id	sid	tid	token	lemma	pos
ejpam-5411	1	1	european	european	PROPN
ejpam-5411	1	2	journal	journal	PROPN
ejpam-5411	1	3	of	of	ADP
ejpam-5411	1	4	pure	pure	ADJ
ejpam-5411	1	5	and	and	CCONJ
ejpam-5411	1	6	applied	apply	VERB
ejpam-5411	1	7	mathematics	mathematic	NOUN
ejpam-5411	1	8	vol	vol	NOUN
ejpam-5411	1	9	.	.	PROPN
ejpam-5411	2	1	17	17	NUM
ejpam-5411	2	2	,	,	PUNCT
ejpam-5411	2	3	no	no	INTJ
ejpam-5411	2	4	.	.	NOUN
ejpam-5411	2	5	4	4	NUM
ejpam-5411	2	6	,	,	PUNCT
ejpam-5411	2	7	2024	2024	NUM
ejpam-5411	2	8	,	,	PUNCT
ejpam-5411	2	9	2405	2405	NUM
ejpam-5411	2	10	-	-	SYM
ejpam-5411	2	11	2430	2430	NUM
ejpam-5411	2	12	issn	issn	PROPN
ejpam-5411	2	13	1307	1307	NUM
ejpam-5411	2	14	-	-	SYM
ejpam-5411	2	15	5543	5543	NUM
ejpam-5411	2	16	–	–	PUNCT
ejpam-5411	2	17	ejpam.com	ejpam.com	X
ejpam-5411	2	18	published	publish	VERB
ejpam-5411	2	19	by	by	ADP
ejpam-5411	2	20	new	new	PROPN
ejpam-5411	2	21	york	york	PROPN
ejpam-5411	2	22	business	business	PROPN
ejpam-5411	2	23	global	global	PROPN
ejpam-5411	2	24	some	some	DET
ejpam-5411	2	25	results	result	NOUN
ejpam-5411	2	26	of	of	ADP
ejpam-5411	2	27	conformable	conformable	ADJ
ejpam-5411	2	28	fourier	fourier	NOUN
ejpam-5411	2	29	transform	transform	NOUN
ejpam-5411	2	30	bahloul	bahloul	PROPN
ejpam-5411	2	31	rachid1	rachid1	PROPN
ejpam-5411	2	32	,	,	PUNCT
ejpam-5411	2	33	,	,	PUNCT
ejpam-5411	2	34	rechdaoui	rechdaoui	ADV
ejpam-5411	2	35	my	my	PRON
ejpam-5411	2	36	soufiane2	soufiane2	NOUN
ejpam-5411	2	37	,	,	PUNCT
ejpam-5411	2	38	,	,	PUNCT
ejpam-5411	2	39	thabet	thabet	ADJ
ejpam-5411	2	40	abdeljawad3,4,5,6,∗	abdeljawad3,4,5,6,∗	PROPN
ejpam-5411	2	41	,	,	PUNCT
ejpam-5411	2	42	bahaaeldin	bahaaeldin	VERB
ejpam-5411	2	43	abdalla3	abdalla3	PROPN
ejpam-5411	2	44	1	1	NUM
ejpam-5411	2	45	limati	limati	NOUN
ejpam-5411	2	46	laboratory	laboratory	NOUN
ejpam-5411	2	47	,	,	PUNCT
ejpam-5411	2	48	department	department	NOUN
ejpam-5411	2	49	of	of	ADP
ejpam-5411	2	50	mathematics	mathematic	NOUN
ejpam-5411	2	51	,	,	PUNCT
ejpam-5411	2	52	polydisciplinary	polydisciplinary	ADJ
ejpam-5411	2	53	faculty	faculty	NOUN
ejpam-5411	2	54	,	,	PUNCT
ejpam-5411	2	55	sultan	sultan	PROPN
ejpam-5411	2	56	moulay	moulay	PROPN
ejpam-5411	2	57	slimane	slimane	PROPN
ejpam-5411	2	58	university	university	PROPN
ejpam-5411	2	59	,	,	PUNCT
ejpam-5411	2	60	beni	beni	ADJ
ejpam-5411	2	61	mellal	mellal	PROPN
ejpam-5411	2	62	,	,	PUNCT
ejpam-5411	2	63	morocco	morocco	PROPN
ejpam-5411	2	64	2	2	NUM
ejpam-5411	2	65	mias	mias	PROPN
ejpam-5411	2	66	laboratory	laboratory	NOUN
ejpam-5411	2	67	,	,	PUNCT
ejpam-5411	2	68	mamcs	mamcs	NOUN
ejpam-5411	2	69	team	team	NOUN
ejpam-5411	2	70	,	,	PUNCT
ejpam-5411	2	71	higher	high	ADJ
ejpam-5411	2	72	school	school	NOUN
ejpam-5411	2	73	of	of	ADP
ejpam-5411	2	74	technology	technology	NOUN
ejpam-5411	2	75	,	,	PUNCT
ejpam-5411	2	76	my	my	PRON
ejpam-5411	2	77	ismail	ismail	NOUN
ejpam-5411	2	78	university	university	PROPN
ejpam-5411	2	79	,	,	PUNCT
ejpam-5411	2	80	meknes	meknes	PROPN
ejpam-5411	2	81	,	,	PUNCT
ejpam-5411	2	82	morocco	morocco	PROPN
ejpam-5411	2	83	3	3	NUM
ejpam-5411	2	84	department	department	NOUN
ejpam-5411	2	85	of	of	ADP
ejpam-5411	2	86	mathematics	mathematic	NOUN
ejpam-5411	2	87	and	and	CCONJ
ejpam-5411	2	88	sciences	science	NOUN
ejpam-5411	2	89	,	,	PUNCT
ejpam-5411	2	90	prince	prince	PROPN
ejpam-5411	2	91	sultan	sultan	PROPN
ejpam-5411	2	92	university	university	PROPN
ejpam-5411	2	93	,	,	PUNCT
ejpam-5411	2	94	riyadh	riyadh	PROPN
ejpam-5411	2	95	11586	11586	NUM
ejpam-5411	2	96	,	,	PUNCT
ejpam-5411	2	97	saudi	saudi	PROPN
ejpam-5411	2	98	arabia	arabia	PROPN
ejpam-5411	2	99	4	4	NUM
ejpam-5411	2	100	department	department	NOUN
ejpam-5411	2	101	of	of	ADP
ejpam-5411	2	102	medical	medical	ADJ
ejpam-5411	2	103	research	research	NOUN
ejpam-5411	2	104	,	,	PUNCT
ejpam-5411	2	105	china	china	PROPN
ejpam-5411	2	106	medical	medical	PROPN
ejpam-5411	2	107	university	university	PROPN
ejpam-5411	2	108	,	,	PUNCT
ejpam-5411	2	109	taichung	taichung	PROPN
ejpam-5411	2	110	40402	40402	NUM
ejpam-5411	2	111	,	,	PUNCT
ejpam-5411	2	112	taiwan	taiwan	PROPN
ejpam-5411	2	113	5	5	NUM
ejpam-5411	2	114	department	department	NOUN
ejpam-5411	2	115	of	of	ADP
ejpam-5411	2	116	mathematics	mathematic	NOUN
ejpam-5411	2	117	and	and	CCONJ
ejpam-5411	2	118	applied	apply	VERB
ejpam-5411	2	119	mathematics	mathematic	NOUN
ejpam-5411	2	120	,	,	PUNCT
ejpam-5411	2	121	school	school	NOUN
ejpam-5411	2	122	of	of	ADP
ejpam-5411	2	123	science	science	NOUN
ejpam-5411	2	124	and	and	CCONJ
ejpam-5411	2	125	technology	technology	NOUN
ejpam-5411	2	126	,	,	PUNCT
ejpam-5411	2	127	sefako	sefako	PROPN
ejpam-5411	2	128	makagatho	makagatho	PROPN
ejpam-5411	2	129	health	health	PROPN
ejpam-5411	2	130	sciences	sciences	PROPN
ejpam-5411	2	131	university	university	PROPN
ejpam-5411	2	132	,	,	PUNCT
ejpam-5411	2	133	ga	ga	PROPN
ejpam-5411	2	134	-	-	PROPN
ejpam-5411	2	135	rankuwa	rankuwa	NOUN
ejpam-5411	2	136	0208	0208	NUM
ejpam-5411	2	137	,	,	PUNCT
ejpam-5411	2	138	south	south	PROPN
ejpam-5411	2	139	africa	africa	PROPN
ejpam-5411	2	140	6	6	NUM
ejpam-5411	2	141	center	center	NOUN
ejpam-5411	2	142	for	for	ADP
ejpam-5411	2	143	applied	applied	ADJ
ejpam-5411	2	144	mathematics	mathematic	NOUN
ejpam-5411	2	145	and	and	CCONJ
ejpam-5411	2	146	bioinformatics	bioinformatics	NOUN
ejpam-5411	2	147	(	(	PUNCT
ejpam-5411	2	148	camb	camb	PROPN
ejpam-5411	2	149	)	)	PUNCT
ejpam-5411	2	150	,	,	PUNCT
ejpam-5411	2	151	gulf	gulf	PROPN
ejpam-5411	2	152	university	university	PROPN
ejpam-5411	2	153	for	for	ADP
ejpam-5411	2	154	science	science	NOUN
ejpam-5411	2	155	and	and	CCONJ
ejpam-5411	2	156	technology	technology	NOUN
ejpam-5411	2	157	,	,	PUNCT
ejpam-5411	2	158	hawally	hawally	ADV
ejpam-5411	2	159	,	,	PUNCT
ejpam-5411	2	160	32093	32093	NUM
ejpam-5411	2	161	,	,	PUNCT
ejpam-5411	2	162	kuwait	kuwait	PROPN
ejpam-5411	2	163	abstract	abstract	NOUN
ejpam-5411	2	164	.	.	PUNCT
ejpam-5411	3	1	based	base	VERB
ejpam-5411	3	2	on	on	ADP
ejpam-5411	3	3	a	a	DET
ejpam-5411	3	4	new	new	ADJ
ejpam-5411	3	5	definition	definition	NOUN
ejpam-5411	3	6	of	of	ADP
ejpam-5411	3	7	α	α	NOUN
ejpam-5411	3	8	-	-	PUNCT
ejpam-5411	3	9	periodicals	periodical	NOUN
ejpam-5411	3	10	functions	function	NOUN
ejpam-5411	3	11	with	with	ADP
ejpam-5411	3	12	0	0	NUM
ejpam-5411	3	13	<	<	X
ejpam-5411	3	14	α	α	PROPN
ejpam-5411	3	15	≤	≤	ADV
ejpam-5411	3	16	1	1	NUM
ejpam-5411	3	17	introduced	introduce	VERB
ejpam-5411	3	18	by	by	ADP
ejpam-5411	3	19	khalil	khalil	PROPN
ejpam-5411	3	20	et	et	PROPN
ejpam-5411	3	21	al	al	PROPN
ejpam-5411	3	22	(	(	PUNCT
ejpam-5411	3	23	2014	2014	NUM
ejpam-5411	3	24	)	)	PUNCT
ejpam-5411	3	25	,	,	PUNCT
ejpam-5411	3	26	we	we	PRON
ejpam-5411	3	27	introduce	introduce	VERB
ejpam-5411	3	28	a	a	DET
ejpam-5411	3	29	new	new	ADJ
ejpam-5411	3	30	definition	definition	NOUN
ejpam-5411	3	31	of	of	ADP
ejpam-5411	3	32	conformable	conformable	ADJ
ejpam-5411	3	33	fourier	fourier	NOUN
ejpam-5411	3	34	transform	transform	NOUN
ejpam-5411	3	35	for	for	ADP
ejpam-5411	3	36	such	such	DET
ejpam-5411	3	37	a	a	DET
ejpam-5411	3	38	class	class	NOUN
ejpam-5411	3	39	of	of	ADP
ejpam-5411	3	40	functions	function	NOUN
ejpam-5411	3	41	.	.	PUNCT
ejpam-5411	4	1	further	far	ADV
ejpam-5411	4	2	,	,	PUNCT
ejpam-5411	4	3	we	we	PRON
ejpam-5411	4	4	establish	establish	VERB
ejpam-5411	4	5	some	some	DET
ejpam-5411	4	6	operational	operational	ADJ
ejpam-5411	4	7	formulas	formula	NOUN
ejpam-5411	4	8	,	,	PUNCT
ejpam-5411	4	9	and	and	CCONJ
ejpam-5411	4	10	we	we	PRON
ejpam-5411	4	11	set	set	VERB
ejpam-5411	4	12	the	the	DET
ejpam-5411	4	13	relation	relation	NOUN
ejpam-5411	4	14	between	between	ADP
ejpam-5411	4	15	the	the	DET
ejpam-5411	4	16	newly	newly	ADV
ejpam-5411	4	17	defined	define	VERB
ejpam-5411	4	18	conformable	conformable	ADJ
ejpam-5411	4	19	fourier	fourier	NOUN
ejpam-5411	4	20	transform	transform	NOUN
ejpam-5411	4	21	and	and	CCONJ
ejpam-5411	4	22	the	the	DET
ejpam-5411	4	23	classical	classical	ADJ
ejpam-5411	4	24	fourier	fourier	NOUN
ejpam-5411	4	25	transform	transform	NOUN
ejpam-5411	4	26	.	.	PUNCT
ejpam-5411	5	1	finally	finally	ADV
ejpam-5411	5	2	,	,	PUNCT
ejpam-5411	5	3	some	some	DET
ejpam-5411	5	4	classical	classical	ADJ
ejpam-5411	5	5	results	result	NOUN
ejpam-5411	5	6	of	of	ADP
ejpam-5411	5	7	periodical	periodical	ADJ
ejpam-5411	5	8	functions	function	NOUN
ejpam-5411	5	9	are	be	AUX
ejpam-5411	5	10	obtained	obtain	VERB
ejpam-5411	5	11	and	and	CCONJ
ejpam-5411	5	12	some	some	DET
ejpam-5411	5	13	illustrative	illustrative	ADJ
ejpam-5411	5	14	examples	example	NOUN
ejpam-5411	5	15	are	be	AUX
ejpam-5411	5	16	constructed	construct	VERB
ejpam-5411	5	17	.	.	PUNCT
ejpam-5411	6	1	2020	2020	NUM
ejpam-5411	6	2	mathematics	mathematics	PROPN
ejpam-5411	6	3	subject	subject	NOUN
ejpam-5411	6	4	classifications	classification	NOUN
ejpam-5411	6	5	:	:	PUNCT
ejpam-5411	6	6	45n05	45n05	NUM
ejpam-5411	6	7	,	,	PUNCT
ejpam-5411	6	8	44a10	44a10	NUM
ejpam-5411	6	9	,	,	PUNCT
ejpam-5411	6	10	43a15	43a15	NUM
ejpam-5411	6	11	,	,	PUNCT
ejpam-5411	6	12	44a35	44a35	NUM
ejpam-5411	6	13	,	,	PUNCT
ejpam-5411	6	14	43a50	43a50	NUM
ejpam-5411	6	15	,	,	PUNCT
ejpam-5411	6	16	45d05	45d05	NUM
ejpam-5411	6	17	key	key	ADJ
ejpam-5411	6	18	words	word	NOUN
ejpam-5411	6	19	and	and	CCONJ
ejpam-5411	6	20	phrases	phrase	NOUN
ejpam-5411	6	21	:	:	PUNCT
ejpam-5411	6	22	α	α	X
ejpam-5411	6	23	-	-	ADJ
ejpam-5411	6	24	periodic	periodic	ADJ
ejpam-5411	6	25	function	function	NOUN
ejpam-5411	6	26	,	,	PUNCT
ejpam-5411	6	27	conformable	conformable	ADJ
ejpam-5411	6	28	derivative	derivative	ADJ
ejpam-5411	6	29	,	,	PUNCT
ejpam-5411	6	30	conformable	conformable	ADJ
ejpam-5411	6	31	fourier	fourier	NOUN
ejpam-5411	6	32	transform	transform	NOUN
ejpam-5411	6	33	,	,	PUNCT
ejpam-5411	6	34	conformable	conformable	ADJ
ejpam-5411	6	35	fractional	fractional	ADJ
ejpam-5411	6	36	integral	integral	ADJ
ejpam-5411	6	37	1	1	NUM
ejpam-5411	6	38	.	.	PUNCT
ejpam-5411	6	39	introduction	introduction	NOUN
ejpam-5411	6	40	the	the	DET
ejpam-5411	6	41	fractional	fractional	ADJ
ejpam-5411	6	42	calculus	calculus	NOUN
ejpam-5411	6	43	[	[	X
ejpam-5411	6	44	11	11	NUM
ejpam-5411	6	45	,	,	PUNCT
ejpam-5411	6	46	14	14	NUM
ejpam-5411	6	47	,	,	PUNCT
ejpam-5411	6	48	17	17	NUM
ejpam-5411	6	49	]	]	PUNCT
ejpam-5411	6	50	attracted	attract	VERB
ejpam-5411	6	51	many	many	ADJ
ejpam-5411	6	52	researches	research	NOUN
ejpam-5411	6	53	in	in	ADP
ejpam-5411	6	54	the	the	DET
ejpam-5411	6	55	last	last	ADJ
ejpam-5411	6	56	and	and	CCONJ
ejpam-5411	6	57	present	present	ADJ
ejpam-5411	6	58	centuries	century	NOUN
ejpam-5411	6	59	.	.	PUNCT
ejpam-5411	7	1	the	the	DET
ejpam-5411	7	2	impact	impact	NOUN
ejpam-5411	7	3	of	of	ADP
ejpam-5411	7	4	this	this	DET
ejpam-5411	7	5	fractional	fractional	ADJ
ejpam-5411	7	6	calculus	calculus	NOUN
ejpam-5411	7	7	in	in	ADP
ejpam-5411	7	8	both	both	CCONJ
ejpam-5411	7	9	pure	pure	ADJ
ejpam-5411	7	10	and	and	CCONJ
ejpam-5411	7	11	applied	applied	ADJ
ejpam-5411	7	12	branches	branch	NOUN
ejpam-5411	7	13	of	of	ADP
ejpam-5411	7	14	science	science	NOUN
ejpam-5411	7	15	and	and	CCONJ
ejpam-5411	7	16	engineering	engineering	NOUN
ejpam-5411	7	17	started	start	VERB
ejpam-5411	7	18	to	to	PART
ejpam-5411	7	19	increase	increase	VERB
ejpam-5411	7	20	substantially	substantially	ADV
ejpam-5411	7	21	during	during	ADP
ejpam-5411	7	22	the	the	DET
ejpam-5411	7	23	last	last	ADJ
ejpam-5411	7	24	two	two	NUM
ejpam-5411	7	25	decades	decade	NOUN
ejpam-5411	7	26	apparently	apparently	ADV
ejpam-5411	7	27	.	.	PUNCT
ejpam-5411	8	1	∗corresponding	∗corresponde	VERB
ejpam-5411	8	2	author	author	NOUN
ejpam-5411	8	3	.	.	PUNCT
ejpam-5411	9	1	doi	doi	NOUN
ejpam-5411	9	2	:	:	PUNCT
ejpam-5411	9	3	https://doi.org/10.29020/nybg.ejpam.v17i4.5411	https://doi.org/10.29020/nybg.ejpam.v17i4.5411	ADJ
ejpam-5411	9	4	email	email	NOUN
ejpam-5411	9	5	addresses	address	VERB
ejpam-5411	9	6	:	:	PUNCT
ejpam-5411	9	7	bahloulrachid363@gmail.com	bahloulrachid363@gmail.com	X
ejpam-5411	9	8	(	(	PUNCT
ejpam-5411	9	9	r.	r.	PROPN
ejpam-5411	9	10	bahloul	bahloul	PROPN
ejpam-5411	9	11	)	)	PUNCT
ejpam-5411	9	12	,	,	PUNCT
ejpam-5411	9	13	m.rechdaoui@umi.ac.ma	m.rechdaoui@umi.ac.ma	X
ejpam-5411	9	14	(	(	PUNCT
ejpam-5411	9	15	m.	m.	NOUN
ejpam-5411	9	16	s.	s.	PROPN
ejpam-5411	9	17	rechdaoui	rechdaoui	PROPN
ejpam-5411	9	18	)	)	PUNCT
ejpam-5411	9	19	,	,	PUNCT
ejpam-5411	9	20	tabdeljawad@psu.edu.sa	tabdeljawad@psu.edu.sa	PROPN
ejpam-5411	9	21	(	(	PUNCT
ejpam-5411	9	22	t.	t.	NOUN
ejpam-5411	9	23	abdeljawad	abdeljawad	PROPN
ejpam-5411	9	24	)	)	PUNCT
ejpam-5411	9	25	,	,	PUNCT
ejpam-5411	9	26	babdallah@psu.edu.sa	babdallah@psu.edu.sa	PROPN
ejpam-5411	9	27	(	(	PUNCT
ejpam-5411	9	28	b.	b.	PROPN
ejpam-5411	9	29	abdalla	abdalla	PROPN
ejpam-5411	9	30	)	)	PUNCT
ejpam-5411	9	31	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5411	9	32	2405	2405	NUM
ejpam-5411	10	1	copyright	copyright	NOUN
ejpam-5411	10	2	:	:	PUNCT
ejpam-5411	10	3	©	©	PROPN
ejpam-5411	10	4	2024	2024	NUM
ejpam-5411	10	5	the	the	DET
ejpam-5411	10	6	author(s	author(s	NOUN
ejpam-5411	10	7	)	)	PUNCT
ejpam-5411	10	8	.	.	PUNCT
ejpam-5411	11	1	(	(	PUNCT
ejpam-5411	11	2	cc	cc	NOUN
ejpam-5411	11	3	by	by	ADP
ejpam-5411	11	4	-	-	PUNCT
ejpam-5411	11	5	nc	nc	PROPN
ejpam-5411	11	6	4.0	4.0	NUM
ejpam-5411	11	7	)	)	PUNCT
ejpam-5411	11	8	t.	t.	NOUN
ejpam-5411	11	9	abdeljawad	abdeljawad	NOUN
ejpam-5411	11	10	et	et	PROPN
ejpam-5411	11	11	al	al	PROPN
ejpam-5411	11	12	.	.	PUNCT
ejpam-5411	11	13	/	/	SYM
ejpam-5411	11	14	eur	eur	PROPN
ejpam-5411	11	15	.	.	PUNCT
ejpam-5411	12	1	j.	j.	PROPN
ejpam-5411	12	2	pure	pure	PROPN
ejpam-5411	12	3	appl	appl	PROPN
ejpam-5411	12	4	.	.	PROPN
ejpam-5411	12	5	math	math	PROPN
ejpam-5411	12	6	,	,	PUNCT
ejpam-5411	12	7	17	17	NUM
ejpam-5411	12	8	(	(	PUNCT
ejpam-5411	12	9	4	4	NUM
ejpam-5411	12	10	)	)	PUNCT
ejpam-5411	12	11	(	(	PUNCT
ejpam-5411	12	12	2024	2024	NUM
ejpam-5411	12	13	)	)	PUNCT
ejpam-5411	12	14	,	,	PUNCT
ejpam-5411	12	15	2405	2405	NUM
ejpam-5411	12	16	-	-	SYM
ejpam-5411	12	17	2430	2430	NUM
ejpam-5411	12	18	2406	2406	NUM
ejpam-5411	12	19	traditionally	traditionally	ADV
ejpam-5411	12	20	,	,	PUNCT
ejpam-5411	12	21	the	the	DET
ejpam-5411	12	22	arbitrary	arbitrary	ADJ
ejpam-5411	12	23	order	order	NOUN
ejpam-5411	12	24	of	of	ADP
ejpam-5411	12	25	integration	integration	NOUN
ejpam-5411	12	26	and	and	CCONJ
ejpam-5411	12	27	differentiation	differentiation	NOUN
ejpam-5411	12	28	has	have	AUX
ejpam-5411	12	29	been	be	AUX
ejpam-5411	12	30	described	describe	VERB
ejpam-5411	12	31	by	by	ADP
ejpam-5411	12	32	nonlocal	nonlocal	ADJ
ejpam-5411	12	33	fractional	fractional	ADJ
ejpam-5411	12	34	operators	operator	NOUN
ejpam-5411	12	35	with	with	ADP
ejpam-5411	12	36	kernels	kernel	NOUN
ejpam-5411	12	37	reflecting	reflect	VERB
ejpam-5411	12	38	their	their	PRON
ejpam-5411	12	39	memories	memory	NOUN
ejpam-5411	12	40	.	.	PUNCT
ejpam-5411	13	1	recently	recently	ADV
ejpam-5411	13	2	,	,	PUNCT
ejpam-5411	13	3	the	the	DET
ejpam-5411	13	4	conformable	conformable	ADJ
ejpam-5411	13	5	derivative	derivative	ADJ
ejpam-5411	13	6	operator	operator	NOUN
ejpam-5411	13	7	t	t	PROPN
ejpam-5411	13	8	(	(	PUNCT
ejpam-5411	13	9	α)(x)(t	α)(x)(t	PROPN
ejpam-5411	13	10	)	)	PUNCT
ejpam-5411	13	11	=	=	SYM
ejpam-5411	13	12	limh→0	limh→0	PROPN
ejpam-5411	13	13	f(t+ht1−α)−f(t	f(t+ht1−α)−f(t	NUM
ejpam-5411	13	14	)	)	PUNCT
ejpam-5411	13	15	h	h	NOUN
ejpam-5411	13	16	was	be	AUX
ejpam-5411	13	17	introduced	introduce	VERB
ejpam-5411	13	18	in	in	ADP
ejpam-5411	13	19	the	the	DET
ejpam-5411	13	20	literature	literature	NOUN
ejpam-5411	13	21	by	by	ADP
ejpam-5411	13	22	khalil	khalil	PROPN
ejpam-5411	14	1	[	[	X
ejpam-5411	14	2	9	9	NUM
ejpam-5411	14	3	]	]	PUNCT
ejpam-5411	14	4	to	to	PART
ejpam-5411	14	5	allow	allow	VERB
ejpam-5411	14	6	integrating	integrating	NOUN
ejpam-5411	14	7	and	and	CCONJ
ejpam-5411	14	8	differentiating	differentiate	VERB
ejpam-5411	14	9	with	with	ADP
ejpam-5411	14	10	respect	respect	NOUN
ejpam-5411	14	11	to	to	ADP
ejpam-5411	14	12	arbitrary	arbitrary	ADJ
ejpam-5411	14	13	order	order	NOUN
ejpam-5411	14	14	without	without	ADP
ejpam-5411	14	15	having	have	VERB
ejpam-5411	14	16	memory	memory	NOUN
ejpam-5411	14	17	in	in	ADP
ejpam-5411	14	18	the	the	DET
ejpam-5411	14	19	structure	structure	NOUN
ejpam-5411	14	20	and	and	CCONJ
ejpam-5411	14	21	hence	hence	ADV
ejpam-5411	14	22	falling	fall	VERB
ejpam-5411	14	23	in	in	ADP
ejpam-5411	14	24	a	a	DET
ejpam-5411	14	25	similar	similar	ADJ
ejpam-5411	14	26	category	category	NOUN
ejpam-5411	14	27	to	to	ADP
ejpam-5411	14	28	local	local	ADJ
ejpam-5411	14	29	fractional	fractional	ADJ
ejpam-5411	14	30	calculus	calculus	NOUN
ejpam-5411	14	31	and	and	CCONJ
ejpam-5411	14	32	fractal	fractal	ADJ
ejpam-5411	14	33	calculus	calculus	NOUN
ejpam-5411	14	34	[	[	X
ejpam-5411	14	35	16	16	NUM
ejpam-5411	14	36	,	,	PUNCT
ejpam-5411	14	37	21	21	NUM
ejpam-5411	14	38	]	]	PUNCT
ejpam-5411	14	39	.	.	PUNCT
ejpam-5411	15	1	since	since	SCONJ
ejpam-5411	15	2	then	then	ADV
ejpam-5411	15	3	,	,	PUNCT
ejpam-5411	15	4	many	many	ADJ
ejpam-5411	15	5	classical	classical	ADJ
ejpam-5411	15	6	problems	problem	NOUN
ejpam-5411	15	7	have	have	AUX
ejpam-5411	15	8	been	be	AUX
ejpam-5411	15	9	generalized	generalize	VERB
ejpam-5411	15	10	to	to	ADP
ejpam-5411	15	11	the	the	DET
ejpam-5411	15	12	conformable	conformable	ADJ
ejpam-5411	15	13	case	case	NOUN
ejpam-5411	15	14	[	[	X
ejpam-5411	15	15	12	12	NUM
ejpam-5411	15	16	,	,	PUNCT
ejpam-5411	15	17	13	13	NUM
ejpam-5411	15	18	]	]	PUNCT
ejpam-5411	15	19	.	.	PUNCT
ejpam-5411	16	1	later	later	ADV
ejpam-5411	16	2	,	,	PUNCT
ejpam-5411	16	3	several	several	ADJ
ejpam-5411	16	4	modification	modification	NOUN
ejpam-5411	16	5	of	of	ADP
ejpam-5411	16	6	conformable	conformable	ADJ
ejpam-5411	16	7	derivatives	derivative	NOUN
ejpam-5411	16	8	have	have	AUX
ejpam-5411	16	9	been	be	AUX
ejpam-5411	16	10	appeared	appear	VERB
ejpam-5411	16	11	such	such	ADJ
ejpam-5411	16	12	as	as	ADP
ejpam-5411	16	13	:	:	PUNCT
ejpam-5411	16	14	the	the	DET
ejpam-5411	16	15	fractional	fractional	ADJ
ejpam-5411	16	16	beta	beta	ADJ
ejpam-5411	16	17	derivative	derivative	NOUN
ejpam-5411	16	18	[	[	X
ejpam-5411	16	19	15	15	NUM
ejpam-5411	16	20	]	]	PUNCT
ejpam-5411	16	21	defined	define	VERB
ejpam-5411	16	22	as	as	ADP
ejpam-5411	16	23	dγ	dγ	PROPN
ejpam-5411	16	24	ρ	ρ	PROPN
ejpam-5411	16	25	(	(	PUNCT
ejpam-5411	16	26	f(ρ	f(ρ	NOUN
ejpam-5411	16	27	)	)	PUNCT
ejpam-5411	16	28	)	)	PUNCT
ejpam-5411	17	1	=	=	PUNCT
ejpam-5411	18	1	limϵ→0	limϵ→0	NOUN
ejpam-5411	18	2	f(ρ+ϵ(ρ+	f(ρ+ϵ(ρ+	VERB
ejpam-5411	18	3	1	1	NUM
ejpam-5411	18	4	γ(γ	γ(γ	NOUN
ejpam-5411	18	5	)	)	PUNCT
ejpam-5411	18	6	)	)	PUNCT
ejpam-5411	18	7	)	)	PUNCT
ejpam-5411	18	8	−f(ρ	−f(ρ	X
ejpam-5411	18	9	)	)	PUNCT
ejpam-5411	18	10	ϵ	ϵ	NOUN
ejpam-5411	18	11	and	and	CCONJ
ejpam-5411	18	12	the	the	DET
ejpam-5411	18	13	m	m	ADV
ejpam-5411	18	14	-	-	ADJ
ejpam-5411	18	15	truncated	truncate	VERB
ejpam-5411	18	16	derivative	derivative	NOUN
ejpam-5411	18	17	[	[	X
ejpam-5411	18	18	19	19	NUM
ejpam-5411	18	19	]	]	PUNCT
ejpam-5411	18	20	defined	define	VERB
ejpam-5411	18	21	as	as	ADP
ejpam-5411	18	22	dm	dm	PROPN
ejpam-5411	18	23	α	α	NUM
ejpam-5411	18	24	,	,	PUNCT
ejpam-5411	18	25	βf(t	βf(t	PUNCT
ejpam-5411	18	26	)	)	PUNCT
ejpam-5411	18	27	=	=	PUNCT
ejpam-5411	19	1	limϵ→0	limϵ→0	PROPN
ejpam-5411	19	2	f(teβ	f(teβ	NOUN
ejpam-5411	19	3	,	,	PUNCT
ejpam-5411	19	4	i(ϵt	i(ϵt	NOUN
ejpam-5411	19	5	−α))−f(t	−α))−f(t	PROPN
ejpam-5411	19	6	)	)	PUNCT
ejpam-5411	19	7	ϵ	ϵ	ADP
ejpam-5411	19	8	where	where	SCONJ
ejpam-5411	19	9	eβ	eβ	NOUN
ejpam-5411	19	10	,	,	PUNCT
ejpam-5411	19	11	i(z	i(z	NOUN
ejpam-5411	19	12	)	)	PUNCT
ejpam-5411	19	13	=	=	SYM
ejpam-5411	19	14	∑i	∑i	NOUN
ejpam-5411	19	15	k=0	k=0	PUNCT
ejpam-5411	19	16	zk	zk	PROPN
ejpam-5411	19	17	γ(βk+1	γ(βk+1	PROPN
ejpam-5411	19	18	)	)	PUNCT
ejpam-5411	19	19	.	.	PUNCT
ejpam-5411	20	1	cauchy	cauchy	PROPN
ejpam-5411	20	2	type	type	NOUN
ejpam-5411	20	3	problems	problem	NOUN
ejpam-5411	20	4	are	be	AUX
ejpam-5411	20	5	very	very	ADV
ejpam-5411	20	6	well	well	ADV
ejpam-5411	20	7	-	-	PUNCT
ejpam-5411	20	8	known	know	VERB
ejpam-5411	20	9	important	important	ADJ
ejpam-5411	20	10	in	in	ADP
ejpam-5411	20	11	many	many	ADJ
ejpam-5411	20	12	fields	field	NOUN
ejpam-5411	20	13	of	of	ADP
ejpam-5411	20	14	science	science	NOUN
ejpam-5411	20	15	and	and	CCONJ
ejpam-5411	20	16	engineering	engineering	NOUN
ejpam-5411	20	17	.	.	PUNCT
ejpam-5411	21	1	several	several	ADJ
ejpam-5411	21	2	results	result	NOUN
ejpam-5411	21	3	regarding	regard	VERB
ejpam-5411	21	4	the	the	DET
ejpam-5411	21	5	capture	capture	NOUN
ejpam-5411	21	6	of	of	ADP
ejpam-5411	21	7	candidate	candidate	NOUN
ejpam-5411	21	8	solutions	solution	NOUN
ejpam-5411	21	9	of	of	ADP
ejpam-5411	21	10	the	the	DET
ejpam-5411	21	11	conformable	conformable	ADJ
ejpam-5411	21	12	differential	differential	NOUN
ejpam-5411	21	13	equations	equation	NOUN
ejpam-5411	21	14	can	can	AUX
ejpam-5411	21	15	be	be	AUX
ejpam-5411	21	16	found	find	VERB
ejpam-5411	21	17	in	in	ADP
ejpam-5411	21	18	[	[	X
ejpam-5411	21	19	18	18	NUM
ejpam-5411	21	20	]	]	PUNCT
ejpam-5411	21	21	.	.	PUNCT
ejpam-5411	22	1	this	this	DET
ejpam-5411	22	2	new	new	ADJ
ejpam-5411	22	3	definition	definition	NOUN
ejpam-5411	22	4	has	have	AUX
ejpam-5411	22	5	been	be	AUX
ejpam-5411	22	6	developed	develop	VERB
ejpam-5411	22	7	by	by	ADP
ejpam-5411	22	8	abdeljawad	abdeljawad	NOUN
ejpam-5411	22	9	[	[	X
ejpam-5411	22	10	1	1	NUM
ejpam-5411	22	11	]	]	PUNCT
ejpam-5411	22	12	and	and	CCONJ
ejpam-5411	22	13	by	by	ADP
ejpam-5411	22	14	el	el	PROPN
ejpam-5411	22	15	-	-	PROPN
ejpam-5411	22	16	ajou	ajou	PROPN
ejpam-5411	23	1	[	[	X
ejpam-5411	23	2	6	6	NUM
ejpam-5411	23	3	]	]	PUNCT
ejpam-5411	23	4	.	.	PUNCT
ejpam-5411	24	1	for	for	ADP
ejpam-5411	24	2	more	more	ADJ
ejpam-5411	24	3	developments	development	NOUN
ejpam-5411	24	4	on	on	ADP
ejpam-5411	24	5	the	the	DET
ejpam-5411	24	6	conformable	conformable	ADJ
ejpam-5411	24	7	differentiation	differentiation	NOUN
ejpam-5411	24	8	,	,	PUNCT
ejpam-5411	24	9	we	we	PRON
ejpam-5411	24	10	refer	refer	VERB
ejpam-5411	24	11	to	to	ADP
ejpam-5411	24	12	[	[	X
ejpam-5411	24	13	3	3	NUM
ejpam-5411	24	14	,	,	PUNCT
ejpam-5411	24	15	5	5	NUM
ejpam-5411	24	16	]	]	PUNCT
ejpam-5411	24	17	.	.	PUNCT
ejpam-5411	25	1	the	the	DET
ejpam-5411	25	2	usability	usability	NOUN
ejpam-5411	25	3	of	of	ADP
ejpam-5411	25	4	the	the	DET
ejpam-5411	25	5	conformable	conformable	ADJ
ejpam-5411	25	6	derivative	derivative	ADJ
ejpam-5411	25	7	notion	notion	NOUN
ejpam-5411	25	8	has	have	VERB
ejpam-5411	25	9	wide	wide	ADJ
ejpam-5411	25	10	areas	area	NOUN
ejpam-5411	25	11	of	of	ADP
ejpam-5411	25	12	interest	interest	NOUN
ejpam-5411	25	13	in	in	ADP
ejpam-5411	25	14	both	both	CCONJ
ejpam-5411	25	15	theoretical	theoretical	ADJ
ejpam-5411	25	16	and	and	CCONJ
ejpam-5411	25	17	practical	practical	ADJ
ejpam-5411	25	18	aspects	aspect	NOUN
ejpam-5411	25	19	(	(	PUNCT
ejpam-5411	25	20	see	see	VERB
ejpam-5411	25	21	[	[	X
ejpam-5411	25	22	10	10	NUM
ejpam-5411	25	23	]	]	PUNCT
ejpam-5411	25	24	,	,	PUNCT
ejpam-5411	25	25	[	[	X
ejpam-5411	25	26	20	20	NUM
ejpam-5411	25	27	]	]	NUM
ejpam-5411	25	28	)	)	PUNCT
ejpam-5411	25	29	.	.	PUNCT
ejpam-5411	26	1	the	the	DET
ejpam-5411	26	2	authors	author	NOUN
ejpam-5411	26	3	of	of	ADP
ejpam-5411	26	4	(	(	PUNCT
ejpam-5411	26	5	[	[	X
ejpam-5411	26	6	2	2	NUM
ejpam-5411	26	7	]	]	PUNCT
ejpam-5411	26	8	,	,	PUNCT
ejpam-5411	26	9	[	[	X
ejpam-5411	26	10	24	24	NUM
ejpam-5411	26	11	]	]	PUNCT
ejpam-5411	26	12	)	)	PUNCT
ejpam-5411	26	13	provided	provide	VERB
ejpam-5411	26	14	some	some	DET
ejpam-5411	26	15	applications	application	NOUN
ejpam-5411	26	16	through	through	ADP
ejpam-5411	26	17	partial	partial	ADJ
ejpam-5411	26	18	differential	differential	ADJ
ejpam-5411	26	19	equations	equation	NOUN
ejpam-5411	26	20	(	(	PUNCT
ejpam-5411	26	21	pdes	pde	NOUN
ejpam-5411	26	22	)	)	PUNCT
ejpam-5411	26	23	in	in	ADP
ejpam-5411	26	24	the	the	DET
ejpam-5411	26	25	conformable	conformable	ADJ
ejpam-5411	26	26	sense	sense	NOUN
ejpam-5411	26	27	.	.	PUNCT
ejpam-5411	27	1	precisely	precisely	ADV
ejpam-5411	27	2	,	,	PUNCT
ejpam-5411	27	3	maxwell	maxwell	PROPN
ejpam-5411	27	4	’s	’s	PART
ejpam-5411	27	5	equations	equation	NOUN
ejpam-5411	27	6	have	have	AUX
ejpam-5411	27	7	been	be	AUX
ejpam-5411	27	8	considered	consider	VERB
ejpam-5411	27	9	in	in	ADP
ejpam-5411	27	10	the	the	DET
ejpam-5411	27	11	conformable	conformable	ADJ
ejpam-5411	27	12	fractional	fractional	ADJ
ejpam-5411	27	13	setting	setting	NOUN
ejpam-5411	27	14	to	to	PART
ejpam-5411	27	15	describe	describe	VERB
ejpam-5411	27	16	electromagnetic	electromagnetic	ADJ
ejpam-5411	27	17	fields	field	NOUN
ejpam-5411	27	18	of	of	ADP
ejpam-5411	27	19	media	medium	NOUN
ejpam-5411	27	20	in	in	ADP
ejpam-5411	27	21	[	[	X
ejpam-5411	27	22	23	23	NUM
ejpam-5411	27	23	]	]	PUNCT
ejpam-5411	27	24	.	.	PUNCT
ejpam-5411	28	1	the	the	DET
ejpam-5411	28	2	conformable	conformable	ADJ
ejpam-5411	28	3	differential	differential	NOUN
ejpam-5411	28	4	equation	equation	NOUN
ejpam-5411	28	5	(	(	PUNCT
ejpam-5411	28	6	cde	cde	PROPN
ejpam-5411	28	7	)	)	PUNCT
ejpam-5411	28	8	has	have	AUX
ejpam-5411	28	9	been	be	AUX
ejpam-5411	28	10	used	use	VERB
ejpam-5411	28	11	for	for	ADP
ejpam-5411	28	12	the	the	DET
ejpam-5411	28	13	description	description	NOUN
ejpam-5411	28	14	of	of	ADP
ejpam-5411	28	15	the	the	DET
ejpam-5411	28	16	subdiffusion	subdiffusion	NOUN
ejpam-5411	28	17	process	process	NOUN
ejpam-5411	28	18	in	in	ADP
ejpam-5411	28	19	[	[	X
ejpam-5411	28	20	24	24	NUM
ejpam-5411	28	21	]	]	PUNCT
ejpam-5411	28	22	.	.	PUNCT
ejpam-5411	29	1	also	also	ADV
ejpam-5411	29	2	,	,	PUNCT
ejpam-5411	29	3	some	some	DET
ejpam-5411	29	4	applications	application	NOUN
ejpam-5411	29	5	in	in	ADP
ejpam-5411	29	6	quantum	quantum	ADJ
ejpam-5411	29	7	mechanics	mechanic	NOUN
ejpam-5411	29	8	have	have	AUX
ejpam-5411	29	9	been	be	AUX
ejpam-5411	29	10	treated	treat	VERB
ejpam-5411	29	11	in	in	ADP
ejpam-5411	29	12	the	the	DET
ejpam-5411	29	13	context	context	NOUN
ejpam-5411	29	14	of	of	ADP
ejpam-5411	29	15	cfd	cfd	NOUN
ejpam-5411	29	16	(	(	PUNCT
ejpam-5411	29	17	see	see	VERB
ejpam-5411	29	18	for	for	ADP
ejpam-5411	29	19	example	example	NOUN
ejpam-5411	29	20	[	[	X
ejpam-5411	29	21	2	2	NUM
ejpam-5411	29	22	]	]	NUM
ejpam-5411	29	23	)	)	PUNCT
ejpam-5411	29	24	.	.	PUNCT
ejpam-5411	30	1	fourier	fourier	PROPN
ejpam-5411	30	2	series	series	PROPN
ejpam-5411	30	3	is	be	AUX
ejpam-5411	30	4	one	one	NUM
ejpam-5411	30	5	of	of	ADP
ejpam-5411	30	6	the	the	DET
ejpam-5411	30	7	most	most	ADV
ejpam-5411	30	8	important	important	ADJ
ejpam-5411	30	9	tools	tool	NOUN
ejpam-5411	30	10	in	in	ADP
ejpam-5411	30	11	applied	applied	ADJ
ejpam-5411	30	12	sciences	science	NOUN
ejpam-5411	30	13	.	.	PUNCT
ejpam-5411	31	1	for	for	ADP
ejpam-5411	31	2	example	example	NOUN
ejpam-5411	31	3	one	one	PRON
ejpam-5411	31	4	can	can	AUX
ejpam-5411	31	5	solve	solve	VERB
ejpam-5411	31	6	partial	partial	ADJ
ejpam-5411	31	7	differential	differential	NOUN
ejpam-5411	31	8	equations	equation	NOUN
ejpam-5411	31	9	using	use	VERB
ejpam-5411	31	10	fourier	fourier	ADJ
ejpam-5411	31	11	series	series	NOUN
ejpam-5411	31	12	.	.	PUNCT
ejpam-5411	32	1	further	further	ADJ
ejpam-5411	32	2	one	one	PRON
ejpam-5411	32	3	can	can	AUX
ejpam-5411	32	4	find	find	VERB
ejpam-5411	32	5	the	the	DET
ejpam-5411	32	6	sum	sum	NOUN
ejpam-5411	32	7	of	of	ADP
ejpam-5411	32	8	certain	certain	ADJ
ejpam-5411	32	9	numerical	numerical	ADJ
ejpam-5411	32	10	series	series	PROPN
ejpam-5411	32	11	using	use	VERB
ejpam-5411	32	12	fourier	fourier	NOUN
ejpam-5411	32	13	series	series	NOUN
ejpam-5411	32	14	.	.	PUNCT
ejpam-5411	33	1	fractional	fractional	ADJ
ejpam-5411	33	2	partial	partial	ADJ
ejpam-5411	33	3	differential	differential	NOUN
ejpam-5411	33	4	equations	equation	NOUN
ejpam-5411	33	5	appeared	appear	VERB
ejpam-5411	33	6	to	to	PART
ejpam-5411	33	7	have	have	VERB
ejpam-5411	33	8	many	many	ADJ
ejpam-5411	33	9	applications	application	NOUN
ejpam-5411	33	10	in	in	ADP
ejpam-5411	33	11	physics	physics	NOUN
ejpam-5411	33	12	and	and	CCONJ
ejpam-5411	33	13	engineering	engineering	NOUN
ejpam-5411	33	14	.	.	PUNCT
ejpam-5411	34	1	there	there	PRON
ejpam-5411	34	2	are	be	VERB
ejpam-5411	34	3	many	many	ADJ
ejpam-5411	34	4	definitions	definition	NOUN
ejpam-5411	34	5	of	of	ADP
ejpam-5411	34	6	fractional	fractional	ADJ
ejpam-5411	34	7	derivative	derivative	NOUN
ejpam-5411	34	8	.	.	PUNCT
ejpam-5411	35	1	the	the	DET
ejpam-5411	35	2	conformable	conformable	ADJ
ejpam-5411	35	3	fractional	fractional	ADJ
ejpam-5411	35	4	fourier	fourier	NOUN
ejpam-5411	35	5	series	series	NOUN
ejpam-5411	35	6	for	for	ADP
ejpam-5411	35	7	α	α	NOUN
ejpam-5411	35	8	-	-	PUNCT
ejpam-5411	35	9	periodical	periodical	ADJ
ejpam-5411	35	10	functions	function	NOUN
ejpam-5411	35	11	is	be	AUX
ejpam-5411	35	12	introduced	introduce	VERB
ejpam-5411	35	13	by	by	ADP
ejpam-5411	35	14	khalil	khalil	PROPN
ejpam-5411	35	15	et	et	PROPN
ejpam-5411	35	16	al	al	PROPN
ejpam-5411	36	1	[	[	X
ejpam-5411	36	2	8	8	NUM
ejpam-5411	36	3	]	]	PUNCT
ejpam-5411	36	4	.	.	PUNCT
ejpam-5411	37	1	they	they	PRON
ejpam-5411	37	2	proved	prove	VERB
ejpam-5411	37	3	that	that	SCONJ
ejpam-5411	37	4	the	the	DET
ejpam-5411	37	5	fractional	fractional	ADJ
ejpam-5411	37	6	fourier	fourier	NOUN
ejpam-5411	37	7	series	series	NOUN
ejpam-5411	37	8	of	of	ADP
ejpam-5411	37	9	a	a	DET
ejpam-5411	37	10	piece	piece	NOUN
ejpam-5411	37	11	wise	wise	ADJ
ejpam-5411	37	12	continuous	continuous	ADJ
ejpam-5411	37	13	α	α	PRON
ejpam-5411	37	14	-	-	ADJ
ejpam-5411	37	15	periodical	periodical	ADJ
ejpam-5411	37	16	function	function	NOUN
ejpam-5411	37	17	converges	converge	VERB
ejpam-5411	37	18	pointwise	pointwise	VERB
ejpam-5411	37	19	to	to	ADP
ejpam-5411	37	20	the	the	DET
ejpam-5411	37	21	average	average	ADJ
ejpam-5411	37	22	limit	limit	NOUN
ejpam-5411	37	23	of	of	ADP
ejpam-5411	37	24	the	the	DET
ejpam-5411	37	25	function	function	NOUN
ejpam-5411	37	26	at	at	ADP
ejpam-5411	37	27	each	each	DET
ejpam-5411	37	28	point	point	NOUN
ejpam-5411	37	29	of	of	ADP
ejpam-5411	37	30	discontinuity	discontinuity	NOUN
ejpam-5411	37	31	,	,	PUNCT
ejpam-5411	37	32	and	and	CCONJ
ejpam-5411	37	33	to	to	ADP
ejpam-5411	37	34	the	the	DET
ejpam-5411	37	35	function	function	NOUN
ejpam-5411	37	36	at	at	ADP
ejpam-5411	37	37	each	each	DET
ejpam-5411	37	38	point	point	NOUN
ejpam-5411	37	39	of	of	ADP
ejpam-5411	37	40	continuity	continuity	NOUN
ejpam-5411	37	41	.	.	PUNCT
ejpam-5411	38	1	the	the	DET
ejpam-5411	38	2	rest	rest	NOUN
ejpam-5411	38	3	of	of	ADP
ejpam-5411	38	4	this	this	DET
ejpam-5411	38	5	paper	paper	NOUN
ejpam-5411	38	6	is	be	AUX
ejpam-5411	38	7	structured	structure	VERB
ejpam-5411	38	8	as	as	SCONJ
ejpam-5411	38	9	follows	follow	VERB
ejpam-5411	38	10	:	:	PUNCT
ejpam-5411	38	11	in	in	ADP
ejpam-5411	38	12	section	section	NOUN
ejpam-5411	38	13	2	2	NUM
ejpam-5411	38	14	,	,	PUNCT
ejpam-5411	38	15	we	we	PRON
ejpam-5411	38	16	introduce	introduce	VERB
ejpam-5411	38	17	the	the	DET
ejpam-5411	38	18	basic	basic	ADJ
ejpam-5411	38	19	definitions	definition	NOUN
ejpam-5411	38	20	and	and	CCONJ
ejpam-5411	38	21	properties	property	NOUN
ejpam-5411	38	22	of	of	ADP
ejpam-5411	38	23	α	α	NOUN
ejpam-5411	38	24	-	-	ADJ
ejpam-5411	38	25	conformable	conformable	ADJ
ejpam-5411	38	26	functional	functional	ADJ
ejpam-5411	38	27	derivative	derivative	ADJ
ejpam-5411	38	28	t	t	NOUN
ejpam-5411	38	29	(	(	PUNCT
ejpam-5411	38	30	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	38	31	)	)	PUNCT
ejpam-5411	38	32	for	for	ADP
ejpam-5411	38	33	0	0	NUM
ejpam-5411	38	34	<	<	X
ejpam-5411	38	35	α	α	PROPN
ejpam-5411	38	36	≤	≤	NUM
ejpam-5411	38	37	1	1	NUM
ejpam-5411	38	38	and	and	CCONJ
ejpam-5411	38	39	f	f	NOUN
ejpam-5411	38	40	:	:	PUNCT
ejpam-5411	39	1	[	[	X
ejpam-5411	39	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	39	3	r	r	NOUN
ejpam-5411	39	4	is	be	AUX
ejpam-5411	39	5	α	α	PRON
ejpam-5411	39	6	-	-	ADJ
ejpam-5411	39	7	periodic	periodic	ADJ
ejpam-5411	39	8	function	function	NOUN
ejpam-5411	39	9	,	,	PUNCT
ejpam-5411	39	10	define	define	VERB
ejpam-5411	39	11	by	by	ADP
ejpam-5411	39	12	khalil	khalil	PROPN
ejpam-5411	39	13	et	et	PROPN
ejpam-5411	39	14	al	al	PROPN
ejpam-5411	40	1	[	[	X
ejpam-5411	40	2	9	9	NUM
ejpam-5411	40	3	]	]	PUNCT
ejpam-5411	40	4	.	.	PUNCT
ejpam-5411	41	1	in	in	ADP
ejpam-5411	41	2	section	section	NOUN
ejpam-5411	41	3	3	3	NUM
ejpam-5411	41	4	,	,	PUNCT
ejpam-5411	41	5	we	we	PRON
ejpam-5411	41	6	prove	prove	VERB
ejpam-5411	41	7	some	some	DET
ejpam-5411	41	8	results	result	NOUN
ejpam-5411	41	9	and	and	CCONJ
ejpam-5411	41	10	examples	example	NOUN
ejpam-5411	41	11	of	of	ADP
ejpam-5411	41	12	α	α	NOUN
ejpam-5411	41	13	-	-	ADJ
ejpam-5411	41	14	periodic	periodic	ADJ
ejpam-5411	41	15	functions	function	NOUN
ejpam-5411	41	16	which	which	PRON
ejpam-5411	41	17	are	be	AUX
ejpam-5411	41	18	important	important	ADJ
ejpam-5411	41	19	for	for	ADP
ejpam-5411	41	20	the	the	DET
ejpam-5411	41	21	next	next	ADJ
ejpam-5411	41	22	section	section	NOUN
ejpam-5411	41	23	.	.	PUNCT
ejpam-5411	42	1	in	in	ADP
ejpam-5411	42	2	section	section	NOUN
ejpam-5411	42	3	4	4	NUM
ejpam-5411	42	4	,	,	PUNCT
ejpam-5411	42	5	we	we	PRON
ejpam-5411	42	6	give	give	VERB
ejpam-5411	42	7	a	a	DET
ejpam-5411	42	8	new	new	ADJ
ejpam-5411	42	9	definition	definition	NOUN
ejpam-5411	42	10	of	of	ADP
ejpam-5411	42	11	conformable	conformable	ADJ
ejpam-5411	42	12	fourier	fourier	NOUN
ejpam-5411	42	13	transform	transform	NOUN
ejpam-5411	42	14	for	for	ADP
ejpam-5411	42	15	αperiodical	αperiodical	ADJ
ejpam-5411	42	16	functions	function	NOUN
ejpam-5411	42	17	.	.	PUNCT
ejpam-5411	43	1	t.	t.	PROPN
ejpam-5411	43	2	abdeljawad	abdeljawad	PROPN
ejpam-5411	43	3	et	et	PROPN
ejpam-5411	43	4	al	al	PROPN
ejpam-5411	43	5	.	.	PUNCT
ejpam-5411	43	6	/	/	SYM
ejpam-5411	43	7	eur	eur	PROPN
ejpam-5411	43	8	.	.	PUNCT
ejpam-5411	44	1	j.	j.	PROPN
ejpam-5411	44	2	pure	pure	PROPN
ejpam-5411	44	3	appl	appl	PROPN
ejpam-5411	44	4	.	.	PROPN
ejpam-5411	44	5	math	math	PROPN
ejpam-5411	44	6	,	,	PUNCT
ejpam-5411	44	7	17	17	NUM
ejpam-5411	44	8	(	(	PUNCT
ejpam-5411	44	9	4	4	NUM
ejpam-5411	44	10	)	)	PUNCT
ejpam-5411	44	11	(	(	PUNCT
ejpam-5411	44	12	2024	2024	NUM
ejpam-5411	44	13	)	)	PUNCT
ejpam-5411	44	14	,	,	PUNCT
ejpam-5411	44	15	2405	2405	NUM
ejpam-5411	44	16	-	-	SYM
ejpam-5411	44	17	2430	2430	NUM
ejpam-5411	44	18	2407	2407	NUM
ejpam-5411	44	19	in	in	ADP
ejpam-5411	44	20	the	the	DET
ejpam-5411	44	21	first	first	ADJ
ejpam-5411	44	22	result	result	NOUN
ejpam-5411	44	23	(	(	PUNCT
ejpam-5411	44	24	theorem	theorem	ADJ
ejpam-5411	44	25	8)	8)	NUM
ejpam-5411	44	26	,	,	PUNCT
ejpam-5411	44	27	we	we	PRON
ejpam-5411	44	28	show	show	VERB
ejpam-5411	44	29	that	that	SCONJ
ejpam-5411	44	30	there	there	PRON
ejpam-5411	44	31	exists	exist	VERB
ejpam-5411	44	32	a	a	DET
ejpam-5411	44	33	relationship	relationship	NOUN
ejpam-5411	44	34	between	between	ADP
ejpam-5411	44	35	the	the	DET
ejpam-5411	44	36	conformable	conformable	ADJ
ejpam-5411	44	37	fourier	fourier	NOUN
ejpam-5411	44	38	transform	transform	NOUN
ejpam-5411	44	39	and	and	CCONJ
ejpam-5411	44	40	the	the	DET
ejpam-5411	44	41	classical	classical	ADJ
ejpam-5411	44	42	fourier	fourier	NOUN
ejpam-5411	44	43	transform	transform	NOUN
ejpam-5411	44	44	as	as	SCONJ
ejpam-5411	44	45	follows	follow	VERB
ejpam-5411	44	46	:	:	PUNCT
ejpam-5411	44	47	fα{f(t)}(k	fα{f(t)}(k	X
ejpam-5411	44	48	)	)	PUNCT
ejpam-5411	44	49	=	=	SYM
ejpam-5411	45	1	f{f((αt	f{f((αt	NOUN
ejpam-5411	45	2	)	)	PUNCT
ejpam-5411	45	3	1	1	NUM
ejpam-5411	45	4	α	α	NOUN
ejpam-5411	45	5	)	)	PUNCT
ejpam-5411	45	6	}	}	PUNCT
ejpam-5411	45	7	(	(	PUNCT
ejpam-5411	45	8	k	k	NOUN
ejpam-5411	45	9	)	)	PUNCT
ejpam-5411	45	10	for	for	ADP
ejpam-5411	45	11	all	all	DET
ejpam-5411	45	12	k	k	PROPN
ejpam-5411	45	13	∈	∈	PROPN
ejpam-5411	45	14	z.	z.	PROPN
ejpam-5411	45	15	in	in	ADP
ejpam-5411	45	16	the	the	DET
ejpam-5411	45	17	second	second	ADJ
ejpam-5411	45	18	and	and	CCONJ
ejpam-5411	45	19	third	third	ADJ
ejpam-5411	45	20	result	result	NOUN
