x	NOUN
ejpam-6146	326	16	)	)	PUNCT
ejpam-6146	326	17	.	.	PUNCT
ejpam-6146	327	1	in	in	ADP
ejpam-6146	327	2	the	the	DET
ejpam-6146	327	3	case	case	NOUN
ejpam-6146	327	4	of	of	ADP
ejpam-6146	327	5	the	the	DET
ejpam-6146	327	6	familiar	familiar	ADJ
ejpam-6146	327	7	equation	equation	NOUN
ejpam-6146	327	8	,	,	PUNCT
ejpam-6146	327	9	y′′+ay	y′′+ay	NOUN
ejpam-6146	328	1	=	=	NOUN
ejpam-6146	328	2	0	0	NUM
ejpam-6146	328	3	with	with	ADP
ejpam-6146	328	4	a	a	DET
ejpam-6146	328	5	>	>	X
ejpam-6146	328	6	0	0	NUM
ejpam-6146	328	7	,	,	PUNCT
ejpam-6146	328	8	we	we	PRON
ejpam-6146	328	9	know	know	VERB
ejpam-6146	328	10	that	that	SCONJ
ejpam-6146	328	11	larger	large	ADJ
ejpam-6146	328	12	the	the	DET
ejpam-6146	328	13	a	a	NOUN
ejpam-6146	328	14	,	,	PUNCT
ejpam-6146	328	15	more	more	ADJ
ejpam-6146	328	16	the	the	DET
ejpam-6146	328	17	number	number	NOUN
ejpam-6146	328	18	of	of	ADP
ejpam-6146	328	19	zeros	zero	NOUN
ejpam-6146	328	20	in	in	ADP
ejpam-6146	328	21	the	the	DET
ejpam-6146	328	22	same	same	ADJ
ejpam-6146	328	23	interval	interval	NOUN
ejpam-6146	328	24	.	.	PUNCT
ejpam-6146	329	1	the	the	DET
ejpam-6146	329	2	following	follow	VERB
ejpam-6146	329	3	theorem	theorem	ADJ
ejpam-6146	329	4	addresses	address	NOUN
ejpam-6146	329	5	this	this	DET
ejpam-6146	329	6	scenario	scenario	NOUN
ejpam-6146	329	7	in	in	ADP
ejpam-6146	329	8	totality	totality	NOUN
ejpam-6146	329	9	.	.	PUNCT
ejpam-6146	330	1	theorem	theorem	ADJ
ejpam-6146	330	2	6	6	NUM
ejpam-6146	330	3	.	.	PUNCT
ejpam-6146	331	1	sturm	sturm	PROPN
ejpam-6146	331	2	comparison	comparison	NOUN
ejpam-6146	331	3	theorem	theorem	VERB
ejpam-6146	331	4	:	:	PUNCT
ejpam-6146	331	5	let	let	VERB
ejpam-6146	331	6	y(x	y(x	NOUN
ejpam-6146	331	7	)	)	PUNCT
ejpam-6146	331	8	and	and	CCONJ
ejpam-6146	331	9	z(x	z(x	NUM
ejpam-6146	331	10	)	)	PUNCT
ejpam-6146	331	11	be	be	VERB
ejpam-6146	331	12	the	the	DET
ejpam-6146	331	13	nontrivial	nontrivial	ADJ
ejpam-6146	331	14	solutions	solution	NOUN
ejpam-6146	331	15	of	of	ADP
ejpam-6146	331	16	y′′	y′′	PROPN
ejpam-6146	331	17	+	+	CCONJ
ejpam-6146	331	18	q(x)y	q(x)y	PROPN
ejpam-6146	331	19	=	=	SYM
ejpam-6146	331	20	0	0	NUM
ejpam-6146	331	21	and	and	CCONJ
ejpam-6146	331	22	z′′	z′′	PROPN
ejpam-6146	331	23	+	+	CCONJ
ejpam-6146	331	24	r(x)z	r(x)z	NOUN
ejpam-6146	331	25	=	=	SYM
ejpam-6146	331	26	0	0	PROPN
ejpam-6146	331	27	,	,	PUNCT
ejpam-6146	331	28	where	where	SCONJ
ejpam-6146	331	29	q(x	q(x	NOUN
ejpam-6146	331	30	)	)	PUNCT
ejpam-6146	331	31	and	and	CCONJ
ejpam-6146	331	32	r(x	r(x	PROPN
ejpam-6146	331	33	)	)	PUNCT
ejpam-6146	331	34	are	be	AUX
ejpam-6146	331	35	positive	positive	ADJ
ejpam-6146	331	36	functions	function	NOUN
ejpam-6146	331	37	such	such	ADJ
ejpam-6146	331	38	that	that	SCONJ
ejpam-6146	331	39	q(x	q(x	PROPN
ejpam-6146	331	40	)	)	PUNCT
ejpam-6146	331	41	>	>	X
ejpam-6146	331	42	r(x	r(x	PROPN
ejpam-6146	331	43	)	)	PUNCT
ejpam-6146	331	44	.	.	PUNCT
ejpam-6146	332	1	then	then	ADV
ejpam-6146	332	2	y(x	y(x	NOUN
ejpam-6146	332	3	)	)	PUNCT
ejpam-6146	332	4	vanishes	vanish	VERB
ejpam-6146	332	5	at	at	ADP
ejpam-6146	332	6	least	least	ADJ
ejpam-6146	332	7	once	once	ADV
ejpam-6146	332	8	between	between	ADP
ejpam-6146	332	9	any	any	DET
ejpam-6146	332	10	two	two	NUM
ejpam-6146	332	11	successive	successive	ADJ
ejpam-6146	332	12	zeros	zero	NOUN
ejpam-6146	332	13	of	of	ADP
ejpam-6146	332	14	z(x	z(x	NUM
ejpam-6146	332	15	)	)	PUNCT
ejpam-6146	332	16	.	.	PUNCT
ejpam-6146	333	1	the	the	DET
ejpam-6146	333	2	proof	proof	NOUN
ejpam-6146	333	3	is	be	AUX
ejpam-6146	333	4	based	base	VERB
ejpam-6146	333	5	on	on	ADP
ejpam-6146	333	6	convexity	convexity	NOUN
ejpam-6146	333	7	arguments	argument	NOUN
ejpam-6146	333	8	along	along	ADP
ejpam-6146	333	9	with	with	ADP
ejpam-6146	333	10	the	the	DET
ejpam-6146	333	11	wronskian	wronskian	NOUN
ejpam-6146	333	12	.	.	PUNCT
ejpam-6146	334	1	from	from	ADP
ejpam-6146	334	2	the	the	DET
ejpam-6146	334	3	theorem	theorem	NOUN
ejpam-6146	334	4	,	,	PUNCT
ejpam-6146	334	5	we	we	PRON
ejpam-6146	334	6	conclude	conclude	VERB
ejpam-6146	334	7	that	that	SCONJ
ejpam-6146	334	8	the	the	DET
ejpam-6146	334	9	solutions	solution	NOUN
ejpam-6146	334	10	of	of	ADP
ejpam-6146	334	11	eq	eq	PROPN
ejpam-6146	334	12	.	.	PUNCT
ejpam-6146	335	1	(	(	PUNCT
ejpam-6146	335	2	21	21	NUM
ejpam-6146	335	3	)	)	PUNCT
ejpam-6146	335	4	oscillate	oscillate	VERB
ejpam-6146	335	5	more	more	ADV
ejpam-6146	335	6	rapidly	rapidly	ADV
ejpam-6146	335	7	as	as	ADP
ejpam-6146	335	8	q(x	q(x	NOUN
ejpam-6146	335	9	)	)	PUNCT
ejpam-6146	335	10	is	be	AUX
ejpam-6146	335	11	increased	increase	VERB
ejpam-6146	335	12	.	.	PUNCT
ejpam-6146	336	1	4	4	X
ejpam-6146	336	2	.	.	X
ejpam-6146	336	3	hyperbolic	hyperbolic	ADJ
ejpam-6146	336	4	functions	function	NOUN
ejpam-6146	336	5	from	from	ADP
ejpam-6146	336	6	y′′	y′′	PROPN
ejpam-6146	336	7	−	−	PROPN
ejpam-6146	336	8	y	y	PROPN
ejpam-6146	336	9	=	=	SYM
ejpam-6146	336	10	0	0	NUM
ejpam-6146	336	11	following	follow	VERB
ejpam-6146	336	12	a	a	DET
ejpam-6146	336	13	very	very	ADV
ejpam-6146	336	14	similar	similar	ADJ
ejpam-6146	336	15	approach	approach	NOUN
ejpam-6146	336	16	,	,	PUNCT
ejpam-6146	336	17	it	it	PRON
ejpam-6146	336	18	is	be	AUX
ejpam-6146	336	19	possible	possible	ADJ
ejpam-6146	336	20	to	to	PART
ejpam-6146	336	21	deduce	deduce	VERB
ejpam-6146	336	22	the	the	DET
ejpam-6146	336	23	properties	property	NOUN
ejpam-6146	336	24	of	of	ADP
ejpam-6146	336	25	the	the	DET
ejpam-6146	336	26	solutions	solution	NOUN
ejpam-6146	336	27	of	of	ADP
ejpam-6146	336	28	the	the	DET
ejpam-6146	336	29	equation	equation	NOUN
ejpam-6146	336	30	y′′	y′′	NOUN
ejpam-6146	336	31	−	−	PROPN
ejpam-6146	336	32	y	y	PROPN
ejpam-6146	336	33	=	=	NOUN
ejpam-6146	336	34	0	0	PROPN
ejpam-6146	336	35	.	.	PUNCT
ejpam-6146	337	1	(	(	PUNCT
ejpam-6146	337	2	22	22	NUM
ejpam-6146	337	3	)	)	PUNCT
ejpam-6146	337	4	from	from	ADP
ejpam-6146	337	5	the	the	DET
ejpam-6146	337	6	theorems	theorem	NOUN
ejpam-6146	337	7	,	,	PUNCT
ejpam-6146	337	8	we	we	PRON
ejpam-6146	337	9	can	can	AUX
ejpam-6146	337	10	conclude	conclude	VERB
ejpam-6146	337	11	that	that	SCONJ
ejpam-6146	337	12	the	the	DET
ejpam-6146	337	13	hyperbolic	hyperbolic	ADJ
ejpam-6146	337	14	solutions	solution	NOUN
ejpam-6146	337	15	do	do	AUX
ejpam-6146	337	16	not	not	PART
ejpam-6146	337	17	have	have	VERB
ejpam-6146	337	18	any	any	DET
ejpam-6146	337	19	oscillation	oscillation	NOUN
ejpam-6146	337	20	.	.	PUNCT
ejpam-6146	338	1	the	the	DET
ejpam-6146	338	2	solution	solution	NOUN
ejpam-6146	338	3	y	y	PROPN
ejpam-6146	338	4	=	=	PUNCT
ejpam-6146	338	5	sinhx	sinhx	PROPN
ejpam-6146	338	6	has	have	VERB
ejpam-6146	338	7	only	only	ADV
ejpam-6146	338	8	one	one	NUM
ejpam-6146	338	9	zero	zero	NUM
ejpam-6146	338	10	at	at	ADP
ejpam-6146	338	11	x	x	X
ejpam-6146	338	12	=	=	NOUN
ejpam-6146	338	13	0	0	NUM
ejpam-6146	338	14	.	.	PUNCT
ejpam-6146	339	1	the	the	DET
ejpam-6146	339	2	equivalent	equivalent	ADJ
ejpam-6146	339	3	pair	pair	NOUN
ejpam-6146	339	4	,	,	PUNCT
ejpam-6146	339	5	e−x	e−x	NOUN
ejpam-6146	339	6	and	and	CCONJ
ejpam-6146	339	7	e+x	e+x	PROPN
ejpam-6146	339	8	also	also	ADV
ejpam-6146	339	9	does	do	AUX
ejpam-6146	339	10	not	not	PART
ejpam-6146	339	11	have	have	VERB
ejpam-6146	339	12	oscillatory	oscillatory	ADJ
ejpam-6146	339	13	solutions	solution	NOUN
ejpam-6146	339	14	and	and	CCONJ
ejpam-6146	339	15	both	both	PRON
ejpam-6146	339	16	of	of	ADP
ejpam-6146	339	17	them	they	PRON
ejpam-6146	339	18	do	do	AUX
ejpam-6146	339	19	not	not	PART
ejpam-6146	339	20	have	have	VERB
ejpam-6146	339	21	any	any	DET
ejpam-6146	339	22	zeros	zero	NOUN
ejpam-6146	339	23	.	.	PUNCT
ejpam-6146	340	1	the	the	DET
ejpam-6146	340	2	two	two	NUM
ejpam-6146	340	3	linearly	linearly	ADV
ejpam-6146	340	4	independent	independent	ADJ
ejpam-6146	340	5	solutions	solution	NOUN
ejpam-6146	340	6	indeed	indeed	ADV
ejpam-6146	340	7	are	be	AUX
ejpam-6146	340	8	y	y	PROPN
ejpam-6146	340	9	=	=	PUNCT
ejpam-6146	340	10	sinhx	sinhx	PROPN
ejpam-6146	340	11	y	y	PROPN
ejpam-6146	340	12	=	=	PUNCT
ejpam-6146	340	13	coshx	coshx	PROPN
ejpam-6146	340	14	.	.	PUNCT
ejpam-6146	341	1	(	(	PUNCT
ejpam-6146	341	2	23	23	NUM
ejpam-6146	341	3	)	)	PUNCT
ejpam-6146	341	4	s.	s.	PROPN
ejpam-6146	341	5	a.	a.	PROPN
ejpam-6146	341	6	khan	khan	PROPN
ejpam-6146	341	7	,	,	PUNCT
ejpam-6146	341	8	m.	m.	NOUN
ejpam-6146	341	9	m.	m.	PROPN
ejpam-6146	341	10	kankarej	kankarej	PROPN
ejpam-6146	341	11	,	,	PUNCT
ejpam-6146	341	12	m.	m.	PROPN
ejpam-6146	341	13	n.	n.	PROPN
ejpam-6146	341	14	i.	i.	PROPN
ejpam-6146	341	15	khan	khan	PROPN
ejpam-6146	341	16	/	/	SYM
ejpam-6146	341	17	eur	eur	PROPN
ejpam-6146	341	18	.	.	PUNCT
ejpam-6146	342	1	j.	j.	PROPN
ejpam-6146	342	2	pure	pure	PROPN
ejpam-6146	342	3	appl	appl	PROPN
ejpam-6146	342	4	.	.	PROPN
ejpam-6146	342	5	math	math	PROPN
ejpam-6146	342	6	,	,	PUNCT
ejpam-6146	342	7	18	18	NUM
ejpam-6146	342	8	(	(	PUNCT
ejpam-6146	342	9	4	4	NUM
ejpam-6146	342	10	)	)	PUNCT
ejpam-6146	342	11	(	(	PUNCT
ejpam-6146	342	12	2025	2025	NUM
ejpam-6146	342	13	)	)	PUNCT
ejpam-6146	342	14	,	,	PUNCT
ejpam-6146	342	15	6146	6146	NUM
ejpam-6146	342	16	11	11	NUM
ejpam-6146	342	17	of	of	ADP
ejpam-6146	342	18	14	14	NUM
ejpam-6146	342	19	5	5	NUM
ejpam-6146	342	20	.	.	PUNCT
ejpam-6146	342	21	bessel	bessel	ADJ
ejpam-6146	342	22	functions	function	NOUN
ejpam-6146	342	23	from	from	ADP
ejpam-6146	342	24	x2y′′	x2y′′	PROPN
ejpam-6146	342	25	+	+	CCONJ
ejpam-6146	342	26	xy′	xy′	NOUN
ejpam-6146	343	1	+	+	CCONJ
ejpam-6146	343	2	(	(	PUNCT
ejpam-6146	343	3	x2	x2	INTJ
ejpam-6146	343	4	−	−	PROPN
ejpam-6146	343	5	p2	p2	PROPN
ejpam-6146	343	6	)	)	PUNCT
ejpam-6146	343	7	y	y	PROPN
ejpam-6146	343	8	=	=	PUNCT
ejpam-6146	343	9	0	0	PUNCT
ejpam-6146	343	10	let	let	VERB
ejpam-6146	343	11	us	we	PRON
