id	sid	tid	token	lemma	pos
ejpam-6557	1	1	european	european	PROPN
ejpam-6557	1	2	journal	journal	PROPN
ejpam-6557	1	3	of	of	ADP
ejpam-6557	1	4	pure	pure	ADJ
ejpam-6557	1	5	and	and	CCONJ
ejpam-6557	1	6	applied	applied	ADJ
ejpam-6557	1	7	mathematics	mathematic	NOUN
ejpam-6557	1	8	2025	2025	NUM
ejpam-6557	1	9	,	,	PUNCT
ejpam-6557	1	10	vol	vol	NOUN
ejpam-6557	1	11	.	.	PROPN
ejpam-6557	1	12	18	18	NUM
ejpam-6557	1	13	,	,	PUNCT
ejpam-6557	1	14	issue	issue	NOUN
ejpam-6557	1	15	4	4	NUM
ejpam-6557	1	16	,	,	PUNCT
ejpam-6557	1	17	article	article	NOUN
ejpam-6557	1	18	number	number	NOUN
ejpam-6557	1	19	6557	6557	NUM
ejpam-6557	1	20	issn	issn	PROPN
ejpam-6557	1	21	1307	1307	NUM
ejpam-6557	1	22	-	-	SYM
ejpam-6557	1	23	5543	5543	NUM
ejpam-6557	1	24	–	–	PUNCT
ejpam-6557	1	25	ejpam.com	ejpam.com	X
ejpam-6557	1	26	published	publish	VERB
ejpam-6557	1	27	by	by	ADP
ejpam-6557	1	28	new	new	PROPN
ejpam-6557	1	29	york	york	PROPN
ejpam-6557	1	30	business	business	PROPN
ejpam-6557	1	31	global	global	PROPN
ejpam-6557	1	32	a	a	DET
ejpam-6557	1	33	new	new	ADJ
ejpam-6557	1	34	definition	definition	NOUN
ejpam-6557	1	35	of	of	ADP
ejpam-6557	1	36	(	(	PUNCT
ejpam-6557	1	37	α	α	X
ejpam-6557	1	38	,	,	PUNCT
ejpam-6557	1	39	β)-fractional	β)-fractional	PUNCT
ejpam-6557	1	40	derivatives	derivative	NOUN
ejpam-6557	1	41	for	for	ADP
ejpam-6557	1	42	complex	complex	ADV
ejpam-6557	1	43	-	-	PUNCT
ejpam-6557	1	44	valued	value	VERB
ejpam-6557	1	45	functions	function	NOUN
ejpam-6557	1	46	and	and	CCONJ
ejpam-6557	1	47	their	their	PRON
ejpam-6557	1	48	analytic	analytic	ADJ
ejpam-6557	1	49	properties	property	NOUN
ejpam-6557	1	50	waseem	waseem	PROPN
ejpam-6557	1	51	ghazi	ghazi	PROPN
ejpam-6557	1	52	alshanti1,∗	alshanti1,∗	PROPN
ejpam-6557	1	53	,	,	PUNCT
ejpam-6557	1	54	ma	ma	PROPN
ejpam-6557	2	1	′	′	NUM
ejpam-6557	3	1	mon	mon	PROPN
ejpam-6557	3	2	abu	abu	PROPN
ejpam-6557	3	3	hammad1	hammad1	PROPN
ejpam-6557	3	4	,	,	PUNCT
ejpam-6557	3	5	roshdi	roshdi	ADJ
ejpam-6557	3	6	khalil2	khalil2	PROPN
ejpam-6557	3	7	1	1	NUM
ejpam-6557	3	8	department	department	NOUN
ejpam-6557	3	9	of	of	ADP
ejpam-6557	3	10	mathematics	mathematic	NOUN
ejpam-6557	3	11	,	,	PUNCT
ejpam-6557	3	12	al	al	PROPN
ejpam-6557	3	13	zaytoonah	zaytoonah	PROPN
ejpam-6557	3	14	university	university	PROPN
ejpam-6557	3	15	of	of	ADP
ejpam-6557	3	16	jordan	jordan	PROPN
ejpam-6557	3	17	,	,	PUNCT
ejpam-6557	3	18	amman	amman	PROPN
ejpam-6557	3	19	,	,	PUNCT
ejpam-6557	3	20	jordan	jordan	PROPN
ejpam-6557	3	21	2	2	NUM
ejpam-6557	3	22	department	department	NOUN
ejpam-6557	3	23	of	of	ADP
ejpam-6557	3	24	mathematics	mathematic	NOUN
ejpam-6557	3	25	,	,	PUNCT
ejpam-6557	3	26	the	the	DET
ejpam-6557	3	27	university	university	PROPN
ejpam-6557	3	28	of	of	ADP
ejpam-6557	3	29	jordan	jordan	PROPN
ejpam-6557	3	30	,	,	PUNCT
ejpam-6557	3	31	amman	amman	PROPN
ejpam-6557	3	32	,	,	PUNCT
ejpam-6557	3	33	jordan	jordan	PROPN
ejpam-6557	3	34	abstract	abstract	PROPN
ejpam-6557	3	35	.	.	PUNCT
ejpam-6557	4	1	in	in	ADP
ejpam-6557	4	2	this	this	DET
ejpam-6557	4	3	paper	paper	NOUN
ejpam-6557	4	4	,	,	PUNCT
ejpam-6557	4	5	we	we	PRON
ejpam-6557	4	6	present	present	VERB
ejpam-6557	4	7	a	a	DET
ejpam-6557	4	8	new	new	ADJ
ejpam-6557	4	9	definition	definition	NOUN
ejpam-6557	4	10	of	of	ADP
ejpam-6557	4	11	fractional	fractional	ADJ
ejpam-6557	4	12	derivatives	derivative	NOUN
ejpam-6557	4	13	of	of	ADP
ejpam-6557	4	14	complex	complex	ADV
ejpam-6557	4	15	-	-	PUNCT
ejpam-6557	4	16	valued	value	VERB
ejpam-6557	4	17	functions	function	NOUN
ejpam-6557	4	18	,	,	PUNCT
ejpam-6557	4	19	defined	define	VERB
ejpam-6557	4	20	via	via	ADP
ejpam-6557	4	21	two	two	NUM
ejpam-6557	4	22	parameters	parameter	NOUN
ejpam-6557	4	23	(	(	PUNCT
ejpam-6557	4	24	α	α	NOUN
ejpam-6557	4	25	,	,	PUNCT
ejpam-6557	4	26	β	β	NOUN
ejpam-6557	4	27	)	)	PUNCT
ejpam-6557	4	28	.	.	PUNCT
ejpam-6557	5	1	we	we	PRON
ejpam-6557	5	2	introduce	introduce	VERB
ejpam-6557	5	3	the	the	DET
ejpam-6557	5	4	concepts	concept	NOUN
ejpam-6557	5	5	of	of	ADP
ejpam-6557	5	6	both	both	DET
ejpam-6557	5	7	(	(	PUNCT
ejpam-6557	5	8	α	α	NOUN
ejpam-6557	5	9	,	,	PUNCT
ejpam-6557	5	10	β)-cauchyriemann	β)-cauchyriemann	PUNCT
ejpam-6557	5	11	equations	equation	NOUN
ejpam-6557	5	12	and	and	CCONJ
ejpam-6557	5	13	(	(	PUNCT
ejpam-6557	5	14	α	α	NOUN
ejpam-6557	5	15	,	,	PUNCT
ejpam-6557	5	16	β)-fractional	β)-fractional	DET
ejpam-6557	5	17	analytic	analytic	ADJ
ejpam-6557	5	18	complex	complex	NOUN
ejpam-6557	5	19	valued	value	VERB
ejpam-6557	5	20	function	function	NOUN
ejpam-6557	5	21	.	.	PUNCT
ejpam-6557	6	1	2020	2020	NUM
ejpam-6557	6	2	mathematics	mathematic	NOUN
ejpam-6557	6	3	subject	subject	NOUN
ejpam-6557	6	4	classifications	classification	NOUN
ejpam-6557	6	5	:	:	PUNCT
ejpam-6557	6	6	32q02	32q02	NUM
ejpam-6557	6	7	,	,	PUNCT
ejpam-6557	6	8	30k99	30k99	NUM
ejpam-6557	6	9	,	,	PUNCT
ejpam-6557	6	10	30j15	30j15	NUM
ejpam-6557	6	11	key	key	ADJ
ejpam-6557	6	12	words	word	NOUN
ejpam-6557	6	13	and	and	CCONJ
ejpam-6557	6	14	phrases	phrase	NOUN
ejpam-6557	6	15	:	:	PUNCT
ejpam-6557	6	16	(	(	PUNCT
ejpam-6557	6	17	α	α	X
ejpam-6557	6	18	,	,	PUNCT
ejpam-6557	6	19	β)-derivative	β)-derivative	ADJ
ejpam-6557	6	20	,	,	PUNCT
ejpam-6557	6	21	(	(	PUNCT
ejpam-6557	6	22	α	α	X
ejpam-6557	6	23	,	,	PUNCT
ejpam-6557	6	24	β)-cauchy	β)-cauchy	ADJ
ejpam-6557	6	25	-	-	PUNCT
ejpam-6557	6	26	riemann	riemann	PROPN
ejpam-6557	6	27	equations	equation	NOUN
ejpam-6557	6	28	,	,	PUNCT
ejpam-6557	6	29	(	(	PUNCT
ejpam-6557	6	30	α	α	X
ejpam-6557	6	31	,	,	PUNCT
ejpam-6557	6	32	β)-analytic	β)-analytic	ADJ
ejpam-6557	6	33	complex	complex	ADJ
ejpam-6557	6	34	valued	value	VERB
ejpam-6557	6	35	function	function	NOUN
ejpam-6557	6	36	1	1	NUM
ejpam-6557	6	37	.	.	PUNCT
ejpam-6557	6	38	introduction	introduction	NOUN
ejpam-6557	6	39	in	in	ADP
ejpam-6557	6	40	mathematics	mathematic	NOUN
ejpam-6557	6	41	,	,	PUNCT
ejpam-6557	6	42	the	the	DET
ejpam-6557	6	43	positive	positive	ADJ
ejpam-6557	6	44	integer	integer	NOUN
ejpam-6557	6	45	ordered	order	VERB
ejpam-6557	6	46	derivative	derivative	ADJ
ejpam-6557	6	47	dy	dy	NOUN
ejpam-6557	6	48	dx	dx	PROPN
ejpam-6557	6	49	=	=	SYM
ejpam-6557	6	50	f	f	PROPN
ejpam-6557	6	51	′(x	′(x	PROPN
ejpam-6557	6	52	)	)	PUNCT
ejpam-6557	6	53	of	of	ADP
ejpam-6557	6	54	a	a	DET
ejpam-6557	6	55	function	function	NOUN
ejpam-6557	6	56	y	y	PROPN
ejpam-6557	6	57	=	=	SYM
ejpam-6557	6	58	f(x	f(x	PROPN
ejpam-6557	6	59	)	)	PUNCT
ejpam-6557	6	60	is	be	AUX
ejpam-6557	6	61	a	a	DET
ejpam-6557	6	62	concept	concept	NOUN
ejpam-6557	6	63	by	by	ADP
ejpam-6557	6	64	which	which	PRON
ejpam-6557	6	65	we	we	PRON
ejpam-6557	6	66	can	can	AUX
ejpam-6557	6	67	find	find	VERB
ejpam-6557	6	68	the	the	DET
ejpam-6557	6	69	instantaneous	instantaneous	ADJ
ejpam-6557	6	70	rate	rate	NOUN
ejpam-6557	6	71	of	of	ADP
ejpam-6557	6	72	change	change	NOUN
ejpam-6557	6	73	of	of	ADP
ejpam-6557	6	74	a	a	DET
ejpam-6557	6	75	vertical	vertical	ADJ
ejpam-6557	6	76	variable	variable	ADJ
ejpam-6557	6	77	y	y	PROPN
ejpam-6557	6	78	with	with	ADP
ejpam-6557	6	79	respect	respect	NOUN
ejpam-6557	6	80	to	to	ADP
ejpam-6557	6	81	a	a	DET
ejpam-6557	6	82	horizontal	horizontal	ADJ
ejpam-6557	6	83	variable	variable	NOUN
ejpam-6557	6	84	x	x	NOUN
ejpam-6557	6	85	,	,	PUNCT
ejpam-6557	6	86	specifically	specifically	ADV
ejpam-6557	6	87	,	,	PUNCT
ejpam-6557	6	88	we	we	PRON
ejpam-6557	6	89	write	write	VERB
ejpam-6557	6	90	f	f	PROPN
ejpam-6557	6	91	′(x	′(x	NOUN
ejpam-6557	6	92	)	)	PUNCT
ejpam-6557	7	1	=	=	SYM
ejpam-6557	7	2	lim	lim	PROPN
ejpam-6557	7	3	△	△	PROPN
ejpam-6557	7	4	x→0	x→0	PROPN
ejpam-6557	7	5	△	△	PROPN
ejpam-6557	7	6	y	y	PROPN
ejpam-6557	7	7	△	△	X
ejpam-6557	7	8	x	x	X
ejpam-6557	7	9	=	=	SYM
ejpam-6557	7	10	lim	lim	PROPN
ejpam-6557	7	11	△	△	PROPN
ejpam-6557	7	12	x→0	x→0	PROPN
ejpam-6557	7	13	f	f	X
ejpam-6557	7	14	(	(	PUNCT
ejpam-6557	7	15	x+	x+	PROPN
ejpam-6557	7	16	△	△	PROPN
ejpam-6557	7	17	x)−	x)−	PROPN
ejpam-6557	7	18	f	f	X
ejpam-6557	7	19	(	(	PUNCT
ejpam-6557	7	20	x	x	X
ejpam-6557	7	21	)	)	PUNCT
ejpam-6557	7	22	△	△	NOUN
ejpam-6557	7	23	x	x	X
ejpam-6557	7	24	,	,	PUNCT
ejpam-6557	7	25	where	where	SCONJ
ejpam-6557	7	26	△	△	NOUN
ejpam-6557	7	27	x	x	X
ejpam-6557	7	28	and	and	CCONJ
ejpam-6557	7	29	△	△	PROPN
ejpam-6557	7	30	y	y	PROPN
ejpam-6557	7	31	are	be	AUX
ejpam-6557	7	32	,	,	PUNCT
ejpam-6557	7	33	respectively	respectively	ADV
ejpam-6557	7	34	,	,	PUNCT
ejpam-6557	7	35	the	the	DET
ejpam-6557	7	36	corresponding	corresponding	ADJ
ejpam-6557	7	37	increments	increment	NOUN
ejpam-6557	7	38	of	of	ADP
ejpam-6557	7	39	the	the	DET
ejpam-6557	7	40	variable	variable	NOUN
ejpam-6557	7	41	x	x	PUNCT
ejpam-6557	7	42	and	and	CCONJ
ejpam-6557	7	43	the	the	DET
ejpam-6557	7	44	variable	variable	ADJ
ejpam-6557	7	45	y	y	PROPN
ejpam-6557	8	1	[	[	X
ejpam-6557	8	2	1	1	NUM
ejpam-6557	8	3	]	]	PUNCT
ejpam-6557	8	4	.	.	PUNCT
ejpam-6557	9	1	dynamical	dynamical	ADJ
ejpam-6557	9	2	systems	system	NOUN
ejpam-6557	9	3	whose	whose	DET
ejpam-6557	9	4	state	state	NOUN
ejpam-6557	9	5	evolves	evolve	VERB
ejpam-6557	9	6	over	over	ADP
ejpam-6557	9	7	time	time	NOUN
ejpam-6557	9	8	,	,	PUNCT
ejpam-6557	9	9	such	such	ADJ
ejpam-6557	9	10	as	as	ADP
ejpam-6557	9	11	population	population	NOUN
ejpam-6557	9	12	growth	growth	NOUN
ejpam-6557	9	13	and	and	CCONJ
ejpam-6557	9	14	the	the	DET
ejpam-6557	9	15	flowing	flowing	NOUN
ejpam-6557	9	16	of	of	ADP
ejpam-6557	9	17	a	a	DET
ejpam-6557	9	18	fluid	fluid	NOUN
ejpam-6557	9	19	through	through	ADP
ejpam-6557	9	20	a	a	DET
ejpam-6557	9	21	pipe	pipe	NOUN
ejpam-6557	9	22	,	,	PUNCT
ejpam-6557	9	23	can	can	AUX
ejpam-6557	9	24	be	be	AUX
ejpam-6557	9	25	described	describe	VERB
ejpam-6557	9	26	and	and	CCONJ
ejpam-6557	9	27	studied	study	VERB
ejpam-6557	9	28	by	by	ADP
ejpam-6557	9	29	differential	differential	ADJ
ejpam-6557	9	30	equations	equation	NOUN
ejpam-6557	9	31	(	(	PUNCT
ejpam-6557	9	32	ordinary	ordinary	ADJ
ejpam-6557	9	33	or	or	CCONJ
ejpam-6557	9	34	partial	partial	ADJ
ejpam-6557	9	35	)	)	PUNCT
ejpam-6557	9	36	that	that	PRON
ejpam-6557	9	37	involve	involve	VERB
ejpam-6557	9	38	derivative	derivative	NOUN
ejpam-6557	9	39	of	of	ADP
ejpam-6557	9	40	a	a	DET
ejpam-6557	9	41	function	function	NOUN
ejpam-6557	9	42	.	.	PUNCT
ejpam-6557	10	1	however	however	ADV
ejpam-6557	10	2	,	,	PUNCT
ejpam-6557	10	3	some	some	PRON
ejpam-6557	10	4	of	of	ADP
ejpam-6557	10	5	these	these	DET
ejpam-6557	10	6	dynamical	dynamical	ADJ
ejpam-6557	10	7	systems	system	NOUN
ejpam-6557	10	8	that	that	PRON
ejpam-6557	10	9	represent	represent	VERB
ejpam-6557	10	10	phenomena	phenomenon	NOUN
ejpam-6557	10	11	like	like	ADP
ejpam-6557	10	12	electromagnetism	electromagnetism	NOUN
ejpam-6557	10	13	,	,	PUNCT
ejpam-6557	10	14	economy	economy	NOUN
ejpam-6557	10	15	and	and	CCONJ
ejpam-6557	10	16	finance	finance	NOUN
ejpam-6557	10	17	,	,	PUNCT
ejpam-6557	10	18	and	and	CCONJ
ejpam-6557	10	19	signal	signal	NOUN
ejpam-6557	10	20	processing	processing	NOUN
ejpam-6557	10	21	can	can	AUX
ejpam-6557	10	22	not	not	PART
ejpam-6557	10	23	be	be	AUX
ejpam-6557	10	24	handled	handle	VERB
ejpam-6557	10	25	accurately	accurately	ADV
ejpam-6557	10	26	using	use	VERB
ejpam-6557	10	27	standard	standard	ADJ
ejpam-6557	10	28	differential	differential	ADJ
ejpam-6557	10	29	equations	equation	NOUN
ejpam-6557	10	30	that	that	PRON
ejpam-6557	10	31	include	include	VERB
ejpam-6557	10	32	positive	positive	ADJ
ejpam-6557	10	33	integer	integer	NOUN
ejpam-6557	10	34	ordered	order	VERB
ejpam-6557	10	35	derivatives	derivative	NOUN
ejpam-6557	10	36	.	.	PUNCT
ejpam-6557	11	1	in	in	ADP
ejpam-6557	11	2	such	such	ADJ
ejpam-6557	11	3	cases	case	NOUN
ejpam-6557	11	4	,	,	PUNCT
ejpam-6557	11	5	differential	differential	ADJ
ejpam-6557	11	6	equations	equation	NOUN
ejpam-6557	11	7	in	in	ADP
ejpam-6557	11	8	which	which	PRON
ejpam-6557	11	9	the	the	DET
ejpam-6557	11	10	included	include	VERB
ejpam-6557	11	11	derivatives	derivative	NOUN
ejpam-6557	11	12	possess	possess	VERB
ejpam-6557	11	13	fractional	fractional	ADJ
ejpam-6557	11	14	orders	order	NOUN
ejpam-6557	11	15	come	come	VERB
ejpam-6557	11	16	into	into	ADP
ejpam-6557	11	17	play	play	NOUN
ejpam-6557	11	18	.	.	PUNCT
ejpam-6557	12	1	it	it	PRON
ejpam-6557	12	2	is	be	AUX
ejpam-6557	12	3	known	know	VERB
ejpam-6557	12	4	that	that	SCONJ
ejpam-6557	12	5	,	,	PUNCT
ejpam-6557	12	6	the	the	DET
ejpam-6557	12	7	most	most	ADV
ejpam-6557	12	8	widely	widely	ADV
ejpam-6557	12	9	used	use	VERB
ejpam-6557	12	10	definitions	definition	NOUN
ejpam-6557	12	11	for	for	ADP
ejpam-6557	12	12	fractional	fractional	ADJ
ejpam-6557	12	13	derivatives	derivative	NOUN
ejpam-6557	12	14	are	be	AUX
ejpam-6557	12	15	the	the	DET
ejpam-6557	12	16	old	old	ADJ
ejpam-6557	12	17	ones	one	NOUN
ejpam-6557	12	18	riemann	riemann	PROPN
ejpam-6557	12	19	–	–	PUNCT
ejpam-6557	12	20	liouville	liouville	PROPN
ejpam-6557	12	21	and	and	CCONJ
ejpam-6557	12	22	caputo	caputo	PROPN
ejpam-6557	12	23	definitions	definition	NOUN
ejpam-6557	12	24	and	and	CCONJ
ejpam-6557	12	25	the	the	DET
ejpam-6557	12	26	latest	late	ADJ
ejpam-6557	12	27	one	one	NOUN
ejpam-6557	13	1	[	[	X
ejpam-6557	13	2	1–4	1–4	NOUN
ejpam-6557	13	3	]	]	X
ejpam-6557	13	4	,	,	PUNCT
ejpam-6557	13	5	the	the	DET
ejpam-6557	13	6	so	so	ADV
ejpam-6557	13	7	called	call	VERB
ejpam-6557	13	8	conformable	conformable	ADJ
ejpam-6557	13	9	fractional	fractional	ADJ
ejpam-6557	13	10	derivative	derivative	NOUN
ejpam-6557	14	1	[	[	X
ejpam-6557	14	2	5	5	NUM
ejpam-6557	14	3	]	]	PUNCT
ejpam-6557	14	4	.	.	PUNCT
ejpam-6557	15	1	these	these	DET
ejpam-6557	15	2	definitions	definition	NOUN
ejpam-6557	15	3	can	can	AUX
ejpam-6557	15	4	be	be	AUX
ejpam-6557	15	5	stated	state	VERB
ejpam-6557	15	6	as	as	SCONJ
ejpam-6557	15	7	follows	follow	VERB
ejpam-6557	15	8	:	:	PUNCT
ejpam-6557	15	9	∗corresponding	∗corresponde	VERB
ejpam-6557	15	10	author	author	NOUN
ejpam-6557	15	11	.	.	PUNCT
ejpam-6557	16	1	doi	doi	NOUN
ejpam-6557	16	2	:	:	PUNCT
ejpam-6557	16	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6557	https://doi.org/10.29020/nybg.ejpam.v18i4.6557	NOUN
ejpam-6557	16	4	email	email	NOUN
ejpam-6557	16	5	addresses	address	NOUN
ejpam-6557	16	6	:	:	PUNCT
ejpam-6557	16	7	w.alshanti@zuj.edu.jo	w.alshanti@zuj.edu.jo	ADP
ejpam-6557	16	8	(	(	PUNCT
ejpam-6557	16	9	w.	w.	PROPN
ejpam-6557	16	10	g.	g.	PROPN
ejpam-6557	16	11	alshanti	alshanti	PROPN
ejpam-6557	16	12	)	)	PUNCT
ejpam-6557	16	13	,	,	PUNCT
ejpam-6557	16	14	m.abuhammad@zuj.edu.jo	m.abuhammad@zuj.edu.jo	NOUN
ejpam-6557	16	15	(	(	PUNCT
ejpam-6557	16	16	m.	m.	NOUN
ejpam-6557	16	17	a.	a.	PROPN
ejpam-6557	16	18	hammad	hammad	PROPN
ejpam-6557	16	19	)	)	PUNCT
ejpam-6557	16	20	,	,	PUNCT
ejpam-6557	17	1	roshdi@ju.edu.jo	roshdi@ju.edu.jo	PROPN
ejpam-6557	17	2	(	(	PUNCT
ejpam-6557	17	3	r.	r.	PROPN
ejpam-6557	17	4	khalil	khalil	PROPN
ejpam-6557	17	5	)	)	PUNCT
ejpam-6557	17	6	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6557	18	1	1	1	NUM
ejpam-6557	18	2	copyright	copyright	NOUN
ejpam-6557	18	3	:	:	PUNCT
ejpam-6557	18	4	©	©	PROPN
ejpam-6557	18	5	2025	2025	NUM
ejpam-6557	18	6	the	the	DET
ejpam-6557	18	7	author(s	author(s	NOUN
ejpam-6557	18	8	)	)	PUNCT
ejpam-6557	18	9	.	.	PUNCT
ejpam-6557	19	1	(	(	PUNCT
ejpam-6557	19	2	cc	cc	NOUN
ejpam-6557	19	3	by	by	ADP
ejpam-6557	19	4	-	-	PUNCT
ejpam-6557	19	5	nc	nc	PROPN
ejpam-6557	19	6	4.0	4.0	NUM
ejpam-6557	19	7	)	)	PUNCT
ejpam-6557	19	8	w.	w.	PROPN
ejpam-6557	19	9	g.	g.	PROPN
ejpam-6557	19	10	alshanti	alshanti	PROPN
ejpam-6557	19	11	,	,	PUNCT
ejpam-6557	19	12	m.	m.	NOUN
ejpam-6557	19	13	a.	a.	PROPN
ejpam-6557	19	14	hammad	hammad	PROPN
ejpam-6557	19	15	,	,	PUNCT
ejpam-6557	19	16	r.	r.	PROPN
ejpam-6557	19	17	khalil	khalil	PROPN
ejpam-6557	19	18	/	/	SYM
ejpam-6557	19	19	eur	eur	PROPN
ejpam-6557	19	20	.	.	PUNCT
ejpam-6557	20	1	j.	j.	PROPN
ejpam-6557	20	2	pure	pure	PROPN
ejpam-6557	20	3	appl	appl	PROPN
ejpam-6557	20	4	.	.	PROPN
ejpam-6557	20	5	math	math	PROPN
ejpam-6557	20	6	,	,	PUNCT
ejpam-6557	20	7	18	18	NUM
ejpam-6557	20	8	(	(	PUNCT
ejpam-6557	20	9	4	4	NUM
ejpam-6557	20	10	)	)	PUNCT
ejpam-6557	20	11	(	(	PUNCT
ejpam-6557	20	12	2025	2025	NUM
ejpam-6557	20	13	)	)	PUNCT
ejpam-6557	20	14	,	,	PUNCT
ejpam-6557	20	15	6557	6557	NUM
ejpam-6557	20	16	2	2	NUM
ejpam-6557	20	17	of	of	ADP
ejpam-6557	20	18	8	8	NUM
ejpam-6557	20	19	(	(	PUNCT
ejpam-6557	20	20	i	i	NOUN
ejpam-6557	20	21	)	)	PUNCT
ejpam-6557	20	22	riemann	riemann	PROPN
ejpam-6557	20	23	–	–	PUNCT
ejpam-6557	20	24	liouville	liouville	NOUN
ejpam-6557	20	25	definition	definition	NOUN
ejpam-6557	20	26	[	[	X
ejpam-6557	20	27	1	1	NUM
ejpam-6557	20	28	]	]	PUNCT
ejpam-6557	20	29	.	.	PUNCT
ejpam-6557	21	1	for	for	ADP
ejpam-6557	21	2	α	α	PRON
ejpam-6557	21	3	∈	∈	PROPN
ejpam-6557	22	1	[	[	X
ejpam-6557	22	2	n−	n−	NOUN
ejpam-6557	22	3	1	1	NUM
ejpam-6557	22	4	,	,	PUNCT
ejpam-6557	22	5	n	n	CCONJ
ejpam-6557	22	6	)	)	PUNCT
ejpam-6557	22	7	,	,	PUNCT
ejpam-6557	22	8	the	the	DET
ejpam-6557	22	9	α	α	PROPN
ejpam-6557	22	10	derivative	derivative	NOUN
ejpam-6557	22	11	for	for	ADP
ejpam-6557	22	12	f	f	PROPN