ejpam-5411	45	21	(	(	PUNCT
ejpam-5411	45	22	theorem	theorem	ADJ
ejpam-5411	45	23	9	9	NUM
ejpam-5411	45	24	and	and	CCONJ
ejpam-5411	45	25	theorem	theorem	VERB
ejpam-5411	45	26	11	11	NUM
ejpam-5411	45	27	)	)	PUNCT
ejpam-5411	45	28	,	,	PUNCT
ejpam-5411	45	29	we	we	PRON
ejpam-5411	45	30	give	give	VERB
ejpam-5411	45	31	the	the	DET
ejpam-5411	45	32	results	result	NOUN
ejpam-5411	45	33	of	of	ADP
ejpam-5411	45	34	the	the	DET
ejpam-5411	45	35	conformable	conformable	ADJ
ejpam-5411	45	36	fourier	fourier	NOUN
ejpam-5411	45	37	transform	transform	NOUN
ejpam-5411	45	38	for	for	ADP
ejpam-5411	45	39	the	the	DET
ejpam-5411	45	40	conformable	conformable	ADJ
ejpam-5411	45	41	fractional	fractional	ADJ
ejpam-5411	45	42	integral	integral	ADJ
ejpam-5411	45	43	iα(f)(t	iα(f)(t	NOUN
ejpam-5411	45	44	)	)	PUNCT
ejpam-5411	45	45	defined	define	VERB
ejpam-5411	45	46	by	by	ADP
ejpam-5411	45	47	abdeljawad	abdeljawad	NOUN
ejpam-5411	45	48	[	[	X
ejpam-5411	45	49	1	1	NUM
ejpam-5411	45	50	]	]	PUNCT
ejpam-5411	45	51	,	,	PUNCT
ejpam-5411	45	52	as	as	SCONJ
ejpam-5411	45	53	follows	follow	VERB
ejpam-5411	45	54	:	:	PUNCT
ejpam-5411	45	55	fα(iα(f)(t))(k	fα(iα(f)(t))(k	NOUN
ejpam-5411	45	56	)	)	PUNCT
ejpam-5411	45	57	=	=	NOUN
ejpam-5411	46	1	(	(	PUNCT
ejpam-5411	46	2	2ikπ	2ikπ	NOUN
ejpam-5411	46	3	α	α	NOUN
ejpam-5411	46	4	pα	pα	NOUN
ejpam-5411	46	5	)	)	PUNCT
ejpam-5411	46	6	fα(f(t)(k	fα(f(t)(k	NOUN
ejpam-5411	46	7	)	)	PUNCT
ejpam-5411	46	8	for	for	ADP
ejpam-5411	46	9	all	all	DET
ejpam-5411	46	10	k	k	PROPN
ejpam-5411	46	11	∈	∈	PROPN
ejpam-5411	46	12	z∗	z∗	PROPN
ejpam-5411	46	13	and	and	CCONJ
ejpam-5411	46	14	for	for	ADP
ejpam-5411	46	15	the	the	DET
ejpam-5411	46	16	conformable	conformable	ADJ
ejpam-5411	46	17	derivative	derivative	NOUN
ejpam-5411	46	18	introduced	introduce	VERB
ejpam-5411	46	19	by	by	ADP
ejpam-5411	46	20	khalil	khalil	PROPN
ejpam-5411	46	21	et	et	PROPN
ejpam-5411	46	22	al	al	PROPN
ejpam-5411	47	1	[	[	X
ejpam-5411	47	2	9	9	NUM
ejpam-5411	47	3	]	]	PUNCT
ejpam-5411	47	4	as	as	SCONJ
ejpam-5411	47	5	follows	follow	VERB
ejpam-5411	47	6	,	,	PUNCT
ejpam-5411	47	7	fα(t	fα(t	X
ejpam-5411	47	8	(	(	PUNCT
ejpam-5411	47	9	α)(f)(t))(k	α)(f)(t))(k	X
ejpam-5411	47	10	)	)	PUNCT
ejpam-5411	47	11	=	=	SYM
ejpam-5411	47	12	(	(	PUNCT
ejpam-5411	47	13	2ikπ	2ikπ	NOUN
ejpam-5411	47	14	α	α	NOUN
ejpam-5411	47	15	pα	pα	NOUN
ejpam-5411	47	16	)	)	PUNCT
ejpam-5411	47	17	fα(f(t)(k	fα(f(t)(k	NOUN
ejpam-5411	47	18	)	)	PUNCT
ejpam-5411	47	19	and	and	CCONJ
ejpam-5411	47	20	in	in	ADP
ejpam-5411	47	21	the	the	DET
ejpam-5411	47	22	general	general	ADJ
ejpam-5411	47	23	case	case	NOUN
ejpam-5411	47	24	for	for	ADP
ejpam-5411	47	25	n	n	PRON
ejpam-5411	47	26	∈	∈	PROPN
ejpam-5411	47	27	n	n	CCONJ
ejpam-5411	47	28	,	,	PUNCT
ejpam-5411	47	29	fα(t	fα(t	X
ejpam-5411	47	30	(	(	PUNCT
ejpam-5411	47	31	jα)(f)(t))(k	jα)(f)(t))(k	PROPN
ejpam-5411	47	32	)	)	PUNCT
ejpam-5411	47	33	=	=	PUNCT
ejpam-5411	48	1	(	(	PUNCT
ejpam-5411	48	2	2ikπ	2ikπ	NOUN
ejpam-5411	48	3	α	α	NOUN
ejpam-5411	48	4	pα	pα	NOUN
ejpam-5411	48	5	)	)	PUNCT
ejpam-5411	48	6	jfα(f(t)(k	jfα(f(t)(k	PROPN
ejpam-5411	48	7	)	)	PUNCT
ejpam-5411	48	8	,	,	PUNCT
ejpam-5411	48	9	∀j	∀j	PROPN
ejpam-5411	48	10	∈	∈	PROPN
ejpam-5411	48	11	{	{	PUNCT
ejpam-5411	48	12	0	0	NUM
ejpam-5411	48	13	,	,	PUNCT
ejpam-5411	48	14	1	1	NUM
ejpam-5411	48	15	,	,	PUNCT
ejpam-5411	48	16	...	...	PUNCT
ejpam-5411	48	17	,	,	PUNCT
ejpam-5411	48	18	n	n	CCONJ
ejpam-5411	48	19	}	}	PUNCT
ejpam-5411	48	20	.	.	PUNCT
ejpam-5411	49	1	a	a	DET
ejpam-5411	49	2	following	follow	VERB
ejpam-5411	49	3	classical	classical	ADJ
ejpam-5411	49	4	result	result	NOUN
ejpam-5411	49	5	is	be	AUX
ejpam-5411	49	6	also	also	ADV
ejpam-5411	49	7	obtained	obtain	VERB
ejpam-5411	49	8	for	for	ADP
ejpam-5411	49	9	α	α	PRON
ejpam-5411	49	10	-	-	PUNCT
ejpam-5411	49	11	periodical	periodical	ADJ
ejpam-5411	49	12	functions	function	NOUN
ejpam-5411	49	13	fα((a	fα((a	PROPN
ejpam-5411	49	14	∗α	∗α	PROPN
ejpam-5411	49	15	f)−∞(t))(k	f)−∞(t))(k	PROPN
ejpam-5411	49	16	)	)	PUNCT
ejpam-5411	50	1	=	=	SYM
ejpam-5411	50	2	lα(a(t))(2ikπ	lα(a(t))(2ikπ	NOUN
ejpam-5411	50	3	α	α	NOUN
ejpam-5411	50	4	pα	pα	NOUN
ejpam-5411	50	5	)	)	PUNCT
ejpam-5411	50	6	fα(f((t))(k	fα(f((t))(k	NOUN
ejpam-5411	50	7	)	)	PUNCT
ejpam-5411	51	1	where	where	SCONJ
ejpam-5411	51	2	(	(	PUNCT
ejpam-5411	51	3	a	a	DET
ejpam-5411	51	4	∗α	∗α	NOUN
ejpam-5411	51	5	f)−∞(t	f)−∞(t	PROPN
ejpam-5411	51	6	)	)	PUNCT
ejpam-5411	51	7	=	=	SYM
ejpam-5411	52	1	∫	∫	PROPN
ejpam-5411	52	2	tα	tα	PROPN
ejpam-5411	52	3	α	α	PRON
ejpam-5411	52	4	−∞	−∞	ADP
ejpam-5411	52	5	a((tα	a((tα	SYM
ejpam-5411	52	6	−	−	PROPN
ejpam-5411	52	7	αs	αs	ADJ
ejpam-5411	52	8	)	)	PUNCT
ejpam-5411	52	9	1	1	NUM
ejpam-5411	52	10	α	α	NOUN
ejpam-5411	52	11	)	)	PUNCT
ejpam-5411	52	12	f((αs	f((αs	NOUN
ejpam-5411	52	13	)	)	PUNCT
ejpam-5411	52	14	1	1	NUM
ejpam-5411	52	15	α	α	NOUN
ejpam-5411	52	16	)	)	PUNCT
ejpam-5411	52	17	ds	ds	ADJ
ejpam-5411	52	18	and	and	CCONJ
ejpam-5411	52	19	lα(a(t))(λ	lα(a(t))(λ	NOUN
ejpam-5411	52	20	)	)	PUNCT
ejpam-5411	52	21	is	be	AUX
ejpam-5411	52	22	the	the	DET
ejpam-5411	52	23	conformable	conformable	ADJ
ejpam-5411	52	24	laplace	laplace	NOUN
ejpam-5411	52	25	transform	transform	NOUN
ejpam-5411	52	26	of	of	ADP
ejpam-5411	52	27	the	the	DET
ejpam-5411	52	28	function	function	NOUN
ejpam-5411	52	29	a(t	a(t	NOUN
ejpam-5411	52	30	)	)	PUNCT
ejpam-5411	52	31	,	,	PUNCT
ejpam-5411	52	32	given	give	VERB
ejpam-5411	52	33	by	by	ADP
ejpam-5411	52	34	z.al	z.al	PROPN
ejpam-5411	52	35	-	-	NOUN
ejpam-5411	52	36	zhouri	zhouri	PROPN
ejpam-5411	52	37	et	et	NOUN
ejpam-5411	52	38	al	al	PROPN
ejpam-5411	53	1	[	[	X
ejpam-5411	53	2	22	22	NUM
ejpam-5411	53	3	]	]	PUNCT
ejpam-5411	53	4	.	.	PUNCT
ejpam-5411	54	1	many	many	ADJ
ejpam-5411	54	2	examples	example	NOUN
ejpam-5411	54	3	are	be	AUX
ejpam-5411	54	4	given	give	VERB
ejpam-5411	54	5	to	to	PART
ejpam-5411	54	6	support	support	VERB
ejpam-5411	54	7	the	the	DET
ejpam-5411	54	8	results	result	NOUN
ejpam-5411	54	9	presented	present	VERB
ejpam-5411	54	10	.	.	PUNCT
ejpam-5411	55	1	finally	finally	ADV
ejpam-5411	55	2	,	,	PUNCT
ejpam-5411	55	3	the	the	DET
ejpam-5411	55	4	conclusion	conclusion	NOUN
ejpam-5411	55	5	is	be	AUX
ejpam-5411	55	6	presented	present	VERB
ejpam-5411	55	7	in	in	ADP
ejpam-5411	55	8	section	section	NOUN
ejpam-5411	55	9	5	5	NUM
ejpam-5411	55	10	.	.	NOUN
ejpam-5411	56	1	2	2	NUM
ejpam-5411	56	2	.	.	NOUN
ejpam-5411	56	3	basic	basic	ADJ
ejpam-5411	56	4	definitions	definition	NOUN
ejpam-5411	56	5	and	and	CCONJ
ejpam-5411	56	6	tools	tool	NOUN
ejpam-5411	56	7	in	in	ADP
ejpam-5411	56	8	this	this	DET
ejpam-5411	56	9	section	section	NOUN
ejpam-5411	56	10	,	,	PUNCT
ejpam-5411	56	11	we	we	PRON
ejpam-5411	56	12	introduce	introduce	VERB
ejpam-5411	56	13	the	the	DET
ejpam-5411	56	14	definition	definition	NOUN
ejpam-5411	56	15	of	of	ADP
ejpam-5411	56	16	conformable	conformable	ADJ
ejpam-5411	56	17	fractional	fractional	ADJ
ejpam-5411	56	18	calculus	calculus	NOUN
ejpam-5411	56	19	and	and	CCONJ
ejpam-5411	56	20	its	its	PRON
ejpam-5411	56	21	important	important	ADJ
ejpam-5411	56	22	properties	property	NOUN
ejpam-5411	56	23	.	.	PUNCT
ejpam-5411	57	1	definition	definition	NOUN
ejpam-5411	57	2	1	1	NUM
ejpam-5411	57	3	.	.	PUNCT
ejpam-5411	58	1	[	[	X
ejpam-5411	58	2	9	9	NUM
ejpam-5411	58	3	]	]	PUNCT
ejpam-5411	58	4	given	give	VERB
ejpam-5411	58	5	a	a	DET
ejpam-5411	58	6	function	function	NOUN
ejpam-5411	58	7	f	f	NOUN
ejpam-5411	58	8	:	:	PUNCT
ejpam-5411	58	9	[	[	X
ejpam-5411	58	10	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	58	11	r	r	NOUN
ejpam-5411	58	12	,	,	PUNCT
ejpam-5411	58	13	the	the	DET
ejpam-5411	58	14	conformable	conformable	ADJ
ejpam-5411	58	15	fractional	fractional	ADJ
ejpam-5411	58	16	derivative	derivative	NOUN
ejpam-5411	58	17	of	of	ADP
ejpam-5411	58	18	order	order	NOUN
ejpam-5411	58	19	α	α	NOUN
ejpam-5411	58	20	is	be	AUX
ejpam-5411	58	21	defined	define	VERB
ejpam-5411	58	22	by	by	ADP
ejpam-5411	58	23	:	:	PUNCT
ejpam-5411	58	24	t	t	PROPN
ejpam-5411	58	25	(	(	PUNCT
ejpam-5411	58	26	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	58	27	)	)	PUNCT
ejpam-5411	58	28	=	=	VERB
ejpam-5411	58	29	lim	lim	PROPN
ejpam-5411	58	30	h→0	h→0	ADV
ejpam-5411	58	31	f(t	f(t	PROPN
ejpam-5411	58	32	+	+	CCONJ
ejpam-5411	58	33	ht1−α	ht1−α	ADJ
ejpam-5411	58	34	)	)	PUNCT
ejpam-5411	58	35	−	−	PRON
ejpam-5411	58	36	f(t	f(t	NOUN
ejpam-5411	58	37	)	)	PUNCT
ejpam-5411	58	38	h	h	NOUN
ejpam-5411	58	39	for	for	ADP
ejpam-5411	58	40	all	all	DET
ejpam-5411	58	41	t	t	NOUN
ejpam-5411	58	42	>	>	X
ejpam-5411	58	43	0	0	PUNCT
ejpam-5411	59	1	and	and	CCONJ
ejpam-5411	59	2	0	0	NUM
ejpam-5411	59	3	<	<	X
ejpam-5411	59	4	α	α	X
ejpam-5411	59	5	≤	≤	NUM
ejpam-5411	59	6	1	1	NUM
ejpam-5411	59	7	.	.	PUNCT
ejpam-5411	60	1	definition	definition	NOUN
ejpam-5411	60	2	2	2	NUM
ejpam-5411	60	3	.	.	PUNCT
ejpam-5411	61	1	let	let	VERB
ejpam-5411	61	2	0	0	NUM
ejpam-5411	61	3	<	<	X
ejpam-5411	61	4	α	α	PROPN
ejpam-5411	61	5	≤	≤	NUM
ejpam-5411	61	6	1	1	NUM
ejpam-5411	61	7	and	and	CCONJ
ejpam-5411	61	8	f	f	NOUN
ejpam-5411	61	9	:	:	PUNCT
ejpam-5411	62	1	[	[	X
ejpam-5411	62	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	62	3	r.	r.	PROPN
ejpam-5411	62	4	(	(	PUNCT
ejpam-5411	62	5	i	i	NOUN
ejpam-5411	62	6	)	)	PUNCT
ejpam-5411	62	7	the	the	DET
ejpam-5411	62	8	function	function	NOUN
ejpam-5411	62	9	f	f	PROPN
ejpam-5411	62	10	is	be	AUX
ejpam-5411	62	11	called	call	VERB
ejpam-5411	62	12	α	α	PRON
ejpam-5411	62	13	-	-	NOUN
ejpam-5411	62	14	differentiable	differentiable	ADJ
ejpam-5411	62	15	on	on	ADP
ejpam-5411	62	16	[	[	X
ejpam-5411	62	17	0,+∞	0,+∞	PROPN
ejpam-5411	62	18	[	[	X
ejpam-5411	62	19	,	,	PUNCT
ejpam-5411	62	20	if	if	SCONJ
ejpam-5411	62	21	f	f	PROPN
ejpam-5411	62	22	is	be	AUX
ejpam-5411	62	23	continuous	continuous	ADJ
ejpam-5411	62	24	.	.	PUNCT
ejpam-5411	63	1	t	t	PROPN
ejpam-5411	63	2	(	(	PUNCT
ejpam-5411	63	3	α)f(t	α)f(t	ADJ
ejpam-5411	63	4	)	)	PUNCT
ejpam-5411	63	5	exists	exist	VERB
ejpam-5411	63	6	for	for	ADP
ejpam-5411	63	7	all	all	DET
ejpam-5411	63	8	t	t	NOUN
ejpam-5411	63	9	∈]0,+∞	∈]0,+∞	PUNCT
ejpam-5411	63	10	[	[	PUNCT
ejpam-5411	63	11	and	and	CCONJ
ejpam-5411	63	12	t	t	PROPN
ejpam-5411	63	13	(	(	PUNCT
ejpam-5411	63	14	α)f(0	α)f(0	NOUN
ejpam-5411	63	15	)	)	PUNCT
ejpam-5411	63	16	=	=	SYM
ejpam-5411	63	17	limt→0	limt→0	PROPN
ejpam-5411	63	18	+	+	SYM
ejpam-5411	63	19	t	t	PROPN
ejpam-5411	63	20	(	(	PUNCT
ejpam-5411	63	21	α)f(t	α)f(t	ADJ
ejpam-5411	63	22	)	)	PUNCT
ejpam-5411	63	23	exists	exist	VERB
ejpam-5411	63	24	.	.	PUNCT
ejpam-5411	64	1	(	(	PUNCT
ejpam-5411	64	2	ii	ii	X
ejpam-5411	64	3	)	)	PUNCT
ejpam-5411	64	4	the	the	DET
ejpam-5411	64	5	function	function	NOUN
ejpam-5411	64	6	f	f	PROPN
ejpam-5411	64	7	is	be	AUX
ejpam-5411	64	8	called	call	VERB
ejpam-5411	64	9	continuously	continuously	ADV
ejpam-5411	64	10	α	α	VERB
ejpam-5411	64	11	-	-	NOUN
ejpam-5411	64	12	differentiable	differentiable	ADJ
ejpam-5411	64	13	on	on	ADP
ejpam-5411	64	14	[	[	X
ejpam-5411	64	15	0,+∞	0,+∞	NUM
ejpam-5411	64	16	)	)	PUNCT
ejpam-5411	64	17	if	if	SCONJ
ejpam-5411	64	18	f	f	PROPN
ejpam-5411	64	19	is	be	AUX
ejpam-5411	64	20	α	α	NOUN
ejpam-5411	64	21	-	-	NOUN
ejpam-5411	64	22	differentiable	differentiable	ADJ
ejpam-5411	64	23	on	on	ADP
ejpam-5411	64	24	[	[	X
ejpam-5411	64	25	0,+∞	0,+∞	NUM
ejpam-5411	64	26	)	)	PUNCT
ejpam-5411	64	27	and	and	CCONJ
ejpam-5411	64	28	t	t	PROPN
ejpam-5411	64	29	(	(	PUNCT
ejpam-5411	64	30	α)f(t	α)f(t	VERB
ejpam-5411	64	31	)	)	PUNCT
ejpam-5411	64	32	is	be	AUX
ejpam-5411	64	33	continuous	continuous	ADJ
ejpam-5411	64	34	on	on	ADP
ejpam-5411	64	35	[	[	X
ejpam-5411	64	36	0,+∞	0,+∞	PROPN
ejpam-5411	64	37	[	[	X
ejpam-5411	64	38	.	.	PUNCT
ejpam-5411	65	1	t.	t.	PROPN
ejpam-5411	65	2	abdeljawad	abdeljawad	PROPN
ejpam-5411	65	3	et	et	PROPN
ejpam-5411	65	4	al	al	PROPN
ejpam-5411	65	5	.	.	PUNCT
ejpam-5411	65	6	/	/	SYM
ejpam-5411	65	7	eur	eur	PROPN
ejpam-5411	65	8	.	.	PUNCT
ejpam-5411	66	1	j.	j.	PROPN
ejpam-5411	66	2	pure	pure	PROPN
ejpam-5411	66	3	appl	appl	PROPN
ejpam-5411	66	4	.	.	PROPN
ejpam-5411	66	5	math	math	PROPN
ejpam-5411	66	6	,	,	PUNCT
ejpam-5411	66	7	17	17	NUM
ejpam-5411	66	8	(	(	PUNCT
ejpam-5411	66	9	4	4	NUM
ejpam-5411	66	10	)	)	PUNCT
ejpam-5411	66	11	(	(	PUNCT
ejpam-5411	66	12	2024	2024	NUM
ejpam-5411	66	13	)	)	PUNCT
ejpam-5411	66	14	,	,	PUNCT
ejpam-5411	66	15	2405	2405	NUM
ejpam-5411	66	16	-	-	SYM
ejpam-5411	66	17	2430	2430	NUM
ejpam-5411	66	18	2408	2408	NUM
ejpam-5411	66	19	definition	definition	NOUN
ejpam-5411	66	20	3	3	NUM
ejpam-5411	66	21	.	.	PUNCT
ejpam-5411	67	1	let	let	VERB
ejpam-5411	67	2	0	0	NUM
ejpam-5411	67	3	<	<	X
ejpam-5411	67	4	α	α	PROPN
ejpam-5411	67	5	≤	≤	NUM
ejpam-5411	67	6	1	1	NUM
ejpam-5411	67	7	,	,	PUNCT
ejpam-5411	67	8	n	n	PRON
ejpam-5411	67	9	∈	∈	PROPN
ejpam-5411	67	10	n	n	NOUN
ejpam-5411	67	11	and	and	CCONJ
ejpam-5411	67	12	f	f	NOUN
ejpam-5411	67	13	:	:	PUNCT
ejpam-5411	68	1	[	[	X
ejpam-5411	68	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	68	3	r.	r.	PROPN
ejpam-5411	68	4	(	(	PUNCT
ejpam-5411	68	5	i	i	NOUN
ejpam-5411	68	6	)	)	PUNCT
ejpam-5411	68	7	the	the	DET
ejpam-5411	68	8	function	function	NOUN
ejpam-5411	68	9	f	f	PROPN
ejpam-5411	68	10	is	be	AUX
ejpam-5411	68	11	called	call	VERB
ejpam-5411	68	12	n	n	PRON
ejpam-5411	68	13	times	time	NOUN
ejpam-5411	68	14	α	α	PRON
ejpam-5411	68	15	-	-	NOUN
ejpam-5411	68	16	differentiable	differentiable	ADJ
ejpam-5411	68	17	on	on	ADP
ejpam-5411	68	18	[	[	X
ejpam-5411	68	19	0,+∞	0,+∞	X
ejpam-5411	68	20	[	[	PUNCT
ejpam-5411	68	21	if	if	SCONJ
ejpam-5411	68	22	f	f	PROPN
ejpam-5411	68	23	is	be	AUX
ejpam-5411	68	24	continuous	continuous	ADJ
ejpam-5411	68	25	,	,	PUNCT
ejpam-5411	68	26	∀j	∀j	PROPN
ejpam-5411	68	27	∈	∈	PROPN
ejpam-5411	68	28	{	{	PUNCT
ejpam-5411	68	29	0	0	NUM
ejpam-5411	68	30	,	,	PUNCT
ejpam-5411	68	31	...	...	PUNCT
ejpam-5411	68	32	n	n	CCONJ
ejpam-5411	68	33	}	}	PUNCT
ejpam-5411	68	34	t	t	PROPN
ejpam-5411	68	35	(	(	PUNCT
ejpam-5411	68	36	jα)f(t	jα)f(t	X
ejpam-5411	68	37	)	)	PUNCT
ejpam-5411	68	38	=	=	SYM
ejpam-5411	68	39	t	t	PROPN
ejpam-5411	68	40	(	(	PUNCT
ejpam-5411	68	41	α)(t	α)(t	PROPN
ejpam-5411	68	42	(	(	PUNCT
ejpam-5411	68	43	α)	α)	NUM
ejpam-5411	68	44	...	...	PUNCT
ejpam-5411	68	45	(t	(t	NOUN
ejpam-5411	68	46	(	(	PUNCT
ejpam-5411	68	47	α)(f)))(t	α)(f)))(t	PROPN
ejpam-5411	68	48	)	)	PUNCT
ejpam-5411	68	49	,	,	PUNCT
ejpam-5411	68	50	j	j	PROPN
ejpam-5411	68	51	times	time	NOUN
ejpam-5411	68	52	,	,	PUNCT
ejpam-5411	68	53	exists	exist	VERB
ejpam-5411	68	54	for	for	ADP
ejpam-5411	68	55	all	all	DET
ejpam-5411	68	56	t	t	NOUN
ejpam-5411	68	57	∈]0,+∞	∈]0,+∞	PUNCT
ejpam-5411	68	58	[	[	PUNCT
ejpam-5411	68	59	and	and	CCONJ
ejpam-5411	68	60	t	t	PROPN
ejpam-5411	68	61	(	(	PUNCT
ejpam-5411	68	62	jα)f(0	jα)f(0	PROPN
ejpam-5411	68	63	)	)	PUNCT
ejpam-5411	68	64	=	=	SYM
ejpam-5411	68	65	limt→0	limt→0	PROPN
ejpam-5411	68	66	+	+	SYM
ejpam-5411	68	67	t	t	PROPN
ejpam-5411	68	68	(	(	PUNCT
ejpam-5411	68	69	jα)f(t	jα)f(t	X
ejpam-5411	68	70	)	)	PUNCT
ejpam-5411	68	71	exists	exist	VERB
ejpam-5411	68	72	.	.	PUNCT
ejpam-5411	69	1	(	(	PUNCT
ejpam-5411	69	2	ii	ii	X
ejpam-5411	69	3	)	)	PUNCT
ejpam-5411	69	4	the	the	DET
ejpam-5411	69	5	function	function	NOUN
ejpam-5411	69	6	f	f	PROPN
ejpam-5411	69	7	is	be	AUX
ejpam-5411	69	8	called	call	VERB
ejpam-5411	69	9	n	n	DET
ejpam-5411	69	10	times	time	NOUN
ejpam-5411	69	11	continuously	continuously	ADV
ejpam-5411	69	12	α	α	VERB
ejpam-5411	69	13	-	-	NOUN
ejpam-5411	69	14	differentiable	differentiable	ADJ
ejpam-5411	69	15	on	on	ADP
ejpam-5411	69	16	[	[	X
ejpam-5411	69	17	0,+∞	0,+∞	NUM
ejpam-5411	69	18	)	)	PUNCT
ejpam-5411	69	19	if	if	SCONJ
ejpam-5411	69	20	f	f	PROPN
ejpam-5411	69	21	is	be	AUX
ejpam-5411	69	22	n	n	PRON
ejpam-5411	69	23	times	time	NOUN
ejpam-5411	69	24	α	α	PRON
ejpam-5411	69	25	-	-	NOUN
ejpam-5411	69	26	differentiable	differentiable	ADJ
ejpam-5411	69	27	on	on	ADP
ejpam-5411	69	28	[	[	X
ejpam-5411	69	29	0,+∞	0,+∞	NUM
ejpam-5411	69	30	)	)	PUNCT
ejpam-5411	69	31	and	and	CCONJ
ejpam-5411	69	32	∀j	∀j	PROPN
ejpam-5411	69	33	∈	∈	PROPN
ejpam-5411	69	34	{	{	PUNCT
ejpam-5411	69	35	0	0	NUM
ejpam-5411	69	36	,	,	PUNCT
ejpam-5411	69	37	.	.	PUNCT
ejpam-5411	69	38	.	.	PUNCT
ejpam-5411	70	1	.	.	PUNCT
ejpam-5411	71	1	,	,	PUNCT
ejpam-5411	71	2	n	n	CCONJ
ejpam-5411	71	3	}	}	PUNCT
ejpam-5411	71	4	t	t	PROPN
ejpam-5411	71	5	(	(	PUNCT
ejpam-5411	71	6	jα)f(t	jα)f(t	NOUN
ejpam-5411	71	7	)	)	PUNCT
ejpam-5411	71	8	is	be	AUX
ejpam-5411	71	9	continuous	continuous	ADJ
ejpam-5411	71	10	on	on	ADP
ejpam-5411	71	11	[	[	X
ejpam-5411	71	12	0,+∞	0,+∞	PROPN
ejpam-5411	71	13	[	[	X
ejpam-5411	71	14	.	.	PUNCT
ejpam-5411	72	1	(	(	PUNCT
ejpam-5411	72	2	iii	iii	X
ejpam-5411	72	3	)	)	PUNCT
ejpam-5411	72	4	the	the	DET
ejpam-5411	72	5	function	function	NOUN
ejpam-5411	72	6	f	f	PROPN
ejpam-5411	72	7	is	be	AUX
ejpam-5411	72	8	called	call	VERB
ejpam-5411	72	9	infinitely	infinitely	ADV
ejpam-5411	72	10	continuously	continuously	ADV
ejpam-5411	72	11	α	α	NOUN
ejpam-5411	72	12	-	-	NOUN
ejpam-5411	72	13	differentiable	differentiable	ADJ
ejpam-5411	72	14	,	,	PUNCT
ejpam-5411	72	15	if	if	SCONJ
ejpam-5411	72	16	f	f	PROPN
ejpam-5411	72	17	is	be	AUX
ejpam-5411	72	18	n	n	PRON
ejpam-5411	72	19	times	time	NOUN
ejpam-5411	72	20	continuously	continuously	ADV
ejpam-5411	72	21	α	α	VERB
ejpam-5411	72	22	-	-	NOUN
ejpam-5411	72	23	differentiable	differentiable	ADJ
ejpam-5411	72	24	for	for	ADP
ejpam-5411	72	25	all	all	DET
ejpam-5411	72	26	n	n	PRON
ejpam-5411	72	27	∈	∈	PROPN
ejpam-5411	72	28	n.	n.	NOUN
ejpam-5411	72	29	note	note	VERB
ejpam-5411	72	30	that	that	SCONJ
ejpam-5411	72	31	for	for	ADP
ejpam-5411	72	32	n	n	NOUN
ejpam-5411	72	33	=	=	SYM
ejpam-5411	72	34	0	0	NUM
ejpam-5411	72	35	,	,	PUNCT
ejpam-5411	72	36	f	f	PROPN
ejpam-5411	72	37	is	be	AUX
ejpam-5411	72	38	n	n	PRON
ejpam-5411	72	39	time	time	NOUN
ejpam-5411	72	40	α	α	NOUN
ejpam-5411	72	41	-	-	NOUN
ejpam-5411	72	42	differentiable	differentiable	ADJ
ejpam-5411	72	43	if	if	SCONJ
ejpam-5411	72	44	there	there	PRON
ejpam-5411	72	45	is	be	VERB
ejpam-5411	72	46	continuous	continuous	ADJ
ejpam-5411	72	47	.	.	PUNCT
ejpam-5411	72	48	example	example	NOUN
ejpam-5411	73	1	1	1	NUM
ejpam-5411	73	2	.	.	PUNCT
ejpam-5411	73	3	let	let	VERB
ejpam-5411	73	4	f(t	f(t	NOUN
ejpam-5411	73	5	)	)	PUNCT
ejpam-5411	73	6	=	=	SYM
ejpam-5411	73	7	et	et	PROPN
ejpam-5411	73	8	,	,	PUNCT
ejpam-5411	73	9	t	t	PROPN
ejpam-5411	73	10	∈	∈	PROPN
ejpam-5411	74	1	[	[	X
ejpam-5411	74	2	0,+∞	0,+∞	NUM
ejpam-5411	74	3	[	[	X
ejpam-5411	74	4	.	.	PUNCT
ejpam-5411	74	5	(	(	PUNCT
ejpam-5411	74	6	i	i	NOUN
ejpam-5411	74	7	)	)	PUNCT
ejpam-5411	74	8	for	for	ADP
ejpam-5411	74	9	all	all	DET
ejpam-5411	74	10	t	t	PROPN
ejpam-5411	74	11	>	>	X
ejpam-5411	74	12	0	0	PUNCT
ejpam-5411	75	1	and	and	CCONJ
ejpam-5411	75	2	0	0	NUM
ejpam-5411	75	3	<	<	X
ejpam-5411	75	4	α	α	PROPN
ejpam-5411	75	5	≤	≤	NUM
ejpam-5411	75	6	1	1	NUM
ejpam-5411	75	7	t	t	NOUN
ejpam-5411	75	8	(	(	PUNCT
ejpam-5411	75	9	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	75	10	)	)	PUNCT
ejpam-5411	76	1	=	=	SYM
ejpam-5411	76	2	lim	lim	PROPN
ejpam-5411	76	3	h→0	h→0	PROPN
ejpam-5411	76	4	e(t+ht1−α	e(t+ht1−α	PROPN
ejpam-5411	76	5	)	)	PUNCT
ejpam-5411	76	6	−	−	PROPN
ejpam-5411	76	7	e(t	e(t	NOUN
ejpam-5411	76	8	)	)	PUNCT
ejpam-5411	76	9	h	h	NOUN
ejpam-5411	76	10	=	=	PUNCT
ejpam-5411	77	1	t1−αet	t1−αet	VERB
ejpam-5411	77	2	lim	lim	PROPN
ejpam-5411	77	3	h→0	h→0	PROPN
ejpam-5411	77	4	eht	eht	PROPN
ejpam-5411	77	5	1−α	1−α	NUM
ejpam-5411	78	1	−	−	NUM
ejpam-5411	78	2	1	1	NUM
ejpam-5411	78	3	ht1−α	ht1−α	NOUN
ejpam-5411	78	4	=	=	PUNCT
ejpam-5411	78	5	t1−αet	t1−αet	VERB
ejpam-5411	78	6	(	(	PUNCT
ejpam-5411	78	7	ii	ii	NOUN
ejpam-5411	78	8	)	)	PUNCT
ejpam-5411	78	9	for	for	ADP
ejpam-5411	78	10	all	all	DET
ejpam-5411	78	11	t	t	PROPN
ejpam-5411	78	12	>	>	X
ejpam-5411	78	13	0	0	PUNCT
ejpam-5411	79	1	and	and	CCONJ
ejpam-5411	79	2	0	0	NUM
ejpam-5411	79	3	<	<	X
ejpam-5411	79	4	α	α	PROPN
ejpam-5411	79	5	≤	≤	NUM
ejpam-5411	79	6	1	1	NUM
ejpam-5411	79	7	t	t	NOUN
ejpam-5411	79	8	(	(	PUNCT
ejpam-5411	79	9	2α)(f)(t	2α)(f)(t	NUM
ejpam-5411	79	10	)	)	PUNCT
ejpam-5411	79	11	=	=	SYM
ejpam-5411	79	12	t	t	PROPN
ejpam-5411	79	13	(	(	PUNCT
ejpam-5411	79	14	α)(t	α)(t	PROPN
ejpam-5411	79	15	(	(	PUNCT
ejpam-5411	79	16	α)(f)(t	α)(f)(t	NUM
ejpam-5411	79	17	)	)	PUNCT
ejpam-5411	79	18	)	)	PUNCT
ejpam-5411	80	1	=	=	SYM
ejpam-5411	80	2	t	t	PROPN
ejpam-5411	80	3	(	(	PUNCT
ejpam-5411	80	4	α)(t1−αet	α)(t1−αet	NUM
ejpam-5411	80	5	)	)	PUNCT
ejpam-5411	81	1	=	=	VERB
ejpam-5411	81	2	lim	lim	PROPN
ejpam-5411	81	3	h→0	h→0	ADV
ejpam-5411	81	4	(	(	PUNCT
ejpam-5411	81	5	t	t	PROPN
ejpam-5411	81	6	+	+	NUM
ejpam-5411	81	7	ht1−α)1−αe(t+ht1−α	ht1−α)1−αe(t+ht1−α	PROPN
ejpam-5411	81	8	)	)	PUNCT
ejpam-5411	81	9	−	−	PROPN
ejpam-5411	81	10	t1−αet	t1−αet	VERB
ejpam-5411	81	11	h	h	NOUN
ejpam-5411	81	12	=	=	PUNCT
ejpam-5411	81	13	t1−αet	t1−αet	VERB
ejpam-5411	81	14	lim	lim	NOUN
ejpam-5411	81	15	h→0	h→0	ADV
ejpam-5411	82	1	(	(	PUNCT
ejpam-5411	82	2	1	1	NUM
ejpam-5411	82	3	−	−	PROPN
ejpam-5411	82	4	ht−α)1−αeht	ht−α)1−αeht	NOUN
ejpam-5411	82	5	1−α	1−α	NUM
ejpam-5411	82	6	−	−	NUM
ejpam-5411	82	7	1	1	NUM
ejpam-5411	82	8	h	h	NOUN
ejpam-5411	82	9	=	=	SYM
ejpam-5411	82	10	t1−αetg′(0	t1−αetg′(0	NOUN
ejpam-5411	82	11	)	)	PUNCT
ejpam-5411	82	12	,	,	PUNCT
ejpam-5411	82	13	where	where	SCONJ
ejpam-5411	82	14	g(t	g(t	NOUN
ejpam-5411	82	15	)	)	PUNCT
ejpam-5411	82	16	=	=	PUNCT
ejpam-5411	82	17	(	(	PUNCT
ejpam-5411	82	18	1	1	NUM
ejpam-5411	82	19	−	−	PROPN
ejpam-5411	82	20	ht−α)1−αeht	ht−α)1−αeht	PROPN
ejpam-5411	82	21	1−α	1−α	NUM
ejpam-5411	82	22	and	and	CCONJ
ejpam-5411	82	23	g′(0	g′(0	PROPN
ejpam-5411	82	24	)	)	PUNCT
ejpam-5411	82	25	=	=	PUNCT
ejpam-5411	82	26	(	(	PUNCT
ejpam-5411	82	27	1	1	NUM
ejpam-5411	82	28	−	−	NUM
ejpam-5411	82	29	α)t−α	α)t−α	NUM
ejpam-5411	83	1	+	+	CCONJ
ejpam-5411	83	2	t1−α	t1−α	NOUN
ejpam-5411	83	3	.	.	PUNCT
ejpam-5411	84	1	then	then	ADV
ejpam-5411	84	2	,	,	PUNCT
ejpam-5411	84	3	we	we	PRON
ejpam-5411	84	4	get	get	VERB
ejpam-5411	84	5	t	t	NOUN
ejpam-5411	84	6	(	(	PUNCT
ejpam-5411	84	7	2α)(et	2α)(et	NUM
ejpam-5411	84	8	)	)	PUNCT
ejpam-5411	84	9	=	=	SYM
ejpam-5411	84	10	t1−αet((1	t1−αet((1	NUM
ejpam-5411	85	1	−	−	PROPN
ejpam-5411	86	1	α)t−α	α)t−α	NUM
ejpam-5411	86	2	+	+	CCONJ
ejpam-5411	86	3	t1−α	t1−α	NOUN
ejpam-5411	86	4	)	)	PUNCT
ejpam-5411	86	5	.	.	PUNCT
ejpam-5411	87	1	theorem	theorem	NOUN
ejpam-5411	87	2	1	1	NUM
ejpam-5411	87	3	.	.	PUNCT
ejpam-5411	88	1	[	[	X
ejpam-5411	88	2	9	9	NUM
ejpam-5411	88	3	]	]	PUNCT
ejpam-5411	88	4	let	let	VERB
ejpam-5411	88	5	α	α	PRON
ejpam-5411	88	6	∈	∈	PROPN
ejpam-5411	88	7	(	(	PUNCT
ejpam-5411	88	8	0	0	NUM
ejpam-5411	88	9	,	,	PUNCT
ejpam-5411	88	10	1	1	NUM
ejpam-5411	88	11	]	]	PUNCT
ejpam-5411	88	12	and	and	CCONJ
ejpam-5411	88	13	f	f	PROPN
ejpam-5411	88	14	is	be	AUX
ejpam-5411	88	15	α	α	NOUN
ejpam-5411	88	16	-	-	NOUN
ejpam-5411	88	17	differentiable	differentiable	ADJ
ejpam-5411	88	18	at	at	ADP
ejpam-5411	88	19	a	a	DET
ejpam-5411	88	20	point	point	NOUN
ejpam-5411	88	21	t	t	NOUN
ejpam-5411	88	22	>	>	X
ejpam-5411	88	23	0	0	X
ejpam-5411	88	24	.	.	PUNCT
ejpam-5411	89	1	then	then	ADV
ejpam-5411	89	2	(	(	PUNCT
ejpam-5411	89	3	i	i	NOUN
ejpam-5411	89	4	)	)	PUNCT
ejpam-5411	89	5	t	t	PROPN
ejpam-5411	89	6	(	(	PUNCT
ejpam-5411	89	7	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	89	8	)	)	PUNCT
ejpam-5411	89	9	=	=	PUNCT
ejpam-5411	89	10	t1−αf	t1−αf	NUM
ejpam-5411	89	11	′(t	′(t	NOUN
ejpam-5411	89	12	)	)	PUNCT
ejpam-5411	89	13	.	.	PUNCT
ejpam-5411	90	1	(	(	PUNCT
ejpam-5411	90	2	ii	ii	PROPN
ejpam-5411	90	3	)	)	PUNCT
ejpam-5411	90	4	t	t	PROPN
ejpam-5411	90	5	(	(	PUNCT
ejpam-5411	90	6	α)(ect	α)(ect	NOUN
ejpam-5411	90	7	)	)	PUNCT
ejpam-5411	90	8	=	=	PUNCT
ejpam-5411	91	1	c	c	NOUN
ejpam-5411	91	2	t1−αect	t1−αect	NOUN
ejpam-5411	91	3	,	,	PUNCT
ejpam-5411	91	4	c	c	PROPN
ejpam-5411	91	5	∈	∈	PROPN
ejpam-5411	91	6	r	r	NOUN
ejpam-5411	91	7	or	or	CCONJ
ejpam-5411	91	8	c.	c.	NOUN
ejpam-5411	91	9	(	(	PUNCT
ejpam-5411	91	10	iii	iii	PROPN
ejpam-5411	91	11	)	)	PUNCT
ejpam-5411	91	12	t	t	NOUN
ejpam-5411	91	13	(	(	PUNCT
ejpam-5411	91	14	α	α	NOUN
ejpam-5411	91	15	)	)	PUNCT
ejpam-5411	91	16	(	(	PUNCT
ejpam-5411	91	17	t	t	PROPN
ejpam-5411	91	18	α	α	PROPN
ejpam-5411	91	19	α	α	NOUN
ejpam-5411	91	20	)	)	PUNCT
ejpam-5411	91	21	=	=	SYM
ejpam-5411	92	1	1	1	X
ejpam-5411	92	2	.	.	PUNCT
ejpam-5411	93	1	t.	t.	PROPN
ejpam-5411	93	2	abdeljawad	abdeljawad	PROPN
ejpam-5411	93	3	et	et	PROPN
ejpam-5411	93	4	al	al	PROPN
ejpam-5411	93	5	.	.	PUNCT
ejpam-5411	93	6	/	/	SYM
ejpam-5411	93	7	eur	eur	PROPN
ejpam-5411	93	8	.	.	PUNCT
ejpam-5411	94	1	j.	j.	PROPN
ejpam-5411	94	2	pure	pure	PROPN
ejpam-5411	94	3	appl	appl	PROPN
ejpam-5411	94	4	.	.	PROPN
ejpam-5411	94	5	math	math	PROPN
ejpam-5411	94	6	,	,	PUNCT
ejpam-5411	94	7	17	17	NUM
ejpam-5411	94	8	(	(	PUNCT
ejpam-5411	94	9	4	4	NUM
ejpam-5411	94	10	)	)	PUNCT
ejpam-5411	94	11	(	(	PUNCT
ejpam-5411	94	12	2024	2024	NUM
ejpam-5411	94	13	)	)	PUNCT
ejpam-5411	94	14	,	,	PUNCT
ejpam-5411	94	15	2405	2405	NUM
ejpam-5411	94	16	-	-	SYM
ejpam-5411	94	17	2430	2430	NUM
ejpam-5411	94	18	2409	2409	NUM
ejpam-5411	94	19	example	example	NOUN
ejpam-5411	94	20	2	2	NUM
ejpam-5411	94	21	.	.	PUNCT
ejpam-5411	95	1	[	[	X
ejpam-5411	95	2	9	9	NUM
ejpam-5411	95	3	]	]	PUNCT
ejpam-5411	95	4	(	(	PUNCT
ejpam-5411	95	5	i	i	NOUN
ejpam-5411	95	6	)	)	PUNCT
ejpam-5411	95	7	t	t	PROPN
ejpam-5411	95	8	(	(	PUNCT
ejpam-5411	95	9	α)(tp	α)(tp	NUM
ejpam-5411	95	10	)	)	PUNCT
ejpam-5411	95	11	=	=	PRON
ejpam-5411	95	12	ptα−p	ptα−p	PROPN
ejpam-5411	95	13	.	.	PUNCT
ejpam-5411	96	1	(	(	PUNCT
ejpam-5411	96	2	ii	ii	PROPN
ejpam-5411	96	3	)	)	PUNCT
ejpam-5411	96	4	t	t	PROPN
ejpam-5411	96	5	(	(	PUNCT
ejpam-5411	96	6	α)(eikt	α)(eikt	NUM
ejpam-5411	96	7	)	)	PUNCT
ejpam-5411	96	8	=	=	SYM
ejpam-5411	97	1	ikt1−αeikt	ikt1−αeikt	PROPN
ejpam-5411	97	2	,	,	PUNCT
ejpam-5411	97	3	k	k	PROPN
ejpam-5411	97	4	∈	∈	PROPN
ejpam-5411	97	5	z.	z.	PROPN
ejpam-5411	97	6	(	(	PUNCT
ejpam-5411	97	7	iii	iii	PROPN
ejpam-5411	97	8	)	)	PUNCT
ejpam-5411	97	9	t	t	PROPN
ejpam-5411	97	10	(	(	PUNCT
ejpam-5411	97	11	α)(sin	α)(sin	PROPN
ejpam-5411	97	12	(	(	PUNCT
ejpam-5411	97	13	1	1	NUM
ejpam-5411	97	14	α	α	NOUN
ejpam-5411	97	15	t	t	NOUN
ejpam-5411	97	16	α	α	NOUN
ejpam-5411	97	17	)	)	PUNCT
ejpam-5411	97	18	)	)	PUNCT
ejpam-5411	98	1	=	=	PUNCT
ejpam-5411	98	2	cos	cos	PROPN
ejpam-5411	98	3	(	(	PUNCT
ejpam-5411	98	4	1	1	NUM
ejpam-5411	98	5	α	α	NOUN
ejpam-5411	98	6	t	t	NOUN
ejpam-5411	98	7	α	α	NOUN
ejpam-5411	98	8	)	)	PUNCT
ejpam-5411	98	9	.	.	PUNCT
ejpam-5411	99	1	(	(	PUNCT
ejpam-5411	99	2	iv	iv	X
ejpam-5411	99	3	)	)	PUNCT
ejpam-5411	99	4	t	t	NOUN
ejpam-5411	99	5	(	(	PUNCT
ejpam-5411	99	6	1	1	NUM
ejpam-5411	99	7	2	2	NUM
ejpam-5411	99	8	)	)	PUNCT
ejpam-5411	99	9	(	(	PUNCT
ejpam-5411	99	10	2	2	NUM
ejpam-5411	99	11	√	√	NUM
ejpam-5411	99	12	t	t	PROPN
ejpam-5411	99	13	)	)	PUNCT
ejpam-5411	99	14	=	=	SYM
ejpam-5411	100	1	1	1	X
ejpam-5411	100	2	.	.	PUNCT
ejpam-5411	100	3	(	(	PUNCT
ejpam-5411	100	4	v	v	NOUN
ejpam-5411	100	5	)	)	PUNCT
ejpam-5411	100	6	t	t	NOUN
ejpam-5411	100	7	(	(	PUNCT
ejpam-5411	100	8	2α)(et	2α)(et	NUM
ejpam-5411	100	9	)	)	PUNCT
ejpam-5411	100	10	=	=	SYM
ejpam-5411	100	11	t	t	PROPN
ejpam-5411	100	12	(	(	PUNCT
ejpam-5411	100	13	α)(t1−αet	α)(t1−αet	NUM
ejpam-5411	100	14	)	)	PUNCT
ejpam-5411	100	15	=	=	SYM
ejpam-5411	101	1	t1−α(t1−αet)′	t1−α(t1−αet)′	NOUN
ejpam-5411	101	2	=	=	SYM
ejpam-5411	102	1	t1−αet((1	t1−αet((1	NUM
ejpam-5411	102	2	−	−	PROPN
ejpam-5411	103	1	α)t−α	α)t−α	NUM
ejpam-5411	103	2	+	+	CCONJ
ejpam-5411	103	3	t1−α	t1−α	NOUN
ejpam-5411	103	4	)	)	PUNCT
ejpam-5411	103	5	.	.	PUNCT
ejpam-5411	104	1	definition	definition	NOUN
ejpam-5411	104	2	4	4	NUM
ejpam-5411	104	3	.	.	PUNCT
ejpam-5411	105	1	[	[	X
ejpam-5411	105	2	1	1	X
ejpam-5411	105	3	]	]	PUNCT
ejpam-5411	105	4	the	the	DET
ejpam-5411	105	5	conformable	conformable	ADJ
ejpam-5411	105	6	fractional	fractional	ADJ
ejpam-5411	105	7	integral	integral	ADJ
ejpam-5411	105	8	of	of	ADP
ejpam-5411	105	9	order	order	NOUN
ejpam-5411	105	10	0	0	PUNCT
ejpam-5411	105	11	<	<	X
ejpam-5411	105	12	α	α	PROPN
ejpam-5411	105	13	≤	≤	NUM
ejpam-5411	105	14	1	1	NUM
ejpam-5411	105	15	is	be	AUX
ejpam-5411	105	16	defined	define	VERB
ejpam-5411	105	17	by	by	ADP
ejpam-5411	105	18	iα(f)(t	iα(f)(t	PRON
ejpam-5411	105	19	)	)	PUNCT
ejpam-5411	106	1	=	=	SYM
ejpam-5411	106	2	∫	∫	PROPN
ejpam-5411	106	3	t	t	PROPN
ejpam-5411	106	4	0	0	NUM
ejpam-5411	106	5	sα−1f(s	sα−1f(	NOUN
ejpam-5411	106	6	)	)	PUNCT
ejpam-5411	106	7	ds	ds	PROPN
ejpam-5411	106	8	,	,	PUNCT
ejpam-5411	106	9	t	t	PROPN
ejpam-5411	106	10	∈	∈	PROPN
ejpam-5411	107	1	[	[	X
ejpam-5411	107	2	0,+∞	0,+∞	NUM
ejpam-5411	107	3	[	[	X
ejpam-5411	107	4	.	.	PUNCT
ejpam-5411	108	1	lemma	lemma	PROPN
ejpam-5411	108	2	1	1	NUM
ejpam-5411	108	3	.	.	PUNCT
ejpam-5411	109	1	[	[	X
ejpam-5411	109	2	1	1	X
ejpam-5411	109	3	]	]	PUNCT
ejpam-5411	109	4	assume	assume	VERB
ejpam-5411	109	5	that	that	SCONJ
ejpam-5411	109	6	f	f	X
ejpam-5411	109	7	:	:	PUNCT
ejpam-5411	110	1	[	[	X
ejpam-5411	110	2	0,+∞	0,+∞	NUM
ejpam-5411	110	3	)	)	PUNCT
ejpam-5411	110	4	→	→	SYM
ejpam-5411	110	5	r	r	NOUN
ejpam-5411	110	6	is	be	AUX
ejpam-5411	110	7	continuous	continuous	ADJ
ejpam-5411	110	8	and	and	CCONJ
ejpam-5411	110	9	0	0	NUM
ejpam-5411	110	10	<	<	X
ejpam-5411	110	11	α	α	PROPN
ejpam-5411	110	12	≤	≤	NUM
ejpam-5411	110	13	1	1	NUM
ejpam-5411	110	14	.	.	PUNCT
ejpam-5411	111	1	then	then	ADV
ejpam-5411	111	2	,	,	PUNCT
ejpam-5411	111	3	for	for	ADP
ejpam-5411	111	4	all	all	DET
ejpam-5411	111	5	t	t	PROPN
ejpam-5411	111	6	>	>	X
ejpam-5411	111	7	0	0	NUM
ejpam-5411	111	8	,	,	PUNCT
ejpam-5411	111	9	we	we	PRON
ejpam-5411	111	10	have	have	VERB
ejpam-5411	111	11	t	t	PROPN
ejpam-5411	111	12	(	(	PUNCT
ejpam-5411	111	13	α)(iα(f))(t	α)(iα(f))(t	PROPN
ejpam-5411	111	14	)	)	PUNCT
ejpam-5411	111	15	=	=	SYM
ejpam-5411	111	16	f(t	f(t	NOUN
ejpam-5411	111	17	)	)	PUNCT
ejpam-5411	111	18	lemma	lemma	PROPN
ejpam-5411	111	19	2	2	NUM
ejpam-5411	111	20	.	.	PUNCT
ejpam-5411	112	1	[	[	X
ejpam-5411	112	2	1	1	X
ejpam-5411	112	3	]	]	PUNCT
ejpam-5411	112	4	let	let	VERB
ejpam-5411	112	5	f	f	NOUN
ejpam-5411	112	6	:	:	PUNCT
ejpam-5411	113	1	[	[	X
ejpam-5411	113	2	0,+∞	0,+∞	NUM
ejpam-5411	113	3	)	)	PUNCT
ejpam-5411	113	4	→	→	PUNCT
ejpam-5411	113	5	r	r	NOUN
ejpam-5411	113	6	be	be	VERB
ejpam-5411	113	7	α	α	NOUN
ejpam-5411	113	8	-	-	ADJ
ejpam-5411	113	9	differentiable	differentiable	ADJ
ejpam-5411	113	10	and	and	CCONJ
ejpam-5411	113	11	0	0	NUM
ejpam-5411	113	12	<	<	X
ejpam-5411	113	13	α	α	X
ejpam-5411	113	14	≤	≤	NUM
ejpam-5411	113	15	1	1	NUM
ejpam-5411	113	16	.	.	PUNCT
ejpam-5411	114	1	then	then	ADV
ejpam-5411	114	2	,	,	PUNCT
ejpam-5411	114	3	for	for	ADP
ejpam-5411	114	4	all	all	DET
ejpam-5411	114	5	t	t	PROPN
ejpam-5411	114	6	>	>	X
ejpam-5411	114	7	0	0	NUM
ejpam-5411	114	8	we	we	PRON
ejpam-5411	114	9	have	have	VERB
ejpam-5411	114	10	iα(t	iα(t	NOUN
ejpam-5411	114	11	(	(	PUNCT
ejpam-5411	114	12	α)(f))(t	α)(f))(t	ADJ
ejpam-5411	114	13	)	)	PUNCT
ejpam-5411	114	14	=	=	SYM
ejpam-5411	114	15	f(t	f(t	NOUN
ejpam-5411	114	16	)	)	PUNCT
ejpam-5411	115	1	−	−	PROPN
ejpam-5411	115	2	f(0	f(0	NOUN
ejpam-5411	115	3	)	)	PUNCT
ejpam-5411	115	4	.	.	PUNCT
ejpam-5411	116	1	let	let	VERB
ejpam-5411	116	2	x	x	PRON
ejpam-5411	116	3	be	be	AUX
ejpam-5411	116	4	a	a	DET
ejpam-5411	116	5	banach	banach	NOUN
ejpam-5411	116	6	space	space	NOUN
ejpam-5411	116	7	,	,	PUNCT
ejpam-5411	116	8	and	and	CCONJ
ejpam-5411	116	9	f	f	PROPN
ejpam-5411	116	10	is	be	AUX
ejpam-5411	116	11	a	a	DET
ejpam-5411	116	12	periodic	periodic	ADJ
ejpam-5411	116	13	function	function	NOUN
ejpam-5411	116	14	with	with	ADP
ejpam-5411	116	15	period	period	NOUN
ejpam-5411	116	16	t	t	NOUN
ejpam-5411	116	17	on	on	ADP
ejpam-5411	116	18	r.	r.	PROPN
ejpam-5411	116	19	for	for	ADP
ejpam-5411	116	20	a	a	DET
ejpam-5411	116	21	function	function	NOUN
ejpam-5411	116	22	f	f	PROPN
ejpam-5411	116	23	∈	∈	PROPN
ejpam-5411	116	24	l1(0	l1(0	PROPN
ejpam-5411	116	25	,	,	PUNCT
ejpam-5411	116	26	t	t	PROPN
ejpam-5411	116	27	;	;	PUNCT
ejpam-5411	116	28	x	x	X
ejpam-5411	116	29	)	)	PUNCT
ejpam-5411	116	30	,	,	PUNCT
ejpam-5411	116	31	the	the	DET
ejpam-5411	116	32	kth	kth	PROPN
ejpam-5411	116	33	fourier	fourier	NOUN
ejpam-5411	116	34	coefficient	coefficient	NOUN
ejpam-5411	116	35	of	of	ADP
ejpam-5411	116	36	f	f	PROPN
ejpam-5411	116	37	is	be	AUX
ejpam-5411	116	38	given	give	VERB
ejpam-5411	116	39	by	by	ADP
ejpam-5411	116	40	f(f(t))(k	f(f(t))(k	NOUN
ejpam-5411	116	41	)	)	PUNCT
ejpam-5411	116	42	=	=	SYM
ejpam-5411	117	1	1	1	NUM
ejpam-5411	117	2	t	t	NOUN
ejpam-5411	117	3	∫	∫	PROPN
ejpam-5411	117	4	t	t	PROPN
ejpam-5411	117	5	0	0	NUM
ejpam-5411	117	6	e−ik	e−ik	PROPN
ejpam-5411	117	7	2π	2π	PROPN
ejpam-5411	117	8	t	t	PROPN
ejpam-5411	117	9	tf(t)dt	tf(t)dt	NOUN
ejpam-5411	117	10	.	.	PUNCT
ejpam-5411	118	1	definition	definition	NOUN
ejpam-5411	118	2	5	5	NUM
ejpam-5411	118	3	.	.	PUNCT
ejpam-5411	119	1	[	[	X
ejpam-5411	119	2	22	22	NUM
ejpam-5411	119	3	]	]	PUNCT
ejpam-5411	119	4	let	let	VERB
ejpam-5411	119	5	f	f	PRON
ejpam-5411	119	6	:	:	PUNCT
ejpam-5411	120	1	[	[	X
ejpam-5411	120	2	0	0	NUM
ejpam-5411	120	3	;	;	PUNCT
ejpam-5411	120	4	+	+	SYM
ejpam-5411	120	5	∞[→	∞[→	ADJ
ejpam-5411	120	6	r	r	NOUN
ejpam-5411	120	7	be	be	VERB
ejpam-5411	120	8	a	a	DET
ejpam-5411	120	9	given	give	VERB
ejpam-5411	120	10	function	function	NOUN
ejpam-5411	120	11	and	and	CCONJ
ejpam-5411	120	12	0	0	NUM
ejpam-5411	120	13	<	<	X
ejpam-5411	120	14	α	α	PROPN
ejpam-5411	120	15	≤	≤	ADJ
ejpam-5411	120	16	1	1	NUM
ejpam-5411	120	17	.	.	PUNCT
ejpam-5411	121	1	then	then	ADV
ejpam-5411	121	2	the	the	DET
ejpam-5411	121	3	conformable	conformable	ADJ
ejpam-5411	121	4	fractional	fractional	ADJ
ejpam-5411	121	5	laplace	laplace	NOUN
ejpam-5411	121	6	transform	transform	NOUN
ejpam-5411	121	7	of	of	ADP
ejpam-5411	121	8	f	f	PROPN
ejpam-5411	121	9	is	be	AUX
ejpam-5411	121	10	defined	define	VERB
ejpam-5411	121	11	as	as	ADP
ejpam-5411	121	12	:	:	PUNCT
ejpam-5411	121	13	lα(f(t))(λ	lα(f(t))(λ	NOUN
ejpam-5411	121	14	)	)	PUNCT
ejpam-5411	122	1	=	=	SYM
ejpam-5411	122	2	∫	∫	PROPN
ejpam-5411	123	1	+	+	NUM
ejpam-5411	123	2	∞	∞	NOUN
ejpam-5411	123	3	0	0	NUM
ejpam-5411	123	4	e−λ	e−λ	NOUN
ejpam-5411	123	5	tα	tα	VERB
ejpam-5411	123	6	α	α	PRON
ejpam-5411	123	7	tα−1f(t)dt	tα−1f(t)dt	PROPN
ejpam-5411	123	8	provided	provide	VERB
ejpam-5411	123	9	the	the	DET
ejpam-5411	123	10	integral	integral	ADJ
ejpam-5411	123	11	exists	exist	NOUN
ejpam-5411	123	12	.	.	PUNCT
ejpam-5411	124	1	theorem	theorem	NOUN
ejpam-5411	124	2	2	2	NUM
ejpam-5411	124	3	.	.	PUNCT
ejpam-5411	125	1	[	[	X
ejpam-5411	125	2	22	22	NUM
ejpam-5411	125	3	]	]	PUNCT
ejpam-5411	125	4	let	let	VERB
ejpam-5411	125	5	a	a	PRON
ejpam-5411	125	6	:	:	PUNCT
ejpam-5411	125	7	[	[	X
ejpam-5411	125	8	0	0	NUM
ejpam-5411	125	9	;	;	PUNCT
ejpam-5411	125	10	+	+	SYM
ejpam-5411	125	11	∞[→	∞[→	ADJ
ejpam-5411	125	12	r	r	NOUN
ejpam-5411	125	13	be	be	VERB
ejpam-5411	125	14	a	a	DET
ejpam-5411	125	15	function	function	NOUN
ejpam-5411	125	16	and	and	CCONJ
ejpam-5411	125	17	0	0	NUM
ejpam-5411	125	18	<	<	X
ejpam-5411	125	19	α	α	PROPN
ejpam-5411	125	20	≤	≤	ADJ
ejpam-5411	125	21	1	1	NUM
ejpam-5411	125	22	.	.	PUNCT
ejpam-5411	125	23	then	then	ADV
ejpam-5411	125	24	lα(a(t))(λ	lα(a(t))(λ	VERB
ejpam-5411	125	25	)	)	PUNCT
ejpam-5411	126	1	=	=	PUNCT
ejpam-5411	126	2	l(a((αt	l(a((αt	PROPN
ejpam-5411	126	3	)	)	PUNCT
ejpam-5411	126	4	1	1	NUM
ejpam-5411	126	5	α	α	NOUN
ejpam-5411	126	6	)	)	PUNCT
ejpam-5411	126	7	)	)	PUNCT
ejpam-5411	126	8	(	(	PUNCT
ejpam-5411	126	9	λ	λ	NOUN
ejpam-5411	126	10	)	)	PUNCT
ejpam-5411	126	11	,	,	PUNCT
ejpam-5411	126	12	λ	λ	PROPN
ejpam-5411	126	13	∈	∈	PROPN
ejpam-5411	126	14	c.	c.	NOUN
ejpam-5411	126	15	where	where	SCONJ
ejpam-5411	126	16	l(a(t))(λ	l(a(t))(λ	NUM
ejpam-5411	126	17	)	)	PUNCT
ejpam-5411	126	18	=	=	SYM
ejpam-5411	127	1	∫	∫	PROPN
ejpam-5411	128	1	+	+	NUM
ejpam-5411	128	2	∞	∞	PROPN