ejpam-6146	343	12	consider	consider	VERB
ejpam-6146	343	13	the	the	DET
ejpam-6146	343	14	bessel	bessel	ADJ
ejpam-6146	343	15	equation	equation	NOUN
ejpam-6146	343	16	x2y′′	x2y′′	PROPN
ejpam-6146	344	1	+	+	CCONJ
ejpam-6146	344	2	xy′	xy′	PROPN
ejpam-6146	345	1	+	+	CCONJ
ejpam-6146	345	2	(	(	PUNCT
ejpam-6146	345	3	x2	x2	INTJ
ejpam-6146	345	4	−	−	PROPN
ejpam-6146	345	5	p2	p2	PROPN
ejpam-6146	345	6	)	)	PUNCT
ejpam-6146	346	1	y	y	PROPN
ejpam-6146	346	2	=	=	NOUN
ejpam-6146	346	3	0	0	PROPN
ejpam-6146	346	4	.	.	PUNCT
ejpam-6146	347	1	(	(	PUNCT
ejpam-6146	347	2	24	24	NUM
ejpam-6146	347	3	)	)	PUNCT
ejpam-6146	347	4	next	next	ADV
ejpam-6146	347	5	,	,	PUNCT
ejpam-6146	347	6	we	we	PRON
ejpam-6146	347	7	write	write	VERB
ejpam-6146	347	8	the	the	DET
ejpam-6146	347	9	bessel	bessel	ADJ
ejpam-6146	347	10	equation	equation	NOUN
ejpam-6146	347	11	in	in	ADP
ejpam-6146	347	12	eq	eq	ADP
ejpam-6146	347	13	.	.	PUNCT
ejpam-6146	348	1	(	(	PUNCT
ejpam-6146	348	2	24	24	NUM
ejpam-6146	348	3	)	)	PUNCT
ejpam-6146	348	4	in	in	ADP
ejpam-6146	348	5	normal	normal	ADJ
ejpam-6146	348	6	form	form	NOUN
ejpam-6146	348	7	using	use	VERB
ejpam-6146	348	8	the	the	DET
ejpam-6146	348	9	transform	transform	NOUN
ejpam-6146	348	10	in	in	ADP
ejpam-6146	348	11	eq	eq	ADP
ejpam-6146	348	12	.	.	PUNCT
ejpam-6146	349	1	(	(	PUNCT
ejpam-6146	349	2	21	21	NUM
ejpam-6146	349	3	)	)	PUNCT
ejpam-6146	349	4	,	,	PUNCT
ejpam-6146	349	5	which	which	PRON
ejpam-6146	349	6	leads	lead	VERB
ejpam-6146	349	7	to	to	ADP
ejpam-6146	349	8	u′′	u′′	PROPN
ejpam-6146	349	9	+	+	CCONJ
ejpam-6146	349	10	(	(	PUNCT
ejpam-6146	349	11	1	1	NUM
ejpam-6146	349	12	+	+	NUM
ejpam-6146	349	13	1−	1−	NUM
ejpam-6146	349	14	4p2	4p2	NUM
ejpam-6146	349	15	4x2	4x2	X
ejpam-6146	349	16	)	)	PUNCT
ejpam-6146	349	17	u	u	NOUN
ejpam-6146	349	18	=	=	NOUN
ejpam-6146	349	19	0	0	PROPN
ejpam-6146	349	20	.	.	PUNCT
ejpam-6146	350	1	(	(	PUNCT
ejpam-6146	350	2	25	25	NUM
ejpam-6146	350	3	)	)	PUNCT
ejpam-6146	350	4	using	use	VERB
ejpam-6146	350	5	the	the	DET
ejpam-6146	350	6	theorems	theorem	NOUN
ejpam-6146	350	7	from	from	ADP
ejpam-6146	350	8	section	section	NOUN
ejpam-6146	350	9	2	2	NUM
ejpam-6146	350	10	,	,	PUNCT
ejpam-6146	350	11	we	we	PRON
ejpam-6146	350	12	arrive	arrive	VERB
ejpam-6146	350	13	at	at	ADP
ejpam-6146	350	14	the	the	DET
ejpam-6146	350	15	following	following	ADJ
ejpam-6146	350	16	result	result	NOUN
ejpam-6146	350	17	stated	state	VERB
ejpam-6146	350	18	in	in	ADP
ejpam-6146	350	19	the	the	DET
ejpam-6146	350	20	form	form	NOUN
ejpam-6146	350	21	of	of	ADP
ejpam-6146	350	22	a	a	DET
ejpam-6146	350	23	theorem	theorem	VERB
ejpam-6146	350	24	.	.	PUNCT
ejpam-6146	351	1	theorem	theorem	ADJ
ejpam-6146	351	2	7	7	NUM
ejpam-6146	351	3	.	.	PUNCT
ejpam-6146	351	4	solutions	solution	NOUN
ejpam-6146	351	5	of	of	ADP
ejpam-6146	351	6	bessel	bessel	ADJ
ejpam-6146	351	7	equation	equation	NOUN
ejpam-6146	351	8	theorem	theorem	VERB
ejpam-6146	351	9	:	:	PUNCT
ejpam-6146	351	10	let	let	VERB
ejpam-6146	351	11	yp(x	yp(x	VERB
ejpam-6146	351	12	)	)	PUNCT
ejpam-6146	352	1	be	be	AUX
ejpam-6146	352	2	a	a	DET
ejpam-6146	352	3	nontrivial	nontrivial	ADJ
ejpam-6146	352	4	solution	solution	NOUN
ejpam-6146	352	5	of	of	ADP
ejpam-6146	352	6	bessel	bessel	ADJ
ejpam-6146	352	7	equation	equation	NOUN
ejpam-6146	352	8	on	on	ADP
ejpam-6146	352	9	the	the	DET
ejpam-6146	352	10	positive	positive	ADJ
ejpam-6146	352	11	x	x	NOUN
ejpam-6146	352	12	-	-	NOUN
ejpam-6146	352	13	axis	axis	ADJ
ejpam-6146	352	14	.	.	PUNCT
ejpam-6146	353	1	if	if	SCONJ
ejpam-6146	353	2	0	0	NUM
ejpam-6146	353	3	≤	≤	NOUN
ejpam-6146	354	1	p	p	X
ejpam-6146	354	2	<	<	X
ejpam-6146	354	3	1/2	1/2	NUM
ejpam-6146	354	4	,	,	PUNCT
ejpam-6146	354	5	then	then	ADV
ejpam-6146	354	6	every	every	DET
ejpam-6146	354	7	interval	interval	NOUN
ejpam-6146	354	8	of	of	ADP
ejpam-6146	354	9	length	length	NOUN
ejpam-6146	354	10	π	π	PROPN
ejpam-6146	354	11	contains	contain	VERB
ejpam-6146	354	12	at	at	ADV
ejpam-6146	354	13	least	least	ADV
ejpam-6146	354	14	one	one	NUM
ejpam-6146	354	15	zero	zero	NUM
ejpam-6146	354	16	of	of	ADP
ejpam-6146	354	17	yp(x	yp(x	PROPN
ejpam-6146	354	18	)	)	PUNCT
ejpam-6146	354	19	;	;	PUNCT
ejpam-6146	354	20	if	if	SCONJ
ejpam-6146	354	21	p	p	NOUN
ejpam-6146	354	22	=	=	SYM
ejpam-6146	354	23	1/2	1/2	NUM
ejpam-6146	354	24	,	,	PUNCT
ejpam-6146	354	25	then	then	ADV
ejpam-6146	354	26	the	the	DET
ejpam-6146	354	27	distance	distance	NOUN
ejpam-6146	354	28	between	between	ADP
ejpam-6146	354	29	successive	successive	ADJ
ejpam-6146	354	30	zeros	zero	NOUN
ejpam-6146	354	31	of	of	ADP
ejpam-6146	354	32	yp(x	yp(x	PROPN
ejpam-6146	354	33	)	)	PUNCT
ejpam-6146	354	34	is	be	AUX
ejpam-6146	354	35	exactly	exactly	ADV
ejpam-6146	354	36	π	π	X
ejpam-6146	354	37	;	;	PUNCT
ejpam-6146	354	38	if	if	SCONJ
ejpam-6146	354	39	p	p	X
ejpam-6146	354	40	>	>	X
ejpam-6146	354	41	1/2	1/2	NUM
ejpam-6146	354	42	,	,	PUNCT
ejpam-6146	354	43	then	then	ADV
ejpam-6146	354	44	every	every	DET
ejpam-6146	354	45	interval	interval	NOUN
ejpam-6146	354	46	of	of	ADP
ejpam-6146	354	47	length	length	NOUN
ejpam-6146	354	48	π	π	PROPN
ejpam-6146	354	49	contains	contain	VERB
ejpam-6146	354	50	at	at	ADP
ejpam-6146	354	51	most	most	ADJ
ejpam-6146	354	52	one	one	NUM
ejpam-6146	354	53	zero	zero	NUM
ejpam-6146	354	54	of	of	ADP
ejpam-6146	354	55	yp(x	yp(x	PROPN
ejpam-6146	354	56	)	)	PUNCT
ejpam-6146	354	57	.	.	PUNCT
ejpam-6146	355	1	we	we	PRON
ejpam-6146	355	2	used	use	VERB
ejpam-6146	355	3	eq	eq	ADP
ejpam-6146	355	4	.	.	PUNCT
ejpam-6146	356	1	(	(	PUNCT
ejpam-6146	356	2	25	25	NUM
ejpam-6146	356	3	)	)	PUNCT
ejpam-6146	356	4	to	to	PART
ejpam-6146	356	5	obtain	obtain	VERB
ejpam-6146	356	6	the	the	DET
ejpam-6146	356	7	results	result	NOUN
ejpam-6146	356	8	stated	state	VERB
ejpam-6146	356	9	in	in	ADP
ejpam-6146	356	10	theorem	theorem	NOUN
ejpam-6146	356	11	(	(	PUNCT
ejpam-6146	356	12	7	7	X
ejpam-6146	356	13	)	)	PUNCT
ejpam-6146	356	14	using	use	VERB
ejpam-6146	356	15	qualitative	qualitative	ADJ
ejpam-6146	356	16	analysis	analysis	NOUN
ejpam-6146	356	17	.	.	PUNCT
ejpam-6146	357	1	now	now	ADV
ejpam-6146	357	2	,	,	PUNCT
ejpam-6146	357	3	we	we	PRON
ejpam-6146	357	4	shall	shall	AUX
ejpam-6146	357	5	revisit	revisit	VERB
ejpam-6146	357	6	these	these	DET
ejpam-6146	357	7	qualitative	qualitative	ADJ
ejpam-6146	357	8	results	result	NOUN
ejpam-6146	357	9	after	after	ADP
ejpam-6146	357	10	noting	note	VERB
ejpam-6146	357	11	the	the	DET
ejpam-6146	357	12	infinite	infinite	ADJ
ejpam-6146	357	13	series	series	NOUN
ejpam-6146	357	14	solutions	solution	NOUN
ejpam-6146	357	15	of	of	ADP
ejpam-6146	357	16	the	the	DET
ejpam-6146	357	17	bessel	bessel	ADJ
ejpam-6146	357	18	equation	equation	NOUN
ejpam-6146	357	19	in	in	ADP
ejpam-6146	357	20	its	its	PRON
ejpam-6146	357	21	standard	standard	ADJ
ejpam-6146	357	22	form	form	NOUN
ejpam-6146	357	23	in	in	ADP
ejpam-6146	357	24	eq	eq	ADP
ejpam-6146	357	25	.	.	PUNCT
ejpam-6146	358	1	(	(	PUNCT
ejpam-6146	358	2	24	24	NUM
ejpam-6146	358	3	)	)	PUNCT
ejpam-6146	358	4	.	.	PUNCT
ejpam-6146	359	1	equation	equation	NOUN
ejpam-6146	359	2	(	(	PUNCT
ejpam-6146	359	3	24	24	NUM
ejpam-6146	359	4	)	)	PUNCT
ejpam-6146	359	5	is	be	AUX
ejpam-6146	359	6	a	a	DET
ejpam-6146	359	7	second	second	ADJ
ejpam-6146	359	8	order	order	NOUN
ejpam-6146	359	9	equation	equation	NOUN
ejpam-6146	359	10	and	and	CCONJ
ejpam-6146	359	11	has	have	VERB
ejpam-6146	359	12	two	two	NUM
ejpam-6146	359	13	linearly	linearly	ADV
ejpam-6146	359	14	independent	independent	ADJ
ejpam-6146	359	15	solutions	solution	NOUN
ejpam-6146	359	16	.	.	PUNCT
ejpam-6146	360	1	one	one	NUM
ejpam-6146	360	2	of	of	ADP
ejpam-6146	360	3	the	the	DET
ejpam-6146	360	4	two	two	NUM
ejpam-6146	360	5	linearly	linearly	ADV
ejpam-6146	360	6	independent	independent	ADJ
ejpam-6146	360	7	solutions	solution	NOUN
ejpam-6146	360	8	known	know	VERB
ejpam-6146	360	9	as	as	ADP
ejpam-6146	360	10	the	the	DET
ejpam-6146	360	11	bessel	bessel	ADJ
ejpam-6146	360	12	function	function	NOUN
ejpam-6146	360	13	of	of	ADP
ejpam-6146	360	14	the	the	DET
ejpam-6146	360	15	first	first	ADJ
ejpam-6146	360	16	kind	kind	NOUN
ejpam-6146	360	17	is	be	AUX
ejpam-6146	360	18	denoted	denote	VERB
ejpam-6146	360	19	by	by	ADP
ejpam-6146	360	20	jp(x	jp(x	NOUN
ejpam-6146	360	21	)	)	PUNCT
ejpam-6146	360	22	and	and	CCONJ
ejpam-6146	360	23	has	have	VERB
ejpam-6146	360	24	the	the	DET
ejpam-6146	360	25	following	follow	VERB
ejpam-6146	360	26	infinite	infinite	ADJ
ejpam-6146	360	27	series	series	NOUN
ejpam-6146	360	28	representation	representation	NOUN
ejpam-6146	360	29	jp	jp	NOUN
ejpam-6146	360	30	=	=	PUNCT
ejpam-6146	361	1	∞∑	∞∑	NUM
ejpam-6146	361	2	m=0	m=0	PROPN
ejpam-6146	361	3	(	(	PUNCT
ejpam-6146	361	4	−1)m	−1)m	PROPN
ejpam-6146	361	5	1	1	NUM
ejpam-6146	361	6	m!γ(m+	m!γ(m+	NOUN
ejpam-6146	361	7	p+	p+	VERB
ejpam-6146	361	8	1	1	NUM
ejpam-6146	361	9	)	)	PUNCT
ejpam-6146	361	10	(	(	PUNCT
ejpam-6146	361	11	x	x	SYM
ejpam-6146	361	12	2	2	NUM
ejpam-6146	361	13	)	)	PUNCT
ejpam-6146	361	14	2m+p	2m+p	NUM
ejpam-6146	361	15	,	,	PUNCT
ejpam-6146	361	16	(	(	PUNCT
ejpam-6146	361	17	26	26	NUM
ejpam-6146	361	18	)	)	PUNCT
ejpam-6146	361	19	where	where	SCONJ
ejpam-6146	361	20	γ(n	γ(n	X
ejpam-6146	361	21	)	)	PUNCT
ejpam-6146	361	22	is	be	AUX
ejpam-6146	361	23	the	the	DET
ejpam-6146	361	24	gamma	gamma	PROPN
ejpam-6146	361	25	function	function	NOUN
ejpam-6146	361	26	.	.	PUNCT
ejpam-6146	362	1	in	in	ADP
ejpam-6146	362	2	case	case	NOUN
ejpam-6146	362	3	n	n	NOUN
ejpam-6146	362	4	is	be	AUX
ejpam-6146	362	5	a	a	DET
ejpam-6146	362	6	positive	positive	ADJ
ejpam-6146	362	7	integer	integer	NOUN
ejpam-6146	362	8	,	,	PUNCT
ejpam-6146	362	9	we	we	PRON
ejpam-6146	362	10	obtain	obtain	VERB
ejpam-6146	362	11	γ(n	γ(n	X
ejpam-6146	362	12	)	)	PUNCT
ejpam-6146	363	1	=	=	SYM