ejpam-6557	22	13	is	be	AUX
ejpam-6557	22	14	dα	dα	ADP
ejpam-6557	22	15	a	a	DET
ejpam-6557	22	16	(	(	PUNCT
ejpam-6557	22	17	f	f	NOUN
ejpam-6557	22	18	)	)	PUNCT
ejpam-6557	22	19	(	(	PUNCT
ejpam-6557	22	20	t	t	NOUN
ejpam-6557	22	21	)	)	PUNCT
ejpam-6557	22	22	=	=	SYM
ejpam-6557	22	23	1	1	NUM
ejpam-6557	22	24	γ	γ	X
ejpam-6557	22	25	(	(	PUNCT
ejpam-6557	22	26	n−	n−	NOUN
ejpam-6557	22	27	α	α	X
ejpam-6557	22	28	)	)	PUNCT
ejpam-6557	22	29	dn	dn	PROPN
ejpam-6557	22	30	dtn	dtn	PROPN
ejpam-6557	22	31	t∫	t∫	PRON
ejpam-6557	22	32	a	a	DET
ejpam-6557	22	33	f	f	X
ejpam-6557	22	34	(	(	PUNCT
ejpam-6557	22	35	x	x	X
ejpam-6557	22	36	)	)	PUNCT
ejpam-6557	22	37	(	(	PUNCT
ejpam-6557	22	38	t−	t−	PROPN
ejpam-6557	22	39	x)α−n+1dx	x)α−n+1dx	PROPN
ejpam-6557	22	40	.	.	PUNCT
ejpam-6557	23	1	(	(	PUNCT
ejpam-6557	23	2	ii	ii	X
ejpam-6557	23	3	)	)	PUNCT
ejpam-6557	23	4	caputo	caputo	PROPN
ejpam-6557	23	5	definition	definition	NOUN
ejpam-6557	24	1	[	[	X
ejpam-6557	24	2	2	2	NUM
ejpam-6557	24	3	]	]	PUNCT
ejpam-6557	24	4	.	.	PUNCT
ejpam-6557	25	1	for	for	ADP
ejpam-6557	25	2	α	α	PRON
ejpam-6557	25	3	∈	∈	PROPN
ejpam-6557	26	1	[	[	X
ejpam-6557	26	2	n−	n−	NOUN
ejpam-6557	26	3	1	1	NUM
ejpam-6557	26	4	,	,	PUNCT
ejpam-6557	26	5	n	n	CCONJ
ejpam-6557	26	6	)	)	PUNCT
ejpam-6557	26	7	,	,	PUNCT
ejpam-6557	26	8	the	the	DET
ejpam-6557	26	9	α	α	PROPN
ejpam-6557	26	10	derivative	derivative	NOUN
ejpam-6557	26	11	of	of	ADP
ejpam-6557	26	12	f	f	PROPN
ejpam-6557	26	13	is	be	AUX
ejpam-6557	26	14	dα	dα	ADP
ejpam-6557	26	15	a	a	DET
ejpam-6557	26	16	(	(	PUNCT
ejpam-6557	26	17	f	f	NOUN
ejpam-6557	26	18	)	)	PUNCT
ejpam-6557	26	19	(	(	PUNCT
ejpam-6557	26	20	t	t	NOUN
ejpam-6557	26	21	)	)	PUNCT
ejpam-6557	26	22	=	=	SYM
ejpam-6557	26	23	1	1	NUM
ejpam-6557	26	24	γ	γ	X
ejpam-6557	26	25	(	(	PUNCT
ejpam-6557	26	26	n−	n−	NOUN
ejpam-6557	26	27	α	α	NOUN
ejpam-6557	26	28	)	)	PUNCT
ejpam-6557	26	29	t∫	t∫	PROPN
ejpam-6557	26	30	a	a	DET
ejpam-6557	26	31	f	f	X
ejpam-6557	26	32	(	(	PUNCT
ejpam-6557	26	33	n	n	CCONJ
ejpam-6557	26	34	)	)	PUNCT
ejpam-6557	26	35	(	(	PUNCT
ejpam-6557	26	36	x	x	X
ejpam-6557	26	37	)	)	PUNCT
ejpam-6557	26	38	(	(	PUNCT
ejpam-6557	26	39	t−	t−	PROPN
ejpam-6557	26	40	x)α−n+1dx	x)α−n+1dx	PROPN
ejpam-6557	26	41	.	.	PUNCT
ejpam-6557	27	1	(	(	PUNCT
ejpam-6557	27	2	iii	iii	X
ejpam-6557	27	3	)	)	PUNCT
ejpam-6557	27	4	khalil	khalil	PROPN
ejpam-6557	27	5	definition	definition	NOUN
ejpam-6557	27	6	[	[	X
ejpam-6557	27	7	5	5	NUM
ejpam-6557	27	8	]	]	PUNCT
ejpam-6557	27	9	.	.	PUNCT
ejpam-6557	28	1	for	for	ADP
ejpam-6557	28	2	α	α	PROPN
ejpam-6557	28	3	∈	∈	PROPN
ejpam-6557	28	4	(	(	PUNCT
ejpam-6557	28	5	0	0	NUM
ejpam-6557	28	6	,	,	PUNCT
ejpam-6557	28	7	1	1	NUM
ejpam-6557	28	8	)	)	PUNCT
ejpam-6557	28	9	,	,	PUNCT
ejpam-6557	28	10	the	the	DET
ejpam-6557	28	11	α	α	NOUN
ejpam-6557	28	12	-	-	ADJ
ejpam-6557	28	13	conformable	conformable	ADJ
ejpam-6557	28	14	derivative	derivative	NOUN
ejpam-6557	28	15	of	of	ADP
ejpam-6557	28	16	f	f	PROPN
ejpam-6557	28	17	is	be	AUX
ejpam-6557	28	18	dαu(x	dαu(x	NOUN
ejpam-6557	28	19	)	)	PUNCT
ejpam-6557	29	1	=	=	SYM
ejpam-6557	29	2	lim	lim	PROPN
ejpam-6557	29	3	ϵ→0	ϵ→0	X
ejpam-6557	29	4	u(x+	u(x+	PROPN
ejpam-6557	29	5	ϵx1−α)−	ϵx1−α)−	X
ejpam-6557	29	6	u(x	u(x	NOUN
ejpam-6557	29	7	)	)	PUNCT
ejpam-6557	29	8	ϵ	ϵ	X
ejpam-6557	29	9	.	.	PUNCT
ejpam-6557	30	1	there	there	PRON
ejpam-6557	30	2	is	be	VERB
ejpam-6557	30	3	still	still	ADV
ejpam-6557	30	4	no	no	DET
ejpam-6557	30	5	general	general	ADJ
ejpam-6557	30	6	consensus	consensus	NOUN
ejpam-6557	30	7	on	on	ADP
ejpam-6557	30	8	the	the	DET
ejpam-6557	30	9	concept	concept	NOUN
ejpam-6557	30	10	of	of	ADP
ejpam-6557	30	11	complex	complex	ADJ
ejpam-6557	30	12	-	-	PUNCT
ejpam-6557	30	13	order	order	NOUN
ejpam-6557	30	14	derivatives	derivative	NOUN
ejpam-6557	30	15	.	.	PUNCT
ejpam-6557	31	1	the	the	DET
ejpam-6557	31	2	past	past	ADJ
ejpam-6557	31	3	century	century	NOUN
ejpam-6557	31	4	has	have	AUX
ejpam-6557	31	5	seen	see	VERB
ejpam-6557	31	6	few	few	ADJ
ejpam-6557	31	7	contributions	contribution	NOUN
ejpam-6557	31	8	related	relate	VERB
ejpam-6557	31	9	to	to	ADP
ejpam-6557	31	10	this	this	DET
ejpam-6557	31	11	concept	concept	NOUN
ejpam-6557	31	12	.	.	PUNCT
ejpam-6557	32	1	for	for	ADP
ejpam-6557	32	2	example	example	NOUN
ejpam-6557	32	3	in	in	ADP
ejpam-6557	32	4	[	[	X
ejpam-6557	32	5	6	6	NUM
ejpam-6557	32	6	]	]	PUNCT
ejpam-6557	32	7	,	,	PUNCT
ejpam-6557	32	8	the	the	DET
ejpam-6557	32	9	author	author	NOUN
ejpam-6557	32	10	assumed	assume	VERB
ejpam-6557	32	11	,	,	PUNCT
ejpam-6557	32	12	together	together	ADV
ejpam-6557	32	13	with	with	ADP
ejpam-6557	32	14	certain	certain	ADJ
ejpam-6557	32	15	assumption	assumption	NOUN
ejpam-6557	32	16	,	,	PUNCT
ejpam-6557	32	17	that	that	SCONJ
ejpam-6557	32	18	the	the	DET
ejpam-6557	32	19	derivative	derivative	NOUN
ejpam-6557	32	20	of	of	ADP
ejpam-6557	32	21	complex	complex	ADJ
ejpam-6557	32	22	order	order	NOUN
ejpam-6557	32	23	w	w	NOUN
ejpam-6557	32	24	of	of	ADP
ejpam-6557	32	25	a	a	DET
ejpam-6557	32	26	complex	complex	ADJ
ejpam-6557	32	27	function	function	NOUN
ejpam-6557	32	28	f	f	NOUN
ejpam-6557	32	29	(	(	PUNCT
ejpam-6557	32	30	z	z	NOUN
ejpam-6557	32	31	)	)	PUNCT
ejpam-6557	32	32	,	,	PUNCT
ejpam-6557	32	33	to	to	PART
ejpam-6557	32	34	be	be	AUX
ejpam-6557	32	35	defined	define	VERB
ejpam-6557	32	36	by	by	ADP
ejpam-6557	32	37	the	the	DET
ejpam-6557	32	38	generalized	generalize	VERB
ejpam-6557	32	39	cauchy	cauchy	ADJ
ejpam-6557	32	40	integral	integral	ADJ
ejpam-6557	32	41	d(w)f	d(w)f	NOUN
ejpam-6557	32	42	(	(	PUNCT
ejpam-6557	32	43	z	z	NOUN
ejpam-6557	32	44	)	)	PUNCT
ejpam-6557	32	45	=	=	SYM
ejpam-6557	32	46	γ	γ	X
ejpam-6557	32	47	(	(	PUNCT
ejpam-6557	32	48	1	1	NUM
ejpam-6557	32	49	+	+	CCONJ
ejpam-6557	32	50	w	w	NOUN
ejpam-6557	32	51	)	)	PUNCT
ejpam-6557	32	52	2πi	2πi	NOUN
ejpam-6557	32	53	∫	∫	PROPN
ejpam-6557	33	1	∂d	∂d	PROPN
ejpam-6557	33	2	f	f	PROPN
ejpam-6557	33	3	(	(	PUNCT
ejpam-6557	33	4	η	η	PROPN
ejpam-6557	33	5	)	)	PUNCT
ejpam-6557	33	6	(	(	PUNCT
ejpam-6557	33	7	η	η	PROPN
ejpam-6557	33	8	−	−	PROPN
ejpam-6557	33	9	z)−w−1	z)−w−1	NUM
ejpam-6557	33	10	dη	dη	NOUN
ejpam-6557	33	11	,	,	PUNCT
ejpam-6557	33	12	where	where	SCONJ
ejpam-6557	33	13	d	d	NOUN
ejpam-6557	33	14	is	be	AUX
ejpam-6557	33	15	a	a	DET
ejpam-6557	33	16	closed	closed	ADJ
ejpam-6557	33	17	region	region	NOUN
ejpam-6557	33	18	within	within	ADP
ejpam-6557	33	19	the	the	DET
ejpam-6557	33	20	complex	complex	ADJ
ejpam-6557	33	21	plane	plane	NOUN
ejpam-6557	33	22	c.	c.	NOUN
ejpam-6557	33	23	moreover	moreover	ADV
ejpam-6557	33	24	,	,	PUNCT
ejpam-6557	33	25	as	as	ADP
ejpam-6557	33	26	an	an	DET
ejpam-6557	33	27	application	application	NOUN
ejpam-6557	33	28	in	in	ADP
ejpam-6557	33	29	solving	solve	VERB
ejpam-6557	33	30	hypergeometric	hypergeometric	ADJ
ejpam-6557	33	31	integral	integral	ADJ
ejpam-6557	33	32	equations	equation	NOUN
ejpam-6557	33	33	,	,	PUNCT
ejpam-6557	33	34	the	the	DET
ejpam-6557	33	35	concept	concept	NOUN
ejpam-6557	33	36	of	of	ADP
ejpam-6557	33	37	derivatives	derivative	NOUN
ejpam-6557	33	38	of	of	ADP
ejpam-6557	33	39	purely	purely	ADV
ejpam-6557	33	40	imaginary	imaginary	ADJ
ejpam-6557	33	41	orders	order	NOUN
ejpam-6557	33	42	were	be	AUX
ejpam-6557	33	43	also	also	ADV
ejpam-6557	33	44	studied	study	VERB
ejpam-6557	33	45	[	[	PUNCT
ejpam-6557	33	46	7	7	NUM
ejpam-6557	33	47	]	]	PUNCT
ejpam-6557	33	48	.	.	PUNCT
ejpam-6557	34	1	recently	recently	ADV
ejpam-6557	34	2	,	,	PUNCT
ejpam-6557	34	3	the	the	DET
ejpam-6557	34	4	concept	concept	NOUN
ejpam-6557	34	5	of	of	ADP
ejpam-6557	34	6	α	α	NOUN
ejpam-6557	34	7	-	-	PUNCT
ejpam-6557	34	8	fractional	fractional	ADJ
ejpam-6557	34	9	analytic	analytic	ADJ
ejpam-6557	34	10	function	function	NOUN
ejpam-6557	34	11	was	be	AUX
ejpam-6557	34	12	carried	carry	VERB
ejpam-6557	34	13	out	out	ADP
ejpam-6557	34	14	with	with	ADP
ejpam-6557	34	15	some	some	DET
ejpam-6557	34	16	nice	nice	ADJ
ejpam-6557	34	17	results	result	NOUN
ejpam-6557	34	18	[	[	X
ejpam-6557	34	19	8	8	NUM
ejpam-6557	34	20	]	]	PUNCT
ejpam-6557	34	21	.	.	PUNCT
ejpam-6557	35	1	in	in	ADP
ejpam-6557	35	2	this	this	DET
ejpam-6557	35	3	paper	paper	NOUN
ejpam-6557	35	4	,	,	PUNCT
ejpam-6557	35	5	we	we	PRON
ejpam-6557	35	6	present	present	VERB
ejpam-6557	35	7	the	the	DET
ejpam-6557	35	8	definition	definition	NOUN
ejpam-6557	35	9	of	of	ADP
ejpam-6557	35	10	(	(	PUNCT
ejpam-6557	35	11	α	α	X
ejpam-6557	35	12	,	,	PUNCT
ejpam-6557	35	13	β)-fractional	β)-fractional	DET
ejpam-6557	35	14	derivative	derivative	NOUN
ejpam-6557	35	15	of	of	ADP
ejpam-6557	35	16	a	a	DET
ejpam-6557	35	17	complex	complex	ADJ
ejpam-6557	35	18	function	function	NOUN
ejpam-6557	35	19	f	f	NOUN
ejpam-6557	35	20	(	(	PUNCT
ejpam-6557	35	21	z	z	NOUN
ejpam-6557	35	22	)	)	PUNCT
ejpam-6557	35	23	defined	define	VERB
ejpam-6557	35	24	on	on	ADP
ejpam-6557	35	25	a	a	DET
ejpam-6557	35	26	region	region	NOUN
ejpam-6557	35	27	g	g	PROPN
ejpam-6557	35	28	⊆	⊆	NUM
ejpam-6557	35	29	c.	c.	NOUN
ejpam-6557	35	30	2	2	NUM
ejpam-6557	35	31	.	.	PUNCT
ejpam-6557	36	1	the	the	DET
ejpam-6557	36	2	definition	definition	NOUN
ejpam-6557	36	3	/	/	SYM
ejpam-6557	36	4	proof	proof	NOUN
ejpam-6557	36	5	of	of	ADP
ejpam-6557	36	6	the	the	DET
ejpam-6557	36	7	basic	basic	ADJ
ejpam-6557	36	8	result	result	NOUN
ejpam-6557	36	9	throughout	throughout	ADP
ejpam-6557	36	10	this	this	DET
ejpam-6557	36	11	paper	paper	NOUN
ejpam-6557	36	12	,	,	PUNCT
ejpam-6557	36	13	let	let	VERB
ejpam-6557	36	14	e	e	NOUN
ejpam-6557	36	15	=	=	PRON
ejpam-6557	36	16	{	{	PUNCT
ejpam-6557	36	17	(	(	PUNCT
ejpam-6557	36	18	x	x	NOUN
ejpam-6557	36	19	,	,	PUNCT
ejpam-6557	36	20	y	y	PROPN
ejpam-6557	36	21	)	)	PUNCT
ejpam-6557	36	22	:	:	PUNCT
ejpam-6557	37	1	x	x	X
ejpam-6557	37	2	,	,	PUNCT
ejpam-6557	37	3	y	y	PROPN
ejpam-6557	37	4	>	>	X
ejpam-6557	37	5	0	0	NUM
ejpam-6557	37	6	}	}	PUNCT
ejpam-6557	37	7	and	and	CCONJ
ejpam-6557	37	8	f	f	X
ejpam-6557	37	9	:	:	PUNCT
ejpam-6557	37	10	g	g	PROPN
ejpam-6557	37	11	⊆	⊆	NUM
ejpam-6557	37	12	e	e	X
ejpam-6557	37	13	→	→	PUNCT
ejpam-6557	37	14	c	c	X
ejpam-6557	37	15	be	be	AUX
ejpam-6557	37	16	a	a	DET
ejpam-6557	37	17	complex	complex	ADJ
ejpam-6557	37	18	valued	value	VERB
ejpam-6557	37	19	function	function	NOUN
ejpam-6557	37	20	defined	define	VERB
ejpam-6557	37	21	on	on	ADP
ejpam-6557	37	22	a	a	DET
ejpam-6557	37	23	region	region	NOUN
ejpam-6557	37	24	g	g	PROPN
ejpam-6557	37	25	⊆	⊆	NUM
ejpam-6557	37	26	c.	c.	NOUN
ejpam-6557	37	27	let	let	VERB
ejpam-6557	37	28	z	z	NOUN
ejpam-6557	37	29	◦	◦	VERB
ejpam-6557	37	30	=	=	SYM
ejpam-6557	37	31	(	(	PUNCT
ejpam-6557	37	32	x	x	NOUN
ejpam-6557	37	33	◦	◦	NOUN
ejpam-6557	37	34	,	,	PUNCT
ejpam-6557	37	35	y	y	NOUN
ejpam-6557	37	36	◦	◦	NOUN
ejpam-6557	37	37	)	)	PUNCT
ejpam-6557	37	38	∈	∈	PROPN
ejpam-6557	37	39	g	g	PROPN
ejpam-6557	37	40	◦	◦	NOUN
ejpam-6557	37	41	,	,	PUNCT
ejpam-6557	37	42	the	the	DET
ejpam-6557	37	43	interior	interior	NOUN
ejpam-6557	37	44	of	of	ADP
ejpam-6557	37	45	g	g	NOUN
ejpam-6557	37	46	,	,	PUNCT
ejpam-6557	37	47	and	and	CCONJ
ejpam-6557	37	48	x	x	PART
ejpam-6557	37	49	◦	◦	NOUN
ejpam-6557	37	50	,	,	PUNCT
ejpam-6557	37	51	y	y	NOUN
ejpam-6557	37	52	◦	◦	NOUN
ejpam-6557	37	53	>	>	X
ejpam-6557	37	54	0	0	X
ejpam-6557	37	55	.	.	PUNCT
ejpam-6557	37	56	definition	definition	NOUN
ejpam-6557	37	57	1	1	NUM
ejpam-6557	37	58	.	.	PUNCT
ejpam-6557	38	1	a	a	DET
ejpam-6557	38	2	function	function	NOUN
ejpam-6557	38	3	f	f	NOUN
ejpam-6557	38	4	:	:	PUNCT
ejpam-6557	38	5	g	g	PROPN
ejpam-6557	38	6	⊆	⊆	NUM
ejpam-6557	38	7	c	c	NOUN
ejpam-6557	38	8	→	→	SYM
ejpam-6557	38	9	c	c	PROPN
ejpam-6557	38	10	is	be	AUX
ejpam-6557	38	11	said	say	VERB
ejpam-6557	38	12	to	to	PART
ejpam-6557	38	13	be	be	AUX
ejpam-6557	38	14	(	(	PUNCT
ejpam-6557	38	15	α	α	NOUN
ejpam-6557	38	16	,	,	PUNCT
ejpam-6557	38	17	β)-differentiable	β)-differentiable	PUNCT
ejpam-6557	38	18	at	at	ADP
ejpam-6557	38	19	z	z	NOUN
ejpam-6557	38	20	◦	◦	NOUN
ejpam-6557	38	21	=	=	SYM
ejpam-6557	39	1	x	x	X
ejpam-6557	39	2	◦	◦	NOUN
ejpam-6557	39	3	+	+	CCONJ
ejpam-6557	39	4	iy	iy	X
ejpam-6557	39	5	◦	◦	NOUN
ejpam-6557	39	6	∈	∈	PROPN
ejpam-6557	39	7	g	g	ADP
ejpam-6557	39	8	◦	◦	NOUN
ejpam-6557	39	9	and	and	CCONJ
ejpam-6557	39	10	denoted	denote	VERB
ejpam-6557	39	11	by	by	ADP
ejpam-6557	39	12	f	f	PROPN
ejpam-6557	39	13	(	(	PUNCT
ejpam-6557	39	14	α	α	X
ejpam-6557	39	15	,	,	PUNCT
ejpam-6557	39	16	β)(z	β)(z	PUNCT
ejpam-6557	39	17	◦	◦	NOUN
ejpam-6557	39	18	)	)	PUNCT
ejpam-6557	39	19	,	,	PUNCT
ejpam-6557	40	1	where	where	SCONJ
ejpam-6557	40	2	|α|	|α|	PROPN
ejpam-6557	40	3	,	,	PUNCT
ejpam-6557	40	4	|β|	|β|	X
ejpam-6557	40	5	<	<	X
ejpam-6557	40	6	1	1	NUM
ejpam-6557	40	7	,	,	PUNCT
ejpam-6557	40	8	if	if	SCONJ
ejpam-6557	40	9	f	f	PROPN
ejpam-6557	40	10	(	(	PUNCT
ejpam-6557	40	11	α	α	X
ejpam-6557	40	12	,	,	PUNCT
ejpam-6557	40	13	β)(z	β)(z	PUNCT
ejpam-6557	40	14	◦	◦	NOUN
ejpam-6557	40	15	)	)	PUNCT
ejpam-6557	40	16	=	=	PUNCT
ejpam-6557	40	17	d(α	d(α	NOUN
ejpam-6557	40	18	,	,	PUNCT
ejpam-6557	40	19	β)f	β)f	PUNCT
ejpam-6557	40	20	(	(	PUNCT
ejpam-6557	40	21	z	z	NOUN
ejpam-6557	40	22	◦	◦	NOUN
ejpam-6557	40	23	)	)	PUNCT
ejpam-6557	40	24	=	=	SYM
ejpam-6557	40	25	lim	lim	PROPN
ejpam-6557	40	26	(	(	PUNCT
ejpam-6557	40	27	ϵ1,ϵ2)→(0,0	ϵ1,ϵ2)→(0,0	NOUN
ejpam-6557	40	28	)	)	PUNCT
ejpam-6557	40	29	f(x	f(x	NOUN
ejpam-6557	40	30	◦	◦	NOUN
ejpam-6557	40	31	+	+	CCONJ
ejpam-6557	40	32	ϵ1x	ϵ1x	NOUN
ejpam-6557	40	33	1−α	1−α	NUM
ejpam-6557	40	34	◦	◦	NOUN
ejpam-6557	40	35	,	,	PUNCT
ejpam-6557	40	36	y	y	NOUN
ejpam-6557	40	37	◦	◦	NOUN
ejpam-6557	40	38	+	+	CCONJ
ejpam-6557	40	39	ϵ2y	ϵ2y	X
ejpam-6557	40	40	1−β	1−β	NUM
ejpam-6557	40	41	◦	◦	NOUN
ejpam-6557	40	42	)	)	PUNCT
ejpam-6557	40	43	−	−	PROPN
ejpam-6557	40	44	f(x	f(x	PROPN
ejpam-6557	40	45	◦	◦	NOUN
ejpam-6557	40	46	,	,	PUNCT
ejpam-6557	40	47	y	y	NOUN
ejpam-6557	40	48	◦	◦	NOUN
ejpam-6557	40	49	)	)	PUNCT
ejpam-6557	40	50	ϵ1	ϵ1	NOUN
ejpam-6557	40	51	+	+	NUM
ejpam-6557	40	52	iϵ2	iϵ2	NOUN
ejpam-6557	40	53	,	,	PUNCT
ejpam-6557	40	54	(	(	PUNCT
ejpam-6557	40	55	1	1	X
ejpam-6557	40	56	)	)	PUNCT
ejpam-6557	40	57	exists	exist	VERB
ejpam-6557	40	58	.	.	PUNCT
ejpam-6557	41	1	w.	w.	PROPN
ejpam-6557	41	2	g.	g.	PROPN
ejpam-6557	41	3	alshanti	alshanti	PROPN
ejpam-6557	41	4	,	,	PUNCT
ejpam-6557	41	5	m.	m.	NOUN
ejpam-6557	41	6	a.	a.	PROPN
ejpam-6557	41	7	hammad	hammad	PROPN
ejpam-6557	41	8	,	,	PUNCT
ejpam-6557	41	9	r.	r.	PROPN
ejpam-6557	41	10	khalil	khalil	PROPN
ejpam-6557	41	11	/	/	SYM
ejpam-6557	41	12	eur	eur	PROPN
ejpam-6557	41	13	.	.	PUNCT
ejpam-6557	42	1	j.	j.	PROPN
ejpam-6557	42	2	pure	pure	PROPN
ejpam-6557	42	3	appl	appl	PROPN
ejpam-6557	42	4	.	.	PROPN
ejpam-6557	42	5	math	math	PROPN
ejpam-6557	42	6	,	,	PUNCT
ejpam-6557	42	7	18	18	NUM
ejpam-6557	42	8	(	(	PUNCT
ejpam-6557	42	9	4	4	NUM
ejpam-6557	42	10	)	)	PUNCT
ejpam-6557	42	11	(	(	PUNCT
ejpam-6557	42	12	2025	2025	NUM
ejpam-6557	42	13	)	)	PUNCT
ejpam-6557	42	14	,	,	PUNCT
ejpam-6557	42	15	6557	6557	NUM
ejpam-6557	42	16	3	3	NUM
ejpam-6557	42	17	of	of	ADP
ejpam-6557	42	18	8	8	NUM
ejpam-6557	42	19	accordingly	accordingly	ADV
ejpam-6557	42	20	,	,	PUNCT
ejpam-6557	42	21	let	let	VERB
ejpam-6557	42	22	us	we	PRON
ejpam-6557	42	23	assume	assume	VERB
ejpam-6557	42	24	f	f	PROPN
ejpam-6557	42	25	(	(	PUNCT
ejpam-6557	42	26	z	z	NOUN
ejpam-6557	42	27	)	)	PUNCT
ejpam-6557	42	28	=	=	SYM
ejpam-6557	42	29	u	u	SYM
ejpam-6557	42	30	(	(	PUNCT
ejpam-6557	42	31	x	x	X
ejpam-6557	42	32	,	,	PUNCT
ejpam-6557	42	33	y)+iv	y)+iv	NOUN
ejpam-6557	42	34	(	(	PUNCT
ejpam-6557	42	35	x	x	NOUN
ejpam-6557	42	36	,	,	PUNCT
ejpam-6557	42	37	y	y	NOUN
ejpam-6557	42	38	)	)	PUNCT
ejpam-6557	42	39	such	such	ADJ
ejpam-6557	42	40	that	that	SCONJ
ejpam-6557	42	41	it	it	PRON
ejpam-6557	42	42	is	be	AUX
ejpam-6557	42	43	(	(	PUNCT
ejpam-6557	42	44	α	α	X
ejpam-6557	42	45	,	,	PUNCT
ejpam-6557	42	46	β)-differentiable	β)-differentiable	PUNCT
ejpam-6557	42	47	at	at	ADP
ejpam-6557	42	48	z	z	NOUN
ejpam-6557	42	49	◦	◦	NOUN
ejpam-6557	42	50	=	=	SYM
ejpam-6557	42	51	(	(	PUNCT
ejpam-6557	42	52	x	x	NOUN
ejpam-6557	42	53	◦	◦	NOUN
ejpam-6557	42	54	,	,	PUNCT
ejpam-6557	42	55	y	y	NOUN
ejpam-6557	42	56	◦	◦	NOUN
ejpam-6557	42	57	)	)	PUNCT
ejpam-6557	42	58	.	.	PUNCT
ejpam-6557	43	1	thus	thus	ADV
ejpam-6557	43	2	,	,	PUNCT
ejpam-6557	43	3	the	the	DET
ejpam-6557	43	4	above	above	ADJ
ejpam-6557	43	5	limit	limit	NOUN
ejpam-6557	43	6	(	(	PUNCT
ejpam-6557	43	7	1	1	X
ejpam-6557	43	8	)	)	PUNCT
ejpam-6557	43	9	exists	exist	VERB
ejpam-6557	43	10	along	along	ADP
ejpam-6557	43	11	any	any	DET
ejpam-6557	43	12	bath	bath	NOUN
ejpam-6557	43	13	toward	toward	ADP
ejpam-6557	43	14	z	z	NOUN
ejpam-6557	43	15	◦	◦	NOUN
ejpam-6557	43	16	and	and	CCONJ
ejpam-6557	43	17	all	all	PRON
ejpam-6557	43	18	are	be	AUX
ejpam-6557	43	19	equal	equal	ADJ
ejpam-6557	43	20	.	.	PUNCT
ejpam-6557	44	1	hence	hence	ADV
ejpam-6557	44	2	,	,	PUNCT
ejpam-6557	44	3	if	if	SCONJ
ejpam-6557	44	4	we	we	PRON
ejpam-6557	44	5	move	move	VERB
ejpam-6557	44	6	along	along	ADV
ejpam-6557	44	7	(	(	PUNCT
ejpam-6557	44	8	x	x	NOUN