ejpam-5411	128	3	0	0	X
ejpam-5411	128	4	e−λta(t)dt	e−λta(t)dt	PROPN
ejpam-5411	128	5	denotes	denote	VERB
ejpam-5411	128	6	the	the	DET
ejpam-5411	128	7	laplace	laplace	NOUN
ejpam-5411	128	8	transform	transform	NOUN
ejpam-5411	128	9	of	of	ADP
ejpam-5411	128	10	a(t	a(t	NOUN
ejpam-5411	128	11	)	)	PUNCT
ejpam-5411	128	12	.	.	PUNCT
ejpam-5411	129	1	theorem	theorem	VERB
ejpam-5411	129	2	3	3	NUM
ejpam-5411	129	3	.	.	PUNCT
ejpam-5411	130	1	[	[	X
ejpam-5411	130	2	7	7	X
ejpam-5411	130	3	]	]	PUNCT
ejpam-5411	130	4	given	give	VERB
ejpam-5411	130	5	a	a	DET
ejpam-5411	130	6	∈	∈	PROPN
ejpam-5411	130	7	l1(r+	l1(r+	PROPN
ejpam-5411	130	8	)	)	PUNCT
ejpam-5411	130	9	and	and	CCONJ
ejpam-5411	130	10	g	g	NOUN
ejpam-5411	130	11	:	:	PUNCT
ejpam-5411	131	1	[	[	X
ejpam-5411	131	2	0	0	NUM
ejpam-5411	131	3	,	,	PUNCT
ejpam-5411	131	4	2π	2π	NOUN
ejpam-5411	131	5	]	]	PUNCT
ejpam-5411	131	6	→	→	PUNCT
ejpam-5411	131	7	x	x	X
ejpam-5411	131	8	is	be	AUX
ejpam-5411	131	9	a	a	DET
ejpam-5411	131	10	periodic	periodic	ADJ
ejpam-5411	131	11	function	function	NOUN
ejpam-5411	131	12	with	with	ADP
ejpam-5411	131	13	period	period	NOUN
ejpam-5411	131	14	2π	2π	PROPN
ejpam-5411	131	15	(	(	PUNCT
ejpam-5411	131	16	extended	extend	VERB
ejpam-5411	131	17	by	by	ADP
ejpam-5411	131	18	periodicity	periodicity	NOUN
ejpam-5411	131	19	to	to	ADP
ejpam-5411	131	20	r	r	NOUN
ejpam-5411	131	21	)	)	PUNCT
ejpam-5411	131	22	,	,	PUNCT
ejpam-5411	131	23	where	where	SCONJ
ejpam-5411	131	24	x	x	PRON
ejpam-5411	131	25	is	be	AUX
ejpam-5411	131	26	a	a	DET
ejpam-5411	131	27	banach	banach	NOUN
ejpam-5411	131	28	space	space	NOUN
ejpam-5411	131	29	.	.	PUNCT
ejpam-5411	132	1	we	we	PRON
ejpam-5411	132	2	find	find	VERB
ejpam-5411	132	3	that	that	SCONJ
ejpam-5411	132	4	f(f	f(f	PROPN
ejpam-5411	132	5	(	(	PUNCT
ejpam-5411	132	6	t))(k	t))(k	NOUN
ejpam-5411	132	7	)	)	PUNCT
ejpam-5411	132	8	=	=	SYM
ejpam-5411	132	9	l(a(t))(ik)f(g(t))(k	l(a(t))(ik)f(g(t))(k	PROPN
ejpam-5411	132	10	)	)	PUNCT
ejpam-5411	132	11	,	,	PUNCT
ejpam-5411	132	12	k	k	PROPN
ejpam-5411	132	13	∈	∈	PROPN
ejpam-5411	132	14	z	z	X
ejpam-5411	132	15	(	(	PUNCT
ejpam-5411	132	16	1	1	NUM
ejpam-5411	132	17	)	)	PUNCT
ejpam-5411	132	18	where	where	SCONJ
ejpam-5411	132	19	the	the	DET
ejpam-5411	132	20	function	function	NOUN
ejpam-5411	132	21	f	f	PROPN
ejpam-5411	132	22	is	be	AUX
ejpam-5411	132	23	defined	define	VERB
ejpam-5411	132	24	by	by	ADP
ejpam-5411	132	25	f	f	PROPN
ejpam-5411	132	26	(	(	PUNCT
ejpam-5411	132	27	t	t	PROPN
ejpam-5411	132	28	)	)	PUNCT
ejpam-5411	132	29	=	=	SYM
ejpam-5411	132	30	∫	∫	PROPN
ejpam-5411	132	31	t	t	PROPN
ejpam-5411	132	32	−∞	−∞	ADP
ejpam-5411	132	33	a(t	a(t	PROPN
ejpam-5411	132	34	−	−	NOUN
ejpam-5411	132	35	s)g(s)ds	s)g(s)ds	NOUN
ejpam-5411	132	36	=	=	SYM
ejpam-5411	132	37	∫	∫	PROPN
ejpam-5411	133	1	+	+	NUM
ejpam-5411	133	2	∞	∞	PROPN
ejpam-5411	133	3	0	0	NUM
ejpam-5411	133	4	a(s)g(t	a(s)g(t	PROPN
ejpam-5411	133	5	−	−	PROPN
ejpam-5411	133	6	s)ds	s)ds	PROPN
ejpam-5411	133	7	is	be	AUX
ejpam-5411	133	8	continuous	continuous	ADJ
ejpam-5411	133	9	and	and	CCONJ
ejpam-5411	133	10	bounded	bound	VERB
ejpam-5411	133	11	on	on	ADP
ejpam-5411	133	12	r.	r.	PROPN
ejpam-5411	133	13	t.	t.	PROPN
ejpam-5411	133	14	abdeljawad	abdeljawad	PROPN
ejpam-5411	133	15	et	et	PROPN
ejpam-5411	133	16	al	al	PROPN
ejpam-5411	133	17	.	.	PUNCT
ejpam-5411	133	18	/	/	SYM
ejpam-5411	133	19	eur	eur	PROPN
ejpam-5411	133	20	.	.	PUNCT
ejpam-5411	134	1	j.	j.	PROPN
ejpam-5411	134	2	pure	pure	PROPN
ejpam-5411	134	3	appl	appl	PROPN
ejpam-5411	134	4	.	.	PROPN
ejpam-5411	134	5	math	math	PROPN
ejpam-5411	134	6	,	,	PUNCT
ejpam-5411	134	7	17	17	NUM
ejpam-5411	134	8	(	(	PUNCT
ejpam-5411	134	9	4	4	NUM
ejpam-5411	134	10	)	)	PUNCT
ejpam-5411	134	11	(	(	PUNCT
ejpam-5411	134	12	2024	2024	NUM
ejpam-5411	134	13	)	)	PUNCT
ejpam-5411	134	14	,	,	PUNCT
ejpam-5411	134	15	2405	2405	NUM
ejpam-5411	134	16	-	-	SYM
ejpam-5411	134	17	2430	2430	NUM
ejpam-5411	134	18	2410	2410	NUM
ejpam-5411	134	19	this	this	DET
ejpam-5411	134	20	theorem	theorem	NOUN
ejpam-5411	134	21	has	have	AUX
ejpam-5411	134	22	been	be	AUX
ejpam-5411	134	23	used	use	VERB
ejpam-5411	134	24	by	by	ADP
ejpam-5411	134	25	many	many	ADJ
ejpam-5411	134	26	authors	author	NOUN
ejpam-5411	134	27	to	to	PART
ejpam-5411	134	28	solve	solve	VERB
ejpam-5411	134	29	some	some	DET
ejpam-5411	134	30	integro	integro	ADJ
ejpam-5411	134	31	-	-	PUNCT
ejpam-5411	134	32	differential	differential	NOUN
ejpam-5411	134	33	equations	equation	NOUN
ejpam-5411	134	34	using	use	VERB
ejpam-5411	134	35	the	the	DET
ejpam-5411	134	36	fourier	fourier	NOUN
ejpam-5411	134	37	transform	transform	NOUN
ejpam-5411	134	38	(	(	PUNCT
ejpam-5411	134	39	[	[	X
ejpam-5411	134	40	7	7	NUM
ejpam-5411	134	41	]	]	PUNCT
ejpam-5411	134	42	,	,	PUNCT
ejpam-5411	134	43	[	[	X
ejpam-5411	134	44	4	4	NUM
ejpam-5411	134	45	]	]	NUM
ejpam-5411	134	46	)	)	PUNCT
ejpam-5411	134	47	.	.	PUNCT
ejpam-5411	135	1	in	in	ADP
ejpam-5411	135	2	the	the	DET
ejpam-5411	135	3	next	next	ADJ
ejpam-5411	135	4	section	section	NOUN
ejpam-5411	135	5	,	,	PUNCT
ejpam-5411	135	6	we	we	PRON
ejpam-5411	135	7	present	present	VERB
ejpam-5411	135	8	some	some	DET
ejpam-5411	135	9	results	result	NOUN
ejpam-5411	135	10	of	of	ADP
ejpam-5411	135	11	α	α	NOUN
ejpam-5411	135	12	-	-	ADJ
ejpam-5411	135	13	periodic	periodic	ADJ
ejpam-5411	135	14	functions	function	NOUN
ejpam-5411	135	15	.	.	PUNCT
ejpam-5411	136	1	3	3	X
ejpam-5411	136	2	.	.	X
ejpam-5411	137	1	some	some	DET
ejpam-5411	137	2	results	result	NOUN
ejpam-5411	137	3	of	of	ADP
ejpam-5411	137	4	α	α	NOUN
ejpam-5411	137	5	-	-	ADJ
ejpam-5411	137	6	periodic	periodic	ADJ
ejpam-5411	137	7	functions	function	NOUN
ejpam-5411	137	8	definition	definition	NOUN
ejpam-5411	137	9	6	6	NUM
ejpam-5411	137	10	.	.	PUNCT
ejpam-5411	138	1	[	[	X
ejpam-5411	138	2	8	8	NUM
ejpam-5411	138	3	]	]	SYM
ejpam-5411	138	4	(	(	PUNCT
ejpam-5411	138	5	α	α	NOUN
ejpam-5411	138	6	-	-	ADJ
ejpam-5411	138	7	periodic	periodic	ADJ
ejpam-5411	138	8	function	function	NOUN
ejpam-5411	138	9	)	)	PUNCT
ejpam-5411	138	10	let	let	VERB
ejpam-5411	138	11	0	0	PUNCT
ejpam-5411	138	12	<	<	X
ejpam-5411	138	13	α	α	X
ejpam-5411	138	14	≤	≤	NUM
ejpam-5411	138	15	1	1	NUM
ejpam-5411	138	16	.	.	PUNCT
ejpam-5411	139	1	the	the	DET
ejpam-5411	139	2	function	function	NOUN
ejpam-5411	139	3	f	f	NOUN
ejpam-5411	139	4	:	:	PUNCT
ejpam-5411	140	1	[	[	X
ejpam-5411	140	2	0,+∞	0,+∞	NUM
ejpam-5411	140	3	)	)	PUNCT
ejpam-5411	140	4	→	→	PUNCT
ejpam-5411	140	5	r	r	NOUN
ejpam-5411	140	6	is	be	AUX
ejpam-5411	140	7	called	call	VERB
ejpam-5411	140	8	α	α	PRON
ejpam-5411	140	9	-	-	NOUN
ejpam-5411	140	10	periodical	periodical	ADJ
ejpam-5411	140	11	with	with	ADP
ejpam-5411	140	12	period	period	NOUN
ejpam-5411	140	13	p	p	X
ejpam-5411	140	14	>	>	X
ejpam-5411	140	15	0	0	NUM
ejpam-5411	140	16	,	,	PUNCT
ejpam-5411	140	17	if	if	SCONJ
ejpam-5411	140	18	there	there	PRON
ejpam-5411	140	19	exists	exist	VERB
ejpam-5411	140	20	a	a	DET
ejpam-5411	140	21	continuous	continuous	ADJ
ejpam-5411	140	22	function	function	NOUN
ejpam-5411	140	23	g	g	NOUN
ejpam-5411	140	24	:	:	PUNCT
ejpam-5411	141	1	[	[	X
ejpam-5411	141	2	0,+∞	0,+∞	NUM
ejpam-5411	141	3	)	)	PUNCT
ejpam-5411	141	4	→	→	PUNCT
ejpam-5411	141	5	r	r	NOUN
ejpam-5411	141	6	such	such	ADJ
ejpam-5411	141	7	that	that	SCONJ
ejpam-5411	141	8	f(t	f(t	NOUN
ejpam-5411	141	9	)	)	PUNCT
ejpam-5411	141	10	=	=	SYM
ejpam-5411	141	11	g	g	PROPN
ejpam-5411	141	12	(	(	PUNCT
ejpam-5411	141	13	tα	tα	PROPN
ejpam-5411	141	14	α	α	PROPN
ejpam-5411	141	15	)	)	PUNCT
ejpam-5411	142	1	=	=	SYM
ejpam-5411	142	2	g	g	PROPN
ejpam-5411	142	3	(	(	PUNCT
ejpam-5411	142	4	tα	tα	PROPN
ejpam-5411	142	5	α	α	PROPN
ejpam-5411	143	1	+	+	CCONJ
ejpam-5411	143	2	pα	pα	PROPN
ejpam-5411	143	3	α	α	NOUN
ejpam-5411	143	4	)	)	PUNCT
ejpam-5411	143	5	for	for	ADP
ejpam-5411	143	6	all	all	DET
ejpam-5411	143	7	t	t	NOUN
ejpam-5411	143	8	∈	∈	PROPN
ejpam-5411	144	1	[	[	X
ejpam-5411	144	2	0,+∞	0,+∞	NUM
ejpam-5411	144	3	)	)	PUNCT
ejpam-5411	144	4	.	.	PUNCT
ejpam-5411	145	1	remark	remark	PROPN
ejpam-5411	145	2	1	1	NUM
ejpam-5411	145	3	.	.	PUNCT
ejpam-5411	146	1	:	:	PUNCT
ejpam-5411	146	2	(	(	PUNCT
ejpam-5411	146	3	i	i	NOUN
ejpam-5411	146	4	)	)	PUNCT
ejpam-5411	146	5	note	note	VERB
ejpam-5411	146	6	that	that	SCONJ
ejpam-5411	146	7	the	the	DET
ejpam-5411	146	8	continuity	continuity	NOUN
ejpam-5411	146	9	of	of	ADP
ejpam-5411	146	10	g	g	PROPN
ejpam-5411	146	11	implies	imply	VERB
ejpam-5411	146	12	that	that	PRON
ejpam-5411	146	13	of	of	ADP
ejpam-5411	146	14	f	f	PROPN
ejpam-5411	146	15	.	.	PUNCT
ejpam-5411	147	1	(	(	PUNCT
ejpam-5411	147	2	ii	ii	NOUN
ejpam-5411	147	3	)	)	PUNCT
ejpam-5411	147	4	the	the	DET
ejpam-5411	147	5	function	function	PROPN
ejpam-5411	147	6	g(t	g(t	PROPN
ejpam-5411	147	7	)	)	PUNCT
ejpam-5411	147	8	=	=	SYM
ejpam-5411	147	9	f((αt	f((αt	NOUN
ejpam-5411	147	10	)	)	PUNCT
ejpam-5411	147	11	1	1	NUM
ejpam-5411	147	12	α	α	NOUN
ejpam-5411	147	13	)	)	PUNCT
ejpam-5411	147	14	is	be	AUX
ejpam-5411	147	15	periodic	periodic	ADJ
ejpam-5411	147	16	with	with	ADP
ejpam-5411	147	17	period	period	NOUN
ejpam-5411	147	18	pα	pα	INTJ
ejpam-5411	147	19	α	α	INTJ
ejpam-5411	147	20	.	.	PUNCT
ejpam-5411	148	1	example	example	NOUN
ejpam-5411	149	1	3	3	X
ejpam-5411	149	2	.	.	PUNCT
ejpam-5411	150	1	let	let	VERB
ejpam-5411	150	2	0	0	NUM
ejpam-5411	150	3	<	<	X
ejpam-5411	150	4	α	α	X
ejpam-5411	150	5	≤	≤	NUM
ejpam-5411	150	6	1	1	NUM
ejpam-5411	150	7	.	.	PUNCT
ejpam-5411	151	1	for	for	ADP
ejpam-5411	151	2	all	all	DET
ejpam-5411	151	3	t	t	NOUN
ejpam-5411	151	4	∈	∈	PROPN
ejpam-5411	152	1	[	[	X
ejpam-5411	152	2	0	0	NUM
ejpam-5411	152	3	,	,	PUNCT
ejpam-5411	152	4	(	(	PUNCT
ejpam-5411	152	5	1	1	NUM
ejpam-5411	152	6	α	α	NOUN
ejpam-5411	152	7	)	)	PUNCT
ejpam-5411	152	8	1	1	NUM
ejpam-5411	152	9	α	α	NOUN
ejpam-5411	152	10	]	]	X
ejpam-5411	152	11	,	,	PUNCT
ejpam-5411	152	12	let	let	VERB
ejpam-5411	152	13	us	we	PRON
ejpam-5411	152	14	consider	consider	VERB
ejpam-5411	152	15	the	the	DET
ejpam-5411	152	16	following	follow	VERB
ejpam-5411	152	17	functions	function	NOUN
ejpam-5411	152	18	f1(t	f1(t	PROPN
ejpam-5411	152	19	)	)	PUNCT
ejpam-5411	152	20	and	and	CCONJ
ejpam-5411	152	21	f2(t	f2(t	PROPN
ejpam-5411	152	22	)	)	PUNCT
ejpam-5411	152	23	f1(t	f1(t	PROPN
ejpam-5411	152	24	)	)	PUNCT
ejpam-5411	153	1	=	=	SYM
ejpam-5411	154	1			PRON
ejpam-5411	154	2	tα	tα	VERB
ejpam-5411	154	3	α	α	PROPN
ejpam-5411	154	4	,	,	PUNCT
ejpam-5411	154	5	0	0	NUM
ejpam-5411	154	6	≤	≤	NUM
ejpam-5411	154	7	t	t	PROPN
ejpam-5411	154	8	≤	≤	NUM
ejpam-5411	154	9	(	(	PUNCT
ejpam-5411	154	10	1	1	NUM
ejpam-5411	154	11	2α	2α	NOUN
ejpam-5411	154	12	)	)	PUNCT
ejpam-5411	154	13	1	1	NUM
ejpam-5411	154	14	α	α	SYM
ejpam-5411	154	15	1	1	NUM
ejpam-5411	154	16	α2	α2	NOUN
ejpam-5411	154	17	−	−	PROPN
ejpam-5411	154	18	tα	tα	PROPN
ejpam-5411	154	19	α	α	PROPN
ejpam-5411	154	20	,	,	PUNCT
ejpam-5411	154	21	(	(	PUNCT
ejpam-5411	154	22	1	1	NUM
ejpam-5411	154	23	2α	2α	NOUN
ejpam-5411	154	24	)	)	PUNCT
ejpam-5411	154	25	1	1	NUM
ejpam-5411	154	26	α	α	NOUN
ejpam-5411	154	27	<	<	X
ejpam-5411	154	28	t	t	X
ejpam-5411	154	29	≤	≤	NUM
ejpam-5411	154	30	(	(	PUNCT
ejpam-5411	154	31	1	1	NUM
ejpam-5411	154	32	α	α	NOUN
ejpam-5411	154	33	)	)	PUNCT
ejpam-5411	154	34	1	1	NUM
ejpam-5411	154	35	α	α	NOUN
ejpam-5411	154	36	(	(	PUNCT
ejpam-5411	154	37	2	2	NUM
ejpam-5411	154	38	)	)	PUNCT
ejpam-5411	154	39	and	and	CCONJ
ejpam-5411	154	40	f2(t	f2(t	NOUN
ejpam-5411	154	41	)	)	PUNCT
ejpam-5411	154	42	=	=	PUNCT
ejpam-5411	154	43			NUM
ejpam-5411	154	44	tα	tα	VERB
ejpam-5411	154	45	α	α	PROPN
ejpam-5411	154	46	,	,	PUNCT
ejpam-5411	154	47	0	0	NUM
ejpam-5411	154	48	≤	≤	NUM
ejpam-5411	154	49	t	t	PROPN
ejpam-5411	154	50	≤	≤	NUM
ejpam-5411	154	51	(	(	PUNCT
ejpam-5411	154	52	1	1	NUM
ejpam-5411	154	53	4α	4α	NOUN
ejpam-5411	154	54	)	)	PUNCT
ejpam-5411	154	55	1	1	NUM
ejpam-5411	154	56	α	α	SYM
ejpam-5411	154	57	1	1	NUM
ejpam-5411	154	58	2α2	2α2	NUM
ejpam-5411	154	59	−	−	NOUN
ejpam-5411	154	60	tα	tα	PROPN
ejpam-5411	154	61	α	α	PROPN
ejpam-5411	154	62	,	,	PUNCT
ejpam-5411	154	63	(	(	PUNCT
ejpam-5411	154	64	1	1	NUM
ejpam-5411	154	65	4α	4α	NOUN
ejpam-5411	154	66	)	)	PUNCT
ejpam-5411	154	67	1	1	NUM
ejpam-5411	154	68	α	α	NOUN
ejpam-5411	154	69	<	<	X
ejpam-5411	154	70	t	t	X
ejpam-5411	154	71	≤	≤	NUM
ejpam-5411	154	72	(	(	PUNCT
ejpam-5411	154	73	3	3	NUM
ejpam-5411	154	74	4α	4α	NOUN
ejpam-5411	154	75	)	)	PUNCT
ejpam-5411	154	76	1	1	NUM
ejpam-5411	154	77	α	α	NOUN
ejpam-5411	154	78	tα	tα	VERB
ejpam-5411	154	79	α	α	PRON
ejpam-5411	154	80	−	−	PROPN
ejpam-5411	154	81	1	1	NUM
ejpam-5411	154	82	α2	α2	ADJ
ejpam-5411	154	83	,	,	PUNCT
ejpam-5411	154	84	(	(	PUNCT
ejpam-5411	154	85	3	3	NUM
ejpam-5411	154	86	4α	4α	NOUN
ejpam-5411	154	87	)	)	PUNCT
ejpam-5411	155	1	1	1	NUM
ejpam-5411	155	2	α	α	NOUN
ejpam-5411	155	3	<	<	X
ejpam-5411	155	4	t	t	X
ejpam-5411	155	5	≤	≤	NUM
ejpam-5411	155	6	(	(	PUNCT
ejpam-5411	155	7	1	1	NUM
ejpam-5411	155	8	α	α	NOUN
ejpam-5411	155	9	)	)	PUNCT
ejpam-5411	155	10	1	1	NUM
ejpam-5411	155	11	α	α	NOUN
ejpam-5411	155	12	(	(	PUNCT
ejpam-5411	155	13	3	3	X
ejpam-5411	155	14	)	)	PUNCT
ejpam-5411	155	15	we	we	PRON
ejpam-5411	155	16	have	have	VERB
ejpam-5411	155	17	f1(t	f1(t	X
ejpam-5411	155	18	)	)	PUNCT
ejpam-5411	155	19	=	=	SYM
ejpam-5411	155	20	g1	g1	PROPN
ejpam-5411	155	21	(	(	PUNCT
ejpam-5411	155	22	tα	tα	PROPN
ejpam-5411	155	23	α	α	PROPN
ejpam-5411	155	24	)	)	PUNCT
ejpam-5411	155	25	and	and	CCONJ
ejpam-5411	155	26	f2(t	f2(t	X
ejpam-5411	155	27	)	)	PUNCT
ejpam-5411	155	28	=	=	SYM
ejpam-5411	155	29	g2	g2	PROPN
ejpam-5411	155	30	(	(	PUNCT
ejpam-5411	155	31	tα	tα	PROPN
ejpam-5411	155	32	α	α	PROPN
ejpam-5411	155	33	)	)	PUNCT
ejpam-5411	155	34	,	,	PUNCT
ejpam-5411	155	35	where	where	SCONJ
ejpam-5411	155	36	g1(t	g1(t	ADP
ejpam-5411	155	37	)	)	PUNCT
ejpam-5411	155	38	=	=	SYM
ejpam-5411	155	39			PROPN
ejpam-5411	155	40	t	t	PROPN
ejpam-5411	155	41	,	,	PUNCT
ejpam-5411	155	42	0	0	NUM
ejpam-5411	155	43	≤	≤	NUM
ejpam-5411	155	44	t	t	NOUN
ejpam-5411	155	45	≤	≤	NUM
ejpam-5411	155	46	1	1	NUM
ejpam-5411	155	47	2α2	2α2	NUM
ejpam-5411	155	48	1	1	NUM
ejpam-5411	155	49	α2	α2	ADJ
ejpam-5411	155	50	−	−	PROPN
ejpam-5411	156	1	t	t	PROPN
ejpam-5411	157	1	,	,	PUNCT
ejpam-5411	157	2	1	1	NUM
ejpam-5411	157	3	2α2	2α2	NUM
ejpam-5411	157	4	<	<	X
ejpam-5411	157	5	t	t	X
ejpam-5411	157	6	≤	≤	NUM
ejpam-5411	157	7	1	1	NUM
ejpam-5411	157	8	α2	α2	ADJ
ejpam-5411	157	9	(	(	PUNCT
ejpam-5411	157	10	4	4	NUM
ejpam-5411	157	11	)	)	PUNCT
ejpam-5411	157	12	and	and	CCONJ
ejpam-5411	157	13	g2(t	g2(t	NOUN
ejpam-5411	157	14	)	)	PUNCT
ejpam-5411	157	15	=	=	PUNCT
ejpam-5411	157	16			PROPN
ejpam-5411	157	17	t	t	PROPN
ejpam-5411	157	18	,	,	PUNCT
ejpam-5411	157	19	0	0	NUM
ejpam-5411	157	20	≤	≤	NUM
ejpam-5411	157	21	t	t	NOUN
ejpam-5411	157	22	≤	≤	NUM
ejpam-5411	157	23	1	1	NUM
ejpam-5411	157	24	4α2	4α2	NUM
ejpam-5411	157	25	1	1	NUM
ejpam-5411	157	26	2α2	2α2	NUM
ejpam-5411	157	27	−	−	PROPN
ejpam-5411	157	28	t	t	PROPN
ejpam-5411	157	29	,	,	PUNCT
ejpam-5411	157	30	1	1	NUM
ejpam-5411	157	31	4α2	4α2	NOUN
ejpam-5411	157	32	<	<	X
ejpam-5411	157	33	t	t	X
ejpam-5411	157	34	≤	≤	NUM
ejpam-5411	157	35	3	3	NUM
ejpam-5411	157	36	4α2	4α2	NOUN
ejpam-5411	157	37	t−	t−	PROPN
ejpam-5411	157	38	1	1	NUM
ejpam-5411	157	39	α2	α2	ADJ
ejpam-5411	157	40	,	,	PUNCT
ejpam-5411	157	41	3	3	NUM
ejpam-5411	157	42	4α2	4α2	NOUN
ejpam-5411	157	43	<	<	X
ejpam-5411	157	44	t	t	X
ejpam-5411	157	45	≤	≤	NUM
ejpam-5411	157	46	1	1	NUM
ejpam-5411	157	47	α2	α2	ADJ
ejpam-5411	157	48	(	(	PUNCT
ejpam-5411	157	49	5	5	NUM
ejpam-5411	157	50	)	)	PUNCT
ejpam-5411	157	51	t.	t.	NOUN
ejpam-5411	157	52	abdeljawad	abdeljawad	NOUN
ejpam-5411	157	53	et	et	PROPN
ejpam-5411	157	54	al	al	PROPN
ejpam-5411	157	55	.	.	PUNCT
ejpam-5411	157	56	/	/	SYM
ejpam-5411	157	57	eur	eur	PROPN
ejpam-5411	157	58	.	.	PUNCT
ejpam-5411	158	1	j.	j.	PROPN
ejpam-5411	158	2	pure	pure	PROPN
ejpam-5411	158	3	appl	appl	PROPN
ejpam-5411	158	4	.	.	PROPN
ejpam-5411	158	5	math	math	PROPN
ejpam-5411	158	6	,	,	PUNCT
ejpam-5411	158	7	17	17	NUM
ejpam-5411	158	8	(	(	PUNCT
ejpam-5411	158	9	4	4	NUM
ejpam-5411	158	10	)	)	PUNCT
ejpam-5411	158	11	(	(	PUNCT
ejpam-5411	158	12	2024	2024	NUM
ejpam-5411	158	13	)	)	PUNCT
ejpam-5411	158	14	,	,	PUNCT
ejpam-5411	158	15	2405	2405	NUM
ejpam-5411	158	16	-	-	SYM
ejpam-5411	158	17	2430	2430	NUM
ejpam-5411	158	18	2411	2411	NUM
ejpam-5411	158	19	for	for	ADP
ejpam-5411	158	20	all	all	DET
ejpam-5411	158	21	t	t	NOUN
ejpam-5411	158	22	∈	∈	PROPN
ejpam-5411	159	1	[	[	X
ejpam-5411	159	2	0	0	NUM
ejpam-5411	159	3	,	,	PUNCT
ejpam-5411	159	4	1	1	NUM
ejpam-5411	159	5	α2	α2	ADJ
ejpam-5411	159	6	]	]	PUNCT
ejpam-5411	159	7	.	.	PUNCT
ejpam-5411	160	1	g1(t	g1(t	X
ejpam-5411	160	2	)	)	PUNCT
ejpam-5411	160	3	and	and	CCONJ
ejpam-5411	160	4	g2(t	g2(t	NOUN
ejpam-5411	160	5	)	)	PUNCT
ejpam-5411	160	6	are	be	AUX
ejpam-5411	160	7	countinuous	countinuous	ADJ
ejpam-5411	160	8	periodic	periodic	ADJ
ejpam-5411	160	9	functions	function	NOUN
ejpam-5411	160	10	with	with	ADP
ejpam-5411	160	11	period	period	NOUN
ejpam-5411	160	12	1	1	NUM
ejpam-5411	160	13	α2	α2	ADJ
ejpam-5411	160	14	for	for	ADP
ejpam-5411	160	15	all	all	DET
ejpam-5411	160	16	t	t	NOUN
ejpam-5411	160	17	∈	∈	PROPN
ejpam-5411	161	1	[	[	X
ejpam-5411	161	2	0,+∞	0,+∞	NUM
ejpam-5411	161	3	[	[	PUNCT
ejpam-5411	161	4	(	(	PUNCT
ejpam-5411	161	5	extended	extend	VERB
ejpam-5411	161	6	by	by	ADP
ejpam-5411	161	7	periodicity	periodicity	NOUN
ejpam-5411	161	8	to	to	ADP
ejpam-5411	161	9	[	[	X
ejpam-5411	161	10	0,+∞	0,+∞	PROPN
ejpam-5411	161	11	[	[	NOUN
ejpam-5411	161	12	)	)	PUNCT
ejpam-5411	161	13	.	.	PUNCT
ejpam-5411	162	1	then	then	ADV
ejpam-5411	162	2	f1(t	f1(t	NUM
ejpam-5411	162	3	)	)	PUNCT
ejpam-5411	162	4	and	and	CCONJ
ejpam-5411	162	5	f2(t	f2(t	PROPN
ejpam-5411	162	6	)	)	PUNCT
ejpam-5411	162	7	are	be	AUX
ejpam-5411	162	8	α	α	X
ejpam-5411	162	9	-	-	NOUN
ejpam-5411	162	10	periodic	periodic	NOUN
ejpam-5411	162	11	with	with	ADP
ejpam-5411	162	12	period	period	NOUN
ejpam-5411	162	13	p	p	X
ejpam-5411	162	14	=	=	PUNCT
ejpam-5411	162	15	(	(	PUNCT
ejpam-5411	162	16	1	1	NUM
ejpam-5411	162	17	α	α	NOUN
ejpam-5411	162	18	)	)	PUNCT
ejpam-5411	162	19	1	1	NUM
ejpam-5411	162	20	α	α	NOUN
ejpam-5411	162	21	for	for	ADP
ejpam-5411	162	22	all	all	DET
ejpam-5411	162	23	t	t	NOUN
ejpam-5411	162	24	∈	∈	PROPN
ejpam-5411	163	1	[	[	X
ejpam-5411	163	2	0,+∞	0,+∞	NUM
ejpam-5411	163	3	[	[	X
ejpam-5411	163	4	.	.	PUNCT
ejpam-5411	163	5	theorem	theorem	NOUN
ejpam-5411	163	6	4	4	NUM
ejpam-5411	163	7	.	.	PUNCT
ejpam-5411	164	1	let	let	VERB
ejpam-5411	164	2	0	0	NUM
ejpam-5411	164	3	<	<	X
ejpam-5411	164	4	α	α	PROPN
ejpam-5411	164	5	≤	≤	NUM
ejpam-5411	164	6	1	1	NUM
ejpam-5411	164	7	and	and	CCONJ
ejpam-5411	164	8	assume	assume	VERB
ejpam-5411	164	9	that	that	SCONJ
ejpam-5411	164	10	f	f	X
ejpam-5411	164	11	:	:	PUNCT
ejpam-5411	165	1	[	[	X
ejpam-5411	165	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	165	3	r	r	NOUN
ejpam-5411	165	4	is	be	AUX
ejpam-5411	165	5	a	a	DET
ejpam-5411	165	6	α	α	ADJ
ejpam-5411	165	7	-	-	ADJ
ejpam-5411	165	8	periodic	periodic	ADJ
ejpam-5411	165	9	function	function	NOUN
ejpam-5411	165	10	with	with	ADP
ejpam-5411	165	11	period	period	NOUN
ejpam-5411	165	12	p	p	PRON
ejpam-5411	165	13	such	such	ADJ
ejpam-5411	165	14	that	that	DET
ejpam-5411	165	15	iα(f)(p	iα(f)(p	NOUN
ejpam-5411	165	16	)	)	PUNCT
ejpam-5411	165	17	=	=	SYM
ejpam-5411	165	18	0	0	NUM
ejpam-5411	166	1	(	(	PUNCT
ejpam-5411	166	2	6	6	NUM
ejpam-5411	166	3	)	)	PUNCT
ejpam-5411	166	4	then	then	ADV
ejpam-5411	166	5	iα(f)(t	iα(f)(t	NUM
ejpam-5411	166	6	)	)	PUNCT
ejpam-5411	166	7	is	be	AUX
ejpam-5411	166	8	a	a	DET
ejpam-5411	166	9	α	α	ADJ
ejpam-5411	166	10	-	-	ADJ
ejpam-5411	166	11	periodic	periodic	ADJ
ejpam-5411	166	12	function	function	NOUN
ejpam-5411	166	13	with	with	ADP
ejpam-5411	166	14	period	period	NOUN
ejpam-5411	166	15	p	p	X
ejpam-5411	166	16	,	,	PUNCT
ejpam-5411	166	17	for	for	ADP
ejpam-5411	166	18	all	all	DET
ejpam-5411	166	19	t	t	NOUN
ejpam-5411	166	20	∈	∈	PROPN
ejpam-5411	167	1	[	[	X
ejpam-5411	167	2	0,+∞	0,+∞	PROPN
ejpam-5411	167	3	[	[	X
ejpam-5411	167	4	.	.	PUNCT
ejpam-5411	168	1	proof	proof	NOUN
ejpam-5411	168	2	.	.	PUNCT
ejpam-5411	169	1	let	let	VERB
ejpam-5411	169	2	0	0	NUM
ejpam-5411	169	3	<	<	X
ejpam-5411	169	4	α	α	PROPN
ejpam-5411	169	5	≤	≤	NUM
ejpam-5411	169	6	1	1	NUM
ejpam-5411	169	7	and	and	CCONJ
ejpam-5411	169	8	assume	assume	VERB
ejpam-5411	169	9	that	that	SCONJ
ejpam-5411	169	10	f	f	X
ejpam-5411	169	11	:	:	PUNCT
ejpam-5411	170	1	[	[	X
ejpam-5411	170	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	170	3	r	r	NOUN
ejpam-5411	170	4	is	be	AUX
ejpam-5411	170	5	a	a	DET
ejpam-5411	170	6	α	α	ADJ
ejpam-5411	170	7	-	-	ADJ
ejpam-5411	170	8	periodic	periodic	ADJ
ejpam-5411	170	9	function	function	NOUN
ejpam-5411	170	10	with	with	ADP
ejpam-5411	170	11	period	period	NOUN
ejpam-5411	170	12	p.	p.	NOUN
ejpam-5411	170	13	by	by	ADP
ejpam-5411	170	14	definition	definition	NOUN
ejpam-5411	170	15	4	4	NUM
ejpam-5411	170	16	and	and	CCONJ
ejpam-5411	170	17	using	use	VERB
ejpam-5411	170	18	variable	variable	ADJ
ejpam-5411	170	19	change	change	NOUN
ejpam-5411	170	20	u	u	NOUN
ejpam-5411	170	21	=	=	NOUN
ejpam-5411	170	22	pα	pα	NOUN
ejpam-5411	170	23	α	α	NOUN
ejpam-5411	170	24	,	,	PUNCT
ejpam-5411	170	25	we	we	PRON
ejpam-5411	170	26	have	have	VERB
ejpam-5411	170	27	for	for	ADP
ejpam-5411	170	28	all	all	DET
ejpam-5411	170	29	t	t	NOUN
ejpam-5411	170	30	∈	∈	PROPN
ejpam-5411	171	1	[	[	X
ejpam-5411	171	2	0,+∞	0,+∞	X
ejpam-5411	171	3	[	[	PUNCT
ejpam-5411	171	4	iα(f)(t	iα(f)(t	ADJ
ejpam-5411	171	5	)	)	PUNCT
ejpam-5411	171	6	=	=	SYM
ejpam-5411	172	1	∫	∫	PROPN
ejpam-5411	172	2	t	t	PROPN
ejpam-5411	172	3	0	0	NUM
ejpam-5411	172	4	sα−1f(s)ds	sα−1f(s)ds	PROPN
ejpam-5411	172	5	=	=	SYM
ejpam-5411	172	6	∫	∫	PROPN
ejpam-5411	172	7	tα	tα	PROPN
ejpam-5411	172	8	α	α	NOUN
ejpam-5411	172	9	0	0	SYM
ejpam-5411	172	10	f((αu	f((αu	NOUN
ejpam-5411	172	11	)	)	PUNCT
ejpam-5411	172	12	1	1	NUM
ejpam-5411	172	13	α	α	NOUN
ejpam-5411	172	14	)	)	PUNCT
ejpam-5411	172	15	du	du	PROPN
ejpam-5411	173	1	=	=	NOUN
ejpam-5411	173	2	:	:	PUNCT
ejpam-5411	173	3	g1	g1	PROPN
ejpam-5411	173	4	(	(	PUNCT
ejpam-5411	173	5	tα	tα	PROPN
ejpam-5411	173	6	α	α	PROPN
ejpam-5411	173	7	)	)	PUNCT
ejpam-5411	173	8	with	with	ADP
ejpam-5411	173	9	g1(t	g1(t	NOUN
ejpam-5411	173	10	)	)	PUNCT
ejpam-5411	173	11	is	be	AUX
ejpam-5411	173	12	the	the	DET
ejpam-5411	173	13	continuous	continuous	ADJ
ejpam-5411	173	14	function	function	NOUN
ejpam-5411	173	15	defined	define	VERB
ejpam-5411	173	16	by	by	ADP
ejpam-5411	173	17	g1(t	g1(t	NOUN
ejpam-5411	173	18	)	)	PUNCT
ejpam-5411	173	19	=	=	SYM
ejpam-5411	173	20	∫	∫	PROPN
ejpam-5411	173	21	t	t	NOUN
ejpam-5411	173	22	0	0	NUM
ejpam-5411	173	23	f((αu	f((αu	X
ejpam-5411	173	24	)	)	PUNCT
ejpam-5411	173	25	1	1	NUM
ejpam-5411	173	26	α	α	NOUN
ejpam-5411	173	27	)	)	PUNCT
ejpam-5411	173	28	du	du	PROPN
ejpam-5411	173	29	.	.	PUNCT
ejpam-5411	174	1	then	then	ADV
ejpam-5411	174	2	,	,	PUNCT
ejpam-5411	174	3	we	we	PRON
ejpam-5411	174	4	have	have	VERB
ejpam-5411	174	5	g1	g1	NOUN
ejpam-5411	174	6	(	(	PUNCT
ejpam-5411	174	7	tα	tα	PROPN
ejpam-5411	174	8	α	α	PROPN
ejpam-5411	175	1	+	+	CCONJ
ejpam-5411	175	2	pα	pα	NOUN
ejpam-5411	175	3	α	α	NOUN
ejpam-5411	175	4	)	)	PUNCT
ejpam-5411	175	5	=	=	SYM
ejpam-5411	175	6	iα(f)(p	iα(f)(p	NOUN
ejpam-5411	175	7	)	)	PUNCT
ejpam-5411	176	1	+	+	CCONJ
ejpam-5411	176	2	g1	g1	PROPN
ejpam-5411	176	3	(	(	PUNCT
ejpam-5411	176	4	tα	tα	PROPN
ejpam-5411	176	5	α	α	PROPN
ejpam-5411	176	6	)	)	PUNCT
ejpam-5411	176	7	using	use	VERB
ejpam-5411	176	8	the	the	DET
ejpam-5411	176	9	condition	condition	NOUN
ejpam-5411	176	10	given	give	VERB
ejpam-5411	176	11	in	in	ADP
ejpam-5411	176	12	equation	equation	NOUN
ejpam-5411	176	13	6	6	NUM
ejpam-5411	176	14	,	,	PUNCT
ejpam-5411	176	15	we	we	PRON
ejpam-5411	176	16	obtain	obtain	VERB
ejpam-5411	176	17	:	:	PUNCT
ejpam-5411	176	18	g1	g1	PROPN
ejpam-5411	176	19	(	(	PUNCT
ejpam-5411	176	20	tα	tα	PROPN
ejpam-5411	176	21	α	α	PROPN
ejpam-5411	177	1	+	+	CCONJ
ejpam-5411	177	2	pα	pα	NOUN
ejpam-5411	177	3	α	α	NOUN
ejpam-5411	177	4	)	)	PUNCT
ejpam-5411	177	5	=	=	SYM
ejpam-5411	177	6	g1	g1	PROPN
ejpam-5411	177	7	(	(	PUNCT
ejpam-5411	177	8	tα	tα	PROPN
ejpam-5411	177	9	α	α	PROPN
ejpam-5411	177	10	)	)	PUNCT
ejpam-5411	177	11	.	.	PUNCT
ejpam-5411	178	1	then	then	ADV
ejpam-5411	178	2	the	the	DET
ejpam-5411	178	3	function	function	NOUN
ejpam-5411	178	4	g1	g1	PROPN
ejpam-5411	178	5	is	be	AUX
ejpam-5411	178	6	a	a	DET
ejpam-5411	178	7	continuous	continuous	ADJ
ejpam-5411	178	8	periodic	periodic	ADJ
ejpam-5411	178	9	function	function	NOUN
ejpam-5411	178	10	with	with	ADP
ejpam-5411	178	11	period	period	NOUN
ejpam-5411	178	12	pα	pα	INTJ
ejpam-5411	178	13	α	α	X
ejpam-5411	178	14	.	.	PUNCT
ejpam-5411	179	1	thus	thus	ADV
ejpam-5411	179	2	iα(f)(t	iα(f)(t	NUM
ejpam-5411	179	3	)	)	PUNCT
ejpam-5411	179	4	is	be	AUX
ejpam-5411	179	5	α	α	X
ejpam-5411	179	6	-	-	NOUN
ejpam-5411	179	7	periodic	periodic	NOUN
ejpam-5411	179	8	with	with	ADP
ejpam-5411	179	9	period	period	NOUN
ejpam-5411	179	10	p	p	NOUN
ejpam-5411	179	11	for	for	ADP
ejpam-5411	179	12	all	all	DET
ejpam-5411	179	13	t	t	NOUN
ejpam-5411	179	14	∈	∈	PROPN
ejpam-5411	180	1	[	[	X
ejpam-5411	180	2	0,+∞	0,+∞	NUM
ejpam-5411	180	3	[	[	NOUN
ejpam-5411	180	4	.	.	PUNCT
ejpam-5411	180	5	example	example	NOUN
ejpam-5411	180	6	4	4	NUM
ejpam-5411	180	7	.	.	NOUN
ejpam-5411	180	8	1	1	NUM
ejpam-5411	180	9	.	.	PUNCT
ejpam-5411	181	1	the	the	DET
ejpam-5411	181	2	function	function	NOUN
ejpam-5411	181	3	f2	f2	PROPN
ejpam-5411	181	4	defined	define	VERB
ejpam-5411	181	5	by	by	ADP
ejpam-5411	181	6	example	example	NOUN
ejpam-5411	181	7	3	3	NUM
ejpam-5411	181	8	is	be	AUX
ejpam-5411	181	9	α	α	NOUN
ejpam-5411	181	10	-	-	NOUN
ejpam-5411	181	11	periodic	periodic	NOUN
ejpam-5411	181	12	with	with	ADP
ejpam-5411	181	13	period	period	NOUN
ejpam-5411	181	14	p	p	X
ejpam-5411	181	15	=	=	PUNCT
ejpam-5411	181	16	(	(	PUNCT
ejpam-5411	181	17	1	1	NUM
ejpam-5411	181	18	α	α	NOUN
ejpam-5411	181	19	)	)	PUNCT
ejpam-5411	181	20	1	1	NUM
ejpam-5411	181	21	α	α	NOUN
ejpam-5411	181	22	and	and	CCONJ
ejpam-5411	181	23	we	we	PRON
ejpam-5411	181	24	have	have	VERB
ejpam-5411	181	25	iα(f2)(t	iα(f2)(t	NOUN
ejpam-5411	181	26	)	)	PUNCT
ejpam-5411	181	27	=	=	PUNCT
ejpam-5411	181	28			NUM
ejpam-5411	181	29	∫	∫	NOUN
ejpam-5411	182	1	tα	tα	PROPN
ejpam-5411	182	2	α	α	PROPN
ejpam-5411	182	3	0	0	NUM
ejpam-5411	182	4	sds	sds	PROPN
ejpam-5411	182	5	,	,	PUNCT
ejpam-5411	182	6	0	0	NUM
ejpam-5411	182	7	≤	≤	NUM
ejpam-5411	182	8	t	t	PROPN
ejpam-5411	182	9	≤	≤	NUM
ejpam-5411	182	10	(	(	PUNCT
ejpam-5411	182	11	1	1	NUM
ejpam-5411	182	12	4α	4α	NOUN
ejpam-5411	182	13	)	)	PUNCT
ejpam-5411	182	14	1	1	NUM
ejpam-5411	183	1	α	α	NUM
ejpam-5411	183	2	∫	∫	PROPN
ejpam-5411	183	3	(	(	PUNCT
ejpam-5411	183	4	1	1	NUM
ejpam-5411	183	5	4α	4α	NOUN
ejpam-5411	183	6	)	)	PUNCT
ejpam-5411	183	7	1	1	NUM
ejpam-5411	183	8	α	α	NOUN
ejpam-5411	183	9	0	0	NUM
ejpam-5411	183	10	sds	sds	PROPN
ejpam-5411	183	11	+	+	CCONJ
ejpam-5411	183	12	∫	∫	PROPN
ejpam-5411	183	13	tα	tα	PROPN
ejpam-5411	183	14	α	α	PROPN
ejpam-5411	183	15	(	(	PUNCT
ejpam-5411	183	16	1	1	NUM
ejpam-5411	183	17	4α	4α	NOUN
ejpam-5411	183	18	)	)	PUNCT
ejpam-5411	183	19	1	1	NUM
ejpam-5411	183	20	α	α	NOUN
ejpam-5411	183	21	(	(	PUNCT
ejpam-5411	183	22	1	1	NUM
ejpam-5411	183	23	2α2	2α2	NUM
ejpam-5411	183	24	−	−	PROPN
ejpam-5411	183	25	s)ds	s)ds	PROPN
ejpam-5411	183	26	,	,	PUNCT
ejpam-5411	183	27	(	(	PUNCT
ejpam-5411	183	28	1	1	NUM
ejpam-5411	183	29	4α	4α	NOUN
ejpam-5411	183	30	)	)	PUNCT
ejpam-5411	183	31	1	1	NUM
ejpam-5411	183	32	α	α	NOUN
ejpam-5411	183	33	<	<	X
ejpam-5411	183	34	t	t	X
ejpam-5411	183	35	≤	≤	NUM
ejpam-5411	183	36	(	(	PUNCT
ejpam-5411	183	37	3	3	NUM
ejpam-5411	183	38	4α	4α	NOUN
ejpam-5411	183	39	)	)	PUNCT
ejpam-5411	183	40	1	1	NUM
ejpam-5411	184	1	α	α	NUM
ejpam-5411	184	2	∫	∫	PROPN
ejpam-5411	184	3	(	(	PUNCT
ejpam-5411	184	4	1	1	NUM
ejpam-5411	184	5	4α	4α	NOUN
ejpam-5411	184	6	)	)	PUNCT
ejpam-5411	184	7	1	1	NUM
ejpam-5411	184	8	α	α	NOUN
ejpam-5411	184	9	0	0	NUM
ejpam-5411	184	10	sds	sds	PROPN
ejpam-5411	184	11	+	+	X
ejpam-5411	184	12	∫	∫	PROPN
ejpam-5411	184	13	(	(	PUNCT
ejpam-5411	184	14	3	3	NUM
ejpam-5411	184	15	4α	4α	NOUN
ejpam-5411	184	16	)	)	PUNCT
ejpam-5411	184	17	1	1	NUM
ejpam-5411	184	18	α	α	NOUN
ejpam-5411	184	19	(	(	PUNCT
ejpam-5411	184	20	1	1	NUM
ejpam-5411	184	21	4α	4α	NOUN
ejpam-5411	184	22	)	)	PUNCT
ejpam-5411	184	23	1	1	NUM
ejpam-5411	184	24	α	α	NOUN
ejpam-5411	184	25	(	(	PUNCT
ejpam-5411	184	26	1	1	NUM
ejpam-5411	184	27	2α2	2α2	NUM
ejpam-5411	184	28	−	−	PROPN
ejpam-5411	185	1	s)ds	s)ds	PROPN
ejpam-5411	185	2	+	+	CCONJ
ejpam-5411	185	3	∫	∫	PROPN
ejpam-5411	185	4	tα	tα	PROPN
ejpam-5411	185	5	α	α	PROPN
ejpam-5411	185	6	(	(	PUNCT
ejpam-5411	185	7	3	3	NUM
ejpam-5411	185	8	4α	4α	NOUN
ejpam-5411	185	9	)	)	PUNCT
ejpam-5411	185	10	1	1	NUM
ejpam-5411	185	11	α	α	NOUN
ejpam-5411	185	12	(	(	PUNCT
ejpam-5411	185	13	s−	s−	PROPN
ejpam-5411	185	14	1	1	NUM
ejpam-5411	185	15	α2	α2	PROPN
ejpam-5411	185	16	)	)	PUNCT
ejpam-5411	185	17	,	,	PUNCT
ejpam-5411	185	18	(	(	PUNCT
ejpam-5411	185	19	3	3	NUM
ejpam-5411	185	20	4α	4α	NOUN
ejpam-5411	185	21	)	)	PUNCT
ejpam-5411	185	22	1	1	NUM
ejpam-5411	185	23	α	α	NOUN
ejpam-5411	185	24	<	<	X
ejpam-5411	185	25	t	t	X
ejpam-5411	185	26	≤	≤	NUM
ejpam-5411	185	27	(	(	PUNCT
ejpam-5411	185	28	1	1	NUM
ejpam-5411	185	29	α	α	NOUN
ejpam-5411	185	30	)	)	PUNCT
ejpam-5411	185	31	1	1	NUM
ejpam-5411	185	32	α	α	NOUN
ejpam-5411	185	33	then	then	ADV
ejpam-5411	185	34	iα(f2)(t	iα(f2)(t	ADJ
ejpam-5411	185	35	)	)	PUNCT
ejpam-5411	185	36	=	=	PUNCT
ejpam-5411	185	37			NUM
ejpam-5411	185	38	1	1	NUM
ejpam-5411	185	39	2	2	NUM
ejpam-5411	185	40	(	(	PUNCT
ejpam-5411	185	41	t	t	PROPN
ejpam-5411	185	42	α	α	PROPN
ejpam-5411	185	43	α	α	NOUN
ejpam-5411	185	44	)	)	PUNCT
ejpam-5411	185	45	2	2	NUM
ejpam-5411	185	46	,	,	PUNCT
ejpam-5411	185	47	0	0	NUM
ejpam-5411	185	48	≤	≤	NUM
ejpam-5411	185	49	t	t	PROPN
ejpam-5411	185	50	≤	≤	NUM
ejpam-5411	185	51	(	(	PUNCT
ejpam-5411	185	52	1	1	NUM
ejpam-5411	185	53	4α	4α	NOUN
ejpam-5411	185	54	)	)	PUNCT
ejpam-5411	185	55	1	1	NUM
ejpam-5411	185	56	α	α	NUM
ejpam-5411	185	57	−1	−1	NOUN
ejpam-5411	185	58	16α4	16α4	NUM
ejpam-5411	185	59	+	+	CCONJ
ejpam-5411	185	60	1	1	NUM
ejpam-5411	185	61	2α2	2α2	NUM
ejpam-5411	185	62	(	(	PUNCT
ejpam-5411	185	63	t	t	PROPN
ejpam-5411	185	64	α	α	PROPN
ejpam-5411	185	65	α	α	NOUN
ejpam-5411	185	66	)	)	PUNCT
ejpam-5411	186	1	−	−	NOUN
ejpam-5411	187	1	1	1	NUM
ejpam-5411	187	2	2	2	NUM
ejpam-5411	187	3	(	(	PUNCT
ejpam-5411	187	4	t	t	PROPN
ejpam-5411	187	5	α	α	PROPN
ejpam-5411	187	6	α	α	NOUN
ejpam-5411	187	7	)	)	PUNCT
ejpam-5411	187	8	2	2	NUM
ejpam-5411	187	9	,	,	PUNCT
ejpam-5411	187	10	(	(	PUNCT
ejpam-5411	187	11	1	1	NUM
ejpam-5411	187	12	4α	4α	NOUN
ejpam-5411	187	13	)	)	PUNCT
ejpam-5411	187	14	1	1	NUM
ejpam-5411	187	15	α	α	NOUN
ejpam-5411	187	16	<	<	X
ejpam-5411	187	17	t	t	X
ejpam-5411	187	18	≤	≤	NUM
ejpam-5411	187	19	(	(	PUNCT
ejpam-5411	187	20	3	3	NUM
ejpam-5411	187	21	4α	4α	NOUN
ejpam-5411	187	22	)	)	PUNCT
ejpam-5411	187	23	1	1	NUM
ejpam-5411	187	24	α	α	NUM
ejpam-5411	187	25	1	1	NUM
ejpam-5411	187	26	2	2	NUM
ejpam-5411	187	27	(	(	PUNCT
ejpam-5411	187	28	t	t	PROPN
ejpam-5411	187	29	α	α	PROPN
ejpam-5411	187	30	α	α	NOUN
ejpam-5411	187	31	)	)	PUNCT
ejpam-5411	187	32	2	2	NUM
ejpam-5411	187	33	−	−	PROPN
ejpam-5411	187	34	1	1	NUM
ejpam-5411	187	35	α2	α2	NOUN
ejpam-5411	187	36	(	(	PUNCT
ejpam-5411	187	37	t	t	PROPN
ejpam-5411	187	38	α	α	PROPN
ejpam-5411	187	39	α	α	NOUN
ejpam-5411	187	40	)	)	PUNCT
ejpam-5411	188	1	+	+	CCONJ
ejpam-5411	188	2	1	1	NUM
ejpam-5411	188	3	2α4	2α4	NUM
ejpam-5411	188	4	,	,	PUNCT
ejpam-5411	188	5	(	(	PUNCT
ejpam-5411	188	6	3	3	NUM
ejpam-5411	188	7	4α	4α	NOUN
ejpam-5411	188	8	)	)	PUNCT
ejpam-5411	188	9	1	1	NUM
ejpam-5411	188	10	α	α	NOUN
ejpam-5411	188	11	<	<	X
ejpam-5411	188	12	t	t	X
ejpam-5411	188	13	≤	≤	NUM
ejpam-5411	188	14	(	(	PUNCT
ejpam-5411	188	15	1	1	NUM
ejpam-5411	188	16	α	α	NOUN
ejpam-5411	188	17	)	)	PUNCT
ejpam-5411	188	18	1	1	NUM
ejpam-5411	188	19	α	α	NOUN
ejpam-5411	188	20	(	(	PUNCT
ejpam-5411	188	21	7	7	NUM
ejpam-5411	188	22	)	)	PUNCT
ejpam-5411	189	1	t.	t.	NOUN
ejpam-5411	189	2	abdeljawad	abdeljawad	NOUN
ejpam-5411	189	3	et	et	PROPN
ejpam-5411	189	4	al	al	PROPN
ejpam-5411	189	5	.	.	PUNCT
ejpam-5411	189	6	/	/	SYM
ejpam-5411	189	7	eur	eur	PROPN
ejpam-5411	189	8	.	.	PUNCT
ejpam-5411	190	1	j.	j.	PROPN
ejpam-5411	190	2	pure	pure	PROPN
ejpam-5411	190	3	appl	appl	PROPN
ejpam-5411	190	4	.	.	PROPN
ejpam-5411	190	5	math	math	PROPN
ejpam-5411	190	6	,	,	PUNCT
ejpam-5411	190	7	17	17	NUM
ejpam-5411	190	8	(	(	PUNCT
ejpam-5411	190	9	4	4	NUM
ejpam-5411	190	10	)	)	PUNCT
ejpam-5411	190	11	(	(	PUNCT
ejpam-5411	190	12	2024	2024	NUM
ejpam-5411	190	13	)	)	PUNCT
ejpam-5411	190	14	,	,	PUNCT
ejpam-5411	190	15	2405	2405	NUM
ejpam-5411	190	16	-	-	SYM
ejpam-5411	190	17	2430	2430	NUM
ejpam-5411	190	18	2412	2412	NUM
ejpam-5411	190	19	and	and	CCONJ
ejpam-5411	190	20	g2(t	g2(t	NOUN
ejpam-5411	190	21	)	)	PUNCT
ejpam-5411	190	22	=	=	SYM
ejpam-5411	190	23	iα(f2)((αt	iα(f2)((αt	NOUN
ejpam-5411	190	24	)	)	PUNCT
ejpam-5411	190	25	1	1	NUM
ejpam-5411	190	26	α	α	NOUN
ejpam-5411	190	27	)	)	PUNCT
ejpam-5411	190	28	=	=	PUNCT
ejpam-5411	191	1			NOUN
ejpam-5411	191	2	1	1	NUM
ejpam-5411	191	3	2	2	NUM
ejpam-5411	191	4	t	t	NOUN
ejpam-5411	191	5	2	2	NUM
ejpam-5411	191	6	,	,	PUNCT
ejpam-5411	191	7	0	0	NUM
ejpam-5411	191	8	≤	≤	NUM
ejpam-5411	191	9	t	t	NOUN
ejpam-5411	191	10	≤	≤	NUM
ejpam-5411	191	11	1	1	NUM
ejpam-5411	191	12	4α2	4α2	NUM
ejpam-5411	191	13	−1	−1	NOUN
ejpam-5411	191	14	16α4	16α4	NUM
ejpam-5411	191	15	+	+	CCONJ
ejpam-5411	191	16	1	1	NUM
ejpam-5411	191	17	2α2	2α2	NUM
ejpam-5411	191	18	t−	t−	DET
ejpam-5411	191	19	1	1	NUM
ejpam-5411	191	20	2	2	NUM
ejpam-5411	191	21	t	t	NOUN
ejpam-5411	191	22	2	2	NUM
ejpam-5411	191	23	,	,	PUNCT
ejpam-5411	191	24	1	1	NUM
ejpam-5411	191	25	4α2	4α2	NOUN
ejpam-5411	191	26	<	<	X
ejpam-5411	191	27	t	t	X
ejpam-5411	191	28	≤	≤	NUM
ejpam-5411	191	29	3	3	NUM
ejpam-5411	191	30	4α2	4α2	NOUN
ejpam-5411	191	31	1	1	NUM
ejpam-5411	191	32	2	2	NUM
ejpam-5411	191	33	t	t	NOUN
ejpam-5411	191	34	2	2	NUM
ejpam-5411	191	35	−	−	NOUN
ejpam-5411	191	36	1	1	NUM
ejpam-5411	191	37	α2	α2	NOUN
ejpam-5411	191	38	t	t	NOUN
ejpam-5411	192	1	+	+	CCONJ
ejpam-5411	192	2	1	1	NUM
ejpam-5411	192	3	2α4	2α4	NUM
ejpam-5411	192	4	,	,	PUNCT
ejpam-5411	192	5	3	3	NUM
ejpam-5411	192	6	4α2	4α2	NOUN
ejpam-5411	192	7	<	<	X
ejpam-5411	192	8	t	t	X
ejpam-5411	192	9	≤	≤	NUM
ejpam-5411	192	10	1	1	NUM
ejpam-5411	192	11	α2	α2	ADJ
ejpam-5411	192	12	(	(	PUNCT
ejpam-5411	192	13	8)	8)	NUM
ejpam-5411	192	14	therefore	therefore	ADV
ejpam-5411	192	15	,	,	PUNCT
ejpam-5411	192	16	we	we	PRON
ejpam-5411	192	17	have	have	AUX
ejpam-5411	192	18	iα(f2)(p	iα(f2)(p	VERB
ejpam-5411	192	19	)	)	PUNCT
ejpam-5411	192	20	=	=	SYM
ejpam-5411	192	21	1	1	NUM
ejpam-5411	192	22	2	2	NUM
ejpam-5411	192	23	(	(	PUNCT
ejpam-5411	192	24	pα	pα	NOUN
ejpam-5411	192	25	α	α	NOUN
ejpam-5411	192	26	)	)	PUNCT
ejpam-5411	192	27	2	2	NUM
ejpam-5411	192	28	−	−	PROPN
ejpam-5411	192	29	1	1	NUM
ejpam-5411	192	30	α2	α2	NOUN
ejpam-5411	192	31	(	(	PUNCT
ejpam-5411	192	32	pα	pα	NOUN
ejpam-5411	192	33	α	α	PROPN
ejpam-5411	192	34	)	)	PUNCT
ejpam-5411	193	1	+	+	CCONJ
ejpam-5411	193	2	1	1	NUM
ejpam-5411	193	3	2α4	2α4	NUM
ejpam-5411	193	4	=	=	SYM
ejpam-5411	193	5	1	1	NUM
ejpam-5411	193	6	2α4	2α4	NUM
ejpam-5411	193	7	−	−	NUM
ejpam-5411	193	8	1	1	NUM
ejpam-5411	193	9	α4	α4	NOUN
ejpam-5411	193	10	+	+	CCONJ
ejpam-5411	193	11	1	1	NUM
ejpam-5411	193	12	2α4	2α4	NUM
ejpam-5411	193	13	=	=	SYM
ejpam-5411	193	14	0	0	NUM
ejpam-5411	193	15	.	.	PUNCT
ejpam-5411	194	1	the	the	DET
ejpam-5411	194	2	condition	condition	NOUN
ejpam-5411	194	3	6	6	NUM
ejpam-5411	194	4	is	be	AUX
ejpam-5411	194	5	satisfied	satisfied	ADJ
ejpam-5411	194	6	,	,	PUNCT
ejpam-5411	194	7	then	then	ADV