ejpam-6146	363	2	(	(	PUNCT
ejpam-6146	363	3	n−1	n−1	PROPN
ejpam-6146	363	4	)	)	PUNCT
ejpam-6146	363	5	!	!	PUNCT
ejpam-6146	364	1	which	which	PRON
ejpam-6146	364	2	is	be	AUX
ejpam-6146	364	3	the	the	DET
ejpam-6146	364	4	factorial	factorial	ADJ
ejpam-6146	364	5	function	function	NOUN
ejpam-6146	364	6	.	.	PUNCT
ejpam-6146	365	1	in	in	ADP
ejpam-6146	365	2	such	such	ADJ
ejpam-6146	365	3	cases	case	NOUN
ejpam-6146	365	4	,	,	PUNCT
ejpam-6146	365	5	the	the	DET
ejpam-6146	365	6	solutions	solution	NOUN
ejpam-6146	365	7	are	be	AUX
ejpam-6146	365	8	jn	jn	PROPN
ejpam-6146	365	9	=	=	PROPN
ejpam-6146	366	1	∞∑	∞∑	NUM
ejpam-6146	366	2	m=0	m=0	PROPN
ejpam-6146	366	3	(	(	PUNCT
ejpam-6146	366	4	−1)m	−1)m	PROPN
ejpam-6146	366	5	1	1	NUM
ejpam-6146	366	6	m!(m+	m!(m+	NOUN
ejpam-6146	366	7	n	n	CCONJ
ejpam-6146	366	8	)	)	PUNCT
ejpam-6146	366	9	!	!	PUNCT
ejpam-6146	367	1	(	(	PUNCT
ejpam-6146	367	2	x	x	SYM
ejpam-6146	367	3	2	2	NUM
ejpam-6146	367	4	)	)	PUNCT
ejpam-6146	367	5	2m+n	2m+n	NUM
ejpam-6146	367	6	.	.	PUNCT
ejpam-6146	368	1	(	(	PUNCT
ejpam-6146	368	2	27	27	NUM
ejpam-6146	368	3	)	)	PUNCT
ejpam-6146	368	4	in	in	ADP
ejpam-6146	368	5	the	the	DET
ejpam-6146	368	6	context	context	NOUN
ejpam-6146	368	7	of	of	ADP
ejpam-6146	368	8	theorem	theorem	NOUN
ejpam-6146	368	9	(	(	PUNCT
ejpam-6146	368	10	7	7	NUM
ejpam-6146	368	11	)	)	PUNCT
ejpam-6146	368	12	,	,	PUNCT
ejpam-6146	368	13	we	we	PRON
ejpam-6146	368	14	note	note	VERB
ejpam-6146	368	15	the	the	DET
ejpam-6146	368	16	following	follow	VERB
ejpam-6146	368	17	properties	property	NOUN
ejpam-6146	368	18	(	(	PUNCT
ejpam-6146	368	19	i	i	NOUN
ejpam-6146	368	20	)	)	PUNCT
ejpam-6146	368	21	bessel	bessel	NOUN
ejpam-6146	368	22	function	function	NOUN
ejpam-6146	368	23	for	for	ADP
ejpam-6146	368	24	negative	negative	ADJ
ejpam-6146	368	25	integers	integer	NOUN
ejpam-6146	368	26	is	be	AUX
ejpam-6146	368	27	j−n(x	j−n(x	ADJ
ejpam-6146	368	28	)	)	PUNCT
ejpam-6146	369	1	=	=	SYM
ejpam-6146	369	2	(	(	PUNCT
ejpam-6146	369	3	−1)njn(x	−1)njn(x	ADJ
ejpam-6146	369	4	)	)	PUNCT
ejpam-6146	369	5	.	.	PUNCT
ejpam-6146	370	1	(	(	PUNCT
ejpam-6146	370	2	ii	ii	NOUN
ejpam-6146	370	3	)	)	PUNCT
ejpam-6146	370	4	in	in	ADP
ejpam-6146	370	5	theorem	theorem	NOUN
ejpam-6146	370	6	7	7	NUM
ejpam-6146	370	7	,	,	PUNCT
ejpam-6146	370	8	there	there	PRON
ejpam-6146	370	9	is	be	VERB
ejpam-6146	370	10	a	a	DET
ejpam-6146	370	11	special	special	ADJ
ejpam-6146	370	12	value	value	NOUN
ejpam-6146	370	13	p	p	NOUN
ejpam-6146	370	14	=	=	NOUN
ejpam-6146	370	15	1/2	1/2	NUM
ejpam-6146	370	16	.	.	PUNCT
ejpam-6146	371	1	recalling	recall	VERB
ejpam-6146	371	2	that	that	PRON
ejpam-6146	371	3	,	,	PUNCT
ejpam-6146	371	4	γ(12	γ(12	ADJ
ejpam-6146	371	5	)	)	PUNCT
ejpam-6146	371	6	=	=	PUNCT
ejpam-6146	371	7	√	√	NUM
ejpam-6146	371	8	π	π	X
ejpam-6146	371	9	,	,	PUNCT
ejpam-6146	371	10	we	we	PRON
ejpam-6146	371	11	note	note	VERB
ejpam-6146	371	12	j	j	PROPN
ejpam-6146	371	13	1	1	NUM
ejpam-6146	371	14	2	2	NUM
ejpam-6146	371	15	(	(	PUNCT
ejpam-6146	371	16	x	x	NOUN
ejpam-6146	371	17	)	)	PUNCT
ejpam-6146	371	18	=	=	SYM
ejpam-6146	371	19	√	√	NUM
ejpam-6146	371	20	2	2	NUM
ejpam-6146	371	21	πx	πx	ADP
ejpam-6146	371	22	sinx	sinx	X
ejpam-6146	371	23	j−	j−	VERB
ejpam-6146	371	24	1	1	NUM
ejpam-6146	371	25	2	2	NUM
ejpam-6146	371	26	(	(	PUNCT
ejpam-6146	371	27	x	x	NOUN
ejpam-6146	371	28	)	)	PUNCT
ejpam-6146	371	29	=	=	SYM
ejpam-6146	371	30	√	√	NUM
ejpam-6146	371	31	2	2	NUM
ejpam-6146	371	32	πx	πx	ADP
ejpam-6146	371	33	cosx	cosx	PROPN
ejpam-6146	371	34	.	.	PUNCT
ejpam-6146	372	1	s.	s.	PROPN
ejpam-6146	372	2	a.	a.	PROPN
ejpam-6146	372	3	khan	khan	PROPN
ejpam-6146	372	4	,	,	PUNCT
ejpam-6146	372	5	m.	m.	NOUN
ejpam-6146	372	6	m.	m.	PROPN
ejpam-6146	372	7	kankarej	kankarej	PROPN
ejpam-6146	372	8	,	,	PUNCT
ejpam-6146	372	9	m.	m.	PROPN
ejpam-6146	372	10	n.	n.	PROPN
ejpam-6146	372	11	i.	i.	PROPN
ejpam-6146	372	12	khan	khan	PROPN
ejpam-6146	372	13	/	/	SYM
ejpam-6146	372	14	eur	eur	PROPN
ejpam-6146	372	15	.	.	PUNCT
ejpam-6146	373	1	j.	j.	PROPN
ejpam-6146	373	2	pure	pure	PROPN
ejpam-6146	373	3	appl	appl	PROPN
ejpam-6146	373	4	.	.	PROPN
ejpam-6146	373	5	math	math	PROPN
ejpam-6146	373	6	,	,	PUNCT
ejpam-6146	373	7	18	18	NUM
ejpam-6146	373	8	(	(	PUNCT
ejpam-6146	373	9	4	4	NUM
ejpam-6146	373	10	)	)	PUNCT
ejpam-6146	373	11	(	(	PUNCT
ejpam-6146	373	12	2025	2025	NUM
ejpam-6146	373	13	)	)	PUNCT
ejpam-6146	373	14	,	,	PUNCT
ejpam-6146	373	15	6146	6146	NUM
ejpam-6146	373	16	12	12	NUM
ejpam-6146	373	17	of	of	ADP
ejpam-6146	373	18	14	14	NUM
ejpam-6146	373	19	6	6	NUM
ejpam-6146	373	20	.	.	PUNCT
ejpam-6146	374	1	orthogonal	orthogonal	ADJ
ejpam-6146	374	2	polynomials	polynomial	NOUN
ejpam-6146	374	3	any	any	DET
ejpam-6146	374	4	sequence	sequence	NOUN
ejpam-6146	374	5	of	of	ADP
ejpam-6146	374	6	polynomials	polynomial	NOUN
ejpam-6146	374	7	{	{	PUNCT
ejpam-6146	374	8	pn(x	pn(x	ADJ
ejpam-6146	374	9	)	)	PUNCT
ejpam-6146	374	10	}	}	PUNCT
ejpam-6146	374	11	in	in	ADP
ejpam-6146	374	12	which	which	PRON
ejpam-6146	374	13	the	the	DET
ejpam-6146	374	14	degree	degree	NOUN
ejpam-6146	374	15	of	of	ADP
ejpam-6146	374	16	pn(x	pn(x	PRON
ejpam-6146	374	17	)	)	PUNCT
ejpam-6146	374	18	is	be	AUX
ejpam-6146	374	19	n	n	PRON
ejpam-6146	374	20	for	for	ADP
ejpam-6146	374	21	all	all	PRON
ejpam-6146	374	22	n	n	PRON
ejpam-6146	374	23	is	be	AUX
ejpam-6146	374	24	said	say	VERB
ejpam-6146	374	25	to	to	PART
ejpam-6146	374	26	be	be	AUX
ejpam-6146	374	27	orthogonal	orthogonal	ADJ
ejpam-6146	374	28	with	with	ADP
ejpam-6146	374	29	respect	respect	NOUN
ejpam-6146	374	30	to	to	ADP
ejpam-6146	374	31	the	the	DET
ejpam-6146	374	32	measure	measure	NOUN
ejpam-6146	374	33	dα(x	dα(x	NOUN
ejpam-6146	374	34	)	)	PUNCT
ejpam-6146	374	35	if∫	if∫	PROPN
ejpam-6146	374	36	b	b	PROPN
ejpam-6146	374	37	a	a	DET
ejpam-6146	374	38	pn(x)pm(x)dα(x	pn(x)pm(x)dα(x	NOUN
ejpam-6146	374	39	)	)	PUNCT
ejpam-6146	374	40	=	=	SYM
ejpam-6146	374	41	0	0	NUM
ejpam-6146	374	42	,	,	PUNCT
ejpam-6146	374	43	n	n	CCONJ
ejpam-6146	375	1	̸=	̸=	PROPN
ejpam-6146	375	2	m,∫	m,∫	NOUN
ejpam-6146	375	3	b	b	PROPN
ejpam-6146	375	4	a	a	DET
ejpam-6146	375	5	pn(x)pn(x)dα(x	pn(x)pn(x)dα(x	PROPN
ejpam-6146	375	6	)	)	PUNCT
ejpam-6146	375	7	̸=	̸=	PROPN
ejpam-6146	375	8	0	0	NUM
ejpam-6146	375	9	,	,	PUNCT
ejpam-6146	375	10	∀n	∀n	NUM
ejpam-6146	375	11	≥	≥	NOUN
ejpam-6146	375	12	0	0	NUM
ejpam-6146	375	13	.	.	PUNCT
ejpam-6146	376	1	(	(	PUNCT
ejpam-6146	376	2	28	28	NUM
ejpam-6146	376	3	)	)	PUNCT
ejpam-6146	376	4	the	the	DET
ejpam-6146	376	5	limits	limit	NOUN
ejpam-6146	376	6	of	of	ADP
ejpam-6146	376	7	integration	integration	NOUN
ejpam-6146	376	8	can	can	AUX
ejpam-6146	376	9	be	be	AUX
ejpam-6146	376	10	finite	finite	ADJ
ejpam-6146	376	11	(	(	PUNCT
ejpam-6146	376	12	as	as	ADP
ejpam-6146	376	13	in	in	ADP
ejpam-6146	376	14	the	the	DET
ejpam-6146	376	15	case	case	NOUN
ejpam-6146	376	16	of	of	ADP
ejpam-6146	376	17	laguerre	laguerre	NOUN
ejpam-6146	376	18	polynomials	polynomial	NOUN
ejpam-6146	376	19	and	and	CCONJ
ejpam-6146	376	20	jacobi	jacobi	PROPN
ejpam-6146	376	21	polynomials	polynomial	NOUN
ejpam-6146	376	22	)	)	PUNCT
ejpam-6146	376	23	or	or	CCONJ
ejpam-6146	376	24	infinite	infinite	ADJ
ejpam-6146	376	25	(	(	PUNCT
ejpam-6146	376	26	as	as	ADP
ejpam-6146	376	27	in	in	ADP
ejpam-6146	376	28	the	the	DET
ejpam-6146	376	29	case	case	NOUN
ejpam-6146	376	30	of	of	ADP
ejpam-6146	376	31	hermite	hermite	ADJ
ejpam-6146	376	32	polynomials	polynomial	NOUN
ejpam-6146	376	33	)	)	PUNCT
ejpam-6146	376	34	.	.	PUNCT
ejpam-6146	377	1	when	when	SCONJ
ejpam-6146	377	2	the	the	DET
ejpam-6146	377	3	measure	measure	NOUN
ejpam-6146	377	4	is	be	AUX
ejpam-6146	377	5	absolutely	absolutely	ADV
ejpam-6146	377	6	continuous	continuous	ADJ
ejpam-6146	377	7	,	,	PUNCT
ejpam-6146	377	8	that	that	PRON
ejpam-6146	377	9	is	be	AUX
ejpam-6146	377	10	dα(x	dα(x	NOUN
ejpam-6146	377	11	)	)	PUNCT
ejpam-6146	377	12	=	=	SYM
ejpam-6146	377	13	w(x	w(x	X
ejpam-6146	377	14	)	)	PUNCT
ejpam-6146	377	15	dx	dx	PROPN
ejpam-6146	377	16	,	,	PUNCT
ejpam-6146	377	17	where	where	SCONJ
ejpam-6146	377	18	w(x	w(x	NOUN
ejpam-6146	377	19	)	)	PUNCT
ejpam-6146	377	20	is	be	AUX
ejpam-6146	377	21	called	call	VERB
ejpam-6146	377	22	the	the	DET
ejpam-6146	377	23	weight	weight	NOUN
ejpam-6146	377	24	function	function	NOUN
ejpam-6146	377	25	,	,	PUNCT
ejpam-6146	377	26	then	then	ADV
ejpam-6146	377	27	the	the	DET
ejpam-6146	377	28	relations	relation	NOUN
ejpam-6146	377	29	in	in	ADP
ejpam-6146	377	30	(	(	PUNCT
ejpam-6146	377	31	28	28	NUM
ejpam-6146	377	32	)	)	PUNCT
ejpam-6146	377	33	become∫	become∫	NOUN
ejpam-6146	377	34	b	b	X
ejpam-6146	377	35	a	a	DET
ejpam-6146	377	36	pn(x)pm(x)w(x)dx	pn(x)pm(x)w(x)dx	NOUN
ejpam-6146	377	37	=	=	SYM
ejpam-6146	377	38	0	0	NUM
ejpam-6146	377	39	,	,	PUNCT
ejpam-6146	377	40	n	n	CCONJ
ejpam-6146	377	41	̸=	̸=	PROPN
ejpam-6146	377	42	m,∫	m,∫	NOUN
ejpam-6146	377	43	b	b	PROPN
ejpam-6146	377	44	a	a	DET
ejpam-6146	377	45	pn(x)pn(x)w(x)dx	pn(x)pn(x)w(x)dx	ADJ
ejpam-6146	377	46	̸=	̸=	NOUN
ejpam-6146	377	47	0	0	NUM
ejpam-6146	377	48	,	,	PUNCT
ejpam-6146	377	49	∀n	∀n	NUM
ejpam-6146	377	50	≥	≥	NOUN
ejpam-6146	377	51	0	0	NUM
ejpam-6146	377	52	.	.	PUNCT
ejpam-6146	378	1	(	(	PUNCT
ejpam-6146	378	2	29	29	NUM
ejpam-6146	378	3	)	)	PUNCT
ejpam-6146	378	4	the	the	DET
ejpam-6146	378	5	classical	classical	ADJ
ejpam-6146	378	6	orthogonal	orthogonal	ADJ
ejpam-6146	378	7	polynomials	polynomial	NOUN
ejpam-6146	378	8	are	be	AUX
ejpam-6146	378	9	hermite	hermite	ADJ
ejpam-6146	378	10	polynomials	polynomial	NOUN
ejpam-6146	378	11	,	,	PUNCT
ejpam-6146	378	12	laguerre	laguerre	NOUN
ejpam-6146	378	13	polynomials	polynomial	NOUN
ejpam-6146	378	14	and	and	CCONJ
ejpam-6146	378	15	jacobi	jacobi	PROPN