ejpam-6557	44	9	,	,	PUNCT
ejpam-6557	44	10	y	y	PROPN
ejpam-6557	44	11	◦	◦	NOUN
ejpam-6557	44	12	)	)	PUNCT
ejpam-6557	44	13	towards	towards	ADP
ejpam-6557	44	14	z	z	NOUN
ejpam-6557	44	15	◦	◦	NOUN
ejpam-6557	44	16	=	=	SYM
ejpam-6557	44	17	(	(	PUNCT
ejpam-6557	44	18	x	x	NOUN
ejpam-6557	44	19	◦	◦	NOUN
ejpam-6557	44	20	,	,	PUNCT
ejpam-6557	44	21	y	y	NOUN
ejpam-6557	44	22	◦	◦	NOUN
ejpam-6557	44	23	)	)	PUNCT
ejpam-6557	44	24	,	,	PUNCT
ejpam-6557	44	25	we	we	PRON
ejpam-6557	44	26	get	get	VERB
ejpam-6557	44	27	d(α	d(α	NOUN
ejpam-6557	44	28	,	,	PUNCT
ejpam-6557	44	29	β)f	β)f	PUNCT
ejpam-6557	44	30	(	(	PUNCT
ejpam-6557	44	31	z	z	NOUN
ejpam-6557	44	32	◦	◦	NOUN
ejpam-6557	44	33	)	)	PUNCT
ejpam-6557	45	1	=	=	SYM
ejpam-6557	45	2	lim	lim	PROPN
ejpam-6557	46	1	ϵ1→0	ϵ1→0	PROPN
ejpam-6557	46	2	[	[	PUNCT
ejpam-6557	46	3	u(x	u(x	VERB
ejpam-6557	46	4	◦	◦	NOUN
ejpam-6557	47	1	+	+	NUM
ejpam-6557	47	2	ϵ1x	ϵ1x	NOUN
ejpam-6557	47	3	1−α	1−α	NUM
ejpam-6557	47	4	◦	◦	NOUN
ejpam-6557	47	5	,	,	PUNCT
ejpam-6557	47	6	y	y	NOUN
ejpam-6557	47	7	◦	◦	NOUN
ejpam-6557	47	8	)	)	PUNCT
ejpam-6557	48	1	+	+	CCONJ
ejpam-6557	48	2	iv(x	iv(x	VERB
ejpam-6557	48	3	◦	◦	NOUN
ejpam-6557	48	4	+	+	NUM
ejpam-6557	48	5	ϵ1x	ϵ1x	NOUN
ejpam-6557	48	6	1−α	1−α	NUM
ejpam-6557	49	1	◦	◦	NOUN
ejpam-6557	49	2	,	,	PUNCT
ejpam-6557	49	3	y	y	NOUN
ejpam-6557	49	4	◦	◦	NOUN
ejpam-6557	49	5	)	)	PUNCT
ejpam-6557	49	6	]	]	PUNCT
ejpam-6557	50	1	−	−	PROPN
ejpam-6557	51	1	[	[	X
ejpam-6557	51	2	u	u	X
ejpam-6557	51	3	(	(	PUNCT
ejpam-6557	51	4	x	x	NOUN
ejpam-6557	51	5	◦	◦	NOUN
ejpam-6557	51	6	,	,	PUNCT
ejpam-6557	51	7	y	y	NOUN
ejpam-6557	51	8	◦	◦	NOUN
ejpam-6557	51	9	)	)	PUNCT
ejpam-6557	51	10	+	+	NUM
ejpam-6557	51	11	iv	iv	X
ejpam-6557	51	12	(	(	PUNCT
ejpam-6557	51	13	x	x	NOUN
ejpam-6557	51	14	◦	◦	NOUN
ejpam-6557	51	15	,	,	PUNCT
ejpam-6557	51	16	y	y	NOUN
ejpam-6557	51	17	◦	◦	NOUN
ejpam-6557	51	18	)	)	PUNCT
ejpam-6557	51	19	]	]	PUNCT
ejpam-6557	51	20	ϵ1	ϵ1	NOUN
ejpam-6557	51	21	=	=	SYM
ejpam-6557	51	22	uαx(x	uαx(x	NOUN
ejpam-6557	51	23	◦	◦	NOUN
ejpam-6557	51	24	,	,	PUNCT
ejpam-6557	51	25	y	y	NOUN
ejpam-6557	51	26	◦	◦	NOUN
ejpam-6557	51	27	)	)	PUNCT
ejpam-6557	51	28	+	+	CCONJ
ejpam-6557	51	29	ivαx	ivαx	ADJ
ejpam-6557	51	30	(	(	PUNCT
ejpam-6557	51	31	x	x	NOUN
ejpam-6557	51	32	◦	◦	NOUN
ejpam-6557	51	33	,	,	PUNCT
ejpam-6557	51	34	y	y	NOUN
ejpam-6557	51	35	◦	◦	NOUN
ejpam-6557	51	36	)	)	PUNCT
ejpam-6557	51	37	,	,	PUNCT
ejpam-6557	51	38	(	(	PUNCT
ejpam-6557	51	39	2	2	X
ejpam-6557	51	40	)	)	PUNCT
ejpam-6557	51	41	where	where	SCONJ
ejpam-6557	51	42	uαx(x	uαx(x	NOUN
ejpam-6557	51	43	◦	◦	NOUN
ejpam-6557	51	44	,	,	PUNCT
ejpam-6557	51	45	y	y	NOUN
ejpam-6557	51	46	◦	◦	NOUN
ejpam-6557	51	47	)	)	PUNCT
ejpam-6557	51	48	is	be	AUX
ejpam-6557	51	49	the	the	DET
ejpam-6557	51	50	α	α	NOUN
ejpam-6557	51	51	-	-	PUNCT
ejpam-6557	51	52	fractional	fractional	ADJ
ejpam-6557	51	53	partial	partial	ADJ
ejpam-6557	51	54	derivative	derivative	NOUN
ejpam-6557	51	55	of	of	ADP
ejpam-6557	51	56	u	u	NOUN
ejpam-6557	51	57	with	with	ADP
ejpam-6557	51	58	respect	respect	NOUN
ejpam-6557	51	59	to	to	ADP
ejpam-6557	51	60	x	x	PUNCT
ejpam-6557	51	61	at	at	ADP
ejpam-6557	51	62	(	(	PUNCT
ejpam-6557	51	63	x	x	NOUN
ejpam-6557	51	64	◦	◦	NOUN
ejpam-6557	51	65	,	,	PUNCT
ejpam-6557	51	66	y	y	NOUN
ejpam-6557	51	67	◦	◦	NOUN
ejpam-6557	51	68	)	)	PUNCT
ejpam-6557	51	69	and	and	CCONJ
ejpam-6557	51	70	vαx	vαx	ADJ
ejpam-6557	51	71	(	(	PUNCT
ejpam-6557	51	72	x	x	NOUN
ejpam-6557	51	73	◦	◦	NOUN
ejpam-6557	51	74	,	,	PUNCT
ejpam-6557	51	75	y	y	NOUN
ejpam-6557	51	76	◦	◦	NOUN
ejpam-6557	51	77	)	)	PUNCT
ejpam-6557	51	78	is	be	AUX
ejpam-6557	51	79	the	the	DET
ejpam-6557	51	80	α	α	NOUN
ejpam-6557	51	81	-	-	PUNCT
ejpam-6557	51	82	fractional	fractional	ADJ
ejpam-6557	51	83	partial	partial	ADJ
ejpam-6557	51	84	derivative	derivative	NOUN
ejpam-6557	51	85	of	of	ADP
ejpam-6557	51	86	v	v	NOUN
ejpam-6557	51	87	with	with	ADP
ejpam-6557	51	88	respect	respect	NOUN
ejpam-6557	51	89	to	to	ADP
ejpam-6557	51	90	x	x	PUNCT
ejpam-6557	51	91	at	at	ADP
ejpam-6557	51	92	(	(	PUNCT
ejpam-6557	51	93	x	x	NOUN
ejpam-6557	51	94	◦	◦	NOUN
ejpam-6557	51	95	,	,	PUNCT
ejpam-6557	51	96	y	y	NOUN
ejpam-6557	51	97	◦	◦	NOUN
ejpam-6557	51	98	)	)	PUNCT
ejpam-6557	51	99	.	.	PUNCT
ejpam-6557	52	1	similarly	similarly	ADV
ejpam-6557	52	2	,	,	PUNCT
ejpam-6557	52	3	if	if	SCONJ
ejpam-6557	52	4	we	we	PRON
ejpam-6557	52	5	move	move	VERB
ejpam-6557	52	6	along	along	ADV
ejpam-6557	52	7	(	(	PUNCT
ejpam-6557	52	8	x	x	NOUN
ejpam-6557	52	9	◦	◦	NOUN
ejpam-6557	52	10	,	,	PUNCT
ejpam-6557	52	11	y	y	NOUN
ejpam-6557	52	12	)	)	PUNCT
ejpam-6557	52	13	towards	towards	ADP
ejpam-6557	52	14	z	z	NOUN
ejpam-6557	52	15	◦	◦	NOUN
ejpam-6557	52	16	=	=	SYM
ejpam-6557	52	17	(	(	PUNCT
ejpam-6557	52	18	x	x	NOUN
ejpam-6557	52	19	◦	◦	NOUN
ejpam-6557	52	20	,	,	PUNCT
ejpam-6557	52	21	y	y	NOUN
ejpam-6557	52	22	◦	◦	NOUN
ejpam-6557	52	23	)	)	PUNCT
ejpam-6557	52	24	,	,	PUNCT
ejpam-6557	52	25	we	we	PRON
ejpam-6557	52	26	get	get	VERB
ejpam-6557	52	27	d(α	d(α	NOUN
ejpam-6557	52	28	,	,	PUNCT
ejpam-6557	52	29	β)f	β)f	PUNCT
ejpam-6557	52	30	(	(	PUNCT
ejpam-6557	52	31	z	z	NOUN
ejpam-6557	52	32	◦	◦	NOUN
ejpam-6557	52	33	)	)	PUNCT
ejpam-6557	53	1	=	=	SYM
ejpam-6557	53	2	lim	lim	PROPN
ejpam-6557	53	3	ϵ2→0	ϵ2→0	PROPN
ejpam-6557	53	4	[	[	PUNCT
ejpam-6557	53	5	u(x	u(x	PROPN
ejpam-6557	53	6	◦	◦	NOUN
ejpam-6557	53	7	,	,	PUNCT
ejpam-6557	53	8	y	y	NOUN
ejpam-6557	53	9	◦	◦	NOUN
ejpam-6557	54	1	+	+	CCONJ
ejpam-6557	54	2	ϵ2y	ϵ2y	X
ejpam-6557	54	3	1−β	1−β	NUM
ejpam-6557	54	4	◦	◦	NOUN
ejpam-6557	54	5	)	)	PUNCT
ejpam-6557	55	1	+	+	CCONJ
ejpam-6557	55	2	iv(x	iv(x	NOUN
ejpam-6557	55	3	◦	◦	NOUN
ejpam-6557	55	4	,	,	PUNCT
ejpam-6557	55	5	y	y	NOUN
ejpam-6557	55	6	◦	◦	NOUN
ejpam-6557	55	7	+	+	CCONJ
ejpam-6557	55	8	ϵ2y	ϵ2y	X
ejpam-6557	55	9	1−β	1−β	NUM
ejpam-6557	55	10	◦	◦	NOUN
ejpam-6557	55	11	)	)	PUNCT
ejpam-6557	55	12	]	]	PUNCT
ejpam-6557	56	1	−	−	PUNCT
ejpam-6557	57	1	[	[	X
ejpam-6557	57	2	u	u	X
ejpam-6557	57	3	(	(	PUNCT
ejpam-6557	57	4	x	x	NOUN
ejpam-6557	57	5	◦	◦	NOUN
ejpam-6557	57	6	,	,	PUNCT
ejpam-6557	57	7	y	y	NOUN
ejpam-6557	57	8	◦	◦	NOUN
ejpam-6557	57	9	)	)	PUNCT
ejpam-6557	57	10	+	+	NUM
ejpam-6557	57	11	iv	iv	X
ejpam-6557	57	12	(	(	PUNCT
ejpam-6557	57	13	x	x	NOUN
ejpam-6557	57	14	◦	◦	NOUN
ejpam-6557	57	15	,	,	PUNCT
ejpam-6557	57	16	y	y	NOUN
ejpam-6557	57	17	◦	◦	NOUN
ejpam-6557	57	18	)	)	PUNCT
ejpam-6557	57	19	]	]	PUNCT
ejpam-6557	57	20	iϵ2	iϵ2	NOUN
ejpam-6557	57	21	=	=	SYM
ejpam-6557	57	22	vβy	vβy	X
ejpam-6557	57	23	(	(	PUNCT
ejpam-6557	57	24	x	x	SYM
ejpam-6557	57	25	◦	◦	NOUN
ejpam-6557	57	26	,	,	PUNCT
ejpam-6557	57	27	y	y	PROPN
ejpam-6557	57	28	◦	◦	NOUN
ejpam-6557	57	29	)−	)−	PUNCT
ejpam-6557	57	30	iuβy	iuβy	ADJ
ejpam-6557	57	31	(	(	PUNCT
ejpam-6557	57	32	x	x	SYM
ejpam-6557	57	33	◦	◦	NOUN
ejpam-6557	57	34	,	,	PUNCT
ejpam-6557	57	35	y	y	NOUN
ejpam-6557	57	36	◦	◦	NOUN
ejpam-6557	57	37	)	)	PUNCT
ejpam-6557	57	38	,	,	PUNCT
ejpam-6557	57	39	(	(	PUNCT
ejpam-6557	57	40	3	3	X
ejpam-6557	57	41	)	)	PUNCT
ejpam-6557	57	42	where	where	SCONJ
ejpam-6557	57	43	vβy	vβy	NOUN
ejpam-6557	57	44	(	(	PUNCT
ejpam-6557	57	45	x	x	NOUN
ejpam-6557	57	46	◦	◦	NOUN
ejpam-6557	57	47	,	,	PUNCT
ejpam-6557	57	48	y	y	NOUN
ejpam-6557	57	49	◦	◦	NOUN
ejpam-6557	57	50	)	)	PUNCT
ejpam-6557	57	51	is	be	AUX
ejpam-6557	57	52	the	the	DET
ejpam-6557	57	53	β	β	NOUN
ejpam-6557	57	54	-	-	ADJ
ejpam-6557	57	55	fractional	fractional	ADJ
ejpam-6557	57	56	partial	partial	ADJ
ejpam-6557	57	57	derivative	derivative	NOUN
ejpam-6557	57	58	of	of	ADP
ejpam-6557	57	59	v	v	NOUN
ejpam-6557	57	60	with	with	ADP
ejpam-6557	57	61	respect	respect	NOUN
ejpam-6557	57	62	to	to	ADP
ejpam-6557	57	63	y	y	PROPN
ejpam-6557	57	64	at	at	ADP
ejpam-6557	57	65	(	(	PUNCT
ejpam-6557	57	66	x	x	NOUN
ejpam-6557	57	67	◦	◦	NOUN
ejpam-6557	57	68	,	,	PUNCT
ejpam-6557	57	69	y	y	NOUN
ejpam-6557	57	70	◦	◦	NOUN
ejpam-6557	57	71	)	)	PUNCT
ejpam-6557	57	72	and	and	CCONJ
ejpam-6557	57	73	uβy	uβy	ADJ
ejpam-6557	57	74	(	(	PUNCT
ejpam-6557	57	75	x	x	NOUN
ejpam-6557	57	76	◦	◦	NOUN
ejpam-6557	57	77	,	,	PUNCT
ejpam-6557	57	78	y	y	NOUN
ejpam-6557	57	79	◦	◦	NOUN
ejpam-6557	57	80	)	)	PUNCT
ejpam-6557	57	81	is	be	AUX
ejpam-6557	57	82	the	the	DET
ejpam-6557	57	83	β	β	NOUN
ejpam-6557	57	84	-	-	ADJ
ejpam-6557	57	85	fractional	fractional	ADJ
ejpam-6557	57	86	partial	partial	ADJ
ejpam-6557	57	87	derivative	derivative	NOUN
ejpam-6557	57	88	of	of	ADP
ejpam-6557	57	89	v	v	NOUN
ejpam-6557	57	90	with	with	ADP
ejpam-6557	57	91	respect	respect	NOUN
ejpam-6557	57	92	to	to	ADP
ejpam-6557	57	93	y	y	PROPN
ejpam-6557	57	94	at	at	ADP
ejpam-6557	57	95	(	(	PUNCT
ejpam-6557	57	96	x	x	NOUN
ejpam-6557	57	97	◦	◦	NOUN
ejpam-6557	57	98	,	,	PUNCT
ejpam-6557	57	99	y	y	NOUN
ejpam-6557	57	100	◦	◦	NOUN
ejpam-6557	57	101	)	)	PUNCT
ejpam-6557	57	102	.	.	PUNCT
ejpam-6557	58	1	therefore	therefore	ADV
ejpam-6557	58	2	,	,	PUNCT
ejpam-6557	58	3	for	for	ADP
ejpam-6557	58	4	(	(	PUNCT
ejpam-6557	58	5	α	α	NOUN
ejpam-6557	58	6	,	,	PUNCT
ejpam-6557	58	7	β)-differentiability	β)-differentiability	PUNCT
ejpam-6557	58	8	of	of	ADP
ejpam-6557	58	9	f	f	PROPN
ejpam-6557	58	10	at	at	ADP
ejpam-6557	58	11	z	z	PROPN
ejpam-6557	58	12	◦	◦	NOUN
ejpam-6557	58	13	=	=	SYM
ejpam-6557	58	14	(	(	PUNCT
ejpam-6557	58	15	x	x	NOUN
ejpam-6557	58	16	◦	◦	NOUN
ejpam-6557	58	17	,	,	PUNCT
ejpam-6557	58	18	y	y	NOUN
ejpam-6557	58	19	◦	◦	NOUN
ejpam-6557	58	20	)	)	PUNCT
ejpam-6557	58	21	,	,	PUNCT
ejpam-6557	58	22	we	we	PRON
ejpam-6557	58	23	must	must	AUX
ejpam-6557	58	24	have	have	VERB
ejpam-6557	58	25	uαx(z	uαx(z	VERB
ejpam-6557	58	26	◦	◦	NOUN
ejpam-6557	58	27	)	)	PUNCT
ejpam-6557	58	28	=	=	VERB
ejpam-6557	59	1	vβy	vβy	NOUN
ejpam-6557	59	2	(	(	PUNCT
ejpam-6557	59	3	z	z	NOUN
ejpam-6557	59	4	◦	◦	NOUN
ejpam-6557	59	5	)	)	PUNCT
ejpam-6557	59	6	,	,	PUNCT
ejpam-6557	59	7	uβy	uβy	PROPN
ejpam-6557	59	8	(	(	PUNCT
ejpam-6557	59	9	z	z	NOUN
ejpam-6557	59	10	◦	◦	NOUN
ejpam-6557	59	11	)	)	PUNCT
ejpam-6557	59	12	=	=	SYM
ejpam-6557	59	13	−vαx	−vαx	NOUN
ejpam-6557	59	14	(	(	PUNCT
ejpam-6557	59	15	z	z	NOUN
ejpam-6557	59	16	◦	◦	NOUN
ejpam-6557	59	17	)	)	PUNCT
ejpam-6557	59	18	.	.	PUNCT
ejpam-6557	60	1	(	(	PUNCT
ejpam-6557	60	2	4	4	X
ejpam-6557	60	3	)	)	PUNCT
ejpam-6557	60	4	equations	equation	NOUN
ejpam-6557	60	5	(	(	PUNCT
ejpam-6557	60	6	4	4	X
ejpam-6557	60	7	)	)	PUNCT
ejpam-6557	60	8	will	will	AUX
ejpam-6557	60	9	be	be	AUX
ejpam-6557	60	10	called	call	VERB
ejpam-6557	60	11	the	the	DET
ejpam-6557	60	12	(	(	PUNCT
ejpam-6557	60	13	α	α	NOUN
ejpam-6557	60	14	,	,	PUNCT
ejpam-6557	60	15	β)-cauchy	β)-cauchy	NOUN
ejpam-6557	60	16	-	-	PUNCT
ejpam-6557	60	17	riemann	riemann	PROPN
ejpam-6557	60	18	equtions	eqution	NOUN
ejpam-6557	60	19	.	.	PUNCT
ejpam-6557	61	1	noting	note	VERB
ejpam-6557	61	2	that	that	SCONJ
ejpam-6557	61	3	if	if	SCONJ
ejpam-6557	61	4	α	α	PRON
ejpam-6557	61	5	=	=	SYM
ejpam-6557	61	6	β	β	NOUN
ejpam-6557	61	7	,	,	PUNCT
ejpam-6557	61	8	then	then	ADV
ejpam-6557	61	9	the	the	DET
ejpam-6557	61	10	two	two	NUM
ejpam-6557	61	11	equations	equation	NOUN
ejpam-6557	61	12	(	(	PUNCT
ejpam-6557	61	13	4	4	X
ejpam-6557	61	14	)	)	PUNCT
ejpam-6557	61	15	represent	represent	VERB
ejpam-6557	61	16	the	the	DET
ejpam-6557	61	17	α	α	PROPN
ejpam-6557	61	18	-	-	PUNCT
ejpam-6557	61	19	cauchy	cauchy	ADJ
ejpam-6557	61	20	-	-	PUNCT
ejpam-6557	61	21	riemann	riemann	PROPN
ejpam-6557	61	22	equations	equation	NOUN
ejpam-6557	61	23	that	that	PRON
ejpam-6557	61	24	are	be	AUX
ejpam-6557	61	25	reported	report	VERB
ejpam-6557	61	26	in	in	ADP
ejpam-6557	61	27	[	[	X
ejpam-6557	61	28	8	8	NUM
ejpam-6557	61	29	]	]	PUNCT
ejpam-6557	61	30	.	.	PUNCT
ejpam-6557	62	1	example	example	NOUN
ejpam-6557	63	1	1	1	X
ejpam-6557	63	2	.	.	X
ejpam-6557	63	3	consider	consider	VERB
ejpam-6557	63	4	f	f	PROPN
ejpam-6557	63	5	(	(	PUNCT
ejpam-6557	63	6	z	z	NOUN
ejpam-6557	63	7	)	)	PUNCT
ejpam-6557	63	8	=	=	SYM
ejpam-6557	63	9	f(x	f(x	PROPN
ejpam-6557	63	10	,	,	PUNCT
ejpam-6557	63	11	y	y	NOUN
ejpam-6557	63	12	)	)	PUNCT
ejpam-6557	63	13	=	=	X
ejpam-6557	63	14	β2x2α−α2y2β	β2x2α−α2y2β	PUNCT
ejpam-6557	64	1	α2β2	α2β2	PUNCT
ejpam-6557	64	2	+	+	X
ejpam-6557	64	3	i2xαyβ	i2xαyβ	ADJ
ejpam-6557	64	4	αβ	αβ	INTJ
ejpam-6557	64	5	,	,	PUNCT
ejpam-6557	64	6	where	where	SCONJ
ejpam-6557	64	7	|α|	|α|	PROPN
ejpam-6557	64	8	,	,	PUNCT
ejpam-6557	64	9	|β|	|β|	X
ejpam-6557	64	10	<	<	X
ejpam-6557	64	11	1	1	X
ejpam-6557	64	12	.	.	PUNCT
ejpam-6557	65	1	then	then	ADV
ejpam-6557	65	2	we	we	PRON
ejpam-6557	65	3	have	have	VERB
ejpam-6557	65	4	uαx	uαx	NOUN
ejpam-6557	65	5	=	=	SYM
ejpam-6557	65	6	vβy	vβy	NOUN
ejpam-6557	65	7	=	=	SYM
ejpam-6557	65	8	2	2	NUM
ejpam-6557	65	9	αx	αx	ADV
ejpam-6557	65	10	α	α	NOUN
ejpam-6557	65	11	,	,	PUNCT
ejpam-6557	65	12	and	and	CCONJ
ejpam-6557	65	13	uβy	uβy	VERB
ejpam-6557	65	14	=	=	SYM
ejpam-6557	65	15	−vαx	−vαx	ADJ
ejpam-6557	65	16	=	=	PUNCT
ejpam-6557	65	17	−	−	PROPN
ejpam-6557	65	18	2	2	NUM
ejpam-6557	65	19	βy	βy	NOUN
ejpam-6557	65	20	β	β	X
ejpam-6557	65	21	[	[	X
ejpam-6557	65	22	5	5	NUM
ejpam-6557	65	23	]	]	PUNCT
ejpam-6557	65	24	.	.	PUNCT
ejpam-6557	66	1	hence	hence	ADV
ejpam-6557	66	2	,	,	PUNCT
ejpam-6557	66	3	uαx	uαx	PROPN
ejpam-6557	66	4	,	,	PUNCT
ejpam-6557	66	5	uβy	uβy	ADJ
ejpam-6557	66	6	,	,	PUNCT
ejpam-6557	66	7	vαx	vαx	ADJ
ejpam-6557	66	8	,	,	PUNCT
ejpam-6557	66	9	and	and	CCONJ
ejpam-6557	66	10	vβy	vβy	NOUN
ejpam-6557	66	11	are	be	AUX
ejpam-6557	66	12	all	all	ADV
ejpam-6557	66	13	exist	exist	VERB
ejpam-6557	66	14	and	and	CCONJ
ejpam-6557	66	15	continuous	continuous	ADJ
ejpam-6557	66	16	at	at	ADP
ejpam-6557	66	17	z	z	NOUN
ejpam-6557	66	18	◦	◦	NOUN
ejpam-6557	66	19	=	=	SYM
ejpam-6557	66	20	x	x	X
ejpam-6557	66	21	◦	◦	NOUN
ejpam-6557	66	22	+	+	CCONJ
ejpam-6557	66	23	iy	iy	X
ejpam-6557	66	24	◦	◦	NOUN
ejpam-6557	66	25	∈	∈	PROPN
ejpam-6557	66	26	g	g	ADP
ejpam-6557	66	27	◦	◦	NOUN
ejpam-6557	66	28	.	.	PUNCT
ejpam-6557	67	1	moreover	moreover	ADV
ejpam-6557	67	2	,	,	PUNCT
ejpam-6557	67	3	they	they	PRON
ejpam-6557	67	4	satisfy	satisfy	VERB
ejpam-6557	67	5	the	the	DET
ejpam-6557	67	6	(	(	PUNCT
ejpam-6557	67	7	α	α	NOUN
ejpam-6557	67	8	,	,	PUNCT
ejpam-6557	67	9	β)-cauchy	β)-cauchy	PUNCT
ejpam-6557	67	10	-	-	PUNCT
ejpam-6557	67	11	riemann	riemann	PROPN
ejpam-6557	67	12	equations	equation	NOUN
ejpam-6557	67	13	(	(	PUNCT
ejpam-6557	67	14	4	4	NUM
ejpam-6557	67	15	)	)	PUNCT
ejpam-6557	67	16	.	.	PUNCT
ejpam-6557	68	1	example	example	NOUN
ejpam-6557	69	1	2	2	NUM
ejpam-6557	69	2	.	.	X
ejpam-6557	69	3	for	for	ADP
ejpam-6557	69	4	f	f	PROPN
ejpam-6557	69	5	(	(	PUNCT
ejpam-6557	69	6	z	z	NOUN
ejpam-6557	69	7	)	)	PUNCT
ejpam-6557	69	8	=	=	SYM
ejpam-6557	69	9	f(x	f(x	PROPN
ejpam-6557	69	10	,	,	PUNCT
ejpam-6557	69	11	y	y	NOUN
ejpam-6557	69	12	)	)	PUNCT
ejpam-6557	70	1	=	=	SYM
ejpam-6557	70	2	e	e	X
ejpam-6557	70	3	(	(	PUNCT
ejpam-6557	70	4	xα	xα	ADP
ejpam-6557	70	5	α	α	PRON
ejpam-6557	71	1	+	+	NOUN
ejpam-6557	71	2	i	i	VERB
ejpam-6557	71	3	y	y	NOUN
ejpam-6557	71	4	β	β	X
ejpam-6557	71	5	β	β	X
ejpam-6557	71	6	)	)	PUNCT
ejpam-6557	72	1	=	=	PUNCT
ejpam-6557	72	2	e	e	X
ejpam-6557	72	3	xα	xα	PROPN
ejpam-6557	72	4	α	α	PROPN
ejpam-6557	72	5	cos	cos	PROPN
ejpam-6557	72	6	yβ	yβ	PROPN
ejpam-6557	72	7	β	β	PROPN
ejpam-6557	72	8	+	+	CCONJ
ejpam-6557	72	9	ie	ie	X
ejpam-6557	72	10	xα	xα	ADJ
ejpam-6557	72	11	α	α	PRON
ejpam-6557	72	12	sin	sin	NOUN
ejpam-6557	72	13	yβ	yβ	NOUN
ejpam-6557	72	14	β	β	NOUN
ejpam-6557	72	15	,	,	PUNCT
ejpam-6557	72	16	where	where	SCONJ
ejpam-6557	72	17	|α|	|α|	PROPN
ejpam-6557	72	18	,	,	PUNCT
ejpam-6557	72	19	|β|	|β|	X
ejpam-6557	72	20	<	<	X
ejpam-6557	72	21	1	1	X
ejpam-6557	72	22	.	.	PUNCT
ejpam-6557	73	1	we	we	PRON
ejpam-6557	73	2	have	have	VERB
ejpam-6557	73	3	uαx	uαx	NOUN
ejpam-6557	73	4	=	=	SYM
ejpam-6557	73	5	vβy	vβy	NOUN
ejpam-6557	73	6	=	=	PUNCT
ejpam-6557	73	7	e	e	X
ejpam-6557	73	8	xα	xα	PROPN
ejpam-6557	73	9	α	α	PROPN
ejpam-6557	73	10	cos	cos	PROPN
ejpam-6557	73	11	yβ	yβ	PROPN
ejpam-6557	73	12	β	β	X
ejpam-6557	73	13	,	,	PUNCT
ejpam-6557	73	14	and	and	CCONJ
ejpam-6557	73	15	uβy	uβy	VERB
ejpam-6557	73	16	=	=	SYM
ejpam-6557	73	17	−vαx	−vαx	NOUN
ejpam-6557	73	18	=	=	PUNCT
ejpam-6557	73	19	−e	−e	NOUN
ejpam-6557	73	20	xα	xα	PUNCT
ejpam-6557	73	21	α	α	PRON
ejpam-6557	73	22	sin	sin	NOUN
ejpam-6557	73	23	yβ	yβ	NOUN
ejpam-6557	73	24	β	β	X
ejpam-6557	74	1	[	[	X
ejpam-6557	74	2	5	5	NUM
ejpam-6557	74	3	]	]	PUNCT
ejpam-6557	74	4	.	.	PUNCT
ejpam-6557	75	1	hence	hence	ADV
ejpam-6557	75	2	,	,	PUNCT
ejpam-6557	75	3	uαx	uαx	PROPN
ejpam-6557	75	4	,	,	PUNCT
ejpam-6557	75	5	u	u	NOUN
ejpam-6557	75	6	β	β	X
ejpam-6557	75	7	y	y	PROPN
ejpam-6557	75	8	,	,	PUNCT
ejpam-6557	75	9	vαx	vαx	ADJ
ejpam-6557	75	10	,	,	PUNCT
ejpam-6557	75	11	and	and	CCONJ
ejpam-6557	75	12	vβy	vβy	NOUN
ejpam-6557	75	13	are	be	AUX
ejpam-6557	75	14	all	all	ADV
ejpam-6557	75	15	exist	exist	VERB
ejpam-6557	75	16	and	and	CCONJ
ejpam-6557	75	17	continuous	continuous	ADJ
ejpam-6557	75	18	at	at	ADP
ejpam-6557	75	19	z	z	NOUN
ejpam-6557	75	20	◦	◦	NOUN