ejpam-5411	194	8	the	the	DET
ejpam-5411	194	9	function	function	NOUN
ejpam-5411	194	10	g2	g2	PROPN
ejpam-5411	194	11	is	be	AUX
ejpam-5411	194	12	continuous	continuous	ADJ
ejpam-5411	194	13	periodic	periodic	NOUN
ejpam-5411	194	14	with	with	ADP
ejpam-5411	194	15	period	period	NOUN
ejpam-5411	194	16	1	1	NUM
ejpam-5411	194	17	α2	α2	ADJ
ejpam-5411	194	18	,	,	PUNCT
ejpam-5411	194	19	thus	thus	ADV
ejpam-5411	194	20	iα(f2	iα(f2	VERB
ejpam-5411	194	21	)	)	PUNCT
ejpam-5411	194	22	is	be	AUX
ejpam-5411	194	23	α	α	DET
ejpam-5411	194	24	-	-	ADJ
ejpam-5411	194	25	periodic	periodic	ADJ
ejpam-5411	194	26	function	function	NOUN
ejpam-5411	194	27	with	with	ADP
ejpam-5411	194	28	period	period	NOUN
ejpam-5411	194	29	(	(	PUNCT
ejpam-5411	194	30	1	1	NUM
ejpam-5411	194	31	α	α	NOUN
ejpam-5411	194	32	)	)	PUNCT
ejpam-5411	194	33	1	1	NUM
ejpam-5411	194	34	α	α	NOUN
ejpam-5411	194	35	.	.	PUNCT
ejpam-5411	195	1	2	2	X
ejpam-5411	195	2	.	.	X
ejpam-5411	195	3	the	the	DET
ejpam-5411	195	4	function	function	NOUN
ejpam-5411	195	5	f1	f1	NOUN
ejpam-5411	195	6	defined	define	VERB
ejpam-5411	195	7	by	by	ADP
ejpam-5411	195	8	example	example	NOUN
ejpam-5411	195	9	3	3	NUM
ejpam-5411	195	10	is	be	AUX
ejpam-5411	195	11	α	α	NOUN
ejpam-5411	195	12	-	-	NOUN
ejpam-5411	195	13	periodic	periodic	NOUN
ejpam-5411	195	14	with	with	ADP
ejpam-5411	195	15	period	period	NOUN
ejpam-5411	195	16	p	p	X
ejpam-5411	195	17	=	=	PUNCT
ejpam-5411	195	18	(	(	PUNCT
ejpam-5411	195	19	1	1	NUM
ejpam-5411	195	20	α	α	NOUN
ejpam-5411	195	21	)	)	PUNCT
ejpam-5411	195	22	1	1	NUM
ejpam-5411	195	23	α	α	NOUN
ejpam-5411	195	24	,	,	PUNCT
ejpam-5411	195	25	and	and	CCONJ
ejpam-5411	195	26	we	we	PRON
ejpam-5411	195	27	have	have	VERB
ejpam-5411	195	28	iα(f1)(t	iα(f1)(t	NOUN
ejpam-5411	195	29	)	)	PUNCT
ejpam-5411	195	30	=	=	PUNCT
ejpam-5411	196	1			PROPN
ejpam-5411	196	2	∫	∫	PROPN
ejpam-5411	196	3	tα	tα	PROPN
ejpam-5411	196	4	α	α	PROPN
ejpam-5411	196	5	0	0	NUM
ejpam-5411	196	6	sds	sds	PROPN
ejpam-5411	196	7	,	,	PUNCT
ejpam-5411	196	8	0	0	NUM
ejpam-5411	196	9	≤	≤	NUM
ejpam-5411	196	10	t	t	PROPN
ejpam-5411	196	11	≤	≤	NUM
ejpam-5411	196	12	(	(	PUNCT
ejpam-5411	196	13	1	1	NUM
ejpam-5411	196	14	2α	2α	NOUN
ejpam-5411	196	15	)	)	PUNCT
ejpam-5411	196	16	1	1	NUM
ejpam-5411	196	17	α	α	NUM
ejpam-5411	196	18	∫	∫	PROPN
ejpam-5411	196	19	(	(	PUNCT
ejpam-5411	196	20	1	1	NUM
ejpam-5411	196	21	2α	2α	NOUN
ejpam-5411	196	22	)	)	PUNCT
ejpam-5411	196	23	1	1	NUM
ejpam-5411	196	24	α	α	NOUN
ejpam-5411	196	25	0	0	NUM
ejpam-5411	196	26	sds	sds	PROPN
ejpam-5411	197	1	+	+	CCONJ
ejpam-5411	197	2	∫	∫	PROPN
ejpam-5411	197	3	tα	tα	PROPN
ejpam-5411	197	4	α	α	PROPN
ejpam-5411	197	5	(	(	PUNCT
ejpam-5411	197	6	1	1	NUM
ejpam-5411	197	7	2α	2α	NOUN
ejpam-5411	197	8	)	)	PUNCT
ejpam-5411	197	9	1	1	NUM
ejpam-5411	197	10	α	α	NOUN
ejpam-5411	197	11	(	(	PUNCT
ejpam-5411	197	12	1	1	NUM
ejpam-5411	197	13	α2	α2	NOUN
ejpam-5411	197	14	−	−	PROPN
ejpam-5411	197	15	s)ds	s)ds	PROPN
ejpam-5411	197	16	,	,	PUNCT
ejpam-5411	197	17	(	(	PUNCT
ejpam-5411	197	18	1	1	NUM
ejpam-5411	197	19	2α	2α	NOUN
ejpam-5411	197	20	)	)	PUNCT
ejpam-5411	197	21	1	1	NUM
ejpam-5411	197	22	α	α	NOUN
ejpam-5411	197	23	<	<	X
ejpam-5411	197	24	t	t	X
ejpam-5411	197	25	≤	≤	NUM
ejpam-5411	197	26	(	(	PUNCT
ejpam-5411	197	27	1	1	NUM
ejpam-5411	197	28	α	α	NOUN
ejpam-5411	197	29	)	)	PUNCT
ejpam-5411	197	30	1	1	NUM
ejpam-5411	197	31	α	α	NOUN
ejpam-5411	197	32	(	(	PUNCT
ejpam-5411	197	33	9	9	NUM
ejpam-5411	197	34	)	)	PUNCT
ejpam-5411	197	35	then	then	ADV
ejpam-5411	197	36	iα(f1)(t	iα(f1)(t	X
ejpam-5411	197	37	)	)	PUNCT
ejpam-5411	197	38	=	=	SYM
ejpam-5411	198	1			NOUN
ejpam-5411	198	2	1	1	NUM
ejpam-5411	198	3	2	2	NUM
ejpam-5411	198	4	(	(	PUNCT
ejpam-5411	198	5	t	t	PROPN
ejpam-5411	198	6	α	α	PROPN
ejpam-5411	198	7	α	α	NOUN
ejpam-5411	198	8	)	)	PUNCT
ejpam-5411	198	9	2	2	NUM
ejpam-5411	198	10	,	,	PUNCT
ejpam-5411	198	11	0	0	NUM
ejpam-5411	198	12	≤	≤	NUM
ejpam-5411	198	13	t	t	PROPN
ejpam-5411	198	14	≤	≤	NUM
ejpam-5411	198	15	(	(	PUNCT
ejpam-5411	198	16	1	1	NUM
ejpam-5411	198	17	2α	2α	NOUN
ejpam-5411	198	18	)	)	PUNCT
ejpam-5411	198	19	1	1	NUM
ejpam-5411	198	20	α	α	NUM
ejpam-5411	198	21	−1	−1	NOUN
ejpam-5411	198	22	2	2	NUM
ejpam-5411	198	23	(	(	PUNCT
ejpam-5411	198	24	t	t	PROPN
ejpam-5411	198	25	α	α	PROPN
ejpam-5411	198	26	α	α	NOUN
ejpam-5411	198	27	)	)	PUNCT
ejpam-5411	198	28	2	2	NUM
ejpam-5411	198	29	+	+	SYM
ejpam-5411	198	30	1	1	NUM
ejpam-5411	198	31	α2	α2	ADJ
ejpam-5411	198	32	tα	tα	VERB
ejpam-5411	198	33	α	α	NOUN
ejpam-5411	198	34	−	−	PROPN
ejpam-5411	198	35	1	1	NUM
ejpam-5411	198	36	4α4	4α4	NUM
ejpam-5411	198	37	,	,	PUNCT
ejpam-5411	198	38	(	(	PUNCT
ejpam-5411	198	39	1	1	NUM
ejpam-5411	198	40	2α	2α	NOUN
ejpam-5411	198	41	)	)	PUNCT
ejpam-5411	198	42	1	1	NUM
ejpam-5411	198	43	α	α	NOUN
ejpam-5411	198	44	<	<	X
ejpam-5411	198	45	t	t	X
ejpam-5411	198	46	≤	≤	NUM
ejpam-5411	199	1	(	(	PUNCT
ejpam-5411	199	2	1	1	NUM
ejpam-5411	199	3	α	α	NOUN
ejpam-5411	199	4	)	)	PUNCT
ejpam-5411	199	5	1	1	NUM
ejpam-5411	199	6	α	α	NOUN
ejpam-5411	199	7	.	.	PUNCT
ejpam-5411	200	1	(	(	PUNCT
ejpam-5411	200	2	10	10	NUM
ejpam-5411	200	3	)	)	PUNCT
ejpam-5411	200	4	and	and	CCONJ
ejpam-5411	200	5	g1(t	g1(t	X
ejpam-5411	200	6	)	)	PUNCT
ejpam-5411	200	7	=	=	SYM
ejpam-5411	200	8	iα(f1)((αt	iα(f1)((αt	PROPN
ejpam-5411	200	9	)	)	PUNCT
ejpam-5411	200	10	1	1	NUM
ejpam-5411	200	11	α	α	NOUN
ejpam-5411	200	12	)	)	PUNCT
ejpam-5411	200	13	=	=	PUNCT
ejpam-5411	201	1			PUNCT
ejpam-5411	201	2	1	1	NUM
ejpam-5411	201	3	2	2	NUM
ejpam-5411	201	4	t	t	NOUN
ejpam-5411	201	5	2	2	NUM
ejpam-5411	201	6	,	,	PUNCT
ejpam-5411	201	7	0	0	NUM
ejpam-5411	201	8	≤	≤	NUM
ejpam-5411	201	9	t	t	NOUN
ejpam-5411	201	10	≤	≤	NUM
ejpam-5411	201	11	1	1	NUM
ejpam-5411	201	12	2α2	2α2	NUM
ejpam-5411	201	13	−1	−1	NOUN
ejpam-5411	201	14	2	2	NUM
ejpam-5411	201	15	t	t	NOUN
ejpam-5411	201	16	2	2	NUM
ejpam-5411	201	17	+	+	CCONJ
ejpam-5411	201	18	1	1	NUM
ejpam-5411	201	19	α2	α2	ADJ
ejpam-5411	201	20	t−	t−	PROPN
ejpam-5411	201	21	1	1	NUM
ejpam-5411	201	22	4α4	4α4	NUM
ejpam-5411	201	23	,	,	PUNCT
ejpam-5411	201	24	1	1	NUM
ejpam-5411	201	25	2α2	2α2	NUM
ejpam-5411	201	26	<	<	X
ejpam-5411	201	27	t	t	X
ejpam-5411	201	28	≤	≤	NOUN
ejpam-5411	201	29	1	1	NUM
ejpam-5411	201	30	α2	α2	ADJ
ejpam-5411	201	31	.	.	PUNCT
ejpam-5411	202	1	(	(	PUNCT
ejpam-5411	202	2	11	11	NUM
ejpam-5411	202	3	)	)	PUNCT
ejpam-5411	202	4	we	we	PRON
ejpam-5411	202	5	have	have	VERB
ejpam-5411	202	6	iα(f1)(p	iα(f1)(p	NUM
ejpam-5411	202	7	)	)	PUNCT
ejpam-5411	202	8	=	=	SYM
ejpam-5411	202	9	1	1	NUM
ejpam-5411	202	10	4α4	4α4	NUM
ejpam-5411	202	11	̸=	̸=	PROPN
ejpam-5411	202	12	0	0	NUM
ejpam-5411	202	13	,	,	PUNCT
ejpam-5411	202	14	then	then	ADV
ejpam-5411	202	15	g1	g1	PROPN
ejpam-5411	202	16	is	be	AUX
ejpam-5411	202	17	not	not	PART
ejpam-5411	202	18	a	a	DET
ejpam-5411	202	19	continuous	continuous	ADJ
ejpam-5411	202	20	periodic	periodic	ADJ
ejpam-5411	202	21	function	function	NOUN
ejpam-5411	202	22	with	with	ADP
ejpam-5411	202	23	period	period	NOUN
ejpam-5411	202	24	1	1	NUM
ejpam-5411	202	25	α2	α2	ADJ
ejpam-5411	202	26	,	,	PUNCT
ejpam-5411	202	27	therfore	therfore	VERB
ejpam-5411	202	28	iα(f1	iα(f1	NOUN
ejpam-5411	202	29	)	)	PUNCT
ejpam-5411	202	30	is	be	AUX
ejpam-5411	202	31	not	not	PART
ejpam-5411	202	32	a	a	DET
ejpam-5411	202	33	α	α	ADJ
ejpam-5411	202	34	-	-	ADJ
ejpam-5411	202	35	periodic	periodic	ADJ
ejpam-5411	202	36	function	function	NOUN
ejpam-5411	202	37	with	with	ADP
ejpam-5411	202	38	period	period	NOUN
ejpam-5411	202	39	(	(	PUNCT
ejpam-5411	202	40	1	1	NUM
ejpam-5411	202	41	α	α	NOUN
ejpam-5411	202	42	)	)	PUNCT
ejpam-5411	202	43	1	1	NUM
ejpam-5411	202	44	α	α	NOUN
ejpam-5411	202	45	.	.	PUNCT
ejpam-5411	203	1	theorem	theorem	ADJ
ejpam-5411	203	2	5	5	NUM
ejpam-5411	203	3	.	.	PUNCT
ejpam-5411	204	1	let	let	VERB
ejpam-5411	204	2	0	0	NUM
ejpam-5411	204	3	<	<	X
ejpam-5411	204	4	α	α	PROPN
ejpam-5411	204	5	≤	≤	NUM
ejpam-5411	204	6	1	1	NUM
ejpam-5411	204	7	and	and	CCONJ
ejpam-5411	204	8	assume	assume	VERB
ejpam-5411	204	9	that	that	SCONJ
ejpam-5411	204	10	the	the	DET
ejpam-5411	204	11	function	function	NOUN
ejpam-5411	204	12	f	f	NOUN
ejpam-5411	204	13	:	:	PUNCT
ejpam-5411	205	1	[	[	X
ejpam-5411	205	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	205	3	r	r	NOUN
ejpam-5411	205	4	is	be	AUX
ejpam-5411	205	5	continuously	continuously	ADV
ejpam-5411	205	6	α	α	PRON
ejpam-5411	205	7	-	-	NOUN
ejpam-5411	205	8	differentiable	differentiable	ADJ
ejpam-5411	205	9	on	on	ADP
ejpam-5411	205	10	[	[	X
ejpam-5411	205	11	0,+∞	0,+∞	PROPN
ejpam-5411	205	12	[	[	X
ejpam-5411	205	13	,	,	PUNCT
ejpam-5411	205	14	and	and	CCONJ
ejpam-5411	205	15	α	α	NOUN
ejpam-5411	205	16	-	-	NOUN
ejpam-5411	205	17	periodic	periodic	NOUN
ejpam-5411	205	18	with	with	ADP
ejpam-5411	205	19	period	period	NOUN
ejpam-5411	205	20	p.	p.	NOUN
ejpam-5411	206	1	then	then	ADV
ejpam-5411	206	2	we	we	PRON
ejpam-5411	206	3	have	have	VERB
ejpam-5411	206	4	(	(	PUNCT
ejpam-5411	206	5	i	i	NOUN
ejpam-5411	206	6	)	)	PUNCT
ejpam-5411	206	7	t	t	PROPN
ejpam-5411	206	8	(	(	PUNCT
ejpam-5411	206	9	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	206	10	)	)	PUNCT
ejpam-5411	206	11	=	=	SYM
ejpam-5411	206	12	g′	g′	NOUN
ejpam-5411	206	13	(	(	PUNCT
ejpam-5411	206	14	t	t	PROPN
ejpam-5411	206	15	α	α	PROPN
ejpam-5411	206	16	α	α	NOUN
ejpam-5411	206	17	)	)	PUNCT
ejpam-5411	206	18	and	and	CCONJ
ejpam-5411	206	19	g	g	PROPN
ejpam-5411	206	20	∈	∈	PROPN
ejpam-5411	206	21	c1([0,+∞	c1([0,+∞	VERB
ejpam-5411	206	22	[	[	X
ejpam-5411	206	23	)	)	PUNCT
ejpam-5411	206	24	,	,	PUNCT
ejpam-5411	206	25	where	where	SCONJ
ejpam-5411	206	26	g(t	g(t	NOUN
ejpam-5411	206	27	)	)	PUNCT
ejpam-5411	206	28	=	=	SYM
ejpam-5411	206	29	f((αt	f((αt	NOUN
ejpam-5411	206	30	)	)	PUNCT
ejpam-5411	206	31	1	1	NUM
ejpam-5411	206	32	α	α	NOUN
ejpam-5411	206	33	)	)	PUNCT
ejpam-5411	206	34	,	,	PUNCT
ejpam-5411	206	35	(	(	PUNCT
ejpam-5411	206	36	ii	ii	NOUN
ejpam-5411	206	37	)	)	PUNCT
ejpam-5411	206	38	t	t	PROPN
ejpam-5411	206	39	(	(	PUNCT
ejpam-5411	206	40	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	206	41	)	)	PUNCT
ejpam-5411	206	42	is	be	AUX
ejpam-5411	206	43	α	α	DET
ejpam-5411	206	44	-	-	ADJ
ejpam-5411	206	45	periodic	periodic	ADJ
ejpam-5411	206	46	function	function	NOUN
ejpam-5411	206	47	with	with	ADP
ejpam-5411	206	48	period	period	NOUN
ejpam-5411	206	49	p	p	NOUN
ejpam-5411	206	50	for	for	ADP
ejpam-5411	206	51	all	all	DET
ejpam-5411	206	52	t	t	NOUN
ejpam-5411	206	53	∈	∈	PROPN
ejpam-5411	207	1	[	[	X
ejpam-5411	207	2	0,+∞	0,+∞	PROPN
ejpam-5411	207	3	[	[	X
ejpam-5411	207	4	.	.	PUNCT
ejpam-5411	208	1	t.	t.	PROPN
ejpam-5411	208	2	abdeljawad	abdeljawad	PROPN
ejpam-5411	208	3	et	et	PROPN
ejpam-5411	208	4	al	al	PROPN
ejpam-5411	208	5	.	.	PUNCT
ejpam-5411	208	6	/	/	SYM
ejpam-5411	208	7	eur	eur	PROPN
ejpam-5411	208	8	.	.	PUNCT
ejpam-5411	209	1	j.	j.	PROPN
ejpam-5411	209	2	pure	pure	PROPN
ejpam-5411	209	3	appl	appl	PROPN
ejpam-5411	209	4	.	.	PROPN
ejpam-5411	209	5	math	math	PROPN
ejpam-5411	209	6	,	,	PUNCT
ejpam-5411	209	7	17	17	NUM
ejpam-5411	209	8	(	(	PUNCT
ejpam-5411	209	9	4	4	NUM
ejpam-5411	209	10	)	)	PUNCT
ejpam-5411	209	11	(	(	PUNCT
ejpam-5411	209	12	2024	2024	NUM
ejpam-5411	209	13	)	)	PUNCT
ejpam-5411	209	14	,	,	PUNCT
ejpam-5411	209	15	2405	2405	NUM
ejpam-5411	209	16	-	-	SYM
ejpam-5411	209	17	2430	2430	NUM
ejpam-5411	209	18	2413	2413	NUM
ejpam-5411	209	19	proof	proof	NOUN
ejpam-5411	209	20	.	.	PUNCT
ejpam-5411	210	1	let	let	VERB
ejpam-5411	210	2	0	0	NUM
ejpam-5411	210	3	<	<	X
ejpam-5411	210	4	α	α	PROPN
ejpam-5411	210	5	≤	≤	NUM
ejpam-5411	210	6	1	1	NUM
ejpam-5411	210	7	and	and	CCONJ
ejpam-5411	210	8	f	f	NOUN
ejpam-5411	210	9	:	:	PUNCT
ejpam-5411	211	1	[	[	X
ejpam-5411	211	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	211	3	r	r	NOUN
ejpam-5411	211	4	is	be	AUX
ejpam-5411	211	5	α	α	DET
ejpam-5411	211	6	-	-	ADJ
ejpam-5411	211	7	periodic	periodic	ADJ
ejpam-5411	211	8	function	function	NOUN
ejpam-5411	211	9	with	with	ADP
ejpam-5411	211	10	period	period	NOUN
ejpam-5411	211	11	p	p	NOUN
ejpam-5411	211	12	and	and	CCONJ
ejpam-5411	211	13	continuously	continuously	ADV
ejpam-5411	211	14	α	α	X
ejpam-5411	211	15	-	-	NOUN
ejpam-5411	211	16	differentiable	differentiable	ADJ
ejpam-5411	211	17	on	on	ADP
ejpam-5411	211	18	[	[	X
ejpam-5411	211	19	0,+∞	0,+∞	PROPN
ejpam-5411	211	20	[	[	X
ejpam-5411	211	21	.	.	PUNCT
ejpam-5411	212	1	then	then	ADV
ejpam-5411	212	2	f(t	f(t	NOUN
ejpam-5411	212	3	)	)	PUNCT
ejpam-5411	212	4	is	be	AUX
ejpam-5411	212	5	α	α	NOUN
ejpam-5411	212	6	-	-	ADJ
ejpam-5411	212	7	differentiable	differentiable	ADJ
ejpam-5411	212	8	and	and	CCONJ
ejpam-5411	212	9	t	t	PROPN
ejpam-5411	212	10	(	(	PUNCT
ejpam-5411	212	11	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	212	12	)	)	PUNCT
ejpam-5411	212	13	is	be	AUX
ejpam-5411	212	14	continuous	continuous	ADJ
ejpam-5411	212	15	,	,	PUNCT
ejpam-5411	212	16	for	for	ADP
ejpam-5411	212	17	all	all	DET
ejpam-5411	212	18	t	t	NOUN
ejpam-5411	212	19	∈	∈	PROPN
ejpam-5411	213	1	[	[	X
ejpam-5411	213	2	0,+∞	0,+∞	NUM
ejpam-5411	213	3	[	[	X
ejpam-5411	213	4	.	.	PUNCT
ejpam-5411	214	1	by	by	ADP
ejpam-5411	214	2	definition	definition	NOUN
ejpam-5411	214	3	6	6	NUM
ejpam-5411	214	4	,	,	PUNCT
ejpam-5411	214	5	there	there	PRON
ejpam-5411	214	6	exists	exist	VERB
ejpam-5411	214	7	a	a	DET
ejpam-5411	214	8	continuous	continuous	ADJ
ejpam-5411	214	9	function	function	NOUN
ejpam-5411	214	10	g	g	NOUN
ejpam-5411	214	11	:	:	PUNCT
ejpam-5411	215	1	[	[	X
ejpam-5411	215	2	0,+∞[→	0,+∞[→	NOUN
ejpam-5411	215	3	r	r	NOUN
ejpam-5411	215	4	such	such	ADJ
ejpam-5411	215	5	that	that	SCONJ
ejpam-5411	215	6	f(t	f(t	NOUN
ejpam-5411	215	7	)	)	PUNCT
ejpam-5411	215	8	=	=	SYM
ejpam-5411	215	9	g	g	PROPN
ejpam-5411	215	10	(	(	PUNCT
ejpam-5411	215	11	tα	tα	PROPN
ejpam-5411	215	12	α	α	PROPN
ejpam-5411	215	13	)	)	PUNCT
ejpam-5411	215	14	=	=	SYM
ejpam-5411	215	15	g	g	NOUN
ejpam-5411	215	16	(	(	PUNCT
ejpam-5411	215	17	tα	tα	PROPN
ejpam-5411	215	18	α	α	PROPN
ejpam-5411	215	19	+	+	CCONJ
ejpam-5411	215	20	pα	pα	NOUN
ejpam-5411	215	21	α	α	NOUN
ejpam-5411	215	22	)	)	PUNCT
ejpam-5411	215	23	case	case	NOUN
ejpam-5411	215	24	1	1	NUM
ejpam-5411	215	25	:	:	PUNCT
ejpam-5411	215	26	t	t	X
ejpam-5411	215	27	>	>	X
ejpam-5411	215	28	0	0	PUNCT
ejpam-5411	216	1	(	(	PUNCT
ejpam-5411	216	2	1	1	NUM
ejpam-5411	216	3	)	)	PUNCT
ejpam-5411	216	4	by	by	ADP
ejpam-5411	216	5	theorem	theorem	NOUN
ejpam-5411	216	6	1	1	NUM
ejpam-5411	216	7	,	,	PUNCT
ejpam-5411	216	8	t	t	PROPN
ejpam-5411	216	9	(	(	PUNCT
ejpam-5411	216	10	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	216	11	)	)	PUNCT
ejpam-5411	216	12	=	=	PUNCT
ejpam-5411	216	13	t1−αf	t1−αf	NUM
ejpam-5411	216	14	′(t	′(t	NOUN
ejpam-5411	216	15	)	)	PUNCT
ejpam-5411	216	16	=	=	SYM
ejpam-5411	217	1	g′	g′	NOUN
ejpam-5411	217	2	(	(	PUNCT
ejpam-5411	217	3	tα	tα	PROPN
ejpam-5411	217	4	α	α	PROPN
ejpam-5411	217	5	)	)	PUNCT
ejpam-5411	217	6	:	:	PUNCT
ejpam-5411	217	7	=	=	X
ejpam-5411	217	8	g1	g1	PROPN
ejpam-5411	217	9	(	(	PUNCT
ejpam-5411	217	10	tα	tα	PROPN
ejpam-5411	217	11	α	α	PROPN
ejpam-5411	217	12	)	)	PUNCT
ejpam-5411	217	13	(	(	PUNCT
ejpam-5411	217	14	12	12	NUM
ejpam-5411	217	15	)	)	PUNCT
ejpam-5411	217	16	with	with	ADP
ejpam-5411	217	17	g1(t	g1(t	NOUN
ejpam-5411	217	18	)	)	PUNCT
ejpam-5411	217	19	=	=	SYM
ejpam-5411	217	20	g′(t	g′(t	NOUN
ejpam-5411	217	21	)	)	PUNCT
ejpam-5411	217	22	.	.	PUNCT
ejpam-5411	218	1	if	if	SCONJ
ejpam-5411	218	2	f(t	f(t	NOUN
ejpam-5411	218	3	)	)	PUNCT
ejpam-5411	218	4	is	be	AUX
ejpam-5411	218	5	α	α	PRON
ejpam-5411	218	6	-	-	NOUN
ejpam-5411	218	7	differentiable	differentiable	ADJ
ejpam-5411	218	8	,	,	PUNCT
ejpam-5411	218	9	then	then	ADV
ejpam-5411	218	10	t	t	PROPN
ejpam-5411	218	11	(	(	PUNCT
ejpam-5411	218	12	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	218	13	)	)	PUNCT
ejpam-5411	218	14	exists	exist	VERB
ejpam-5411	218	15	.	.	PUNCT
ejpam-5411	219	1	therefore	therefore	ADV
ejpam-5411	219	2	g(t	g(t	PROPN
ejpam-5411	219	3	)	)	PUNCT
ejpam-5411	219	4	is	be	AUX
ejpam-5411	219	5	differentiable	differentiable	ADJ
ejpam-5411	219	6	and	and	CCONJ
ejpam-5411	219	7	g′(t	g′(t	PROPN
ejpam-5411	219	8	)	)	PUNCT
ejpam-5411	220	1	=	=	SYM
ejpam-5411	220	2	t	t	PROPN
ejpam-5411	220	3	(	(	PUNCT
ejpam-5411	220	4	α)(f)((αt	α)(f)((αt	PROPN
ejpam-5411	220	5	)	)	PUNCT
ejpam-5411	220	6	1	1	NUM
ejpam-5411	220	7	α	α	NOUN
ejpam-5411	220	8	)	)	PUNCT
ejpam-5411	220	9	.	.	PUNCT
ejpam-5411	221	1	on	on	ADP
ejpam-5411	221	2	the	the	DET
ejpam-5411	221	3	other	other	ADJ
ejpam-5411	221	4	hand	hand	NOUN
ejpam-5411	221	5	,	,	PUNCT
ejpam-5411	221	6	if	if	SCONJ
ejpam-5411	221	7	t	t	PROPN
ejpam-5411	221	8	(	(	PUNCT
ejpam-5411	221	9	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	221	10	)	)	PUNCT
ejpam-5411	221	11	is	be	AUX
ejpam-5411	221	12	continuous	continuous	ADJ
ejpam-5411	221	13	,	,	PUNCT
ejpam-5411	221	14	then	then	ADV
ejpam-5411	221	15	g	g	PROPN
ejpam-5411	221	16	∈	∈	PROPN
ejpam-5411	221	17	c1(]0,+∞	c1(]0,+∞	X
ejpam-5411	221	18	[	[	X
ejpam-5411	221	19	)	)	PUNCT
ejpam-5411	221	20	.	.	PUNCT
ejpam-5411	222	1	case	case	NOUN
ejpam-5411	222	2	2	2	NUM
ejpam-5411	222	3	:	:	PUNCT
ejpam-5411	222	4	t	t	NOUN
ejpam-5411	222	5	=	=	PUNCT
ejpam-5411	222	6	0	0	PUNCT
ejpam-5411	222	7	if	if	SCONJ
ejpam-5411	222	8	f(t	f(t	NOUN
ejpam-5411	222	9	)	)	PUNCT
ejpam-5411	222	10	is	be	AUX
ejpam-5411	222	11	α	α	NOUN
ejpam-5411	222	12	-	-	NOUN
ejpam-5411	222	13	differentiable	differentiable	ADJ
ejpam-5411	222	14	for	for	ADP
ejpam-5411	222	15	all	all	DET
ejpam-5411	222	16	t	t	NOUN
ejpam-5411	222	17	∈	∈	PROPN
ejpam-5411	223	1	[	[	X
ejpam-5411	223	2	0,+∞	0,+∞	NUM
ejpam-5411	223	3	[	[	PUNCT
ejpam-5411	223	4	especially	especially	ADV
ejpam-5411	223	5	for	for	ADP
ejpam-5411	223	6	t	t	NOUN
ejpam-5411	223	7	=	=	SYM
ejpam-5411	223	8	0	0	NUM
ejpam-5411	223	9	,	,	PUNCT
ejpam-5411	223	10	then	then	ADV
ejpam-5411	223	11	t	t	PROPN
ejpam-5411	223	12	(	(	PUNCT
ejpam-5411	223	13	α)(f)(0	α)(f)(0	NUM
ejpam-5411	223	14	)	)	PUNCT
ejpam-5411	223	15	=	=	SYM
ejpam-5411	224	1	limt→0+t	limt→0+t	PROPN
ejpam-5411	224	2	(	(	PUNCT
ejpam-5411	224	3	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	224	4	)	)	PUNCT
ejpam-5411	224	5	exists	exist	VERB
ejpam-5411	224	6	and	and	CCONJ
ejpam-5411	224	7	by	by	ADP
ejpam-5411	224	8	continuity	continuity	NOUN
ejpam-5411	224	9	of	of	ADP
ejpam-5411	224	10	t	t	PROPN
ejpam-5411	224	11	(	(	PUNCT
ejpam-5411	224	12	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	224	13	)	)	PUNCT
ejpam-5411	224	14	and	and	CCONJ
ejpam-5411	224	15	g′(t	g′(t	PROPN
ejpam-5411	224	16	)	)	PUNCT
ejpam-5411	224	17	we	we	PRON
ejpam-5411	224	18	have	have	VERB
ejpam-5411	224	19	lim	lim	NOUN
ejpam-5411	224	20	t→0	t→0	PROPN
ejpam-5411	224	21	+	+	CCONJ
ejpam-5411	224	22	g′(t	g′(t	ADJ
ejpam-5411	224	23	)	)	PUNCT
ejpam-5411	225	1	=	=	SYM
ejpam-5411	225	2	lim	lim	PROPN
ejpam-5411	225	3	t→0	t→0	PROPN
ejpam-5411	225	4	+	+	PROPN
ejpam-5411	225	5	t	t	PROPN
ejpam-5411	225	6	(	(	PUNCT
ejpam-5411	225	7	α)(f)((αt	α)(f)((αt	PROPN
ejpam-5411	225	8	)	)	PUNCT
ejpam-5411	225	9	1	1	NUM
ejpam-5411	225	10	α	α	NOUN
ejpam-5411	225	11	)	)	PUNCT
ejpam-5411	226	1	=	=	SYM
ejpam-5411	226	2	t	t	PROPN
ejpam-5411	226	3	(	(	PUNCT
ejpam-5411	226	4	α)(f)(0	α)(f)(0	NUM
ejpam-5411	226	5	)	)	PUNCT
ejpam-5411	226	6	=	=	SYM
ejpam-5411	226	7	g′(0	g′(0	PROPN
ejpam-5411	226	8	)	)	PUNCT
ejpam-5411	226	9	.	.	PUNCT
ejpam-5411	227	1	finally	finally	ADV
ejpam-5411	227	2	t	t	X
ejpam-5411	227	3	(	(	PUNCT
ejpam-5411	227	4	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	227	5	)	)	PUNCT
ejpam-5411	227	6	=	=	SYM
ejpam-5411	227	7	g′	g′	NOUN
ejpam-5411	227	8	(	(	PUNCT
ejpam-5411	227	9	tα	tα	PROPN
ejpam-5411	227	10	α	α	PROPN
ejpam-5411	227	11	)	)	PUNCT
ejpam-5411	227	12	for	for	ADP
ejpam-5411	227	13	all	all	DET
ejpam-5411	227	14	t	t	NOUN
ejpam-5411	227	15	∈	∈	PROPN
ejpam-5411	228	1	[	[	X
ejpam-5411	228	2	0,+∞	0,+∞	NUM
ejpam-5411	228	3	[	[	PUNCT
ejpam-5411	228	4	and	and	CCONJ
ejpam-5411	228	5	g	g	PROPN
ejpam-5411	228	6	∈	∈	PROPN
ejpam-5411	228	7	c1([0,+∞	c1([0,+∞	VERB
ejpam-5411	228	8	[	[	X
ejpam-5411	228	9	)	)	PUNCT
ejpam-5411	228	10	(	(	PUNCT
ejpam-5411	228	11	2	2	X
ejpam-5411	228	12	)	)	PUNCT
ejpam-5411	228	13	if	if	SCONJ
ejpam-5411	228	14	f	f	PROPN
ejpam-5411	228	15	is	be	AUX
ejpam-5411	228	16	α	α	NOUN
ejpam-5411	228	17	-	-	NOUN
ejpam-5411	228	18	periodic	periodic	ADJ
ejpam-5411	228	19	then	then	ADV
ejpam-5411	228	20	g	g	PROPN
ejpam-5411	228	21	(	(	PUNCT
ejpam-5411	228	22	t	t	PROPN
ejpam-5411	228	23	α	α	PROPN
ejpam-5411	228	24	α	α	PROPN
ejpam-5411	229	1	+	+	CCONJ
ejpam-5411	229	2	pα	pα	NOUN
ejpam-5411	229	3	α	α	NOUN
ejpam-5411	229	4	)	)	PUNCT
ejpam-5411	230	1	=	=	SYM
ejpam-5411	230	2	g	g	PROPN
ejpam-5411	230	3	(	(	PUNCT
ejpam-5411	230	4	t	t	PROPN
ejpam-5411	230	5	α	α	PROPN
ejpam-5411	230	6	α	α	PROPN
ejpam-5411	230	7	)	)	PUNCT
ejpam-5411	230	8	for	for	ADP
ejpam-5411	230	9	all	all	DET
ejpam-5411	230	10	t	t	NOUN
ejpam-5411	230	11	∈	∈	PROPN
ejpam-5411	231	1	[	[	X
ejpam-5411	231	2	0,+∞	0,+∞	NUM
ejpam-5411	232	1	[	[	X
ejpam-5411	232	2	.	.	PUNCT
ejpam-5411	233	1	if	if	SCONJ
ejpam-5411	233	2	g	g	PROPN
ejpam-5411	233	3	∈	∈	PROPN
ejpam-5411	233	4	c1([0,+∞	c1([0,+∞	VERB
ejpam-5411	233	5	[	[	X
ejpam-5411	233	6	)	)	PUNCT
ejpam-5411	233	7	,	,	PUNCT
ejpam-5411	233	8	then	then	ADV
ejpam-5411	233	9	g′	g′	PROPN
ejpam-5411	233	10	(	(	PUNCT
ejpam-5411	233	11	t	t	PROPN
ejpam-5411	233	12	α	α	PROPN
ejpam-5411	233	13	α	α	PROPN
ejpam-5411	234	1	+	+	CCONJ
ejpam-5411	234	2	pα	pα	NOUN
ejpam-5411	234	3	α	α	NOUN
ejpam-5411	234	4	)	)	PUNCT
ejpam-5411	235	1	=	=	SYM
ejpam-5411	235	2	g′	g′	NOUN
ejpam-5411	235	3	(	(	PUNCT
ejpam-5411	235	4	t	t	PROPN
ejpam-5411	235	5	α	α	PROPN
ejpam-5411	235	6	α	α	PROPN
ejpam-5411	235	7	)	)	PUNCT
ejpam-5411	235	8	.	.	PUNCT
ejpam-5411	236	1	thus	thus	ADV
ejpam-5411	236	2	g1	g1	PROPN
ejpam-5411	236	3	(	(	PUNCT
ejpam-5411	236	4	tα	tα	PROPN
ejpam-5411	236	5	α	α	PROPN
ejpam-5411	237	1	+	+	CCONJ
ejpam-5411	237	2	pα	pα	NOUN
ejpam-5411	237	3	α	α	NOUN
ejpam-5411	237	4	)	)	PUNCT
ejpam-5411	237	5	=	=	SYM
ejpam-5411	237	6	g1	g1	PROPN
ejpam-5411	237	7	(	(	PUNCT
ejpam-5411	237	8	tα	tα	PROPN
ejpam-5411	237	9	α	α	PROPN
ejpam-5411	237	10	)	)	PUNCT
ejpam-5411	237	11	,	,	PUNCT
ejpam-5411	237	12	for	for	ADP
ejpam-5411	237	13	all	all	DET
ejpam-5411	237	14	t	t	NOUN
ejpam-5411	237	15	∈	∈	PROPN
ejpam-5411	238	1	[	[	X
ejpam-5411	238	2	0,+∞	0,+∞	NUM
ejpam-5411	239	1	[	[	X
ejpam-5411	239	2	.	.	PUNCT
ejpam-5411	240	1	finally	finally	ADV
ejpam-5411	240	2	t	t	X
ejpam-5411	240	3	(	(	PUNCT
ejpam-5411	240	4	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	240	5	)	)	PUNCT
ejpam-5411	240	6	is	be	AUX
ejpam-5411	240	7	α	α	X
ejpam-5411	240	8	-	-	NOUN
ejpam-5411	240	9	periodic	periodic	NOUN
ejpam-5411	240	10	with	with	ADP
ejpam-5411	240	11	period	period	NOUN
ejpam-5411	240	12	p	p	NOUN
ejpam-5411	240	13	for	for	ADP
ejpam-5411	240	14	all	all	DET
ejpam-5411	240	15	t	t	NOUN
ejpam-5411	240	16	∈	∈	PROPN
ejpam-5411	241	1	[	[	X
ejpam-5411	241	2	0,+∞	0,+∞	NUM
ejpam-5411	241	3	[	[	NOUN
ejpam-5411	241	4	.	.	PUNCT
ejpam-5411	241	5	example	example	NOUN
ejpam-5411	241	6	5	5	NUM
ejpam-5411	241	7	.	.	PUNCT
ejpam-5411	242	1	let	let	VERB
ejpam-5411	242	2	0	0	NUM
ejpam-5411	242	3	<	<	X
ejpam-5411	242	4	α	α	PROPN
ejpam-5411	242	5	≤	≤	NUM
ejpam-5411	242	6	1	1	NUM
ejpam-5411	242	7	and	and	CCONJ
ejpam-5411	242	8	t	t	NOUN
ejpam-5411	242	9	∈	∈	PROPN
ejpam-5411	243	1	[	[	X
ejpam-5411	243	2	0	0	NUM
ejpam-5411	243	3	,	,	PUNCT
ejpam-5411	243	4	(	(	PUNCT
ejpam-5411	243	5	3π2α	3π2α	NUM
ejpam-5411	243	6	)	)	PUNCT
ejpam-5411	243	7	1	1	NUM
ejpam-5411	243	8	α	α	NOUN
ejpam-5411	243	9	]	]	PUNCT
ejpam-5411	243	10	.	.	PUNCT
ejpam-5411	244	1	let	let	VERB
ejpam-5411	244	2	us	we	PRON
ejpam-5411	244	3	consider	consider	VERB
ejpam-5411	244	4	the	the	DET
ejpam-5411	244	5	function	function	NOUN
ejpam-5411	244	6	f(t	f(t	NOUN
ejpam-5411	244	7	)	)	PUNCT
ejpam-5411	245	1	=	=	SYM
ejpam-5411	246	1			PRON
ejpam-5411	246	2	f1(t	f1(t	X
ejpam-5411	246	3	)	)	PUNCT
ejpam-5411	246	4	=	=	SYM
ejpam-5411	246	5	sin(αtα	sin(αtα	NOUN
ejpam-5411	246	6	)	)	PUNCT
ejpam-5411	246	7	,	,	PUNCT
ejpam-5411	246	8	0	0	NUM
ejpam-5411	246	9	≤	≤	NUM
ejpam-5411	246	10	t	t	X
ejpam-5411	246	11	<	<	X
ejpam-5411	246	12	(	(	PUNCT
ejpam-5411	246	13	πα	πα	PROPN
ejpam-5411	246	14	)	)	PUNCT
ejpam-5411	246	15	1	1	NUM
ejpam-5411	246	16	α	α	NOUN
ejpam-5411	246	17	f2(t	f2(t	NOUN
ejpam-5411	246	18	)	)	PUNCT
ejpam-5411	246	19	=	=	SYM
ejpam-5411	246	20	−1	−1	NOUN
ejpam-5411	246	21	2	2	NUM
ejpam-5411	246	22	sin(2αtα	sin(2αtα	NOUN
ejpam-5411	246	23	)	)	PUNCT
ejpam-5411	246	24	,	,	PUNCT
ejpam-5411	246	25	(	(	PUNCT
ejpam-5411	246	26	πα	πα	X
ejpam-5411	246	27	)	)	PUNCT
ejpam-5411	246	28	1	1	NUM
ejpam-5411	246	29	α	α	PROPN
ejpam-5411	246	30	≤	≤	PUNCT
ejpam-5411	246	31	t	t	PROPN
ejpam-5411	246	32	≤	≤	NUM
ejpam-5411	246	33	(	(	PUNCT
ejpam-5411	246	34	3π2α	3π2α	NUM
ejpam-5411	246	35	)	)	PUNCT
ejpam-5411	246	36	1	1	NUM
ejpam-5411	246	37	α	α	NOUN
ejpam-5411	246	38	(	(	PUNCT
ejpam-5411	246	39	13	13	NUM
ejpam-5411	246	40	)	)	PUNCT
ejpam-5411	246	41	with	with	ADP
ejpam-5411	246	42	g(t	g(t	PROPN
ejpam-5411	246	43	)	)	PUNCT
ejpam-5411	246	44	=	=	SYM
ejpam-5411	246	45	f((αt	f((αt	NOUN
ejpam-5411	246	46	)	)	PUNCT
ejpam-5411	246	47	1	1	NUM
ejpam-5411	246	48	α	α	NOUN
ejpam-5411	246	49	)	)	PUNCT
ejpam-5411	246	50	=	=	PUNCT
ejpam-5411	246	51			PUNCT
ejpam-5411	246	52	g1(t	g1(t	PROPN
ejpam-5411	246	53	)	)	PUNCT
ejpam-5411	246	54	=	=	SYM
ejpam-5411	246	55	sin(α2	sin(α2	PROPN
ejpam-5411	246	56	t	t	PROPN
ejpam-5411	246	57	)	)	PUNCT
ejpam-5411	246	58	,	,	PUNCT
ejpam-5411	246	59	0	0	NUM
ejpam-5411	246	60	≤	≤	NUM
ejpam-5411	246	61	t	t	X
ejpam-5411	246	62	<	<	X
ejpam-5411	246	63	π	π	X
ejpam-5411	246	64	α2	α2	PROPN
ejpam-5411	246	65	g2(t	g2(t	PROPN
ejpam-5411	246	66	)	)	PUNCT
ejpam-5411	246	67	=	=	SYM
ejpam-5411	246	68	−1	−1	NOUN
ejpam-5411	246	69	2	2	NUM
ejpam-5411	246	70	sin(2α2	sin(2α2	NOUN
ejpam-5411	246	71	t	t	NOUN
ejpam-5411	246	72	)	)	PUNCT
ejpam-5411	246	73	,	,	PUNCT
ejpam-5411	246	74	π	π	PROPN
ejpam-5411	246	75	α2	α2	PROPN
ejpam-5411	246	76	≤	≤	PROPN
ejpam-5411	246	77	t	t	X
ejpam-5411	246	78	≤	≤	ADJ
ejpam-5411	246	79	3π	3π	NOUN
ejpam-5411	246	80	2α2	2α2	NUM
ejpam-5411	246	81	(	(	PUNCT
ejpam-5411	246	82	14	14	NUM
ejpam-5411	246	83	)	)	PUNCT
ejpam-5411	246	84	the	the	DET
ejpam-5411	246	85	function	function	NOUN
ejpam-5411	246	86	g(t	g(t	PROPN
ejpam-5411	246	87	)	)	PUNCT
ejpam-5411	246	88	is	be	AUX
ejpam-5411	246	89	continuous	continuous	ADJ
ejpam-5411	246	90	periodic	periodic	NOUN
ejpam-5411	246	91	with	with	ADP
ejpam-5411	246	92	period	period	NOUN
ejpam-5411	246	93	3π	3π	NOUN
ejpam-5411	246	94	2α2	2α2	NUM
ejpam-5411	246	95	for	for	ADP
ejpam-5411	246	96	all	all	DET
ejpam-5411	246	97	t	t	NOUN
ejpam-5411	246	98	∈	∈	PROPN
ejpam-5411	247	1	[	[	X
ejpam-5411	247	2	0,+∞	0,+∞	NUM
ejpam-5411	247	3	[	[	PUNCT
ejpam-5411	247	4	(	(	PUNCT
ejpam-5411	247	5	extended	extend	VERB
ejpam-5411	247	6	by	by	ADP
ejpam-5411	247	7	periodicity	periodicity	NOUN
ejpam-5411	247	8	to	to	ADP
ejpam-5411	247	9	[	[	X
ejpam-5411	247	10	0,+∞	0,+∞	PROPN
ejpam-5411	247	11	[	[	X
ejpam-5411	247	12	)	)	PUNCT
ejpam-5411	247	13	and	and	CCONJ
ejpam-5411	247	14	f(t	f(t	NOUN
ejpam-5411	247	15	)	)	PUNCT
ejpam-5411	247	16	is	be	AUX
ejpam-5411	247	17	α	α	X
ejpam-5411	247	18	-	-	NOUN
ejpam-5411	247	19	periodic	periodic	NOUN
ejpam-5411	247	20	with	with	ADP
ejpam-5411	247	21	period	period	NOUN
ejpam-5411	247	22	(	(	PUNCT
ejpam-5411	247	23	3π2α	3π2α	NUM
ejpam-5411	247	24	)	)	PUNCT
ejpam-5411	247	25	1	1	NUM
ejpam-5411	247	26	α	α	NOUN
ejpam-5411	247	27	for	for	ADP
ejpam-5411	247	28	all	all	DET
ejpam-5411	247	29	t	t	NOUN
ejpam-5411	247	30	∈	∈	PROPN
ejpam-5411	248	1	[	[	X
ejpam-5411	248	2	0,+∞	0,+∞	NUM
ejpam-5411	249	1	[	[	X
ejpam-5411	249	2	.	.	PUNCT
ejpam-5411	250	1	therefore	therefore	ADV
ejpam-5411	250	2	,	,	PUNCT
ejpam-5411	250	3	we	we	PRON
ejpam-5411	250	4	have	have	VERB
ejpam-5411	250	5	t	t	X
ejpam-5411	250	6	(	(	PUNCT
ejpam-5411	250	7	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	250	8	)	)	PUNCT
ejpam-5411	250	9	=	=	PUNCT
ejpam-5411	251	1			PRON
ejpam-5411	251	2	t	t	X
ejpam-5411	251	3	(	(	PUNCT
ejpam-5411	251	4	α)(f1)(t	α)(f1)(t	NUM
ejpam-5411	251	5	)	)	PUNCT
ejpam-5411	251	6	=	=	SYM
ejpam-5411	251	7	α2	α2	ADJ
ejpam-5411	251	8	cos(αtα	cos(αtα	NOUN
ejpam-5411	251	9	)	)	PUNCT
ejpam-5411	251	10	,	,	PUNCT
ejpam-5411	251	11	0	0	NUM
ejpam-5411	251	12	<	<	X
ejpam-5411	251	13	t	t	X
ejpam-5411	251	14	<	<	X
ejpam-5411	251	15	(	(	PUNCT
ejpam-5411	251	16	πα	πα	PROPN
ejpam-5411	251	17	)	)	PUNCT
ejpam-5411	251	18	1	1	NUM
ejpam-5411	252	1	α	α	NOUN
ejpam-5411	252	2	t	t	PROPN
ejpam-5411	252	3	(	(	PUNCT
ejpam-5411	252	4	α)(f2)(t	α)(f2)(t	PROPN
ejpam-5411	252	5	)	)	PUNCT
ejpam-5411	252	6	=	=	SYM
ejpam-5411	252	7	−α2	−α2	PROPN
ejpam-5411	252	8	cos(2αtα	cos(2αtα	X
ejpam-5411	252	9	)	)	PUNCT
ejpam-5411	252	10	,	,	PUNCT
ejpam-5411	252	11	(	(	PUNCT
ejpam-5411	252	12	πα	πα	X
ejpam-5411	252	13	)	)	PUNCT
ejpam-5411	252	14	1	1	NUM
ejpam-5411	252	15	α	α	NOUN
ejpam-5411	252	16	<	<	X
ejpam-5411	252	17	t	t	X
ejpam-5411	252	18	<	<	X
ejpam-5411	252	19	(	(	PUNCT
ejpam-5411	252	20	3π2α	3π2α	NUM
ejpam-5411	252	21	)	)	PUNCT
ejpam-5411	252	22	1	1	NUM
ejpam-5411	252	23	α	α	NOUN
ejpam-5411	252	24	(	(	PUNCT
ejpam-5411	252	25	15	15	NUM
ejpam-5411	252	26	)	)	PUNCT
ejpam-5411	252	27	t.	t.	NOUN
ejpam-5411	252	28	abdeljawad	abdeljawad	NOUN
ejpam-5411	252	29	et	et	PROPN
ejpam-5411	252	30	al	al	PROPN
ejpam-5411	252	31	.	.	PUNCT
ejpam-5411	252	32	/	/	SYM
ejpam-5411	252	33	eur	eur	PROPN
ejpam-5411	252	34	.	.	PUNCT
ejpam-5411	253	1	j.	j.	PROPN
ejpam-5411	253	2	pure	pure	PROPN
ejpam-5411	253	3	appl	appl	PROPN
ejpam-5411	253	4	.	.	PROPN
ejpam-5411	253	5	math	math	PROPN
ejpam-5411	253	6	,	,	PUNCT
ejpam-5411	253	7	17	17	NUM
ejpam-5411	253	8	(	(	PUNCT
ejpam-5411	253	9	4	4	NUM
ejpam-5411	253	10	)	)	PUNCT
ejpam-5411	253	11	(	(	PUNCT
ejpam-5411	253	12	2024	2024	NUM
ejpam-5411	253	13	)	)	PUNCT
ejpam-5411	253	14	,	,	PUNCT
ejpam-5411	253	15	2405	2405	NUM
ejpam-5411	253	16	-	-	SYM
ejpam-5411	253	17	2430	2430	NUM
ejpam-5411	253	18	2414	2414	NUM
ejpam-5411	253	19	and	and	CCONJ
ejpam-5411	253	20	g′(t	g′(t	PROPN
ejpam-5411	253	21	)	)	PUNCT
ejpam-5411	253	22	=	=	SYM
ejpam-5411	253	23	t	t	PROPN
ejpam-5411	253	24	(	(	PUNCT
ejpam-5411	253	25	α)(f)((αt	α)(f)((αt	PROPN
ejpam-5411	253	26	)	)	PUNCT
ejpam-5411	253	27	1	1	NUM
ejpam-5411	253	28	α	α	NOUN
ejpam-5411	253	29	)	)	PUNCT
ejpam-5411	254	1	=	=	PUNCT
ejpam-5411	254	2			PUNCT
ejpam-5411	254	3	g′1(t	g′1(t	NUM
ejpam-5411	254	4	)	)	PUNCT
ejpam-5411	254	5	=	=	VERB
ejpam-5411	254	6	α2	α2	PROPN
ejpam-5411	254	7	cos(α2	cos(α2	PROPN
ejpam-5411	254	8	t	t	PROPN
ejpam-5411	254	9	)	)	PUNCT
ejpam-5411	254	10	,	,	PUNCT
ejpam-5411	255	1	0	0	NUM
ejpam-5411	255	2	<	<	X
ejpam-5411	255	3	t	t	X
ejpam-5411	255	4	<	<	X
ejpam-5411	255	5	π	π	X
ejpam-5411	255	6	α2	α2	NOUN
ejpam-5411	255	7	g′2(t	g′2(t	NOUN
ejpam-5411	255	8	)	)	PUNCT
ejpam-5411	256	1	=	=	PUNCT
ejpam-5411	257	1	−α2	−α2	PROPN
ejpam-5411	257	2	cos(2α2	cos(2α2	PROPN
ejpam-5411	257	3	t	t	PROPN
ejpam-5411	257	4	)	)	PUNCT
ejpam-5411	257	5	,	,	PUNCT
ejpam-5411	257	6	π	π	PROPN
ejpam-5411	257	7	α2	α2	PROPN
ejpam-5411	257	8	<	<	X
ejpam-5411	257	9	t	t	X
ejpam-5411	257	10	<	<	X
ejpam-5411	257	11	3π	3π	NUM
ejpam-5411	257	12	2α2	2α2	NUM
ejpam-5411	257	13	(	(	PUNCT
ejpam-5411	257	14	16	16	NUM
ejpam-5411	257	15	)	)	PUNCT
ejpam-5411	257	16	the	the	DET
ejpam-5411	257	17	function	function	NOUN
ejpam-5411	257	18	f1	f1	NOUN
ejpam-5411	257	19	is	be	AUX
ejpam-5411	257	20	continuously	continuously	ADV
ejpam-5411	257	21	α	α	PRON
ejpam-5411	257	22	-	-	NOUN
ejpam-5411	257	23	differentiable	differentiable	ADJ
ejpam-5411	257	24	on	on	ADP
ejpam-5411	257	25	[	[	X
ejpam-5411	257	26	0	0	NUM
ejpam-5411	257	27	,	,	PUNCT
ejpam-5411	257	28	(	(	PUNCT
ejpam-5411	257	29	πα	πα	X
ejpam-5411	257	30	)	)	PUNCT
ejpam-5411	257	31	1	1	NUM
ejpam-5411	257	32	α	α	NOUN
ejpam-5411	257	33	]	]	PUNCT
ejpam-5411	257	34	and	and	CCONJ
ejpam-5411	257	35	g1	g1	PROPN
ejpam-5411	257	36	∈	∈	PROPN
ejpam-5411	257	37	c1([0	c1([0	PROPN
ejpam-5411	257	38	,	,	PUNCT
ejpam-5411	257	39	π	π	PROPN
ejpam-5411	257	40	α2	α2	PROPN
ejpam-5411	257	41	]	]	PUNCT
ejpam-5411	257	42	)	)	PUNCT
ejpam-5411	257	43	.	.	PUNCT
ejpam-5411	258	1	the	the	DET
ejpam-5411	258	2	function	function	NOUN
ejpam-5411	258	3	f2	f2	PROPN
ejpam-5411	258	4	is	be	AUX
ejpam-5411	258	5	continuously	continuously	ADV
ejpam-5411	258	6	α	α	PRON
ejpam-5411	258	7	-	-	NOUN
ejpam-5411	258	8	differentiable	differentiable	ADJ
ejpam-5411	258	9	on	on	ADP
ejpam-5411	258	10	[	[	X
ejpam-5411	258	11	(	(	PUNCT
ejpam-5411	258	12	πα	πα	ADJ
ejpam-5411	258	13	)	)	PUNCT
ejpam-5411	258	14	1	1	NUM
ejpam-5411	258	15	α	α	NOUN
ejpam-5411	258	16	,	,	PUNCT
ejpam-5411	258	17	(	(	PUNCT
ejpam-5411	258	18	3π2α	3π2α	NUM
ejpam-5411	258	19	)	)	PUNCT
ejpam-5411	258	20	1	1	NUM
ejpam-5411	258	21	α	α	NOUN
ejpam-5411	258	22	]	]	PUNCT
ejpam-5411	258	23	and	and	CCONJ
ejpam-5411	258	24	g2	g2	PROPN
ejpam-5411	258	25	∈	∈	PROPN
ejpam-5411	258	26	c1	c1	PROPN
ejpam-5411	258	27	(	(	PUNCT
ejpam-5411	258	28	[	[	PUNCT
ejpam-5411	258	29	π	π	X
ejpam-5411	258	30	α2	α2	PROPN
ejpam-5411	258	31	,	,	PUNCT
ejpam-5411	258	32	3π	3π	NOUN
ejpam-5411	258	33	2α2	2α2	NUM
ejpam-5411	258	34	]	]	PUNCT
ejpam-5411	258	35	)	)	PUNCT
ejpam-5411	258	36	.	.	PUNCT
ejpam-5411	259	1	on	on	ADP
ejpam-5411	259	2	the	the	DET
ejpam-5411	259	3	other	other	ADJ
ejpam-5411	259	4	hand	hand	NOUN
ejpam-5411	259	5	,	,	PUNCT
ejpam-5411	259	6	we	we	PRON
ejpam-5411	259	7	have	have	VERB
ejpam-5411	259	8	t	t	PROPN
ejpam-5411	259	9	(	(	PUNCT
ejpam-5411	259	10	α)(f1	α)(f1	PROPN
ejpam-5411	259	11	)	)	PUNCT
ejpam-5411	259	12	(	(	PUNCT
ejpam-5411	259	13	(	(	PUNCT
ejpam-5411	259	14	π	π	NOUN
ejpam-5411	259	15	α	α	NOUN
ejpam-5411	259	16	)	)	PUNCT
ejpam-5411	259	17	1	1	NUM
ejpam-5411	259	18	α	α	NOUN
ejpam-5411	259	19	)	)	PUNCT
ejpam-5411	260	1	=	=	SYM
ejpam-5411	260	2	g′1	g′1	PROPN
ejpam-5411	260	3	(	(	PUNCT
ejpam-5411	260	4	π	π	NOUN
ejpam-5411	260	5	α2	α2	PROPN
ejpam-5411	260	6	)	)	PUNCT
ejpam-5411	261	1	=	=	SYM
ejpam-5411	261	2	t	t	PROPN
ejpam-5411	261	3	(	(	PUNCT
ejpam-5411	261	4	α)(f2	α)(f2	PROPN
ejpam-5411	261	5	)	)	PUNCT
ejpam-5411	261	6	(	(	PUNCT
ejpam-5411	261	7	(	(	PUNCT
ejpam-5411	261	8	π	π	NOUN
ejpam-5411	261	9	α	α	NOUN
ejpam-5411	261	10	)	)	PUNCT
ejpam-5411	261	11	1	1	NUM
ejpam-5411	261	12	α	α	NOUN
ejpam-5411	261	13	)	)	PUNCT
ejpam-5411	262	1	=	=	SYM
ejpam-5411	262	2	g′2	g′2	NOUN
ejpam-5411	262	3	(	(	PUNCT
ejpam-5411	262	4	π	π	NOUN
ejpam-5411	262	5	α2	α2	PROPN
ejpam-5411	262	6	)	)	PUNCT
ejpam-5411	263	1	=	=	SYM
ejpam-5411	263	2	−α2	−α2	PROPN
ejpam-5411	263	3	and	and	CCONJ
ejpam-5411	263	4	t	t	PROPN
ejpam-5411	263	5	(	(	PUNCT
ejpam-5411	263	6	α)(f1)(0	α)(f1)(0	PROPN
ejpam-5411	263	7	)	)	PUNCT
ejpam-5411	263	8	=	=	PUNCT
ejpam-5411	263	9	g′1(0	g′1(0	PROPN
ejpam-5411	263	10	)	)	PUNCT
ejpam-5411	264	1	=	=	SYM
ejpam-5411	264	2	t	t	PROPN
ejpam-5411	264	3	(	(	PUNCT
ejpam-5411	264	4	α)(f1	α)(f1	PROPN
ejpam-5411	264	5	)	)	PUNCT
ejpam-5411	264	6	(	(	PUNCT
ejpam-5411	264	7	(	(	PUNCT
ejpam-5411	264	8	3π	3π	NOUN
ejpam-5411	264	9	2α	2α	NOUN
ejpam-5411	264	10	)	)	PUNCT
ejpam-5411	264	11	1	1	NUM
ejpam-5411	264	12	α	α	NOUN
ejpam-5411	264	13	)	)	PUNCT
ejpam-5411	265	1	=	=	PUNCT
ejpam-5411	265	2	g′2	g′2	NOUN
ejpam-5411	265	3	(	(	PUNCT
ejpam-5411	265	4	3π	3π	NOUN
ejpam-5411	265	5	2α2	2α2	NUM
ejpam-5411	265	6	)	)	PUNCT
ejpam-5411	265	7	=	=	SYM
ejpam-5411	266	1	α2	α2	ADJ
ejpam-5411	266	2	.	.	PUNCT
ejpam-5411	267	1	then	then	ADV
ejpam-5411	267	2	f	f	PROPN
ejpam-5411	267	3	is	be	AUX
ejpam-5411	267	4	continuously	continuously	ADV
ejpam-5411	267	5	α	α	PRON
ejpam-5411	267	6	-	-	NOUN
ejpam-5411	267	7	differentiable	differentiable	ADJ
ejpam-5411	267	8	on	on	ADP
ejpam-5411	267	9	[	[	X
ejpam-5411	267	10	0	0	NUM
ejpam-5411	267	11	,	,	PUNCT