ejpam-6146	378	16	polynomials	polynomial	NOUN
ejpam-6146	378	17	(	(	PUNCT
ejpam-6146	378	18	also	also	ADV
ejpam-6146	378	19	known	know	VERB
ejpam-6146	378	20	as	as	ADP
ejpam-6146	378	21	hypergeometric	hypergeometric	ADJ
ejpam-6146	378	22	polynomials	polynomial	NOUN
ejpam-6146	378	23	and	and	CCONJ
ejpam-6146	378	24	the	the	DET
ejpam-6146	378	25	special	special	ADJ
ejpam-6146	378	26	cases	case	NOUN
ejpam-6146	378	27	include	include	VERB
ejpam-6146	378	28	,	,	PUNCT
ejpam-6146	378	29	gegenbauer	gegenbauer	PROPN
ejpam-6146	378	30	,	,	PUNCT
ejpam-6146	378	31	legendre	legendre	PROPN
ejpam-6146	378	32	,	,	PUNCT
ejpam-6146	378	33	zernike	zernike	PROPN
ejpam-6146	378	34	and	and	CCONJ
ejpam-6146	378	35	chebyshev	chebyshev	NOUN
ejpam-6146	378	36	polynomials	polynomial	NOUN
ejpam-6146	378	37	)	)	PUNCT
ejpam-6146	378	38	.	.	PUNCT
ejpam-6146	379	1	the	the	DET
ejpam-6146	379	2	orthogonal	orthogonal	ADJ
ejpam-6146	379	3	polynomials	polynomial	NOUN
ejpam-6146	379	4	occur	occur	VERB
ejpam-6146	379	5	across	across	ADP
ejpam-6146	379	6	mathematics	mathematic	NOUN
ejpam-6146	379	7	and	and	CCONJ
ejpam-6146	379	8	sciences	science	NOUN
ejpam-6146	379	9	.	.	PUNCT
ejpam-6146	380	1	all	all	DET
ejpam-6146	380	2	the	the	DET
ejpam-6146	380	3	classical	classical	ADJ
ejpam-6146	380	4	orthogonal	orthogonal	ADJ
ejpam-6146	380	5	polynomials	polynomial	NOUN
ejpam-6146	380	6	are	be	AUX
ejpam-6146	380	7	governed	govern	VERB
ejpam-6146	380	8	by	by	ADP
ejpam-6146	380	9	the	the	DET
ejpam-6146	380	10	second	second	ADJ
ejpam-6146	380	11	-	-	PUNCT
ejpam-6146	380	12	order	order	NOUN
ejpam-6146	380	13	linear	linear	ADJ
ejpam-6146	380	14	differential	differential	NOUN
ejpam-6146	380	15	equation	equation	NOUN
ejpam-6146	380	16	of	of	ADP
ejpam-6146	380	17	the	the	DET
ejpam-6146	380	18	sturm	sturm	NOUN
ejpam-6146	380	19	-	-	PUNCT
ejpam-6146	380	20	liouville	liouville	NOUN
ejpam-6146	380	21	type	type	NOUN
ejpam-6146	380	22	q(x)y′′	q(x)y′′	PROPN
ejpam-6146	381	1	+	+	CCONJ
ejpam-6146	381	2	l(x)y′	l(x)y′	ADJ
ejpam-6146	381	3	+	+	CCONJ
ejpam-6146	381	4	λny	λny	NOUN
ejpam-6146	381	5	=	=	SYM
ejpam-6146	381	6	0	0	NUM
ejpam-6146	381	7	,	,	PUNCT
ejpam-6146	381	8	(	(	PUNCT
ejpam-6146	381	9	30	30	NUM
ejpam-6146	381	10	)	)	PUNCT
ejpam-6146	381	11	where	where	SCONJ
ejpam-6146	381	12	q(x	q(x	NOUN
ejpam-6146	381	13	)	)	PUNCT
ejpam-6146	381	14	is	be	AUX
ejpam-6146	381	15	a	a	DET
ejpam-6146	381	16	polynomial	polynomial	NOUN
ejpam-6146	381	17	in	in	ADP
ejpam-6146	381	18	x	x	PUNCT
ejpam-6146	381	19	of	of	ADP
ejpam-6146	381	20	degree	degree	NOUN
ejpam-6146	381	21	≤	≤	ADV
ejpam-6146	381	22	2	2	NUM
ejpam-6146	381	23	(	(	PUNCT
ejpam-6146	381	24	i.e.	i.e.	X
ejpam-6146	381	25	,	,	PUNCT
ejpam-6146	381	26	at	at	ADP
ejpam-6146	381	27	most	most	ADJ
ejpam-6146	381	28	a	a	DET
ejpam-6146	381	29	quadratic	quadratic	ADJ
ejpam-6146	381	30	polynomial	polynomial	NOUN
ejpam-6146	381	31	)	)	PUNCT
ejpam-6146	381	32	,	,	PUNCT
ejpam-6146	381	33	l(x	l(x	PROPN
ejpam-6146	381	34	)	)	PUNCT
ejpam-6146	381	35	is	be	AUX
ejpam-6146	381	36	a	a	DET
ejpam-6146	381	37	polynomial	polynomial	NOUN
ejpam-6146	381	38	in	in	ADP
ejpam-6146	381	39	x	x	PUNCT
ejpam-6146	381	40	of	of	ADP
ejpam-6146	381	41	degree	degree	NOUN
ejpam-6146	381	42	≤	≤	ADV
ejpam-6146	381	43	1	1	NUM
ejpam-6146	381	44	(	(	PUNCT
ejpam-6146	381	45	i.e.	i.e.	X
ejpam-6146	381	46	,	,	PUNCT
ejpam-6146	381	47	at	at	ADP
ejpam-6146	381	48	most	most	ADJ
ejpam-6146	381	49	a	a	DET
ejpam-6146	381	50	linear	linear	ADJ
ejpam-6146	381	51	polynomial	polynomial	NOUN
ejpam-6146	381	52	)	)	PUNCT
ejpam-6146	381	53	,	,	PUNCT
ejpam-6146	381	54	and	and	CCONJ
ejpam-6146	381	55	λn	λn	PROPN
ejpam-6146	381	56	is	be	AUX
ejpam-6146	381	57	independent	independent	ADJ
ejpam-6146	381	58	of	of	ADP
ejpam-6146	381	59	x	x	X
ejpam-6146	381	60	(	(	PUNCT
ejpam-6146	381	61	i.e.	i.e.	X
ejpam-6146	381	62	,	,	PUNCT
ejpam-6146	381	63	a	a	DET
ejpam-6146	381	64	number	number	NOUN
ejpam-6146	381	65	)	)	PUNCT
ejpam-6146	381	66	.	.	PUNCT
ejpam-6146	382	1	both	both	DET
ejpam-6146	382	2	q(x	q(x	NOUN
ejpam-6146	382	3	)	)	PUNCT
ejpam-6146	382	4	and	and	CCONJ
ejpam-6146	382	5	l(x	l(x	PROPN
ejpam-6146	382	6	)	)	PUNCT
ejpam-6146	382	7	are	be	AUX
ejpam-6146	382	8	independent	independent	ADJ
ejpam-6146	382	9	of	of	ADP
ejpam-6146	382	10	n.	n.	NOUN
ejpam-6146	382	11	for	for	ADP
ejpam-6146	382	12	example	example	NOUN
ejpam-6146	382	13	,	,	PUNCT
ejpam-6146	382	14	the	the	DET
ejpam-6146	382	15	hermite	hermite	ADJ
ejpam-6146	382	16	polynomials	polynomial	NOUN
ejpam-6146	382	17	satisfy	satisfy	VERB
ejpam-6146	382	18	the	the	DET
ejpam-6146	382	19	differential	differential	ADJ
ejpam-6146	382	20	equation	equation	NOUN
ejpam-6146	382	21	,	,	PUNCT
ejpam-6146	382	22	y′′	y′′	PROPN
ejpam-6146	382	23	−	−	PROPN
ejpam-6146	382	24	2xy′	2xy′	NUM
ejpam-6146	382	25	+	+	CCONJ
ejpam-6146	382	26	2ny	2ny	ADJ
ejpam-6146	382	27	=	=	ADJ
ejpam-6146	382	28	0	0	X
ejpam-6146	382	29	.	.	PUNCT
ejpam-6146	383	1	for	for	ADP
ejpam-6146	383	2	each	each	DET
ejpam-6146	383	3	value	value	NOUN
ejpam-6146	383	4	of	of	ADP
ejpam-6146	383	5	n	n	NOUN
ejpam-6146	383	6	=	=	SYM
ejpam-6146	383	7	0	0	NUM
ejpam-6146	383	8	,	,	PUNCT
ejpam-6146	383	9	1	1	NUM
ejpam-6146	383	10	,	,	PUNCT
ejpam-6146	383	11	2	2	NUM
ejpam-6146	383	12	,	,	PUNCT
ejpam-6146	383	13	3	3	NUM
ejpam-6146	383	14	,	,	PUNCT
ejpam-6146	383	15	.	.	PUNCT
ejpam-6146	383	16	.	.	PUNCT
ejpam-6146	383	17	.	.	PUNCT
ejpam-6146	384	1	,	,	PUNCT
ejpam-6146	384	2	the	the	DET
ejpam-6146	384	3	solutions	solution	NOUN
ejpam-6146	384	4	are	be	AUX
ejpam-6146	384	5	n	n	PRON
ejpam-6146	384	6	-	-	PUNCT
ejpam-6146	384	7	th	th	VERB
ejpam-6146	384	8	degree	degree	NOUN
ejpam-6146	384	9	polynomials	polynomial	NOUN
ejpam-6146	384	10	.	.	PUNCT
ejpam-6146	385	1	the	the	DET
ejpam-6146	385	2	equation	equation	NOUN
ejpam-6146	385	3	for	for	ADP
ejpam-6146	385	4	the	the	DET
ejpam-6146	385	5	laguerre	laguerre	NOUN
ejpam-6146	385	6	polynomials	polynomial	NOUN
ejpam-6146	385	7	is	be	AUX
ejpam-6146	385	8	xy′′	xy′′	PROPN
ejpam-6146	386	1	+	+	CCONJ
ejpam-6146	386	2	(	(	PUNCT
ejpam-6146	386	3	1−	1−	NUM
ejpam-6146	386	4	x)y′	x)y′	PROPN
ejpam-6146	387	1	+	+	CCONJ
ejpam-6146	387	2	ny	ny	PROPN
ejpam-6146	387	3	=	=	SYM
ejpam-6146	387	4	0	0	PROPN
ejpam-6146	387	5	.	.	PUNCT
ejpam-6146	387	6	orthogonal	orthogonal	ADJ
ejpam-6146	387	7	polynomials	polynomial	NOUN
ejpam-6146	387	8	have	have	VERB
ejpam-6146	387	9	many	many	ADJ
ejpam-6146	387	10	properties	property	NOUN
ejpam-6146	387	11	.	.	PUNCT
ejpam-6146	388	1	derivatives	derivative	NOUN
ejpam-6146	388	2	of	of	ADP
ejpam-6146	388	3	the	the	DET
ejpam-6146	388	4	orthogonal	orthogonal	ADJ
ejpam-6146	388	5	polynomials	polynomial	NOUN
ejpam-6146	388	6	sets	set	NOUN
ejpam-6146	388	7	also	also	ADV
ejpam-6146	388	8	form	form	VERB
ejpam-6146	388	9	orthogonal	orthogonal	ADJ
ejpam-6146	388	10	polynomials	polynomial	NOUN
ejpam-6146	388	11	sets	set	NOUN
ejpam-6146	388	12	.	.	PUNCT
ejpam-6146	389	1	the	the	DET
ejpam-6146	389	2	roots	root	NOUN
ejpam-6146	389	3	of	of	ADP
ejpam-6146	389	4	the	the	DET
ejpam-6146	389	5	orthogonal	orthogonal	ADJ
ejpam-6146	389	6	polynomials	polynomial	NOUN
ejpam-6146	389	7	are	be	AUX
ejpam-6146	389	8	all	all	ADV
ejpam-6146	389	9	real	real	ADJ
ejpam-6146	389	10	,	,	PUNCT
ejpam-6146	389	11	distinct	distinct	ADJ
ejpam-6146	389	12	and	and	CCONJ
ejpam-6146	389	13	lie	lie	VERB
ejpam-6146	389	14	in	in	ADP
ejpam-6146	389	15	the	the	DET
ejpam-6146	389	16	interval	interval	NOUN
ejpam-6146	389	17	of	of	ADP
ejpam-6146	389	18	orthogonality	orthogonality	NOUN
ejpam-6146	389	19	.	.	PUNCT
ejpam-6146	390	1	in	in	ADP
ejpam-6146	390	2	the	the	DET
ejpam-6146	390	3	present	present	ADJ
ejpam-6146	390	4	context	context	NOUN
ejpam-6146	390	5	,	,	PUNCT
ejpam-6146	390	6	we	we	PRON
ejpam-6146	390	7	note	note	VERB
ejpam-6146	390	8	the	the	DET
ejpam-6146	390	9	property	property	NOUN
ejpam-6146	390	10	of	of	ADP
ejpam-6146	390	11	the	the	DET
ejpam-6146	390	12	interlacing	interlacing	NOUN
ejpam-6146	390	13	of	of	ADP
ejpam-6146	390	14	zeros	zero	NOUN
ejpam-6146	390	15	of	of	ADP
ejpam-6146	390	16	the	the	DET
ejpam-6146	390	17	orthogonal	orthogonal	ADJ
ejpam-6146	390	18	polynomials	polynomial	NOUN
ejpam-6146	390	19	.	.	PUNCT
ejpam-6146	391	1	if	if	SCONJ
ejpam-6146	391	2	{	{	PUNCT
ejpam-6146	391	3	xn	xn	PROPN
ejpam-6146	391	4	,	,	PUNCT
ejpam-6146	391	5	k}nk=1	k}nk=1	PROPN
ejpam-6146	391	6	and	and	CCONJ
ejpam-6146	391	7	{	{	PUNCT
ejpam-6146	391	8	xn+1	xn+1	PROPN
ejpam-6146	391	9	,	,	PUNCT
ejpam-6146	391	10	k}n+1	k}n+1	PROPN
ejpam-6146	391	11	k=1	k=1	PROPN
ejpam-6146	391	12	denote	denote	VERB
ejpam-6146	391	13	the	the	DET
ejpam-6146	391	14	consecutive	consecutive	ADJ
ejpam-6146	391	15	zeros	zero	NOUN
ejpam-6146	391	16	of	of	ADP
ejpam-6146	391	17	pn(x	pn(x	X
ejpam-6146	391	18	)	)	PUNCT
ejpam-6146	391	19	and	and	CCONJ
ejpam-6146	391	20	pn+1(x	pn+1(x	NOUN
ejpam-6146	391	21	)	)	PUNCT
ejpam-6146	391	22	respectively	respectively	ADV
ejpam-6146	391	23	,	,	PUNCT
ejpam-6146	391	24	then	then	ADV
ejpam-6146	391	25	we	we	PRON
ejpam-6146	391	26	have	have	VERB
ejpam-6146	391	27	a	a	DET
ejpam-6146	391	28	<	<	X
ejpam-6146	391	29	xn+1	xn+1	PROPN
ejpam-6146	391	30	,	,	PUNCT
ejpam-6146	391	31	1	1	NUM
ejpam-6146	391	32	<	<	X
ejpam-6146	391	33	xn	xn	PROPN
ejpam-6146	391	34	,	,	PUNCT
ejpam-6146	391	35	1	1	NUM
ejpam-6146	391	36	<	<	X
ejpam-6146	391	37	xn+1	xn+1	PROPN
ejpam-6146	391	38	,	,	PUNCT
ejpam-6146	391	39	2	2	NUM
ejpam-6146	391	40	<	<	X
ejpam-6146	391	41	xn	xn	PROPN
ejpam-6146	391	42	,	,	PUNCT
ejpam-6146	391	43	2	2	NUM
ejpam-6146	391	44	<	<	X
ejpam-6146	391	45	.	.	PUNCT
ejpam-6146	391	46	.	.	PUNCT
ejpam-6146	391	47	.	.	PUNCT
ejpam-6146	392	1	<	<	X
ejpam-6146	392	2	xn+1	xn+1	PROPN