ejpam-6557	75	21	=	=	SYM
ejpam-6557	75	22	x	x	SYM
ejpam-6557	75	23	◦	◦	NOUN
ejpam-6557	75	24	+	+	CCONJ
ejpam-6557	75	25	iy	iy	NOUN
ejpam-6557	75	26	◦	◦	NOUN
ejpam-6557	75	27	∈	∈	PROPN
ejpam-6557	75	28	g	g	ADP
ejpam-6557	75	29	◦	◦	NOUN
ejpam-6557	75	30	.	.	PUNCT
ejpam-6557	76	1	moreover	moreover	ADV
ejpam-6557	76	2	,	,	PUNCT
ejpam-6557	76	3	they	they	PRON
ejpam-6557	76	4	satisfy	satisfy	VERB
ejpam-6557	76	5	the	the	DET
ejpam-6557	76	6	(	(	PUNCT
ejpam-6557	76	7	α	α	NOUN
ejpam-6557	76	8	,	,	PUNCT
ejpam-6557	76	9	β)-cauchy	β)-cauchy	PUNCT
ejpam-6557	76	10	-	-	PUNCT
ejpam-6557	76	11	riemann	riemann	PROPN
ejpam-6557	76	12	equations	equation	NOUN
ejpam-6557	76	13	(	(	PUNCT
ejpam-6557	76	14	4	4	NUM
ejpam-6557	76	15	)	)	PUNCT
ejpam-6557	76	16	.	.	PUNCT
ejpam-6557	77	1	clearly	clearly	ADV
ejpam-6557	77	2	,	,	PUNCT
ejpam-6557	77	3	as	as	ADP
ejpam-6557	77	4	in	in	ADP
ejpam-6557	77	5	the	the	DET
ejpam-6557	77	6	classical	classical	ADJ
ejpam-6557	77	7	case	case	NOUN
ejpam-6557	77	8	,	,	PUNCT
ejpam-6557	77	9	(	(	PUNCT
ejpam-6557	77	10	α	α	X
ejpam-6557	77	11	,	,	PUNCT
ejpam-6557	77	12	β)-cauchy	β)-cauchy	PUNCT
ejpam-6557	77	13	-	-	PUNCT
ejpam-6557	77	14	riemann	riemann	PROPN
ejpam-6557	77	15	equations	equation	NOUN
ejpam-6557	77	16	are	be	AUX
ejpam-6557	77	17	necessary	necessary	ADJ
ejpam-6557	77	18	condition	condition	NOUN
ejpam-6557	77	19	for	for	ADP
ejpam-6557	77	20	(	(	PUNCT
ejpam-6557	77	21	α	α	NOUN
ejpam-6557	77	22	,	,	PUNCT
ejpam-6557	77	23	β)-differentiability	β)-differentiability	PUNCT
ejpam-6557	77	24	of	of	ADP
ejpam-6557	77	25	f	f	PROPN
ejpam-6557	77	26	(	(	PUNCT
ejpam-6557	77	27	z	z	NOUN
ejpam-6557	77	28	)	)	PUNCT
ejpam-6557	77	29	at	at	ADP
ejpam-6557	77	30	z	z	NOUN
ejpam-6557	77	31	◦	◦	NOUN
ejpam-6557	77	32	but	but	CCONJ
ejpam-6557	77	33	not	not	PART
ejpam-6557	77	34	a	a	DET
ejpam-6557	77	35	sufficient	sufficient	ADJ
ejpam-6557	77	36	one	one	NOUN
ejpam-6557	77	37	.	.	PUNCT
ejpam-6557	78	1	this	this	PRON
ejpam-6557	78	2	leads	lead	VERB
ejpam-6557	78	3	to	to	ADP
ejpam-6557	78	4	the	the	DET
ejpam-6557	78	5	following	following	ADJ
ejpam-6557	78	6	result	result	NOUN
ejpam-6557	78	7	.	.	PUNCT
ejpam-6557	79	1	w.	w.	PROPN
ejpam-6557	79	2	g.	g.	PROPN
ejpam-6557	79	3	alshanti	alshanti	PROPN
ejpam-6557	79	4	,	,	PUNCT
ejpam-6557	79	5	m.	m.	NOUN
ejpam-6557	79	6	a.	a.	PROPN
ejpam-6557	79	7	hammad	hammad	PROPN
ejpam-6557	79	8	,	,	PUNCT
ejpam-6557	79	9	r.	r.	PROPN
ejpam-6557	79	10	khalil	khalil	PROPN
ejpam-6557	79	11	/	/	SYM
ejpam-6557	79	12	eur	eur	PROPN
ejpam-6557	79	13	.	.	PUNCT
ejpam-6557	80	1	j.	j.	PROPN
ejpam-6557	80	2	pure	pure	PROPN
ejpam-6557	80	3	appl	appl	PROPN
ejpam-6557	80	4	.	.	PROPN
ejpam-6557	80	5	math	math	PROPN
ejpam-6557	80	6	,	,	PUNCT
ejpam-6557	80	7	18	18	NUM
ejpam-6557	80	8	(	(	PUNCT
ejpam-6557	80	9	4	4	NUM
ejpam-6557	80	10	)	)	PUNCT
ejpam-6557	80	11	(	(	PUNCT
ejpam-6557	80	12	2025	2025	NUM
ejpam-6557	80	13	)	)	PUNCT
ejpam-6557	80	14	,	,	PUNCT
ejpam-6557	80	15	6557	6557	NUM
ejpam-6557	80	16	4	4	NUM
ejpam-6557	80	17	of	of	ADP
ejpam-6557	80	18	8	8	NUM
ejpam-6557	80	19	theorem	theorem	NOUN
ejpam-6557	80	20	1	1	NUM
ejpam-6557	80	21	.	.	PUNCT
ejpam-6557	81	1	(	(	PUNCT
ejpam-6557	81	2	sufficient	sufficient	ADJ
ejpam-6557	81	3	conditions	condition	NOUN
ejpam-6557	81	4	for	for	ADP
ejpam-6557	81	5	(	(	PUNCT
ejpam-6557	81	6	α	α	NOUN
ejpam-6557	81	7	,	,	PUNCT
ejpam-6557	81	8	β)-differentiability	β)-differentiability	PUNCT
ejpam-6557	81	9	)	)	PUNCT
ejpam-6557	81	10	let	let	VERB
ejpam-6557	81	11	f	f	NOUN
ejpam-6557	81	12	:	:	PUNCT
ejpam-6557	81	13	g	g	PROPN
ejpam-6557	81	14	⊆	⊆	NUM
ejpam-6557	81	15	c	c	NOUN
ejpam-6557	81	16	→	→	SYM
ejpam-6557	81	17	c	c	X
ejpam-6557	81	18	be	be	AUX
ejpam-6557	81	19	a	a	DET
ejpam-6557	81	20	complex	complex	ADJ
ejpam-6557	81	21	valued	value	VERB
ejpam-6557	81	22	function	function	NOUN
ejpam-6557	81	23	defined	define	VERB
ejpam-6557	81	24	on	on	ADP
ejpam-6557	81	25	a	a	DET
ejpam-6557	81	26	region	region	NOUN
ejpam-6557	81	27	g	g	PROPN
ejpam-6557	81	28	⊆	⊆	NUM
ejpam-6557	81	29	c.	c.	NOUN
ejpam-6557	81	30	let	let	VERB
ejpam-6557	81	31	z	z	NOUN
ejpam-6557	81	32	◦	◦	VERB
ejpam-6557	81	33	=	=	SYM
ejpam-6557	81	34	(	(	PUNCT
ejpam-6557	81	35	x	x	NOUN
ejpam-6557	81	36	◦	◦	NOUN
ejpam-6557	81	37	,	,	PUNCT
ejpam-6557	81	38	y	y	NOUN
ejpam-6557	81	39	◦	◦	NOUN
ejpam-6557	81	40	)	)	PUNCT
ejpam-6557	81	41	∈	∈	NOUN
ejpam-6557	82	1	g	g	NOUN
ejpam-6557	82	2	◦	◦	NOUN
ejpam-6557	82	3	the	the	DET
ejpam-6557	82	4	interior	interior	NOUN
ejpam-6557	82	5	of	of	ADP
ejpam-6557	82	6	g.	g.	PROPN
ejpam-6557	82	7	if	if	SCONJ
ejpam-6557	82	8	i	i	PROPN
ejpam-6557	82	9	)	)	PUNCT
ejpam-6557	82	10	uαx	uαx	PROPN
ejpam-6557	82	11	,	,	PUNCT
ejpam-6557	82	12	uβy	uβy	ADJ
ejpam-6557	82	13	,	,	PUNCT
ejpam-6557	82	14	vαx	vαx	ADJ
ejpam-6557	82	15	,	,	PUNCT
ejpam-6557	82	16	and	and	CCONJ
ejpam-6557	82	17	vβy	vβy	NOUN
ejpam-6557	82	18	all	all	PRON
ejpam-6557	82	19	exist	exist	VERB
ejpam-6557	82	20	and	and	CCONJ
ejpam-6557	82	21	are	be	AUX
ejpam-6557	82	22	continuous	continuous	ADJ
ejpam-6557	82	23	at	at	ADP
ejpam-6557	82	24	z	z	PROPN
ejpam-6557	82	25	◦	◦	NOUN
ejpam-6557	82	26	,	,	PUNCT
ejpam-6557	82	27	ii	ii	NOUN
ejpam-6557	82	28	)	)	PUNCT
ejpam-6557	82	29	(	(	PUNCT
ejpam-6557	82	30	α	α	X
ejpam-6557	82	31	,	,	PUNCT
ejpam-6557	82	32	β	β	NOUN
ejpam-6557	82	33	)	)	PUNCT
ejpam-6557	82	34	-cauchy	-cauchy	PROPN
ejpam-6557	82	35	-	-	PUNCT
ejpam-6557	82	36	riemann	riemann	PROPN
ejpam-6557	82	37	equations	equation	NOUN
ejpam-6557	82	38	(	(	PUNCT
ejpam-6557	82	39	4	4	X
ejpam-6557	82	40	)	)	PUNCT
ejpam-6557	82	41	are	be	AUX
ejpam-6557	82	42	satisfied	satisfied	ADJ
ejpam-6557	82	43	at	at	ADP
ejpam-6557	82	44	z	z	NOUN
ejpam-6557	82	45	◦	◦	NOUN
ejpam-6557	82	46	,	,	PUNCT
ejpam-6557	82	47	then	then	ADV
ejpam-6557	82	48	,	,	PUNCT
ejpam-6557	82	49	f	f	PROPN
ejpam-6557	82	50	is	be	AUX
ejpam-6557	82	51	(	(	PUNCT
ejpam-6557	82	52	α	α	NOUN
ejpam-6557	82	53	,	,	PUNCT
ejpam-6557	82	54	β)-differentiable	β)-differentiable	PUNCT
ejpam-6557	82	55	at	at	ADP
ejpam-6557	82	56	z	z	NOUN
ejpam-6557	82	57	◦	◦	NOUN
ejpam-6557	82	58	.	.	PUNCT
ejpam-6557	83	1	proof	proof	NOUN
ejpam-6557	83	2	.	.	PUNCT
ejpam-6557	84	1	suppose	suppose	VERB
ejpam-6557	84	2	that	that	SCONJ
ejpam-6557	84	3	the	the	DET
ejpam-6557	84	4	two	two	NUM
ejpam-6557	84	5	conditions	condition	NOUN
ejpam-6557	84	6	of	of	ADP
ejpam-6557	84	7	the	the	DET
ejpam-6557	84	8	theorem	theorem	ADJ
ejpam-6557	84	9	1	1	NUM
ejpam-6557	84	10	hold	hold	NOUN
ejpam-6557	84	11	.	.	PUNCT
ejpam-6557	85	1	now	now	ADV
ejpam-6557	85	2	,	,	PUNCT
ejpam-6557	85	3	f(x	f(x	PROPN
ejpam-6557	85	4	◦	◦	NOUN
ejpam-6557	85	5	+	+	CCONJ
ejpam-6557	85	6	x−x	x−x	AUX
ejpam-6557	85	7	◦	◦	VERB
ejpam-6557	85	8	x1−α	x1−α	ADJ
ejpam-6557	85	9	◦	◦	NOUN
ejpam-6557	85	10	x1−α	x1−α	ADJ
ejpam-6557	85	11	◦	◦	NOUN
ejpam-6557	86	1	+	+	CCONJ
ejpam-6557	86	2	i	i	PRON
ejpam-6557	86	3	[	[	PUNCT
ejpam-6557	86	4	y	y	NOUN
ejpam-6557	86	5	◦	◦	NOUN
ejpam-6557	86	6	+	+	NUM
ejpam-6557	86	7	y−y	y−y	NOUN
ejpam-6557	86	8	◦	◦	VERB
ejpam-6557	86	9	y1−β	y1−β	PROPN
ejpam-6557	86	10	◦	◦	NOUN
ejpam-6557	86	11	y1−β	y1−β	PROPN
ejpam-6557	86	12	◦	◦	NOUN
ejpam-6557	86	13	]	]	PUNCT
ejpam-6557	86	14	)	)	PUNCT
ejpam-6557	86	15	−	−	PROPN
ejpam-6557	86	16	f(x	f(x	PROPN
ejpam-6557	86	17	◦	◦	NOUN
ejpam-6557	86	18	,	,	PUNCT
ejpam-6557	86	19	y	y	NOUN
ejpam-6557	86	20	◦	◦	NOUN
ejpam-6557	86	21	)	)	PUNCT
ejpam-6557	86	22	x−x	x−x	VERB
ejpam-6557	86	23	◦	◦	VERB
ejpam-6557	86	24	x1−α	x1−α	ADJ
ejpam-6557	86	25	◦	◦	NOUN
ejpam-6557	86	26	+	+	CCONJ
ejpam-6557	86	27	iy−y	iy−y	NOUN
ejpam-6557	86	28	◦	◦	VERB
ejpam-6557	86	29	y1−β	y1−β	PROPN
ejpam-6557	86	30	◦	◦	NOUN
ejpam-6557	86	31	=	=	PUNCT
ejpam-6557	86	32	u(x	u(x	VERB
ejpam-6557	86	33	◦	◦	NOUN
ejpam-6557	86	34	+	+	CCONJ
ejpam-6557	86	35	x−x	x−x	AUX
ejpam-6557	86	36	◦	◦	VERB
ejpam-6557	86	37	x1−α	x1−α	ADJ
ejpam-6557	86	38	◦	◦	NOUN
ejpam-6557	86	39	x1−α	x1−α	ADJ
ejpam-6557	86	40	◦	◦	NOUN
ejpam-6557	86	41	,	,	PUNCT
ejpam-6557	86	42	y	y	NOUN
ejpam-6557	86	43	◦	◦	NOUN
ejpam-6557	86	44	+	+	NUM
ejpam-6557	86	45	y−y	y−y	NOUN
ejpam-6557	86	46	◦	◦	VERB
ejpam-6557	86	47	y1−β	y1−β	PROPN
ejpam-6557	86	48	◦	◦	NOUN
ejpam-6557	86	49	y1−β	y1−β	PROPN
ejpam-6557	86	50	◦	◦	NOUN
ejpam-6557	86	51	)	)	PUNCT
ejpam-6557	86	52	−	−	PROPN
ejpam-6557	87	1	u(x	u(x	PROPN
ejpam-6557	87	2	◦	◦	NOUN
ejpam-6557	87	3	,	,	PUNCT
ejpam-6557	87	4	y	y	NOUN
ejpam-6557	87	5	◦	◦	NOUN
ejpam-6557	87	6	)	)	PUNCT
ejpam-6557	87	7	x−x	x−x	VERB
ejpam-6557	87	8	◦	◦	VERB
ejpam-6557	87	9	x1−α	x1−α	ADJ
ejpam-6557	87	10	◦	◦	NOUN
ejpam-6557	87	11	+	+	CCONJ
ejpam-6557	87	12	iy−y	iy−y	NOUN
ejpam-6557	87	13	◦	◦	VERB
ejpam-6557	87	14	y1−β	y1−β	PROPN
ejpam-6557	87	15	◦	◦	NOUN
ejpam-6557	88	1	+	+	ADV
ejpam-6557	88	2	i	i	PRON
ejpam-6557	88	3	v(x	v(x	VERB
ejpam-6557	88	4	◦	◦	NOUN
ejpam-6557	88	5	+	+	CCONJ
ejpam-6557	88	6	x−x	x−x	AUX
ejpam-6557	88	7	◦	◦	VERB
ejpam-6557	88	8	x1−α	x1−α	ADJ
ejpam-6557	88	9	◦	◦	NOUN
ejpam-6557	88	10	x1−α	x1−α	ADJ
ejpam-6557	88	11	◦	◦	NOUN
ejpam-6557	88	12	,	,	PUNCT
ejpam-6557	88	13	y	y	NOUN
ejpam-6557	88	14	◦	◦	NOUN
ejpam-6557	88	15	+	+	NUM
ejpam-6557	88	16	y−y	y−y	NOUN
ejpam-6557	88	17	◦	◦	VERB
ejpam-6557	88	18	y1−β	y1−β	PROPN
ejpam-6557	88	19	◦	◦	NOUN
ejpam-6557	88	20	y1−β	y1−β	PROPN
ejpam-6557	88	21	◦	◦	NOUN
ejpam-6557	88	22	)	)	PUNCT
ejpam-6557	88	23	−	−	ADP
ejpam-6557	89	1	v(x	v(x	PROPN
ejpam-6557	89	2	◦	◦	NOUN
ejpam-6557	89	3	,	,	PUNCT
ejpam-6557	89	4	y	y	NOUN
ejpam-6557	89	5	◦	◦	NOUN
ejpam-6557	89	6	)	)	PUNCT
ejpam-6557	89	7	x−x	x−x	VERB
ejpam-6557	89	8	◦	◦	VERB
ejpam-6557	89	9	x1−α	x1−α	ADJ
ejpam-6557	89	10	◦	◦	NOUN
ejpam-6557	89	11	+	+	CCONJ
ejpam-6557	89	12	iy−y	iy−y	NOUN
ejpam-6557	89	13	◦	◦	VERB
ejpam-6557	89	14	y1−β	y1−β	PROPN
ejpam-6557	89	15	◦	◦	NOUN
ejpam-6557	89	16	.	.	PUNCT
ejpam-6557	90	1	but	but	CCONJ
ejpam-6557	90	2	,	,	PUNCT
ejpam-6557	90	3	u(x	u(x	VERB
ejpam-6557	90	4	◦	◦	NOUN
ejpam-6557	90	5	+	+	NUM
ejpam-6557	90	6	x−	x−	PROPN
ejpam-6557	90	7	x	x	ADJ
ejpam-6557	90	8	◦	◦	NOUN
ejpam-6557	90	9	x1−α	x1−α	ADJ
ejpam-6557	90	10	◦	◦	NOUN
ejpam-6557	90	11	x1−α	x1−α	ADJ
ejpam-6557	90	12	◦	◦	NOUN
ejpam-6557	90	13	,	,	PUNCT
ejpam-6557	90	14	y	y	NOUN
ejpam-6557	90	15	◦	◦	NOUN
ejpam-6557	91	1	+	+	CCONJ
ejpam-6557	91	2	y	y	PROPN
ejpam-6557	91	3	−	−	PROPN
ejpam-6557	91	4	y	y	PROPN
ejpam-6557	91	5	◦	◦	NOUN
ejpam-6557	91	6	y1−β	y1−β	PROPN
ejpam-6557	91	7	◦	◦	NOUN
ejpam-6557	91	8	y1−β	y1−β	PROPN
ejpam-6557	91	9	◦	◦	NOUN
ejpam-6557	91	10	)	)	PUNCT
ejpam-6557	91	11	−	−	PROPN
ejpam-6557	91	12	u(x	u(x	PROPN
ejpam-6557	91	13	◦	◦	NOUN
ejpam-6557	91	14	,	,	PUNCT
ejpam-6557	91	15	y	y	NOUN
ejpam-6557	91	16	◦	◦	NOUN
ejpam-6557	91	17	)	)	PUNCT
ejpam-6557	91	18	=	=	SYM
ejpam-6557	91	19	u	u	NOUN
ejpam-6557	91	20	(	(	PUNCT
ejpam-6557	91	21	x	x	NOUN
ejpam-6557	91	22	,	,	PUNCT
ejpam-6557	91	23	y)−	y)−	PROPN
ejpam-6557	91	24	u(x	u(x	NOUN
ejpam-6557	91	25	◦	◦	NOUN
ejpam-6557	91	26	,	,	PUNCT
ejpam-6557	91	27	y	y	NOUN
ejpam-6557	91	28	◦	◦	NOUN
ejpam-6557	91	29	)	)	PUNCT
ejpam-6557	91	30	=	=	SYM
ejpam-6557	91	31	x−	x−	PROPN
ejpam-6557	91	32	x	x	PUNCT
ejpam-6557	91	33	◦	◦	NOUN
ejpam-6557	91	34	x1−α	x1−α	ADJ
ejpam-6557	91	35	◦	◦	VERB
ejpam-6557	91	36	uαx(x	uαx(x	NOUN
ejpam-6557	91	37	◦	◦	NOUN
ejpam-6557	91	38	,	,	PUNCT
ejpam-6557	91	39	y	y	NOUN
ejpam-6557	91	40	◦	◦	NOUN
ejpam-6557	91	41	)	)	PUNCT
ejpam-6557	92	1	+	+	CCONJ
ejpam-6557	92	2	y	y	PROPN
ejpam-6557	92	3	−	−	PROPN
ejpam-6557	92	4	y	y	PROPN
ejpam-6557	92	5	◦	◦	NOUN
ejpam-6557	92	6	y1−β	y1−β	PROPN
ejpam-6557	92	7	◦	◦	NOUN
ejpam-6557	92	8	uβy	uβy	ADJ
ejpam-6557	92	9	(	(	PUNCT
ejpam-6557	92	10	x	x	NOUN
ejpam-6557	92	11	◦	◦	NOUN
ejpam-6557	92	12	,	,	PUNCT
ejpam-6557	92	13	y	y	NOUN
ejpam-6557	92	14	◦	◦	NOUN
ejpam-6557	92	15	)	)	PUNCT
ejpam-6557	93	1	+	+	CCONJ
ejpam-6557	93	2	√	√	NUM
ejpam-6557	93	3	(	(	PUNCT
ejpam-6557	93	4	x−	x−	PROPN
ejpam-6557	93	5	x	x	SYM
ejpam-6557	93	6	◦	◦	NOUN
ejpam-6557	93	7	)	)	PUNCT
ejpam-6557	93	8	2	2	NUM
ejpam-6557	93	9	+	+	CCONJ
ejpam-6557	93	10	(	(	PUNCT
ejpam-6557	93	11	y	y	PROPN
ejpam-6557	93	12	−	−	PROPN
ejpam-6557	93	13	y	y	PROPN
ejpam-6557	93	14	◦	◦	NOUN
ejpam-6557	93	15	)	)	PUNCT
ejpam-6557	93	16	2δ1	2δ1	NUM
ejpam-6557	93	17	(	(	PUNCT
ejpam-6557	93	18	z	z	NOUN
ejpam-6557	93	19	)	)	PUNCT
ejpam-6557	93	20	,	,	PUNCT
ejpam-6557	93	21	and	and	CCONJ
ejpam-6557	93	22	v(x	v(x	PROPN
ejpam-6557	93	23	◦	◦	NOUN
ejpam-6557	93	24	+	+	X
ejpam-6557	93	25	x−	x−	PROPN
ejpam-6557	93	26	x	x	ADJ
ejpam-6557	93	27	◦	◦	NOUN
ejpam-6557	93	28	x1−α	x1−α	ADJ
ejpam-6557	93	29	◦	◦	NOUN
ejpam-6557	93	30	x1−α	x1−α	ADJ
ejpam-6557	93	31	◦	◦	NOUN
ejpam-6557	93	32	,	,	PUNCT
ejpam-6557	93	33	y	y	NOUN
ejpam-6557	93	34	◦	◦	NOUN
ejpam-6557	93	35	+	+	CCONJ
ejpam-6557	93	36	y	y	PROPN
ejpam-6557	93	37	−	−	PROPN
ejpam-6557	93	38	y	y	PROPN
ejpam-6557	93	39	◦	◦	NOUN
ejpam-6557	93	40	y1−β	y1−β	PROPN
ejpam-6557	93	41	◦	◦	NOUN
ejpam-6557	93	42	y1−β	y1−β	PROPN
ejpam-6557	93	43	◦	◦	NOUN
ejpam-6557	93	44	)	)	PUNCT
ejpam-6557	93	45	−	−	ADP
ejpam-6557	93	46	v(x	v(x	PROPN
ejpam-6557	93	47	◦	◦	NOUN
ejpam-6557	93	48	,	,	PUNCT
ejpam-6557	93	49	y	y	NOUN
ejpam-6557	93	50	◦	◦	NOUN
ejpam-6557	93	51	)	)	PUNCT
ejpam-6557	93	52	=	=	SYM
ejpam-6557	93	53	v	v	X
ejpam-6557	93	54	(	(	PUNCT
ejpam-6557	93	55	x	x	X
ejpam-6557	93	56	,	,	PUNCT
ejpam-6557	93	57	y)−	y)−	PROPN
ejpam-6557	93	58	v(x	v(x	PROPN
ejpam-6557	93	59	◦	◦	NOUN
ejpam-6557	93	60	,	,	PUNCT
ejpam-6557	93	61	y	y	NOUN
ejpam-6557	93	62	◦	◦	NOUN
ejpam-6557	93	63	)	)	PUNCT
ejpam-6557	93	64	=	=	SYM
ejpam-6557	93	65	x−	x−	PROPN
ejpam-6557	93	66	x	x	PUNCT
ejpam-6557	93	67	◦	◦	NOUN
ejpam-6557	93	68	x1−α	x1−α	ADJ
ejpam-6557	93	69	◦	◦	NOUN
ejpam-6557	93	70	vαx	vαx	ADJ
ejpam-6557	93	71	(	(	PUNCT
ejpam-6557	93	72	x	x	NOUN
ejpam-6557	93	73	◦	◦	NOUN
ejpam-6557	93	74	,	,	PUNCT
ejpam-6557	93	75	y	y	NOUN
ejpam-6557	93	76	◦	◦	NOUN
ejpam-6557	93	77	)	)	PUNCT
ejpam-6557	93	78	+	+	CCONJ
ejpam-6557	93	79	y	y	PROPN
ejpam-6557	93	80	−	−	PROPN
ejpam-6557	93	81	y	y	PROPN
ejpam-6557	93	82	◦	◦	NOUN
ejpam-6557	93	83	y1−β	y1−β	PROPN
ejpam-6557	93	84	◦	◦	NOUN
ejpam-6557	93	85	vβy	vβy	NOUN
ejpam-6557	93	86	(	(	PUNCT
ejpam-6557	93	87	x	x	SYM
ejpam-6557	93	88	◦	◦	NOUN
ejpam-6557	93	89	,	,	PUNCT
ejpam-6557	93	90	y	y	NOUN
ejpam-6557	93	91	◦	◦	NOUN
ejpam-6557	93	92	)	)	PUNCT
ejpam-6557	94	1	+	+	CCONJ
ejpam-6557	94	2	√	√	NUM
ejpam-6557	94	3	(	(	PUNCT
ejpam-6557	94	4	x−	x−	PROPN
ejpam-6557	94	5	x	x	SYM
ejpam-6557	94	6	◦	◦	NOUN
ejpam-6557	94	7	)	)	PUNCT
ejpam-6557	94	8	2	2	NUM
ejpam-6557	94	9	+	+	CCONJ
ejpam-6557	94	10	(	(	PUNCT
ejpam-6557	94	11	y	y	PROPN
ejpam-6557	94	12	−	−	PROPN
ejpam-6557	94	13	y	y	PROPN
ejpam-6557	94	14	◦	◦	NOUN
ejpam-6557	94	15	)	)	PUNCT
ejpam-6557	94	16	2δ2	2δ2	NUM
ejpam-6557	94	17	(	(	PUNCT
ejpam-6557	94	18	z	z	NOUN
ejpam-6557	94	19	)	)	PUNCT
ejpam-6557	94	20	,	,	PUNCT
ejpam-6557	94	21	where	where	SCONJ
ejpam-6557	94	22	z	z	NOUN
ejpam-6557	94	23	=	=	SYM
ejpam-6557	94	24	(	(	PUNCT
ejpam-6557	94	25	x	x	X
ejpam-6557	94	26	,	,	PUNCT
ejpam-6557	94	27	y	y	PROPN
ejpam-6557	94	28	)	)	PUNCT
ejpam-6557	94	29	.	.	PUNCT
ejpam-6557	95	1	by	by	ADP
ejpam-6557	95	2	the	the	DET
ejpam-6557	95	3	first	first	ADJ
ejpam-6557	95	4	condition	condition	NOUN
ejpam-6557	95	5	(	(	PUNCT
ejpam-6557	95	6	i	i	NOUN
ejpam-6557	95	7	)	)	PUNCT
ejpam-6557	95	8	of	of	ADP
ejpam-6557	95	9	theorem	theorem	NOUN
ejpam-6557	95	10	1	1	NUM
ejpam-6557	95	11	,	,	PUNCT
ejpam-6557	95	12	that	that	PRON
ejpam-6557	95	13	is	be	AUX
ejpam-6557	95	14	uαx	uαx	ADJ
ejpam-6557	95	15	,	,	PUNCT
ejpam-6557	95	16	u	u	NOUN
ejpam-6557	95	17	β	β	X
ejpam-6557	95	18	y	y	PROPN
ejpam-6557	95	19	,	,	PUNCT
ejpam-6557	95	20	vαx	vαx	ADJ
ejpam-6557	95	21	,	,	PUNCT
ejpam-6557	95	22	and	and	CCONJ
ejpam-6557	95	23	vβy	vβy	NOUN
ejpam-6557	95	24	exist	exist	VERB
ejpam-6557	95	25	and	and	CCONJ
ejpam-6557	95	26	continuous	continuous	ADJ
ejpam-6557	95	27	at	at	ADP
ejpam-6557	95	28	z	z	NOUN
ejpam-6557	95	29	◦	◦	NOUN
ejpam-6557	95	30	,	,	PUNCT
ejpam-6557	95	31	we	we	PRON
ejpam-6557	95	32	deduce	deduce	VERB
ejpam-6557	95	33	that	that	SCONJ
ejpam-6557	95	34	both	both	DET
ejpam-6557	95	35	lim	lim	PROPN
ejpam-6557	95	36	z→z	z→z	NUM
ejpam-6557	95	37	◦	◦	NOUN
ejpam-6557	95	38	δ1	δ1	NOUN
ejpam-6557	95	39	(	(	PUNCT
ejpam-6557	95	40	z	z	NOUN
ejpam-6557	95	41	)	)	PUNCT
ejpam-6557	95	42	=	=	SYM
ejpam-6557	96	1	lim	lim	PROPN
ejpam-6557	96	2	z→z	z→z	NUM
ejpam-6557	96	3	◦	◦	NOUN
ejpam-6557	96	4	δ2	δ2	VERB
ejpam-6557	96	5	(	(	PUNCT
ejpam-6557	96	6	z	z	NOUN
ejpam-6557	96	7	)	)	PUNCT
ejpam-6557	96	8	=	=	SYM
ejpam-6557	96	9	0	0	X
ejpam-6557	96	10	.	.	PUNCT
ejpam-6557	97	1	in	in	ADP
ejpam-6557	97	2	other	other	ADJ
ejpam-6557	97	3	words	word	NOUN
ejpam-6557	97	4	,	,	PUNCT
ejpam-6557	97	5	as	as	SCONJ
ejpam-6557	97	6	(	(	PUNCT
ejpam-6557	97	7	x−x	x−x	PRON
ejpam-6557	97	8	◦	◦	VERB
ejpam-6557	97	9	x1−α	x1−α	ADJ
ejpam-6557	97	10	◦	◦	NOUN
ejpam-6557	97	11	+	+	CCONJ
ejpam-6557	97	12	iy−y	iy−y	NOUN
ejpam-6557	97	13	◦	◦	VERB
ejpam-6557	97	14	y1−β	y1−β	PROPN