ejpam-5411	267	12	(	(	PUNCT
ejpam-5411	267	13	3π2α	3π2α	NUM
ejpam-5411	267	14	)	)	PUNCT
ejpam-5411	267	15	1	1	NUM
ejpam-5411	267	16	α	α	NOUN
ejpam-5411	267	17	]	]	PUNCT
ejpam-5411	267	18	and	and	CCONJ
ejpam-5411	267	19	g	g	PROPN
ejpam-5411	267	20	∈	∈	PROPN
ejpam-5411	267	21	c1([0	c1([0	PROPN
ejpam-5411	267	22	,	,	PUNCT
ejpam-5411	267	23	3π	3π	NOUN
ejpam-5411	267	24	2α2	2α2	NUM
ejpam-5411	267	25	]	]	PUNCT
ejpam-5411	267	26	)	)	PUNCT
ejpam-5411	267	27	.	.	PUNCT
ejpam-5411	268	1	therefore	therefore	ADV
ejpam-5411	268	2	f	f	PROPN
ejpam-5411	268	3	is	be	AUX
ejpam-5411	268	4	continuously	continuously	ADV
ejpam-5411	268	5	α	α	PRON
ejpam-5411	268	6	-	-	NOUN
ejpam-5411	268	7	differentiable	differentiable	ADJ
ejpam-5411	268	8	on	on	ADP
ejpam-5411	268	9	[	[	X
ejpam-5411	268	10	0,+∞	0,+∞	NUM
ejpam-5411	268	11	[	[	PUNCT
ejpam-5411	268	12	and	and	CCONJ
ejpam-5411	268	13	g	g	PROPN
ejpam-5411	268	14	∈	∈	PROPN
ejpam-5411	268	15	c1([0,+∞	c1([0,+∞	VERB
ejpam-5411	268	16	[	[	X
ejpam-5411	268	17	)	)	PUNCT
ejpam-5411	268	18	.	.	PUNCT
ejpam-5411	269	1	so	so	ADV
ejpam-5411	269	2	,	,	PUNCT
ejpam-5411	269	3	we	we	PRON
ejpam-5411	269	4	have	have	AUX
ejpam-5411	269	5	g′(t	g′(t	NOUN
ejpam-5411	269	6	)	)	PUNCT
ejpam-5411	269	7	is	be	AUX
ejpam-5411	269	8	periodic	periodic	ADJ
ejpam-5411	269	9	with	with	ADP
ejpam-5411	269	10	period	period	NOUN
ejpam-5411	269	11	3π	3π	NOUN
ejpam-5411	269	12	2α2	2α2	NUM
ejpam-5411	269	13	for	for	ADP
ejpam-5411	269	14	all	all	DET
ejpam-5411	269	15	t	t	NOUN
ejpam-5411	269	16	∈	∈	PROPN
ejpam-5411	270	1	[	[	X
ejpam-5411	270	2	0,+∞	0,+∞	NUM
ejpam-5411	270	3	[	[	PUNCT
ejpam-5411	270	4	(	(	PUNCT
ejpam-5411	270	5	extended	extend	VERB
ejpam-5411	270	6	by	by	ADP
ejpam-5411	270	7	periodicity	periodicity	NOUN
ejpam-5411	270	8	to	to	ADP
ejpam-5411	270	9	[	[	X
ejpam-5411	270	10	0,+∞	0,+∞	PROPN
ejpam-5411	270	11	[	[	NOUN
ejpam-5411	270	12	)	)	PUNCT
ejpam-5411	270	13	and	and	CCONJ
ejpam-5411	270	14	tα(f)(t	tα(f)(t	NUM
ejpam-5411	270	15	)	)	PUNCT
ejpam-5411	270	16	is	be	AUX
ejpam-5411	270	17	α	α	X
ejpam-5411	270	18	-	-	NOUN
ejpam-5411	270	19	periodic	periodic	NOUN
ejpam-5411	270	20	with	with	ADP
ejpam-5411	270	21	period	period	NOUN
ejpam-5411	270	22	(	(	PUNCT
ejpam-5411	270	23	3π2α	3π2α	NUM
ejpam-5411	270	24	)	)	PUNCT
ejpam-5411	270	25	1	1	NUM
ejpam-5411	270	26	α	α	NOUN
ejpam-5411	270	27	for	for	ADP
ejpam-5411	270	28	all	all	DET
ejpam-5411	270	29	t	t	NOUN
ejpam-5411	270	30	∈	∈	PROPN
ejpam-5411	271	1	[	[	X
ejpam-5411	271	2	0,+∞	0,+∞	NUM
ejpam-5411	271	3	[	[	X
ejpam-5411	271	4	.	.	PUNCT
ejpam-5411	271	5	theorem	theorem	NOUN
ejpam-5411	271	6	6	6	NUM
ejpam-5411	271	7	.	.	PUNCT
ejpam-5411	272	1	let	let	VERB
ejpam-5411	272	2	0	0	NUM
ejpam-5411	272	3	<	<	X
ejpam-5411	272	4	α	α	X
ejpam-5411	272	5	≤	≤	NUM
ejpam-5411	272	6	1	1	NUM
ejpam-5411	272	7	.	.	PUNCT
ejpam-5411	273	1	assume	assume	VERB
ejpam-5411	273	2	that	that	SCONJ
ejpam-5411	273	3	the	the	DET
ejpam-5411	273	4	function	function	NOUN
ejpam-5411	273	5	f	f	NOUN
ejpam-5411	273	6	:	:	PUNCT
ejpam-5411	274	1	[	[	X
ejpam-5411	274	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	274	3	r	r	NOUN
ejpam-5411	274	4	is	be	AUX
ejpam-5411	274	5	n	n	PRON
ejpam-5411	274	6	times	time	NOUN
ejpam-5411	274	7	continuously	continuously	ADV
ejpam-5411	274	8	α	α	VERB
ejpam-5411	274	9	-	-	NOUN
ejpam-5411	274	10	differentiable	differentiable	ADJ
ejpam-5411	274	11	on	on	ADP
ejpam-5411	274	12	[	[	X
ejpam-5411	274	13	0,+∞	0,+∞	NUM
ejpam-5411	274	14	[	[	PUNCT
ejpam-5411	274	15	for	for	ADP
ejpam-5411	274	16	n	n	PRON
ejpam-5411	274	17	∈	∈	PROPN
ejpam-5411	274	18	n	n	NOUN
ejpam-5411	274	19	and	and	CCONJ
ejpam-5411	274	20	α	α	NOUN
ejpam-5411	274	21	-	-	NOUN
ejpam-5411	274	22	periodic	periodic	NOUN
ejpam-5411	274	23	with	with	ADP
ejpam-5411	274	24	period	period	NOUN
ejpam-5411	274	25	p.	p.	NOUN
ejpam-5411	274	26	then	then	ADV
ejpam-5411	274	27	for	for	ADP
ejpam-5411	274	28	all	all	DET
ejpam-5411	274	29	j	j	PROPN
ejpam-5411	274	30	∈	∈	PROPN
ejpam-5411	274	31	{	{	PUNCT
ejpam-5411	274	32	0	0	NUM
ejpam-5411	274	33	,	,	PUNCT
ejpam-5411	274	34	.	.	PUNCT
ejpam-5411	274	35	.	.	PUNCT
ejpam-5411	274	36	.	.	PUNCT
ejpam-5411	275	1	,	,	PUNCT
ejpam-5411	275	2	n	n	CCONJ
ejpam-5411	275	3	}	}	PUNCT
ejpam-5411	276	1	and	and	CCONJ
ejpam-5411	276	2	for	for	ADP
ejpam-5411	276	3	all	all	DET
ejpam-5411	276	4	t	t	NOUN
ejpam-5411	276	5	∈	∈	PROPN
ejpam-5411	277	1	[	[	X
ejpam-5411	277	2	0,+∞	0,+∞	NUM
ejpam-5411	277	3	[	[	X
ejpam-5411	277	4	,	,	PUNCT
ejpam-5411	277	5	we	we	PRON
ejpam-5411	277	6	have	have	VERB
ejpam-5411	277	7	(	(	PUNCT
ejpam-5411	277	8	i	i	NOUN
ejpam-5411	277	9	)	)	PUNCT
ejpam-5411	277	10	t	t	PROPN
ejpam-5411	277	11	(	(	PUNCT
ejpam-5411	277	12	jα)(f)(t	jα)(f)(t	NOUN
ejpam-5411	277	13	)	)	PUNCT
ejpam-5411	277	14	=	=	SYM
ejpam-5411	277	15	g(j	g(j	PROPN
ejpam-5411	277	16	)	)	PUNCT
ejpam-5411	277	17	(	(	PUNCT
ejpam-5411	277	18	t	t	PROPN
ejpam-5411	277	19	α	α	PROPN
ejpam-5411	277	20	α	α	NOUN
ejpam-5411	277	21	)	)	PUNCT
ejpam-5411	277	22	and	and	CCONJ
ejpam-5411	277	23	g	g	NOUN
ejpam-5411	277	24	∈	∈	PROPN
ejpam-5411	278	1	cj([0,+∞	cj([0,+∞	X
ejpam-5411	278	2	[	[	X
ejpam-5411	278	3	)	)	PUNCT
ejpam-5411	278	4	where	where	SCONJ
ejpam-5411	278	5	g(t	g(t	NOUN
ejpam-5411	278	6	)	)	PUNCT
ejpam-5411	278	7	=	=	SYM
ejpam-5411	278	8	f((αt	f((αt	NOUN
ejpam-5411	278	9	)	)	PUNCT
ejpam-5411	278	10	1	1	NUM
ejpam-5411	278	11	α	α	NOUN
ejpam-5411	278	12	)	)	PUNCT
ejpam-5411	278	13	.	.	PUNCT
ejpam-5411	278	14	(	(	PUNCT
ejpam-5411	278	15	ii	ii	PROPN
ejpam-5411	278	16	)	)	PUNCT
ejpam-5411	278	17	t	t	PROPN
ejpam-5411	278	18	(	(	PUNCT
ejpam-5411	278	19	jα)(f)(t	jα)(f)(t	PROPN
ejpam-5411	278	20	)	)	PUNCT
ejpam-5411	278	21	is	be	AUX
ejpam-5411	278	22	α	α	DET
ejpam-5411	278	23	-	-	ADJ
ejpam-5411	278	24	periodic	periodic	ADJ
ejpam-5411	278	25	function	function	NOUN
ejpam-5411	278	26	with	with	ADP
ejpam-5411	278	27	period	period	NOUN
ejpam-5411	279	1	p.	p.	NOUN
ejpam-5411	279	2	note	note	VERB
ejpam-5411	279	3	that	that	SCONJ
ejpam-5411	279	4	t	t	PROPN
ejpam-5411	279	5	(	(	PUNCT
ejpam-5411	279	6	0)(f)(t	0)(f)(t	NUM
ejpam-5411	279	7	)	)	PUNCT
ejpam-5411	279	8	=	=	SYM
ejpam-5411	279	9	f(t	f(t	NOUN
ejpam-5411	279	10	)	)	PUNCT
ejpam-5411	279	11	and	and	CCONJ
ejpam-5411	279	12	g(0)(t	g(0)(t	ADJ
ejpam-5411	279	13	)	)	PUNCT
ejpam-5411	279	14	=	=	SYM
ejpam-5411	279	15	g(t	g(t	PROPN
ejpam-5411	279	16	)	)	PUNCT
ejpam-5411	279	17	.	.	PUNCT
ejpam-5411	280	1	proof	proof	NOUN
ejpam-5411	280	2	.	.	PUNCT
ejpam-5411	281	1	let	let	VERB
ejpam-5411	281	2	0	0	NUM
ejpam-5411	281	3	<	<	X
ejpam-5411	281	4	α	α	PROPN
ejpam-5411	281	5	≤	≤	NUM
ejpam-5411	281	6	1	1	NUM
ejpam-5411	281	7	and	and	CCONJ
ejpam-5411	281	8	f	f	PROPN
ejpam-5411	281	9	is	be	AUX
ejpam-5411	281	10	n	n	PRON
ejpam-5411	281	11	times	time	NOUN
ejpam-5411	281	12	continuously	continuously	ADV
ejpam-5411	281	13	α	α	VERB
ejpam-5411	281	14	-	-	NOUN
ejpam-5411	281	15	differentiable	differentiable	ADJ
ejpam-5411	281	16	on	on	ADP
ejpam-5411	281	17	[	[	X
ejpam-5411	281	18	0,+∞	0,+∞	NUM
ejpam-5411	281	19	[	[	PUNCT
ejpam-5411	281	20	for	for	ADP
ejpam-5411	281	21	n	n	PRON
ejpam-5411	281	22	∈	∈	PROPN
ejpam-5411	281	23	n	n	NOUN
ejpam-5411	281	24	and	and	CCONJ
ejpam-5411	281	25	α	α	NOUN
ejpam-5411	281	26	-	-	NOUN
ejpam-5411	281	27	periodic	periodic	NOUN
ejpam-5411	281	28	with	with	ADP
ejpam-5411	281	29	period	period	NOUN
ejpam-5411	281	30	p.	p.	NOUN
ejpam-5411	281	31	let	let	VERB
ejpam-5411	281	32	j	j	PROPN
ejpam-5411	281	33	∈	∈	PROPN
ejpam-5411	281	34	{	{	PUNCT
ejpam-5411	281	35	0	0	NUM
ejpam-5411	281	36	,	,	PUNCT
ejpam-5411	281	37	.	.	PUNCT
ejpam-5411	281	38	.	.	PUNCT
ejpam-5411	282	1	.	.	PUNCT
ejpam-5411	282	2	,	,	PUNCT
ejpam-5411	283	1	n	n	CCONJ
ejpam-5411	283	2	}	}	PUNCT
ejpam-5411	283	3	and	and	CCONJ
ejpam-5411	283	4	by	by	ADP
ejpam-5411	283	5	recurrence	recurrence	NOUN
ejpam-5411	283	6	,	,	PUNCT
ejpam-5411	283	7	we	we	PRON
ejpam-5411	283	8	have	have	VERB
ejpam-5411	283	9	the	the	DET
ejpam-5411	283	10	following	following	NOUN
ejpam-5411	283	11	:	:	PUNCT
ejpam-5411	283	12	for	for	ADP
ejpam-5411	283	13	j	j	PROPN
ejpam-5411	283	14	=	=	SYM
ejpam-5411	283	15	0	0	PROPN
ejpam-5411	283	16	,	,	PUNCT
ejpam-5411	283	17	f	f	PROPN
ejpam-5411	283	18	is	be	AUX
ejpam-5411	283	19	α	α	NOUN
ejpam-5411	283	20	-	-	NOUN
ejpam-5411	283	21	periodic	periodic	ADJ
ejpam-5411	283	22	,	,	PUNCT
ejpam-5411	283	23	then	then	ADV
ejpam-5411	283	24	by	by	ADP
ejpam-5411	283	25	definition	definition	NOUN
ejpam-5411	283	26	6	6	NUM
ejpam-5411	283	27	there	there	ADV
ejpam-5411	283	28	exists	exist	VERB
ejpam-5411	283	29	a	a	DET
ejpam-5411	283	30	continuous	continuous	ADJ
ejpam-5411	283	31	function	function	NOUN
ejpam-5411	283	32	g	g	NOUN
ejpam-5411	283	33	:	:	PUNCT
ejpam-5411	284	1	[	[	X
ejpam-5411	284	2	0,+∞[→	0,+∞[→	NOUN
ejpam-5411	284	3	r	r	NOUN
ejpam-5411	284	4	such	such	ADJ
ejpam-5411	284	5	that	that	SCONJ
ejpam-5411	284	6	f(t	f(t	NOUN
ejpam-5411	284	7	)	)	PUNCT
ejpam-5411	284	8	=	=	SYM
ejpam-5411	284	9	g	g	PROPN
ejpam-5411	284	10	(	(	PUNCT
ejpam-5411	284	11	tα	tα	PROPN
ejpam-5411	284	12	α	α	PROPN
ejpam-5411	284	13	)	)	PUNCT
ejpam-5411	285	1	=	=	SYM
ejpam-5411	285	2	g	g	PROPN
ejpam-5411	285	3	(	(	PUNCT
ejpam-5411	285	4	tα	tα	PROPN
ejpam-5411	285	5	α	α	PROPN
ejpam-5411	286	1	+	+	CCONJ
ejpam-5411	286	2	pα	pα	NOUN
ejpam-5411	286	3	α	α	NOUN
ejpam-5411	286	4	)	)	PUNCT
ejpam-5411	286	5	.	.	PUNCT
ejpam-5411	287	1	thus	thus	ADV
ejpam-5411	287	2	(	(	PUNCT
ejpam-5411	287	3	1	1	X
ejpam-5411	287	4	)	)	PUNCT
ejpam-5411	287	5	and	and	CCONJ
ejpam-5411	287	6	(	(	PUNCT
ejpam-5411	287	7	2	2	X
ejpam-5411	287	8	)	)	PUNCT
ejpam-5411	287	9	are	be	AUX
ejpam-5411	287	10	satisfied	satisfied	ADJ
ejpam-5411	287	11	.	.	PUNCT
ejpam-5411	288	1	for	for	ADP
ejpam-5411	288	2	j	j	PROPN
ejpam-5411	288	3	=	=	SYM
ejpam-5411	288	4	1	1	NUM
ejpam-5411	288	5	,	,	PUNCT
ejpam-5411	288	6	see	see	VERB
ejpam-5411	288	7	theorem	theorem	NOUN
ejpam-5411	288	8	5	5	NUM
ejpam-5411	288	9	.	.	PUNCT
ejpam-5411	288	10	suppose	suppose	VERB
ejpam-5411	288	11	that	that	SCONJ
ejpam-5411	288	12	for	for	ADP
ejpam-5411	288	13	all	all	DET
ejpam-5411	288	14	j	j	PROPN
ejpam-5411	288	15	∈	∈	PROPN
ejpam-5411	288	16	{	{	PUNCT
ejpam-5411	288	17	2	2	NUM
ejpam-5411	288	18	,	,	PUNCT
ejpam-5411	288	19	...	...	PUNCT
ejpam-5411	288	20	,	,	PUNCT
ejpam-5411	288	21	n	n	CCONJ
ejpam-5411	288	22	}	}	PUNCT
ejpam-5411	288	23	and	and	CCONJ
ejpam-5411	288	24	for	for	ADP
ejpam-5411	288	25	all	all	DET
ejpam-5411	288	26	t	t	NOUN
ejpam-5411	288	27	∈	∈	PROPN
ejpam-5411	288	28	[	[	X
ejpam-5411	288	29	0,+∞	0,+∞	PROPN
ejpam-5411	288	30	[	[	X
ejpam-5411	288	31	,	,	PUNCT
ejpam-5411	288	32	(	(	PUNCT
ejpam-5411	288	33	*	*	NOUN
ejpam-5411	288	34	)	)	PUNCT
ejpam-5411	288	35	t	t	NOUN
ejpam-5411	288	36	(	(	PUNCT
ejpam-5411	288	37	(	(	PUNCT
ejpam-5411	288	38	j−1)α)(f)(t	j−1)α)(f)(t	PROPN
ejpam-5411	288	39	)	)	PUNCT
ejpam-5411	288	40	=	=	SYM
ejpam-5411	288	41	g(j−1	g(j−1	NOUN
ejpam-5411	288	42	)	)	PUNCT
ejpam-5411	288	43	(	(	PUNCT
ejpam-5411	288	44	t	t	PROPN
ejpam-5411	288	45	α	α	PROPN
ejpam-5411	288	46	α	α	NOUN
ejpam-5411	288	47	)	)	PUNCT
ejpam-5411	288	48	and	and	CCONJ
ejpam-5411	288	49	g	g	PROPN
ejpam-5411	288	50	∈	∈	PROPN
ejpam-5411	288	51	cj−1([0,+∞	cj−1([0,+∞	PROPN
ejpam-5411	288	52	[	[	X
ejpam-5411	288	53	)	)	PUNCT
ejpam-5411	288	54	(	(	PUNCT
ejpam-5411	288	55	*	*	NOUN
ejpam-5411	288	56	)	)	PUNCT
ejpam-5411	288	57	t	t	NOUN
ejpam-5411	288	58	(	(	PUNCT
ejpam-5411	288	59	(	(	PUNCT
ejpam-5411	288	60	j−1)α)(f)(t	j−1)α)(f)(t	PROPN
ejpam-5411	288	61	)	)	PUNCT
ejpam-5411	288	62	is	be	AUX
ejpam-5411	288	63	α	α	NOUN
ejpam-5411	288	64	-	-	NOUN
ejpam-5411	288	65	periodic	periodic	NOUN
ejpam-5411	288	66	with	with	ADP
ejpam-5411	288	67	period	period	NOUN
ejpam-5411	289	1	p.	p.	NOUN
ejpam-5411	289	2	(	(	PUNCT
ejpam-5411	289	3	1	1	NUM
ejpam-5411	289	4	)	)	PUNCT
ejpam-5411	289	5	for	for	ADP
ejpam-5411	289	6	all	all	DET
ejpam-5411	289	7	t	t	NOUN
ejpam-5411	289	8	∈	∈	PROPN
ejpam-5411	290	1	[	[	X
ejpam-5411	290	2	0,+∞	0,+∞	NUM
ejpam-5411	290	3	[	[	X
ejpam-5411	290	4	,	,	PUNCT
ejpam-5411	290	5	we	we	PRON
ejpam-5411	290	6	have	have	VERB
ejpam-5411	290	7	f(t	f(t	NOUN
ejpam-5411	290	8	)	)	PUNCT
ejpam-5411	290	9	is	be	AUX
ejpam-5411	290	10	j	j	PROPN
ejpam-5411	290	11	times	time	NOUN
ejpam-5411	290	12	continuously	continuously	ADV
ejpam-5411	290	13	α	α	VERB
ejpam-5411	290	14	-	-	NOUN
ejpam-5411	290	15	differentiable	differentiable	ADJ
ejpam-5411	290	16	,	,	PUNCT
ejpam-5411	290	17	then	then	ADV
ejpam-5411	290	18	t	t	PROPN
ejpam-5411	290	19	(	(	PUNCT
ejpam-5411	290	20	jα)(f)(t	jα)(f)(t	NOUN
ejpam-5411	290	21	)	)	PUNCT
ejpam-5411	290	22	exists	exist	VERB
ejpam-5411	290	23	and	and	CCONJ
ejpam-5411	290	24	continuous	continuous	ADJ
ejpam-5411	290	25	.	.	PUNCT
ejpam-5411	291	1	case	case	NOUN
ejpam-5411	291	2	1	1	NUM
ejpam-5411	291	3	:	:	PUNCT
ejpam-5411	291	4	t	t	PROPN
ejpam-5411	291	5	>	>	X
ejpam-5411	291	6	0	0	PUNCT
ejpam-5411	292	1	by	by	ADP
ejpam-5411	292	2	hypothesis	hypothesis	NOUN
ejpam-5411	292	3	t	t	PROPN
ejpam-5411	292	4	(	(	PUNCT
ejpam-5411	292	5	(	(	PUNCT
ejpam-5411	292	6	j−1)α)(f)(t	j−1)α)(f)(t	PROPN
ejpam-5411	292	7	)	)	PUNCT
ejpam-5411	292	8	is	be	AUX
ejpam-5411	292	9	α	α	NOUN
ejpam-5411	292	10	-	-	NOUN
ejpam-5411	292	11	periodic	periodic	NOUN
ejpam-5411	292	12	and	and	CCONJ
ejpam-5411	292	13	g	g	NOUN
ejpam-5411	292	14	∈	∈	PROPN
ejpam-5411	292	15	cj−1([0,+∞	cj−1([0,+∞	PROPN
ejpam-5411	292	16	[	[	X
ejpam-5411	292	17	)	)	PUNCT
ejpam-5411	292	18	,	,	PUNCT
ejpam-5411	292	19	then	then	ADV
ejpam-5411	292	20	t	t	PROPN
ejpam-5411	292	21	(	(	PUNCT
ejpam-5411	292	22	jα)(f)(t	jα)(f)(t	PROPN
ejpam-5411	292	23	)	)	PUNCT
ejpam-5411	292	24	:	:	PUNCT
ejpam-5411	293	1	=	=	SYM
ejpam-5411	293	2	t	t	X
ejpam-5411	293	3	(	(	PUNCT
ejpam-5411	293	4	α)(t	α)(t	PROPN
ejpam-5411	293	5	(	(	PUNCT
ejpam-5411	293	6	(	(	PUNCT
ejpam-5411	293	7	j−1)α)(f))(t	j−1)α)(f))(t	X
ejpam-5411	293	8	)	)	PUNCT
ejpam-5411	293	9	=	=	SYM
ejpam-5411	293	10	t	t	PROPN
ejpam-5411	293	11	(	(	PUNCT
ejpam-5411	293	12	α)(g(j−1	α)(g(j−1	NOUN
ejpam-5411	293	13	)	)	PUNCT
ejpam-5411	293	14	)	)	PUNCT
ejpam-5411	293	15	(	(	PUNCT
ejpam-5411	293	16	tα	tα	PROPN
ejpam-5411	293	17	α	α	PROPN
ejpam-5411	293	18	)	)	PUNCT
ejpam-5411	294	1	t.	t.	PROPN
ejpam-5411	294	2	abdeljawad	abdeljawad	NOUN
ejpam-5411	294	3	et	et	PROPN
ejpam-5411	294	4	al	al	PROPN
ejpam-5411	294	5	.	.	PUNCT
ejpam-5411	294	6	/	/	SYM
ejpam-5411	294	7	eur	eur	PROPN
ejpam-5411	294	8	.	.	PUNCT
ejpam-5411	295	1	j.	j.	PROPN
ejpam-5411	295	2	pure	pure	PROPN
ejpam-5411	295	3	appl	appl	PROPN
ejpam-5411	295	4	.	.	PROPN
ejpam-5411	295	5	math	math	PROPN
ejpam-5411	295	6	,	,	PUNCT
ejpam-5411	295	7	17	17	NUM
ejpam-5411	295	8	(	(	PUNCT
ejpam-5411	295	9	4	4	NUM
ejpam-5411	295	10	)	)	PUNCT
ejpam-5411	295	11	(	(	PUNCT
ejpam-5411	295	12	2024	2024	NUM
ejpam-5411	295	13	)	)	PUNCT
ejpam-5411	295	14	,	,	PUNCT
ejpam-5411	295	15	2405	2405	NUM
ejpam-5411	295	16	-	-	SYM
ejpam-5411	295	17	2430	2430	NUM
ejpam-5411	295	18	2415	2415	NUM
ejpam-5411	295	19	=	=	SYM
ejpam-5411	295	20	t1−α(g(j−1	t1−α(g(j−1	NUM
ejpam-5411	295	21	)	)	PUNCT
ejpam-5411	295	22	(	(	PUNCT
ejpam-5411	295	23	tα	tα	PROPN
ejpam-5411	295	24	α	α	PROPN
ejpam-5411	295	25	)	)	PUNCT
ejpam-5411	295	26	)	)	PUNCT
ejpam-5411	295	27	′	′	NUM
ejpam-5411	296	1	=	=	PUNCT
ejpam-5411	296	2	g(j	g(j	PROPN
ejpam-5411	296	3	)	)	PUNCT
ejpam-5411	296	4	(	(	PUNCT
ejpam-5411	296	5	tα	tα	PROPN
ejpam-5411	296	6	α	α	PROPN
ejpam-5411	296	7	)	)	PUNCT
ejpam-5411	296	8	and	and	CCONJ
ejpam-5411	296	9	the	the	DET
ejpam-5411	296	10	function	function	NOUN
ejpam-5411	296	11	g(j)(t	g(j)(t	NUM
ejpam-5411	296	12	)	)	PUNCT
ejpam-5411	296	13	=	=	SYM
ejpam-5411	296	14	t	t	PROPN
ejpam-5411	296	15	(	(	PUNCT
ejpam-5411	296	16	jα)(f)((αt	jα)(f)((αt	PROPN
ejpam-5411	296	17	)	)	PUNCT
ejpam-5411	296	18	1	1	NUM
ejpam-5411	296	19	α	α	NOUN
ejpam-5411	296	20	)	)	PUNCT
ejpam-5411	296	21	exists	exist	VERB
ejpam-5411	296	22	and	and	CCONJ
ejpam-5411	296	23	continuous	continuous	ADJ
ejpam-5411	296	24	for	for	ADP
ejpam-5411	296	25	all	all	DET
ejpam-5411	296	26	t	t	NOUN
ejpam-5411	296	27	∈]0,+∞	∈]0,+∞	PUNCT
ejpam-5411	296	28	[	[	X
ejpam-5411	296	29	.	.	PUNCT
ejpam-5411	297	1	thus	thus	ADV
ejpam-5411	297	2	g	g	PROPN
ejpam-5411	297	3	∈	∈	PROPN
ejpam-5411	297	4	cj(]0,+∞	cj(]0,+∞	X
ejpam-5411	297	5	[	[	X
ejpam-5411	297	6	)	)	PUNCT
ejpam-5411	297	7	case	case	NOUN
ejpam-5411	297	8	2	2	NUM
ejpam-5411	297	9	:	:	PUNCT
ejpam-5411	297	10	t	t	NOUN
ejpam-5411	297	11	=	=	SYM
ejpam-5411	297	12	0	0	NUM
ejpam-5411	298	1	we	we	PRON
ejpam-5411	298	2	have	have	VERB
ejpam-5411	298	3	t	t	X
ejpam-5411	298	4	(	(	PUNCT
ejpam-5411	298	5	jα)(f)(0	jα)(f)(0	ADJ
ejpam-5411	298	6	)	)	PUNCT
ejpam-5411	298	7	=	=	SYM
ejpam-5411	298	8	limt→0	limt→0	PROPN
ejpam-5411	298	9	+	+	SYM
ejpam-5411	298	10	t	t	PROPN
ejpam-5411	298	11	(	(	PUNCT
ejpam-5411	298	12	jα)(f)(t	jα)(f)(t	NOUN
ejpam-5411	298	13	)	)	PUNCT
ejpam-5411	298	14	exists	exist	VERB
ejpam-5411	298	15	.	.	PUNCT
ejpam-5411	299	1	the	the	DET
ejpam-5411	299	2	functions	function	NOUN
ejpam-5411	299	3	t	t	X
ejpam-5411	299	4	(	(	PUNCT
ejpam-5411	299	5	jα)(f)(t	jα)(f)(t	NOUN
ejpam-5411	299	6	)	)	PUNCT
ejpam-5411	299	7	and	and	CCONJ
ejpam-5411	299	8	g(j)(t	g(j)(t	NUM
ejpam-5411	299	9	)	)	PUNCT
ejpam-5411	299	10	are	be	AUX
ejpam-5411	299	11	continuous	continuous	ADJ
ejpam-5411	299	12	,	,	PUNCT
ejpam-5411	299	13	then	then	ADV
ejpam-5411	299	14	lim	lim	PROPN
ejpam-5411	299	15	t→0	t→0	AUX
ejpam-5411	299	16	+	+	CCONJ
ejpam-5411	299	17	g(j)(t	g(j)(t	NUM
ejpam-5411	299	18	)	)	PUNCT
ejpam-5411	299	19	=	=	SYM
ejpam-5411	299	20	lim	lim	PROPN
ejpam-5411	299	21	t→0	t→0	PROPN
ejpam-5411	299	22	+	+	PROPN
ejpam-5411	299	23	t	t	PROPN
ejpam-5411	299	24	(	(	PUNCT
ejpam-5411	299	25	jα)(f)(t	jα)(f)(t	NOUN
ejpam-5411	299	26	)	)	PUNCT
ejpam-5411	299	27	=	=	SYM
ejpam-5411	299	28	t	t	PROPN
ejpam-5411	299	29	(	(	PUNCT
ejpam-5411	299	30	jα)(f)(0	jα)(f)(0	ADJ
ejpam-5411	299	31	)	)	PUNCT
ejpam-5411	299	32	=	=	SYM
ejpam-5411	299	33	g(j)(0	g(j)(0	PROPN
ejpam-5411	299	34	)	)	PUNCT
ejpam-5411	299	35	.	.	PUNCT
ejpam-5411	300	1	finally	finally	ADV
ejpam-5411	300	2	,	,	PUNCT
ejpam-5411	300	3	t	t	PROPN
ejpam-5411	300	4	(	(	PUNCT
ejpam-5411	300	5	jα)(f)(t	jα)(f)(t	PROPN
ejpam-5411	300	6	)	)	PUNCT
ejpam-5411	300	7	=	=	SYM
ejpam-5411	300	8	g(j	g(j	PROPN
ejpam-5411	300	9	)	)	PUNCT
ejpam-5411	300	10	(	(	PUNCT
ejpam-5411	300	11	t	t	PROPN
ejpam-5411	300	12	α	α	PROPN
ejpam-5411	300	13	α	α	PROPN
ejpam-5411	300	14	)	)	PUNCT
ejpam-5411	300	15	for	for	ADP
ejpam-5411	300	16	all	all	DET
ejpam-5411	300	17	t	t	NOUN
ejpam-5411	300	18	∈	∈	PROPN
ejpam-5411	301	1	[	[	X
ejpam-5411	301	2	0,+∞	0,+∞	NUM
ejpam-5411	301	3	[	[	PUNCT
ejpam-5411	301	4	and	and	CCONJ
ejpam-5411	301	5	g	g	PROPN
ejpam-5411	301	6	∈	∈	PROPN
ejpam-5411	301	7	cj([0,+∞	cj([0,+∞	X
ejpam-5411	301	8	[	[	X
ejpam-5411	301	9	)	)	PUNCT
ejpam-5411	301	10	.	.	PUNCT
ejpam-5411	302	1	(	(	PUNCT
ejpam-5411	302	2	2	2	X
ejpam-5411	302	3	)	)	PUNCT
ejpam-5411	302	4	we	we	PRON
ejpam-5411	302	5	have	have	VERB
ejpam-5411	302	6	g(j−1	g(j−1	NOUN
ejpam-5411	302	7	)	)	PUNCT
ejpam-5411	302	8	(	(	PUNCT
ejpam-5411	302	9	t	t	PROPN
ejpam-5411	302	10	α	α	PROPN
ejpam-5411	302	11	α	α	NOUN
ejpam-5411	302	12	)	)	PUNCT
ejpam-5411	302	13	=	=	SYM
ejpam-5411	302	14	g(j−1	g(j−1	NOUN
ejpam-5411	302	15	)	)	PUNCT
ejpam-5411	302	16	(	(	PUNCT
ejpam-5411	302	17	t	t	PROPN
ejpam-5411	302	18	α	α	PROPN
ejpam-5411	302	19	α	α	PROPN
ejpam-5411	303	1	+	+	CCONJ
ejpam-5411	303	2	pα	pα	PROPN
ejpam-5411	303	3	α	α	NOUN
ejpam-5411	303	4	)	)	PUNCT
ejpam-5411	303	5	for	for	ADP
ejpam-5411	303	6	all	all	DET
ejpam-5411	303	7	t	t	NOUN
ejpam-5411	303	8	∈	∈	PROPN
ejpam-5411	304	1	[	[	X
ejpam-5411	304	2	0,+∞	0,+∞	NUM
ejpam-5411	304	3	[	[	PUNCT
ejpam-5411	304	4	and	and	CCONJ
ejpam-5411	304	5	g	g	PROPN
ejpam-5411	304	6	∈	∈	PROPN
ejpam-5411	304	7	cj([0,+∞	cj([0,+∞	X
ejpam-5411	304	8	[	[	X
ejpam-5411	304	9	)	)	PUNCT
ejpam-5411	304	10	,	,	PUNCT
ejpam-5411	304	11	then	then	ADV
ejpam-5411	304	12	g(j	g(j	PROPN
ejpam-5411	304	13	)	)	PUNCT
ejpam-5411	304	14	(	(	PUNCT
ejpam-5411	304	15	t	t	PROPN
ejpam-5411	304	16	α	α	PROPN
ejpam-5411	304	17	α	α	NOUN
ejpam-5411	304	18	)	)	PUNCT
ejpam-5411	305	1	=	=	SYM
ejpam-5411	305	2	g(j	g(j	PROPN
ejpam-5411	305	3	)	)	PUNCT
ejpam-5411	305	4	(	(	PUNCT
ejpam-5411	305	5	t	t	PROPN
ejpam-5411	305	6	α	α	PROPN
ejpam-5411	305	7	α	α	PROPN
ejpam-5411	306	1	+	+	CCONJ
ejpam-5411	306	2	pα	pα	NOUN
ejpam-5411	306	3	α	α	NOUN
ejpam-5411	306	4	)	)	PUNCT
ejpam-5411	306	5	.	.	PUNCT
ejpam-5411	307	1	thus	thus	ADV
ejpam-5411	307	2	t	t	X
ejpam-5411	307	3	(	(	PUNCT
ejpam-5411	307	4	jα)(f)(t	jα)(f)(t	PROPN
ejpam-5411	307	5	)	)	PUNCT
ejpam-5411	307	6	is	be	AUX
ejpam-5411	307	7	α	α	X
ejpam-5411	307	8	-	-	NOUN
ejpam-5411	307	9	periodic	periodic	NOUN
ejpam-5411	307	10	with	with	ADP
ejpam-5411	307	11	period	period	NOUN
ejpam-5411	307	12	p	p	NOUN
ejpam-5411	307	13	for	for	ADP
ejpam-5411	307	14	all	all	DET
ejpam-5411	307	15	t	t	NOUN
ejpam-5411	307	16	∈	∈	PROPN
ejpam-5411	308	1	[	[	X
ejpam-5411	308	2	0,+∞	0,+∞	NUM
ejpam-5411	308	3	[	[	NOUN
ejpam-5411	308	4	.	.	PUNCT
ejpam-5411	308	5	example	example	NOUN
ejpam-5411	308	6	6	6	NUM
ejpam-5411	308	7	.	.	PUNCT
ejpam-5411	309	1	let	let	VERB
ejpam-5411	309	2	us	we	PRON
ejpam-5411	309	3	consider	consider	VERB
ejpam-5411	309	4	the	the	DET
ejpam-5411	309	5	example	example	NOUN
ejpam-5411	309	6	5	5	NUM
ejpam-5411	309	7	.	.	PUNCT
ejpam-5411	310	1	the	the	DET
ejpam-5411	310	2	function	function	NOUN
ejpam-5411	310	3	f	f	PROPN
ejpam-5411	310	4	is	be	AUX
ejpam-5411	310	5	α	α	NOUN
ejpam-5411	310	6	-	-	NOUN
ejpam-5411	310	7	periodic	periodic	NOUN
ejpam-5411	310	8	with	with	ADP
ejpam-5411	310	9	period	period	NOUN
ejpam-5411	310	10	(	(	PUNCT
ejpam-5411	310	11	3π2α	3π2α	NUM
ejpam-5411	310	12	)	)	PUNCT
ejpam-5411	310	13	1	1	NUM
ejpam-5411	310	14	α	α	NOUN
ejpam-5411	310	15	and	and	CCONJ
ejpam-5411	310	16	g	g	PROPN
ejpam-5411	310	17	is	be	AUX
ejpam-5411	310	18	continuous	continuous	ADJ
ejpam-5411	310	19	periodic	periodic	NOUN
ejpam-5411	310	20	with	with	ADP
ejpam-5411	310	21	period	period	NOUN
ejpam-5411	310	22	3π	3π	NOUN
ejpam-5411	310	23	2α2	2α2	NUM
ejpam-5411	310	24	.	.	PUNCT
ejpam-5411	311	1	then	then	ADV
ejpam-5411	311	2	,	,	PUNCT
ejpam-5411	311	3	we	we	PRON
ejpam-5411	311	4	have	have	VERB
ejpam-5411	311	5	for	for	ADP
ejpam-5411	311	6	n	n	PRON
ejpam-5411	311	7	∈	∈	PROPN
ejpam-5411	311	8	n	n	NOUN
ejpam-5411	311	9	and	and	CCONJ
ejpam-5411	311	10	t	t	PROPN
ejpam-5411	311	11	∈	∈	PROPN
ejpam-5411	312	1	[	[	X
ejpam-5411	312	2	0	0	NUM
ejpam-5411	312	3	,	,	PUNCT
ejpam-5411	312	4	(	(	PUNCT
ejpam-5411	312	5	3π2α	3π2α	NUM
ejpam-5411	312	6	)	)	PUNCT
ejpam-5411	312	7	1	1	NUM
ejpam-5411	312	8	α	α	NOUN
ejpam-5411	312	9	]	]	X
ejpam-5411	312	10	t	t	PROPN
ejpam-5411	312	11	(	(	PUNCT
ejpam-5411	312	12	nα)(f)(t	nα)(f)(t	PROPN
ejpam-5411	312	13	)	)	PUNCT
ejpam-5411	312	14	=	=	SYM
ejpam-5411	313	1			PRON
ejpam-5411	313	2	t	t	X
ejpam-5411	313	3	(	(	PUNCT
ejpam-5411	313	4	nα)(f1)(t	nα)(f1)(t	SYM
ejpam-5411	313	5	)	)	PUNCT
ejpam-5411	313	6	=	=	SYM
ejpam-5411	313	7	α2n	α2n	PROPN
ejpam-5411	313	8	sin(αtα	sin(αtα	NOUN
ejpam-5411	313	9	+	+	CCONJ
ejpam-5411	313	10	nπ	nπ	NOUN
ejpam-5411	313	11	2	2	NUM
ejpam-5411	313	12	)	)	PUNCT
ejpam-5411	313	13	,	,	PUNCT
ejpam-5411	313	14	0	0	NUM
ejpam-5411	313	15	<	<	X
ejpam-5411	313	16	t	t	X
ejpam-5411	313	17	<	<	X
ejpam-5411	313	18	(	(	PUNCT
ejpam-5411	313	19	πα	πα	PROPN
ejpam-5411	313	20	)	)	PUNCT
ejpam-5411	313	21	1	1	NUM
ejpam-5411	313	22	α	α	NOUN
ejpam-5411	313	23	t	t	NOUN
ejpam-5411	313	24	(	(	PUNCT
ejpam-5411	313	25	nα)(f2)(t	nα)(f2)(t	PROPN
ejpam-5411	313	26	)	)	PUNCT
ejpam-5411	313	27	=	=	SYM
ejpam-5411	313	28	−2n−1α2n	−2n−1α2n	NOUN
ejpam-5411	313	29	sin(2αtα	sin(2αtα	X
ejpam-5411	313	30	+	+	CCONJ
ejpam-5411	313	31	nπ	nπ	NOUN
ejpam-5411	313	32	2	2	NUM
ejpam-5411	313	33	)	)	PUNCT
ejpam-5411	313	34	,	,	PUNCT
ejpam-5411	313	35	(	(	PUNCT
ejpam-5411	313	36	πα	πα	X
ejpam-5411	313	37	)	)	PUNCT
ejpam-5411	313	38	1	1	NUM
ejpam-5411	313	39	α	α	NOUN
ejpam-5411	313	40	<	<	X
ejpam-5411	313	41	t	t	X
ejpam-5411	313	42	<	<	X
ejpam-5411	313	43	(	(	PUNCT
ejpam-5411	313	44	3π2α	3π2α	NUM
ejpam-5411	313	45	)	)	PUNCT
ejpam-5411	313	46	1	1	NUM
ejpam-5411	313	47	α	α	NOUN
ejpam-5411	313	48	and	and	CCONJ
ejpam-5411	313	49	for	for	ADP
ejpam-5411	313	50	t	t	PROPN
ejpam-5411	313	51	∈	∈	PROPN
ejpam-5411	314	1	[	[	X
ejpam-5411	314	2	0	0	NUM
ejpam-5411	314	3	,	,	PUNCT
ejpam-5411	314	4	3π	3π	NOUN
ejpam-5411	314	5	2α2	2α2	NUM
ejpam-5411	314	6	]	]	PUNCT
ejpam-5411	314	7	g(n)(t	g(n)(t	NOUN
ejpam-5411	314	8	)	)	PUNCT
ejpam-5411	314	9	=	=	PUNCT
ejpam-5411	315	1			PRON
ejpam-5411	315	2	g	g	PROPN
ejpam-5411	315	3	(	(	PUNCT
ejpam-5411	315	4	n	n	CCONJ
ejpam-5411	315	5	)	)	PUNCT
ejpam-5411	315	6	1	1	NUM
ejpam-5411	315	7	(	(	PUNCT
ejpam-5411	315	8	t	t	NOUN
ejpam-5411	315	9	)	)	PUNCT
ejpam-5411	315	10	=	=	SYM
ejpam-5411	315	11	α2n	α2n	PROPN
ejpam-5411	315	12	sin(α2	sin(α2	PROPN
ejpam-5411	315	13	t	t	PROPN
ejpam-5411	315	14	+	+	CCONJ
ejpam-5411	315	15	nπ	nπ	NOUN
ejpam-5411	315	16	2	2	NUM
ejpam-5411	315	17	)	)	PUNCT
ejpam-5411	315	18	,	,	PUNCT
ejpam-5411	315	19	0	0	NUM
ejpam-5411	315	20	<	<	X
ejpam-5411	315	21	t	t	X
ejpam-5411	315	22	<	<	X
ejpam-5411	315	23	π	π	PROPN
ejpam-5411	315	24	α2	α2	PROPN
ejpam-5411	315	25	g	g	PROPN
ejpam-5411	315	26	(	(	PUNCT
ejpam-5411	315	27	n	n	CCONJ
ejpam-5411	315	28	)	)	PUNCT
ejpam-5411	315	29	2	2	NUM
ejpam-5411	315	30	(	(	PUNCT
ejpam-5411	315	31	t	t	NOUN
ejpam-5411	315	32	)	)	PUNCT
ejpam-5411	315	33	=	=	SYM
ejpam-5411	316	1	−2n−1α2n	−2n−1α2n	NOUN
ejpam-5411	316	2	sin(2α2	sin(2α2	PROPN
ejpam-5411	316	3	t	t	NOUN
ejpam-5411	316	4	+	+	NUM
ejpam-5411	316	5	nπ	nπ	NOUN
ejpam-5411	316	6	2	2	NUM
ejpam-5411	316	7	)	)	PUNCT
ejpam-5411	316	8	,	,	PUNCT
ejpam-5411	316	9	π	π	PROPN
ejpam-5411	316	10	α2	α2	PROPN
ejpam-5411	316	11	<	<	X
ejpam-5411	316	12	t	t	X
ejpam-5411	316	13	<	<	X
ejpam-5411	316	14	3π	3π	NUM
ejpam-5411	316	15	2α2	2α2	NUM
ejpam-5411	317	1	the	the	DET
ejpam-5411	317	2	function	function	NOUN
ejpam-5411	317	3	f1	f1	NOUN
ejpam-5411	317	4	is	be	AUX
ejpam-5411	317	5	n	n	DET
ejpam-5411	317	6	times	time	NOUN
ejpam-5411	317	7	continuously	continuously	ADV
ejpam-5411	317	8	α	α	VERB
ejpam-5411	317	9	-	-	NOUN
ejpam-5411	317	10	differentiable	differentiable	ADJ
ejpam-5411	317	11	on	on	ADP
ejpam-5411	317	12	[	[	X
ejpam-5411	317	13	0	0	NUM
ejpam-5411	317	14	,	,	PUNCT
ejpam-5411	317	15	(	(	PUNCT
ejpam-5411	317	16	πα	πα	X
ejpam-5411	317	17	)	)	PUNCT
ejpam-5411	317	18	1	1	NUM
ejpam-5411	317	19	α	α	NOUN
ejpam-5411	317	20	]	]	PUNCT
ejpam-5411	317	21	and	and	CCONJ
ejpam-5411	317	22	g1	g1	PROPN
ejpam-5411	317	23	∈	∈	PROPN
ejpam-5411	317	24	cn([0	cn([0	NOUN
ejpam-5411	317	25	,	,	PUNCT
ejpam-5411	317	26	π	π	PROPN
ejpam-5411	317	27	α2	α2	PROPN
ejpam-5411	317	28	]	]	PUNCT
ejpam-5411	317	29	)	)	PUNCT
ejpam-5411	317	30	.	.	PUNCT
ejpam-5411	318	1	the	the	DET
ejpam-5411	318	2	function	function	NOUN
ejpam-5411	318	3	f2	f2	PROPN
ejpam-5411	318	4	is	be	AUX
ejpam-5411	318	5	n	n	DET
ejpam-5411	318	6	times	time	NOUN
ejpam-5411	318	7	continuously	continuously	ADV
ejpam-5411	318	8	α	α	VERB
ejpam-5411	318	9	-	-	NOUN
ejpam-5411	318	10	differentiable	differentiable	ADJ
ejpam-5411	318	11	on	on	ADP
ejpam-5411	318	12	[	[	X
ejpam-5411	318	13	(	(	PUNCT
ejpam-5411	318	14	πα	πα	ADJ
ejpam-5411	318	15	)	)	PUNCT
ejpam-5411	318	16	1	1	NUM
ejpam-5411	318	17	α	α	NOUN
ejpam-5411	318	18	,	,	PUNCT
ejpam-5411	318	19	(	(	PUNCT
ejpam-5411	318	20	3π2α	3π2α	NUM
ejpam-5411	318	21	)	)	PUNCT
ejpam-5411	318	22	1	1	NUM
ejpam-5411	318	23	α	α	NOUN
ejpam-5411	318	24	]	]	PUNCT
ejpam-5411	318	25	and	and	CCONJ
ejpam-5411	318	26	g2	g2	PROPN
ejpam-5411	318	27	∈	∈	PROPN
ejpam-5411	318	28	cn	cn	PROPN
ejpam-5411	318	29	(	(	PUNCT
ejpam-5411	318	30	[	[	PUNCT
ejpam-5411	318	31	π	π	X
ejpam-5411	318	32	α2	α2	PROPN
ejpam-5411	318	33	,	,	PUNCT
ejpam-5411	318	34	3π	3π	NOUN
ejpam-5411	318	35	2α2	2α2	NUM
ejpam-5411	318	36	]	]	PUNCT
ejpam-5411	318	37	)	)	PUNCT
ejpam-5411	318	38	.	.	PUNCT
ejpam-5411	319	1	to	to	PART
ejpam-5411	319	2	study	study	VERB
ejpam-5411	319	3	the	the	DET
ejpam-5411	319	4	continuity	continuity	NOUN
ejpam-5411	319	5	of	of	ADP
ejpam-5411	319	6	t	t	PROPN
ejpam-5411	319	7	(	(	PUNCT
ejpam-5411	319	8	nα)(f	nα)(f	PROPN
ejpam-5411	319	9	)	)	PUNCT
ejpam-5411	319	10	and	and	CCONJ
ejpam-5411	319	11	of	of	ADP
ejpam-5411	319	12	g(n	g(n	PROPN
ejpam-5411	319	13	)	)	PUNCT
ejpam-5411	319	14	on	on	ADP
ejpam-5411	319	15	[	[	X
ejpam-5411	319	16	0,+∞	0,+∞	PROPN
ejpam-5411	319	17	[	[	X
ejpam-5411	319	18	,	,	PUNCT
ejpam-5411	319	19	we	we	PRON
ejpam-5411	319	20	put	put	VERB
ejpam-5411	319	21	∆α	∆α	PROPN
ejpam-5411	319	22	n	n	PROPN
ejpam-5411	319	23	=	=	SYM
ejpam-5411	319	24	t	t	PROPN
ejpam-5411	319	25	(	(	PUNCT
ejpam-5411	319	26	nα)(f1)(0	nα)(f1)(0	ADV
ejpam-5411	319	27	)	)	PUNCT
ejpam-5411	319	28	−	−	PROPN
ejpam-5411	319	29	t	t	PROPN
ejpam-5411	319	30	(	(	PUNCT
ejpam-5411	319	31	nα)(f2	nα)(f2	NOUN
ejpam-5411	319	32	)	)	PUNCT
ejpam-5411	319	33	(	(	PUNCT
ejpam-5411	319	34	(	(	PUNCT
ejpam-5411	319	35	3π	3π	NOUN
ejpam-5411	319	36	2α	2α	NOUN
ejpam-5411	319	37	)	)	PUNCT
ejpam-5411	319	38	1	1	NUM
ejpam-5411	319	39	α	α	NOUN
ejpam-5411	319	40	)	)	PUNCT
ejpam-5411	320	1	=	=	SYM
ejpam-5411	320	2	g	g	PROPN
ejpam-5411	320	3	(	(	PUNCT
ejpam-5411	320	4	n	n	CCONJ
ejpam-5411	320	5	)	)	PUNCT
ejpam-5411	320	6	1	1	NUM
ejpam-5411	320	7	(	(	PUNCT
ejpam-5411	320	8	0	0	NUM
ejpam-5411	320	9	)	)	PUNCT
ejpam-5411	320	10	−	−	PROPN
ejpam-5411	320	11	g	g	PROPN
ejpam-5411	320	12	(	(	PUNCT
ejpam-5411	320	13	n	n	CCONJ
ejpam-5411	320	14	)	)	PUNCT
ejpam-5411	320	15	2	2	NUM
ejpam-5411	320	16	(	(	PUNCT
ejpam-5411	320	17	3π	3π	NOUN
ejpam-5411	320	18	2α2	2α2	NUM
ejpam-5411	320	19	)	)	PUNCT
ejpam-5411	320	20	and	and	CCONJ
ejpam-5411	320	21	δαn	δαn	NUM
ejpam-5411	320	22	=	=	SYM
ejpam-5411	320	23	t	t	PROPN
ejpam-5411	320	24	(	(	PUNCT
ejpam-5411	320	25	nα)(f1	nα)(f1	NOUN
ejpam-5411	320	26	)	)	PUNCT
ejpam-5411	320	27	(	(	PUNCT
ejpam-5411	320	28	(	(	PUNCT
ejpam-5411	320	29	π	π	NOUN
ejpam-5411	320	30	α	α	NOUN
ejpam-5411	320	31	)	)	PUNCT
ejpam-5411	320	32	1	1	NUM
ejpam-5411	320	33	α	α	NOUN
ejpam-5411	320	34	)	)	PUNCT
ejpam-5411	320	35	−	−	PROPN
ejpam-5411	320	36	t	t	PROPN
ejpam-5411	320	37	(	(	PUNCT
ejpam-5411	320	38	nα)(f2	nα)(f2	NOUN
ejpam-5411	320	39	)	)	PUNCT
ejpam-5411	320	40	(	(	PUNCT
ejpam-5411	320	41	(	(	PUNCT
ejpam-5411	320	42	π	π	NOUN
ejpam-5411	320	43	α	α	NOUN
ejpam-5411	320	44	)	)	PUNCT
ejpam-5411	320	45	1	1	NUM
ejpam-5411	320	46	α	α	NOUN
ejpam-5411	320	47	)	)	PUNCT
ejpam-5411	321	1	=	=	SYM
ejpam-5411	321	2	g	g	PROPN
ejpam-5411	321	3	(	(	PUNCT
ejpam-5411	321	4	n	n	CCONJ
ejpam-5411	321	5	)	)	PUNCT
ejpam-5411	321	6	1	1	NUM
ejpam-5411	321	7	(	(	PUNCT
ejpam-5411	321	8	π	π	NOUN
ejpam-5411	321	9	α2	α2	PROPN
ejpam-5411	321	10	)	)	PUNCT
ejpam-5411	322	1	−	−	PROPN
ejpam-5411	322	2	g	g	PROPN
ejpam-5411	322	3	(	(	PUNCT
ejpam-5411	322	4	n	n	CCONJ
ejpam-5411	322	5	)	)	PUNCT
ejpam-5411	322	6	2	2	NUM
ejpam-5411	322	7	(	(	PUNCT
ejpam-5411	322	8	π	π	NOUN
ejpam-5411	322	9	α2	α2	PROPN
ejpam-5411	322	10	)	)	PUNCT
ejpam-5411	322	11	.	.	PUNCT
ejpam-5411	323	1	now	now	ADV
ejpam-5411	323	2	,	,	PUNCT
ejpam-5411	323	3	we	we	PRON
ejpam-5411	323	4	have	have	VERB
ejpam-5411	323	5	∆α	∆α	PROPN
ejpam-5411	323	6	n	n	PROPN
ejpam-5411	323	7	=	=	SYM
ejpam-5411	323	8	α2n(1	α2n(1	PROPN
ejpam-5411	323	9	−	−	PROPN
ejpam-5411	323	10	2n−1	2n−1	NUM
ejpam-5411	323	11	)	)	PUNCT
ejpam-5411	324	1	sin(n	sin(n	PROPN
ejpam-5411	324	2	π	π	PROPN
ejpam-5411	324	3	2	2	X
ejpam-5411	324	4	)	)	PUNCT
ejpam-5411	324	5	t.	t.	NOUN
ejpam-5411	324	6	abdeljawad	abdeljawad	NOUN
ejpam-5411	324	7	et	et	PROPN
ejpam-5411	324	8	al	al	PROPN
ejpam-5411	324	9	.	.	PUNCT
ejpam-5411	324	10	/	/	SYM
ejpam-5411	324	11	eur	eur	PROPN
ejpam-5411	324	12	.	.	PUNCT
ejpam-5411	325	1	j.	j.	PROPN
ejpam-5411	325	2	pure	pure	PROPN
ejpam-5411	325	3	appl	appl	PROPN
ejpam-5411	325	4	.	.	PROPN
ejpam-5411	325	5	math	math	PROPN
ejpam-5411	325	6	,	,	PUNCT
ejpam-5411	325	7	17	17	NUM
ejpam-5411	325	8	(	(	PUNCT
ejpam-5411	325	9	4	4	NUM
ejpam-5411	325	10	)	)	PUNCT
ejpam-5411	325	11	(	(	PUNCT
ejpam-5411	325	12	2024	2024	NUM
ejpam-5411	325	13	)	)	PUNCT
ejpam-5411	325	14	,	,	PUNCT
ejpam-5411	325	15	2405	2405	NUM
ejpam-5411	325	16	-	-	SYM
ejpam-5411	325	17	2430	2430	NUM
ejpam-5411	325	18	2416	2416	NUM
ejpam-5411	325	19	and	and	CCONJ
ejpam-5411	325	20	δαn	δαn	NOUN
ejpam-5411	325	21	=	=	SYM
ejpam-5411	325	22	−α2n(1	−α2n(1	NOUN
ejpam-5411	325	23	−	−	NUM
ejpam-5411	325	24	2n−1	2n−1	NUM
ejpam-5411	325	25	)	)	PUNCT
ejpam-5411	325	26	sin(n	sin(n	PROPN
ejpam-5411	325	27	π	π	PROPN
ejpam-5411	325	28	2	2	NUM
ejpam-5411	325	29	)	)	PUNCT
ejpam-5411	325	30	.	.	PUNCT
ejpam-5411	326	1	the	the	DET
ejpam-5411	326	2	continuity	continuity	NOUN
ejpam-5411	326	3	conditions	condition	NOUN
ejpam-5411	326	4	of	of	ADP
ejpam-5411	326	5	t	t	PROPN
ejpam-5411	326	6	(	(	PUNCT
ejpam-5411	326	7	nα)(f	nα)(f	PROPN
ejpam-5411	326	8	)	)	PUNCT
ejpam-5411	326	9	on	on	ADP
ejpam-5411	326	10	[	[	X
ejpam-5411	326	11	0	0	NUM
ejpam-5411	326	12	,	,	PUNCT
ejpam-5411	326	13	(	(	PUNCT
ejpam-5411	326	14	3π2α	3π2α	NUM
ejpam-5411	326	15	)	)	PUNCT
ejpam-5411	326	16	1	1	NUM
ejpam-5411	326	17	α	α	NOUN
ejpam-5411	326	18	]	]	PUNCT
ejpam-5411	326	19	of	of	ADP
ejpam-5411	326	20	g(n	g(n	PROPN
ejpam-5411	326	21	)	)	PUNCT
ejpam-5411	326	22	on	on	ADP
ejpam-5411	326	23	[	[	X
ejpam-5411	326	24	0	0	NUM
ejpam-5411	326	25	,	,	PUNCT
ejpam-5411	326	26	3π	3π	NOUN
ejpam-5411	326	27	2α2	2α2	NUM
ejpam-5411	326	28	]	]	PUNCT
ejpam-5411	326	29	and	and	CCONJ
ejpam-5411	326	30	their	their	PRON
ejpam-5411	326	31	extention	extention	NOUN
ejpam-5411	326	32	by	by	ADP
ejpam-5411	326	33	periodicity	periodicity	NOUN
ejpam-5411	326	34	to	to	ADP
ejpam-5411	326	35	[	[	X
ejpam-5411	326	36	0,+∞	0,+∞	NUM
ejpam-5411	326	37	[	[	PUNCT
ejpam-5411	326	38	are	be	AUX
ejpam-5411	326	39	∆α	∆α	PROPN
ejpam-5411	326	40	n	n	NOUN
ejpam-5411	326	41	=	=	SYM
ejpam-5411	326	42	δαn	δαn	NOUN
ejpam-5411	326	43	=	=	SYM
ejpam-5411	326	44	0	0	X
ejpam-5411	326	45	.	.	PUNCT
ejpam-5411	327	1	therefore	therefore	ADV
ejpam-5411	327	2	under	under	ADP
ejpam-5411	327	3	this	this	DET
ejpam-5411	327	4	condition	condition	NOUN
ejpam-5411	327	5	,	,	PUNCT
ejpam-5411	327	6	f	f	PROPN
ejpam-5411	327	7	is	be	AUX
ejpam-5411	327	8	n	n	PRON
ejpam-5411	327	9	times	time	NOUN
ejpam-5411	327	10	continuously	continuously	ADV
ejpam-5411	327	11	α	α	VERB
ejpam-5411	327	12	-	-	NOUN
ejpam-5411	327	13	differentiable	differentiable	ADJ
ejpam-5411	327	14	on	on	ADP
ejpam-5411	327	15	[	[	X
ejpam-5411	327	16	0,+∞	0,+∞	NUM
ejpam-5411	327	17	[	[	PUNCT
ejpam-5411	327	18	and	and	CCONJ
ejpam-5411	327	19	g	g	PROPN
ejpam-5411	327	20	∈	∈	PROPN
ejpam-5411	327	21	cn([0,+∞	cn([0,+∞	PROPN
ejpam-5411	327	22	[	[	X
ejpam-5411	327	23	)	)	PUNCT
ejpam-5411	327	24	.	.	PUNCT
ejpam-5411	328	1	on	on	ADP
ejpam-5411	328	2	the	the	DET
ejpam-5411	328	3	other	other	ADJ
ejpam-5411	328	4	hand	hand	NOUN
ejpam-5411	328	5	∆α	∆α	PROPN
ejpam-5411	328	6	n	n	NOUN
ejpam-5411	328	7	=	=	SYM
ejpam-5411	328	8	δαn	δαn	X
ejpam-5411	328	9	=	=	SYM
ejpam-5411	328	10	0	0	PROPN
ejpam-5411	328	11	⇔	⇔	PROPN
ejpam-5411	328	12	n	n	CCONJ
ejpam-5411	328	13	∈	∈	PROPN
ejpam-5411	328	14	{	{	PUNCT
ejpam-5411	328	15	0	0	NUM
ejpam-5411	328	16	,	,	PUNCT
ejpam-5411	328	17	1	1	NUM
ejpam-5411	328	18	,	,	PUNCT
ejpam-5411	328	19	2	2	NUM
ejpam-5411	328	20	}	}	PUNCT
ejpam-5411	328	21	then	then	ADV
ejpam-5411	328	22	the	the	DET
ejpam-5411	328	23	function	function	NOUN
ejpam-5411	328	24	f	f	PROPN
ejpam-5411	328	25	is	be	AUX
ejpam-5411	328	26	twice	twice	ADV
ejpam-5411	328	27	continuously	continuously	ADV
ejpam-5411	328	28	α	α	VERB
ejpam-5411	328	29	-	-	NOUN
ejpam-5411	328	30	differentiable	differentiable	ADJ
ejpam-5411	328	31	on	on	ADP
ejpam-5411	328	32	[	[	X