ejpam-6146	392	3	,	,	PUNCT
ejpam-6146	392	4	n	n	CCONJ
ejpam-6146	392	5	<	<	X
ejpam-6146	392	6	xn	xn	PROPN
ejpam-6146	392	7	,	,	PUNCT
ejpam-6146	392	8	n	n	CCONJ
ejpam-6146	392	9	<	<	X
ejpam-6146	392	10	xn+1	xn+1	PROPN
ejpam-6146	392	11	,	,	PUNCT
ejpam-6146	392	12	n+1	n+1	PROPN
ejpam-6146	392	13	<	<	X
ejpam-6146	392	14	b	b	X
ejpam-6146	392	15	.	.	PUNCT
ejpam-6146	393	1	(	(	PUNCT
ejpam-6146	393	2	31	31	NUM
ejpam-6146	393	3	)	)	PUNCT
ejpam-6146	393	4	these	these	DET
ejpam-6146	393	5	properties	property	NOUN
ejpam-6146	393	6	acquire	acquire	VERB
ejpam-6146	393	7	an	an	DET
ejpam-6146	393	8	extra	extra	ADJ
ejpam-6146	393	9	significance	significance	NOUN
ejpam-6146	393	10	as	as	SCONJ
ejpam-6146	393	11	no	no	DET
ejpam-6146	393	12	formula	formula	NOUN
ejpam-6146	393	13	is	be	AUX
ejpam-6146	393	14	known	know	VERB
ejpam-6146	393	15	for	for	ADP
ejpam-6146	393	16	the	the	DET
ejpam-6146	393	17	roots	root	NOUN
ejpam-6146	393	18	of	of	ADP
ejpam-6146	393	19	any	any	PRON
ejpam-6146	393	20	of	of	ADP
ejpam-6146	393	21	the	the	DET
ejpam-6146	393	22	orthogonal	orthogonal	ADJ
ejpam-6146	393	23	polynomials	polynomial	NOUN
ejpam-6146	393	24	with	with	ADP
ejpam-6146	393	25	the	the	DET
ejpam-6146	393	26	exception	exception	NOUN
ejpam-6146	393	27	of	of	ADP
ejpam-6146	393	28	the	the	DET
ejpam-6146	393	29	chebyshev	chebyshev	NOUN
ejpam-6146	393	30	polynomials	polynomial	NOUN
ejpam-6146	393	31	.	.	PUNCT
ejpam-6146	394	1	the	the	DET
ejpam-6146	394	2	s.	s.	PROPN
ejpam-6146	394	3	a.	a.	PROPN
ejpam-6146	394	4	khan	khan	PROPN
ejpam-6146	394	5	,	,	PUNCT
ejpam-6146	394	6	m.	m.	NOUN
ejpam-6146	394	7	m.	m.	PROPN
ejpam-6146	394	8	kankarej	kankarej	PROPN
ejpam-6146	394	9	,	,	PUNCT
ejpam-6146	394	10	m.	m.	PROPN
ejpam-6146	394	11	n.	n.	PROPN
ejpam-6146	394	12	i.	i.	PROPN
ejpam-6146	394	13	khan	khan	PROPN
ejpam-6146	394	14	/	/	SYM
ejpam-6146	394	15	eur	eur	PROPN
ejpam-6146	394	16	.	.	PUNCT
ejpam-6146	395	1	j.	j.	PROPN
ejpam-6146	395	2	pure	pure	PROPN
ejpam-6146	395	3	appl	appl	PROPN
ejpam-6146	395	4	.	.	PROPN
ejpam-6146	395	5	math	math	PROPN
ejpam-6146	395	6	,	,	PUNCT
ejpam-6146	395	7	18	18	NUM
ejpam-6146	395	8	(	(	PUNCT
ejpam-6146	395	9	4	4	NUM
ejpam-6146	395	10	)	)	PUNCT
ejpam-6146	395	11	(	(	PUNCT
ejpam-6146	395	12	2025	2025	NUM
ejpam-6146	395	13	)	)	PUNCT
ejpam-6146	395	14	,	,	PUNCT
ejpam-6146	395	15	6146	6146	NUM
ejpam-6146	395	16	13	13	NUM
ejpam-6146	395	17	of	of	ADP
ejpam-6146	395	18	14	14	NUM
ejpam-6146	395	19	chebyshev	chebyshev	NOUN
ejpam-6146	395	20	polynomials	polynomial	NOUN
ejpam-6146	395	21	are	be	AUX
ejpam-6146	395	22	one	one	NUM
ejpam-6146	395	23	of	of	ADP
ejpam-6146	395	24	the	the	DET
ejpam-6146	395	25	special	special	ADJ
ejpam-6146	395	26	cases	case	NOUN
ejpam-6146	395	27	of	of	ADP
ejpam-6146	395	28	the	the	DET
ejpam-6146	395	29	jacobi	jacobi	PROPN
ejpam-6146	395	30	polynomials	polynomial	NOUN
ejpam-6146	395	31	.	.	PUNCT
ejpam-6146	396	1	chebyshev	chebyshev	NOUN
ejpam-6146	396	2	polynomials	polynomial	NOUN
ejpam-6146	396	3	of	of	ADP
ejpam-6146	396	4	the	the	DET
ejpam-6146	396	5	first	first	ADJ
ejpam-6146	396	6	kind	kind	NOUN
ejpam-6146	396	7	occur	occur	VERB
ejpam-6146	396	8	in	in	ADP
ejpam-6146	396	9	the	the	DET
ejpam-6146	396	10	expansion	expansion	NOUN
ejpam-6146	396	11	of	of	ADP
ejpam-6146	396	12	cos(nθ	cos(nθ	NOUN
ejpam-6146	396	13	)	)	PUNCT
ejpam-6146	396	14	=	=	SYM
ejpam-6146	396	15	tn(cos	tn(cos	PROPN
ejpam-6146	396	16	θ	θ	PROPN
ejpam-6146	396	17	)	)	PUNCT
ejpam-6146	396	18	.	.	PUNCT
ejpam-6146	397	1	chebyshev	chebyshev	PROPN
ejpam-6146	397	2	polynomials	polynomial	NOUN
ejpam-6146	397	3	of	of	ADP
ejpam-6146	397	4	the	the	DET
ejpam-6146	397	5	second	second	ADJ
ejpam-6146	397	6	kind	kind	NOUN
ejpam-6146	397	7	occur	occur	VERB
ejpam-6146	397	8	in	in	ADP
ejpam-6146	397	9	the	the	DET
ejpam-6146	397	10	expansion	expansion	NOUN
ejpam-6146	397	11	of	of	ADP
ejpam-6146	397	12	sin(nθ	sin(nθ	NOUN
ejpam-6146	397	13	)	)	PUNCT
ejpam-6146	397	14	=	=	PUNCT
ejpam-6146	397	15	sin	sin	NOUN
ejpam-6146	397	16	θ	θ	PROPN
ejpam-6146	397	17	un−1(cos	un−1(cos	PROPN
ejpam-6146	397	18	θ	θ	PROPN
ejpam-6146	397	19	)	)	PUNCT
ejpam-6146	397	20	.	.	PUNCT
ejpam-6146	398	1	a	a	DET
ejpam-6146	398	2	chebyshev	chebyshev	NOUN
ejpam-6146	398	3	polynomial	polynomial	NOUN
ejpam-6146	398	4	of	of	ADP
ejpam-6146	398	5	either	either	DET
ejpam-6146	398	6	kind	kind	NOUN
ejpam-6146	398	7	of	of	ADP
ejpam-6146	398	8	degree	degree	NOUN
ejpam-6146	398	9	n	n	PRON
ejpam-6146	398	10	has	have	VERB
ejpam-6146	398	11	n	n	NUM
ejpam-6146	398	12	different	different	ADJ
ejpam-6146	398	13	simple	simple	ADJ
ejpam-6146	398	14	roots	root	NOUN
ejpam-6146	398	15	,	,	PUNCT
ejpam-6146	398	16	called	call	VERB
ejpam-6146	398	17	chebyshev	chebyshev	NOUN
ejpam-6146	398	18	roots	root	NOUN
ejpam-6146	398	19	,	,	PUNCT
ejpam-6146	398	20	in	in	ADP
ejpam-6146	398	21	the	the	DET
ejpam-6146	398	22	interval	interval	NOUN
ejpam-6146	398	23	of	of	ADP
ejpam-6146	398	24	orthogonality	orthogonality	NOUN
ejpam-6146	398	25	[	[	X
ejpam-6146	398	26	−1	−1	NOUN
ejpam-6146	398	27	,	,	PUNCT
ejpam-6146	398	28	1	1	NUM
ejpam-6146	398	29	]	]	PUNCT
ejpam-6146	398	30	.	.	PUNCT
ejpam-6146	399	1	the	the	DET
ejpam-6146	399	2	n	n	PRON
ejpam-6146	399	3	roots	root	NOUN
ejpam-6146	399	4	of	of	ADP
ejpam-6146	399	5	tn(x	tn(x	PUNCT
ejpam-6146	399	6	)	)	PUNCT
ejpam-6146	399	7	are	be	AUX
ejpam-6146	399	8	xk	xk	PROPN
ejpam-6146	399	9	=	=	PUNCT
ejpam-6146	399	10	cos	cos	PROPN
ejpam-6146	399	11	(	(	PUNCT
ejpam-6146	399	12	(	(	PUNCT
ejpam-6146	399	13	k	k	X
ejpam-6146	399	14	+	+	NOUN
ejpam-6146	399	15	1	1	NUM
ejpam-6146	399	16	2	2	NUM
ejpam-6146	399	17	)	)	PUNCT
ejpam-6146	399	18	n	n	NUM
ejpam-6146	399	19	π	π	NOUN
ejpam-6146	399	20	)	)	PUNCT
ejpam-6146	399	21	,	,	PUNCT
ejpam-6146	399	22	k	k	X
ejpam-6146	399	23	=	=	PUNCT
ejpam-6146	399	24	0	0	PROPN
ejpam-6146	399	25	,	,	PUNCT
ejpam-6146	399	26	.	.	PUNCT
ejpam-6146	399	27	.	.	PUNCT
ejpam-6146	400	1	.	.	PUNCT
ejpam-6146	401	1	,	,	PUNCT
ejpam-6146	401	2	n−	n−	NOUN
ejpam-6146	401	3	1	1	NUM
ejpam-6146	401	4	.	.	PUNCT
ejpam-6146	402	1	(	(	PUNCT
ejpam-6146	402	2	32	32	NUM
ejpam-6146	402	3	)	)	PUNCT
ejpam-6146	402	4	the	the	DET
ejpam-6146	402	5	n	n	PRON
ejpam-6146	402	6	roots	root	NOUN
ejpam-6146	402	7	of	of	ADP
ejpam-6146	402	8	un(x	un(x	NOUN
ejpam-6146	402	9	)	)	PUNCT
ejpam-6146	402	10	are	be	AUX
ejpam-6146	402	11	xk	xk	PROPN
ejpam-6146	402	12	=	=	PUNCT
ejpam-6146	402	13	cos	cos	PROPN
ejpam-6146	402	14	(	(	PUNCT
ejpam-6146	402	15	k	k	X
ejpam-6146	402	16	n+	n+	ADP
ejpam-6146	402	17	1	1	NUM
ejpam-6146	402	18	π	π	NOUN
ejpam-6146	402	19	)	)	PUNCT
ejpam-6146	402	20	,	,	PUNCT
ejpam-6146	402	21	k	k	X
ejpam-6146	403	1	=	=	SYM
ejpam-6146	403	2	1	1	NUM
ejpam-6146	403	3	,	,	PUNCT
ejpam-6146	403	4	.	.	PUNCT
ejpam-6146	403	5	.	.	PUNCT
ejpam-6146	404	1	.	.	PUNCT
ejpam-6146	405	1	,	,	PUNCT
ejpam-6146	405	2	n	n	X
ejpam-6146	405	3	.	.	PUNCT
ejpam-6146	406	1	(	(	PUNCT
ejpam-6146	406	2	33	33	NUM
ejpam-6146	406	3	)	)	PUNCT
ejpam-6146	406	4	as	as	ADP
ejpam-6146	406	5	no	no	DET
ejpam-6146	406	6	such	such	ADJ
ejpam-6146	406	7	formulae	formulae	NOUN
ejpam-6146	406	8	are	be	AUX
ejpam-6146	406	9	known	know	VERB
ejpam-6146	406	10	for	for	ADP
ejpam-6146	406	11	any	any	PRON
ejpam-6146	406	12	of	of	ADP
ejpam-6146	406	13	the	the	DET
ejpam-6146	406	14	other	other	ADJ
ejpam-6146	406	15	orthogonal	orthogonal	ADJ
ejpam-6146	406	16	polynomials	polynomial	NOUN
ejpam-6146	406	17	,	,	PUNCT
ejpam-6146	406	18	the	the	DET
ejpam-6146	406	19	topic	topic	NOUN
ejpam-6146	406	20	is	be	AUX
ejpam-6146	406	21	of	of	ADP
ejpam-6146	406	22	active	active	ADJ
ejpam-6146	406	23	research	research	NOUN
ejpam-6146	406	24	interest	interest	NOUN
ejpam-6146	406	25	.	.	PUNCT
ejpam-6146	407	1	in	in	ADP
ejpam-6146	407	2	passing	pass	VERB
ejpam-6146	407	3	,	,	PUNCT
ejpam-6146	407	4	we	we	PRON
ejpam-6146	407	5	note	note	VERB
ejpam-6146	407	6	that	that	SCONJ
ejpam-6146	407	7	all	all	DET
ejpam-6146	407	8	these	these	DET
ejpam-6146	407	9	polynomials	polynomial	NOUN
ejpam-6146	407	10	have	have	AUX
ejpam-6146	407	11	been	be	AUX
ejpam-6146	407	12	generalized	generalize	VERB
ejpam-6146	407	13	in	in	ADP
ejpam-6146	407	14	various	various	ADJ
ejpam-6146	407	15	ways	way	NOUN
ejpam-6146	407	16	.	.	PUNCT
ejpam-6146	408	1	one	one	NUM
ejpam-6146	408	2	possibility	possibility	NOUN
ejpam-6146	408	3	of	of	ADP
ejpam-6146	408	4	the	the	DET
ejpam-6146	408	5	generalizations	generalization	NOUN
ejpam-6146	408	6	is	be	AUX
ejpam-6146	408	7	through	through	ADP
ejpam-6146	408	8	the	the	DET
ejpam-6146	408	9	quantum	quantum	ADJ
ejpam-6146	408	10	algebras	algebra	NOUN
ejpam-6146	408	11	[	[	X
ejpam-6146	408	12	20–23	20–23	NOUN
ejpam-6146	408	13	]	]	X
ejpam-6146	408	14	.	.	PUNCT
ejpam-6146	409	1	7	7	X
ejpam-6146	409	2	.	.	X
ejpam-6146	409	3	concluding	conclude	VERB
ejpam-6146	409	4	remarks	remark	NOUN
ejpam-6146	409	5	starting	start	VERB
ejpam-6146	409	6	with	with	ADP
ejpam-6146	409	7	the	the	DET
ejpam-6146	409	8	‘	'	PUNCT
ejpam-6146	409	9	simple	simple	ADJ
ejpam-6146	409	10	’	'	PUNCT
ejpam-6146	409	11	differential	differential	ADJ
ejpam-6146	409	12	equation	equation	NOUN
ejpam-6146	409	13	y′′	y′′	PROPN
ejpam-6146	409	14	+	+	CCONJ
ejpam-6146	409	15	y	y	PROPN
ejpam-6146	409	16	=	=	SYM
ejpam-6146	409	17	0	0	PROPN
ejpam-6146	409	18	,	,	PUNCT
ejpam-6146	409	19	we	we	PRON
ejpam-6146	409	20	deduced	deduce	VERB
ejpam-6146	409	21	that	that	SCONJ
ejpam-6146	409	22	the	the	DET
ejpam-6146	409	23	solutions	solution	NOUN
ejpam-6146	409	24	are	be	AUX
ejpam-6146	409	25	sinx	sinx	NOUN
ejpam-6146	409	26	and	and	CCONJ
ejpam-6146	409	27	cosx	cosx	PROPN
ejpam-6146	409	28	,	,	PUNCT
ejpam-6146	409	29	using	use	VERB
ejpam-6146	409	30	qualitative	qualitative	ADJ