ejpam-6557	97	15	◦	◦	NOUN
ejpam-6557	97	16	)	)	PUNCT
ejpam-6557	98	1	→	→	SYM
ejpam-6557	98	2	0	0	NUM
ejpam-6557	98	3	,	,	PUNCT
ejpam-6557	98	4	we	we	PRON
ejpam-6557	98	5	have	have	VERB
ejpam-6557	98	6	δ1	δ1	NOUN
ejpam-6557	98	7	(	(	PUNCT
ejpam-6557	98	8	z	z	NOUN
ejpam-6557	98	9	)	)	PUNCT
ejpam-6557	98	10	→	→	SYM
ejpam-6557	98	11	0	0	NUM
ejpam-6557	98	12	and	and	CCONJ
ejpam-6557	98	13	δ2	δ2	VERB
ejpam-6557	98	14	(	(	PUNCT
ejpam-6557	98	15	z	z	NOUN
ejpam-6557	98	16	)	)	PUNCT
ejpam-6557	98	17	→	→	SYM
ejpam-6557	98	18	0	0	X
ejpam-6557	98	19	.	.	PUNCT
ejpam-6557	99	1	moreover	moreover	ADV
ejpam-6557	99	2	,	,	PUNCT
ejpam-6557	99	3	by	by	ADP
ejpam-6557	99	4	the	the	DET
ejpam-6557	99	5	second	second	ADJ
ejpam-6557	99	6	condition	condition	NOUN
ejpam-6557	99	7	(	(	PUNCT
ejpam-6557	99	8	ii	ii	NOUN
ejpam-6557	99	9	)	)	PUNCT
ejpam-6557	99	10	of	of	ADP
ejpam-6557	99	11	theorem	theorem	NOUN
ejpam-6557	99	12	1	1	NUM
ejpam-6557	99	13	,	,	PUNCT
ejpam-6557	99	14	we	we	PRON
ejpam-6557	99	15	have	have	VERB
ejpam-6557	99	16	u(x	u(x	NOUN
ejpam-6557	99	17	◦	◦	NOUN
ejpam-6557	99	18	+	+	NUM
ejpam-6557	99	19	x−	x−	PROPN
ejpam-6557	99	20	x	x	ADJ
ejpam-6557	99	21	◦	◦	NOUN
ejpam-6557	99	22	x1−α	x1−α	ADJ
ejpam-6557	99	23	◦	◦	NOUN
ejpam-6557	99	24	x1−α	x1−α	ADJ
ejpam-6557	99	25	◦	◦	NOUN
ejpam-6557	99	26	,	,	PUNCT
ejpam-6557	99	27	y	y	NOUN
ejpam-6557	99	28	◦	◦	NOUN
ejpam-6557	100	1	+	+	CCONJ
ejpam-6557	100	2	y	y	PROPN
ejpam-6557	100	3	−	−	PROPN
ejpam-6557	100	4	y	y	PROPN
ejpam-6557	100	5	◦	◦	NOUN
ejpam-6557	100	6	y1−β	y1−β	PROPN
ejpam-6557	100	7	◦	◦	NOUN
ejpam-6557	100	8	y1−β	y1−β	PROPN
ejpam-6557	100	9	◦	◦	NOUN
ejpam-6557	100	10	)	)	PUNCT
ejpam-6557	100	11	−	−	PROPN
ejpam-6557	100	12	u(x	u(x	PROPN
ejpam-6557	100	13	◦	◦	NOUN
ejpam-6557	100	14	,	,	PUNCT
ejpam-6557	100	15	y	y	NOUN
ejpam-6557	100	16	◦	◦	NOUN
ejpam-6557	100	17	)	)	PUNCT
ejpam-6557	100	18	w.	w.	PROPN
ejpam-6557	100	19	g.	g.	PROPN
ejpam-6557	100	20	alshanti	alshanti	PROPN
ejpam-6557	100	21	,	,	PUNCT
ejpam-6557	101	1	m.	m.	NOUN
ejpam-6557	101	2	a.	a.	PROPN
ejpam-6557	101	3	hammad	hammad	PROPN
ejpam-6557	101	4	,	,	PUNCT
ejpam-6557	101	5	r.	r.	PROPN
ejpam-6557	101	6	khalil	khalil	PROPN
ejpam-6557	101	7	/	/	SYM
ejpam-6557	101	8	eur	eur	PROPN
ejpam-6557	101	9	.	.	PUNCT
ejpam-6557	102	1	j.	j.	PROPN
ejpam-6557	102	2	pure	pure	PROPN
ejpam-6557	102	3	appl	appl	PROPN
ejpam-6557	102	4	.	.	PROPN
ejpam-6557	102	5	math	math	PROPN
ejpam-6557	102	6	,	,	PUNCT
ejpam-6557	102	7	18	18	NUM
ejpam-6557	102	8	(	(	PUNCT
ejpam-6557	102	9	4	4	NUM
ejpam-6557	102	10	)	)	PUNCT
ejpam-6557	102	11	(	(	PUNCT
ejpam-6557	102	12	2025	2025	NUM
ejpam-6557	102	13	)	)	PUNCT
ejpam-6557	102	14	,	,	PUNCT
ejpam-6557	102	15	6557	6557	NUM
ejpam-6557	102	16	5	5	NUM
ejpam-6557	102	17	of	of	ADP
ejpam-6557	102	18	8	8	NUM
ejpam-6557	103	1	+	+	NOUN
ejpam-6557	103	2	i	i	PRON
ejpam-6557	103	3	[	[	PUNCT
ejpam-6557	103	4	v(x	v(x	VERB
ejpam-6557	103	5	◦	◦	NOUN
ejpam-6557	103	6	+	+	X
ejpam-6557	103	7	x−	x−	PROPN
ejpam-6557	103	8	x	x	ADJ
ejpam-6557	103	9	◦	◦	NOUN
ejpam-6557	103	10	x1−α	x1−α	ADJ
ejpam-6557	103	11	◦	◦	NOUN
ejpam-6557	103	12	x1−α	x1−α	ADJ
ejpam-6557	103	13	◦	◦	NOUN
ejpam-6557	103	14	,	,	PUNCT
ejpam-6557	103	15	y	y	NOUN
ejpam-6557	103	16	◦	◦	NOUN
ejpam-6557	104	1	+	+	CCONJ
ejpam-6557	104	2	y	y	PROPN
ejpam-6557	104	3	−	−	PROPN
ejpam-6557	104	4	y	y	PROPN
ejpam-6557	104	5	◦	◦	NOUN
ejpam-6557	104	6	y1−β	y1−β	PROPN
ejpam-6557	104	7	◦	◦	NOUN
ejpam-6557	104	8	y1−β	y1−β	PROPN
ejpam-6557	104	9	◦	◦	NOUN
ejpam-6557	104	10	)	)	PUNCT
ejpam-6557	104	11	−	−	ADP
ejpam-6557	105	1	v(x	v(x	PROPN
ejpam-6557	105	2	◦	◦	NOUN
ejpam-6557	105	3	,	,	PUNCT
ejpam-6557	105	4	y	y	NOUN
ejpam-6557	105	5	◦	◦	NOUN
ejpam-6557	105	6	)	)	PUNCT
ejpam-6557	105	7	]	]	PUNCT
ejpam-6557	106	1	=	=	PUNCT
ejpam-6557	106	2	x−	x−	PROPN
ejpam-6557	106	3	x	x	PUNCT
ejpam-6557	106	4	◦	◦	NOUN
ejpam-6557	106	5	x1−α	x1−α	ADJ
ejpam-6557	106	6	◦	◦	NOUN
ejpam-6557	106	7	(	(	PUNCT
ejpam-6557	106	8	uαx(x	uαx(x	NOUN
ejpam-6557	106	9	◦	◦	NOUN
ejpam-6557	106	10	,	,	PUNCT
ejpam-6557	106	11	y	y	NOUN
ejpam-6557	106	12	◦	◦	NOUN
ejpam-6557	106	13	)	)	PUNCT
ejpam-6557	106	14	+	+	CCONJ
ejpam-6557	106	15	ivαx	ivαx	ADJ
ejpam-6557	106	16	(	(	PUNCT
ejpam-6557	106	17	x	x	NOUN
ejpam-6557	106	18	◦	◦	NOUN
ejpam-6557	106	19	,	,	PUNCT
ejpam-6557	106	20	y	y	NOUN
ejpam-6557	106	21	◦	◦	NOUN
ejpam-6557	106	22	)	)	PUNCT
ejpam-6557	106	23	)	)	PUNCT
ejpam-6557	107	1	+	+	CCONJ
ejpam-6557	108	1	y	y	PROPN
ejpam-6557	108	2	−	−	PROPN
ejpam-6557	108	3	y	y	PROPN
ejpam-6557	108	4	◦	◦	NOUN
ejpam-6557	108	5	y1−β	y1−β	PROPN
ejpam-6557	108	6	◦	◦	NOUN
ejpam-6557	108	7	(	(	PUNCT
ejpam-6557	108	8	uβy	uβy	PROPN
ejpam-6557	108	9	(	(	PUNCT
ejpam-6557	108	10	x	x	NOUN
ejpam-6557	108	11	◦	◦	NOUN
ejpam-6557	108	12	,	,	PUNCT
ejpam-6557	108	13	y	y	NOUN
ejpam-6557	108	14	◦	◦	NOUN
ejpam-6557	108	15	)	)	PUNCT
ejpam-6557	109	1	+	+	CCONJ
ejpam-6557	109	2	ivβy	ivβy	ADJ
ejpam-6557	109	3	(	(	PUNCT
ejpam-6557	109	4	x	x	NOUN
ejpam-6557	109	5	◦	◦	NOUN
ejpam-6557	109	6	,	,	PUNCT
ejpam-6557	109	7	y	y	NOUN
ejpam-6557	109	8	◦	◦	NOUN
ejpam-6557	109	9	)	)	PUNCT
ejpam-6557	109	10	)	)	PUNCT
ejpam-6557	110	1	+	+	CCONJ
ejpam-6557	110	2	√	√	NUM
ejpam-6557	110	3	(	(	PUNCT
ejpam-6557	110	4	x−	x−	PROPN
ejpam-6557	110	5	x	x	SYM
ejpam-6557	110	6	◦	◦	NOUN
ejpam-6557	110	7	)	)	PUNCT
ejpam-6557	110	8	2	2	NUM
ejpam-6557	110	9	+	+	CCONJ
ejpam-6557	110	10	(	(	PUNCT
ejpam-6557	110	11	y	y	PROPN
ejpam-6557	110	12	−	−	PROPN
ejpam-6557	110	13	y	y	PROPN
ejpam-6557	110	14	◦	◦	NOUN
ejpam-6557	110	15	)	)	PUNCT
ejpam-6557	110	16	2	2	NUM
ejpam-6557	110	17	(	(	PUNCT
ejpam-6557	110	18	δ1	δ1	NOUN
ejpam-6557	110	19	(	(	PUNCT
ejpam-6557	110	20	z	z	NOUN
ejpam-6557	110	21	)	)	PUNCT
ejpam-6557	110	22	+	+	CCONJ
ejpam-6557	111	1	iδ2	iδ2	ADJ
ejpam-6557	111	2	(	(	PUNCT
ejpam-6557	111	3	z	z	NOUN
ejpam-6557	111	4	)	)	PUNCT
ejpam-6557	111	5	)	)	PUNCT
ejpam-6557	112	1	=	=	PUNCT
ejpam-6557	112	2	x−	x−	PROPN
ejpam-6557	112	3	x	x	PUNCT
ejpam-6557	112	4	◦	◦	NOUN
ejpam-6557	112	5	x1−α	x1−α	ADJ
ejpam-6557	112	6	◦	◦	NOUN
ejpam-6557	112	7	(	(	PUNCT
ejpam-6557	112	8	uαx(x	uαx(x	NOUN
ejpam-6557	112	9	◦	◦	NOUN
ejpam-6557	112	10	,	,	PUNCT
ejpam-6557	112	11	y	y	NOUN
ejpam-6557	112	12	◦	◦	NOUN
ejpam-6557	112	13	)	)	PUNCT
ejpam-6557	113	1	+	+	CCONJ
ejpam-6557	113	2	ivαx	ivαx	ADJ
ejpam-6557	113	3	(	(	PUNCT
ejpam-6557	113	4	x	x	NOUN
ejpam-6557	113	5	◦	◦	NOUN
ejpam-6557	113	6	,	,	PUNCT
ejpam-6557	113	7	y	y	NOUN
ejpam-6557	113	8	◦	◦	NOUN
ejpam-6557	113	9	)	)	PUNCT
ejpam-6557	113	10	)	)	PUNCT
ejpam-6557	114	1	+	+	CCONJ
ejpam-6557	114	2	y	y	PROPN
ejpam-6557	114	3	−	−	PROPN
ejpam-6557	114	4	y	y	PROPN
ejpam-6557	114	5	◦	◦	NOUN
ejpam-6557	114	6	y1−β	y1−β	PROPN
ejpam-6557	114	7	◦	◦	NOUN
ejpam-6557	114	8	(	(	PUNCT
ejpam-6557	114	9	−vαx	−vαx	NOUN
ejpam-6557	114	10	(	(	PUNCT
ejpam-6557	114	11	x	x	NOUN
ejpam-6557	114	12	◦	◦	NOUN
ejpam-6557	114	13	,	,	PUNCT
ejpam-6557	114	14	y	y	NOUN
ejpam-6557	114	15	◦	◦	NOUN
ejpam-6557	114	16	)	)	PUNCT
ejpam-6557	114	17	+	+	CCONJ
ejpam-6557	115	1	iuαx(x	iuαx(x	PROPN
ejpam-6557	115	2	◦	◦	NOUN
ejpam-6557	115	3	,	,	PUNCT
ejpam-6557	115	4	y	y	NOUN
ejpam-6557	115	5	◦	◦	NOUN
ejpam-6557	115	6	)	)	PUNCT
ejpam-6557	115	7	)	)	PUNCT
ejpam-6557	116	1	+	+	CCONJ
ejpam-6557	116	2	√	√	NUM
ejpam-6557	116	3	(	(	PUNCT
ejpam-6557	116	4	x−	x−	PROPN
ejpam-6557	116	5	x	x	SYM
ejpam-6557	116	6	◦	◦	NOUN
ejpam-6557	116	7	)	)	PUNCT
ejpam-6557	116	8	2	2	NUM
ejpam-6557	116	9	+	+	CCONJ
ejpam-6557	116	10	(	(	PUNCT
ejpam-6557	116	11	y	y	PROPN
ejpam-6557	116	12	−	−	PROPN
ejpam-6557	116	13	y	y	PROPN
ejpam-6557	116	14	◦	◦	NOUN
ejpam-6557	116	15	)	)	PUNCT
ejpam-6557	116	16	2	2	NUM
ejpam-6557	116	17	(	(	PUNCT
ejpam-6557	116	18	δ1	δ1	NOUN
ejpam-6557	116	19	(	(	PUNCT
ejpam-6557	116	20	z	z	NOUN
ejpam-6557	116	21	)	)	PUNCT
ejpam-6557	116	22	+	+	CCONJ
ejpam-6557	117	1	iδ2	iδ2	ADJ
ejpam-6557	117	2	(	(	PUNCT
ejpam-6557	117	3	z	z	NOUN
ejpam-6557	117	4	)	)	PUNCT
ejpam-6557	117	5	)	)	PUNCT
ejpam-6557	118	1	=	=	PRON
ejpam-6557	118	2	(	(	PUNCT
ejpam-6557	118	3	x−	x−	PROPN
ejpam-6557	118	4	x	x	X
ejpam-6557	118	5	◦	◦	NOUN
ejpam-6557	118	6	x1−α	x1−α	ADJ
ejpam-6557	119	1	◦	◦	NOUN
ejpam-6557	120	1	+	+	CCONJ
ejpam-6557	120	2	i	i	VERB
ejpam-6557	120	3	y	y	NOUN
ejpam-6557	120	4	−	−	PROPN
ejpam-6557	120	5	y	y	PROPN
ejpam-6557	120	6	◦	◦	NOUN
ejpam-6557	120	7	y1−β	y1−β	PROPN
ejpam-6557	120	8	◦	◦	NOUN
ejpam-6557	120	9	)	)	PUNCT
ejpam-6557	120	10	(	(	PUNCT
ejpam-6557	120	11	uαx(x	uαx(x	NOUN
ejpam-6557	120	12	◦	◦	NOUN
ejpam-6557	120	13	,	,	PUNCT
ejpam-6557	120	14	y	y	NOUN
ejpam-6557	120	15	◦	◦	NOUN
ejpam-6557	120	16	)	)	PUNCT
ejpam-6557	120	17	+	+	CCONJ
ejpam-6557	120	18	ivαx	ivαx	ADJ
ejpam-6557	120	19	(	(	PUNCT
ejpam-6557	120	20	x	x	NOUN
ejpam-6557	120	21	◦	◦	NOUN
ejpam-6557	120	22	,	,	PUNCT
ejpam-6557	120	23	y	y	NOUN
ejpam-6557	120	24	◦	◦	NOUN
ejpam-6557	120	25	)	)	PUNCT
ejpam-6557	120	26	)	)	PUNCT
ejpam-6557	121	1	+	+	CCONJ
ejpam-6557	121	2	√	√	NUM
ejpam-6557	121	3	(	(	PUNCT
ejpam-6557	121	4	x−	x−	PROPN
ejpam-6557	121	5	x	x	SYM
ejpam-6557	121	6	◦	◦	NOUN
ejpam-6557	121	7	)	)	PUNCT
ejpam-6557	121	8	2	2	NUM
ejpam-6557	121	9	+	+	CCONJ
ejpam-6557	121	10	(	(	PUNCT
ejpam-6557	121	11	y	y	PROPN
ejpam-6557	121	12	−	−	PROPN
ejpam-6557	121	13	y	y	PROPN
ejpam-6557	121	14	◦	◦	NOUN
ejpam-6557	121	15	)	)	PUNCT
ejpam-6557	121	16	2	2	NUM
ejpam-6557	121	17	(	(	PUNCT
ejpam-6557	121	18	δ1	δ1	NOUN
ejpam-6557	121	19	(	(	PUNCT
ejpam-6557	121	20	z	z	NOUN
ejpam-6557	121	21	)	)	PUNCT
ejpam-6557	121	22	+	+	CCONJ
ejpam-6557	122	1	iδ2	iδ2	ADJ
ejpam-6557	122	2	(	(	PUNCT
ejpam-6557	122	3	z	z	NOUN
ejpam-6557	122	4	)	)	PUNCT
ejpam-6557	122	5	)	)	PUNCT
ejpam-6557	122	6	.	.	PUNCT
ejpam-6557	123	1	therefore	therefore	ADV
ejpam-6557	123	2	,	,	PUNCT
ejpam-6557	123	3	uαx(x	uαx(x	PROPN
ejpam-6557	123	4	◦	◦	NOUN
ejpam-6557	123	5	,	,	PUNCT
ejpam-6557	123	6	y	y	NOUN
ejpam-6557	123	7	◦	◦	NOUN
ejpam-6557	123	8	)	)	PUNCT
ejpam-6557	124	1	+	+	CCONJ
ejpam-6557	124	2	ivαx	ivαx	ADJ
ejpam-6557	124	3	(	(	PUNCT
ejpam-6557	124	4	x	x	NOUN
ejpam-6557	124	5	◦	◦	NOUN
ejpam-6557	124	6	,	,	PUNCT
ejpam-6557	124	7	y	y	NOUN
ejpam-6557	124	8	◦	◦	NOUN
ejpam-6557	124	9	)	)	PUNCT
ejpam-6557	124	10	+	+	CCONJ
ejpam-6557	124	11	√	√	NUM
ejpam-6557	124	12	(	(	PUNCT
ejpam-6557	124	13	x−	x−	PROPN
ejpam-6557	124	14	x	x	SYM
ejpam-6557	124	15	◦	◦	NOUN
ejpam-6557	124	16	)	)	PUNCT
ejpam-6557	124	17	2	2	NUM
ejpam-6557	125	1	+	+	CCONJ
ejpam-6557	125	2	(	(	PUNCT
ejpam-6557	125	3	y	y	PROPN
ejpam-6557	125	4	−	−	PROPN
ejpam-6557	125	5	y	y	PROPN
ejpam-6557	125	6	◦	◦	NOUN
ejpam-6557	125	7	)	)	PUNCT
ejpam-6557	125	8	2	2	NUM
ejpam-6557	125	9	(	(	PUNCT
ejpam-6557	125	10	δ1	δ1	NOUN
ejpam-6557	125	11	(	(	PUNCT
ejpam-6557	125	12	z	z	NOUN
ejpam-6557	125	13	)	)	PUNCT
ejpam-6557	125	14	+	+	CCONJ
ejpam-6557	125	15	iδ2	iδ2	ADJ
ejpam-6557	125	16	(	(	PUNCT
ejpam-6557	125	17	z	z	NOUN
ejpam-6557	125	18	)	)	PUNCT
ejpam-6557	125	19	)	)	PUNCT
ejpam-6557	126	1	x−x	x−x	AUX
ejpam-6557	126	2	◦	◦	VERB
ejpam-6557	126	3	x1−α	x1−α	ADJ
ejpam-6557	126	4	◦	◦	NOUN
ejpam-6557	126	5	+	+	CCONJ
ejpam-6557	126	6	iy−y	iy−y	NOUN
ejpam-6557	126	7	◦	◦	VERB
ejpam-6557	126	8	y1−β	y1−β	PROPN
ejpam-6557	126	9	◦	◦	NOUN
ejpam-6557	126	10	.	.	PUNCT
ejpam-6557	127	1	which	which	PRON
ejpam-6557	127	2	,	,	PUNCT
ejpam-6557	127	3	as	as	SCONJ
ejpam-6557	127	4	(	(	PUNCT
ejpam-6557	127	5	x−x	x−x	VERB
ejpam-6557	127	6	◦	◦	VERB
ejpam-6557	127	7	x1−α	x1−α	ADJ
ejpam-6557	127	8	◦	◦	NOUN
ejpam-6557	127	9	+	+	CCONJ
ejpam-6557	127	10	iy−y	iy−y	NOUN
ejpam-6557	127	11	◦	◦	VERB
ejpam-6557	127	12	y1−β	y1−β	PROPN
ejpam-6557	127	13	◦	◦	NOUN
ejpam-6557	127	14	)	)	PUNCT
ejpam-6557	127	15	→	→	SYM
ejpam-6557	127	16	0	0	NUM
ejpam-6557	127	17	,	,	PUNCT
ejpam-6557	127	18	implies	imply	VERB
ejpam-6557	127	19	d(α	d(α	NOUN
ejpam-6557	127	20	,	,	PUNCT
ejpam-6557	127	21	β)f	β)f	PUNCT
ejpam-6557	127	22	(	(	PUNCT
ejpam-6557	127	23	z	z	NOUN
ejpam-6557	127	24	◦	◦	NOUN
ejpam-6557	127	25	)	)	PUNCT
ejpam-6557	127	26	=	=	SYM
ejpam-6557	128	1	uαx(x	uαx(x	NOUN
ejpam-6557	128	2	◦	◦	NOUN
ejpam-6557	128	3	,	,	PUNCT
ejpam-6557	128	4	y	y	NOUN
ejpam-6557	128	5	◦	◦	NOUN
ejpam-6557	128	6	)	)	PUNCT
ejpam-6557	129	1	+	+	CCONJ
ejpam-6557	129	2	ivαx	ivαx	ADJ
ejpam-6557	129	3	(	(	PUNCT
ejpam-6557	129	4	x	x	NOUN
ejpam-6557	129	5	◦	◦	NOUN
ejpam-6557	129	6	,	,	PUNCT
ejpam-6557	129	7	y	y	NOUN
ejpam-6557	129	8	◦	◦	NOUN
ejpam-6557	129	9	)	)	PUNCT
ejpam-6557	129	10	=	=	VERB
ejpam-6557	129	11	vβy	vβy	NOUN
ejpam-6557	129	12	(	(	PUNCT
ejpam-6557	129	13	x	x	NOUN
ejpam-6557	129	14	◦	◦	NOUN
ejpam-6557	129	15	,	,	PUNCT
ejpam-6557	129	16	y	y	PROPN
ejpam-6557	129	17	◦	◦	NOUN
ejpam-6557	129	18	)−	)−	PUNCT
ejpam-6557	129	19	iuβy	iuβy	ADJ
ejpam-6557	129	20	(	(	PUNCT
ejpam-6557	129	21	x	x	SYM
ejpam-6557	129	22	◦	◦	NOUN
ejpam-6557	129	23	,	,	PUNCT
ejpam-6557	129	24	y	y	NOUN
ejpam-6557	129	25	◦	◦	NOUN
ejpam-6557	129	26	)	)	PUNCT
ejpam-6557	129	27	.	.	PUNCT
ejpam-6557	130	1	example	example	NOUN
ejpam-6557	131	1	3	3	X
ejpam-6557	131	2	.	.	PUNCT
ejpam-6557	131	3	let	let	VERB
ejpam-6557	131	4	f	f	PROPN
ejpam-6557	131	5	(	(	PUNCT
ejpam-6557	131	6	z	z	NOUN
ejpam-6557	131	7	)	)	PUNCT
ejpam-6557	131	8	=	=	SYM
ejpam-6557	131	9	f(x	f(x	PROPN
ejpam-6557	131	10	,	,	PUNCT
ejpam-6557	131	11	y	y	NOUN
ejpam-6557	131	12	)	)	PUNCT
ejpam-6557	131	13	=	=	X
ejpam-6557	131	14	β2x2α−α2y2β	β2x2α−α2y2β	NOUN
ejpam-6557	131	15	α2β2	α2β2	X
ejpam-6557	131	16	,	,	PUNCT
ejpam-6557	131	17	where	where	SCONJ
ejpam-6557	131	18	|α|	|α|	PROPN
ejpam-6557	131	19	,	,	PUNCT
ejpam-6557	131	20	|β|	|β|	X
ejpam-6557	131	21	<	<	X
ejpam-6557	131	22	1	1	X
ejpam-6557	131	23	.	.	PUNCT
ejpam-6557	132	1	then	then	ADV
ejpam-6557	132	2	(	(	PUNCT
ejpam-6557	132	3	α	α	X
ejpam-6557	132	4	,	,	PUNCT
ejpam-6557	132	5	β)-cauchyriemann	β)-cauchyriemann	PUNCT
ejpam-6557	132	6	equations	equation	NOUN
ejpam-6557	132	7	are	be	AUX
ejpam-6557	132	8	not	not	PART
ejpam-6557	132	9	satisfied	satisfied	ADJ
ejpam-6557	132	10	for	for	ADP
ejpam-6557	132	11	all	all	DET
ejpam-6557	132	12	z	z	NOUN
ejpam-6557	132	13	=	=	PUNCT
ejpam-6557	133	1	x	x	PUNCT
ejpam-6557	134	1	+	+	NUM
ejpam-6557	134	2	iy	iy	PRON
ejpam-6557	134	3	such	such	ADJ
ejpam-6557	134	4	that	that	SCONJ
ejpam-6557	134	5	x	x	NOUN
ejpam-6557	134	6	,	,	PUNCT
ejpam-6557	134	7	y	y	PROPN
ejpam-6557	134	8	̸=	̸=	PROPN
ejpam-6557	134	9	0	0	PUNCT
ejpam-6557	134	10	since	since	SCONJ
ejpam-6557	134	11	uαx	uαx	NOUN
ejpam-6557	134	12	=	=	SYM
ejpam-6557	134	13	2	2	NUM
ejpam-6557	134	14	αx	αx	NOUN
ejpam-6557	134	15	α	α	NOUN
ejpam-6557	134	16	̸=	̸=	PROPN
ejpam-6557	134	17	0	0	NUM
ejpam-6557	134	18	=	=	PUNCT
ejpam-6557	134	19	vβy	vβy	NOUN
ejpam-6557	134	20	,	,	PUNCT
ejpam-6557	134	21	and	and	CCONJ
ejpam-6557	134	22	uβy	uβy	VERB
ejpam-6557	134	23	=	=	SYM
ejpam-6557	134	24	−	−	PROPN
ejpam-6557	134	25	2	2	NUM
ejpam-6557	134	26	β	β	X
ejpam-6557	134	27	y	y	PROPN
ejpam-6557	134	28	β	β	X
ejpam-6557	134	29	̸=	̸=	PROPN
ejpam-6557	134	30	0	0	NUM
ejpam-6557	134	31	=	=	SYM
ejpam-6557	134	32	−vαx	−vαx	NOUN
ejpam-6557	134	33	.	.	PUNCT
ejpam-6557	135	1	consequently	consequently	ADV
ejpam-6557	135	2	,	,	PUNCT
ejpam-6557	135	3	by	by	ADP
ejpam-6557	135	4	theorem	theorem	NOUN
ejpam-6557	135	5	1	1	NUM
ejpam-6557	135	6	,	,	PUNCT
ejpam-6557	135	7	f	f	PROPN
ejpam-6557	135	8	is	be	AUX
ejpam-6557	135	9	not	not	PART
ejpam-6557	135	10	(	(	PUNCT
ejpam-6557	135	11	α	α	NOUN
ejpam-6557	135	12	,	,	PUNCT
ejpam-6557	135	13	β)-differentiable	β)-differentiable	PUNCT
ejpam-6557	135	14	at	at	ADP
ejpam-6557	135	15	any	any	DET
ejpam-6557	135	16	non	non	ADJ
ejpam-6557	135	17	-	-	ADJ
ejpam-6557	135	18	zero	zero	NUM
ejpam-6557	135	19	point	point	NOUN
ejpam-6557	135	20	.	.	PUNCT
ejpam-6557	136	1	however	however	ADV
ejpam-6557	136	2	,	,	PUNCT
ejpam-6557	136	3	at	at	ADP
ejpam-6557	136	4	z	z	NOUN
ejpam-6557	136	5	=	=	SYM
ejpam-6557	136	6	0	0	NUM
ejpam-6557	137	1	the	the	DET
ejpam-6557	137	2	two	two	NUM
ejpam-6557	137	3	conditions	condition	NOUN
ejpam-6557	137	4	of	of	ADP
ejpam-6557	137	5	theorem	theorem	NOUN
ejpam-6557	137	6	1	1	NUM
ejpam-6557	137	7	are	be	AUX
ejpam-6557	137	8	satisfied	satisfied	ADJ
ejpam-6557	137	9	.	.	PUNCT
ejpam-6557	138	1	so	so	ADV
ejpam-6557	138	2	,	,	PUNCT
ejpam-6557	138	3	f	f	PROPN
ejpam-6557	138	4	(	(	PUNCT
ejpam-6557	138	5	α	α	NOUN
ejpam-6557	138	6	,	,	PUNCT
ejpam-6557	138	7	β)(0	β)(0	NUM
ejpam-6557	138	8	)	)	PUNCT
ejpam-6557	138	9	exists	exist	VERB
ejpam-6557	138	10	such	such	ADJ
ejpam-6557	138	11	that	that	SCONJ
ejpam-6557	138	12	f	f	PROPN
ejpam-6557	138	13	(	(	PUNCT
ejpam-6557	138	14	α	α	NOUN
ejpam-6557	138	15	,	,	PUNCT
ejpam-6557	138	16	β)(0	β)(0	NUM
ejpam-6557	138	17	)	)	PUNCT
ejpam-6557	138	18	=	=	PUNCT
ejpam-6557	138	19	uαx(0	uαx(0	ADJ
ejpam-6557	138	20	,	,	PUNCT
ejpam-6557	138	21	0	0	NUM
ejpam-6557	138	22	)	)	PUNCT
ejpam-6557	139	1	+	+	CCONJ
ejpam-6557	139	2	ivαx	ivαx	ADJ
ejpam-6557	139	3	(	(	PUNCT
ejpam-6557	139	4	0	0	NUM
ejpam-6557	139	5	,	,	PUNCT
ejpam-6557	139	6	0	0	NUM