ejpam-5411	328	33	0,+∞	0,+∞	NUM
ejpam-5411	328	34	[	[	PUNCT
ejpam-5411	328	35	and	and	CCONJ
ejpam-5411	328	36	we	we	PRON
ejpam-5411	328	37	have	have	VERB
ejpam-5411	328	38	for	for	ADP
ejpam-5411	328	39	all	all	DET
ejpam-5411	328	40	j	j	PROPN
ejpam-5411	328	41	∈	∈	PROPN
ejpam-5411	328	42	{	{	PUNCT
ejpam-5411	328	43	0	0	NUM
ejpam-5411	328	44	,	,	PUNCT
ejpam-5411	328	45	1	1	NUM
ejpam-5411	328	46	,	,	PUNCT
ejpam-5411	328	47	2	2	NUM
ejpam-5411	328	48	}	}	PUNCT
ejpam-5411	328	49	and	and	CCONJ
ejpam-5411	328	50	t	t	PROPN
ejpam-5411	328	51	∈	∈	PROPN
ejpam-5411	329	1	[	[	X
ejpam-5411	329	2	0,+∞	0,+∞	PROPN
ejpam-5411	329	3	[	[	X
ejpam-5411	329	4	,	,	PUNCT
ejpam-5411	329	5	t	t	PROPN
ejpam-5411	329	6	(	(	PUNCT
ejpam-5411	329	7	jα)f(t	jα)f(t	X
ejpam-5411	329	8	)	)	PUNCT
ejpam-5411	329	9	=	=	SYM
ejpam-5411	329	10	g(j	g(j	PROPN
ejpam-5411	329	11	)	)	PUNCT
ejpam-5411	329	12	(	(	PUNCT
ejpam-5411	329	13	t	t	PROPN
ejpam-5411	329	14	α	α	PROPN
ejpam-5411	329	15	α	α	PROPN
ejpam-5411	329	16	)	)	PUNCT
ejpam-5411	329	17	,	,	PUNCT
ejpam-5411	329	18	g	g	PROPN
ejpam-5411	329	19	∈	∈	PROPN
ejpam-5411	329	20	c(j)([0,+∞	c(j)([0,+∞	VERB
ejpam-5411	329	21	[	[	X
ejpam-5411	329	22	)	)	PUNCT
ejpam-5411	329	23	and	and	CCONJ
ejpam-5411	329	24	t	t	PROPN
ejpam-5411	329	25	(	(	PUNCT
ejpam-5411	329	26	jα)f(t	jα)f(t	NOUN
ejpam-5411	329	27	)	)	PUNCT
ejpam-5411	329	28	is	be	AUX
ejpam-5411	329	29	α	α	DET
ejpam-5411	329	30	-	-	ADJ
ejpam-5411	329	31	periodic	periodic	ADJ
ejpam-5411	329	32	function	function	NOUN
ejpam-5411	329	33	with	with	ADP
ejpam-5411	329	34	period	period	NOUN
ejpam-5411	329	35	(	(	PUNCT
ejpam-5411	329	36	3π2α	3π2α	NUM
ejpam-5411	329	37	)	)	PUNCT
ejpam-5411	329	38	1	1	NUM
ejpam-5411	329	39	α	α	NOUN
ejpam-5411	329	40	.	.	PUNCT
ejpam-5411	330	1	we	we	PRON
ejpam-5411	330	2	conclude	conclude	VERB
ejpam-5411	330	3	this	this	DET
ejpam-5411	330	4	section	section	NOUN
ejpam-5411	330	5	with	with	ADP
ejpam-5411	330	6	the	the	DET
ejpam-5411	330	7	following	follow	VERB
ejpam-5411	330	8	theorem	theorem	NOUN
ejpam-5411	330	9	.	.	PUNCT
ejpam-5411	330	10	theorem	theorem	PROPN
ejpam-5411	330	11	7	7	NUM
ejpam-5411	330	12	.	.	PUNCT
ejpam-5411	331	1	let	let	VERB
ejpam-5411	331	2	0	0	NUM
ejpam-5411	331	3	<	<	X
ejpam-5411	331	4	α	α	X
ejpam-5411	331	5	≤	≤	NUM
ejpam-5411	331	6	1	1	NUM
ejpam-5411	331	7	.	.	PUNCT
ejpam-5411	332	1	assume	assume	VERB
ejpam-5411	332	2	that	that	SCONJ
ejpam-5411	332	3	f	f	PROPN
ejpam-5411	332	4	∈	∈	PROPN
ejpam-5411	332	5	l1(r+,r	l1(r+,r	PROPN
ejpam-5411	332	6	)	)	PUNCT
ejpam-5411	332	7	is	be	AUX
ejpam-5411	332	8	α	α	DET
ejpam-5411	332	9	-	-	ADJ
ejpam-5411	332	10	periodic	periodic	ADJ
ejpam-5411	332	11	function	function	NOUN
ejpam-5411	332	12	with	with	ADP
ejpam-5411	332	13	period	period	NOUN
ejpam-5411	332	14	p	p	NOUN
ejpam-5411	332	15	and	and	CCONJ
ejpam-5411	332	16	a(t	a(t	NOUN
ejpam-5411	332	17	)	)	PUNCT
ejpam-5411	332	18	=	=	SYM
ejpam-5411	332	19	a1	a1	PROPN
ejpam-5411	332	20	(	(	PUNCT
ejpam-5411	332	21	tα	tα	PROPN
ejpam-5411	332	22	α	α	PROPN
ejpam-5411	332	23	)	)	PUNCT
ejpam-5411	332	24	with	with	ADP
ejpam-5411	332	25	a1	a1	NOUN
ejpam-5411	332	26	∈	∈	PROPN
ejpam-5411	332	27	l1(r+	l1(r+	PROPN
ejpam-5411	332	28	)	)	PUNCT
ejpam-5411	332	29	.	.	PUNCT
ejpam-5411	333	1	the	the	DET
ejpam-5411	333	2	function	function	NOUN
ejpam-5411	333	3	(	(	PUNCT
ejpam-5411	333	4	a	a	DET
ejpam-5411	333	5	∗α	∗α	PROPN
ejpam-5411	333	6	f)−∞(t	f)−∞(t	PROPN
ejpam-5411	333	7	)	)	PUNCT
ejpam-5411	333	8	is	be	AUX
ejpam-5411	333	9	defined	define	VERB
ejpam-5411	333	10	by	by	ADP
ejpam-5411	333	11	(	(	PUNCT
ejpam-5411	333	12	a	a	DET
ejpam-5411	333	13	∗α	∗α	NOUN
ejpam-5411	333	14	f)−∞(t	f)−∞(t	PROPN
ejpam-5411	333	15	)	)	PUNCT
ejpam-5411	334	1	=	=	SYM
ejpam-5411	334	2	∫	∫	PROPN
ejpam-5411	334	3	tα	tα	PROPN
ejpam-5411	334	4	α	α	PRON
ejpam-5411	334	5	−∞	−∞	ADP
ejpam-5411	334	6	a((tα	a((tα	SYM
ejpam-5411	334	7	−	−	PROPN
ejpam-5411	334	8	αs	αs	ADJ
ejpam-5411	334	9	)	)	PUNCT
ejpam-5411	334	10	1	1	NUM
ejpam-5411	334	11	α	α	NOUN
ejpam-5411	334	12	)	)	PUNCT
ejpam-5411	334	13	f((αs	f((αs	NOUN
ejpam-5411	334	14	)	)	PUNCT
ejpam-5411	334	15	1	1	NUM
ejpam-5411	334	16	α	α	NOUN
ejpam-5411	334	17	)	)	PUNCT
ejpam-5411	334	18	ds	ds	PROPN
ejpam-5411	334	19	,	,	PUNCT
ejpam-5411	334	20	t	t	PROPN
ejpam-5411	334	21	∈	∈	PROPN
ejpam-5411	335	1	[	[	X
ejpam-5411	335	2	0,+∞	0,+∞	NUM
ejpam-5411	335	3	[	[	PUNCT
ejpam-5411	335	4	is	be	AUX
ejpam-5411	335	5	α	α	X
ejpam-5411	335	6	-	-	NOUN
ejpam-5411	335	7	periodic	periodic	NOUN
ejpam-5411	335	8	with	with	ADP
ejpam-5411	335	9	period	period	NOUN
ejpam-5411	335	10	p.	p.	NOUN
ejpam-5411	335	11	proof	proof	NOUN
ejpam-5411	335	12	.	.	PUNCT
ejpam-5411	336	1	let	let	VERB
ejpam-5411	336	2	0	0	NUM
ejpam-5411	336	3	<	<	X
ejpam-5411	336	4	α	α	PROPN
ejpam-5411	336	5	≤	≤	NUM
ejpam-5411	336	6	1	1	NUM
ejpam-5411	336	7	and	and	CCONJ
ejpam-5411	336	8	f	f	PROPN
ejpam-5411	336	9	∈	∈	PROPN
ejpam-5411	336	10	l1(r+,r	l1(r+,r	PROPN
ejpam-5411	336	11	)	)	PUNCT
ejpam-5411	336	12	is	be	AUX
ejpam-5411	336	13	α	α	DET
ejpam-5411	336	14	-	-	ADJ
ejpam-5411	336	15	periodic	periodic	ADJ
ejpam-5411	336	16	function	function	NOUN
ejpam-5411	336	17	with	with	ADP
ejpam-5411	336	18	period	period	NOUN
ejpam-5411	336	19	p	p	NOUN
ejpam-5411	336	20	and	and	CCONJ
ejpam-5411	336	21	a(t	a(t	NOUN
ejpam-5411	336	22	)	)	PUNCT
ejpam-5411	336	23	=	=	SYM
ejpam-5411	336	24	a1	a1	PROPN
ejpam-5411	336	25	(	(	PUNCT
ejpam-5411	336	26	tα	tα	PROPN
ejpam-5411	336	27	α	α	PROPN
ejpam-5411	336	28	)	)	PUNCT
ejpam-5411	336	29	with	with	ADP
ejpam-5411	336	30	a1	a1	NOUN
ejpam-5411	336	31	∈	∈	PROPN
ejpam-5411	336	32	l1(r+	l1(r+	PROPN
ejpam-5411	336	33	)	)	PUNCT
ejpam-5411	336	34	.	.	PUNCT
ejpam-5411	337	1	for	for	ADP
ejpam-5411	337	2	all	all	DET
ejpam-5411	337	3	t	t	NOUN
ejpam-5411	337	4	∈	∈	PRON
ejpam-5411	337	5	r+	r+	ADV
ejpam-5411	337	6	,	,	PUNCT
ejpam-5411	337	7	we	we	PRON
ejpam-5411	337	8	have	have	VERB
ejpam-5411	337	9	(	(	PUNCT
ejpam-5411	337	10	a	a	DET
ejpam-5411	337	11	∗α	∗α	NOUN
ejpam-5411	337	12	f)−∞(t	f)−∞(t	PROPN
ejpam-5411	337	13	)	)	PUNCT
ejpam-5411	337	14	=	=	SYM
ejpam-5411	338	1	∫	∫	PROPN
ejpam-5411	338	2	tα	tα	PROPN
ejpam-5411	338	3	α	α	PRON
ejpam-5411	338	4	−∞	−∞	ADP
ejpam-5411	338	5	a((tα	a((tα	SYM
ejpam-5411	338	6	−	−	PROPN
ejpam-5411	338	7	αs	αs	ADJ
ejpam-5411	338	8	)	)	PUNCT
ejpam-5411	338	9	1	1	NUM
ejpam-5411	338	10	α	α	NOUN
ejpam-5411	338	11	)	)	PUNCT
ejpam-5411	338	12	f((αs	f((αs	NOUN
ejpam-5411	338	13	)	)	PUNCT
ejpam-5411	338	14	1	1	NUM
ejpam-5411	338	15	α	α	NOUN
ejpam-5411	338	16	)	)	PUNCT
ejpam-5411	338	17	ds	ds	PROPN
ejpam-5411	338	18	=	=	SYM
ejpam-5411	338	19	∫	∫	PROPN
ejpam-5411	338	20	tα	tα	PROPN
ejpam-5411	338	21	α	α	PROPN
ejpam-5411	338	22	−∞	−∞	ADP
ejpam-5411	338	23	a1	a1	NOUN
ejpam-5411	338	24	(	(	PUNCT
ejpam-5411	338	25	tα	tα	PROPN
ejpam-5411	338	26	α	α	NOUN
ejpam-5411	338	27	−	−	PROPN
ejpam-5411	338	28	s	s	PART
ejpam-5411	338	29	)	)	PUNCT
ejpam-5411	338	30	g(s)ds	g(s)ds	PROPN
ejpam-5411	338	31	=	=	SYM
ejpam-5411	338	32	f	f	PROPN
ejpam-5411	338	33	(	(	PUNCT
ejpam-5411	338	34	tα	tα	PROPN
ejpam-5411	338	35	α	α	PROPN
ejpam-5411	338	36	)	)	PUNCT
ejpam-5411	338	37	where	where	SCONJ
ejpam-5411	338	38	f	f	PROPN
ejpam-5411	338	39	is	be	AUX
ejpam-5411	338	40	the	the	DET
ejpam-5411	338	41	continuous	continuous	ADJ
ejpam-5411	338	42	function	function	NOUN
ejpam-5411	338	43	given	give	VERB
ejpam-5411	338	44	by	by	ADP
ejpam-5411	338	45	theorem	theorem	ADJ
ejpam-5411	338	46	3	3	NUM
ejpam-5411	338	47	f	f	PROPN
ejpam-5411	338	48	(	(	PUNCT
ejpam-5411	338	49	t	t	PROPN
ejpam-5411	338	50	)	)	PUNCT
ejpam-5411	338	51	=	=	SYM
ejpam-5411	339	1	∫	∫	PROPN
ejpam-5411	339	2	t	t	PROPN
ejpam-5411	339	3	−∞	−∞	ADP
ejpam-5411	339	4	a1(t−	a1(t−	NOUN
ejpam-5411	339	5	s)g(s)ds	s)g(s)ds	PROPN
ejpam-5411	339	6	.	.	PUNCT
ejpam-5411	340	1	on	on	ADP
ejpam-5411	340	2	the	the	DET
ejpam-5411	340	3	other	other	ADJ
ejpam-5411	340	4	hand	hand	NOUN
ejpam-5411	340	5	f	f	PROPN
ejpam-5411	340	6	(	(	PUNCT
ejpam-5411	340	7	tα	tα	PROPN
ejpam-5411	340	8	α	α	PROPN
ejpam-5411	340	9	+	+	CCONJ
ejpam-5411	340	10	pα	pα	NOUN
ejpam-5411	340	11	α	α	NOUN
ejpam-5411	340	12	)	)	PUNCT
ejpam-5411	341	1	=	=	SYM
ejpam-5411	341	2	∫	∫	PROPN
ejpam-5411	341	3	tα	tα	PROPN
ejpam-5411	341	4	α	α	PROPN
ejpam-5411	342	1	+	+	CCONJ
ejpam-5411	342	2	pα	pα	NOUN
ejpam-5411	342	3	α	α	NOUN
ejpam-5411	342	4	−∞	−∞	ADP
ejpam-5411	342	5	a1	a1	PROPN
ejpam-5411	342	6	(	(	PUNCT
ejpam-5411	342	7	pα	pα	NOUN
ejpam-5411	342	8	α	α	NOUN
ejpam-5411	343	1	+	+	CCONJ
ejpam-5411	343	2	tα	tα	PROPN
ejpam-5411	343	3	α	α	NOUN
ejpam-5411	343	4	−	−	NOUN
ejpam-5411	343	5	s)g(s)ds	s)g(s)ds	NOUN
ejpam-5411	343	6	.	.	PUNCT
ejpam-5411	344	1	by	by	ADP
ejpam-5411	344	2	making	make	VERB
ejpam-5411	344	3	a	a	DET
ejpam-5411	344	4	change	change	NOUN
ejpam-5411	344	5	of	of	ADP
ejpam-5411	344	6	variable	variable	ADJ
ejpam-5411	344	7	u	u	NOUN
ejpam-5411	344	8	=	=	NOUN
ejpam-5411	344	9	s−	s−	PROPN
ejpam-5411	344	10	pα	pα	VERB
ejpam-5411	344	11	α	α	PROPN
ejpam-5411	344	12	,	,	PUNCT
ejpam-5411	344	13	we	we	PRON
ejpam-5411	344	14	obtain	obtain	VERB
ejpam-5411	344	15	f	f	PROPN
ejpam-5411	344	16	(	(	PUNCT
ejpam-5411	344	17	tα	tα	PROPN
ejpam-5411	344	18	α	α	PROPN
ejpam-5411	345	1	+	+	CCONJ
ejpam-5411	345	2	pα	pα	NOUN
ejpam-5411	345	3	α	α	NOUN
ejpam-5411	345	4	)	)	PUNCT
ejpam-5411	346	1	=	=	SYM
ejpam-5411	346	2	∫	∫	PROPN
ejpam-5411	347	1	tα	tα	PROPN
ejpam-5411	347	2	α	α	PROPN
ejpam-5411	347	3	−∞	−∞	ADP
ejpam-5411	347	4	a1	a1	NOUN
ejpam-5411	347	5	(	(	PUNCT
ejpam-5411	347	6	tα	tα	PROPN
ejpam-5411	347	7	α	α	NOUN
ejpam-5411	347	8	−	−	PROPN
ejpam-5411	347	9	u	u	PROPN
ejpam-5411	347	10	)	)	PUNCT
ejpam-5411	347	11	g	g	PROPN
ejpam-5411	347	12	(	(	PUNCT
ejpam-5411	347	13	u	u	NOUN
ejpam-5411	348	1	+	+	CCONJ
ejpam-5411	348	2	pα	pα	PROPN
ejpam-5411	348	3	α	α	NOUN
ejpam-5411	348	4	)	)	PUNCT
ejpam-5411	348	5	ds	ds	ADJ
ejpam-5411	348	6	t.	t.	NOUN
ejpam-5411	348	7	abdeljawad	abdeljawad	NOUN
ejpam-5411	348	8	et	et	PROPN
ejpam-5411	348	9	al	al	PROPN
ejpam-5411	348	10	.	.	PUNCT
ejpam-5411	348	11	/	/	SYM
ejpam-5411	348	12	eur	eur	PROPN
ejpam-5411	348	13	.	.	PUNCT
ejpam-5411	349	1	j.	j.	PROPN
ejpam-5411	349	2	pure	pure	PROPN
ejpam-5411	349	3	appl	appl	PROPN
ejpam-5411	349	4	.	.	PROPN
ejpam-5411	349	5	math	math	PROPN
ejpam-5411	349	6	,	,	PUNCT
ejpam-5411	349	7	17	17	NUM
ejpam-5411	349	8	(	(	PUNCT
ejpam-5411	349	9	4	4	NUM
ejpam-5411	349	10	)	)	PUNCT
ejpam-5411	349	11	(	(	PUNCT
ejpam-5411	349	12	2024	2024	NUM
ejpam-5411	349	13	)	)	PUNCT
ejpam-5411	349	14	,	,	PUNCT
ejpam-5411	349	15	2405	2405	NUM
ejpam-5411	349	16	-	-	SYM
ejpam-5411	349	17	2430	2430	NUM
ejpam-5411	349	18	2417	2417	NUM
ejpam-5411	349	19	the	the	DET
ejpam-5411	349	20	function	function	NOUN
ejpam-5411	349	21	g	g	NOUN
ejpam-5411	349	22	is	be	AUX
ejpam-5411	349	23	continuous	continuous	ADJ
ejpam-5411	349	24	periodic	periodic	NOUN
ejpam-5411	349	25	with	with	ADP
ejpam-5411	349	26	period	period	NOUN
ejpam-5411	349	27	pα	pα	INTJ
ejpam-5411	349	28	α	α	INTJ
ejpam-5411	349	29	,	,	PUNCT
ejpam-5411	349	30	then	then	ADV
ejpam-5411	349	31	f	f	PROPN
ejpam-5411	349	32	(	(	PUNCT
ejpam-5411	349	33	tα	tα	PROPN
ejpam-5411	349	34	α	α	PROPN
ejpam-5411	350	1	+	+	CCONJ
ejpam-5411	350	2	pα	pα	NOUN
ejpam-5411	350	3	α	α	NOUN
ejpam-5411	350	4	)	)	PUNCT
ejpam-5411	351	1	=	=	SYM
ejpam-5411	351	2	∫	∫	PROPN
ejpam-5411	351	3	tα	tα	PROPN
ejpam-5411	351	4	α	α	PROPN
ejpam-5411	351	5	−∞	−∞	ADP
ejpam-5411	351	6	a1	a1	NOUN
ejpam-5411	351	7	(	(	PUNCT
ejpam-5411	351	8	tα	tα	PROPN
ejpam-5411	351	9	α	α	NOUN
ejpam-5411	351	10	−	−	PROPN
ejpam-5411	351	11	u	u	NOUN
ejpam-5411	351	12	)	)	PUNCT
ejpam-5411	351	13	g(u)ds	g(u)ds	PROPN
ejpam-5411	351	14	=	=	SYM
ejpam-5411	351	15	f	f	PROPN
ejpam-5411	351	16	(	(	PUNCT
ejpam-5411	351	17	tα	tα	PROPN
ejpam-5411	351	18	α	α	PROPN
ejpam-5411	351	19	)	)	PUNCT
ejpam-5411	351	20	and	and	CCONJ
ejpam-5411	351	21	(	(	PUNCT
ejpam-5411	351	22	a	a	DET
ejpam-5411	351	23	∗α	∗α	PROPN
ejpam-5411	351	24	f)−∞(t	f)−∞(t	PROPN
ejpam-5411	351	25	)	)	PUNCT
ejpam-5411	351	26	is	be	AUX
ejpam-5411	351	27	α	α	DET
ejpam-5411	351	28	-	-	ADJ
ejpam-5411	351	29	periodic	periodic	ADJ
ejpam-5411	351	30	function	function	NOUN
ejpam-5411	351	31	with	with	ADP
ejpam-5411	351	32	period	period	NOUN
ejpam-5411	351	33	p	p	NOUN
ejpam-5411	351	34	for	for	ADP
ejpam-5411	351	35	all	all	DET
ejpam-5411	351	36	t	t	NOUN
ejpam-5411	351	37	∈	∈	PROPN
ejpam-5411	352	1	[	[	X
ejpam-5411	352	2	0,+∞	0,+∞	NUM
ejpam-5411	352	3	[	[	NOUN
ejpam-5411	352	4	.	.	PUNCT
ejpam-5411	352	5	example	example	NOUN
ejpam-5411	352	6	7	7	X
ejpam-5411	352	7	.	.	PUNCT
ejpam-5411	353	1	let	let	VERB
ejpam-5411	353	2	f1	f1	PROPN
ejpam-5411	353	3	defined	define	VERB
ejpam-5411	353	4	in	in	ADP
ejpam-5411	353	5	example	example	NOUN
ejpam-5411	353	6	3	3	NUM
ejpam-5411	353	7	and	and	CCONJ
ejpam-5411	353	8	a(t	a(t	NOUN
ejpam-5411	353	9	)	)	PUNCT
ejpam-5411	354	1	=	=	SYM
ejpam-5411	354	2	a1	a1	PROPN
ejpam-5411	354	3	(	(	PUNCT
ejpam-5411	354	4	tα	tα	PROPN
ejpam-5411	354	5	α	α	NOUN
ejpam-5411	354	6	)	)	PUNCT
ejpam-5411	354	7	such	such	ADJ
ejpam-5411	354	8	that	that	SCONJ
ejpam-5411	354	9	a1(t	a1(t	ADV
ejpam-5411	354	10	)	)	PUNCT
ejpam-5411	354	11	=	=	SYM
ejpam-5411	354	12	e−t	e−t	NOUN
ejpam-5411	354	13	∈	∈	PROPN
ejpam-5411	354	14	l1(r+	l1(r+	PROPN
ejpam-5411	354	15	)	)	PUNCT
ejpam-5411	354	16	.	.	PUNCT
ejpam-5411	355	1	the	the	DET
ejpam-5411	355	2	function	function	NOUN
ejpam-5411	355	3	f1	f1	NOUN
ejpam-5411	355	4	is	be	AUX
ejpam-5411	355	5	α	α	NOUN
ejpam-5411	355	6	-	-	ADJ
ejpam-5411	355	7	periodic	periodic	ADJ
ejpam-5411	355	8	function	function	NOUN
ejpam-5411	355	9	with	with	ADP
ejpam-5411	355	10	period	period	NOUN
ejpam-5411	355	11	(	(	PUNCT
ejpam-5411	355	12	1	1	NUM
ejpam-5411	355	13	α	α	NOUN
ejpam-5411	355	14	)	)	PUNCT
ejpam-5411	355	15	1	1	NUM
ejpam-5411	355	16	α	α	NOUN
ejpam-5411	355	17	and	and	CCONJ
ejpam-5411	355	18	we	we	PRON
ejpam-5411	355	19	have	have	VERB
ejpam-5411	355	20	for	for	ADP
ejpam-5411	355	21	all	all	DET
ejpam-5411	355	22	t	t	NOUN
ejpam-5411	355	23	∈	∈	PROPN
ejpam-5411	356	1	[	[	X
ejpam-5411	356	2	0,+∞	0,+∞	NUM
ejpam-5411	356	3	[	[	PUNCT
ejpam-5411	356	4	(	(	PUNCT
ejpam-5411	356	5	a	a	DET
ejpam-5411	356	6	∗α	∗α	NOUN
ejpam-5411	356	7	f)−∞(t	f)−∞(t	PROPN
ejpam-5411	356	8	)	)	PUNCT
ejpam-5411	356	9	=	=	SYM
ejpam-5411	357	1	∫	∫	PROPN
ejpam-5411	357	2	tα	tα	PROPN
ejpam-5411	357	3	α	α	PROPN
ejpam-5411	357	4	−∞	−∞	ADP
ejpam-5411	357	5	a1	a1	NOUN
ejpam-5411	357	6	(	(	PUNCT
ejpam-5411	357	7	tα	tα	PROPN
ejpam-5411	357	8	α	α	NOUN
ejpam-5411	357	9	−	−	PROPN
ejpam-5411	357	10	s	s	PART
ejpam-5411	357	11	)	)	PUNCT
ejpam-5411	357	12	f((αs	f((αs	PROPN
ejpam-5411	357	13	)	)	PUNCT
ejpam-5411	357	14	1	1	NUM
ejpam-5411	357	15	α	α	NOUN
ejpam-5411	357	16	)	)	PUNCT
ejpam-5411	357	17	ds	ds	PROPN
ejpam-5411	357	18	=	=	SYM
ejpam-5411	357	19	∫	∫	PROPN
ejpam-5411	357	20	tα	tα	PROPN
ejpam-5411	357	21	α	α	PROPN
ejpam-5411	357	22	−∞	−∞	ADP
ejpam-5411	357	23	e−	e−	PROPN
ejpam-5411	357	24	tα	tα	PROPN
ejpam-5411	358	1	α	α	NOUN
ejpam-5411	359	1	+	+	NOUN
ejpam-5411	359	2	sg(s)ds	sg(s)ds	ADJ
ejpam-5411	359	3	=	=	PUNCT
ejpam-5411	359	4	f	f	X
ejpam-5411	359	5	(	(	PUNCT
ejpam-5411	359	6	tα	tα	PROPN
ejpam-5411	359	7	α	α	PROPN
ejpam-5411	359	8	)	)	PUNCT
ejpam-5411	359	9	.	.	PUNCT
ejpam-5411	360	1	where	where	SCONJ
ejpam-5411	360	2	f	f	PROPN
ejpam-5411	360	3	(	(	PUNCT
ejpam-5411	360	4	t	t	PROPN
ejpam-5411	360	5	)	)	PUNCT
ejpam-5411	360	6	=	=	PUNCT
ejpam-5411	360	7	e−t	e−t	PROPN
ejpam-5411	360	8	∫	∫	PROPN
ejpam-5411	360	9	t	t	PROPN
ejpam-5411	360	10	−∞	−∞	PUNCT
ejpam-5411	360	11	esg(s))ds	esg(s))ds	NOUN
ejpam-5411	360	12	is	be	AUX
ejpam-5411	360	13	a	a	DET
ejpam-5411	360	14	continuous	continuous	ADJ
ejpam-5411	360	15	function	function	NOUN
ejpam-5411	360	16	.	.	PUNCT
ejpam-5411	361	1	we	we	PRON
ejpam-5411	361	2	have	have	VERB
ejpam-5411	361	3	f	f	PROPN
ejpam-5411	361	4	(	(	PUNCT
ejpam-5411	361	5	tα	tα	PROPN
ejpam-5411	361	6	α	α	PROPN
ejpam-5411	361	7	+	+	CCONJ
ejpam-5411	361	8	1	1	NUM
ejpam-5411	361	9	α2	α2	ADJ
ejpam-5411	361	10	)	)	PUNCT
ejpam-5411	362	1	=	=	SYM
ejpam-5411	362	2	∫	∫	PROPN
ejpam-5411	363	1	tα	tα	PROPN
ejpam-5411	363	2	α	α	PROPN
ejpam-5411	364	1	+	+	CCONJ
ejpam-5411	364	2	1	1	NUM
ejpam-5411	364	3	α2	α2	ADJ
ejpam-5411	364	4	−∞	−∞	ADP
ejpam-5411	364	5	e−	e−	PROPN
ejpam-5411	364	6	tα	tα	PROPN
ejpam-5411	364	7	α	α	NOUN
ejpam-5411	365	1	+	+	NOUN
ejpam-5411	365	2	s−	s−	PROPN
ejpam-5411	365	3	1	1	NUM
ejpam-5411	365	4	α2	α2	ADJ
ejpam-5411	365	5	g(s)ds	g(s)ds	NOUN
ejpam-5411	365	6	=	=	SYM
ejpam-5411	365	7	∫	∫	PROPN
ejpam-5411	365	8	tα	tα	PROPN
ejpam-5411	365	9	α	α	PROPN
ejpam-5411	365	10	−∞	−∞	ADP
ejpam-5411	365	11	e−	e−	PROPN
ejpam-5411	365	12	tα	tα	PROPN
ejpam-5411	366	1	α	α	PROPN
ejpam-5411	367	1	+	+	NOUN
ejpam-5411	367	2	sg	sg	PROPN
ejpam-5411	367	3	(	(	PUNCT
ejpam-5411	367	4	s	s	VERB
ejpam-5411	367	5	+	+	CCONJ
ejpam-5411	367	6	1	1	NUM
ejpam-5411	367	7	α2	α2	ADJ
ejpam-5411	367	8	)	)	PUNCT
ejpam-5411	367	9	ds	ds	PROPN
ejpam-5411	367	10	=	=	SYM
ejpam-5411	367	11	f	f	PROPN
ejpam-5411	367	12	(	(	PUNCT
ejpam-5411	367	13	tα	tα	PROPN
ejpam-5411	367	14	α	α	PROPN
ejpam-5411	367	15	)	)	PUNCT
ejpam-5411	367	16	then	then	ADV
ejpam-5411	367	17	(	(	PUNCT
ejpam-5411	367	18	a	a	DET
ejpam-5411	367	19	∗α	∗α	PROPN
ejpam-5411	367	20	f)−∞(t	f)−∞(t	PROPN
ejpam-5411	367	21	)	)	PUNCT
ejpam-5411	367	22	is	be	AUX
ejpam-5411	367	23	α	α	DET
ejpam-5411	367	24	-	-	ADJ
ejpam-5411	367	25	periodic	periodic	ADJ
ejpam-5411	367	26	function	function	NOUN
ejpam-5411	367	27	with	with	ADP
ejpam-5411	367	28	period	period	NOUN
ejpam-5411	367	29	(	(	PUNCT
ejpam-5411	367	30	1	1	NUM
ejpam-5411	367	31	α	α	NOUN
ejpam-5411	367	32	)	)	PUNCT
ejpam-5411	367	33	1	1	NUM
ejpam-5411	367	34	α	α	NOUN
ejpam-5411	367	35	.	.	PUNCT
ejpam-5411	368	1	in	in	ADP
ejpam-5411	368	2	the	the	DET
ejpam-5411	368	3	next	next	ADJ
ejpam-5411	368	4	section	section	NOUN
ejpam-5411	368	5	,	,	PUNCT
ejpam-5411	368	6	we	we	PRON
ejpam-5411	368	7	present	present	VERB
ejpam-5411	368	8	some	some	DET
ejpam-5411	368	9	results	result	NOUN
ejpam-5411	368	10	of	of	ADP
ejpam-5411	368	11	conformable	conformable	ADJ
ejpam-5411	368	12	fourier	fourier	NOUN
ejpam-5411	368	13	transforms	transform	VERB
ejpam-5411	368	14	.	.	PUNCT
ejpam-5411	369	1	4	4	X
ejpam-5411	369	2	.	.	X
ejpam-5411	369	3	result	result	NOUN
ejpam-5411	369	4	of	of	ADP
ejpam-5411	369	5	conformable	conformable	ADJ
ejpam-5411	369	6	fourier	fourier	NOUN
ejpam-5411	369	7	transform	transform	NOUN
ejpam-5411	369	8	for	for	ADP
ejpam-5411	369	9	investigating	investigate	VERB
ejpam-5411	369	10	the	the	DET
ejpam-5411	369	11	property	property	NOUN
ejpam-5411	369	12	of	of	ADP
ejpam-5411	369	13	the	the	DET
ejpam-5411	369	14	classical	classical	ADJ
ejpam-5411	369	15	fourier	fourier	NOUN
ejpam-5411	369	16	transform	transform	NOUN
ejpam-5411	369	17	,	,	PUNCT
ejpam-5411	369	18	the	the	DET
ejpam-5411	369	19	following	follow	VERB
ejpam-5411	369	20	new	new	ADJ
ejpam-5411	369	21	definition	definition	NOUN
ejpam-5411	369	22	of	of	ADP
ejpam-5411	369	23	the	the	DET
ejpam-5411	369	24	conformable	conformable	ADJ
ejpam-5411	369	25	fourier	fourier	NOUN
ejpam-5411	369	26	transform	transform	NOUN
ejpam-5411	369	27	for	for	ADP
ejpam-5411	369	28	α	α	PRON
ejpam-5411	369	29	-	-	ADJ
ejpam-5411	369	30	periodic	periodic	ADJ
ejpam-5411	369	31	function	function	NOUN
ejpam-5411	369	32	is	be	AUX
ejpam-5411	369	33	introduced	introduce	VERB
ejpam-5411	369	34	.	.	PUNCT
ejpam-5411	370	1	definition	definition	NOUN
ejpam-5411	370	2	7	7	NUM
ejpam-5411	370	3	.	.	PUNCT
ejpam-5411	371	1	(	(	PUNCT
ejpam-5411	371	2	conformable	conformable	ADJ
ejpam-5411	371	3	fourier	fourier	NOUN
ejpam-5411	371	4	transform	transform	NOUN
ejpam-5411	371	5	)	)	PUNCT
ejpam-5411	371	6	assume	assume	VERB
ejpam-5411	371	7	that	that	SCONJ
ejpam-5411	371	8	f	f	X
ejpam-5411	371	9	:	:	PUNCT
ejpam-5411	372	1	[	[	X
ejpam-5411	372	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	372	3	r	r	NOUN
ejpam-5411	372	4	is	be	AUX
ejpam-5411	372	5	α	α	DET
ejpam-5411	372	6	-	-	ADJ
ejpam-5411	372	7	periodic	periodic	ADJ
ejpam-5411	372	8	function	function	NOUN
ejpam-5411	372	9	with	with	ADP
ejpam-5411	372	10	period	period	NOUN
ejpam-5411	372	11	p	p	NOUN
ejpam-5411	372	12	and	and	CCONJ
ejpam-5411	372	13	0	0	NUM
ejpam-5411	372	14	<	<	X
ejpam-5411	372	15	α	α	PROPN
ejpam-5411	372	16	≤	≤	NUM
ejpam-5411	372	17	1	1	NUM
ejpam-5411	372	18	.	.	PUNCT
ejpam-5411	373	1	the	the	DET
ejpam-5411	373	2	k	k	NOUN
ejpam-5411	373	3	-	-	PUNCT
ejpam-5411	373	4	th	th	VERB
ejpam-5411	373	5	conformable	conformable	ADJ
ejpam-5411	373	6	fourier	fourier	NOUN
ejpam-5411	373	7	coefficient	coefficient	NOUN
ejpam-5411	373	8	of	of	ADP
ejpam-5411	373	9	f	f	PROPN
ejpam-5411	373	10	denoted	denote	VERB
ejpam-5411	373	11	by	by	ADP
ejpam-5411	373	12	fα(f(t))(k	fα(f(t))(k	NOUN
ejpam-5411	373	13	)	)	PUNCT
ejpam-5411	373	14	is	be	AUX
ejpam-5411	373	15	defined	define	VERB
ejpam-5411	373	16	by	by	ADP
ejpam-5411	373	17	fα(f(t))(k	fα(f(t))(k	NOUN
ejpam-5411	373	18	)	)	PUNCT
ejpam-5411	374	1	=	=	PUNCT
ejpam-5411	375	1	α	α	PRON
ejpam-5411	375	2	pα	pα	INTJ
ejpam-5411	375	3	∫	∫	PROPN
ejpam-5411	376	1	p	p	NOUN
ejpam-5411	376	2	0	0	NUM
ejpam-5411	376	3	e	e	X
ejpam-5411	376	4	−ik	−ik	NOUN
ejpam-5411	376	5	2π	2π	PROPN
ejpam-5411	376	6	pα	pα	INTJ
ejpam-5411	376	7	tα	tα	PROPN
ejpam-5411	376	8	f(t)tα−1dt	f(t)tα−1dt	PROPN
ejpam-5411	376	9	,	,	PUNCT
ejpam-5411	376	10	∀k	∀k	X
ejpam-5411	376	11	∈	∈	PROPN
ejpam-5411	376	12	z	z	NOUN
ejpam-5411	376	13	remark	remark	NOUN
ejpam-5411	376	14	2	2	NUM
ejpam-5411	376	15	.	.	PUNCT
ejpam-5411	376	16	:	:	PUNCT
ejpam-5411	377	1	for	for	ADP
ejpam-5411	377	2	k	k	PROPN
ejpam-5411	377	3	=	=	SYM
ejpam-5411	377	4	0	0	NUM
ejpam-5411	377	5	,	,	PUNCT
ejpam-5411	377	6	fα(f(t))(0	fα(f(t))(0	X
ejpam-5411	377	7	)	)	PUNCT
ejpam-5411	377	8	=	=	PUNCT
ejpam-5411	377	9	α	α	PRON
ejpam-5411	377	10	pα	pα	NOUN
ejpam-5411	377	11	iα(f)(p	iα(f)(p	NOUN
ejpam-5411	377	12	)	)	PUNCT
ejpam-5411	377	13	the	the	DET
ejpam-5411	377	14	next	next	ADJ
ejpam-5411	377	15	theorem	theorem	NOUN
ejpam-5411	377	16	gives	give	VERB
ejpam-5411	377	17	a	a	DET
ejpam-5411	377	18	relationship	relationship	NOUN
ejpam-5411	377	19	between	between	ADP
ejpam-5411	377	20	fourier	fourier	NOUN
ejpam-5411	377	21	conformable	conformable	ADJ
ejpam-5411	377	22	transform	transform	NOUN
ejpam-5411	377	23	and	and	CCONJ
ejpam-5411	377	24	classical	classical	ADJ
ejpam-5411	377	25	fourier	fourier	NOUN
ejpam-5411	377	26	transform	transform	NOUN
ejpam-5411	377	27	applied	apply	VERB
ejpam-5411	377	28	to	to	ADP
ejpam-5411	377	29	α	α	NOUN
ejpam-5411	377	30	-	-	PUNCT
ejpam-5411	377	31	periodic	periodic	ADJ
ejpam-5411	377	32	functions	function	NOUN
ejpam-5411	377	33	.	.	PUNCT
ejpam-5411	378	1	t.	t.	PROPN
ejpam-5411	378	2	abdeljawad	abdeljawad	PROPN
ejpam-5411	378	3	et	et	PROPN
ejpam-5411	378	4	al	al	PROPN
ejpam-5411	378	5	.	.	PUNCT
ejpam-5411	378	6	/	/	SYM
ejpam-5411	378	7	eur	eur	PROPN
ejpam-5411	378	8	.	.	PUNCT
ejpam-5411	379	1	j.	j.	PROPN
ejpam-5411	379	2	pure	pure	PROPN
ejpam-5411	379	3	appl	appl	PROPN
ejpam-5411	379	4	.	.	PROPN
ejpam-5411	379	5	math	math	PROPN
ejpam-5411	379	6	,	,	PUNCT
ejpam-5411	379	7	17	17	NUM
ejpam-5411	379	8	(	(	PUNCT
ejpam-5411	379	9	4	4	NUM
ejpam-5411	379	10	)	)	PUNCT
ejpam-5411	379	11	(	(	PUNCT
ejpam-5411	379	12	2024	2024	NUM
ejpam-5411	379	13	)	)	PUNCT
ejpam-5411	379	14	,	,	PUNCT
ejpam-5411	379	15	2405	2405	NUM
ejpam-5411	379	16	-	-	SYM
ejpam-5411	379	17	2430	2430	NUM
ejpam-5411	379	18	2418	2418	NUM
ejpam-5411	379	19	theorem	theorem	VERB
ejpam-5411	379	20	8	8	NUM
ejpam-5411	379	21	.	.	PUNCT
ejpam-5411	379	22	assume	assume	VERB
ejpam-5411	379	23	that	that	SCONJ
ejpam-5411	379	24	f	f	X
ejpam-5411	379	25	:	:	PUNCT
ejpam-5411	380	1	[	[	X
ejpam-5411	380	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	380	3	r	r	NOUN
ejpam-5411	380	4	is	be	AUX
ejpam-5411	380	5	α	α	DET
ejpam-5411	380	6	-	-	ADJ
ejpam-5411	380	7	periodic	periodic	ADJ
ejpam-5411	380	8	function	function	NOUN
ejpam-5411	380	9	with	with	ADP
ejpam-5411	380	10	period	period	NOUN
ejpam-5411	380	11	p	p	NOUN
ejpam-5411	380	12	and	and	CCONJ
ejpam-5411	380	13	0	0	NUM
ejpam-5411	380	14	<	<	X
ejpam-5411	380	15	α	α	PROPN
ejpam-5411	380	16	≤	≤	ADJ
ejpam-5411	380	17	1	1	NUM
ejpam-5411	380	18	.	.	PUNCT
ejpam-5411	381	1	then	then	ADV
ejpam-5411	381	2	for	for	ADP
ejpam-5411	381	3	all	all	DET
ejpam-5411	381	4	k	k	PROPN
ejpam-5411	381	5	∈	∈	PROPN
ejpam-5411	381	6	z	z	PROPN
ejpam-5411	381	7	,	,	PUNCT
ejpam-5411	381	8	fα{f(t)}(k	fα{f(t)}(k	NOUN
ejpam-5411	381	9	)	)	PUNCT
ejpam-5411	381	10	=	=	SYM
ejpam-5411	381	11	f{f((αt	f{f((αt	NOUN
ejpam-5411	381	12	)	)	PUNCT
ejpam-5411	381	13	1	1	NUM
ejpam-5411	381	14	α	α	NOUN
ejpam-5411	381	15	)	)	PUNCT
ejpam-5411	381	16	}	}	PUNCT
ejpam-5411	381	17	(	(	PUNCT
ejpam-5411	381	18	k	k	X
ejpam-5411	381	19	)	)	PUNCT
ejpam-5411	381	20	proof	proof	NOUN
ejpam-5411	381	21	.	.	PUNCT
ejpam-5411	382	1	let	let	VERB
ejpam-5411	382	2	0	0	NUM
ejpam-5411	382	3	<	<	X
ejpam-5411	382	4	α	α	PROPN
ejpam-5411	382	5	≤	≤	NUM
ejpam-5411	382	6	1	1	NUM
ejpam-5411	382	7	and	and	CCONJ
ejpam-5411	382	8	f	f	NOUN
ejpam-5411	382	9	:	:	PUNCT
ejpam-5411	383	1	[	[	X
ejpam-5411	383	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	383	3	r	r	NOUN
ejpam-5411	383	4	is	be	AUX
ejpam-5411	383	5	α	α	DET
ejpam-5411	383	6	-	-	ADJ
ejpam-5411	383	7	periodic	periodic	ADJ
ejpam-5411	383	8	function	function	NOUN
ejpam-5411	383	9	with	with	ADP
ejpam-5411	383	10	period	period	NOUN
ejpam-5411	383	11	p	p	NOUN
ejpam-5411	383	12	,	,	PUNCT
ejpam-5411	383	13	then	then	ADV
ejpam-5411	383	14	by	by	ADP
ejpam-5411	383	15	remark	remark	NOUN
ejpam-5411	383	16	1	1	NUM
ejpam-5411	383	17	,	,	PUNCT
ejpam-5411	383	18	f((αt	f((αt	NOUN
ejpam-5411	383	19	)	)	PUNCT
ejpam-5411	383	20	1	1	NUM
ejpam-5411	383	21	α	α	NOUN
ejpam-5411	383	22	)	)	PUNCT
ejpam-5411	383	23	is	be	AUX
ejpam-5411	383	24	periodic	periodic	ADJ
ejpam-5411	383	25	with	with	ADP
ejpam-5411	383	26	period	period	NOUN
ejpam-5411	384	1	pp	pp	ADP
ejpam-5411	384	2	α	α	PROPN
ejpam-5411	384	3	and	and	CCONJ
ejpam-5411	384	4	for	for	ADP
ejpam-5411	384	5	all	all	DET
ejpam-5411	384	6	k	k	PROPN
ejpam-5411	384	7	∈	∈	PROPN
ejpam-5411	384	8	z	z	PROPN
ejpam-5411	384	9	,	,	PUNCT
ejpam-5411	384	10	fα{f(t)}(k	fα{f(t)}(k	NOUN
ejpam-5411	384	11	)	)	PUNCT
ejpam-5411	384	12	=	=	SYM
ejpam-5411	385	1	α	α	PRON
ejpam-5411	385	2	pα	pα	INTJ
ejpam-5411	385	3	∫	∫	PROPN
ejpam-5411	386	1	p	p	NOUN
ejpam-5411	386	2	0	0	NUM
ejpam-5411	386	3	e	e	X
ejpam-5411	386	4	−ik	−ik	NOUN
ejpam-5411	386	5	2π	2π	PROPN
ejpam-5411	386	6	pα	pα	INTJ
ejpam-5411	386	7	tα	tα	PROPN
ejpam-5411	386	8	f(t)tα−1dt	f(t)tα−1dt	PROPN
ejpam-5411	386	9	.	.	PUNCT
ejpam-5411	387	1	by	by	ADP
ejpam-5411	387	2	variable	variable	ADJ
ejpam-5411	387	3	change	change	NOUN
ejpam-5411	387	4	tα	tα	ADP
ejpam-5411	387	5	α	α	NOUN
ejpam-5411	387	6	,	,	PUNCT
ejpam-5411	387	7	we	we	PRON
ejpam-5411	387	8	obtain	obtain	VERB
ejpam-5411	387	9	fα{f(t)}(k	fα{f(t)}(k	NOUN
ejpam-5411	387	10	)	)	PUNCT
ejpam-5411	387	11	=	=	SYM
ejpam-5411	388	1	α	α	PRON
ejpam-5411	388	2	pα	pα	INTJ
ejpam-5411	388	3	∫	∫	PROPN
ejpam-5411	388	4	pα	pα	INTJ
ejpam-5411	388	5	α	α	PROPN
ejpam-5411	388	6	0	0	PUNCT
ejpam-5411	389	1	e	e	NOUN
ejpam-5411	389	2	−ik	−ik	NOUN
ejpam-5411	389	3	2πα	2πα	NOUN
ejpam-5411	389	4	pα	pα	VERB
ejpam-5411	389	5	t	t	PROPN
ejpam-5411	389	6	f((αt	f((αt	NOUN
ejpam-5411	389	7	)	)	PUNCT
ejpam-5411	389	8	1	1	NUM
ejpam-5411	389	9	α	α	NOUN
ejpam-5411	389	10	)	)	PUNCT
ejpam-5411	389	11	dt	dt	PROPN
ejpam-5411	389	12	,	,	PUNCT
ejpam-5411	389	13	the	the	DET
ejpam-5411	389	14	function	function	NOUN
ejpam-5411	389	15	t	t	PROPN
ejpam-5411	389	16	∈	∈	PROPN
ejpam-5411	390	1	[	[	X
ejpam-5411	390	2	0,+∞[→	0,+∞[→	NOUN
ejpam-5411	390	3	e	e	X
ejpam-5411	390	4	−ik	−ik	NOUN
ejpam-5411	390	5	2πα	2πα	NOUN
ejpam-5411	390	6	pα	pα	VERB
ejpam-5411	390	7	t	t	PROPN
ejpam-5411	390	8	f((αt	f((αt	NOUN
ejpam-5411	390	9	)	)	PUNCT
ejpam-5411	390	10	1	1	NUM
ejpam-5411	390	11	α	α	NOUN
ejpam-5411	390	12	)	)	PUNCT
ejpam-5411	390	13	is	be	AUX
ejpam-5411	390	14	periodic	periodic	ADJ
ejpam-5411	390	15	with	with	ADP
ejpam-5411	390	16	period	period	NOUN
ejpam-5411	390	17	pα	pα	INTJ
ejpam-5411	390	18	α	α	INTJ
ejpam-5411	390	19	,	,	PUNCT
ejpam-5411	390	20	then	then	ADV
ejpam-5411	390	21	fα{f(t)}(k	fα{f(t)}(k	ADJ
ejpam-5411	390	22	)	)	PUNCT
ejpam-5411	390	23	=	=	SYM
ejpam-5411	391	1	f{f((αt	f{f((αt	NOUN
ejpam-5411	391	2	)	)	PUNCT
ejpam-5411	391	3	1	1	NUM
ejpam-5411	391	4	α	α	NOUN
ejpam-5411	391	5	)	)	PUNCT
ejpam-5411	391	6	}	}	PUNCT
ejpam-5411	391	7	(	(	PUNCT
ejpam-5411	391	8	k	k	NOUN
ejpam-5411	391	9	)	)	PUNCT
ejpam-5411	391	10	.	.	PUNCT
ejpam-5411	392	1	example	example	NOUN
ejpam-5411	392	2	8	8	NUM
ejpam-5411	392	3	.	.	PUNCT
ejpam-5411	393	1	the	the	DET
ejpam-5411	393	2	functions	function	NOUN
ejpam-5411	393	3	f1	f1	NOUN
ejpam-5411	393	4	and	and	CCONJ
ejpam-5411	393	5	f2	f2	PROPN
ejpam-5411	393	6	defined	define	VERB
ejpam-5411	393	7	in	in	ADP
ejpam-5411	393	8	example	example	NOUN
ejpam-5411	393	9	3	3	NUM
ejpam-5411	393	10	are	be	AUX
ejpam-5411	393	11	α	α	X
ejpam-5411	393	12	-	-	NOUN
ejpam-5411	393	13	periodic	periodic	NOUN
ejpam-5411	393	14	with	with	ADP
ejpam-5411	393	15	period	period	NOUN
ejpam-5411	393	16	p	p	X
ejpam-5411	393	17	=	=	PUNCT
ejpam-5411	393	18	(	(	PUNCT
ejpam-5411	393	19	1	1	NUM
ejpam-5411	393	20	α	α	NOUN
ejpam-5411	393	21	)	)	PUNCT
ejpam-5411	393	22	1	1	NUM
ejpam-5411	393	23	α	α	NOUN
ejpam-5411	393	24	.	.	PUNCT
ejpam-5411	394	1	for	for	ADP
ejpam-5411	394	2	k	k	PROPN
ejpam-5411	394	3	̸=	̸=	PROPN
ejpam-5411	394	4	0	0	NUM
ejpam-5411	394	5	,	,	PUNCT
ejpam-5411	394	6	fα(f1(t))(k	fα(f1(t))(k	X
ejpam-5411	394	7	)	)	PUNCT
ejpam-5411	394	8	=	=	SYM
ejpam-5411	394	9	f(f1((αt	f(f1((αt	PROPN
ejpam-5411	394	10	)	)	PUNCT
ejpam-5411	394	11	1	1	NUM
ejpam-5411	394	12	α	α	NOUN
ejpam-5411	394	13	)	)	PUNCT
ejpam-5411	394	14	)	)	PUNCT
ejpam-5411	395	1	(	(	PUNCT
ejpam-5411	395	2	k	k	X
ejpam-5411	395	3	)	)	PUNCT
ejpam-5411	395	4	=	=	VERB
ejpam-5411	396	1	α2	α2	ADJ
ejpam-5411	396	2	∫	∫	PROPN
ejpam-5411	396	3	1	1	NUM
ejpam-5411	396	4	α2	α2	ADJ
ejpam-5411	396	5	0	0	PUNCT
ejpam-5411	396	6	e−2ikπα2tf1((αt	e−2ikπα2tf1((αt	PROPN
ejpam-5411	396	7	)	)	PUNCT
ejpam-5411	396	8	1	1	NUM
ejpam-5411	396	9	α	α	NOUN
ejpam-5411	396	10	)	)	PUNCT
ejpam-5411	396	11	dt	dt	X
ejpam-5411	396	12	=	=	SYM
ejpam-5411	396	13	α2	α2	PROPN
ejpam-5411	396	14	[	[	PUNCT
ejpam-5411	396	15	∫	∫	PROPN
ejpam-5411	396	16	1	1	NUM
ejpam-5411	396	17	2α2	2α2	NUM
ejpam-5411	396	18	0	0	NUM
ejpam-5411	397	1	te−2ikπα2tdt	te−2ikπα2tdt	NOUN
ejpam-5411	398	1	+	+	NUM
ejpam-5411	398	2	∫	∫	PROPN
ejpam-5411	398	3	1	1	NUM
ejpam-5411	398	4	α2	α2	ADJ
ejpam-5411	398	5	1	1	NUM
ejpam-5411	398	6	2α2	2α2	NUM
ejpam-5411	398	7	(	(	PUNCT
ejpam-5411	398	8	1	1	NUM
ejpam-5411	398	9	α2	α2	ADJ
ejpam-5411	398	10	−	−	NOUN
ejpam-5411	398	11	t)e−2ikπα2tdt	t)e−2ikπα2tdt	NOUN
ejpam-5411	398	12	]	]	X
ejpam-5411	398	13	=	=	SYM
ejpam-5411	398	14	(	(	PUNCT
ejpam-5411	398	15	−1)k	−1)k	PROPN
ejpam-5411	398	16	−	−	PROPN
ejpam-5411	398	17	1	1	NUM
ejpam-5411	398	18	2π2k2α2	2π2k2α2	NUM
ejpam-5411	398	19	.	.	PUNCT
ejpam-5411	399	1	and	and	CCONJ
ejpam-5411	399	2	fα(f2(t))(k	fα(f2(t))(k	NOUN
ejpam-5411	399	3	)	)	PUNCT
ejpam-5411	399	4	=	=	PUNCT
ejpam-5411	399	5	f(f2((αt	f(f2((αt	X
ejpam-5411	399	6	)	)	PUNCT
ejpam-5411	399	7	1	1	NUM
ejpam-5411	399	8	α	α	NOUN
ejpam-5411	399	9	)	)	PUNCT
ejpam-5411	399	10	)	)	PUNCT
ejpam-5411	400	1	(	(	PUNCT
ejpam-5411	400	2	k	k	X
ejpam-5411	400	3	)	)	PUNCT
ejpam-5411	400	4	=	=	VERB
ejpam-5411	401	1	α2	α2	ADJ
ejpam-5411	401	2	∫	∫	PROPN
ejpam-5411	401	3	1	1	NUM
ejpam-5411	401	4	α2	α2	NOUN
ejpam-5411	401	5	0	0	NUM
ejpam-5411	402	1	e−2ikπα2tf2((αt	e−2ikπα2tf2((αt	NOUN
ejpam-5411	402	2	)	)	PUNCT
ejpam-5411	402	3	1	1	NUM
ejpam-5411	402	4	α	α	NOUN
ejpam-5411	402	5	)	)	PUNCT
ejpam-5411	402	6	dt	dt	X
ejpam-5411	403	1	=	=	SYM
ejpam-5411	403	2	α2	α2	PROPN
ejpam-5411	403	3	[	[	PUNCT
ejpam-5411	403	4	∫	∫	PROPN
ejpam-5411	403	5	1	1	NUM
ejpam-5411	403	6	4α2	4α2	NOUN
ejpam-5411	403	7	0	0	PUNCT
ejpam-5411	404	1	te−2ikπα2tdt	te−2ikπα2tdt	PROPN
ejpam-5411	405	1	+	+	NUM
ejpam-5411	405	2	∫	∫	PROPN
ejpam-5411	405	3	3	3	NUM
ejpam-5411	405	4	4α2	4α2	NOUN
ejpam-5411	405	5	1	1	NUM
ejpam-5411	405	6	4α2	4α2	NOUN
ejpam-5411	405	7	(	(	PUNCT
ejpam-5411	405	8	1	1	NUM
ejpam-5411	405	9	α2	α2	ADJ
ejpam-5411	405	10	−	−	NOUN
ejpam-5411	405	11	t)e−2ikπα2tdt	t)e−2ikπα2tdt	ADJ
ejpam-5411	405	12	+	+	X
ejpam-5411	405	13	∫	∫	PROPN
ejpam-5411	405	14	1	1	NUM
ejpam-5411	405	15	α2	α2	ADJ
ejpam-5411	405	16	3	3	NUM
ejpam-5411	405	17	4α2	4α2	NOUN
ejpam-5411	405	18	(	(	PUNCT
ejpam-5411	405	19	t−	t−	PROPN
ejpam-5411	405	20	1	1	NUM
ejpam-5411	405	21	α2	α2	ADJ
ejpam-5411	405	22	)	)	PUNCT
ejpam-5411	406	1	e−2ikπα2tdt	e−2ikπα2tdt	PROPN
ejpam-5411	406	2	]	]	X
ejpam-5411	406	3	=	=	PUNCT
ejpam-5411	407	1	(	(	PUNCT
ejpam-5411	407	2	−1)−	−1)−	NOUN
ejpam-5411	407	3	k	k	NOUN
ejpam-5411	407	4	2	2	NUM
ejpam-5411	407	5	(	(	PUNCT
ejpam-5411	407	6	1	1	NUM
ejpam-5411	407	7	−	−	PROPN
ejpam-5411	407	8	(	(	PUNCT
ejpam-5411	407	9	−1)−k	−1)−k	NOUN
ejpam-5411	407	10	)	)	PUNCT
ejpam-5411	407	11	2α2k2π2	2α2k2π2	NUM
ejpam-5411	407	12	.	.	PUNCT
ejpam-5411	408	1	t.	t.	PROPN
ejpam-5411	408	2	abdeljawad	abdeljawad	PROPN
ejpam-5411	408	3	et	et	PROPN
ejpam-5411	408	4	al	al	PROPN
ejpam-5411	408	5	.	.	PUNCT
ejpam-5411	408	6	/	/	SYM
ejpam-5411	408	7	eur	eur	PROPN
ejpam-5411	408	8	.	.	PUNCT
ejpam-5411	409	1	j.	j.	PROPN
ejpam-5411	409	2	pure	pure	PROPN
ejpam-5411	409	3	appl	appl	PROPN
ejpam-5411	409	4	.	.	PROPN
ejpam-5411	409	5	math	math	PROPN
ejpam-5411	409	6	,	,	PUNCT
ejpam-5411	409	7	17	17	NUM
ejpam-5411	409	8	(	(	PUNCT
ejpam-5411	409	9	4	4	NUM
ejpam-5411	409	10	)	)	PUNCT
ejpam-5411	409	11	(	(	PUNCT
ejpam-5411	409	12	2024	2024	NUM
ejpam-5411	409	13	)	)	PUNCT
ejpam-5411	409	14	,	,	PUNCT
ejpam-5411	409	15	2405	2405	NUM
ejpam-5411	409	16	-	-	SYM
ejpam-5411	409	17	2430	2430	NUM
ejpam-5411	409	18	2419	2419	NUM
ejpam-5411	409	19	for	for	ADP
ejpam-5411	409	20	k	k	PROPN
ejpam-5411	409	21	=	=	SYM
ejpam-5411	409	22	0	0	PROPN
ejpam-5411	409	23	,	,	PUNCT
ejpam-5411	409	24	we	we	PRON
ejpam-5411	409	25	have	have	AUX
ejpam-5411	409	26	fα(f1(t))(0	fα(f1(t))(0	NOUN
ejpam-5411	409	27	)	)	PUNCT
ejpam-5411	409	28	=	=	VERB
ejpam-5411	410	1	α2	α2	ADJ
ejpam-5411	410	2	∫	∫	PROPN
ejpam-5411	410	3	1	1	NUM
ejpam-5411	410	4	α2	α2	ADJ
ejpam-5411	410	5	0	0	NUM
ejpam-5411	410	6	f1((αt	f1((αt	NOUN
ejpam-5411	410	7	)	)	PUNCT
ejpam-5411	410	8	1	1	NUM
ejpam-5411	410	9	α	α	NOUN
ejpam-5411	410	10	)	)	PUNCT
ejpam-5411	410	11	)	)	PUNCT
ejpam-5411	410	12	dt	dt	PUNCT
ejpam-5411	411	1	=	=	SYM
ejpam-5411	411	2	1	1	NUM
ejpam-5411	411	3	4α2	4α2	NUM
ejpam-5411	411	4	and	and	CCONJ
ejpam-5411	411	5	fα(f2(t))(0	fα(f2(t))(0	NOUN
ejpam-5411	411	6	)	)	PUNCT
ejpam-5411	411	7	=	=	VERB
ejpam-5411	411	8	α2	α2	ADJ
ejpam-5411	411	9	∫	∫	PROPN
ejpam-5411	411	10	1	1	NUM
ejpam-5411	411	11	α2	α2	NOUN
ejpam-5411	411	12	0	0	NUM
ejpam-5411	411	13	f2((αt	f2((αt	NOUN
ejpam-5411	411	14	)	)	PUNCT
ejpam-5411	411	15	1	1	NUM
ejpam-5411	411	16	α	α	NOUN
ejpam-5411	411	17	)	)	PUNCT
ejpam-5411	411	18	)	)	PUNCT
ejpam-5411	411	19	dt	dt	PROPN
ejpam-5411	412	1	=	=	PUNCT
ejpam-5411	412	2	0	0	X
ejpam-5411	412	3	.	.	PUNCT
ejpam-5411	412	4	then	then	ADV
ejpam-5411	412	5	fα(f1(t))(k	fα(f1(t))(k	VERB
ejpam-5411	412	6	)	)	PUNCT
ejpam-5411	412	7	=	=	PUNCT
ejpam-5411	412	8			PUNCT
ejpam-5411	412	9	(	(	PUNCT
ejpam-5411	412	10	−1)k−1	−1)k−1	NUM
ejpam-5411	412	11	2π2k2α2	2π2k2α2	NUM
ejpam-5411	412	12	,	,	PUNCT
ejpam-5411	412	13	∀k	∀k	NOUN
ejpam-5411	412	14	∈	∈	PROPN
ejpam-5411	412	15	z∗	z∗	PROPN
ejpam-5411	412	16	1	1	NUM
ejpam-5411	412	17	4α2	4α2	NOUN
ejpam-5411	412	18	,	,	PUNCT
ejpam-5411	412	19	k	k	PROPN
ejpam-5411	412	20	=	=	PUNCT
ejpam-5411	412	21	0	0	PROPN
ejpam-5411	412	22	.	.	PUNCT