ejpam-6146	409	31	arguments	argument	NOUN
ejpam-6146	409	32	.	.	PUNCT
ejpam-6146	410	1	we	we	PRON
ejpam-6146	410	2	also	also	ADV
ejpam-6146	410	3	derived	derive	VERB
ejpam-6146	410	4	some	some	PRON
ejpam-6146	410	5	of	of	ADP
ejpam-6146	410	6	their	their	PRON
ejpam-6146	410	7	properties	property	NOUN
ejpam-6146	410	8	including	include	VERB
ejpam-6146	410	9	,	,	PUNCT
ejpam-6146	410	10	periodicity	periodicity	NOUN
ejpam-6146	410	11	and	and	CCONJ
ejpam-6146	410	12	the	the	DET
ejpam-6146	410	13	series	series	NOUN
ejpam-6146	410	14	expansions	expansion	VERB
ejpam-6146	410	15	.	.	PUNCT
ejpam-6146	411	1	the	the	DET
ejpam-6146	411	2	connection	connection	NOUN
ejpam-6146	411	3	to	to	ADP
ejpam-6146	411	4	the	the	DET
ejpam-6146	411	5	geometric	geometric	ADJ
ejpam-6146	411	6	origins	origin	NOUN
ejpam-6146	411	7	of	of	ADP
ejpam-6146	411	8	sinx	sinx	PROPN
ejpam-6146	411	9	and	and	CCONJ
ejpam-6146	411	10	cosx	cosx	PROPN
ejpam-6146	411	11	was	be	AUX
ejpam-6146	411	12	established	establish	VERB
ejpam-6146	411	13	through	through	ADP
ejpam-6146	411	14	the	the	DET
ejpam-6146	411	15	unit	unit	NOUN
ejpam-6146	411	16	circle	circle	NOUN
ejpam-6146	411	17	.	.	PUNCT
ejpam-6146	412	1	some	some	DET
ejpam-6146	412	2	properties	property	NOUN
ejpam-6146	412	3	such	such	ADJ
ejpam-6146	412	4	as	as	ADP
ejpam-6146	412	5	the	the	DET
ejpam-6146	412	6	sine	sine	NOUN
ejpam-6146	412	7	/	/	SYM
ejpam-6146	412	8	cosine	cosine	NOUN
ejpam-6146	412	9	of	of	ADP
ejpam-6146	412	10	sums	sum	NOUN
ejpam-6146	412	11	and	and	CCONJ
ejpam-6146	412	12	differences	difference	NOUN
ejpam-6146	412	13	of	of	ADP
ejpam-6146	412	14	two	two	NUM
ejpam-6146	412	15	angles	angle	NOUN
ejpam-6146	412	16	are	be	AUX
ejpam-6146	412	17	also	also	ADV
ejpam-6146	412	18	derived	derive	VERB
ejpam-6146	412	19	from	from	ADP
ejpam-6146	412	20	the	the	DET
ejpam-6146	412	21	defining	define	VERB
ejpam-6146	412	22	differential	differential	NOUN
ejpam-6146	412	23	equation	equation	NOUN
ejpam-6146	412	24	.	.	PUNCT
ejpam-6146	413	1	we	we	PRON
ejpam-6146	413	2	also	also	ADV
ejpam-6146	413	3	derived	derive	VERB
ejpam-6146	413	4	the	the	DET
ejpam-6146	413	5	series	series	NOUN
ejpam-6146	413	6	expansion	expansion	NOUN
ejpam-6146	413	7	of	of	ADP
ejpam-6146	413	8	the	the	DET
ejpam-6146	413	9	sine	sine	NOUN
ejpam-6146	413	10	and	and	CCONJ
ejpam-6146	413	11	cosine	cosine	NOUN
ejpam-6146	413	12	functions	function	NOUN
ejpam-6146	413	13	.	.	PUNCT
ejpam-6146	414	1	in	in	ADP
ejpam-6146	414	2	fact	fact	NOUN
ejpam-6146	414	3	,	,	PUNCT
ejpam-6146	414	4	the	the	DET
ejpam-6146	414	5	series	series	NOUN
ejpam-6146	414	6	expansion	expansion	NOUN
ejpam-6146	414	7	was	be	AUX
ejpam-6146	414	8	also	also	ADV
ejpam-6146	414	9	derived	derive	VERB
ejpam-6146	414	10	by	by	ADP
ejpam-6146	414	11	the	the	DET
ejpam-6146	414	12	alternate	alternate	ADJ
ejpam-6146	414	13	procedure	procedure	NOUN
ejpam-6146	414	14	of	of	ADP
ejpam-6146	414	15	a	a	DET
ejpam-6146	414	16	matrix	matrix	NOUN
ejpam-6146	414	17	differential	differential	NOUN
ejpam-6146	414	18	equation	equation	NOUN
ejpam-6146	414	19	.	.	PUNCT
ejpam-6146	415	1	it	it	PRON
ejpam-6146	415	2	was	be	AUX
ejpam-6146	415	3	further	far	ADV
ejpam-6146	415	4	shown	show	VERB
ejpam-6146	415	5	that	that	SCONJ
ejpam-6146	415	6	under	under	ADP
ejpam-6146	415	7	certain	certain	ADJ
ejpam-6146	415	8	conditions	condition	NOUN
ejpam-6146	415	9	,	,	PUNCT
ejpam-6146	415	10	the	the	DET
ejpam-6146	415	11	solutions	solution	NOUN
ejpam-6146	415	12	of	of	ADP
ejpam-6146	415	13	the	the	DET
ejpam-6146	415	14	second	second	ADJ
ejpam-6146	415	15	-	-	PUNCT
ejpam-6146	415	16	order	order	NOUN
ejpam-6146	415	17	linear	linear	PROPN
ejpam-6146	415	18	differential	differential	NOUN
ejpam-6146	415	19	equations	equation	NOUN
ejpam-6146	415	20	oscillate	oscillate	VERB
ejpam-6146	415	21	.	.	PUNCT
ejpam-6146	416	1	in	in	ADP
ejpam-6146	416	2	which	which	PRON
ejpam-6146	416	3	a	a	DET
ejpam-6146	416	4	case	case	NOUN
ejpam-6146	416	5	,	,	PUNCT
ejpam-6146	416	6	the	the	DET
ejpam-6146	416	7	zeros	zero	NOUN
ejpam-6146	416	8	of	of	ADP
ejpam-6146	416	9	the	the	DET
ejpam-6146	416	10	two	two	NUM
ejpam-6146	416	11	solutions	solution	NOUN
ejpam-6146	416	12	are	be	AUX
ejpam-6146	416	13	interlaced	interlaced	ADJ
ejpam-6146	416	14	.	.	PUNCT
ejpam-6146	417	1	we	we	PRON
ejpam-6146	417	2	also	also	ADV
ejpam-6146	417	3	looked	look	VERB
ejpam-6146	417	4	at	at	ADP
ejpam-6146	417	5	the	the	DET
ejpam-6146	417	6	sturm	sturm	PROPN
ejpam-6146	417	7	comparison	comparison	NOUN
ejpam-6146	417	8	theorem	theorem	VERB
ejpam-6146	417	9	,	,	PUNCT
ejpam-6146	417	10	which	which	PRON
ejpam-6146	417	11	covers	cover	VERB
ejpam-6146	417	12	the	the	DET
ejpam-6146	417	13	amount	amount	NOUN
ejpam-6146	417	14	of	of	ADP
ejpam-6146	417	15	oscillation	oscillation	NOUN
ejpam-6146	417	16	for	for	ADP
ejpam-6146	417	17	different	different	ADJ
ejpam-6146	417	18	choices	choice	NOUN
ejpam-6146	417	19	of	of	ADP
ejpam-6146	417	20	the	the	DET
ejpam-6146	417	21	decisive	decisive	ADJ
ejpam-6146	417	22	function	function	NOUN
ejpam-6146	417	23	.	.	PUNCT
ejpam-6146	418	1	we	we	PRON
ejpam-6146	418	2	briefly	briefly	ADV
ejpam-6146	418	3	looked	look	VERB
ejpam-6146	418	4	at	at	ADP
ejpam-6146	418	5	the	the	DET
ejpam-6146	418	6	hyperbolic	hyperbolic	ADJ
ejpam-6146	418	7	and	and	CCONJ
ejpam-6146	418	8	bessel	bessel	ADJ
ejpam-6146	418	9	functions	function	NOUN
ejpam-6146	418	10	.	.	PUNCT
ejpam-6146	419	1	we	we	PRON
ejpam-6146	419	2	further	far	ADV
ejpam-6146	419	3	looked	look	VERB
ejpam-6146	419	4	at	at	ADP
ejpam-6146	419	5	the	the	DET
ejpam-6146	419	6	case	case	NOUN
ejpam-6146	419	7	of	of	ADP
ejpam-6146	419	8	classical	classical	ADJ
ejpam-6146	419	9	orthogonal	orthogonal	ADJ
ejpam-6146	419	10	polynomials	polynomial	NOUN
ejpam-6146	419	11	.	.	PUNCT
ejpam-6146	420	1	the	the	DET
ejpam-6146	420	2	approach	approach	NOUN
ejpam-6146	420	3	presented	present	VERB
ejpam-6146	420	4	in	in	ADP
ejpam-6146	420	5	this	this	DET
ejpam-6146	420	6	article	article	NOUN
ejpam-6146	420	7	will	will	AUX
ejpam-6146	420	8	be	be	AUX
ejpam-6146	420	9	useful	useful	ADJ
ejpam-6146	420	10	to	to	PART
ejpam-6146	420	11	see	see	VERB
ejpam-6146	420	12	the	the	DET
ejpam-6146	420	13	connection	connection	NOUN
ejpam-6146	420	14	between	between	ADP
ejpam-6146	420	15	the	the	DET
ejpam-6146	420	16	geometric	geometric	ADJ
ejpam-6146	420	17	approach	approach	NOUN
ejpam-6146	420	18	to	to	ADP
ejpam-6146	420	19	trigonometry	trigonometry	NOUN
ejpam-6146	420	20	and	and	CCONJ
ejpam-6146	420	21	the	the	DET
ejpam-6146	420	22	calculus	calculus	NOUN
ejpam-6146	420	23	based	base	VERB
ejpam-6146	420	24	approach	approach	NOUN
ejpam-6146	420	25	presented	present	VERB
ejpam-6146	420	26	here	here	ADV
ejpam-6146	420	27	.	.	PUNCT
ejpam-6146	421	1	references	reference	NOUN
ejpam-6146	421	2	[	[	X
ejpam-6146	421	3	1	1	NUM
ejpam-6146	421	4	]	]	PUNCT
ejpam-6146	421	5	v.	v.	ADP
ejpam-6146	421	6	katter	katter	PROPN
ejpam-6146	421	7	.	.	PUNCT
ejpam-6146	422	1	historische	historische	PROPN
ejpam-6146	422	2	,	,	PUNCT
ejpam-6146	422	3	logische	logische	PROPN
ejpam-6146	422	4	und	und	NOUN
ejpam-6146	422	5	individuelle	individuelle	PROPN
ejpam-6146	422	6	genese	genese	PROPN
ejpam-6146	422	7	der	der	NOUN
ejpam-6146	422	8	trigonometrie	trigonometrie	PROPN
ejpam-6146	422	9	aus	aus	PROPN
ejpam-6146	422	10	didaktischer	didaktischer	PROPN
ejpam-6146	422	11	sicht	sicht	PROPN
ejpam-6146	422	12	.	.	PUNCT
ejpam-6146	423	1	springer	springer	NOUN
ejpam-6146	423	2	,	,	PUNCT
ejpam-6146	423	3	2023	2023	NUM
ejpam-6146	423	4	.	.	PUNCT
ejpam-6146	424	1	[	[	X
ejpam-6146	424	2	2	2	NUM
ejpam-6146	424	3	]	]	PUNCT
ejpam-6146	424	4	l.	l.	PROPN
ejpam-6146	424	5	rico	rico	PROPN
ejpam-6146	424	6	e.	e.	PROPN
ejpam-6146	424	7	mart́ın	mart́ın	PROPN
ejpam-6146	424	8	-	-	PUNCT
ejpam-6146	424	9	fernández	fernández	PROPN
ejpam-6146	424	10	,	,	PUNCT
ejpam-6146	424	11	j.	j.	PROPN
ejpam-6146	424	12	f.	f.	PROPN
ejpam-6146	424	13	ruiz	ruiz	PROPN
ejpam-6146	424	14	-	-	PUNCT
ejpam-6146	424	15	hidalgo	hidalgo	PROPN
ejpam-6146	424	16	.	.	PUNCT
ejpam-6146	425	1	meaning	meaning	NOUN
ejpam-6146	425	2	and	and	CCONJ
ejpam-6146	425	3	understanding	understanding	NOUN
ejpam-6146	425	4	of	of	ADP
ejpam-6146	425	5	school	school	NOUN
ejpam-6146	425	6	mathematical	mathematical	ADJ
ejpam-6146	425	7	concepts	concept	NOUN
ejpam-6146	425	8	by	by	ADP
ejpam-6146	425	9	secondary	secondary	ADJ
ejpam-6146	425	10	students	student	NOUN
ejpam-6146	425	11	:	:	PUNCT
ejpam-6146	425	12	the	the	DET
ejpam-6146	425	13	study	study	NOUN
ejpam-6146	425	14	of	of	ADP
ejpam-6146	425	15	sine	sine	NOUN
ejpam-6146	425	16	and	and	CCONJ
ejpam-6146	425	17	cosine	cosine	PROPN
ejpam-6146	425	18	.	.	PUNCT
ejpam-6146	426	1	eurasia	eurasia	PROPN
ejpam-6146	426	2	journal	journal	PROPN
ejpam-6146	426	3	of	of	ADP
ejpam-6146	426	4	mathematics	mathematics	PROPN
ejpam-6146	426	5	,	,	PUNCT
ejpam-6146	426	6	science	science	NOUN
ejpam-6146	426	7	and	and	CCONJ
ejpam-6146	426	8	technology	technology	NOUN
ejpam-6146	426	9	education	education	NOUN
ejpam-6146	426	10	,	,	PUNCT
ejpam-6146	426	11	15(12):em1782	15(12):em1782	NUM
ejpam-6146	426	12	,	,	PUNCT
ejpam-6146	426	13	2019	2019	NUM
ejpam-6146	426	14	.	.	PUNCT
ejpam-6146	427	1	s.	s.	PROPN
ejpam-6146	427	2	a.	a.	PROPN
ejpam-6146	427	3	khan	khan	PROPN
ejpam-6146	427	4	,	,	PUNCT
ejpam-6146	427	5	m.	m.	NOUN
ejpam-6146	427	6	m.	m.	PROPN
ejpam-6146	427	7	kankarej	kankarej	PROPN
ejpam-6146	427	8	,	,	PUNCT
ejpam-6146	427	9	m.	m.	PROPN
ejpam-6146	427	10	n.	n.	PROPN
ejpam-6146	427	11	i.	i.	PROPN
ejpam-6146	427	12	khan	khan	PROPN
ejpam-6146	427	13	/	/	SYM
ejpam-6146	427	14	eur	eur	PROPN
ejpam-6146	427	15	.	.	PUNCT
ejpam-6146	428	1	j.	j.	PROPN