ejpam-6557	139	7	)	)	PUNCT
ejpam-6557	139	8	=	=	VERB
ejpam-6557	139	9	vβy	vβy	NOUN
ejpam-6557	139	10	(	(	PUNCT
ejpam-6557	139	11	0	0	NUM
ejpam-6557	139	12	,	,	PUNCT
ejpam-6557	139	13	0)−	0)−	NOUN
ejpam-6557	139	14	iuβy	iuβy	X
ejpam-6557	139	15	(	(	PUNCT
ejpam-6557	139	16	0	0	NUM
ejpam-6557	139	17	,	,	PUNCT
ejpam-6557	139	18	0	0	NUM
ejpam-6557	139	19	)	)	PUNCT
ejpam-6557	139	20	=	=	SYM
ejpam-6557	140	1	0	0	X
ejpam-6557	140	2	.	.	PUNCT
ejpam-6557	140	3	definition	definition	NOUN
ejpam-6557	140	4	2	2	NUM
ejpam-6557	140	5	.	.	PUNCT
ejpam-6557	141	1	a	a	DET
ejpam-6557	141	2	function	function	NOUN
ejpam-6557	141	3	f	f	NOUN
ejpam-6557	141	4	:	:	PUNCT
ejpam-6557	141	5	g	g	PROPN
ejpam-6557	141	6	⊆	⊆	NUM
ejpam-6557	141	7	e	e	X
ejpam-6557	141	8	→	→	SYM
ejpam-6557	141	9	c	c	PROPN
ejpam-6557	141	10	is	be	AUX
ejpam-6557	141	11	said	say	VERB
ejpam-6557	141	12	to	to	PART
ejpam-6557	141	13	be	be	AUX
ejpam-6557	141	14	(	(	PUNCT
ejpam-6557	141	15	α	α	NOUN
ejpam-6557	141	16	,	,	PUNCT
ejpam-6557	141	17	β)-differentiable	β)-differentiable	PUNCT
ejpam-6557	141	18	at	at	ADP
ejpam-6557	141	19	on	on	ADP
ejpam-6557	141	20	a	a	DET
ejpam-6557	141	21	domain	domain	NOUN
ejpam-6557	141	22	g	g	NOUN
ejpam-6557	141	23	if	if	SCONJ
ejpam-6557	141	24	f	f	PROPN
ejpam-6557	141	25	is	be	AUX
ejpam-6557	141	26	(	(	PUNCT
ejpam-6557	141	27	α	α	NOUN
ejpam-6557	141	28	,	,	PUNCT
ejpam-6557	141	29	β)-differentiable	β)-differentiable	PUNCT
ejpam-6557	141	30	at	at	ADP
ejpam-6557	141	31	every	every	DET
ejpam-6557	141	32	z	z	PROPN
ejpam-6557	141	33	∈	∈	PROPN
ejpam-6557	141	34	g.	g.	NOUN
ejpam-6557	141	35	clearly	clearly	ADV
ejpam-6557	141	36	,	,	PUNCT
ejpam-6557	141	37	that	that	SCONJ
ejpam-6557	141	38	both	both	DET
ejpam-6557	141	39	functions	function	NOUN
ejpam-6557	141	40	in	in	ADP
ejpam-6557	141	41	examples	example	NOUN
ejpam-6557	141	42	1	1	NUM
ejpam-6557	141	43	,	,	PUNCT
ejpam-6557	141	44	2	2	NUM
ejpam-6557	141	45	are	be	AUX
ejpam-6557	141	46	(	(	PUNCT
ejpam-6557	141	47	α	α	NOUN
ejpam-6557	141	48	,	,	PUNCT
ejpam-6557	141	49	β)-differentiable	β)-differentiable	PUNCT
ejpam-6557	141	50	over	over	ADP
ejpam-6557	141	51	c.	c.	PROPN
ejpam-6557	141	52	now	now	ADV
ejpam-6557	141	53	,	,	PUNCT
ejpam-6557	141	54	to	to	PART
ejpam-6557	141	55	restate	restate	VERB
ejpam-6557	141	56	the	the	DET
ejpam-6557	141	57	sufficient	sufficient	ADJ
ejpam-6557	141	58	conditions	condition	NOUN
ejpam-6557	141	59	theorem	theorem	VERB
ejpam-6557	141	60	for	for	ADP
ejpam-6557	141	61	(	(	PUNCT
ejpam-6557	141	62	α	α	NOUN
ejpam-6557	141	63	,	,	PUNCT
ejpam-6557	141	64	β)-differentiability	β)-differentiability	PUNCT
ejpam-6557	141	65	in	in	ADP
ejpam-6557	141	66	polar	polar	ADJ
ejpam-6557	141	67	coordinates	coordinate	NOUN
ejpam-6557	141	68	,	,	PUNCT
ejpam-6557	141	69	we	we	PRON
ejpam-6557	141	70	proceed	proceed	VERB
ejpam-6557	141	71	as	as	SCONJ
ejpam-6557	141	72	follows	follow	VERB
ejpam-6557	141	73	:	:	PUNCT
ejpam-6557	141	74	let	let	VERB
ejpam-6557	141	75	x	x	PUNCT
ejpam-6557	141	76	=	=	PUNCT
ejpam-6557	141	77	r	r	NOUN
ejpam-6557	141	78	cos	cos	PROPN
ejpam-6557	141	79	θ	θ	PROPN
ejpam-6557	141	80	and	and	CCONJ
ejpam-6557	141	81	y	y	PROPN
ejpam-6557	141	82	=	=	NOUN
ejpam-6557	141	83	r	r	NOUN
ejpam-6557	141	84	sin	sin	NOUN
ejpam-6557	141	85	θ	θ	PROPN
ejpam-6557	141	86	.	.	PUNCT
ejpam-6557	142	1	then	then	ADV
ejpam-6557	142	2	z	z	NOUN
ejpam-6557	143	1	=	=	PUNCT
ejpam-6557	143	2	x+	x+	PUNCT
ejpam-6557	144	1	iy	iy	X
ejpam-6557	144	2	=	=	PUNCT
ejpam-6557	144	3	r	r	NOUN
ejpam-6557	144	4	(	(	PUNCT
ejpam-6557	144	5	cos	cos	PROPN
ejpam-6557	144	6	θ	θ	PROPN
ejpam-6557	144	7	+	+	CCONJ
ejpam-6557	144	8	i	i	PRON
ejpam-6557	144	9	sin	sin	VERB
ejpam-6557	144	10	θ	θ	NOUN
ejpam-6557	144	11	)	)	PUNCT
ejpam-6557	144	12	=	=	SYM
ejpam-6557	145	1	reiθ	reiθ	PROPN
ejpam-6557	145	2	and	and	CCONJ
ejpam-6557	145	3	f	f	PROPN
ejpam-6557	145	4	(	(	PUNCT
ejpam-6557	145	5	z	z	NOUN
ejpam-6557	145	6	)	)	PUNCT
ejpam-6557	145	7	=	=	PUNCT
ejpam-6557	146	1	f(reiθ	f(reiθ	X
ejpam-6557	146	2	)	)	PUNCT
ejpam-6557	146	3	=	=	SYM
ejpam-6557	146	4	u	u	NOUN
ejpam-6557	146	5	(	(	PUNCT
ejpam-6557	146	6	r	r	NOUN
ejpam-6557	146	7	,	,	PUNCT
ejpam-6557	146	8	θ	θ	NOUN
ejpam-6557	146	9	)	)	PUNCT
ejpam-6557	147	1	+	+	NUM
ejpam-6557	147	2	iv	iv	NUM
ejpam-6557	147	3	(	(	PUNCT
ejpam-6557	147	4	r	r	NOUN
ejpam-6557	147	5	,	,	PUNCT
ejpam-6557	147	6	θ	θ	NOUN
ejpam-6557	147	7	)	)	PUNCT
ejpam-6557	147	8	.	.	PUNCT
ejpam-6557	148	1	(	(	PUNCT
ejpam-6557	148	2	5	5	X
ejpam-6557	148	3	)	)	PUNCT
ejpam-6557	148	4	the	the	DET
ejpam-6557	148	5	(	(	PUNCT
ejpam-6557	148	6	α	α	NOUN
ejpam-6557	148	7	,	,	PUNCT
ejpam-6557	148	8	β)-derivative	β)-derivative	PUNCT
ejpam-6557	148	9	of	of	ADP
ejpam-6557	148	10	(	(	PUNCT
ejpam-6557	148	11	5	5	NUM
ejpam-6557	148	12	)	)	PUNCT
ejpam-6557	148	13	,	,	PUNCT
ejpam-6557	148	14	with	with	ADP
ejpam-6557	148	15	respect	respect	NOUN
ejpam-6557	148	16	to	to	ADP
ejpam-6557	148	17	r	r	NOUN
ejpam-6557	148	18	and	and	CCONJ
ejpam-6557	148	19	with	with	ADP
ejpam-6557	148	20	respect	respect	NOUN
ejpam-6557	148	21	to	to	ADP
ejpam-6557	148	22	θ	θ	PROPN
ejpam-6557	148	23	are	be	AUX
ejpam-6557	148	24	,	,	PUNCT
ejpam-6557	148	25	respectively	respectively	ADV
ejpam-6557	148	26	,	,	PUNCT
ejpam-6557	148	27	f	f	PROPN
ejpam-6557	148	28	(	(	PUNCT
ejpam-6557	148	29	α	α	NOUN
ejpam-6557	148	30	,	,	PUNCT
ejpam-6557	148	31	β)(reiθ)r1−αeiθ	β)(reiθ)r1−αeiθ	NOUN
ejpam-6557	148	32	=	=	PUNCT
ejpam-6557	149	1	uαr	uαr	NOUN
ejpam-6557	149	2	(	(	PUNCT
ejpam-6557	149	3	r	r	NOUN
ejpam-6557	149	4	,	,	PUNCT
ejpam-6557	149	5	θ	θ	NOUN
ejpam-6557	149	6	)	)	PUNCT
ejpam-6557	150	1	+	+	NUM
ejpam-6557	150	2	ivαr	ivαr	NOUN
ejpam-6557	150	3	(	(	PUNCT
ejpam-6557	150	4	r	r	NOUN
ejpam-6557	150	5	,	,	PUNCT
ejpam-6557	150	6	θ	θ	NOUN
ejpam-6557	150	7	)	)	PUNCT
ejpam-6557	150	8	,	,	PUNCT
ejpam-6557	150	9	w.	w.	PROPN
ejpam-6557	150	10	g.	g.	PROPN
ejpam-6557	150	11	alshanti	alshanti	PROPN
ejpam-6557	150	12	,	,	PUNCT
ejpam-6557	150	13	m.	m.	NOUN
ejpam-6557	150	14	a.	a.	PROPN
ejpam-6557	150	15	hammad	hammad	PROPN
ejpam-6557	150	16	,	,	PUNCT
ejpam-6557	150	17	r.	r.	PROPN
ejpam-6557	150	18	khalil	khalil	PROPN
ejpam-6557	150	19	/	/	SYM
ejpam-6557	150	20	eur	eur	PROPN
ejpam-6557	150	21	.	.	PUNCT
ejpam-6557	151	1	j.	j.	PROPN
ejpam-6557	151	2	pure	pure	PROPN
ejpam-6557	151	3	appl	appl	PROPN
ejpam-6557	151	4	.	.	PROPN
ejpam-6557	151	5	math	math	PROPN
ejpam-6557	151	6	,	,	PUNCT
ejpam-6557	151	7	18	18	NUM
ejpam-6557	151	8	(	(	PUNCT
ejpam-6557	151	9	4	4	NUM
ejpam-6557	151	10	)	)	PUNCT
ejpam-6557	151	11	(	(	PUNCT
ejpam-6557	151	12	2025	2025	NUM
ejpam-6557	151	13	)	)	PUNCT
ejpam-6557	151	14	,	,	PUNCT
ejpam-6557	151	15	6557	6557	NUM
ejpam-6557	151	16	6	6	NUM
ejpam-6557	151	17	of	of	ADP
ejpam-6557	151	18	8	8	NUM
ejpam-6557	151	19	f	f	NOUN
ejpam-6557	151	20	(	(	PUNCT
ejpam-6557	151	21	α	α	NOUN
ejpam-6557	151	22	,	,	PUNCT
ejpam-6557	151	23	β)(reiθ)irθ1−βeiθ	β)(reiθ)irθ1−βeiθ	PUNCT
ejpam-6557	151	24	=	=	PUNCT
ejpam-6557	152	1	uβθ	uβθ	NOUN
ejpam-6557	152	2	(	(	PUNCT
ejpam-6557	152	3	r	r	NOUN
ejpam-6557	152	4	,	,	PUNCT
ejpam-6557	152	5	θ	θ	NOUN
ejpam-6557	152	6	)	)	PUNCT
ejpam-6557	152	7	+	+	NUM
ejpam-6557	152	8	ivβθ	ivβθ	NOUN
ejpam-6557	152	9	(	(	PUNCT
ejpam-6557	152	10	r	r	NOUN
ejpam-6557	152	11	,	,	PUNCT
ejpam-6557	152	12	θ	θ	NOUN
ejpam-6557	152	13	)	)	PUNCT
ejpam-6557	152	14	.	.	PUNCT
ejpam-6557	153	1	(	(	PUNCT
ejpam-6557	153	2	6	6	NUM
ejpam-6557	153	3	)	)	PUNCT
ejpam-6557	153	4	hence	hence	ADV
ejpam-6557	153	5	,	,	PUNCT
ejpam-6557	153	6	by	by	ADP
ejpam-6557	153	7	equating	equate	VERB
ejpam-6557	153	8	both	both	DET
ejpam-6557	153	9	equations	equation	NOUN
ejpam-6557	153	10	in	in	ADP
ejpam-6557	153	11	(	(	PUNCT
ejpam-6557	153	12	6	6	NUM
ejpam-6557	153	13	)	)	PUNCT
ejpam-6557	153	14	,	,	PUNCT
ejpam-6557	153	15	we	we	PRON
ejpam-6557	153	16	get	get	VERB
ejpam-6557	153	17	the	the	DET
ejpam-6557	153	18	polar	polar	ADJ
ejpam-6557	153	19	version	version	NOUN
ejpam-6557	153	20	of	of	ADP
ejpam-6557	153	21	the	the	DET
ejpam-6557	153	22	(	(	PUNCT
ejpam-6557	153	23	α	α	NOUN
ejpam-6557	153	24	,	,	PUNCT
ejpam-6557	153	25	β)-cauchyriemann	β)-cauchyriemann	PUNCT
ejpam-6557	153	26	equations	equation	NOUN
ejpam-6557	153	27	,	,	PUNCT
ejpam-6557	153	28	namely	namely	ADV
ejpam-6557	153	29	,	,	PUNCT
ejpam-6557	153	30	uαr	uαr	NOUN
ejpam-6557	153	31	=	=	PUNCT
ejpam-6557	153	32	θβ−1	θβ−1	PROPN
ejpam-6557	153	33	rα	rα	ADJ
ejpam-6557	153	34	vβθ	vβθ	NOUN
ejpam-6557	153	35	,	,	PUNCT
ejpam-6557	153	36	uβθ	uβθ	NOUN
ejpam-6557	153	37	=	=	PUNCT
ejpam-6557	153	38	−	−	PROPN
ejpam-6557	153	39	rα	rα	ADJ
ejpam-6557	153	40	θβ−1	θβ−1	PROPN
ejpam-6557	153	41	vαr	vαr	NOUN
ejpam-6557	153	42	.	.	PUNCT
ejpam-6557	154	1	(	(	PUNCT
ejpam-6557	154	2	7	7	X
ejpam-6557	154	3	)	)	PUNCT
ejpam-6557	154	4	definition	definition	NOUN
ejpam-6557	154	5	3	3	NUM
ejpam-6557	154	6	.	.	PUNCT
ejpam-6557	155	1	a	a	DET
ejpam-6557	155	2	function	function	NOUN
ejpam-6557	155	3	f	f	NOUN
ejpam-6557	155	4	:	:	PUNCT
ejpam-6557	155	5	g	g	PROPN
ejpam-6557	155	6	⊆	⊆	NUM
ejpam-6557	155	7	e	e	X
ejpam-6557	155	8	→	→	SYM
ejpam-6557	155	9	c	c	PROPN
ejpam-6557	155	10	is	be	AUX
ejpam-6557	155	11	said	say	VERB
ejpam-6557	155	12	to	to	PART
ejpam-6557	155	13	be	be	AUX
ejpam-6557	155	14	(	(	PUNCT
ejpam-6557	155	15	α	α	NOUN
ejpam-6557	155	16	,	,	PUNCT
ejpam-6557	155	17	β)-analytic	β)-analytic	PUNCT
ejpam-6557	155	18	at	at	ADP
ejpam-6557	155	19	z	z	NOUN
ejpam-6557	155	20	◦	◦	NOUN
ejpam-6557	155	21	=	=	SYM
ejpam-6557	156	1	x	x	X
ejpam-6557	156	2	◦	◦	NOUN
ejpam-6557	156	3	+iy	+iy	NOUN
ejpam-6557	156	4	◦	◦	NOUN
ejpam-6557	156	5	∈	∈	NOUN
ejpam-6557	156	6	g	g	ADP
ejpam-6557	156	7	◦	◦	NOUN
ejpam-6557	156	8	where	where	SCONJ
ejpam-6557	156	9	|α|	|α|	PROPN
ejpam-6557	156	10	,	,	PUNCT
ejpam-6557	156	11	|β|	|β|	X
ejpam-6557	156	12	<	<	X
ejpam-6557	156	13	1	1	NUM
ejpam-6557	156	14	,	,	PUNCT
ejpam-6557	156	15	if	if	SCONJ
ejpam-6557	156	16	i	i	PRON
ejpam-6557	156	17	)	)	PUNCT
ejpam-6557	156	18	f	f	PROPN
ejpam-6557	156	19	is	be	AUX
ejpam-6557	156	20	(	(	PUNCT
ejpam-6557	156	21	α	α	NOUN
ejpam-6557	156	22	,	,	PUNCT
ejpam-6557	156	23	β	β	NOUN
ejpam-6557	156	24	)	)	PUNCT
ejpam-6557	156	25	-differentiable	-differentiable	ADJ
ejpam-6557	156	26	at	at	ADP
ejpam-6557	156	27	z	z	NOUN
ejpam-6557	156	28	◦	◦	NOUN
ejpam-6557	156	29	,	,	PUNCT
ejpam-6557	156	30	ii	ii	NOUN
ejpam-6557	156	31	)	)	PUNCT
ejpam-6557	156	32	there	there	PRON
ejpam-6557	156	33	exists	exist	VERB
ejpam-6557	156	34	ϵ	ϵ	X
ejpam-6557	156	35	>	>	X
ejpam-6557	156	36	0	0	NUM
ejpam-6557	157	1	such	such	ADJ
ejpam-6557	157	2	that	that	SCONJ
ejpam-6557	157	3	f	f	PROPN
ejpam-6557	157	4	is	be	AUX
ejpam-6557	157	5	(	(	PUNCT
ejpam-6557	157	6	α	α	NOUN
ejpam-6557	157	7	,	,	PUNCT
ejpam-6557	157	8	β	β	NOUN
ejpam-6557	157	9	)	)	PUNCT
ejpam-6557	157	10	-differentiable	-differentiable	ADJ
ejpam-6557	157	11	for	for	ADP
ejpam-6557	157	12	all	all	DET
ejpam-6557	157	13	z	z	NOUN
ejpam-6557	157	14	∈	∈	PROPN
ejpam-6557	157	15	d	d	X
ejpam-6557	157	16	(	(	PUNCT
ejpam-6557	157	17	ϵ	ϵ	NOUN
ejpam-6557	157	18	,	,	PUNCT
ejpam-6557	157	19	z	z	NOUN
ejpam-6557	157	20	◦	◦	NOUN
ejpam-6557	157	21	)	)	PUNCT
ejpam-6557	157	22	.	.	PUNCT
ejpam-6557	158	1	clearly	clearly	ADV
ejpam-6557	158	2	,	,	PUNCT
ejpam-6557	158	3	both	both	DET
ejpam-6557	158	4	functions	function	NOUN
ejpam-6557	158	5	in	in	ADP
ejpam-6557	158	6	examples	example	NOUN
ejpam-6557	158	7	1	1	NUM
ejpam-6557	158	8	and	and	CCONJ
ejpam-6557	158	9	2	2	NUM
ejpam-6557	158	10	are	be	AUX
ejpam-6557	158	11	(	(	PUNCT
ejpam-6557	158	12	α	α	NOUN
ejpam-6557	158	13	,	,	PUNCT
ejpam-6557	158	14	β)-analytic	β)-analytic	ADJ
ejpam-6557	158	15	.	.	PUNCT
ejpam-6557	159	1	definition	definition	NOUN
ejpam-6557	159	2	4	4	NUM
ejpam-6557	159	3	.	.	PUNCT
ejpam-6557	160	1	a	a	DET
ejpam-6557	160	2	function	function	NOUN
ejpam-6557	160	3	f	f	NOUN
ejpam-6557	160	4	:	:	PUNCT
ejpam-6557	160	5	g	g	PROPN
ejpam-6557	160	6	⊆	⊆	NUM
ejpam-6557	160	7	c	c	NOUN
ejpam-6557	160	8	→	→	SYM
ejpam-6557	160	9	c	c	PROPN
ejpam-6557	160	10	is	be	AUX
ejpam-6557	160	11	said	say	VERB
ejpam-6557	160	12	to	to	PART
ejpam-6557	160	13	be	be	AUX
ejpam-6557	160	14	(	(	PUNCT
ejpam-6557	160	15	α	α	NOUN
ejpam-6557	160	16	,	,	PUNCT
ejpam-6557	160	17	β)-analytic	β)-analytic	PUNCT
ejpam-6557	160	18	the	the	DET
ejpam-6557	160	19	domain	domain	NOUN
ejpam-6557	160	20	g	g	NOUN
ejpam-6557	160	21	where	where	SCONJ
ejpam-6557	160	22	|α|	|α|	PROPN
ejpam-6557	160	23	,	,	PUNCT
ejpam-6557	160	24	|β|	|β|	X
ejpam-6557	160	25	<	<	X
ejpam-6557	160	26	1	1	NUM
ejpam-6557	160	27	,	,	PUNCT
ejpam-6557	160	28	if	if	SCONJ
ejpam-6557	160	29	it	it	PRON
ejpam-6557	160	30	is	be	AUX
ejpam-6557	160	31	(	(	PUNCT
ejpam-6557	160	32	α	α	X
ejpam-6557	160	33	,	,	PUNCT
ejpam-6557	160	34	β)-analytic	β)-analytic	ADJ
ejpam-6557	160	35	at	at	ADP
ejpam-6557	160	36	every	every	DET
ejpam-6557	160	37	z	z	PROPN
ejpam-6557	160	38	∈	∈	PROPN
ejpam-6557	160	39	g.	g.	NOUN
ejpam-6557	160	40	3	3	X
ejpam-6557	160	41	.	.	PUNCT
ejpam-6557	160	42	applications	application	NOUN
ejpam-6557	160	43	,	,	PUNCT
ejpam-6557	160	44	discussion	discussion	NOUN
ejpam-6557	160	45	,	,	PUNCT
ejpam-6557	160	46	and	and	CCONJ
ejpam-6557	160	47	implications	implication	VERB
ejpam-6557	160	48	the	the	DET
ejpam-6557	160	49	concept	concept	NOUN
ejpam-6557	160	50	of	of	ADP
ejpam-6557	160	51	the	the	DET
ejpam-6557	160	52	(	(	PUNCT
ejpam-6557	160	53	α	α	NOUN
ejpam-6557	160	54	,	,	PUNCT
ejpam-6557	160	55	β)-fractional	β)-fractional	PUNCT
ejpam-6557	160	56	derivative	derivative	NOUN
ejpam-6557	160	57	offers	offer	VERB
ejpam-6557	160	58	promising	promise	VERB
ejpam-6557	160	59	potential	potential	NOUN
ejpam-6557	160	60	in	in	ADP
ejpam-6557	160	61	a	a	DET
ejpam-6557	160	62	variety	variety	NOUN
ejpam-6557	160	63	of	of	ADP
ejpam-6557	160	64	scientific	scientific	ADJ
ejpam-6557	160	65	and	and	CCONJ
ejpam-6557	160	66	engineering	engineering	NOUN
ejpam-6557	160	67	domains	domain	NOUN
ejpam-6557	160	68	.	.	PUNCT
ejpam-6557	161	1	in	in	ADP
ejpam-6557	161	2	the	the	DET
ejpam-6557	161	3	area	area	NOUN
ejpam-6557	161	4	of	of	ADP
ejpam-6557	161	5	signal	signal	NOUN
ejpam-6557	161	6	and	and	CCONJ
ejpam-6557	161	7	image	image	NOUN
ejpam-6557	161	8	processing	processing	NOUN
ejpam-6557	161	9	,	,	PUNCT
ejpam-6557	161	10	classical	classical	ADJ
ejpam-6557	161	11	derivatives	derivative	NOUN
ejpam-6557	161	12	often	often	ADV
ejpam-6557	161	13	fail	fail	VERB
ejpam-6557	161	14	to	to	PART
ejpam-6557	161	15	capture	capture	VERB
ejpam-6557	161	16	the	the	DET
ejpam-6557	161	17	memory	memory	NOUN
ejpam-6557	161	18	and	and	CCONJ
ejpam-6557	161	19	hereditary	hereditary	ADJ
ejpam-6557	161	20	properties	property	NOUN
ejpam-6557	161	21	inherent	inherent	ADJ
ejpam-6557	161	22	in	in	ADP
ejpam-6557	161	23	certain	certain	ADJ
ejpam-6557	161	24	signals	signal	NOUN
ejpam-6557	161	25	and	and	CCONJ
ejpam-6557	161	26	textures	texture	NOUN
ejpam-6557	161	27	.	.	PUNCT
ejpam-6557	162	1	by	by	ADP
ejpam-6557	162	2	introducing	introduce	VERB
ejpam-6557	162	3	two	two	NUM
ejpam-6557	162	4	parameters	parameter	NOUN
ejpam-6557	162	5	that	that	PRON
ejpam-6557	162	6	independently	independently	ADV
ejpam-6557	162	7	influence	influence	VERB
ejpam-6557	162	8	the	the	DET
ejpam-6557	162	9	phase	phase	NOUN
ejpam-6557	162	10	and	and	CCONJ
ejpam-6557	162	11	magnitude	magnitude	NOUN
ejpam-6557	162	12	in	in	ADP
ejpam-6557	162	13	two	two	NUM
ejpam-6557	162	14	dimensions	dimension	NOUN
ejpam-6557	162	15	,	,	PUNCT
ejpam-6557	162	16	the	the	DET
ejpam-6557	162	17	(	(	PUNCT
ejpam-6557	162	18	α	α	NOUN
ejpam-6557	162	19	,	,	PUNCT
ejpam-6557	162	20	β)-fractional	β)-fractional	PRON
ejpam-6557	162	21	derivative	derivative	NOUN
ejpam-6557	162	22	provides	provide	VERB
ejpam-6557	162	23	finer	fine	ADJ
ejpam-6557	162	24	control	control	NOUN
ejpam-6557	162	25	over	over	ADP
ejpam-6557	162	26	2d	2d	NUM
ejpam-6557	162	27	signal	signal	NOUN
ejpam-6557	162	28	representations	representation	NOUN
ejpam-6557	162	29	.	.	PUNCT
ejpam-6557	163	1	this	this	DET
ejpam-6557	163	2	approach	approach	NOUN
ejpam-6557	163	3	can	can	AUX
ejpam-6557	163	4	be	be	AUX
ejpam-6557	163	5	leveraged	leverage	VERB
ejpam-6557	163	6	for	for	ADP
ejpam-6557	163	7	more	more	ADV
ejpam-6557	163	8	precise	precise	ADJ
ejpam-6557	163	9	techniques	technique	NOUN
ejpam-6557	163	10	in	in	ADP
ejpam-6557	163	11	edge	edge	NOUN
ejpam-6557	163	12	detection	detection	NOUN
ejpam-6557	163	13	,	,	PUNCT
ejpam-6557	163	14	texture	texture	ADJ
ejpam-6557	163	15	segmentation	segmentation	NOUN
ejpam-6557	163	16	,	,	PUNCT
ejpam-6557	163	17	and	and	CCONJ
ejpam-6557	163	18	in	in	ADP
ejpam-6557	163	19	adaptations	adaptation	NOUN
ejpam-6557	163	20	of	of	ADP
ejpam-6557	163	21	the	the	DET
ejpam-6557	163	22	fractional	fractional	ADJ
ejpam-6557	163	23	fourier	fourier	NOUN
ejpam-6557	163	24	transform	transform	NOUN
ejpam-6557	163	25	.	.	PUNCT
ejpam-6557	164	1	in	in	ADP
ejpam-6557	164	2	fractional	fractional	ADJ