ejpam-5411	413	1	(	(	PUNCT
ejpam-5411	413	2	17	17	NUM
ejpam-5411	413	3	)	)	PUNCT
ejpam-5411	413	4	and	and	CCONJ
ejpam-5411	413	5	fα(f2(t))(k	fα(f2(t))(k	NOUN
ejpam-5411	413	6	)	)	PUNCT
ejpam-5411	413	7	=	=	SYM
ejpam-5411	414	1			PROPN
ejpam-5411	414	2	(	(	PUNCT
ejpam-5411	414	3	−1)−	−1)−	NOUN
ejpam-5411	414	4	k	k	NOUN
ejpam-5411	414	5	2	2	NUM
ejpam-5411	414	6	(	(	PUNCT
ejpam-5411	414	7	1−(−1)−k	1−(−1)−k	NUM
ejpam-5411	414	8	)	)	PUNCT
ejpam-5411	414	9	2α2k2π2	2α2k2π2	NUM
ejpam-5411	414	10	,	,	PUNCT
ejpam-5411	414	11	∀k	∀k	NOUN
ejpam-5411	414	12	∈	∈	PROPN
ejpam-5411	414	13	z∗	z∗	PROPN
ejpam-5411	414	14	0	0	NUM
ejpam-5411	414	15	,	,	PUNCT
ejpam-5411	414	16	k	k	NOUN
ejpam-5411	414	17	=	=	PUNCT
ejpam-5411	414	18	0	0	PROPN
ejpam-5411	414	19	.	.	PUNCT
ejpam-5411	414	20	(	(	PUNCT
ejpam-5411	414	21	18	18	NUM
ejpam-5411	414	22	)	)	PUNCT
ejpam-5411	414	23	as	as	ADP
ejpam-5411	414	24	a	a	DET
ejpam-5411	414	25	classical	classical	ADJ
ejpam-5411	414	26	fourier	fourier	NOUN
ejpam-5411	414	27	transform	transform	NOUN
ejpam-5411	414	28	,	,	PUNCT
ejpam-5411	414	29	we	we	PRON
ejpam-5411	414	30	apply	apply	VERB
ejpam-5411	414	31	the	the	DET
ejpam-5411	414	32	conformable	conformable	ADJ
ejpam-5411	414	33	fourier	fourier	NOUN
ejpam-5411	414	34	transform	transform	NOUN
ejpam-5411	414	35	to	to	ADP
ejpam-5411	414	36	the	the	DET
ejpam-5411	414	37	conformable	conformable	ADJ
ejpam-5411	414	38	fractional	fractional	ADJ
ejpam-5411	414	39	integral	integral	ADJ
ejpam-5411	414	40	given	give	VERB
ejpam-5411	414	41	by	by	ADP
ejpam-5411	414	42	definition	definition	NOUN
ejpam-5411	414	43	4	4	NUM
ejpam-5411	414	44	.	.	PUNCT
ejpam-5411	415	1	the	the	DET
ejpam-5411	415	2	following	follow	VERB
ejpam-5411	415	3	theorem	theorem	NOUN
ejpam-5411	415	4	is	be	AUX
ejpam-5411	415	5	obtained	obtain	VERB
ejpam-5411	415	6	.	.	PUNCT
ejpam-5411	416	1	theorem	theorem	NOUN
ejpam-5411	416	2	9	9	NUM
ejpam-5411	416	3	.	.	PUNCT
ejpam-5411	417	1	assume	assume	VERB
ejpam-5411	417	2	that	that	SCONJ
ejpam-5411	417	3	f	f	X
ejpam-5411	417	4	:	:	PUNCT
ejpam-5411	418	1	[	[	X
ejpam-5411	418	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	418	3	r	r	NOUN
ejpam-5411	418	4	is	be	AUX
ejpam-5411	418	5	α	α	DET
ejpam-5411	418	6	-	-	ADJ
ejpam-5411	418	7	periodic	periodic	ADJ
ejpam-5411	418	8	function	function	NOUN
ejpam-5411	418	9	with	with	ADP
ejpam-5411	418	10	period	period	NOUN
ejpam-5411	418	11	p	p	PRON
ejpam-5411	418	12	such	such	ADJ
ejpam-5411	418	13	that	that	DET
ejpam-5411	418	14	iα(f)(p	iα(f)(p	NOUN
ejpam-5411	418	15	)	)	PUNCT
ejpam-5411	418	16	=	=	SYM
ejpam-5411	418	17	0	0	NUM
ejpam-5411	418	18	and	and	CCONJ
ejpam-5411	418	19	0	0	NUM
ejpam-5411	418	20	<	<	X
ejpam-5411	418	21	α	α	X
ejpam-5411	418	22	≤	≤	ADJ
ejpam-5411	418	23	1	1	NUM
ejpam-5411	418	24	.	.	PUNCT
ejpam-5411	419	1	then	then	ADV
ejpam-5411	419	2	for	for	ADP
ejpam-5411	419	3	all	all	DET
ejpam-5411	419	4	t	t	NOUN
ejpam-5411	419	5	∈	∈	PROPN
ejpam-5411	420	1	[	[	X
ejpam-5411	420	2	0,+∞	0,+∞	PROPN
ejpam-5411	420	3	[	[	X
ejpam-5411	420	4	,	,	PUNCT
ejpam-5411	420	5	iα(f)(t	iα(f)(t	NUM
ejpam-5411	420	6	)	)	PUNCT
ejpam-5411	420	7	is	be	AUX
ejpam-5411	420	8	α	α	X
ejpam-5411	420	9	-	-	NOUN
ejpam-5411	420	10	periodic	periodic	NOUN
ejpam-5411	420	11	with	with	ADP
ejpam-5411	420	12	period	period	NOUN
ejpam-5411	420	13	p	p	NOUN
ejpam-5411	420	14	and	and	CCONJ
ejpam-5411	420	15	fα(iα(f)(t))(k	fα(iα(f)(t))(k	NOUN
ejpam-5411	420	16	)	)	PUNCT
ejpam-5411	420	17	=	=	PUNCT
ejpam-5411	421	1			PUNCT
ejpam-5411	421	2	pα	pα	NOUN
ejpam-5411	421	3	2ikπαfα(f(t))(k	2ikπαfα(f(t))(k	NUM
ejpam-5411	421	4	)	)	PUNCT
ejpam-5411	421	5	,	,	PUNCT
ejpam-5411	421	6	∀k	∀k	NOUN
ejpam-5411	421	7	∈	∈	PROPN
ejpam-5411	421	8	z∗	z∗	PROPN
ejpam-5411	421	9	f(fα((αt	f(fα((αt	PROPN
ejpam-5411	421	10	)	)	PUNCT
ejpam-5411	421	11	1	1	NUM
ejpam-5411	421	12	α	α	NOUN
ejpam-5411	421	13	)	)	PUNCT
ejpam-5411	421	14	)	)	PUNCT
ejpam-5411	421	15	(	(	PUNCT
ejpam-5411	421	16	0	0	NUM
ejpam-5411	421	17	)	)	PUNCT
ejpam-5411	421	18	,	,	PUNCT
ejpam-5411	421	19	k	k	X
ejpam-5411	422	1	=	=	PUNCT
ejpam-5411	422	2	0	0	X
ejpam-5411	422	3	.	.	PUNCT
ejpam-5411	423	1	where	where	SCONJ
ejpam-5411	423	2	fα(t	fα(t	ADV
ejpam-5411	423	3	)	)	PUNCT
ejpam-5411	423	4	=	=	SYM
ejpam-5411	423	5	−	−	PROPN
ejpam-5411	423	6	tα	tα	PROPN
ejpam-5411	423	7	α	α	PRON
ejpam-5411	423	8	f(t	f(t	NOUN
ejpam-5411	423	9	)	)	PUNCT
ejpam-5411	423	10	.	.	PUNCT
ejpam-5411	424	1	proof	proof	NOUN
ejpam-5411	424	2	.	.	PUNCT
ejpam-5411	425	1	let	let	VERB
ejpam-5411	425	2	0	0	NUM
ejpam-5411	425	3	<	<	X
ejpam-5411	425	4	α	α	X
ejpam-5411	425	5	≤	≤	NUM
ejpam-5411	425	6	1	1	NUM
ejpam-5411	425	7	.	.	PUNCT
ejpam-5411	426	1	f	f	PROPN
ejpam-5411	426	2	is	be	AUX
ejpam-5411	426	3	α	α	DET
ejpam-5411	426	4	-	-	ADJ
ejpam-5411	426	5	periodic	periodic	ADJ
ejpam-5411	426	6	function	function	NOUN
ejpam-5411	426	7	with	with	ADP
ejpam-5411	426	8	period	period	NOUN
ejpam-5411	426	9	p	p	PRON
ejpam-5411	426	10	such	such	ADJ
ejpam-5411	426	11	that	that	DET
ejpam-5411	426	12	iα(f(p	iα(f(p	NOUN
ejpam-5411	426	13	)	)	PUNCT
ejpam-5411	426	14	)	)	PUNCT
ejpam-5411	427	1	=	=	SYM
ejpam-5411	427	2	0	0	NUM
ejpam-5411	427	3	,	,	PUNCT
ejpam-5411	427	4	then	then	ADV
ejpam-5411	427	5	by	by	ADP
ejpam-5411	427	6	theorem	theorem	NOUN
ejpam-5411	427	7	4	4	NUM
ejpam-5411	427	8	,	,	PUNCT
ejpam-5411	427	9	for	for	ADP
ejpam-5411	427	10	all	all	DET
ejpam-5411	427	11	t	t	NOUN
ejpam-5411	427	12	∈	∈	PROPN
ejpam-5411	428	1	[	[	X
ejpam-5411	428	2	0,+∞	0,+∞	PROPN
ejpam-5411	428	3	[	[	X
ejpam-5411	428	4	,	,	PUNCT
ejpam-5411	428	5	iα(f)(t	iα(f)(t	NUM
ejpam-5411	428	6	)	)	PUNCT
ejpam-5411	428	7	is	be	AUX
ejpam-5411	428	8	α	α	X
ejpam-5411	428	9	-	-	NOUN
ejpam-5411	428	10	periodic	periodic	NOUN
ejpam-5411	428	11	with	with	ADP
ejpam-5411	428	12	period	period	NOUN
ejpam-5411	428	13	p.	p.	NOUN
ejpam-5411	428	14	for	for	ADP
ejpam-5411	428	15	k	k	PROPN
ejpam-5411	428	16	̸=	̸=	PROPN
ejpam-5411	428	17	0	0	NUM
ejpam-5411	428	18	,	,	PUNCT
ejpam-5411	428	19	fα(iα(f)(t))(k	fα(iα(f)(t))(k	NOUN
ejpam-5411	428	20	)	)	PUNCT
ejpam-5411	428	21	=	=	SYM
ejpam-5411	428	22	f(iα(f)((αt	f(iα(f)((αt	PROPN
ejpam-5411	428	23	)	)	PUNCT
ejpam-5411	428	24	1	1	NUM
ejpam-5411	428	25	α	α	NOUN
ejpam-5411	428	26	)	)	PUNCT
ejpam-5411	428	27	)	)	PUNCT
ejpam-5411	428	28	(	(	PUNCT
ejpam-5411	428	29	k	k	X
ejpam-5411	428	30	)	)	PUNCT
ejpam-5411	429	1	=	=	SYM
ejpam-5411	429	2	α	α	PRON
ejpam-5411	429	3	pα	pα	INTJ
ejpam-5411	429	4	∫	∫	PROPN
ejpam-5411	429	5	pα	pα	INTJ
ejpam-5411	429	6	α	α	PROPN
ejpam-5411	429	7	0	0	PUNCT
ejpam-5411	430	1	e	e	NOUN
ejpam-5411	430	2	−2ikπ	−2ikπ	NOUN
ejpam-5411	430	3	α	α	PROPN
ejpam-5411	430	4	pα	pα	PROPN
ejpam-5411	430	5	t	t	PROPN
ejpam-5411	430	6	iα(f)((αt	iα(f)((αt	PROPN
ejpam-5411	430	7	)	)	PUNCT
ejpam-5411	430	8	1	1	NUM
ejpam-5411	430	9	α	α	NOUN
ejpam-5411	430	10	)	)	PUNCT
ejpam-5411	430	11	dt	dt	PUNCT
ejpam-5411	430	12	by	by	ADP
ejpam-5411	430	13	definition	definition	NOUN
ejpam-5411	430	14	4	4	NUM
ejpam-5411	430	15	,	,	PUNCT
ejpam-5411	430	16	we	we	PRON
ejpam-5411	430	17	have	have	VERB
ejpam-5411	430	18	iα(f)(t	iα(f)(t	NUM
ejpam-5411	430	19	)	)	PUNCT
ejpam-5411	431	1	=	=	SYM
ejpam-5411	432	1	∫	∫	PROPN
ejpam-5411	432	2	t	t	PROPN
ejpam-5411	432	3	0	0	NUM
ejpam-5411	432	4	sα−1f(s)ds	sα−1f(s)ds	PROPN
ejpam-5411	432	5	.	.	PUNCT
ejpam-5411	433	1	t.	t.	PROPN
ejpam-5411	433	2	abdeljawad	abdeljawad	PROPN
ejpam-5411	433	3	et	et	PROPN
ejpam-5411	433	4	al	al	PROPN
ejpam-5411	433	5	.	.	PUNCT
ejpam-5411	433	6	/	/	SYM
ejpam-5411	433	7	eur	eur	PROPN
ejpam-5411	433	8	.	.	PUNCT
ejpam-5411	434	1	j.	j.	PROPN
ejpam-5411	434	2	pure	pure	PROPN
ejpam-5411	434	3	appl	appl	PROPN
ejpam-5411	434	4	.	.	PROPN
ejpam-5411	434	5	math	math	PROPN
ejpam-5411	434	6	,	,	PUNCT
ejpam-5411	434	7	17	17	NUM
ejpam-5411	434	8	(	(	PUNCT
ejpam-5411	434	9	4	4	NUM
ejpam-5411	434	10	)	)	PUNCT
ejpam-5411	434	11	(	(	PUNCT
ejpam-5411	434	12	2024	2024	NUM
ejpam-5411	434	13	)	)	PUNCT
ejpam-5411	434	14	,	,	PUNCT
ejpam-5411	434	15	2405	2405	NUM
ejpam-5411	434	16	-	-	SYM
ejpam-5411	434	17	2430	2430	NUM
ejpam-5411	434	18	2420	2420	NUM
ejpam-5411	434	19	using	use	VERB
ejpam-5411	434	20	variable	variable	ADJ
ejpam-5411	434	21	change	change	NOUN
ejpam-5411	434	22	tα	tα	ADP
ejpam-5411	434	23	α	α	NOUN
ejpam-5411	434	24	,	,	PUNCT
ejpam-5411	434	25	we	we	PRON
ejpam-5411	434	26	obtain	obtain	VERB
ejpam-5411	434	27	iα(f)(t	iα(f)(t	PRON
ejpam-5411	434	28	)	)	PUNCT
ejpam-5411	435	1	=	=	SYM
ejpam-5411	435	2	∫	∫	PROPN
ejpam-5411	436	1	tα	tα	PROPN
ejpam-5411	436	2	α	α	NOUN
ejpam-5411	436	3	0	0	SYM
ejpam-5411	436	4	f((αs	f((αs	NOUN
ejpam-5411	436	5	)	)	PUNCT
ejpam-5411	436	6	1	1	NUM
ejpam-5411	436	7	α	α	NOUN
ejpam-5411	436	8	)	)	PUNCT
ejpam-5411	436	9	ds	ds	NOUN
ejpam-5411	436	10	then	then	ADV
ejpam-5411	436	11	iα(f)((αt	iα(f)((αt	NOUN
ejpam-5411	436	12	)	)	PUNCT
ejpam-5411	436	13	1	1	NUM
ejpam-5411	436	14	α	α	NOUN
ejpam-5411	436	15	)	)	PUNCT
ejpam-5411	437	1	=	=	SYM
ejpam-5411	438	1	∫	∫	PROPN
ejpam-5411	438	2	t	t	PROPN
ejpam-5411	438	3	0	0	NUM
ejpam-5411	438	4	f((αs	f((αs	X
ejpam-5411	438	5	)	)	PUNCT
ejpam-5411	438	6	1	1	NUM
ejpam-5411	438	7	α	α	NOUN
ejpam-5411	438	8	)	)	PUNCT
ejpam-5411	438	9	ds	ds	PROPN
ejpam-5411	438	10	.	.	X
ejpam-5411	438	11	by	by	ADP
ejpam-5411	438	12	integrating	integrate	VERB
ejpam-5411	438	13	by	by	ADP
ejpam-5411	438	14	parts	part	NOUN
ejpam-5411	438	15	,	,	PUNCT
ejpam-5411	438	16	we	we	PRON
ejpam-5411	438	17	find	find	VERB
ejpam-5411	438	18	that	that	SCONJ
ejpam-5411	438	19	fα(iα(f)(t))(k	fα(iα(f)(t))(k	NOUN
ejpam-5411	438	20	)	)	PUNCT
ejpam-5411	438	21	=	=	SYM
ejpam-5411	439	1	−	−	PROPN
ejpam-5411	439	2	1	1	NUM
ejpam-5411	439	3	2ikπ	2ikπ	PROPN
ejpam-5411	439	4	[	[	X
ejpam-5411	439	5	iα(f)(p	iα(f)(p	NOUN
ejpam-5411	439	6	)	)	PUNCT
ejpam-5411	440	1	−	−	NOUN
ejpam-5411	441	1	∫	∫	INTJ
ejpam-5411	441	2	pα	pα	INTJ
ejpam-5411	441	3	α	α	PROPN
ejpam-5411	441	4	0	0	PUNCT
ejpam-5411	442	1	e	e	NOUN
ejpam-5411	442	2	−2ikπ	−2ikπ	NOUN
ejpam-5411	442	3	α	α	PROPN
ejpam-5411	442	4	pα	pα	PROPN
ejpam-5411	442	5	t	t	PROPN
ejpam-5411	442	6	f((αt	f((αt	NOUN
ejpam-5411	442	7	)	)	PUNCT
ejpam-5411	442	8	1	1	NUM
ejpam-5411	442	9	α	α	NOUN
ejpam-5411	442	10	)	)	PUNCT
ejpam-5411	442	11	dt	dt	X
ejpam-5411	442	12	]	]	PUNCT
ejpam-5411	442	13	and	and	CCONJ
ejpam-5411	442	14	using	use	VERB
ejpam-5411	442	15	the	the	DET
ejpam-5411	442	16	condition	condition	NOUN
ejpam-5411	442	17	iα(f)(p	iα(f)(p	NOUN
ejpam-5411	442	18	)	)	PUNCT
ejpam-5411	442	19	=	=	SYM
ejpam-5411	442	20	0	0	NUM
ejpam-5411	442	21	,	,	PUNCT
ejpam-5411	442	22	the	the	DET
ejpam-5411	442	23	result	result	NOUN
ejpam-5411	442	24	is	be	AUX
ejpam-5411	442	25	obtained	obtain	VERB
ejpam-5411	442	26	.	.	PUNCT
ejpam-5411	443	1	for	for	ADP
ejpam-5411	443	2	k	k	PROPN
ejpam-5411	443	3	=	=	SYM
ejpam-5411	443	4	0	0	PROPN
ejpam-5411	443	5	,	,	PUNCT
ejpam-5411	443	6	using	use	VERB
ejpam-5411	443	7	remark	remark	NOUN
ejpam-5411	443	8	2	2	NUM
ejpam-5411	443	9	,	,	PUNCT
ejpam-5411	443	10	we	we	PRON
ejpam-5411	443	11	have	have	VERB
ejpam-5411	443	12	fα(iα(f(t)))(0	fα(iα(f(t)))(0	NOUN
ejpam-5411	443	13	)	)	PUNCT
ejpam-5411	443	14	=	=	PUNCT
ejpam-5411	444	1	α	α	PROPN
ejpam-5411	444	2	pα	pα	NOUN
ejpam-5411	444	3	iα(iα(f))(p	iα(iα(f))(p	NOUN
ejpam-5411	444	4	)	)	PUNCT
ejpam-5411	445	1	=	=	SYM
ejpam-5411	445	2	α	α	PRON
ejpam-5411	445	3	pα	pα	INTJ
ejpam-5411	445	4	∫	∫	PROPN
ejpam-5411	445	5	pα	pα	INTJ
ejpam-5411	445	6	α	α	PROPN
ejpam-5411	445	7	0	0	PUNCT
ejpam-5411	445	8	iα(f)((αt	iα(f)((αt	NUM
ejpam-5411	445	9	)	)	PUNCT
ejpam-5411	445	10	1	1	NUM
ejpam-5411	445	11	α	α	NOUN
ejpam-5411	445	12	)	)	PUNCT
ejpam-5411	445	13	dt	dt	NOUN
ejpam-5411	445	14	.	.	PUNCT
ejpam-5411	446	1	using	use	VERB
ejpam-5411	446	2	integration	integration	NOUN
ejpam-5411	446	3	by	by	ADP
ejpam-5411	446	4	parts	part	NOUN
ejpam-5411	446	5	and	and	CCONJ
ejpam-5411	446	6	the	the	DET
ejpam-5411	446	7	condition	condition	NOUN
ejpam-5411	446	8	iα(f)(p	iα(f)(p	NOUN
ejpam-5411	446	9	)	)	PUNCT
ejpam-5411	446	10	=	=	SYM
ejpam-5411	446	11	0	0	NUM
ejpam-5411	446	12	,	,	PUNCT
ejpam-5411	446	13	we	we	PRON
ejpam-5411	446	14	find	find	VERB
ejpam-5411	446	15	that	that	SCONJ
ejpam-5411	446	16	fα(iα(f)(t))(0	fα(iα(f)(t))(0	PROPN
ejpam-5411	446	17	)	)	PUNCT
ejpam-5411	446	18	=	=	SYM
ejpam-5411	447	1	α	α	PRON
ejpam-5411	447	2	pα	pα	INTJ
ejpam-5411	447	3	∫	∫	PROPN
ejpam-5411	447	4	pα	pα	INTJ
ejpam-5411	447	5	α	α	PROPN
ejpam-5411	447	6	0	0	PUNCT
ejpam-5411	448	1	−tf((αt	−tf((αt	NOUN
ejpam-5411	448	2	)	)	PUNCT
ejpam-5411	448	3	1	1	NUM
ejpam-5411	448	4	α	α	NOUN
ejpam-5411	448	5	)	)	PUNCT
ejpam-5411	448	6	dt	dt	X
ejpam-5411	449	1	=	=	SYM
ejpam-5411	449	2	f(fα((αt	f(fα((αt	PROPN
ejpam-5411	449	3	)	)	PUNCT
ejpam-5411	449	4	1	1	NUM
ejpam-5411	449	5	α	α	NOUN
ejpam-5411	449	6	)	)	PUNCT
ejpam-5411	449	7	)	)	PUNCT
ejpam-5411	449	8	(	(	PUNCT
ejpam-5411	449	9	0	0	NUM
ejpam-5411	449	10	)	)	PUNCT
ejpam-5411	449	11	where	where	SCONJ
ejpam-5411	449	12	fα(t	fα(t	ADV
ejpam-5411	449	13	)	)	PUNCT
ejpam-5411	449	14	=	=	SYM
ejpam-5411	450	1	−	−	PROPN
ejpam-5411	450	2	tα	tα	PROPN
ejpam-5411	450	3	α	α	PRON
ejpam-5411	450	4	f(t	f(t	PROPN
ejpam-5411	450	5	)	)	PUNCT
ejpam-5411	450	6	.	.	PUNCT
ejpam-5411	451	1	example	example	NOUN
ejpam-5411	451	2	9	9	NUM
ejpam-5411	451	3	.	.	PUNCT
ejpam-5411	451	4	consider	consider	VERB
ejpam-5411	451	5	the	the	DET
ejpam-5411	451	6	function	function	NOUN
ejpam-5411	451	7	f2	f2	PRON
ejpam-5411	451	8	defined	define	VERB
ejpam-5411	451	9	by	by	ADP
ejpam-5411	451	10	example	example	NOUN
ejpam-5411	451	11	3	3	X
ejpam-5411	451	12	.	.	PUNCT
ejpam-5411	452	1	we	we	PRON
ejpam-5411	452	2	have	have	AUX
ejpam-5411	452	3	showed	show	VERB
ejpam-5411	452	4	in	in	ADP
ejpam-5411	452	5	example	example	NOUN
ejpam-5411	452	6	4	4	NUM
ejpam-5411	452	7	that	that	DET
ejpam-5411	452	8	iα(f2	iα(f2	VERB
ejpam-5411	452	9	)	)	PUNCT
ejpam-5411	452	10	(	(	PUNCT
ejpam-5411	452	11	1	1	NUM
ejpam-5411	452	12	α2	α2	ADJ
ejpam-5411	452	13	)	)	PUNCT
ejpam-5411	452	14	=	=	SYM
ejpam-5411	452	15	0	0	NUM
ejpam-5411	452	16	and	and	CCONJ
ejpam-5411	452	17	iα(f2	iα(f2	PROPN
ejpam-5411	452	18	)	)	PUNCT
ejpam-5411	452	19	is	be	AUX
ejpam-5411	452	20	α	α	X
ejpam-5411	452	21	-	-	NOUN
ejpam-5411	452	22	periodic	periodic	NOUN
ejpam-5411	452	23	with	with	ADP
ejpam-5411	452	24	period	period	NOUN
ejpam-5411	452	25	1	1	NUM
ejpam-5411	452	26	α2	α2	ADJ
ejpam-5411	452	27	.	.	PUNCT
ejpam-5411	453	1	for	for	ADP
ejpam-5411	453	2	k	k	PROPN
ejpam-5411	453	3	∈	∈	PROPN
ejpam-5411	453	4	z∗	z∗	PROPN
ejpam-5411	453	5	,	,	PUNCT
ejpam-5411	453	6	fα(iα(f2)(t))(k	fα(iα(f2)(t))(k	NOUN
ejpam-5411	453	7	)	)	PUNCT
ejpam-5411	453	8	=	=	PUNCT
ejpam-5411	453	9	f(iα(f2)((αt	f(iα(f2)((αt	ADJ
ejpam-5411	453	10	)	)	PUNCT
ejpam-5411	453	11	1	1	NUM
ejpam-5411	453	12	α	α	NOUN
ejpam-5411	453	13	)	)	PUNCT
ejpam-5411	453	14	(	(	PUNCT
ejpam-5411	453	15	k	k	X
ejpam-5411	453	16	)	)	PUNCT
ejpam-5411	453	17	=	=	VERB
ejpam-5411	453	18	α2	α2	ADJ
ejpam-5411	453	19	∫	∫	PROPN
ejpam-5411	453	20	1	1	NUM
ejpam-5411	453	21	α2	α2	NOUN
ejpam-5411	453	22	0	0	PUNCT
ejpam-5411	454	1	e−2ikπα2tiα(f2)((αt	e−2ikπα2tiα(f2)((αt	PROPN
ejpam-5411	454	2	)	)	PUNCT
ejpam-5411	454	3	1	1	NUM
ejpam-5411	454	4	α	α	NOUN
ejpam-5411	454	5	)	)	PUNCT
ejpam-5411	454	6	dt	dt	NOUN
ejpam-5411	454	7	.	.	PUNCT
ejpam-5411	455	1	using	use	VERB
ejpam-5411	455	2	integration	integration	NOUN
ejpam-5411	455	3	by	by	ADP
ejpam-5411	455	4	parts	part	NOUN
ejpam-5411	455	5	and	and	CCONJ
ejpam-5411	455	6	the	the	DET
ejpam-5411	455	7	condition	condition	NOUN
ejpam-5411	455	8	iα(f2	iα(f2	NOUN
ejpam-5411	455	9	)	)	PUNCT
ejpam-5411	455	10	(	(	PUNCT
ejpam-5411	455	11	1	1	NUM
ejpam-5411	455	12	α2	α2	ADJ
ejpam-5411	455	13	)	)	PUNCT
ejpam-5411	456	1	=	=	SYM
ejpam-5411	456	2	0	0	NUM
ejpam-5411	456	3	,	,	PUNCT
ejpam-5411	456	4	we	we	PRON
ejpam-5411	456	5	have	have	VERB
ejpam-5411	456	6	fα(iα(f2)(t))(k	fα(iα(f2)(t))(k	NOUN
ejpam-5411	456	7	)	)	PUNCT
ejpam-5411	456	8	=	=	SYM
ejpam-5411	456	9	1	1	NUM
ejpam-5411	456	10	2ikπ	2ikπ	NUM
ejpam-5411	456	11	∫	∫	NOUN
ejpam-5411	456	12	1	1	NUM
ejpam-5411	456	13	α2	α2	NOUN
ejpam-5411	456	14	0	0	NUM
ejpam-5411	457	1	e−2ikπα2tf2((αt	e−2ikπα2tf2((αt	NOUN
ejpam-5411	457	2	)	)	PUNCT
ejpam-5411	457	3	1	1	NUM
ejpam-5411	457	4	α	α	NOUN
ejpam-5411	457	5	)	)	PUNCT
ejpam-5411	457	6	dt	dt	X
ejpam-5411	458	1	=	=	SYM
ejpam-5411	458	2	1	1	NUM
ejpam-5411	458	3	2ikπ	2ikπ	PROPN
ejpam-5411	458	4	[	[	PUNCT
ejpam-5411	458	5	∫	∫	PROPN
ejpam-5411	458	6	1	1	NUM
ejpam-5411	458	7	4α2	4α2	NOUN
ejpam-5411	458	8	0	0	PUNCT
ejpam-5411	459	1	te−2ikπα2tdt	te−2ikπα2tdt	PROPN
ejpam-5411	460	1	+	+	NUM
ejpam-5411	460	2	∫	∫	PROPN
ejpam-5411	460	3	3	3	NUM
ejpam-5411	460	4	4α2	4α2	NOUN
ejpam-5411	460	5	1	1	NUM
ejpam-5411	460	6	4α2	4α2	NOUN
ejpam-5411	460	7	(	(	PUNCT
ejpam-5411	460	8	1	1	NUM
ejpam-5411	460	9	2α2	2α2	NUM
ejpam-5411	460	10	−	−	NOUN
ejpam-5411	461	1	t)e−2ikπα2tdt	t)e−2ikπα2tdt	NOUN
ejpam-5411	461	2	]	]	X
ejpam-5411	462	1	+	+	CCONJ
ejpam-5411	462	2	1	1	NUM
ejpam-5411	462	3	2ikπ	2ikπ	NUM
ejpam-5411	462	4	∫	∫	NOUN
ejpam-5411	462	5	1	1	NUM
ejpam-5411	462	6	α2	α2	ADJ
ejpam-5411	462	7	3	3	NUM
ejpam-5411	462	8	4α2	4α2	NOUN
ejpam-5411	462	9	(	(	PUNCT
ejpam-5411	462	10	t−	t−	PROPN
ejpam-5411	462	11	1	1	NUM
ejpam-5411	462	12	α2	α2	ADJ
ejpam-5411	462	13	)	)	PUNCT
ejpam-5411	463	1	e−2ikπα2tdt	e−2ikπα2tdt	PROPN
ejpam-5411	463	2	=	=	PRON
ejpam-5411	464	1	(	(	PUNCT
ejpam-5411	464	2	−1)−	−1)−	NOUN
ejpam-5411	464	3	k	k	NOUN
ejpam-5411	464	4	2	2	NUM
ejpam-5411	464	5	(	(	PUNCT
ejpam-5411	464	6	1	1	NUM
ejpam-5411	464	7	−	−	PROPN
ejpam-5411	464	8	(	(	PUNCT
ejpam-5411	464	9	−1)−k	−1)−k	NOUN
ejpam-5411	464	10	)	)	PUNCT
ejpam-5411	464	11	4iα4k3π3	4iα4k3π3	NOUN
ejpam-5411	464	12	t.	t.	NOUN
ejpam-5411	464	13	abdeljawad	abdeljawad	PROPN
ejpam-5411	464	14	et	et	PROPN
ejpam-5411	464	15	al	al	PROPN
ejpam-5411	464	16	.	.	PUNCT
ejpam-5411	464	17	/	/	SYM
ejpam-5411	464	18	eur	eur	PROPN
ejpam-5411	464	19	.	.	PUNCT
ejpam-5411	465	1	j.	j.	PROPN
ejpam-5411	465	2	pure	pure	PROPN
ejpam-5411	465	3	appl	appl	PROPN
ejpam-5411	465	4	.	.	PROPN
ejpam-5411	465	5	math	math	PROPN
ejpam-5411	465	6	,	,	PUNCT
ejpam-5411	465	7	17	17	NUM
ejpam-5411	465	8	(	(	PUNCT
ejpam-5411	465	9	4	4	NUM
ejpam-5411	465	10	)	)	PUNCT
ejpam-5411	465	11	(	(	PUNCT
ejpam-5411	465	12	2024	2024	NUM
ejpam-5411	465	13	)	)	PUNCT
ejpam-5411	465	14	,	,	PUNCT
ejpam-5411	465	15	2405	2405	NUM
ejpam-5411	465	16	-	-	SYM
ejpam-5411	465	17	2430	2430	NUM
ejpam-5411	465	18	2421	2421	NUM
ejpam-5411	465	19	on	on	ADP
ejpam-5411	465	20	the	the	DET
ejpam-5411	465	21	other	other	ADJ
ejpam-5411	465	22	hand	hand	NOUN
ejpam-5411	465	23	,	,	PUNCT
ejpam-5411	465	24	by	by	ADP
ejpam-5411	465	25	example	example	NOUN
ejpam-5411	465	26	8	8	NUM
ejpam-5411	465	27	,	,	PUNCT
ejpam-5411	465	28	we	we	PRON
ejpam-5411	465	29	have	have	AUX
ejpam-5411	465	30	f(f2((αt	f(f2((αt	PROPN
ejpam-5411	465	31	)	)	PUNCT
ejpam-5411	465	32	1	1	NUM
ejpam-5411	465	33	α	α	NOUN
ejpam-5411	465	34	)	)	PUNCT
ejpam-5411	465	35	)	)	PUNCT
ejpam-5411	465	36	(	(	PUNCT
ejpam-5411	465	37	k	k	X
ejpam-5411	465	38	)	)	PUNCT
ejpam-5411	465	39	=	=	SYM
ejpam-5411	466	1	(	(	PUNCT
ejpam-5411	466	2	−1)−	−1)−	NOUN
ejpam-5411	466	3	k	k	NOUN
ejpam-5411	466	4	2	2	NUM
ejpam-5411	466	5	(	(	PUNCT
ejpam-5411	466	6	1	1	NUM
ejpam-5411	466	7	−	−	PROPN
ejpam-5411	466	8	(	(	PUNCT
ejpam-5411	466	9	−1)−k	−1)−k	NOUN
ejpam-5411	466	10	)	)	PUNCT
ejpam-5411	466	11	2α2k2π2	2α2k2π2	NUM
ejpam-5411	466	12	,	,	PUNCT
ejpam-5411	466	13	then	then	ADV
ejpam-5411	466	14	fα(iα(f2)(t))(k	fα(iα(f2)(t))(k	NOUN
ejpam-5411	466	15	)	)	PUNCT
ejpam-5411	466	16	=	=	SYM
ejpam-5411	466	17	1	1	NUM
ejpam-5411	466	18	2iα2kπ	2iα2kπ	NUM
ejpam-5411	466	19	f(f2((αt	f(f2((αt	NOUN
ejpam-5411	466	20	)	)	PUNCT
ejpam-5411	466	21	1	1	NUM
ejpam-5411	466	22	α	α	NOUN
ejpam-5411	466	23	)	)	PUNCT
ejpam-5411	466	24	)	)	PUNCT
ejpam-5411	466	25	(	(	PUNCT
ejpam-5411	466	26	k	k	NOUN
ejpam-5411	466	27	)	)	PUNCT
ejpam-5411	466	28	.	.	PUNCT
ejpam-5411	467	1	for	for	ADP
ejpam-5411	467	2	k	k	PROPN
ejpam-5411	467	3	=	=	SYM
ejpam-5411	467	4	0	0	PROPN
ejpam-5411	467	5	,	,	PUNCT
ejpam-5411	467	6	we	we	PRON
ejpam-5411	467	7	have	have	VERB
ejpam-5411	467	8	f((f2)α((αt	f((f2)α((αt	NOUN
ejpam-5411	467	9	)	)	PUNCT
ejpam-5411	467	10	1	1	NUM
ejpam-5411	467	11	α	α	NOUN
ejpam-5411	467	12	)	)	PUNCT
ejpam-5411	467	13	)	)	PUNCT
ejpam-5411	468	1	(	(	PUNCT
ejpam-5411	468	2	0	0	NUM
ejpam-5411	468	3	)	)	PUNCT
ejpam-5411	468	4	=	=	SYM
ejpam-5411	469	1	−α2	−α2	PROPN
ejpam-5411	469	2	∫	∫	PROPN
ejpam-5411	469	3	1	1	NUM
ejpam-5411	469	4	α2	α2	PROPN
ejpam-5411	469	5	0	0	NUM
ejpam-5411	469	6	tf2((αt	tf2((αt	NOUN
ejpam-5411	469	7	)	)	PUNCT
ejpam-5411	469	8	1	1	NUM
ejpam-5411	469	9	α	α	NOUN
ejpam-5411	469	10	)	)	PUNCT
ejpam-5411	469	11	dt	dt	PROPN
ejpam-5411	470	1	=	=	SYM
ejpam-5411	470	2	−α2	−α2	PROPN
ejpam-5411	470	3	[	[	PUNCT
ejpam-5411	470	4	∫	∫	PROPN
ejpam-5411	470	5	1	1	NUM
ejpam-5411	470	6	4α2	4α2	NOUN
ejpam-5411	470	7	0	0	NUM
ejpam-5411	470	8	t2dt	t2dt	PUNCT
ejpam-5411	471	1	+	+	CCONJ
ejpam-5411	471	2	∫	∫	PROPN
ejpam-5411	471	3	3	3	NUM
ejpam-5411	471	4	4α2	4α2	NUM
ejpam-5411	471	5	1	1	NUM
ejpam-5411	471	6	4α2	4α2	NOUN
ejpam-5411	471	7	t	t	PROPN
ejpam-5411	471	8	(	(	PUNCT
ejpam-5411	471	9	1	1	NUM
ejpam-5411	471	10	2α2	2α2	NUM
ejpam-5411	471	11	−	−	PROPN
ejpam-5411	472	1	t)dt	t)dt	PROPN
ejpam-5411	472	2	+	+	CCONJ
ejpam-5411	472	3	∫	∫	PROPN
ejpam-5411	472	4	1	1	NUM
ejpam-5411	472	5	α2	α2	ADJ
ejpam-5411	472	6	3	3	NUM
ejpam-5411	472	7	4α2	4α2	NOUN
ejpam-5411	472	8	t(t−	t(t−	NOUN
ejpam-5411	472	9	1	1	NUM
ejpam-5411	472	10	α2	α2	ADJ
ejpam-5411	472	11	)	)	PUNCT
ejpam-5411	472	12	dt	dt	X
ejpam-5411	472	13	]	]	X
ejpam-5411	472	14	=	=	SYM
ejpam-5411	472	15	1	1	NUM
ejpam-5411	472	16	32α4	32α4	NUM
ejpam-5411	472	17	.	.	PUNCT
ejpam-5411	473	1	on	on	ADP
ejpam-5411	473	2	the	the	DET
ejpam-5411	473	3	other	other	ADJ
ejpam-5411	473	4	hand	hand	NOUN
ejpam-5411	473	5	,	,	PUNCT
ejpam-5411	473	6	by	by	ADP
ejpam-5411	473	7	example	example	NOUN
ejpam-5411	473	8	4	4	NUM
ejpam-5411	473	9	,	,	PUNCT
ejpam-5411	473	10	we	we	PRON
ejpam-5411	473	11	have	have	VERB
ejpam-5411	473	12	fα(iα(f2)(t))(0	fα(iα(f2)(t))(0	ADJ
ejpam-5411	473	13	)	)	PUNCT
ejpam-5411	473	14	=	=	VERB
ejpam-5411	474	1	α2	α2	ADJ
ejpam-5411	474	2	∫	∫	PROPN
ejpam-5411	474	3	1	1	NUM
ejpam-5411	474	4	α2	α2	ADJ
ejpam-5411	474	5	0	0	NUM
ejpam-5411	474	6	iα(f2)((αt	iα(f2)((αt	NOUN
ejpam-5411	474	7	)	)	PUNCT
ejpam-5411	474	8	1	1	NUM
ejpam-5411	474	9	α	α	NOUN
ejpam-5411	474	10	)	)	PUNCT
ejpam-5411	474	11	dt	dt	NOUN
ejpam-5411	475	1	=	=	PUNCT
ejpam-5411	475	2	α2	α2	PROPN
ejpam-5411	475	3	{	{	PUNCT
ejpam-5411	475	4	∫	∫	PROPN
ejpam-5411	475	5	1	1	NUM
ejpam-5411	475	6	4α2	4α2	NOUN
ejpam-5411	475	7	0	0	NUM
ejpam-5411	475	8	t2	t2	PROPN
ejpam-5411	475	9	2	2	NUM
ejpam-5411	475	10	+	+	CCONJ
ejpam-5411	475	11	∫	∫	PROPN
ejpam-5411	475	12	3	3	NUM
ejpam-5411	475	13	4α2	4α2	NOUN
ejpam-5411	475	14	1	1	NUM
ejpam-5411	475	15	4α2	4α2	NOUN
ejpam-5411	475	16	(	(	PUNCT
ejpam-5411	475	17	−	−	PROPN
ejpam-5411	475	18	1	1	NUM
ejpam-5411	475	19	16α4	16α4	NUM
ejpam-5411	475	20	+	+	CCONJ
ejpam-5411	475	21	1	1	NUM
ejpam-5411	475	22	2α2	2α2	NUM
ejpam-5411	475	23	−	−	NOUN
ejpam-5411	475	24	t2	t2	NOUN
ejpam-5411	475	25	2	2	NUM
ejpam-5411	475	26	)	)	PUNCT
ejpam-5411	475	27	dt	dt	PUNCT
ejpam-5411	476	1	+	+	CCONJ
ejpam-5411	476	2	∫	∫	PROPN
ejpam-5411	476	3	1	1	NUM
ejpam-5411	476	4	α2	α2	ADJ
ejpam-5411	476	5	3	3	NUM
ejpam-5411	476	6	4α2	4α2	NOUN
ejpam-5411	476	7	(	(	PUNCT
ejpam-5411	476	8	t2	t2	PROPN
ejpam-5411	476	9	2	2	NUM
ejpam-5411	476	10	−	−	NOUN
ejpam-5411	476	11	t	t	NOUN
ejpam-5411	476	12	α2	α2	NOUN
ejpam-5411	476	13	+	+	CCONJ
ejpam-5411	476	14	1	1	NUM
ejpam-5411	476	15	2α4	2α4	NUM
ejpam-5411	476	16	)	)	PUNCT
ejpam-5411	476	17	dt	dt	PUNCT
ejpam-5411	476	18	}	}	PUNCT
ejpam-5411	476	19	=	=	SYM
ejpam-5411	476	20	1	1	NUM
ejpam-5411	476	21	32α4	32α4	NUM
ejpam-5411	476	22	.	.	PUNCT
ejpam-5411	477	1	then	then	ADV
ejpam-5411	477	2	fα(iα(f2)(t))(0	fα(iα(f2)(t))(0	ADJ
ejpam-5411	477	3	)	)	PUNCT
ejpam-5411	478	1	=	=	SYM
ejpam-5411	478	2	f(fα((αt	f(fα((αt	PROPN
ejpam-5411	478	3	)	)	PUNCT
ejpam-5411	478	4	1	1	NUM
ejpam-5411	478	5	α	α	NOUN
ejpam-5411	478	6	)	)	PUNCT
ejpam-5411	478	7	)	)	PUNCT
ejpam-5411	478	8	(	(	PUNCT
ejpam-5411	478	9	0	0	NUM
ejpam-5411	478	10	)	)	PUNCT
ejpam-5411	478	11	where	where	SCONJ
ejpam-5411	478	12	fα(t	fα(t	ADV
ejpam-5411	478	13	)	)	PUNCT
ejpam-5411	478	14	=	=	SYM
ejpam-5411	479	1	−	−	PROPN
ejpam-5411	479	2	tα	tα	PROPN
ejpam-5411	479	3	α	α	PRON
ejpam-5411	479	4	f(t	f(t	NOUN
ejpam-5411	479	5	)	)	PUNCT
ejpam-5411	479	6	.	.	PUNCT
ejpam-5411	480	1	in	in	ADP
ejpam-5411	480	2	order	order	NOUN
ejpam-5411	480	3	to	to	PART
ejpam-5411	480	4	establish	establish	VERB
ejpam-5411	480	5	a	a	DET
ejpam-5411	480	6	similar	similar	ADJ
ejpam-5411	480	7	relationship	relationship	NOUN
ejpam-5411	480	8	between	between	ADP
ejpam-5411	480	9	conformable	conformable	ADJ
ejpam-5411	480	10	fourier	fourier	NOUN
ejpam-5411	480	11	transform	transform	NOUN
ejpam-5411	480	12	and	and	CCONJ
ejpam-5411	480	13	conformable	conformable	ADJ
ejpam-5411	480	14	fractional	fractional	ADJ
ejpam-5411	480	15	derivative	derivative	NOUN
ejpam-5411	480	16	as	as	ADP
ejpam-5411	480	17	a	a	DET
ejpam-5411	480	18	classical	classical	ADJ
ejpam-5411	480	19	fourier	fourier	NOUN
ejpam-5411	480	20	transform	transform	NOUN
ejpam-5411	480	21	of	of	ADP
ejpam-5411	480	22	order	order	NOUN
ejpam-5411	480	23	α	α	NOUN
ejpam-5411	480	24	,	,	PUNCT
ejpam-5411	480	25	the	the	DET
ejpam-5411	480	26	following	follow	VERB
ejpam-5411	480	27	two	two	NUM
ejpam-5411	480	28	theorems	theorem	NOUN
ejpam-5411	480	29	are	be	AUX
ejpam-5411	480	30	obtained	obtain	VERB
ejpam-5411	480	31	.	.	PUNCT
ejpam-5411	481	1	theorem	theorem	ADJ
ejpam-5411	481	2	10	10	NUM
ejpam-5411	481	3	.	.	PUNCT
ejpam-5411	482	1	let	let	VERB
ejpam-5411	482	2	0	0	NUM
ejpam-5411	482	3	<	<	X
ejpam-5411	482	4	α	α	PROPN
ejpam-5411	482	5	≤	≤	NUM
ejpam-5411	482	6	1	1	NUM
ejpam-5411	482	7	,	,	PUNCT
ejpam-5411	482	8	and	and	CCONJ
ejpam-5411	482	9	assume	assume	VERB
ejpam-5411	482	10	that	that	SCONJ
ejpam-5411	482	11	f	f	X
ejpam-5411	482	12	:	:	PUNCT
ejpam-5411	483	1	[	[	X
ejpam-5411	483	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	483	3	r	r	NOUN
ejpam-5411	483	4	is	be	AUX
ejpam-5411	483	5	α	α	DET
ejpam-5411	483	6	-	-	ADJ
ejpam-5411	483	7	periodic	periodic	ADJ
ejpam-5411	483	8	function	function	NOUN
ejpam-5411	483	9	with	with	ADP
ejpam-5411	483	10	period	period	NOUN
ejpam-5411	483	11	p	p	NOUN
ejpam-5411	483	12	and	and	CCONJ
ejpam-5411	483	13	continuously	continuously	ADV
ejpam-5411	483	14	α	α	X
ejpam-5411	483	15	-	-	NOUN
ejpam-5411	483	16	differentiable	differentiable	ADJ
ejpam-5411	483	17	on	on	ADP
ejpam-5411	483	18	[	[	X
ejpam-5411	483	19	0,+∞	0,+∞	PROPN
ejpam-5411	483	20	[	[	X
ejpam-5411	483	21	.	.	PUNCT
ejpam-5411	484	1	then	then	ADV
ejpam-5411	484	2	t	t	PROPN
ejpam-5411	484	3	(	(	PUNCT
ejpam-5411	484	4	α)(f	α)(f	PROPN
ejpam-5411	484	5	)	)	PUNCT
ejpam-5411	484	6	is	be	AUX
ejpam-5411	484	7	α	α	DET
ejpam-5411	484	8	-	-	ADJ
ejpam-5411	484	9	periodic	periodic	ADJ
ejpam-5411	484	10	function	function	NOUN
ejpam-5411	484	11	with	with	ADP
ejpam-5411	484	12	period	period	NOUN
ejpam-5411	484	13	p	p	NOUN
ejpam-5411	484	14	and	and	CCONJ
ejpam-5411	484	15	for	for	ADP
ejpam-5411	484	16	all	all	DET
ejpam-5411	484	17	k	k	PROPN
ejpam-5411	484	18	∈	∈	PROPN
ejpam-5411	484	19	z	z	NOUN
ejpam-5411	484	20	:	:	PUNCT
ejpam-5411	484	21	fα(t	fα(t	X
ejpam-5411	484	22	(	(	PUNCT
ejpam-5411	484	23	α)(f)(t))(k	α)(f)(t))(k	X
ejpam-5411	484	24	)	)	PUNCT
ejpam-5411	484	25	=	=	SYM
ejpam-5411	485	1	(	(	PUNCT
ejpam-5411	485	2	2ikπ	2ikπ	NOUN
ejpam-5411	485	3	α	α	NOUN
ejpam-5411	485	4	pα	pα	NOUN
ejpam-5411	485	5	)	)	PUNCT
ejpam-5411	485	6	fα(f(t))(k	fα(f(t))(k	PROPN
ejpam-5411	485	7	)	)	PUNCT
ejpam-5411	485	8	proof	proof	NOUN
ejpam-5411	485	9	.	.	PUNCT
ejpam-5411	486	1	let	let	VERB
ejpam-5411	486	2	0	0	NUM
ejpam-5411	486	3	<	<	X
ejpam-5411	486	4	α	α	PROPN
ejpam-5411	486	5	≤	≤	NUM
ejpam-5411	486	6	1	1	NUM
ejpam-5411	486	7	,	,	PUNCT
ejpam-5411	486	8	f	f	PROPN
ejpam-5411	486	9	is	be	AUX
ejpam-5411	486	10	α	α	DET
ejpam-5411	486	11	-	-	ADJ
ejpam-5411	486	12	periodic	periodic	ADJ
ejpam-5411	486	13	function	function	NOUN
ejpam-5411	486	14	with	with	ADP
ejpam-5411	486	15	period	period	NOUN
ejpam-5411	486	16	p	p	NOUN
ejpam-5411	486	17	and	and	CCONJ
ejpam-5411	486	18	continuously	continuously	ADV
ejpam-5411	486	19	αdifferentiable	αdifferentiable	ADJ
ejpam-5411	486	20	on	on	ADP
ejpam-5411	486	21	[	[	X
ejpam-5411	486	22	0,+∞	0,+∞	PROPN
ejpam-5411	486	23	[	[	X
ejpam-5411	486	24	.	.	PUNCT
ejpam-5411	487	1	by	by	ADP
ejpam-5411	487	2	theorem	theorem	NOUN
ejpam-5411	487	3	5	5	NUM
ejpam-5411	487	4	,	,	PUNCT
ejpam-5411	487	5	t	t	PROPN
ejpam-5411	487	6	(	(	PUNCT
ejpam-5411	487	7	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	487	8	)	)	PUNCT
ejpam-5411	487	9	=	=	SYM
ejpam-5411	487	10	g′	g′	NOUN
ejpam-5411	487	11	(	(	PUNCT
ejpam-5411	487	12	t	t	PROPN
ejpam-5411	487	13	α	α	PROPN
ejpam-5411	487	14	α	α	PROPN
ejpam-5411	487	15	)	)	PUNCT
ejpam-5411	487	16	,	,	PUNCT
ejpam-5411	487	17	g	g	PROPN
ejpam-5411	487	18	∈	∈	PROPN
ejpam-5411	487	19	c1([0,+∞	c1([0,+∞	VERB
ejpam-5411	487	20	[	[	X
ejpam-5411	487	21	)	)	PUNCT
ejpam-5411	487	22	where	where	SCONJ
ejpam-5411	487	23	g(t	g(t	NOUN
ejpam-5411	487	24	)	)	PUNCT
ejpam-5411	487	25	=	=	SYM
ejpam-5411	487	26	f((αt	f((αt	NOUN
ejpam-5411	487	27	)	)	PUNCT
ejpam-5411	487	28	1	1	NUM
ejpam-5411	487	29	α	α	NOUN
ejpam-5411	487	30	)	)	PUNCT
ejpam-5411	487	31	and	and	CCONJ
ejpam-5411	487	32	t	t	PROPN
ejpam-5411	487	33	(	(	PUNCT
ejpam-5411	487	34	α)(f	α)(f	PROPN
ejpam-5411	487	35	)	)	PUNCT
ejpam-5411	487	36	is	be	AUX
ejpam-5411	487	37	α	α	DET
ejpam-5411	487	38	-	-	ADJ
ejpam-5411	487	39	periodic	periodic	ADJ
ejpam-5411	487	40	function	function	NOUN
ejpam-5411	487	41	with	with	ADP
ejpam-5411	487	42	period	period	NOUN
ejpam-5411	487	43	p.	p.	NOUN
ejpam-5411	487	44	for	for	ADP
ejpam-5411	487	45	k	k	PROPN
ejpam-5411	487	46	∈	∈	PROPN
ejpam-5411	487	47	z	z	PROPN
ejpam-5411	487	48	,	,	PUNCT
ejpam-5411	487	49	we	we	PRON
ejpam-5411	487	50	have	have	VERB
ejpam-5411	487	51	fα(t	fα(t	NOUN
ejpam-5411	487	52	(	(	PUNCT
ejpam-5411	487	53	α)(f)(t))(k	α)(f)(t))(k	X
ejpam-5411	487	54	)	)	PUNCT
ejpam-5411	487	55	=	=	SYM
ejpam-5411	488	1	f(t	f(t	PROPN
ejpam-5411	488	2	(	(	PUNCT
ejpam-5411	488	3	α)(f)((αt	α)(f)((αt	PROPN
ejpam-5411	488	4	)	)	PUNCT
ejpam-5411	488	5	1	1	NUM
ejpam-5411	488	6	α	α	NOUN
ejpam-5411	488	7	)	)	PUNCT
ejpam-5411	488	8	)	)	PUNCT
ejpam-5411	488	9	(	(	PUNCT
ejpam-5411	488	10	k	k	X
ejpam-5411	488	11	)	)	PUNCT
ejpam-5411	488	12	t.	t.	NOUN
ejpam-5411	488	13	abdeljawad	abdeljawad	NOUN
ejpam-5411	488	14	et	et	PROPN
ejpam-5411	488	15	al	al	PROPN
ejpam-5411	488	16	.	.	PUNCT
ejpam-5411	488	17	/	/	SYM
ejpam-5411	488	18	eur	eur	PROPN
ejpam-5411	488	19	.	.	PUNCT
ejpam-5411	489	1	j.	j.	PROPN
ejpam-5411	489	2	pure	pure	PROPN
ejpam-5411	489	3	appl	appl	PROPN
ejpam-5411	489	4	.	.	PROPN
ejpam-5411	489	5	math	math	PROPN
ejpam-5411	489	6	,	,	PUNCT
ejpam-5411	489	7	17	17	NUM
ejpam-5411	489	8	(	(	PUNCT
ejpam-5411	489	9	4	4	NUM
ejpam-5411	489	10	)	)	PUNCT
ejpam-5411	489	11	(	(	PUNCT
ejpam-5411	489	12	2024	2024	NUM
ejpam-5411	489	13	)	)	PUNCT
ejpam-5411	489	14	,	,	PUNCT
ejpam-5411	489	15	2405	2405	NUM
ejpam-5411	489	16	-	-	SYM
ejpam-5411	489	17	2430	2430	NUM
ejpam-5411	489	18	2422	2422	NUM
ejpam-5411	489	19	=	=	SYM
ejpam-5411	490	1	α	α	PRON
ejpam-5411	490	2	pα	pα	INTJ
ejpam-5411	490	3	∫	∫	PROPN
ejpam-5411	490	4	pα	pα	INTJ
ejpam-5411	490	5	α	α	PROPN
ejpam-5411	490	6	0	0	PUNCT
ejpam-5411	491	1	e	e	NOUN
ejpam-5411	491	2	−2ikπ	−2ikπ	NOUN
ejpam-5411	491	3	α	α	PROPN
ejpam-5411	491	4	pα	pα	INTJ
ejpam-5411	491	5	t	t	PROPN
ejpam-5411	491	6	t	t	PROPN
ejpam-5411	491	7	(	(	PUNCT
ejpam-5411	491	8	α)(f)((αt	α)(f)((αt	PROPN
ejpam-5411	491	9	)	)	PUNCT
ejpam-5411	491	10	1	1	NUM
ejpam-5411	491	11	α	α	NOUN
ejpam-5411	491	12	)	)	PUNCT
ejpam-5411	491	13	dt	dt	NOUN
ejpam-5411	492	1	=	=	SYM
ejpam-5411	492	2	α	α	INTJ
ejpam-5411	492	3	pα	pα	INTJ
ejpam-5411	492	4	∫	∫	PROPN
ejpam-5411	492	5	pα	pα	INTJ
ejpam-5411	492	6	α	α	PROPN
ejpam-5411	492	7	0	0	PUNCT
ejpam-5411	493	1	e	e	NOUN
ejpam-5411	493	2	−2ikπ	−2ikπ	NOUN
ejpam-5411	493	3	α	α	PROPN
ejpam-5411	493	4	pα	pα	NOUN
ejpam-5411	493	5	t	t	PROPN
ejpam-5411	493	6	g′(t)dt	g′(t)dt	PROPN
ejpam-5411	493	7	using	use	VERB
ejpam-5411	493	8	integration	integration	NOUN
ejpam-5411	493	9	by	by	ADP
ejpam-5411	493	10	parts	part	NOUN
ejpam-5411	493	11	the	the	DET
ejpam-5411	493	12	periodicity	periodicity	NOUN
ejpam-5411	493	13	of	of	ADP
ejpam-5411	493	14	g	g	PROPN
ejpam-5411	493	15	,	,	PUNCT
ejpam-5411	493	16	we	we	PRON
ejpam-5411	493	17	have	have	AUX
ejpam-5411	493	18	fα(t	fα(t	NOUN
ejpam-5411	493	19	(	(	PUNCT
ejpam-5411	493	20	α)(f)(t)(k	α)(f)(t)(k	ADJ
ejpam-5411	493	21	)	)	PUNCT
ejpam-5411	493	22	=	=	SYM
ejpam-5411	494	1	(	(	PUNCT
ejpam-5411	494	2	2ikπ	2ikπ	NOUN
ejpam-5411	494	3	α	α	NOUN
ejpam-5411	494	4	pα	pα	NOUN
ejpam-5411	494	5	)	)	PUNCT
ejpam-5411	494	6	fα(f(t))(k	fα(f(t))(k	PROPN
ejpam-5411	494	7	)	)	PUNCT
ejpam-5411	494	8	.	.	PUNCT
ejpam-5411	495	1	example	example	NOUN
ejpam-5411	496	1	10	10	NUM
ejpam-5411	496	2	.	.	PUNCT
ejpam-5411	497	1	consider	consider	VERB
ejpam-5411	497	2	the	the	DET
ejpam-5411	497	3	same	same	ADJ
ejpam-5411	497	4	function	function	NOUN
ejpam-5411	497	5	from	from	ADP
ejpam-5411	497	6	example	example	NOUN
ejpam-5411	497	7	5	5	NUM
ejpam-5411	497	8	,	,	PUNCT
ejpam-5411	497	9	then	then	ADV
ejpam-5411	497	10	f	f	PROPN
ejpam-5411	497	11	is	be	AUX
ejpam-5411	497	12	α	α	DET
ejpam-5411	497	13	-	-	ADJ
ejpam-5411	497	14	periodic	periodic	ADJ
ejpam-5411	497	15	function	function	NOUN
ejpam-5411	497	16	with	with	ADP
ejpam-5411	497	17	period	period	NOUN
ejpam-5411	497	18	(	(	PUNCT
ejpam-5411	497	19	3π2α	3π2α	NUM
ejpam-5411	497	20	)	)	PUNCT
ejpam-5411	497	21	1	1	NUM
ejpam-5411	497	22	α	α	NOUN
ejpam-5411	497	23	and	and	CCONJ
ejpam-5411	497	24	continuously	continuously	ADV
ejpam-5411	497	25	α	α	X
ejpam-5411	497	26	-	-	NOUN
ejpam-5411	497	27	differentiable	differentiable	ADJ
ejpam-5411	497	28	on	on	ADP
ejpam-5411	497	29	[	[	X
ejpam-5411	497	30	0,+∞	0,+∞	PROPN
ejpam-5411	497	31	[	[	X
ejpam-5411	497	32	.	.	PUNCT
ejpam-5411	498	1	by	by	ADP
ejpam-5411	498	2	theorem	theorem	NOUN
ejpam-5411	498	3	5	5	NUM
ejpam-5411	498	4	,	,	PUNCT
ejpam-5411	498	5	t	t	PROPN
ejpam-5411	498	6	(	(	PUNCT
ejpam-5411	498	7	α)(f)(t	α)(f)(t	PROPN
ejpam-5411	498	8	)	)	PUNCT
ejpam-5411	498	9	=	=	SYM
ejpam-5411	498	10	g′	g′	NOUN
ejpam-5411	498	11	(	(	PUNCT
ejpam-5411	498	12	t	t	PROPN
ejpam-5411	498	13	α	α	PROPN
ejpam-5411	498	14	α	α	PROPN
ejpam-5411	498	15	)	)	PUNCT
ejpam-5411	498	16	,	,	PUNCT
ejpam-5411	498	17	g	g	PROPN
ejpam-5411	498	18	∈	∈	PROPN
ejpam-5411	498	19	c1([0,+∞	c1([0,+∞	VERB
ejpam-5411	498	20	[	[	X
ejpam-5411	498	21	)	)	PUNCT
ejpam-5411	498	22	where	where	SCONJ
ejpam-5411	498	23	g(t	g(t	NOUN
ejpam-5411	498	24	)	)	PUNCT
ejpam-5411	498	25	=	=	SYM
ejpam-5411	498	26	f((αt	f((αt	NOUN
ejpam-5411	498	27	)	)	PUNCT
ejpam-5411	498	28	1	1	NUM
ejpam-5411	498	29	α	α	NOUN
ejpam-5411	498	30	)	)	PUNCT
ejpam-5411	498	31	and	and	CCONJ
ejpam-5411	498	32	t	t	PROPN
ejpam-5411	498	33	(	(	PUNCT
ejpam-5411	498	34	α)(f	α)(f	PROPN
ejpam-5411	498	35	)	)	PUNCT
ejpam-5411	498	36	is	be	AUX
ejpam-5411	498	37	α	α	DET
ejpam-5411	498	38	-	-	ADJ
ejpam-5411	498	39	periodic	periodic	ADJ
ejpam-5411	498	40	function	function	NOUN
ejpam-5411	498	41	with	with	ADP
ejpam-5411	498	42	period	period	NOUN
ejpam-5411	498	43	(	(	PUNCT
ejpam-5411	498	44	3π2α	3π2α	NUM
ejpam-5411	498	45	)	)	PUNCT
ejpam-5411	498	46	1	1	NUM
ejpam-5411	498	47	α	α	NOUN
ejpam-5411	498	48	.	.	PUNCT
ejpam-5411	499	1	for	for	ADP
ejpam-5411	499	2	all	all	DET
ejpam-5411	499	3	k	k	PROPN