ejpam-6146	428	2	pure	pure	PROPN
ejpam-6146	428	3	appl	appl	PROPN
ejpam-6146	428	4	.	.	PROPN
ejpam-6146	428	5	math	math	PROPN
ejpam-6146	428	6	,	,	PUNCT
ejpam-6146	428	7	18	18	NUM
ejpam-6146	428	8	(	(	PUNCT
ejpam-6146	428	9	4	4	NUM
ejpam-6146	428	10	)	)	PUNCT
ejpam-6146	428	11	(	(	PUNCT
ejpam-6146	428	12	2025	2025	NUM
ejpam-6146	428	13	)	)	PUNCT
ejpam-6146	428	14	,	,	PUNCT
ejpam-6146	428	15	6146	6146	NUM
ejpam-6146	428	16	14	14	NUM
ejpam-6146	428	17	of	of	ADP
ejpam-6146	428	18	14	14	NUM
ejpam-6146	428	19	[	[	X
ejpam-6146	428	20	3	3	X
ejpam-6146	428	21	]	]	PUNCT
ejpam-6146	428	22	s.	s.	PROPN
ejpam-6146	428	23	l.	l.	PROPN
ejpam-6146	428	24	loney	loney	PROPN
ejpam-6146	428	25	.	.	PUNCT
ejpam-6146	429	1	plane	plane	NOUN
ejpam-6146	429	2	trigonometry	trigonometry	NOUN
ejpam-6146	429	3	part-1	part-1	PROPN
ejpam-6146	429	4	.	.	PUNCT
ejpam-6146	429	5	aitbs	aitbs	PROPN
ejpam-6146	429	6	publishers	publisher	NOUN
ejpam-6146	429	7	india	india	PROPN
ejpam-6146	429	8	,	,	PUNCT
ejpam-6146	429	9	2019	2019	NUM
ejpam-6146	429	10	.	.	PUNCT
ejpam-6146	430	1	[	[	X
ejpam-6146	430	2	4	4	X
ejpam-6146	430	3	]	]	PUNCT
ejpam-6146	430	4	s.	s.	PROPN
ejpam-6146	430	5	l.	l.	PROPN
ejpam-6146	430	6	loney	loney	PROPN
ejpam-6146	430	7	.	.	PUNCT
ejpam-6146	431	1	plane	plane	NOUN
ejpam-6146	431	2	trigonometry	trigonometry	NOUN
ejpam-6146	431	3	part-2	part-2	PROPN
ejpam-6146	431	4	.	.	PUNCT
ejpam-6146	432	1	aitbs	aitbs	PROPN
ejpam-6146	432	2	publishers	publisher	NOUN
ejpam-6146	432	3	india	india	PROPN
ejpam-6146	432	4	,	,	PUNCT
ejpam-6146	432	5	2019	2019	NUM
ejpam-6146	432	6	.	.	PUNCT
ejpam-6146	433	1	[	[	X
ejpam-6146	433	2	5	5	NUM
ejpam-6146	433	3	]	]	X
ejpam-6146	433	4	eli	eli	PROPN
ejpam-6146	433	5	maor	maor	PROPN
ejpam-6146	433	6	.	.	PUNCT
ejpam-6146	434	1	trigonometric	trigonometric	PROPN
ejpam-6146	434	2	delights	delight	VERB
ejpam-6146	434	3	.	.	PUNCT
ejpam-6146	435	1	princeton	princeton	PROPN
ejpam-6146	435	2	university	university	PROPN
ejpam-6146	435	3	press	press	NOUN
ejpam-6146	435	4	,	,	PUNCT
ejpam-6146	435	5	2013	2013	NUM
ejpam-6146	435	6	.	.	PUNCT
ejpam-6146	436	1	[	[	X
ejpam-6146	436	2	6	6	NUM
ejpam-6146	436	3	]	]	PUNCT
ejpam-6146	436	4	g.	g.	PROPN
ejpam-6146	436	5	f.	f.	PROPN
ejpam-6146	436	6	simmons	simmons	PROPN
ejpam-6146	436	7	.	.	PUNCT
ejpam-6146	437	1	differential	differential	ADJ
ejpam-6146	437	2	equations	equation	NOUN
ejpam-6146	437	3	with	with	ADP
ejpam-6146	437	4	applications	application	NOUN
ejpam-6146	437	5	and	and	CCONJ
ejpam-6146	437	6	historical	historical	ADJ
ejpam-6146	437	7	notes	note	NOUN
ejpam-6146	437	8	.	.	PUNCT
ejpam-6146	438	1	taylor	taylor	PROPN
ejpam-6146	438	2	&	&	CCONJ
ejpam-6146	438	3	francis	francis	PROPN
ejpam-6146	438	4	,	,	PUNCT
ejpam-6146	438	5	2016	2016	NUM
ejpam-6146	438	6	.	.	PUNCT
ejpam-6146	439	1	[	[	X
ejpam-6146	439	2	7	7	X
ejpam-6146	439	3	]	]	X
ejpam-6146	439	4	p.	p.	NOUN
ejpam-6146	439	5	bedient	bedient	PROPN
ejpam-6146	439	6	e.	e.	PROPN
ejpam-6146	439	7	rainville	rainville	PROPN
ejpam-6146	439	8	and	and	CCONJ
ejpam-6146	439	9	r.	r.	PROPN
ejpam-6146	439	10	bedient	bedient	PROPN
ejpam-6146	439	11	.	.	PUNCT
ejpam-6146	440	1	elementary	elementary	PROPN
ejpam-6146	440	2	differential	differential	PROPN
ejpam-6146	440	3	equations	equation	NOUN
ejpam-6146	440	4	.	.	PUNCT
ejpam-6146	441	1	pearson	pearson	PROPN
ejpam-6146	441	2	,	,	PUNCT
ejpam-6146	441	3	2014	2014	NUM
ejpam-6146	441	4	.	.	PUNCT
ejpam-6146	442	1	[	[	X
ejpam-6146	442	2	8	8	NUM
ejpam-6146	442	3	]	]	PUNCT
ejpam-6146	442	4	m.	m.	PROPN
ejpam-6146	442	5	d.	d.	PROPN
ejpam-6146	442	6	weir	weir	PROPN
ejpam-6146	442	7	j.	j.	PROPN
ejpam-6146	442	8	r.	r.	PROPN
ejpam-6146	442	9	hass	hass	PROPN
ejpam-6146	442	10	,	,	PUNCT
ejpam-6146	442	11	c.	c.	PROPN
ejpam-6146	442	12	e.	e.	PROPN
ejpam-6146	442	13	heil	heil	PROPN
ejpam-6146	442	14	and	and	CCONJ
ejpam-6146	442	15	p.	p.	PROPN
ejpam-6146	442	16	bogacki	bogacki	PROPN
ejpam-6146	442	17	.	.	PUNCT
ejpam-6146	443	1	thomas	thomas	PROPN
ejpam-6146	443	2	’	'	PUNCT
ejpam-6146	443	3	calculus	calculus	PROPN
ejpam-6146	443	4	,	,	PUNCT
ejpam-6146	443	5	15th	15th	NOUN
ejpam-6146	443	6	edition	edition	NOUN
ejpam-6146	443	7	.	.	PUNCT
ejpam-6146	444	1	pearson	pearson	PROPN
ejpam-6146	444	2	,	,	PUNCT
ejpam-6146	444	3	2022	2022	NUM
ejpam-6146	444	4	.	.	PUNCT
ejpam-6146	445	1	[	[	X
ejpam-6146	445	2	9	9	NUM
ejpam-6146	445	3	]	]	PUNCT
ejpam-6146	445	4	h.	h.	PROPN
ejpam-6146	445	5	j.	j.	PROPN
ejpam-6146	445	6	weber	weber	PROPN
ejpam-6146	445	7	g.	g.	PROPN
ejpam-6146	445	8	b.	b.	PROPN
ejpam-6146	445	9	arfken	arfken	PROPN
ejpam-6146	445	10	and	and	CCONJ
ejpam-6146	445	11	f.	f.	PROPN
ejpam-6146	445	12	e.	e.	PROPN
ejpam-6146	445	13	harris	harris	PROPN
ejpam-6146	445	14	.	.	PUNCT
ejpam-6146	446	1	mathematical	mathematical	ADJ
ejpam-6146	446	2	methods	method	NOUN
ejpam-6146	446	3	for	for	ADP
ejpam-6146	446	4	physicists	physicist	NOUN
ejpam-6146	446	5	.	.	PUNCT
ejpam-6146	447	1	academic	academic	ADJ
ejpam-6146	447	2	press	press	PROPN
ejpam-6146	447	3	,	,	PUNCT
ejpam-6146	447	4	london	london	PROPN
ejpam-6146	447	5	,	,	PUNCT
ejpam-6146	447	6	uk	uk	PROPN
ejpam-6146	447	7	,	,	PUNCT
ejpam-6146	447	8	2012	2012	NUM
ejpam-6146	447	9	.	.	PUNCT
ejpam-6146	448	1	[	[	X
ejpam-6146	448	2	10	10	NUM
ejpam-6146	448	3	]	]	X
ejpam-6146	448	4	b.	b.	PROPN
ejpam-6146	448	5	agheli	agheli	PROPN
ejpam-6146	448	6	.	.	PUNCT
ejpam-6146	448	7	approximate	approximate	ADJ
ejpam-6146	448	8	solution	solution	NOUN
ejpam-6146	448	9	of	of	ADP
ejpam-6146	448	10	bratu	bratu	PROPN
ejpam-6146	448	11	differential	differential	ADJ
ejpam-6146	448	12	equations	equation	NOUN
ejpam-6146	448	13	using	use	VERB
ejpam-6146	448	14	trigonometric	trigonometric	ADJ
ejpam-6146	448	15	basic	basic	ADJ
ejpam-6146	448	16	functions	function	NOUN
ejpam-6146	448	17	.	.	PUNCT
ejpam-6146	449	1	kragujevac	kragujevac	PROPN
ejpam-6146	449	2	journal	journal	PROPN
ejpam-6146	449	3	of	of	ADP
ejpam-6146	449	4	mathematics	mathematic	NOUN
ejpam-6146	449	5	,	,	PUNCT
ejpam-6146	449	6	45(2):203–214	45(2):203–214	ADJ
ejpam-6146	449	7	,	,	PUNCT
ejpam-6146	449	8	2021	2021	NUM
ejpam-6146	449	9	.	.	PUNCT
ejpam-6146	450	1	[	[	X
ejpam-6146	450	2	11	11	NUM
ejpam-6146	450	3	]	]	PUNCT
ejpam-6146	450	4	m.	m.	NOUN
ejpam-6146	450	5	pakdemirli	pakdemirli	NOUN
ejpam-6146	450	6	.	.	PUNCT
ejpam-6146	451	1	complex	complex	ADJ
ejpam-6146	451	2	exponential	exponential	ADJ
ejpam-6146	451	3	methodfor	methodfor	NOUN
ejpam-6146	451	4	solving	solve	VERB
ejpam-6146	451	5	partial	partial	ADJ
ejpam-6146	451	6	differential	differential	ADJ
ejpam-6146	451	7	equations	equation	NOUN
ejpam-6146	451	8	.	.	PUNCT
ejpam-6146	452	1	engineering	engineering	NOUN
ejpam-6146	452	2	transactions	transaction	NOUN
ejpam-6146	452	3	,	,	PUNCT
ejpam-6146	452	4	72(4):461–474	72(4):461–474	NOUN
ejpam-6146	452	5	,	,	PUNCT
ejpam-6146	452	6	2024	2024	NUM
ejpam-6146	452	7	.	.	PUNCT
ejpam-6146	453	1	[	[	X
ejpam-6146	453	2	12	12	NUM
ejpam-6146	453	3	]	]	PUNCT
ejpam-6146	453	4	h.	h.	PROPN
ejpam-6146	453	5	k.	k.	PROPN
ejpam-6146	453	6	sevindir	sevindir	PROPN
ejpam-6146	453	7	s.	s.	PROPN
ejpam-6146	453	8	çetinkaya	çetinkaya	PROPN
ejpam-6146	453	9	,	,	PUNCT
ejpam-6146	453	10	a.	a.	PROPN
ejpam-6146	453	11	demir	demir	PROPN
ejpam-6146	453	12	.	.	PUNCT
ejpam-6146	454	1	the	the	DET
ejpam-6146	454	2	analytic	analytic	ADJ
ejpam-6146	454	3	solution	solution	NOUN
ejpam-6146	454	4	of	of	ADP
ejpam-6146	454	5	initial	initial	ADJ
ejpam-6146	454	6	boundary	boundary	ADJ
ejpam-6146	454	7	valueproblem	valueproblem	NOUN
ejpam-6146	454	8	including	include	VERB
ejpam-6146	454	9	time	time	NOUN
ejpam-6146	454	10	fractional	fractional	ADJ
ejpam-6146	454	11	diffusion	diffusion	NOUN
ejpam-6146	454	12	equation	equation	NOUN
ejpam-6146	454	13	.	.	PUNCT
ejpam-6146	455	1	facta	facta	PROPN
ejpam-6146	455	2	universitatis	universitatis	PROPN
ejpam-6146	455	3	,	,	PUNCT
ejpam-6146	455	4	series	series	NOUN
ejpam-6146	455	5	:	:	PUNCT
ejpam-6146	455	6	mathematics	mathematic	NOUN
ejpam-6146	455	7	and	and	CCONJ
ejpam-6146	455	8	informatics	informatic	NOUN
ejpam-6146	455	9	,	,	PUNCT
ejpam-6146	455	10	35(1):243–252	35(1):243–252	PROPN
ejpam-6146	455	11	,	,	PUNCT
ejpam-6146	455	12	2020	2020	NUM
ejpam-6146	455	13	.	.	PUNCT
ejpam-6146	456	1	[	[	X
ejpam-6146	456	2	13	13	NUM
ejpam-6146	456	3	]	]	X
ejpam-6146	456	4	c.	c.	PROPN
ejpam-6146	456	5	valls	valls	PROPN
ejpam-6146	456	6	.	.	PUNCT
ejpam-6146	457	1	trigonometric	trigonometric	ADJ
ejpam-6146	457	2	polynomial	polynomial	ADJ
ejpam-6146	457	3	solutions	solution	NOUN
ejpam-6146	457	4	of	of	ADP
ejpam-6146	457	5	bernoulli	bernoulli	PROPN
ejpam-6146	457	6	trigonometric	trigonometric	PROPN
ejpam-6146	457	7	polynomial	polynomial	ADJ
ejpam-6146	457	8	differential	differential	NOUN
ejpam-6146	457	9	equations	equation	NOUN
ejpam-6146	457	10	.	.	PUNCT
ejpam-6146	458	1	mathematics	mathematic	NOUN
ejpam-6146	458	2	,	,	PUNCT
ejpam-6146	458	3	10:4022	10:4022	NOUN
ejpam-6146	458	4	,	,	PUNCT
ejpam-6146	458	5	2022	2022	NUM
ejpam-6146	458	6	.	.	PUNCT
ejpam-6146	459	1	[	[	X
ejpam-6146	459	2	14	14	NUM
ejpam-6146	459	3	]	]	X
ejpam-6146	459	4	c.	c.	PROPN
ejpam-6146	459	5	o.	o.	PROPN
ejpam-6146	459	6	alakofa	alakofa	PROPN
ejpam-6146	459	7	o.	o.	PROPN
ejpam-6146	459	8	o.	o.	PROPN
ejpam-6146	459	9	enoch	enoch	PROPN
ejpam-6146	459	10	.	.	PUNCT
ejpam-6146	460	1	a	a	DET
ejpam-6146	460	2	fifth	fifth	ADJ
ejpam-6146	460	3	order	order	NOUN
ejpam-6146	460	4	block	block	NOUN
ejpam-6146	460	5	methods	method	NOUN
ejpam-6146	460	6	for	for	ADP
ejpam-6146	460	7	solving	solve	VERB
ejpam-6146	460	8	second	second	ADJ
ejpam-6146	460	9	-	-	PUNCT
ejpam-6146	460	10	order	order	NOUN
ejpam-6146	460	11	stiff	stiff	ADJ
ejpam-6146	460	12	ordinary	ordinary	ADJ