ejpam-6557	164	3	control	control	NOUN
ejpam-6557	164	4	theory	theory	NOUN
ejpam-6557	164	5	,	,	PUNCT
ejpam-6557	164	6	modern	modern	ADJ
ejpam-6557	164	7	controllers	controller	NOUN
ejpam-6557	164	8	such	such	ADJ
ejpam-6557	164	9	as	as	ADP
ejpam-6557	164	10	fractional	fractional	ADJ
ejpam-6557	164	11	pid	pid	NOUN
ejpam-6557	164	12	systems	system	NOUN
ejpam-6557	164	13	already	already	ADV
ejpam-6557	164	14	benefit	benefit	VERB
ejpam-6557	164	15	from	from	ADP
ejpam-6557	164	16	generalized	generalized	ADJ
ejpam-6557	164	17	derivatives	derivative	NOUN
ejpam-6557	164	18	to	to	PART
ejpam-6557	164	19	describe	describe	VERB
ejpam-6557	164	20	systems	system	NOUN
ejpam-6557	164	21	with	with	ADP
ejpam-6557	164	22	memory	memory	NOUN
ejpam-6557	164	23	(	(	PUNCT
ejpam-6557	164	24	see	see	VERB
ejpam-6557	164	25	[	[	X
ejpam-6557	164	26	9	9	NUM
ejpam-6557	164	27	]	]	PUNCT
ejpam-6557	164	28	,	,	PUNCT
ejpam-6557	164	29	[	[	X
ejpam-6557	164	30	10	10	NUM
ejpam-6557	164	31	]	]	PUNCT
ejpam-6557	164	32	)	)	PUNCT
ejpam-6557	164	33	.	.	PUNCT
ejpam-6557	165	1	extending	extend	VERB
ejpam-6557	165	2	these	these	DET
ejpam-6557	165	3	ideas	idea	NOUN
ejpam-6557	165	4	to	to	ADP
ejpam-6557	165	5	the	the	DET
ejpam-6557	165	6	complex	complex	ADJ
ejpam-6557	165	7	domain	domain	NOUN
ejpam-6557	165	8	,	,	PUNCT
ejpam-6557	165	9	the	the	DET
ejpam-6557	165	10	(	(	PUNCT
ejpam-6557	165	11	α	α	NOUN
ejpam-6557	165	12	,	,	PUNCT
ejpam-6557	165	13	β)-fractional	β)-fractional	PRON
ejpam-6557	165	14	derivative	derivative	NOUN
ejpam-6557	165	15	could	could	AUX
ejpam-6557	165	16	be	be	AUX
ejpam-6557	165	17	incorporated	incorporate	VERB
ejpam-6557	165	18	into	into	ADP
ejpam-6557	165	19	the	the	DET
ejpam-6557	165	20	design	design	NOUN
ejpam-6557	165	21	of	of	ADP
ejpam-6557	165	22	complex	complex	ADJ
ejpam-6557	165	23	transfer	transfer	NOUN
ejpam-6557	165	24	functions	function	NOUN
ejpam-6557	165	25	and	and	CCONJ
ejpam-6557	165	26	impulse	impulse	ADJ
ejpam-6557	165	27	responses	response	NOUN
ejpam-6557	165	28	for	for	ADP
ejpam-6557	165	29	systems	system	NOUN
ejpam-6557	165	30	with	with	ADP
ejpam-6557	165	31	asymmetrical	asymmetrical	ADJ
ejpam-6557	165	32	time	time	NOUN
ejpam-6557	165	33	scales	scale	NOUN
ejpam-6557	165	34	—	—	PUNCT
ejpam-6557	165	35	where	where	SCONJ
ejpam-6557	165	36	,	,	PUNCT
ejpam-6557	165	37	for	for	ADP
ejpam-6557	165	38	example	example	NOUN
ejpam-6557	165	39	,	,	PUNCT
ejpam-6557	165	40	position	position	NOUN
ejpam-6557	165	41	and	and	CCONJ
ejpam-6557	165	42	velocity	velocity	NOUN
ejpam-6557	165	43	evolve	evolve	VERB
ejpam-6557	165	44	under	under	ADP
ejpam-6557	165	45	different	different	ADJ
ejpam-6557	165	46	fractional	fractional	ADJ
ejpam-6557	165	47	dynamics	dynamic	NOUN
ejpam-6557	165	48	.	.	PUNCT
ejpam-6557	166	1	in	in	ADP
ejpam-6557	166	2	the	the	DET
ejpam-6557	166	3	study	study	NOUN
ejpam-6557	166	4	of	of	ADP
ejpam-6557	166	5	complex	complex	ADJ
ejpam-6557	166	6	dynamics	dynamic	NOUN
ejpam-6557	166	7	and	and	CCONJ
ejpam-6557	166	8	fractals	fractal	NOUN
ejpam-6557	166	9	,	,	PUNCT
ejpam-6557	166	10	the	the	DET
ejpam-6557	166	11	generalized	generalized	ADJ
ejpam-6557	166	12	derivative	derivative	NOUN
ejpam-6557	166	13	opens	open	VERB
ejpam-6557	166	14	a	a	DET
ejpam-6557	166	15	pathway	pathway	NOUN
ejpam-6557	166	16	to	to	ADP
ejpam-6557	166	17	modeling	model	VERB
ejpam-6557	166	18	non	non	ADJ
ejpam-6557	166	19	-	-	ADJ
ejpam-6557	166	20	local	local	ADJ
ejpam-6557	166	21	interactions	interaction	NOUN
ejpam-6557	166	22	and	and	CCONJ
ejpam-6557	166	23	generating	generate	VERB
ejpam-6557	166	24	fractional	fractional	ADJ
ejpam-6557	166	25	julia	julia	PROPN
ejpam-6557	166	26	or	or	CCONJ
ejpam-6557	166	27	mandelbrot	mandelbrot	PROPN
ejpam-6557	166	28	sets	set	NOUN
ejpam-6557	166	29	.	.	PUNCT
ejpam-6557	167	1	the	the	DET
ejpam-6557	167	2	flexibility	flexibility	NOUN
ejpam-6557	167	3	of	of	ADP
ejpam-6557	167	4	the	the	DET
ejpam-6557	167	5	parameters	parameter	NOUN
ejpam-6557	167	6	α	α	PROPN
ejpam-6557	167	7	and	and	CCONJ
ejpam-6557	167	8	β	β	X
ejpam-6557	167	9	may	may	AUX
ejpam-6557	167	10	lead	lead	VERB
ejpam-6557	167	11	to	to	ADP
ejpam-6557	167	12	the	the	DET
ejpam-6557	167	13	creation	creation	NOUN
ejpam-6557	167	14	of	of	ADP
ejpam-6557	167	15	entirely	entirely	ADV
ejpam-6557	167	16	new	new	ADJ
ejpam-6557	167	17	families	family	NOUN
ejpam-6557	167	18	of	of	ADP
ejpam-6557	167	19	fractals	fractal	NOUN
ejpam-6557	167	20	whose	whose	DET
ejpam-6557	167	21	dimensions	dimension	NOUN
ejpam-6557	167	22	and	and	CCONJ
ejpam-6557	167	23	symmetries	symmetry	NOUN
ejpam-6557	167	24	can	can	AUX
ejpam-6557	167	25	be	be	AUX
ejpam-6557	167	26	tuned	tune	VERB
ejpam-6557	167	27	,	,	PUNCT
ejpam-6557	167	28	with	with	ADP
ejpam-6557	167	29	possible	possible	ADJ
ejpam-6557	167	30	applications	application	NOUN
ejpam-6557	167	31	in	in	ADP
ejpam-6557	167	32	data	data	NOUN
ejpam-6557	167	33	compression	compression	NOUN
ejpam-6557	167	34	,	,	PUNCT
ejpam-6557	167	35	encryption	encryption	NOUN
ejpam-6557	167	36	,	,	PUNCT
ejpam-6557	167	37	and	and	CCONJ
ejpam-6557	167	38	chaotic	chaotic	ADJ
ejpam-6557	167	39	modeling	modeling	NOUN
ejpam-6557	167	40	.	.	PUNCT
ejpam-6557	168	1	w.	w.	PROPN
ejpam-6557	168	2	g.	g.	PROPN
ejpam-6557	168	3	alshanti	alshanti	PROPN
ejpam-6557	168	4	,	,	PUNCT
ejpam-6557	168	5	m.	m.	NOUN
ejpam-6557	168	6	a.	a.	PROPN
ejpam-6557	168	7	hammad	hammad	PROPN
ejpam-6557	168	8	,	,	PUNCT
ejpam-6557	168	9	r.	r.	PROPN
ejpam-6557	168	10	khalil	khalil	PROPN
ejpam-6557	168	11	/	/	SYM
ejpam-6557	168	12	eur	eur	PROPN
ejpam-6557	168	13	.	.	PUNCT
ejpam-6557	169	1	j.	j.	PROPN
ejpam-6557	169	2	pure	pure	PROPN
ejpam-6557	169	3	appl	appl	PROPN
ejpam-6557	169	4	.	.	PROPN
ejpam-6557	169	5	math	math	PROPN
ejpam-6557	169	6	,	,	PUNCT
ejpam-6557	169	7	18	18	NUM
ejpam-6557	169	8	(	(	PUNCT
ejpam-6557	169	9	4	4	NUM
ejpam-6557	169	10	)	)	PUNCT
ejpam-6557	169	11	(	(	PUNCT
ejpam-6557	169	12	2025	2025	NUM
ejpam-6557	169	13	)	)	PUNCT
ejpam-6557	169	14	,	,	PUNCT
ejpam-6557	169	15	6557	6557	NUM
ejpam-6557	169	16	7	7	NUM
ejpam-6557	169	17	of	of	ADP
ejpam-6557	169	18	8	8	NUM
ejpam-6557	169	19	physical	physical	ADJ
ejpam-6557	169	20	modeling	modeling	NOUN
ejpam-6557	169	21	in	in	ADP
ejpam-6557	169	22	areas	area	NOUN
ejpam-6557	169	23	such	such	ADJ
ejpam-6557	169	24	as	as	ADP
ejpam-6557	169	25	fluid	fluid	ADJ
ejpam-6557	169	26	flow	flow	NOUN
ejpam-6557	169	27	or	or	CCONJ
ejpam-6557	169	28	electromagnetic	electromagnetic	ADJ
ejpam-6557	169	29	field	field	NOUN
ejpam-6557	169	30	theory	theory	NOUN
ejpam-6557	169	31	may	may	AUX
ejpam-6557	169	32	also	also	ADV
ejpam-6557	169	33	benefit	benefit	VERB
ejpam-6557	169	34	from	from	ADP
ejpam-6557	169	35	the	the	DET
ejpam-6557	169	36	(	(	PUNCT
ejpam-6557	169	37	α	α	NOUN
ejpam-6557	169	38	,	,	PUNCT
ejpam-6557	169	39	β)-cauchy	β)-cauchy	NOUN
ejpam-6557	169	40	-	-	PUNCT
ejpam-6557	169	41	riemann	riemann	PROPN
ejpam-6557	169	42	framework	framework	NOUN
ejpam-6557	169	43	.	.	PUNCT
ejpam-6557	170	1	in	in	ADP
ejpam-6557	170	2	inhomogeneous	inhomogeneous	ADJ
ejpam-6557	170	3	or	or	CCONJ
ejpam-6557	170	4	anisotropic	anisotropic	NOUN
ejpam-6557	170	5	media	medium	NOUN
ejpam-6557	170	6	,	,	PUNCT
ejpam-6557	170	7	conventional	conventional	ADJ
ejpam-6557	170	8	calculus	calculus	NOUN
ejpam-6557	170	9	often	often	ADV
ejpam-6557	170	10	can	can	AUX
ejpam-6557	170	11	not	not	PART
ejpam-6557	170	12	account	account	VERB
ejpam-6557	170	13	for	for	ADP
ejpam-6557	170	14	asymmetric	asymmetric	ADJ
ejpam-6557	170	15	diffusion	diffusion	NOUN
ejpam-6557	170	16	or	or	CCONJ
ejpam-6557	170	17	non	non	ADJ
ejpam-6557	170	18	-	-	ADJ
ejpam-6557	170	19	local	local	ADJ
ejpam-6557	170	20	interactions	interaction	NOUN
ejpam-6557	170	21	.	.	PUNCT
ejpam-6557	171	1	the	the	DET
ejpam-6557	171	2	polar	polar	ADJ
ejpam-6557	171	3	-	-	PUNCT
ejpam-6557	171	4	coordinate	coordinate	NOUN
ejpam-6557	171	5	form	form	NOUN
ejpam-6557	171	6	derived	derive	VERB
ejpam-6557	171	7	in	in	ADP
ejpam-6557	171	8	this	this	DET
ejpam-6557	171	9	paper	paper	NOUN
ejpam-6557	171	10	suggests	suggest	VERB
ejpam-6557	171	11	that	that	SCONJ
ejpam-6557	171	12	the	the	DET
ejpam-6557	171	13	new	new	ADJ
ejpam-6557	171	14	derivative	derivative	NOUN
ejpam-6557	171	15	could	could	AUX
ejpam-6557	171	16	capture	capture	VERB
ejpam-6557	171	17	these	these	DET
ejpam-6557	171	18	effects	effect	NOUN
ejpam-6557	171	19	with	with	ADP
ejpam-6557	171	20	greater	great	ADJ
ejpam-6557	171	21	accuracy	accuracy	NOUN
ejpam-6557	171	22	.	.	PUNCT
ejpam-6557	172	1	finally	finally	ADV
ejpam-6557	172	2	,	,	PUNCT
ejpam-6557	172	3	from	from	ADP
ejpam-6557	172	4	a	a	DET
ejpam-6557	172	5	purely	purely	ADV
ejpam-6557	172	6	mathematical	mathematical	ADJ
ejpam-6557	172	7	perspective	perspective	NOUN
ejpam-6557	172	8	,	,	PUNCT
ejpam-6557	172	9	the	the	DET
ejpam-6557	172	10	new	new	ADJ
ejpam-6557	172	11	definition	definition	NOUN
ejpam-6557	172	12	unifies	unify	VERB
ejpam-6557	172	13	conformable	conformable	ADJ
ejpam-6557	172	14	derivatives	derivative	NOUN
ejpam-6557	172	15	with	with	ADP
ejpam-6557	172	16	the	the	DET
ejpam-6557	172	17	classical	classical	ADJ
ejpam-6557	172	18	cauchy	cauchy	PROPN
ejpam-6557	172	19	-	-	PUNCT
ejpam-6557	172	20	riemann	riemann	PROPN
ejpam-6557	172	21	theory	theory	NOUN
ejpam-6557	172	22	,	,	PUNCT
ejpam-6557	172	23	thereby	thereby	ADV
ejpam-6557	172	24	bridging	bridge	VERB
ejpam-6557	172	25	real	real	ADJ
ejpam-6557	172	26	and	and	CCONJ
ejpam-6557	172	27	complex	complex	ADJ
ejpam-6557	172	28	fractional	fractional	ADJ
ejpam-6557	172	29	calculus	calculus	NOUN
ejpam-6557	172	30	.	.	PUNCT
ejpam-6557	173	1	it	it	PRON
ejpam-6557	173	2	invites	invite	VERB
ejpam-6557	173	3	further	further	ADJ
ejpam-6557	173	4	investigation	investigation	NOUN
ejpam-6557	173	5	into	into	ADP
ejpam-6557	173	6	its	its	PRON
ejpam-6557	173	7	relationships	relationship	NOUN
ejpam-6557	173	8	with	with	ADP
ejpam-6557	173	9	other	other	ADJ
ejpam-6557	173	10	fractional	fractional	ADJ
ejpam-6557	173	11	analytic	analytic	ADJ
ejpam-6557	173	12	function	function	NOUN
ejpam-6557	173	13	classes	class	NOUN
ejpam-6557	173	14	,	,	PUNCT
ejpam-6557	173	15	such	such	ADJ
ejpam-6557	173	16	as	as	ADP
ejpam-6557	173	17	hadamard	hadamard	ADJ
ejpam-6557	173	18	-	-	PUNCT
ejpam-6557	173	19	type	type	NOUN
ejpam-6557	173	20	or	or	CCONJ
ejpam-6557	173	21	caputo	caputo	NOUN
ejpam-6557	173	22	-	-	PUNCT
ejpam-6557	173	23	type	type	NOUN
ejpam-6557	173	24	,	,	PUNCT
ejpam-6557	173	25	potentially	potentially	ADV
ejpam-6557	173	26	contributing	contribute	VERB
ejpam-6557	173	27	to	to	ADP
ejpam-6557	173	28	a	a	DET
ejpam-6557	173	29	more	more	ADV
ejpam-6557	173	30	comprehensive	comprehensive	ADJ
ejpam-6557	173	31	theory	theory	NOUN
ejpam-6557	173	32	of	of	ADP
ejpam-6557	173	33	fractional	fractional	ADJ
ejpam-6557	173	34	complex	complex	ADJ
ejpam-6557	173	35	analysis	analysis	NOUN
ejpam-6557	173	36	.	.	PUNCT
ejpam-6557	174	1	4	4	X
ejpam-6557	174	2	.	.	X
ejpam-6557	174	3	conclusions	conclusion	NOUN
ejpam-6557	174	4	this	this	DET
ejpam-6557	174	5	work	work	NOUN
ejpam-6557	174	6	has	have	AUX
ejpam-6557	174	7	introduced	introduce	VERB
ejpam-6557	174	8	and	and	CCONJ
ejpam-6557	174	9	developed	develop	VERB
ejpam-6557	174	10	the	the	DET
ejpam-6557	174	11	concept	concept	NOUN
ejpam-6557	174	12	of	of	ADP
ejpam-6557	174	13	the	the	DET
ejpam-6557	174	14	(	(	PUNCT
ejpam-6557	174	15	α	α	NOUN
ejpam-6557	174	16	,	,	PUNCT
ejpam-6557	174	17	β)-derivative	β)-derivative	PUNCT
ejpam-6557	174	18	for	for	ADP
ejpam-6557	174	19	complexvalued	complexvalue	VERB
ejpam-6557	174	20	functions	function	NOUN
ejpam-6557	174	21	,	,	PUNCT
ejpam-6557	174	22	along	along	ADP
ejpam-6557	174	23	with	with	ADP
ejpam-6557	174	24	its	its	PRON
ejpam-6557	174	25	associated	associate	VERB
ejpam-6557	174	26	(	(	PUNCT
ejpam-6557	174	27	α	α	X
ejpam-6557	174	28	,	,	PUNCT
ejpam-6557	174	29	β)-cauchy	β)-cauchy	PUNCT
ejpam-6557	174	30	-	-	PUNCT
ejpam-6557	174	31	riemann	riemann	PROPN
ejpam-6557	174	32	equations	equation	NOUN
ejpam-6557	174	33	and	and	CCONJ
ejpam-6557	174	34	the	the	DET
ejpam-6557	174	35	definition	definition	NOUN
ejpam-6557	174	36	of	of	ADP
ejpam-6557	174	37	(	(	PUNCT
ejpam-6557	174	38	α	α	X
ejpam-6557	174	39	,	,	PUNCT
ejpam-6557	174	40	β)-analytic	β)-analytic	ADJ
ejpam-6557	174	41	functions	function	NOUN
ejpam-6557	174	42	.	.	PUNCT
ejpam-6557	175	1	by	by	ADP
ejpam-6557	175	2	generalizing	generalize	VERB
ejpam-6557	175	3	the	the	DET
ejpam-6557	175	4	idea	idea	NOUN
ejpam-6557	175	5	of	of	ADP
ejpam-6557	175	6	fractional	fractional	ADJ
ejpam-6557	175	7	differentiation	differentiation	NOUN
ejpam-6557	175	8	to	to	PART
ejpam-6557	175	9	allow	allow	VERB
ejpam-6557	175	10	independent	independent	ADJ
ejpam-6557	175	11	fractional	fractional	ADJ
ejpam-6557	175	12	orders	order	NOUN
ejpam-6557	175	13	in	in	ADP
ejpam-6557	175	14	two	two	NUM
ejpam-6557	175	15	orthogonal	orthogonal	ADJ
ejpam-6557	175	16	directions	direction	NOUN
ejpam-6557	175	17	,	,	PUNCT
ejpam-6557	175	18	we	we	PRON
ejpam-6557	175	19	have	have	AUX
ejpam-6557	175	20	extended	extend	VERB
ejpam-6557	175	21	both	both	CCONJ
ejpam-6557	175	22	the	the	DET
ejpam-6557	175	23	theoretical	theoretical	ADJ
ejpam-6557	175	24	framework	framework	NOUN
ejpam-6557	175	25	and	and	CCONJ
ejpam-6557	175	26	the	the	DET
ejpam-6557	175	27	potential	potential	ADJ
ejpam-6557	175	28	applicability	applicability	NOUN
ejpam-6557	175	29	of	of	ADP
ejpam-6557	175	30	fractional	fractional	ADJ
ejpam-6557	175	31	complex	complex	ADJ
ejpam-6557	175	32	analysis	analysis	NOUN
ejpam-6557	175	33	.	.	PUNCT
ejpam-6557	176	1	the	the	DET
ejpam-6557	176	2	proposed	propose	VERB
ejpam-6557	176	3	derivative	derivative	NOUN
ejpam-6557	176	4	not	not	PART
ejpam-6557	176	5	only	only	ADV
ejpam-6557	176	6	preserves	preserve	VERB
ejpam-6557	176	7	many	many	ADJ
ejpam-6557	176	8	of	of	ADP
ejpam-6557	176	9	the	the	DET
ejpam-6557	176	10	desirable	desirable	ADJ
ejpam-6557	176	11	properties	property	NOUN
ejpam-6557	176	12	of	of	ADP
ejpam-6557	176	13	classical	classical	ADJ
ejpam-6557	176	14	complex	complex	ADJ
ejpam-6557	176	15	derivatives	derivative	NOUN
ejpam-6557	176	16	but	but	CCONJ
ejpam-6557	176	17	also	also	ADV
ejpam-6557	176	18	adds	add	VERB
ejpam-6557	176	19	new	new	ADJ
ejpam-6557	176	20	degrees	degree	NOUN
ejpam-6557	176	21	of	of	ADP
ejpam-6557	176	22	freedom	freedom	NOUN
ejpam-6557	176	23	for	for	ADP
ejpam-6557	176	24	modeling	model	VERB
ejpam-6557	176	25	phenomena	phenomenon	NOUN
ejpam-6557	176	26	where	where	SCONJ
ejpam-6557	176	27	asymmetry	asymmetry	NOUN
ejpam-6557	176	28	,	,	PUNCT
ejpam-6557	176	29	non	non	ADJ
ejpam-6557	176	30	-	-	ADJ
ejpam-6557	176	31	locality	locality	ADJ
ejpam-6557	176	32	,	,	PUNCT
ejpam-6557	176	33	and	and	CCONJ
ejpam-6557	176	34	memory	memory	NOUN
ejpam-6557	176	35	effects	effect	NOUN
ejpam-6557	176	36	are	be	AUX
ejpam-6557	176	37	present	present	ADJ
ejpam-6557	176	38	.	.	PUNCT
ejpam-6557	177	1	its	its	PRON
ejpam-6557	177	2	polar	polar	ADJ
ejpam-6557	177	3	-	-	PUNCT
ejpam-6557	177	4	coordinate	coordinate	NOUN
ejpam-6557	177	5	formulation	formulation	NOUN
ejpam-6557	177	6	further	far	ADV
ejpam-6557	177	7	enriches	enrich	VERB
ejpam-6557	177	8	the	the	DET
ejpam-6557	177	9	theory	theory	NOUN
ejpam-6557	177	10	by	by	ADP
ejpam-6557	177	11	connecting	connect	VERB
ejpam-6557	177	12	it	it	PRON
ejpam-6557	177	13	naturally	naturally	ADV
ejpam-6557	177	14	to	to	ADP
ejpam-6557	177	15	problems	problem	NOUN
ejpam-6557	177	16	with	with	ADP
ejpam-6557	177	17	radial	radial	ADJ
ejpam-6557	177	18	and	and	CCONJ
ejpam-6557	177	19	angular	angular	ADJ
ejpam-6557	177	20	components	component	NOUN
ejpam-6557	177	21	.	.	PUNCT
ejpam-6557	178	1	looking	look	VERB
ejpam-6557	178	2	ahead	ahead	ADV
ejpam-6557	178	3	,	,	PUNCT
ejpam-6557	178	4	there	there	PRON
ejpam-6557	178	5	are	be	VERB
ejpam-6557	178	6	several	several	ADJ
ejpam-6557	178	7	promising	promising	ADJ
ejpam-6557	178	8	directions	direction	NOUN
ejpam-6557	178	9	for	for	ADP
ejpam-6557	178	10	future	future	ADJ
ejpam-6557	178	11	work	work	NOUN
ejpam-6557	178	12	.	.	PUNCT
ejpam-6557	179	1	analytically	analytically	ADV
ejpam-6557	179	2	,	,	PUNCT
ejpam-6557	179	3	it	it	PRON
ejpam-6557	179	4	would	would	AUX
ejpam-6557	179	5	be	be	AUX
ejpam-6557	179	6	valuable	valuable	ADJ
ejpam-6557	179	7	to	to	PART
ejpam-6557	179	8	explore	explore	VERB
ejpam-6557	179	9	the	the	DET
ejpam-6557	179	10	connections	connection	NOUN
ejpam-6557	179	11	between	between	ADP
ejpam-6557	179	12	(	(	PUNCT
ejpam-6557	179	13	α	α	NOUN
ejpam-6557	179	14	,	,	PUNCT
ejpam-6557	179	15	β)-analyticity	β)-analyticity	NOUN
ejpam-6557	179	16	and	and	CCONJ
ejpam-6557	179	17	other	other	ADJ
ejpam-6557	179	18	generalized	generalized	ADJ
ejpam-6557	179	19	analytic	analytic	ADJ
ejpam-6557	179	20	classes	class	NOUN
ejpam-6557	179	21	,	,	PUNCT
ejpam-6557	179	22	as	as	ADV
ejpam-6557	179	23	well	well	ADV
ejpam-6557	179	24	as	as	ADP
ejpam-6557	179	25	to	to	PART