ejpam-6146	460	13	differential	differential	ADJ
ejpam-6146	460	14	equations	equation	NOUN
ejpam-6146	460	15	using	use	VERB
ejpam-6146	460	16	trigonometric	trigonometric	ADJ
ejpam-6146	460	17	functions	function	NOUN
ejpam-6146	460	18	and	and	CCONJ
ejpam-6146	460	19	polynomial	polynomial	ADJ
ejpam-6146	460	20	function	function	NOUN
ejpam-6146	460	21	as	as	ADP
ejpam-6146	460	22	the	the	DET
ejpam-6146	460	23	basis	basis	NOUN
ejpam-6146	460	24	function	function	NOUN
ejpam-6146	460	25	.	.	PUNCT
ejpam-6146	461	1	african	african	ADJ
ejpam-6146	461	2	scientific	scientific	ADJ
ejpam-6146	461	3	reports	report	NOUN
ejpam-6146	461	4	,	,	PUNCT
ejpam-6146	461	5	3(2):156	3(2):156	NUM
ejpam-6146	461	6	,	,	PUNCT
ejpam-6146	461	7	2024	2024	NUM
ejpam-6146	461	8	.	.	PUNCT
ejpam-6146	462	1	[	[	X
ejpam-6146	462	2	15	15	NUM
ejpam-6146	462	3	]	]	X
ejpam-6146	462	4	b.	b.	PROPN
ejpam-6146	462	5	v.	v.	PROPN
ejpam-6146	462	6	rao	rao	PROPN
ejpam-6146	462	7	.	.	PUNCT
ejpam-6146	463	1	fun	fun	PROPN
ejpam-6146	463	2	with	with	ADP
ejpam-6146	463	3	differential	differential	ADJ
ejpam-6146	463	4	equations	equation	NOUN
ejpam-6146	463	5	.	.	PUNCT
ejpam-6146	464	1	resonance	resonance	NOUN
ejpam-6146	464	2	–	–	PUNCT
ejpam-6146	464	3	journal	journal	NOUN
ejpam-6146	464	4	of	of	ADP
ejpam-6146	464	5	science	science	NOUN
ejpam-6146	464	6	education	education	NOUN
ejpam-6146	464	7	,	,	PUNCT
ejpam-6146	464	8	18(6):543–557	18(6):543–557	PROPN
ejpam-6146	464	9	,	,	PUNCT
ejpam-6146	464	10	2013	2013	NUM
ejpam-6146	464	11	.	.	PUNCT
ejpam-6146	465	1	[	[	X
ejpam-6146	465	2	16	16	NUM
ejpam-6146	465	3	]	]	PUNCT
ejpam-6146	465	4	s.	s.	PROPN
ejpam-6146	465	5	a.	a.	PROPN
ejpam-6146	465	6	khan	khan	PROPN
ejpam-6146	465	7	.	.	PUNCT
ejpam-6146	466	1	trigonometric	trigonometric	ADJ
ejpam-6146	466	2	ratios	ratio	NOUN
ejpam-6146	466	3	using	use	VERB
ejpam-6146	466	4	geometric	geometric	ADJ
ejpam-6146	466	5	methods	method	NOUN
ejpam-6146	466	6	.	.	PUNCT
ejpam-6146	467	1	advances	advance	NOUN
ejpam-6146	467	2	in	in	ADP
ejpam-6146	467	3	mathematics	mathematic	NOUN
ejpam-6146	467	4	:	:	PUNCT
ejpam-6146	467	5	scientific	scientific	ADJ
ejpam-6146	467	6	journal	journal	NOUN
ejpam-6146	467	7	,	,	PUNCT
ejpam-6146	467	8	10(9):8685–8702	10(9):8685–8702	NUM
ejpam-6146	467	9	,	,	PUNCT
ejpam-6146	467	10	2020	2020	NUM
ejpam-6146	467	11	.	.	PUNCT
ejpam-6146	468	1	[	[	X
ejpam-6146	468	2	17	17	NUM
ejpam-6146	468	3	]	]	PUNCT
ejpam-6146	468	4	s.	s.	PROPN
ejpam-6146	468	5	a.	a.	PROPN
ejpam-6146	468	6	khan	khan	PROPN
ejpam-6146	468	7	.	.	PUNCT
ejpam-6146	469	1	trigonometric	trigonometric	ADJ
ejpam-6146	469	2	ratios	ratio	NOUN
ejpam-6146	469	3	using	use	VERB
ejpam-6146	469	4	algebraic	algebraic	ADJ
ejpam-6146	469	5	methods	method	NOUN
ejpam-6146	469	6	.	.	PUNCT
ejpam-6146	470	1	mathematics	mathematic	NOUN
ejpam-6146	470	2	and	and	CCONJ
ejpam-6146	470	3	statistics	statistic	NOUN
ejpam-6146	470	4	,	,	PUNCT
ejpam-6146	470	5	9(6):899–907	9(6):899–907	NUM
ejpam-6146	470	6	,	,	PUNCT
ejpam-6146	470	7	2021	2021	NUM
ejpam-6146	470	8	.	.	PUNCT
ejpam-6146	471	1	[	[	X
ejpam-6146	471	2	18	18	NUM
ejpam-6146	471	3	]	]	PUNCT
ejpam-6146	471	4	s.	s.	PROPN
ejpam-6146	471	5	a.	a.	PROPN
ejpam-6146	471	6	khan	khan	PROPN
ejpam-6146	471	7	.	.	PUNCT
ejpam-6146	472	1	teaching	teach	VERB
ejpam-6146	472	2	irrational	irrational	ADJ
ejpam-6146	472	3	numbers	number	NOUN
ejpam-6146	472	4	through	through	ADP
ejpam-6146	472	5	trigonometry	trigonometry	NOUN
ejpam-6146	472	6	.	.	PUNCT
ejpam-6146	473	1	resonance	resonance	NOUN
ejpam-6146	473	2	journal	journal	PROPN
ejpam-6146	473	3	of	of	ADP
ejpam-6146	473	4	science	science	PROPN
ejpam-6146	473	5	education	education	NOUN
ejpam-6146	473	6	,	,	PUNCT
ejpam-6146	473	7	26(6):813–827	26(6):813–827	PROPN
ejpam-6146	473	8	,	,	PUNCT
ejpam-6146	473	9	2021	2021	NUM
ejpam-6146	473	10	.	.	PUNCT
ejpam-6146	474	1	[	[	X
ejpam-6146	474	2	19	19	NUM
ejpam-6146	474	3	]	]	PUNCT
ejpam-6146	474	4	j.	j.	PROPN
ejpam-6146	474	5	barrow	barrow	PROPN
ejpam-6146	474	6	-	-	PUNCT
ejpam-6146	474	7	green	green	PROPN
ejpam-6146	474	8	.	.	PUNCT
ejpam-6146	475	1	oscar	oscar	PROPN
ejpam-6146	475	2	ii	ii	PROPN
ejpam-6146	475	3	’s	’s	PART
ejpam-6146	475	4	prize	prize	NOUN
ejpam-6146	475	5	competition	competition	NOUN
ejpam-6146	475	6	and	and	CCONJ
ejpam-6146	475	7	the	the	DET
ejpam-6146	475	8	error	error	NOUN
ejpam-6146	475	9	in	in	ADP
ejpam-6146	475	10	poincaré	poincaré	ADJ
ejpam-6146	475	11	’s	’s	PART
ejpam-6146	475	12	memoir	memoir	NOUN
ejpam-6146	475	13	on	on	ADP
ejpam-6146	475	14	the	the	DET
ejpam-6146	475	15	three	three	NUM
ejpam-6146	475	16	body	body	NOUN
ejpam-6146	475	17	problem	problem	NOUN
ejpam-6146	475	18	.	.	PUNCT
ejpam-6146	476	1	archive	archive	NOUN
ejpam-6146	476	2	for	for	ADP
ejpam-6146	476	3	history	history	NOUN
ejpam-6146	476	4	of	of	ADP
ejpam-6146	476	5	exact	exact	ADJ
ejpam-6146	476	6	sciences	science	NOUN
ejpam-6146	476	7	,	,	PUNCT
ejpam-6146	476	8	48:107–131	48:107–131	PROPN
ejpam-6146	476	9	,	,	PUNCT
ejpam-6146	476	10	1994	1994	NUM
ejpam-6146	476	11	.	.	PUNCT
ejpam-6146	477	1	[	[	X
ejpam-6146	477	2	20	20	NUM
ejpam-6146	477	3	]	]	PUNCT
ejpam-6146	477	4	r.	r.	PROPN
ejpam-6146	477	5	chakrabarti	chakrabarti	PROPN
ejpam-6146	477	6	and	and	CCONJ
ejpam-6146	477	7	r.	r.	PROPN
ejpam-6146	477	8	jagannathan	jagannathan	PROPN
ejpam-6146	477	9	.	.	PUNCT
ejpam-6146	478	1	a	a	DET
ejpam-6146	478	2	(	(	PUNCT
ejpam-6146	478	3	p	p	NOUN
ejpam-6146	478	4	,	,	PUNCT
ejpam-6146	478	5	q)-oscillator	q)-oscillator	NOUN
ejpam-6146	478	6	realization	realization	NOUN
ejpam-6146	478	7	of	of	ADP
ejpam-6146	478	8	two	two	NUM
ejpam-6146	478	9	-	-	PUNCT
ejpam-6146	478	10	parameter	parameter	NOUN
ejpam-6146	478	11	quantum	quantum	NOUN
ejpam-6146	478	12	algebras	algebra	NOUN
ejpam-6146	478	13	.	.	PUNCT
ejpam-6146	479	1	j.	j.	PROPN
ejpam-6146	479	2	phys	phys	PROPN
ejpam-6146	479	3	.	.	PUNCT
ejpam-6146	480	1	a	a	DET
ejpam-6146	480	2	:	:	PUNCT
ejpam-6146	480	3	math	math	NOUN
ejpam-6146	480	4	.	.	PUNCT
ejpam-6146	481	1	gen	gen	PROPN
ejpam-6146	481	2	.	.	PROPN
ejpam-6146	481	3	,	,	PUNCT
ejpam-6146	481	4	24	24	NUM
ejpam-6146	481	5	:	:	PUNCT
ejpam-6146	481	6	l7711	l7711	NOUN
ejpam-6146	481	7	–	–	PUNCT
ejpam-6146	481	8	l718	l718	NUM
ejpam-6146	481	9	,	,	PUNCT
ejpam-6146	481	10	1991	1991	NUM
ejpam-6146	481	11	.	.	PUNCT
ejpam-6146	482	1	[	[	X
ejpam-6146	482	2	21	21	NUM
ejpam-6146	482	3	]	]	X
ejpam-6146	482	4	r.	r.	PROPN
ejpam-6146	482	5	jaganathan	jaganathan	PROPN
ejpam-6146	482	6	and	and	CCONJ
ejpam-6146	482	7	s.	s.	PROPN
ejpam-6146	482	8	sinha	sinha	PROPN
ejpam-6146	482	9	.	.	PUNCT
ejpam-6146	483	1	a	a	DET
ejpam-6146	483	2	q	q	ADV
ejpam-6146	483	3	-	-	PUNCT
ejpam-6146	483	4	deformed	deform	VERB
ejpam-6146	483	5	nonlinear	nonlinear	ADJ
ejpam-6146	483	6	map	map	NOUN
ejpam-6146	483	7	.	.	PUNCT
ejpam-6146	484	1	phys	phy	NOUN
ejpam-6146	484	2	.	.	PUNCT
ejpam-6146	485	1	lett	lett	PROPN
ejpam-6146	485	2	.	.	PUNCT
ejpam-6146	486	1	a.	a.	PROPN
ejpam-6146	486	2	,	,	PUNCT
ejpam-6146	486	3	338:277	338:277	NOUN
ejpam-6146	486	4	–	–	PUNCT
ejpam-6146	486	5	287	287	NUM
ejpam-6146	486	6	,	,	PUNCT
ejpam-6146	486	7	2005	2005	NUM
ejpam-6146	486	8	.	.	PUNCT
ejpam-6146	487	1	[	[	X
ejpam-6146	487	2	22	22	NUM
ejpam-6146	487	3	]	]	X
ejpam-6146	487	4	r.	r.	NOUN
ejpam-6146	487	5	jagannathan	jagannathan	PROPN
ejpam-6146	487	6	and	and	CCONJ
ejpam-6146	487	7	s.	s.	PROPN
ejpam-6146	487	8	a.	a.	PROPN
ejpam-6146	487	9	khan	khan	PROPN
ejpam-6146	487	10	.	.	PUNCT
ejpam-6146	488	1	on	on	ADP
ejpam-6146	488	2	the	the	DET
ejpam-6146	488	3	deformed	deform	VERB
ejpam-6146	488	4	oscillator	oscillator	NOUN
ejpam-6146	488	5	and	and	CCONJ
ejpam-6146	488	6	the	the	DET
ejpam-6146	488	7	deformed	deform	VERB
ejpam-6146	488	8	derivative	derivative	NOUN
ejpam-6146	488	9	associated	associate	VERB
ejpam-6146	488	10	with	with	ADP
ejpam-6146	488	11	the	the	DET
ejpam-6146	488	12	tsallis	tsallis	ADJ
ejpam-6146	488	13	q	q	NOUN
ejpam-6146	488	14	-	-	NOUN
ejpam-6146	488	15	exponential	exponential	ADJ
ejpam-6146	488	16	.	.	PUNCT
ejpam-6146	489	1	international	international	ADJ
ejpam-6146	489	2	journal	journal	NOUN
ejpam-6146	489	3	of	of	ADP
ejpam-6146	489	4	theoretical	theoretical	ADJ
ejpam-6146	489	5	physics	physics	NOUN
ejpam-6146	489	6	,	,	PUNCT
ejpam-6146	489	7	59(8):2647–2669	59(8):2647–2669	NUM
ejpam-6146	489	8	,	,	PUNCT
ejpam-6146	489	9	2020	2020	NUM
ejpam-6146	489	10	.	.	PUNCT
ejpam-6146	490	1	[	[	X
ejpam-6146	490	2	23	23	NUM
ejpam-6146	490	3	]	]	PUNCT
ejpam-6146	490	4	s.	s.	PROPN
ejpam-6146	490	5	a.	a.	PROPN
ejpam-6146	490	6	khan	khan	PROPN
ejpam-6146	490	7	and	and	CCONJ
ejpam-6146	490	8	r.	r.	PROPN
ejpam-6146	490	9	jagannathan	jagannathan	PROPN
ejpam-6146	490	10	.	.	PUNCT
ejpam-6146	491	1	on	on	ADP
ejpam-6146	491	2	certain	certain	ADJ
ejpam-6146	491	3	appell	appell	ADJ
ejpam-6146	491	4	polynomials	polynomial	NOUN
ejpam-6146	491	5	and	and	CCONJ
ejpam-6146	491	6	their	their	PRON
ejpam-6146	491	7	generalizations	generalization	NOUN
ejpam-6146	491	8	based	base	VERB
ejpam-6146	491	9	on	on	ADP
ejpam-6146	491	10	the	the	DET
ejpam-6146	491	11	tsallis	tsallis	ADJ
ejpam-6146	491	12	q	q	NOUN
ejpam-6146	491	13	-	-	NOUN
ejpam-6146	491	14	exponential	exponential	ADJ
ejpam-6146	491	15	.	.	PUNCT
ejpam-6146	492	1	bulletin	bulletin	NOUN
ejpam-6146	492	2	of	of	ADP
ejpam-6146	492	3	the	the	DET
ejpam-6146	492	4	malaysian	malaysian	PROPN
ejpam-6146	492	5	mathematical	mathematical	PROPN
ejpam-6146	492	6	sciences	sciences	PROPN
ejpam-6146	492	7	society	society	NOUN
ejpam-6146	492	8	,	,	PUNCT
ejpam-6146	492	9	45(4):145–1472	45(4):145–1472	NOUN
ejpam-6146	492	10	,	,	PUNCT
ejpam-6146	492	11	2022	2022	NUM
ejpam-6146	492	12	.	.	PUNCT