ejpam-6557	179	26	investigate	investigate	VERB
ejpam-6557	179	27	the	the	DET
ejpam-6557	179	28	spectral	spectral	ADJ
ejpam-6557	179	29	and	and	CCONJ
ejpam-6557	179	30	mapping	mapping	NOUN
ejpam-6557	179	31	properties	property	NOUN
ejpam-6557	179	32	of	of	ADP
ejpam-6557	179	33	operators	operator	NOUN
ejpam-6557	179	34	defined	define	VERB
ejpam-6557	179	35	via	via	ADP
ejpam-6557	179	36	this	this	DET
ejpam-6557	179	37	derivative	derivative	NOUN
ejpam-6557	179	38	.	.	PUNCT
ejpam-6557	180	1	numerically	numerically	ADV
ejpam-6557	180	2	,	,	PUNCT
ejpam-6557	180	3	developing	develop	VERB
ejpam-6557	180	4	efficient	efficient	ADJ
ejpam-6557	180	5	algorithms	algorithm	NOUN
ejpam-6557	180	6	for	for	ADP
ejpam-6557	180	7	computing	compute	VERB
ejpam-6557	180	8	(	(	PUNCT
ejpam-6557	180	9	α	α	NOUN
ejpam-6557	180	10	,	,	PUNCT
ejpam-6557	180	11	β)-derivatives	β)-derivative	NOUN
ejpam-6557	180	12	could	could	AUX
ejpam-6557	180	13	enable	enable	VERB
ejpam-6557	180	14	applications	application	NOUN
ejpam-6557	180	15	in	in	ADP
ejpam-6557	180	16	image	image	NOUN
ejpam-6557	180	17	analysis	analysis	NOUN
ejpam-6557	180	18	,	,	PUNCT
ejpam-6557	180	19	control	control	NOUN
ejpam-6557	180	20	system	system	NOUN
ejpam-6557	180	21	simulation	simulation	NOUN
ejpam-6557	180	22	,	,	PUNCT
ejpam-6557	180	23	and	and	CCONJ
ejpam-6557	180	24	computational	computational	ADJ
ejpam-6557	180	25	physics	physics	NOUN
ejpam-6557	180	26	.	.	PUNCT
ejpam-6557	181	1	from	from	ADP
ejpam-6557	181	2	an	an	DET
ejpam-6557	181	3	applied	applied	ADJ
ejpam-6557	181	4	standpoint	standpoint	NOUN
ejpam-6557	181	5	,	,	PUNCT
ejpam-6557	181	6	the	the	DET
ejpam-6557	181	7	flexibility	flexibility	NOUN
ejpam-6557	181	8	of	of	ADP
ejpam-6557	181	9	the	the	DET
ejpam-6557	181	10	new	new	ADJ
ejpam-6557	181	11	framework	framework	NOUN
ejpam-6557	181	12	could	could	AUX
ejpam-6557	181	13	be	be	AUX
ejpam-6557	181	14	harnessed	harness	VERB
ejpam-6557	181	15	to	to	AUX
ejpam-6557	181	16	model	model	VERB
ejpam-6557	181	17	asymmetric	asymmetric	ADJ
ejpam-6557	181	18	processes	process	NOUN
ejpam-6557	181	19	in	in	ADP
ejpam-6557	181	20	engineering	engineering	NOUN
ejpam-6557	181	21	,	,	PUNCT
ejpam-6557	181	22	physics	physics	NOUN
ejpam-6557	181	23	,	,	PUNCT
ejpam-6557	181	24	and	and	CCONJ
ejpam-6557	181	25	finance	finance	NOUN
ejpam-6557	181	26	,	,	PUNCT
ejpam-6557	181	27	where	where	SCONJ
ejpam-6557	181	28	traditional	traditional	ADJ
ejpam-6557	181	29	integer	integer	NOUN
ejpam-6557	181	30	-	-	PUNCT
ejpam-6557	181	31	order	order	NOUN
ejpam-6557	181	32	calculus	calculus	NOUN
ejpam-6557	181	33	fails	fail	VERB
ejpam-6557	181	34	to	to	PART
ejpam-6557	181	35	capture	capture	VERB
ejpam-6557	181	36	the	the	DET
ejpam-6557	181	37	observed	observed	ADJ
ejpam-6557	181	38	behavior	behavior	NOUN
ejpam-6557	181	39	.	.	PUNCT
ejpam-6557	182	1	in	in	ADP
ejpam-6557	182	2	essence	essence	NOUN
ejpam-6557	182	3	,	,	PUNCT
ejpam-6557	182	4	the	the	DET
ejpam-6557	182	5	(	(	PUNCT
ejpam-6557	182	6	α	α	NOUN
ejpam-6557	182	7	,	,	PUNCT
ejpam-6557	182	8	β)-fractional	β)-fractional	PRON
ejpam-6557	182	9	derivative	derivative	NOUN
ejpam-6557	182	10	provides	provide	VERB
ejpam-6557	182	11	a	a	DET
ejpam-6557	182	12	robust	robust	ADJ
ejpam-6557	182	13	and	and	CCONJ
ejpam-6557	182	14	adaptable	adaptable	ADJ
ejpam-6557	182	15	mathematical	mathematical	ADJ
ejpam-6557	182	16	tool	tool	NOUN
ejpam-6557	182	17	that	that	PRON
ejpam-6557	182	18	has	have	VERB
ejpam-6557	182	19	the	the	DET
ejpam-6557	182	20	potential	potential	NOUN
ejpam-6557	182	21	to	to	PART
ejpam-6557	182	22	deepen	deepen	VERB
ejpam-6557	182	23	our	our	PRON
ejpam-6557	182	24	understanding	understanding	NOUN
ejpam-6557	182	25	of	of	ADP
ejpam-6557	182	26	complex	complex	ADJ
ejpam-6557	182	27	-	-	PUNCT
ejpam-6557	182	28	valued	value	VERB
ejpam-6557	182	29	functions	function	NOUN
ejpam-6557	182	30	and	and	CCONJ
ejpam-6557	182	31	to	to	PART
ejpam-6557	182	32	inspire	inspire	VERB
ejpam-6557	182	33	innovative	innovative	ADJ
ejpam-6557	182	34	solutions	solution	NOUN
ejpam-6557	182	35	across	across	ADP
ejpam-6557	182	36	multiple	multiple	ADJ
ejpam-6557	182	37	disciplines	discipline	NOUN
ejpam-6557	182	38	.	.	PUNCT
ejpam-6557	183	1	acknowledgements	acknowledgement	NOUN
ejpam-6557	183	2	the	the	DET
ejpam-6557	183	3	authors	author	NOUN
ejpam-6557	183	4	would	would	AUX
ejpam-6557	183	5	like	like	VERB
ejpam-6557	183	6	to	to	PART
ejpam-6557	183	7	thank	thank	VERB
ejpam-6557	183	8	the	the	DET
ejpam-6557	183	9	editor	editor	NOUN
ejpam-6557	183	10	of	of	ADP
ejpam-6557	183	11	ejpam	ejpam	NOUN
ejpam-6557	183	12	as	as	ADV
ejpam-6557	183	13	well	well	ADV
ejpam-6557	183	14	as	as	ADP
ejpam-6557	183	15	the	the	DET
ejpam-6557	183	16	anonymous	anonymous	ADJ
ejpam-6557	183	17	reviewers	reviewer	NOUN
ejpam-6557	183	18	for	for	ADP
ejpam-6557	183	19	their	their	PRON
ejpam-6557	183	20	valuable	valuable	ADJ
ejpam-6557	183	21	comments	comment	NOUN
ejpam-6557	183	22	and	and	CCONJ
ejpam-6557	183	23	suggestions	suggestion	NOUN
ejpam-6557	183	24	.	.	PUNCT
ejpam-6557	184	1	w.	w.	PROPN
ejpam-6557	184	2	g.	g.	PROPN
ejpam-6557	184	3	alshanti	alshanti	PROPN
ejpam-6557	184	4	,	,	PUNCT
ejpam-6557	184	5	m.	m.	NOUN
ejpam-6557	184	6	a.	a.	PROPN
ejpam-6557	184	7	hammad	hammad	PROPN
ejpam-6557	184	8	,	,	PUNCT
ejpam-6557	184	9	r.	r.	PROPN
ejpam-6557	184	10	khalil	khalil	PROPN
ejpam-6557	184	11	/	/	SYM
ejpam-6557	184	12	eur	eur	PROPN
ejpam-6557	184	13	.	.	PUNCT
ejpam-6557	185	1	j.	j.	PROPN
ejpam-6557	185	2	pure	pure	PROPN
ejpam-6557	185	3	appl	appl	PROPN
ejpam-6557	185	4	.	.	PROPN
ejpam-6557	185	5	math	math	PROPN
ejpam-6557	185	6	,	,	PUNCT
ejpam-6557	185	7	18	18	NUM
ejpam-6557	185	8	(	(	PUNCT
ejpam-6557	185	9	4	4	NUM
ejpam-6557	185	10	)	)	PUNCT
ejpam-6557	185	11	(	(	PUNCT
ejpam-6557	185	12	2025	2025	NUM
ejpam-6557	185	13	)	)	PUNCT
ejpam-6557	185	14	,	,	PUNCT
ejpam-6557	185	15	6557	6557	NUM
ejpam-6557	185	16	8	8	NUM
ejpam-6557	185	17	of	of	ADP
ejpam-6557	185	18	8	8	NUM
ejpam-6557	185	19	references	reference	NOUN
ejpam-6557	185	20	[	[	X
ejpam-6557	185	21	1	1	NUM
ejpam-6557	185	22	]	]	PUNCT
ejpam-6557	185	23	k.	k.	PROPN
ejpam-6557	185	24	s.	s.	PROPN
ejpam-6557	185	25	miller	miller	PROPN
ejpam-6557	185	26	.	.	PUNCT
ejpam-6557	186	1	an	an	DET
ejpam-6557	186	2	introduction	introduction	NOUN
ejpam-6557	186	3	to	to	ADP
ejpam-6557	186	4	fractional	fractional	ADJ
ejpam-6557	186	5	calculus	calculus	NOUN
ejpam-6557	186	6	and	and	CCONJ
ejpam-6557	186	7	fractional	fractional	ADJ
ejpam-6557	186	8	differential	differential	ADJ
ejpam-6557	186	9	equations	equation	NOUN
ejpam-6557	186	10	.	.	PUNCT
ejpam-6557	187	1	j.	j.	PROPN
ejpam-6557	187	2	wiley	wiley	PROPN
ejpam-6557	187	3	and	and	CCONJ
ejpam-6557	187	4	sons	son	NOUN
ejpam-6557	187	5	,	,	PUNCT
ejpam-6557	187	6	new	new	PROPN
ejpam-6557	187	7	york	york	PROPN
ejpam-6557	187	8	,	,	PUNCT
ejpam-6557	187	9	usa	usa	PROPN
ejpam-6557	187	10	,	,	PUNCT
ejpam-6557	187	11	1993	1993	NUM
ejpam-6557	187	12	.	.	PUNCT
ejpam-6557	188	1	[	[	X
ejpam-6557	188	2	2	2	NUM
ejpam-6557	188	3	]	]	PUNCT
ejpam-6557	188	4	k.	k.	PROPN
ejpam-6557	188	5	oldham	oldham	PROPN
ejpam-6557	188	6	and	and	CCONJ
ejpam-6557	188	7	j.	j.	PROPN
ejpam-6557	188	8	spanier	spanier	PROPN
ejpam-6557	188	9	.	.	PUNCT
ejpam-6557	189	1	the	the	DET
ejpam-6557	189	2	fractional	fractional	ADJ
ejpam-6557	189	3	calculus	calculus	NOUN
ejpam-6557	189	4	:	:	PUNCT
ejpam-6557	189	5	theory	theory	NOUN
ejpam-6557	189	6	and	and	CCONJ
ejpam-6557	189	7	applications	application	NOUN
ejpam-6557	189	8	of	of	ADP
ejpam-6557	189	9	differentiation	differentiation	NOUN
ejpam-6557	189	10	and	and	CCONJ
ejpam-6557	189	11	integration	integration	NOUN
ejpam-6557	189	12	of	of	ADP
ejpam-6557	189	13	arbitrary	arbitrary	ADJ
ejpam-6557	189	14	order	order	NOUN
ejpam-6557	189	15	.	.	PUNCT
ejpam-6557	190	1	elsevier	elsevier	PROPN
ejpam-6557	190	2	,	,	PUNCT
ejpam-6557	190	3	massachusetts	massachusetts	PROPN
ejpam-6557	190	4	,	,	PUNCT
ejpam-6557	190	5	usa	usa	PROPN
ejpam-6557	190	6	,	,	PUNCT
ejpam-6557	190	7	1974	1974	NUM
ejpam-6557	190	8	.	.	PUNCT
ejpam-6557	191	1	[	[	X
ejpam-6557	191	2	3	3	NUM
ejpam-6557	191	3	]	]	PUNCT
ejpam-6557	191	4	a.	a.	NOUN
ejpam-6557	191	5	kilbas	kilbas	PROPN
ejpam-6557	191	6	,	,	PUNCT
ejpam-6557	191	7	h.	h.	PROPN
ejpam-6557	191	8	srivastava	srivastava	PROPN
ejpam-6557	191	9	,	,	PUNCT
ejpam-6557	191	10	and	and	CCONJ
ejpam-6557	191	11	j.	j.	PROPN
ejpam-6557	191	12	trujillo	trujillo	PROPN
ejpam-6557	191	13	.	.	PUNCT
ejpam-6557	191	14	theory	theory	NOUN
ejpam-6557	191	15	and	and	CCONJ
ejpam-6557	191	16	applications	application	NOUN
ejpam-6557	191	17	of	of	ADP
ejpam-6557	191	18	fractional	fractional	ADJ
ejpam-6557	191	19	differential	differential	ADJ
ejpam-6557	191	20	equations	equation	NOUN
ejpam-6557	191	21	.	.	PUNCT
ejpam-6557	192	1	elsevier	elsevier	PROPN
ejpam-6557	192	2	,	,	PUNCT
ejpam-6557	192	3	massachusetts	massachusetts	PROPN
ejpam-6557	192	4	,	,	PUNCT
ejpam-6557	192	5	usa	usa	PROPN
ejpam-6557	192	6	,	,	PUNCT
ejpam-6557	192	7	2006	2006	NUM
ejpam-6557	192	8	.	.	PUNCT
ejpam-6557	193	1	[	[	X
ejpam-6557	193	2	4	4	NUM
ejpam-6557	193	3	]	]	PUNCT
ejpam-6557	193	4	i.	i.	NOUN
ejpam-6557	193	5	podlubny	podlubny	PROPN
ejpam-6557	193	6	.	.	PUNCT
ejpam-6557	194	1	fractional	fractional	ADJ
ejpam-6557	194	2	differential	differential	ADJ
ejpam-6557	194	3	equations	equation	NOUN
ejpam-6557	194	4	.	.	PUNCT
ejpam-6557	195	1	elsevier	elsevier	PROPN
ejpam-6557	195	2	,	,	PUNCT
ejpam-6557	195	3	philadelphia	philadelphia	PROPN
ejpam-6557	195	4	,	,	PUNCT
ejpam-6557	195	5	usa	usa	PROPN
ejpam-6557	195	6	,	,	PUNCT
ejpam-6557	195	7	1999	1999	NUM
ejpam-6557	195	8	.	.	PUNCT
ejpam-6557	196	1	[	[	X
ejpam-6557	196	2	5	5	NUM
ejpam-6557	196	3	]	]	PUNCT
ejpam-6557	196	4	r.	r.	PROPN
ejpam-6557	196	5	khalil	khalil	PROPN
ejpam-6557	196	6	,	,	PUNCT
ejpam-6557	196	7	m.	m.	PROPN
ejpam-6557	196	8	al	al	PROPN
ejpam-6557	196	9	horani	horani	PROPN
ejpam-6557	196	10	,	,	PUNCT
ejpam-6557	196	11	a.	a.	NOUN
ejpam-6557	196	12	yousef	yousef	PROPN
ejpam-6557	196	13	,	,	PUNCT
ejpam-6557	196	14	and	and	CCONJ
ejpam-6557	196	15	m.	m.	NOUN
ejpam-6557	196	16	sababheh	sababheh	NOUN
ejpam-6557	196	17	.	.	PUNCT
ejpam-6557	197	1	a	a	DET
ejpam-6557	197	2	new	new	ADJ
ejpam-6557	197	3	definition	definition	NOUN
ejpam-6557	197	4	of	of	ADP
ejpam-6557	197	5	fractional	fractional	ADJ
ejpam-6557	197	6	derivative	derivative	NOUN
ejpam-6557	197	7	.	.	PUNCT
ejpam-6557	198	1	journal	journal	PROPN
ejpam-6557	198	2	of	of	ADP
ejpam-6557	198	3	computational	computational	ADJ
ejpam-6557	198	4	and	and	CCONJ
ejpam-6557	198	5	applied	applied	ADJ
ejpam-6557	198	6	mathematics	mathematic	NOUN
ejpam-6557	198	7	,	,	PUNCT
ejpam-6557	198	8	264:65–70	264:65–70	NUM
ejpam-6557	198	9	,	,	PUNCT
ejpam-6557	198	10	2014	2014	NUM
ejpam-6557	198	11	.	.	PUNCT
ejpam-6557	199	1	[	[	X
ejpam-6557	199	2	6	6	NUM
ejpam-6557	199	3	]	]	PUNCT
ejpam-6557	199	4	l.	l.	PROPN
ejpam-6557	199	5	campos	campos	PROPN
ejpam-6557	199	6	.	.	PUNCT
ejpam-6557	200	1	on	on	ADP
ejpam-6557	200	2	a	a	DET
ejpam-6557	200	3	concept	concept	NOUN
ejpam-6557	200	4	of	of	ADP
ejpam-6557	200	5	derivative	derivative	NOUN
ejpam-6557	200	6	of	of	ADP
ejpam-6557	200	7	complex	complex	ADJ
ejpam-6557	200	8	order	order	NOUN
ejpam-6557	200	9	with	with	ADP
ejpam-6557	200	10	applications	application	NOUN
ejpam-6557	200	11	to	to	ADP
ejpam-6557	200	12	special	special	ADJ
ejpam-6557	200	13	functions	function	NOUN
ejpam-6557	200	14	.	.	PUNCT
ejpam-6557	201	1	i	i	PRON
ejpam-6557	201	2	m	m	VERB
ejpam-6557	201	3	a	a	DET
ejpam-6557	201	4	journal	journal	NOUN
ejpam-6557	201	5	of	of	ADP
ejpam-6557	201	6	applied	apply	VERB
ejpam-6557	201	7	mathematics	mathematic	NOUN
ejpam-6557	201	8	,	,	PUNCT
ejpam-6557	201	9	33(2):109–133	33(2):109–133	PROPN
ejpam-6557	201	10	,	,	PUNCT
ejpam-6557	201	11	1984	1984	NUM
ejpam-6557	201	12	.	.	PUNCT
ejpam-6557	202	1	[	[	X
ejpam-6557	202	2	7	7	X
ejpam-6557	202	3	]	]	X
ejpam-6557	202	4	e.	e.	PROPN
ejpam-6557	202	5	love	love	PROPN
ejpam-6557	202	6	.	.	PUNCT
ejpam-6557	203	1	fractional	fractional	ADJ
ejpam-6557	203	2	derivatives	derivative	NOUN
ejpam-6557	203	3	of	of	ADP
ejpam-6557	203	4	imaginary	imaginary	ADJ
ejpam-6557	203	5	order	order	NOUN
ejpam-6557	203	6	.	.	PUNCT
ejpam-6557	204	1	journal	journal	NOUN
ejpam-6557	204	2	of	of	ADP
ejpam-6557	204	3	the	the	DET
ejpam-6557	204	4	london	london	PROPN
ejpam-6557	204	5	mathematical	mathematical	ADJ
ejpam-6557	204	6	society	society	NOUN
ejpam-6557	204	7	,	,	PUNCT
ejpam-6557	204	8	2(2):241–259	2(2):241–259	NOUN
ejpam-6557	204	9	,	,	PUNCT
ejpam-6557	204	10	1972	1972	NUM
ejpam-6557	204	11	.	.	PUNCT
ejpam-6557	205	1	[	[	X
ejpam-6557	205	2	8	8	NUM
ejpam-6557	205	3	]	]	PUNCT
ejpam-6557	205	4	m.	m.	NOUN
ejpam-6557	205	5	adm	adm	PROPN
ejpam-6557	205	6	and	and	CCONJ
ejpam-6557	205	7	r.	r.	PROPN
ejpam-6557	205	8	khalil	khalil	PROPN
ejpam-6557	205	9	.	.	PUNCT
ejpam-6557	206	1	new	new	ADJ
ejpam-6557	206	2	definition	definition	NOUN
ejpam-6557	206	3	of	of	ADP
ejpam-6557	206	4	fractional	fractional	ADJ
ejpam-6557	206	5	analytic	analytic	ADJ
ejpam-6557	206	6	functions	function	NOUN
ejpam-6557	206	7	.	.	PUNCT
ejpam-6557	207	1	missouri	missouri	PROPN
ejpam-6557	207	2	journal	journal	PROPN
ejpam-6557	207	3	of	of	ADP
ejpam-6557	207	4	mathematical	mathematical	ADJ
ejpam-6557	207	5	sciences	science	NOUN
ejpam-6557	207	6	,	,	PUNCT
ejpam-6557	207	7	35(2):194–209	35(2):194–209	NOUN
ejpam-6557	207	8	,	,	PUNCT
ejpam-6557	207	9	2023	2023	NUM
ejpam-6557	207	10	.	.	PUNCT
ejpam-6557	208	1	[	[	X
ejpam-6557	208	2	9	9	NUM
ejpam-6557	208	3	]	]	PUNCT
ejpam-6557	208	4	i.	i.	NOUN
ejpam-6557	208	5	batiha	batiha	PROPN
ejpam-6557	208	6	,	,	PUNCT
ejpam-6557	208	7	s.	s.	PROPN
ejpam-6557	208	8	njadat	njadat	PROPN
ejpam-6557	208	9	,	,	PUNCT
ejpam-6557	208	10	r.	r.	PROPN
ejpam-6557	208	11	batyha	batyha	PROPN
ejpam-6557	208	12	,	,	PUNCT
ejpam-6557	208	13	a.	a.	NOUN
ejpam-6557	208	14	zraiqat	zraiqat	PROPN
ejpam-6557	208	15	,	,	PUNCT
ejpam-6557	208	16	a.	a.	NOUN
ejpam-6557	208	17	dababneh	dababneh	PROPN
ejpam-6557	208	18	,	,	PUNCT
ejpam-6557	208	19	and	and	CCONJ
ejpam-6557	208	20	s.	s.	PROPN
ejpam-6557	208	21	momani	momani	PROPN
ejpam-6557	208	22	.	.	PUNCT
ejpam-6557	209	1	design	design	NOUN
ejpam-6557	209	2	fractional	fractional	ADJ
ejpam-6557	209	3	-	-	PUNCT
ejpam-6557	209	4	order	order	NOUN
ejpam-6557	209	5	pid	pid	NOUN
ejpam-6557	209	6	controllers	controller	NOUN
ejpam-6557	209	7	for	for	ADP
ejpam-6557	209	8	single	single	ADJ
ejpam-6557	209	9	-	-	PUNCT
ejpam-6557	209	10	joint	joint	ADJ
ejpam-6557	209	11	robot	robot	NOUN
ejpam-6557	209	12	arm	arm	NOUN
ejpam-6557	209	13	model	model	NOUN
ejpam-6557	209	14	.	.	PUNCT
ejpam-6557	210	1	international	international	ADJ
ejpam-6557	210	2	journal	journal	NOUN
ejpam-6557	210	3	of	of	ADP
ejpam-6557	210	4	advances	advance	NOUN
ejpam-6557	210	5	in	in	ADP
ejpam-6557	210	6	soft	soft	ADJ
ejpam-6557	210	7	computing	computing	NOUN
ejpam-6557	210	8	and	and	CCONJ
ejpam-6557	210	9	its	its	PRON
ejpam-6557	210	10	applications	application	NOUN
ejpam-6557	210	11	,	,	PUNCT
ejpam-6557	210	12	14(2):241–259	14(2):241–259	NUM
ejpam-6557	210	13	,	,	PUNCT
ejpam-6557	210	14	2022	2022	NUM
ejpam-6557	210	15	.	.	PUNCT
ejpam-6557	211	1	[	[	X
ejpam-6557	211	2	10	10	NUM
ejpam-6557	211	3	]	]	X
ejpam-6557	211	4	i.	i.	NOUN
ejpam-6557	211	5	batiha	batiha	PROPN
ejpam-6557	211	6	,	,	PUNCT
ejpam-6557	211	7	j.	j.	PROPN
ejpam-6557	211	8	oudetallah	oudetallah	PROPN
ejpam-6557	211	9	,	,	PUNCT
ejpam-6557	211	10	a.	a.	NOUN
ejpam-6557	211	11	ouannas	ouannas	PROPN
ejpam-6557	211	12	,	,	PUNCT
ejpam-6557	211	13	a.	a.	PROPN
ejpam-6557	211	14	al	al	PROPN
ejpam-6557	211	15	-	-	PUNCT
ejpam-6557	211	16	nana	nana	PROPN
ejpam-6557	211	17	,	,	PUNCT
ejpam-6557	211	18	and	and	CCONJ
ejpam-6557	211	19	i.	i.	PROPN
ejpam-6557	211	20	jebril	jebril	NOUN
ejpam-6557	211	21	.	.	PUNCT
ejpam-6557	212	1	tuning	tune	VERB
ejpam-6557	212	2	the	the	DET
ejpam-6557	212	3	fractionalorder	fractionalorder	ADJ
ejpam-6557	212	4	pid	pid	NOUN
ejpam-6557	212	5	-	-	NOUN
ejpam-6557	212	6	controller	controller	NOUN
ejpam-6557	212	7	for	for	ADP
ejpam-6557	212	8	blood	blood	NOUN
ejpam-6557	212	9	glucose	glucose	NOUN
ejpam-6557	212	10	level	level	NOUN
ejpam-6557	212	11	of	of	ADP
ejpam-6557	212	12	diabetic	diabetic	ADJ
ejpam-6557	212	13	patients	patient	NOUN
ejpam-6557	212	14	.	.	PUNCT
ejpam-6557	213	1	international	international	ADJ
ejpam-6557	213	2	journal	journal	NOUN
ejpam-6557	213	3	of	of	ADP
ejpam-6557	213	4	advances	advance	NOUN
ejpam-6557	213	5	in	in	ADP
ejpam-6557	213	6	soft	soft	ADJ
ejpam-6557	213	7	computing	computing	NOUN
ejpam-6557	213	8	and	and	CCONJ
ejpam-6557	213	9	its	its	PRON
ejpam-6557	213	10	applications	application	NOUN
ejpam-6557	213	11	,	,	PUNCT
ejpam-6557	213	12	13(2):1–10	13(2):1–10	NOUN
ejpam-6557	213	13	,	,	PUNCT
ejpam-6557	213	14	2021	2021	NUM
ejpam-6557	213	15	.	.	PUNCT