ejpam-5411	499	4	∈	∈	PROPN
ejpam-5411	499	5	z	z	PROPN
ejpam-5411	499	6	,	,	PUNCT
ejpam-5411	499	7	we	we	PRON
ejpam-5411	499	8	have	have	VERB
ejpam-5411	499	9	fα(f(t))(k	fα(f(t))(k	NOUN
ejpam-5411	499	10	)	)	PUNCT
ejpam-5411	499	11	=	=	PUNCT
ejpam-5411	500	1	f(f((αt	f(f((αt	PROPN
ejpam-5411	500	2	)	)	PUNCT
ejpam-5411	500	3	1	1	NUM
ejpam-5411	500	4	α	α	NOUN
ejpam-5411	500	5	)	)	PUNCT
ejpam-5411	500	6	)	)	PUNCT
ejpam-5411	501	1	(	(	PUNCT
ejpam-5411	501	2	k	k	X
ejpam-5411	501	3	)	)	PUNCT
ejpam-5411	501	4	=	=	SYM
ejpam-5411	501	5	2α2	2α2	NUM
ejpam-5411	501	6	3π	3π	NOUN
ejpam-5411	501	7	∫	∫	PROPN
ejpam-5411	501	8	3π	3π	NOUN
ejpam-5411	501	9	2α2	2α2	NUM
ejpam-5411	501	10	0	0	NUM
ejpam-5411	502	1	e−	e−	PROPN
ejpam-5411	502	2	4	4	NUM
ejpam-5411	502	3	3	3	NUM
ejpam-5411	502	4	ikα2tf((αt	ikα2tf((αt	NOUN
ejpam-5411	502	5	)	)	PUNCT
ejpam-5411	502	6	1	1	NUM
ejpam-5411	502	7	α	α	NOUN
ejpam-5411	502	8	)	)	PUNCT
ejpam-5411	502	9	dt	dt	X
ejpam-5411	503	1	=	=	NOUN
ejpam-5411	503	2	2α2	2α2	NUM
ejpam-5411	503	3	3π	3π	NOUN
ejpam-5411	503	4	[	[	PUNCT
ejpam-5411	503	5	∫	∫	PROPN
ejpam-5411	503	6	π	π	PROPN
ejpam-5411	503	7	α2	α2	PROPN
ejpam-5411	503	8	0	0	PUNCT
ejpam-5411	504	1	sin(α2t)e−	sin(α2t)e−	NUM
ejpam-5411	504	2	4	4	NUM
ejpam-5411	504	3	3	3	NUM
ejpam-5411	504	4	ikα2tdt−	ikα2tdt−	ADJ
ejpam-5411	504	5	1	1	NUM
ejpam-5411	504	6	2	2	NUM
ejpam-5411	504	7	∫	∫	NOUN
ejpam-5411	504	8	3π	3π	NOUN
ejpam-5411	504	9	2α2	2α2	NUM
ejpam-5411	504	10	π	π	PROPN
ejpam-5411	504	11	α2	α2	PROPN
ejpam-5411	504	12	sin(2α2t)e−	sin(2α2t)e−	PROPN
ejpam-5411	504	13	4	4	NUM
ejpam-5411	504	14	3	3	NUM
ejpam-5411	504	15	ikα2tdt	ikα2tdt	NOUN
ejpam-5411	504	16	]	]	X
ejpam-5411	504	17	=	=	SYM
ejpam-5411	504	18	81((−1)−	81((−1)−	NUM
ejpam-5411	504	19	4k	4k	NOUN
ejpam-5411	504	20	3	3	NUM
ejpam-5411	504	21	+	+	CCONJ
ejpam-5411	504	22	1	1	NUM
ejpam-5411	504	23	)	)	PUNCT
ejpam-5411	504	24	2π(64k4	2π(64k4	NUM
ejpam-5411	504	25	−	−	PROPN
ejpam-5411	504	26	180k2	180k2	NUM
ejpam-5411	504	27	+	+	NUM
ejpam-5411	504	28	81	81	NUM
ejpam-5411	504	29	)	)	PUNCT
ejpam-5411	504	30	.	.	PUNCT
ejpam-5411	505	1	on	on	ADP
ejpam-5411	505	2	the	the	DET
ejpam-5411	505	3	other	other	ADJ
ejpam-5411	505	4	hand	hand	NOUN
ejpam-5411	505	5	fα(t	fα(t	X
ejpam-5411	505	6	(	(	PUNCT
ejpam-5411	505	7	α)(f)(t))(k	α)(f)(t))(k	X
ejpam-5411	505	8	)	)	PUNCT
ejpam-5411	505	9	=	=	SYM
ejpam-5411	506	1	f(t	f(t	PROPN
ejpam-5411	506	2	(	(	PUNCT
ejpam-5411	506	3	α)(f)((αt	α)(f)((αt	PROPN
ejpam-5411	506	4	)	)	PUNCT
ejpam-5411	506	5	1	1	NUM
ejpam-5411	506	6	α	α	NOUN
ejpam-5411	506	7	)	)	PUNCT
ejpam-5411	506	8	(	(	PUNCT
ejpam-5411	506	9	k	k	X
ejpam-5411	506	10	)	)	PUNCT
ejpam-5411	506	11	=	=	SYM
ejpam-5411	506	12	2α2	2α2	NUM
ejpam-5411	506	13	3π	3π	NOUN
ejpam-5411	506	14	∫	∫	PROPN
ejpam-5411	506	15	3π	3π	NOUN
ejpam-5411	506	16	2α2	2α2	NUM
ejpam-5411	506	17	0	0	NUM
ejpam-5411	507	1	e−	e−	PROPN
ejpam-5411	507	2	4	4	NUM
ejpam-5411	507	3	3	3	NUM
ejpam-5411	507	4	ikα2tt	ikα2tt	NOUN
ejpam-5411	507	5	(	(	PUNCT
ejpam-5411	507	6	α)(f)((αt	α)(f)((αt	PROPN
ejpam-5411	507	7	)	)	PUNCT
ejpam-5411	507	8	1	1	NUM
ejpam-5411	507	9	α	α	NOUN
ejpam-5411	507	10	)	)	PUNCT
ejpam-5411	507	11	dt	dt	X
ejpam-5411	508	1	=	=	NOUN
ejpam-5411	508	2	2α2	2α2	NUM
ejpam-5411	508	3	3π	3π	NOUN
ejpam-5411	508	4	[	[	PUNCT
ejpam-5411	508	5	∫	∫	PROPN
ejpam-5411	508	6	π	π	PROPN
ejpam-5411	508	7	α2	α2	PROPN
ejpam-5411	508	8	0	0	NUM
ejpam-5411	508	9	α2	α2	PROPN
ejpam-5411	508	10	cos(α2t)e−	cos(α2t)e−	PROPN
ejpam-5411	508	11	4	4	NUM
ejpam-5411	508	12	3	3	NUM
ejpam-5411	508	13	ikα2tdt−	ikα2tdt−	ADJ
ejpam-5411	508	14	∫	∫	PROPN
ejpam-5411	508	15	3π	3π	NOUN
ejpam-5411	508	16	2α2	2α2	NUM
ejpam-5411	508	17	π	π	PROPN
ejpam-5411	508	18	α2	α2	ADJ
ejpam-5411	508	19	α2	α2	PROPN
ejpam-5411	508	20	cos(2α2t)e−	cos(2α2t)e−	PROPN
ejpam-5411	508	21	4	4	NUM
ejpam-5411	508	22	3	3	NUM
ejpam-5411	508	23	ikα2tdt	ikα2tdt	NOUN
ejpam-5411	508	24	]	]	X
ejpam-5411	508	25	=	=	SYM
ejpam-5411	508	26	54ikα2((−1)−	54ikα2((−1)−	NUM
ejpam-5411	508	27	4k	4k	NOUN
ejpam-5411	508	28	3	3	NUM
ejpam-5411	508	29	+	+	CCONJ
ejpam-5411	508	30	1	1	NUM
ejpam-5411	508	31	)	)	PUNCT
ejpam-5411	508	32	π(64k4	π(64k4	NOUN
ejpam-5411	508	33	−	−	PROPN
ejpam-5411	508	34	180k2	180k2	NUM
ejpam-5411	508	35	+	+	SYM
ejpam-5411	508	36	81	81	NUM
ejpam-5411	508	37	)	)	PUNCT
ejpam-5411	508	38	then	then	ADV
ejpam-5411	508	39	fα(t	fα(t	X
ejpam-5411	508	40	(	(	PUNCT
ejpam-5411	508	41	α)(f)(t))(k	α)(f)(t))(k	X
ejpam-5411	508	42	)	)	PUNCT
ejpam-5411	508	43	=	=	PUNCT
ejpam-5411	508	44	(	(	PUNCT
ejpam-5411	508	45	4	4	NUM
ejpam-5411	508	46	3	3	NUM
ejpam-5411	508	47	ikα2)fα(f(t))(k	ikα2)fα(f(t))(k	NOUN
ejpam-5411	508	48	)	)	PUNCT
ejpam-5411	508	49	.	.	PUNCT
ejpam-5411	509	1	theorem	theorem	VERB
ejpam-5411	509	2	11	11	NUM
ejpam-5411	509	3	.	.	PUNCT
ejpam-5411	510	1	let	let	VERB
ejpam-5411	510	2	0	0	NUM
ejpam-5411	510	3	<	<	X
ejpam-5411	510	4	α	α	PROPN
ejpam-5411	510	5	≤	≤	NUM
ejpam-5411	510	6	1	1	NUM
ejpam-5411	510	7	,	,	PUNCT
ejpam-5411	510	8	n	n	PRON
ejpam-5411	510	9	∈	∈	NOUN
ejpam-5411	510	10	n	n	NOUN
ejpam-5411	510	11	and	and	CCONJ
ejpam-5411	510	12	assume	assume	VERB
ejpam-5411	510	13	that	that	SCONJ
ejpam-5411	510	14	the	the	DET
ejpam-5411	510	15	function	function	NOUN
ejpam-5411	510	16	f	f	NOUN
ejpam-5411	510	17	:	:	PUNCT
ejpam-5411	511	1	[	[	X
ejpam-5411	511	2	0,+∞[→	0,+∞[→	ADJ
ejpam-5411	511	3	r	r	NOUN
ejpam-5411	511	4	is	be	AUX
ejpam-5411	511	5	α	α	NOUN
ejpam-5411	511	6	-	-	NOUN
ejpam-5411	511	7	periodic	periodic	NOUN
ejpam-5411	511	8	with	with	ADP
ejpam-5411	511	9	period	period	NOUN
ejpam-5411	511	10	p	p	NOUN
ejpam-5411	511	11	and	and	CCONJ
ejpam-5411	511	12	n	n	PRON
ejpam-5411	511	13	times	time	NOUN
ejpam-5411	511	14	continuously	continuously	ADV
ejpam-5411	511	15	α	α	VERB
ejpam-5411	511	16	-	-	NOUN
ejpam-5411	511	17	differentiable	differentiable	ADJ
ejpam-5411	511	18	on	on	ADP
ejpam-5411	511	19	[	[	X
ejpam-5411	511	20	0,+∞	0,+∞	PROPN
ejpam-5411	511	21	[	[	X
ejpam-5411	511	22	.	.	PUNCT
ejpam-5411	512	1	then	then	ADV
ejpam-5411	512	2	for	for	ADP
ejpam-5411	512	3	all	all	DET
ejpam-5411	512	4	j	j	PROPN
ejpam-5411	512	5	∈	∈	PROPN
ejpam-5411	512	6	{	{	PUNCT
ejpam-5411	512	7	0	0	NUM
ejpam-5411	512	8	,	,	PUNCT
ejpam-5411	512	9	.	.	PUNCT
ejpam-5411	512	10	.	.	PUNCT
ejpam-5411	512	11	.	.	PUNCT
ejpam-5411	512	12	,	,	PUNCT
ejpam-5411	512	13	n	n	CCONJ
ejpam-5411	512	14	}	}	PUNCT
ejpam-5411	512	15	,	,	PUNCT
ejpam-5411	512	16	t	t	PROPN
ejpam-5411	512	17	(	(	PUNCT
ejpam-5411	512	18	jα)(f	jα)(f	PROPN
ejpam-5411	512	19	)	)	PUNCT
ejpam-5411	512	20	is	be	AUX
ejpam-5411	512	21	α	α	X
ejpam-5411	512	22	-	-	NOUN
ejpam-5411	512	23	periodic	periodic	NOUN
ejpam-5411	512	24	with	with	ADP
ejpam-5411	512	25	period	period	NOUN
ejpam-5411	512	26	p	p	NOUN
ejpam-5411	512	27	and	and	CCONJ
ejpam-5411	512	28	for	for	ADP
ejpam-5411	512	29	k	k	PROPN
ejpam-5411	512	30	∈	∈	PROPN
ejpam-5411	512	31	z	z	PROPN
ejpam-5411	512	32	fα(t	fα(t	X
ejpam-5411	512	33	(	(	PUNCT
ejpam-5411	512	34	jα)(f)(t))(k	jα)(f)(t))(k	PROPN
ejpam-5411	512	35	)	)	PUNCT
ejpam-5411	512	36	=	=	PUNCT
ejpam-5411	513	1	(	(	PUNCT
ejpam-5411	513	2	2ikπ	2ikπ	NOUN
ejpam-5411	513	3	α	α	NOUN
ejpam-5411	513	4	pα	pα	NOUN
ejpam-5411	513	5	)	)	PUNCT
ejpam-5411	513	6	jfα(f(t))(k	jfα(f(t))(k	PROPN
ejpam-5411	513	7	)	)	PUNCT
ejpam-5411	513	8	.	.	PUNCT
ejpam-5411	514	1	note	note	VERB
ejpam-5411	514	2	that	that	SCONJ
ejpam-5411	514	3	t	t	PROPN
ejpam-5411	514	4	(	(	PUNCT
ejpam-5411	514	5	0)f(t	0)f(t	PROPN
ejpam-5411	514	6	)	)	PUNCT
ejpam-5411	514	7	=	=	SYM
ejpam-5411	514	8	f(t	f(t	NOUN
ejpam-5411	514	9	)	)	PUNCT
ejpam-5411	514	10	.	.	PUNCT
ejpam-5411	515	1	t.	t.	PROPN
ejpam-5411	515	2	abdeljawad	abdeljawad	PROPN
ejpam-5411	515	3	et	et	PROPN
ejpam-5411	515	4	al	al	PROPN
ejpam-5411	515	5	.	.	PUNCT
ejpam-5411	515	6	/	/	SYM
ejpam-5411	515	7	eur	eur	PROPN
ejpam-5411	515	8	.	.	PUNCT
ejpam-5411	516	1	j.	j.	PROPN
ejpam-5411	516	2	pure	pure	PROPN
ejpam-5411	516	3	appl	appl	PROPN
ejpam-5411	516	4	.	.	PROPN
ejpam-5411	516	5	math	math	PROPN
ejpam-5411	516	6	,	,	PUNCT
ejpam-5411	516	7	17	17	NUM
ejpam-5411	516	8	(	(	PUNCT
ejpam-5411	516	9	4	4	NUM
ejpam-5411	516	10	)	)	PUNCT
ejpam-5411	516	11	(	(	PUNCT
ejpam-5411	516	12	2024	2024	NUM
ejpam-5411	516	13	)	)	PUNCT
ejpam-5411	516	14	,	,	PUNCT
ejpam-5411	516	15	2405	2405	NUM
ejpam-5411	516	16	-	-	SYM
ejpam-5411	516	17	2430	2430	NUM
ejpam-5411	516	18	2423	2423	NUM
ejpam-5411	516	19	proof	proof	NOUN
ejpam-5411	516	20	.	.	PUNCT
ejpam-5411	517	1	let	let	VERB
ejpam-5411	517	2	0	0	NUM
ejpam-5411	517	3	<	<	X
ejpam-5411	517	4	α	α	PROPN
ejpam-5411	517	5	≤	≤	NUM
ejpam-5411	517	6	1	1	NUM
ejpam-5411	517	7	,	,	PUNCT
ejpam-5411	517	8	n	n	PRON
ejpam-5411	517	9	∈	∈	NOUN
ejpam-5411	517	10	n	n	NOUN
ejpam-5411	517	11	and	and	CCONJ
ejpam-5411	517	12	assume	assume	VERB
ejpam-5411	517	13	that	that	SCONJ
ejpam-5411	517	14	the	the	DET
ejpam-5411	517	15	function	function	NOUN
ejpam-5411	517	16	f	f	PROPN
ejpam-5411	517	17	is	be	AUX
ejpam-5411	517	18	α	α	NOUN
ejpam-5411	517	19	-	-	NOUN
ejpam-5411	517	20	periodic	periodic	NOUN
ejpam-5411	517	21	with	with	ADP
ejpam-5411	517	22	period	period	NOUN
ejpam-5411	517	23	p	p	NOUN
ejpam-5411	517	24	and	and	CCONJ
ejpam-5411	517	25	n	n	PRON
ejpam-5411	517	26	times	time	NOUN
ejpam-5411	517	27	continuously	continuously	ADV
ejpam-5411	517	28	α	α	VERB
ejpam-5411	517	29	-	-	NOUN
ejpam-5411	517	30	differentiable	differentiable	ADJ
ejpam-5411	517	31	on	on	ADP
ejpam-5411	517	32	[	[	X
ejpam-5411	517	33	0,+∞	0,+∞	PROPN
ejpam-5411	517	34	[	[	X
ejpam-5411	517	35	.	.	PUNCT
ejpam-5411	518	1	then	then	ADV
ejpam-5411	518	2	by	by	ADP
ejpam-5411	518	3	theorem	theorem	NOUN
ejpam-5411	518	4	6	6	NUM
ejpam-5411	518	5	,	,	PUNCT
ejpam-5411	518	6	we	we	PRON
ejpam-5411	518	7	have	have	VERB
ejpam-5411	518	8	for	for	ADP
ejpam-5411	518	9	all	all	DET
ejpam-5411	518	10	j	j	PROPN
ejpam-5411	518	11	∈	∈	PROPN
ejpam-5411	518	12	{	{	PUNCT
ejpam-5411	518	13	0	0	NUM
ejpam-5411	518	14	,	,	PUNCT
ejpam-5411	518	15	.	.	PUNCT
ejpam-5411	518	16	.	.	PUNCT
ejpam-5411	519	1	.	.	PUNCT
ejpam-5411	520	1	,	,	PUNCT
ejpam-5411	521	1	n	n	CCONJ
ejpam-5411	521	2	}	}	PUNCT
ejpam-5411	521	3	,	,	PUNCT
ejpam-5411	521	4	t	t	PROPN
ejpam-5411	521	5	(	(	PUNCT
ejpam-5411	521	6	jα)(f)(t	jα)(f)(t	PROPN
ejpam-5411	521	7	)	)	PUNCT
ejpam-5411	521	8	=	=	SYM
ejpam-5411	521	9	g(j	g(j	PROPN
ejpam-5411	521	10	)	)	PUNCT
ejpam-5411	521	11	(	(	PUNCT
ejpam-5411	521	12	t	t	PROPN
ejpam-5411	521	13	α	α	PROPN
ejpam-5411	521	14	α	α	PROPN
ejpam-5411	521	15	)	)	PUNCT
ejpam-5411	521	16	,	,	PUNCT
ejpam-5411	521	17	g	g	PROPN
ejpam-5411	521	18	∈	∈	PROPN
ejpam-5411	522	1	cj([0,+∞	cj([0,+∞	ADJ
ejpam-5411	522	2	[	[	X
ejpam-5411	522	3	)	)	PUNCT
ejpam-5411	522	4	where	where	SCONJ
ejpam-5411	522	5	g(t	g(t	NOUN
ejpam-5411	522	6	)	)	PUNCT
ejpam-5411	522	7	=	=	SYM
ejpam-5411	522	8	f((αt	f((αt	NOUN
ejpam-5411	522	9	)	)	PUNCT
ejpam-5411	522	10	1	1	NUM
ejpam-5411	522	11	α	α	NOUN
ejpam-5411	522	12	)	)	PUNCT
ejpam-5411	522	13	,	,	PUNCT
ejpam-5411	522	14	and	and	CCONJ
ejpam-5411	522	15	t	t	PROPN
ejpam-5411	522	16	(	(	PUNCT
ejpam-5411	522	17	jα)(f	jα)(f	PROPN
ejpam-5411	522	18	)	)	PUNCT
ejpam-5411	522	19	is	be	AUX
ejpam-5411	522	20	α	α	DET
ejpam-5411	522	21	-	-	ADJ
ejpam-5411	522	22	periodic	periodic	ADJ
ejpam-5411	522	23	function	function	NOUN
ejpam-5411	522	24	with	with	ADP
ejpam-5411	522	25	period	period	NOUN
ejpam-5411	523	1	p.	p.	NOUN
ejpam-5411	523	2	let	let	VERB
ejpam-5411	523	3	j	j	PROPN
ejpam-5411	523	4	∈	∈	PROPN
ejpam-5411	523	5	{	{	PUNCT
ejpam-5411	523	6	0	0	NUM
ejpam-5411	523	7	,	,	PUNCT
ejpam-5411	523	8	.	.	PUNCT
ejpam-5411	523	9	.	.	PUNCT
ejpam-5411	524	1	.	.	PUNCT
ejpam-5411	525	1	,	,	PUNCT
ejpam-5411	525	2	n	n	CCONJ
ejpam-5411	525	3	}	}	PUNCT
ejpam-5411	525	4	,	,	PUNCT
ejpam-5411	525	5	by	by	ADP
ejpam-5411	525	6	recurence	recurence	NOUN
ejpam-5411	525	7	.	.	PUNCT
ejpam-5411	526	1	for	for	ADP
ejpam-5411	526	2	j	j	PROPN
ejpam-5411	526	3	=	=	SYM
ejpam-5411	526	4	0	0	PROPN
ejpam-5411	526	5	,	,	PUNCT
ejpam-5411	526	6	the	the	DET
ejpam-5411	526	7	property	property	NOUN
ejpam-5411	526	8	is	be	AUX
ejpam-5411	526	9	true	true	ADJ
ejpam-5411	526	10	.	.	PUNCT
ejpam-5411	527	1	for	for	ADP
ejpam-5411	527	2	j	j	PROPN
ejpam-5411	527	3	=	=	SYM
ejpam-5411	527	4	1	1	NUM
ejpam-5411	527	5	,	,	PUNCT
ejpam-5411	527	6	the	the	DET
ejpam-5411	527	7	property	property	NOUN
ejpam-5411	527	8	is	be	AUX
ejpam-5411	527	9	true	true	ADJ
ejpam-5411	527	10	(	(	PUNCT
ejpam-5411	527	11	see	see	VERB
ejpam-5411	527	12	theorem	theorem	NOUN
ejpam-5411	527	13	10	10	NUM
ejpam-5411	527	14	)	)	PUNCT
ejpam-5411	527	15	.	.	PUNCT
ejpam-5411	528	1	suppose	suppose	VERB
ejpam-5411	528	2	that	that	SCONJ
ejpam-5411	528	3	fα(t	fα(t	NOUN
ejpam-5411	528	4	(	(	PUNCT
ejpam-5411	528	5	(	(	PUNCT
ejpam-5411	528	6	j−1)α)(f)(t))(k	j−1)α)(f)(t))(k	NOUN
ejpam-5411	528	7	)	)	PUNCT
ejpam-5411	528	8	=	=	NOUN
ejpam-5411	528	9	(	(	PUNCT
ejpam-5411	528	10	2ikπ	2ikπ	NOUN
ejpam-5411	528	11	α	α	NOUN
ejpam-5411	528	12	pα	pα	NOUN
ejpam-5411	528	13	)	)	PUNCT
ejpam-5411	528	14	j−1f(f((αt	j−1f(f((αt	PROPN
ejpam-5411	528	15	)	)	PUNCT
ejpam-5411	528	16	1	1	NUM
ejpam-5411	528	17	α	α	NOUN
ejpam-5411	528	18	)	)	PUNCT
ejpam-5411	528	19	)	)	PUNCT
ejpam-5411	528	20	(	(	PUNCT
ejpam-5411	528	21	k	k	X
ejpam-5411	528	22	)	)	PUNCT
ejpam-5411	528	23	and	and	CCONJ
ejpam-5411	528	24	we	we	PRON
ejpam-5411	528	25	show	show	VERB
ejpam-5411	528	26	that	that	SCONJ
ejpam-5411	528	27	fα(t	fα(t	NOUN
ejpam-5411	528	28	(	(	PUNCT
ejpam-5411	528	29	jα)(f)(t))(k	jα)(f)(t))(k	PROPN
ejpam-5411	528	30	)	)	PUNCT
ejpam-5411	528	31	=	=	PUNCT
ejpam-5411	529	1	(	(	PUNCT
ejpam-5411	529	2	2ikπ	2ikπ	NOUN
ejpam-5411	529	3	α	α	NOUN
ejpam-5411	529	4	pα	pα	NOUN
ejpam-5411	529	5	)	)	PUNCT
ejpam-5411	529	6	jf(f((αt	jf(f((αt	PROPN
ejpam-5411	529	7	)	)	PUNCT
ejpam-5411	529	8	1	1	NUM
ejpam-5411	529	9	α	α	NOUN
ejpam-5411	529	10	)	)	PUNCT
ejpam-5411	529	11	)	)	PUNCT
ejpam-5411	529	12	(	(	PUNCT
ejpam-5411	529	13	k	k	NOUN
ejpam-5411	529	14	)	)	PUNCT
ejpam-5411	529	15	.	.	PUNCT
ejpam-5411	530	1	the	the	DET
ejpam-5411	530	2	function	function	NOUN
ejpam-5411	530	3	f	f	PROPN
ejpam-5411	530	4	is	be	AUX
ejpam-5411	530	5	n	n	PRON
ejpam-5411	530	6	times	time	NOUN
ejpam-5411	530	7	continuously	continuously	ADV
ejpam-5411	530	8	α	α	VERB
ejpam-5411	530	9	-	-	NOUN
ejpam-5411	530	10	differentiable	differentiable	ADJ
ejpam-5411	530	11	on	on	ADP
ejpam-5411	530	12	[	[	X
ejpam-5411	530	13	0,+∞	0,+∞	NUM
ejpam-5411	530	14	[	[	PUNCT
ejpam-5411	530	15	implies	imply	VERB
ejpam-5411	530	16	that	that	SCONJ
ejpam-5411	530	17	t	t	PROPN
ejpam-5411	530	18	(	(	PUNCT
ejpam-5411	530	19	(	(	PUNCT
ejpam-5411	530	20	j−1)α)(f	j−1)α)(f	PROPN
ejpam-5411	530	21	)	)	PUNCT
ejpam-5411	530	22	is	be	AUX
ejpam-5411	530	23	continuously	continuously	ADV
ejpam-5411	530	24	α	α	PRON
ejpam-5411	530	25	-	-	NOUN
ejpam-5411	530	26	differentiable	differentiable	ADJ
ejpam-5411	530	27	.	.	PUNCT
ejpam-5411	531	1	moreover	moreover	ADV
ejpam-5411	531	2	,	,	PUNCT
ejpam-5411	531	3	by	by	ADP
ejpam-5411	531	4	theorem	theorem	NOUN
ejpam-5411	531	5	10	10	NUM
ejpam-5411	531	6	and	and	CCONJ
ejpam-5411	531	7	the	the	DET
ejpam-5411	531	8	recurrence	recurrence	NOUN
ejpam-5411	531	9	hypothesis	hypothesis	NOUN
ejpam-5411	531	10	fα(t	fα(t	X
ejpam-5411	531	11	(	(	PUNCT
ejpam-5411	531	12	jα)(f)(t))(k	jα)(f)(t))(k	PROPN
ejpam-5411	531	13	)	)	PUNCT
ejpam-5411	531	14	=	=	SYM
ejpam-5411	531	15	fα(tα(t	fα(tα(t	NOUN
ejpam-5411	531	16	(	(	PUNCT
ejpam-5411	531	17	(	(	PUNCT
ejpam-5411	531	18	j−1)α)(f)(t)))(k	j−1)α)(f)(t)))(k	PROPN
ejpam-5411	531	19	)	)	PUNCT
ejpam-5411	531	20	=	=	PUNCT
ejpam-5411	532	1	(	(	PUNCT
ejpam-5411	532	2	2ikπ	2ikπ	NOUN
ejpam-5411	532	3	α	α	NOUN
ejpam-5411	532	4	pα	pα	NOUN
ejpam-5411	532	5	)	)	PUNCT
ejpam-5411	532	6	fα(t	fα(t	X
ejpam-5411	532	7	(	(	PUNCT
ejpam-5411	532	8	(	(	PUNCT
ejpam-5411	532	9	j−1)α)(f)(t))(k	j−1)α)(f)(t))(k	NOUN
ejpam-5411	532	10	)	)	PUNCT
ejpam-5411	532	11	=	=	NOUN
ejpam-5411	533	1	(	(	PUNCT
ejpam-5411	533	2	2ikπ	2ikπ	NOUN
ejpam-5411	533	3	α	α	NOUN
ejpam-5411	533	4	pα	pα	NOUN
ejpam-5411	533	5	)	)	PUNCT
ejpam-5411	533	6	jfα(f(t))(k	jfα(f(t))(k	PROPN
ejpam-5411	533	7	)	)	PUNCT
ejpam-5411	533	8	.	.	PUNCT
ejpam-5411	534	1	then	then	ADV
ejpam-5411	534	2	the	the	DET
ejpam-5411	534	3	property	property	NOUN
ejpam-5411	534	4	is	be	AUX
ejpam-5411	534	5	true	true	ADJ
ejpam-5411	534	6	for	for	ADP
ejpam-5411	534	7	all	all	DET
ejpam-5411	534	8	j	j	PROPN
ejpam-5411	534	9	∈	∈	PROPN
ejpam-5411	534	10	{	{	PUNCT
ejpam-5411	534	11	0	0	NUM
ejpam-5411	534	12	,	,	PUNCT
ejpam-5411	534	13	.	.	PUNCT
ejpam-5411	534	14	.	.	PUNCT
ejpam-5411	535	1	.	.	PUNCT
ejpam-5411	535	2	,	,	PUNCT
ejpam-5411	535	3	n	n	CCONJ
ejpam-5411	535	4	}	}	PUNCT
ejpam-5411	535	5	.	.	PUNCT
ejpam-5411	536	1	example	example	NOUN
ejpam-5411	536	2	11	11	NUM
ejpam-5411	536	3	.	.	PUNCT
ejpam-5411	537	1	consider	consider	VERB
ejpam-5411	537	2	the	the	DET
ejpam-5411	537	3	same	same	ADJ
ejpam-5411	537	4	function	function	NOUN
ejpam-5411	537	5	from	from	ADP
ejpam-5411	537	6	example	example	NOUN
ejpam-5411	537	7	6	6	NUM
ejpam-5411	537	8	,	,	PUNCT
ejpam-5411	537	9	we	we	PRON
ejpam-5411	537	10	have	have	VERB
ejpam-5411	537	11	f	f	PROPN
ejpam-5411	537	12	is	be	AUX
ejpam-5411	537	13	α	α	DET
ejpam-5411	537	14	-	-	ADJ
ejpam-5411	537	15	periodic	periodic	ADJ
ejpam-5411	537	16	function	function	NOUN
ejpam-5411	537	17	with	with	ADP
ejpam-5411	537	18	period	period	NOUN
ejpam-5411	537	19	(	(	PUNCT
ejpam-5411	537	20	3π2α	3π2α	NUM
ejpam-5411	537	21	)	)	PUNCT
ejpam-5411	537	22	1	1	NUM
ejpam-5411	537	23	α	α	NOUN
ejpam-5411	537	24	.	.	PUNCT
ejpam-5411	538	1	we	we	PRON
ejpam-5411	538	2	showed	show	VERB
ejpam-5411	538	3	that	that	SCONJ
ejpam-5411	538	4	f	f	PROPN
ejpam-5411	538	5	is	be	AUX
ejpam-5411	538	6	twice	twice	ADV
ejpam-5411	538	7	continuously	continuously	ADV
ejpam-5411	538	8	α	α	VERB
ejpam-5411	538	9	-	-	NOUN
ejpam-5411	538	10	differentiable	differentiable	ADJ
ejpam-5411	538	11	on	on	ADP
ejpam-5411	538	12	[	[	X
ejpam-5411	538	13	0,+∞	0,+∞	PROPN
ejpam-5411	539	1	[	[	X
ejpam-5411	539	2	,	,	PUNCT
ejpam-5411	539	3	g	g	PROPN
ejpam-5411	539	4	∈	∈	PROPN
ejpam-5411	539	5	c2([0,+∞	c2([0,+∞	ADJ
ejpam-5411	539	6	[	[	X
ejpam-5411	539	7	)	)	PUNCT
ejpam-5411	539	8	where	where	SCONJ
ejpam-5411	539	9	g(t	g(t	NOUN
ejpam-5411	539	10	)	)	PUNCT
ejpam-5411	539	11	=	=	SYM
ejpam-5411	539	12	f((αt	f((αt	NOUN
ejpam-5411	539	13	)	)	PUNCT
ejpam-5411	539	14	1	1	NUM
ejpam-5411	539	15	α	α	NOUN
ejpam-5411	539	16	)	)	PUNCT
ejpam-5411	539	17	and	and	CCONJ
ejpam-5411	539	18	t	t	PROPN
ejpam-5411	539	19	(	(	PUNCT
ejpam-5411	539	20	jα)(f	jα)(f	PROPN
ejpam-5411	539	21	)	)	PUNCT
ejpam-5411	539	22	is	be	AUX
ejpam-5411	539	23	α	α	DET
ejpam-5411	539	24	-	-	ADJ
ejpam-5411	539	25	periodic	periodic	ADJ
ejpam-5411	539	26	function	function	NOUN
ejpam-5411	539	27	with	with	ADP
ejpam-5411	539	28	period	period	NOUN
ejpam-5411	539	29	(	(	PUNCT
ejpam-5411	539	30	3π2α	3π2α	NUM
ejpam-5411	539	31	)	)	PUNCT
ejpam-5411	539	32	1	1	NUM
ejpam-5411	539	33	α	α	NOUN
ejpam-5411	539	34	for	for	ADP
ejpam-5411	539	35	j	j	PROPN
ejpam-5411	539	36	∈	∈	PROPN
ejpam-5411	539	37	{	{	PUNCT
ejpam-5411	539	38	0	0	NUM
ejpam-5411	539	39	,	,	PUNCT
ejpam-5411	539	40	1	1	NUM
ejpam-5411	539	41	,	,	PUNCT
ejpam-5411	539	42	2	2	NUM
ejpam-5411	539	43	}	}	PUNCT
ejpam-5411	539	44	.	.	PUNCT
ejpam-5411	540	1	for	for	ADP
ejpam-5411	540	2	k	k	PROPN
ejpam-5411	540	3	in	in	ADP
ejpam-5411	540	4	z	z	PROPN
ejpam-5411	540	5	,	,	PUNCT
ejpam-5411	540	6	we	we	PRON
ejpam-5411	540	7	have	have	VERB
ejpam-5411	540	8	1	1	NUM
ejpam-5411	540	9	.	.	X
ejpam-5411	541	1	for	for	ADP
ejpam-5411	541	2	j	j	PROPN
ejpam-5411	541	3	=	=	SYM
ejpam-5411	541	4	0	0	PROPN
ejpam-5411	541	5	,	,	PUNCT
ejpam-5411	541	6	the	the	DET
ejpam-5411	541	7	property	property	NOUN
ejpam-5411	541	8	is	be	AUX
ejpam-5411	541	9	true	true	ADJ
ejpam-5411	541	10	.	.	PUNCT
ejpam-5411	542	1	2	2	X
ejpam-5411	542	2	.	.	X
ejpam-5411	542	3	for	for	ADP
ejpam-5411	542	4	j	j	PROPN
ejpam-5411	542	5	=	=	SYM
ejpam-5411	542	6	1	1	NUM
ejpam-5411	542	7	,	,	PUNCT
ejpam-5411	542	8	the	the	DET
ejpam-5411	542	9	property	property	NOUN
ejpam-5411	542	10	is	be	AUX
ejpam-5411	542	11	true	true	ADJ
ejpam-5411	542	12	by	by	ADP
ejpam-5411	542	13	example	example	NOUN
ejpam-5411	542	14	10	10	NUM
ejpam-5411	542	15	.	.	PUNCT
ejpam-5411	543	1	3	3	X
ejpam-5411	543	2	.	.	X
ejpam-5411	543	3	for	for	ADP
ejpam-5411	543	4	j	j	PROPN
ejpam-5411	543	5	=	=	SYM
ejpam-5411	543	6	2	2	NUM
ejpam-5411	543	7	,	,	PUNCT
ejpam-5411	543	8	the	the	DET
ejpam-5411	543	9	function	function	NOUN
ejpam-5411	543	10	t	t	PROPN
ejpam-5411	543	11	(	(	PUNCT
ejpam-5411	543	12	α)(f	α)(f	PROPN
ejpam-5411	543	13	)	)	PUNCT
ejpam-5411	543	14	is	be	AUX
ejpam-5411	543	15	α	α	X
ejpam-5411	543	16	-	-	ADJ
ejpam-5411	543	17	differentiable	differentiable	ADJ
ejpam-5411	543	18	and	and	CCONJ
ejpam-5411	543	19	α	α	NOUN
ejpam-5411	543	20	-	-	NOUN
ejpam-5411	543	21	periodic	periodic	NOUN
ejpam-5411	543	22	with	with	ADP
ejpam-5411	543	23	period	period	NOUN
ejpam-5411	543	24	p	p	X
ejpam-5411	543	25	=	=	X
ejpam-5411	543	26	(	(	PUNCT
ejpam-5411	543	27	3π2α	3π2α	NUM
ejpam-5411	543	28	)	)	PUNCT
ejpam-5411	543	29	1	1	NUM
ejpam-5411	543	30	α	α	NOUN
ejpam-5411	543	31	.	.	PUNCT
ejpam-5411	544	1	then	then	ADV
ejpam-5411	544	2	,	,	PUNCT
ejpam-5411	544	3	we	we	PRON
ejpam-5411	544	4	have	have	VERB
ejpam-5411	544	5	fα(t	fα(t	NOUN
ejpam-5411	544	6	(	(	PUNCT
ejpam-5411	544	7	2α)(f)(t))(k	2α)(f)(t))(k	NUM
ejpam-5411	544	8	)	)	PUNCT
ejpam-5411	544	9	=	=	PUNCT
ejpam-5411	545	1	−72α4k2[(−1)−	−72α4k2[(−1)−	ADJ
ejpam-5411	545	2	4k	4k	NOUN
ejpam-5411	545	3	3	3	NUM
ejpam-5411	545	4	+	+	CCONJ
ejpam-5411	545	5	1	1	NUM
ejpam-5411	545	6	]	]	PUNCT
ejpam-5411	545	7	π(64k4	π(64k4	NOUN
ejpam-5411	545	8	−	−	PROPN
ejpam-5411	545	9	180k2	180k2	NUM
ejpam-5411	545	10	+	+	SYM
ejpam-5411	545	11	81	81	NUM
ejpam-5411	545	12	)	)	PUNCT
ejpam-5411	545	13	and	and	CCONJ
ejpam-5411	545	14	by	by	ADP
ejpam-5411	545	15	example	example	NOUN
ejpam-5411	545	16	10	10	NUM
ejpam-5411	545	17	,	,	PUNCT
ejpam-5411	545	18	we	we	PRON
ejpam-5411	545	19	have	have	VERB
ejpam-5411	545	20	fα(f(t))(k	fα(f(t))(k	NOUN
ejpam-5411	545	21	)	)	PUNCT
ejpam-5411	545	22	=	=	SYM
ejpam-5411	545	23	81((−1)−	81((−1)−	NUM
ejpam-5411	545	24	4k	4k	NUM
ejpam-5411	545	25	3	3	NUM
ejpam-5411	545	26	+	+	CCONJ
ejpam-5411	545	27	1	1	NUM
ejpam-5411	545	28	)	)	PUNCT
ejpam-5411	545	29	2π(64k4	2π(64k4	NUM
ejpam-5411	545	30	−	−	PROPN
ejpam-5411	545	31	180k2	180k2	NUM
ejpam-5411	545	32	+	+	NUM
ejpam-5411	545	33	81	81	NUM
ejpam-5411	545	34	)	)	PUNCT
ejpam-5411	545	35	.	.	PUNCT
ejpam-5411	546	1	then	then	ADV
ejpam-5411	546	2	fα(t	fα(t	X
ejpam-5411	546	3	(	(	PUNCT
ejpam-5411	546	4	2α)(f)(t))(k	2α)(f)(t))(k	NUM
ejpam-5411	546	5	)	)	PUNCT
ejpam-5411	546	6	=	=	PUNCT
ejpam-5411	546	7	(	(	PUNCT
ejpam-5411	546	8	4	4	NUM
ejpam-5411	546	9	3	3	NUM
ejpam-5411	546	10	ikα2)2fα(f(t))(k	ikα2)2fα(f(t))(k	NOUN
ejpam-5411	546	11	)	)	PUNCT
ejpam-5411	546	12	.	.	PUNCT
ejpam-5411	547	1	t.	t.	PROPN
ejpam-5411	547	2	abdeljawad	abdeljawad	PROPN
ejpam-5411	547	3	et	et	PROPN
ejpam-5411	547	4	al	al	PROPN
ejpam-5411	547	5	.	.	PUNCT
ejpam-5411	547	6	/	/	SYM
ejpam-5411	547	7	eur	eur	PROPN
ejpam-5411	547	8	.	.	PUNCT
ejpam-5411	548	1	j.	j.	PROPN
ejpam-5411	548	2	pure	pure	PROPN
ejpam-5411	548	3	appl	appl	PROPN
ejpam-5411	548	4	.	.	PROPN
ejpam-5411	548	5	math	math	PROPN
ejpam-5411	548	6	,	,	PUNCT
ejpam-5411	548	7	17	17	NUM
ejpam-5411	548	8	(	(	PUNCT
ejpam-5411	548	9	4	4	NUM
ejpam-5411	548	10	)	)	PUNCT
ejpam-5411	548	11	(	(	PUNCT
ejpam-5411	548	12	2024	2024	NUM
ejpam-5411	548	13	)	)	PUNCT
ejpam-5411	548	14	,	,	PUNCT
ejpam-5411	548	15	2405	2405	NUM
ejpam-5411	548	16	-	-	SYM
ejpam-5411	548	17	2430	2430	NUM
ejpam-5411	548	18	2424	2424	NUM
ejpam-5411	548	19	corollary	corollary	NOUN
ejpam-5411	548	20	1	1	NUM
ejpam-5411	548	21	.	.	PUNCT
ejpam-5411	549	1	if	if	SCONJ
ejpam-5411	549	2	f	f	PROPN
ejpam-5411	549	3	is	be	AUX
ejpam-5411	549	4	α	α	DET
ejpam-5411	549	5	-	-	ADJ
ejpam-5411	549	6	periodic	periodic	ADJ
ejpam-5411	549	7	function	function	NOUN
ejpam-5411	549	8	with	with	ADP
ejpam-5411	549	9	period	period	NOUN
ejpam-5411	549	10	(	(	PUNCT
ejpam-5411	549	11	2πα	2πα	NOUN
ejpam-5411	549	12	)	)	PUNCT
ejpam-5411	549	13	1	1	NUM
ejpam-5411	549	14	α	α	NOUN
ejpam-5411	549	15	,	,	PUNCT
ejpam-5411	549	16	then	then	ADV
ejpam-5411	549	17	we	we	PRON
ejpam-5411	549	18	obtain	obtain	VERB
ejpam-5411	549	19	the	the	DET
ejpam-5411	549	20	following	following	ADJ
ejpam-5411	549	21	classical	classical	ADJ
ejpam-5411	549	22	fourier	fourier	NOUN
ejpam-5411	549	23	property	property	NOUN
ejpam-5411	549	24	f(f	f(f	PROPN
ejpam-5411	549	25	(	(	PUNCT
ejpam-5411	549	26	n)((αt	n)((αt	PROPN
ejpam-5411	549	27	)	)	PUNCT
ejpam-5411	549	28	1	1	NUM
ejpam-5411	549	29	α	α	NOUN
ejpam-5411	549	30	)	)	PUNCT
ejpam-5411	549	31	)	)	PUNCT
ejpam-5411	549	32	(	(	PUNCT
ejpam-5411	549	33	k	k	X
ejpam-5411	549	34	)	)	PUNCT
ejpam-5411	549	35	=	=	SYM
ejpam-5411	549	36	(	(	PUNCT
ejpam-5411	549	37	ik)nf(f((αt	ik)nf(f((αt	PROPN
ejpam-5411	549	38	)	)	PUNCT
ejpam-5411	549	39	1	1	NUM
ejpam-5411	549	40	α	α	NOUN
ejpam-5411	549	41	)	)	PUNCT
ejpam-5411	549	42	)	)	PUNCT
ejpam-5411	549	43	(	(	PUNCT
ejpam-5411	549	44	k	k	NOUN
ejpam-5411	549	45	)	)	PUNCT
ejpam-5411	549	46	.	.	PUNCT
ejpam-5411	550	1	we	we	PRON
ejpam-5411	550	2	conclude	conclude	VERB
ejpam-5411	550	3	this	this	DET
ejpam-5411	550	4	section	section	NOUN
ejpam-5411	550	5	with	with	ADP
ejpam-5411	550	6	a	a	DET
ejpam-5411	550	7	result	result	NOUN
ejpam-5411	550	8	which	which	PRON
ejpam-5411	550	9	has	have	AUX
ejpam-5411	550	10	been	be	AUX
ejpam-5411	550	11	used	use	VERB
ejpam-5411	550	12	by	by	ADP
ejpam-5411	550	13	several	several	ADJ
ejpam-5411	550	14	authors	author	NOUN
ejpam-5411	550	15	to	to	PART
ejpam-5411	550	16	solve	solve	VERB
ejpam-5411	550	17	certain	certain	ADJ
ejpam-5411	550	18	integro	integro	ADJ
ejpam-5411	550	19	-	-	PUNCT
ejpam-5411	550	20	differential	differential	NOUN
ejpam-5411	550	21	equations	equation	NOUN
ejpam-5411	550	22	.	.	PUNCT
ejpam-5411	551	1	lemma	lemma	PROPN
ejpam-5411	551	2	3	3	X
ejpam-5411	551	3	.	.	PROPN
ejpam-5411	551	4	assume	assume	VERB
ejpam-5411	551	5	that	that	SCONJ
ejpam-5411	551	6	g	g	PROPN
ejpam-5411	551	7	∈	∈	PROPN
ejpam-5411	551	8	l1(r+,r	l1(r+,r	NOUN
ejpam-5411	551	9	)	)	PUNCT
ejpam-5411	551	10	is	be	AUX
ejpam-5411	551	11	a	a	DET
ejpam-5411	551	12	continuous	continuous	ADJ
ejpam-5411	551	13	periodic	periodic	ADJ
ejpam-5411	551	14	function	function	NOUN
ejpam-5411	551	15	with	with	ADP
ejpam-5411	551	16	period	period	NOUN
ejpam-5411	551	17	t	t	NOUN
ejpam-5411	551	18	and	and	CCONJ
ejpam-5411	551	19	a1	a1	NOUN
ejpam-5411	551	20	∈	∈	PROPN
ejpam-5411	551	21	l1(r+	l1(r+	PROPN
ejpam-5411	551	22	)	)	PUNCT
ejpam-5411	551	23	.	.	PUNCT
ejpam-5411	552	1	then	then	ADV
ejpam-5411	552	2	for	for	ADP
ejpam-5411	552	3	t	t	PROPN
ejpam-5411	552	4	∈	∈	PROPN
ejpam-5411	552	5	[	[	X
ejpam-5411	552	6	0,+∞[∫	0,+∞[∫	X
ejpam-5411	552	7	0	0	NUM
ejpam-5411	552	8	−∞	−∞	X
ejpam-5411	552	9	a1(t−	a1(t−	NOUN
ejpam-5411	552	10	s)g(s)ds	s)g(s)ds	NOUN
ejpam-5411	552	11	=	=	PROPN
ejpam-5411	553	1	+	+	ADJ
ejpam-5411	553	2	∞∑	∞∑	NUM
ejpam-5411	553	3	n=1	n=1	NUM
ejpam-5411	553	4	∫	∫	PROPN
ejpam-5411	553	5	t	t	PROPN
ejpam-5411	553	6	0	0	NUM
ejpam-5411	553	7	a1(t−	a1(t−	NOUN
ejpam-5411	553	8	u	u	NOUN
ejpam-5411	553	9	+	+	NOUN
ejpam-5411	553	10	nt	not	PART
ejpam-5411	553	11	)	)	PUNCT
ejpam-5411	553	12	g(u)du	g(u)du	PROPN
ejpam-5411	553	13	.	.	PUNCT
ejpam-5411	554	1	(	(	PUNCT
ejpam-5411	554	2	19	19	NUM
ejpam-5411	554	3	)	)	PUNCT
ejpam-5411	554	4	proof	proof	NOUN
ejpam-5411	554	5	.	.	PUNCT
ejpam-5411	555	1	let	let	VERB
ejpam-5411	555	2	g	g	PROPN
ejpam-5411	555	3	∈	∈	PROPN
ejpam-5411	555	4	l1(r+,r	l1(r+,r	NOUN
ejpam-5411	555	5	)	)	PUNCT
ejpam-5411	555	6	is	be	AUX
ejpam-5411	555	7	a	a	DET
ejpam-5411	555	8	continuous	continuous	ADJ
ejpam-5411	555	9	periodic	periodic	ADJ
ejpam-5411	555	10	function	function	NOUN
ejpam-5411	555	11	with	with	ADP
ejpam-5411	555	12	period	period	NOUN
ejpam-5411	555	13	t	t	NOUN
ejpam-5411	555	14	.	.	PUNCT
ejpam-5411	556	1	we	we	PRON
ejpam-5411	556	2	have∫	have∫	VERB
ejpam-5411	556	3	0	0	PUNCT
ejpam-5411	557	1	−∞	−∞	X
ejpam-5411	557	2	a1(t−	a1(t−	NOUN
ejpam-5411	557	3	s)g(s)ds	s)g(s)ds	NOUN
ejpam-5411	558	1	=	=	PROPN
ejpam-5411	559	1	+	+	ADJ
ejpam-5411	559	2	∞∑	∞∑	NUM
ejpam-5411	559	3	n=1	n=1	ADP
ejpam-5411	559	4	∫	∫	PROPN
ejpam-5411	560	1	−(n−1)t	−(n−1)t	PRON
ejpam-5411	560	2	−nt	−nt	PROPN
ejpam-5411	560	3	a1(t−	a1(t−	NOUN
ejpam-5411	560	4	s)g(s)ds	s)g(s)ds	NOUN
ejpam-5411	560	5	=	=	PUNCT
ejpam-5411	561	1	+	+	ADJ
ejpam-5411	561	2	∞∑	∞∑	NUM
ejpam-5411	561	3	n=1	n=1	ADP
ejpam-5411	561	4	∫	∫	PROPN
ejpam-5411	561	5	−(n−1)t	−(n−1)t	PRON
ejpam-5411	561	6	−nt	−nt	PROPN
ejpam-5411	561	7	a1(t−	a1(t−	NOUN
ejpam-5411	561	8	s)g(s	s)g(s	ADJ
ejpam-5411	561	9	+	+	CCONJ
ejpam-5411	561	10	nt	not	PART
ejpam-5411	561	11	)	)	PUNCT
ejpam-5411	561	12	ds	ds	NOUN
ejpam-5411	561	13	=	=	PUNCT
ejpam-5411	562	1	+	+	NOUN
ejpam-5411	562	2	∞∑	∞∑	NUM
ejpam-5411	562	3	n=1	n=1	NUM
ejpam-5411	562	4	∫	∫	PROPN
ejpam-5411	562	5	t	t	PROPN
ejpam-5411	562	6	0	0	NUM
ejpam-5411	562	7	a1(t−	a1(t−	NOUN
ejpam-5411	562	8	u	u	NOUN
ejpam-5411	562	9	+	+	NOUN
ejpam-5411	562	10	nt	not	PART
ejpam-5411	562	11	)	)	PUNCT
ejpam-5411	562	12	g(u)du	g(u)du	PROPN
ejpam-5411	562	13	.	.	PUNCT
ejpam-5411	562	14	example	example	NOUN
ejpam-5411	563	1	12	12	NUM
ejpam-5411	563	2	.	.	PUNCT
ejpam-5411	564	1	let	let	VERB
ejpam-5411	564	2	f1	f1	NOUN
ejpam-5411	564	3	defined	define	VERB
ejpam-5411	564	4	by	by	ADP
ejpam-5411	564	5	example	example	NOUN
ejpam-5411	564	6	3	3	NUM
ejpam-5411	564	7	and	and	CCONJ
ejpam-5411	564	8	a1(t	a1(t	ADJ
ejpam-5411	564	9	)	)	PUNCT
ejpam-5411	564	10	=	=	SYM
ejpam-5411	564	11	e−t	e−t	NOUN
ejpam-5411	564	12	∈	∈	PROPN
ejpam-5411	564	13	l1(r+	l1(r+	PROPN
ejpam-5411	564	14	)	)	PUNCT
ejpam-5411	564	15	.	.	PUNCT
ejpam-5411	565	1	the	the	DET
ejpam-5411	565	2	function	function	NOUN
ejpam-5411	565	3	f1	f1	NOUN
ejpam-5411	565	4	is	be	AUX
ejpam-5411	565	5	α	α	NOUN
ejpam-5411	565	6	-	-	NOUN
ejpam-5411	565	7	periodic	periodic	NOUN
ejpam-5411	565	8	with	with	ADP
ejpam-5411	565	9	period	period	NOUN
ejpam-5411	565	10	(	(	PUNCT
ejpam-5411	565	11	1	1	NUM
ejpam-5411	565	12	α	α	NOUN
ejpam-5411	565	13	)	)	PUNCT
ejpam-5411	565	14	1	1	NUM
ejpam-5411	565	15	α	α	NOUN
ejpam-5411	565	16	for	for	ADP
ejpam-5411	565	17	all	all	DET
ejpam-5411	565	18	t	t	NOUN
ejpam-5411	565	19	∈	∈	PROPN
ejpam-5411	566	1	[	[	X
ejpam-5411	566	2	0,+∞	0,+∞	NUM
ejpam-5411	566	3	[	[	X
ejpam-5411	566	4	,	,	PUNCT
ejpam-5411	566	5	and	and	CCONJ
ejpam-5411	566	6	the	the	DET
ejpam-5411	566	7	associated	associated	ADJ
ejpam-5411	566	8	function	function	NOUN
ejpam-5411	566	9	g	g	NOUN
ejpam-5411	566	10	satisfies	satisfie	NOUN
ejpam-5411	566	11	g1(t	g1(t	ADP
ejpam-5411	566	12	)	)	PUNCT
ejpam-5411	566	13	=	=	SYM
ejpam-5411	566	14	f1((αt	f1((αt	NOUN
ejpam-5411	566	15	)	)	PUNCT
ejpam-5411	566	16	1	1	NUM
ejpam-5411	566	17	α	α	NOUN
ejpam-5411	566	18	)	)	PUNCT
ejpam-5411	566	19	=	=	PUNCT
ejpam-5411	566	20			PROPN
ejpam-5411	566	21	t	t	PROPN
ejpam-5411	566	22	,	,	PUNCT
ejpam-5411	566	23	0	0	NUM
ejpam-5411	566	24	≤	≤	NUM
ejpam-5411	566	25	t	t	NOUN
ejpam-5411	566	26	≤	≤	NUM
ejpam-5411	566	27	1	1	NUM
ejpam-5411	566	28	2α2	2α2	NUM
ejpam-5411	566	29	1	1	NUM
ejpam-5411	566	30	α2	α2	ADJ
ejpam-5411	566	31	−	−	PROPN
ejpam-5411	567	1	t	t	PROPN
ejpam-5411	567	2	,	,	PUNCT
ejpam-5411	567	3	1	1	NUM
ejpam-5411	567	4	2α2	2α2	NUM
ejpam-5411	567	5	<	<	X
ejpam-5411	567	6	t	t	X
ejpam-5411	567	7	≤	≤	NOUN
ejpam-5411	567	8	1	1	NUM
ejpam-5411	567	9	α2	α2	NOUN
ejpam-5411	567	10	is	be	AUX
ejpam-5411	567	11	periodic	periodic	ADJ
ejpam-5411	567	12	with	with	ADP
ejpam-5411	567	13	period	period	NOUN
ejpam-5411	567	14	1	1	NUM
ejpam-5411	567	15	α2	α2	ADJ
ejpam-5411	567	16	and	and	CCONJ
ejpam-5411	567	17	continuous	continuous	ADJ
ejpam-5411	567	18	for	for	ADP
ejpam-5411	567	19	all	all	DET
ejpam-5411	567	20	t	t	NOUN
ejpam-5411	567	21	∈	∈	PROPN
ejpam-5411	568	1	[	[	X
ejpam-5411	568	2	0,+∞	0,+∞	NUM
ejpam-5411	569	1	[	[	X
ejpam-5411	569	2	.	.	PUNCT
ejpam-5411	570	1	then	then	ADV
ejpam-5411	570	2	,	,	PUNCT
ejpam-5411	570	3	we	we	PRON
ejpam-5411	570	4	have∫	have∫	VERB
ejpam-5411	570	5	0	0	PUNCT
ejpam-5411	571	1	−∞	−∞	NOUN
ejpam-5411	571	2	a1(t−	a1(t−	NOUN
ejpam-5411	571	3	s)g1(s)ds	s)g1(s)ds	NOUN
ejpam-5411	571	4	=	=	PUNCT
ejpam-5411	572	1	+	+	ADJ
ejpam-5411	572	2	∞∑	∞∑	NUM
ejpam-5411	572	3	n=1	n=1	ADJ
ejpam-5411	572	4	∫	∫	PROPN
ejpam-5411	572	5	1	1	NUM
ejpam-5411	572	6	α2	α2	PROPN
ejpam-5411	572	7	0	0	NUM
ejpam-5411	572	8	e−(t−u−	e−(t−u−	PROPN
ejpam-5411	572	9	n	n	X
ejpam-5411	572	10	α2	α2	ADJ
ejpam-5411	572	11	)	)	PUNCT
ejpam-5411	572	12	g1(u)du	g1(u)du	PROPN
ejpam-5411	572	13	=	=	SYM
ejpam-5411	572	14	e−t	e−t	PROPN
ejpam-5411	572	15	(	(	PUNCT
ejpam-5411	572	16	+	+	ADP
ejpam-5411	572	17	∞∑	∞∑	NUM
ejpam-5411	572	18	n=1	n=1	ADP
ejpam-5411	572	19	e−	e−	PROPN
ejpam-5411	572	20	n	n	CCONJ
ejpam-5411	572	21	α2	α2	PROPN
ejpam-5411	572	22	)	)	PUNCT
ejpam-5411	572	23	∫	∫	PROPN
ejpam-5411	572	24	1	1	NUM
ejpam-5411	572	25	α2	α2	NOUN
ejpam-5411	572	26	0	0	NUM
ejpam-5411	573	1	eug1(u)du	eug1(u)du	PROPN
ejpam-5411	574	1	=	=	PUNCT
ejpam-5411	575	1	(	(	PUNCT
ejpam-5411	575	2	e−t	e−t	NOUN
ejpam-5411	575	3	e	e	NOUN
ejpam-5411	575	4	1	1	NUM
ejpam-5411	575	5	α2	α2	ADJ
ejpam-5411	575	6	−	−	PROPN
ejpam-5411	575	7	1	1	NUM
ejpam-5411	575	8	)	)	PUNCT
ejpam-5411	575	9	∫	∫	PROPN
ejpam-5411	575	10	1	1	NUM
ejpam-5411	575	11	α2	α2	NOUN
ejpam-5411	575	12	0	0	NUM
ejpam-5411	576	1	eug1(u)du	eug1(u)du	PROPN
ejpam-5411	576	2	=	=	PUNCT
ejpam-5411	576	3	(	(	PUNCT
ejpam-5411	576	4	e−t	e−t	NOUN
ejpam-5411	576	5	e	e	NOUN
ejpam-5411	576	6	1	1	NUM
ejpam-5411	576	7	α2	α2	ADJ
ejpam-5411	576	8	−	−	NOUN
ejpam-5411	576	9	1	1	NUM
ejpam-5411	576	10	)	)	PUNCT
ejpam-5411	576	11	[	[	PUNCT
ejpam-5411	576	12	∫	∫	PROPN
ejpam-5411	576	13	1	1	NUM
ejpam-5411	576	14	2α2	2α2	NUM
ejpam-5411	576	15	0	0	NUM
ejpam-5411	576	16	ueudu	ueudu	NOUN
ejpam-5411	576	17	+	+	CCONJ
ejpam-5411	576	18	∫	∫	PROPN
ejpam-5411	576	19	1	1	NUM
ejpam-5411	576	20	α2	α2	ADJ
ejpam-5411	576	21	1	1	NUM
ejpam-5411	576	22	2α2	2α2	NUM
ejpam-5411	576	23	(	(	PUNCT
ejpam-5411	576	24	1	1	NUM
ejpam-5411	576	25	α2	α2	ADJ
ejpam-5411	576	26	−	−	PROPN
ejpam-5411	577	1	u)eudu	u)eudu	ADV
ejpam-5411	577	2	]	]	PUNCT
ejpam-5411	577	3	t.	t.	PROPN
ejpam-5411	577	4	abdeljawad	abdeljawad	PROPN
ejpam-5411	577	5	et	et	PROPN
ejpam-5411	577	6	al	al	PROPN
ejpam-5411	577	7	.	.	PUNCT
ejpam-5411	577	8	/	/	SYM
ejpam-5411	577	9	eur	eur	PROPN
ejpam-5411	577	10	.	.	PUNCT
ejpam-5411	578	1	j.	j.	PROPN
ejpam-5411	578	2	pure	pure	PROPN
ejpam-5411	578	3	appl	appl	PROPN
ejpam-5411	578	4	.	.	PROPN
ejpam-5411	578	5	math	math	PROPN
ejpam-5411	578	6	,	,	PUNCT
ejpam-5411	578	7	17	17	NUM
ejpam-5411	578	8	(	(	PUNCT
ejpam-5411	578	9	4	4	NUM
ejpam-5411	578	10	)	)	PUNCT
ejpam-5411	578	11	(	(	PUNCT
ejpam-5411	578	12	2024	2024	NUM
ejpam-5411	578	13	)	)	PUNCT
ejpam-5411	578	14	,	,	PUNCT
ejpam-5411	578	15	2405	2405	NUM
ejpam-5411	578	16	-	-	SYM
ejpam-5411	578	17	2430	2430	NUM
ejpam-5411	578	18	2425	2425	NUM
ejpam-5411	578	19	=	=	SYM
ejpam-5411	578	20	(	(	PUNCT
ejpam-5411	578	21	e	e	NOUN
ejpam-5411	578	22	1	1	NUM
ejpam-5411	578	23	2α2	2α2	NUM
ejpam-5411	578	24	−	−	NUM
ejpam-5411	578	25	1	1	NUM
ejpam-5411	578	26	e	e	NOUN
ejpam-5411	578	27	1	1	NUM
ejpam-5411	578	28	2α2	2α2	NUM
ejpam-5411	578	29	+	+	CCONJ
ejpam-5411	578	30	1	1	NUM
ejpam-5411	578	31	)	)	PUNCT
ejpam-5411	578	32	e−t	e−t	NOUN
ejpam-5411	578	33	.	.	PUNCT
ejpam-5411	579	1	theorem	theorem	NOUN
ejpam-5411	579	2	12	12	NUM
ejpam-5411	579	3	.	.	PUNCT
ejpam-5411	580	1	let	let	VERB
ejpam-5411	580	2	0	0	NUM
ejpam-5411	580	3	<	<	X
ejpam-5411	580	4	α	α	PROPN
ejpam-5411	580	5	≤	≤	NUM
ejpam-5411	580	6	1	1	NUM
ejpam-5411	580	7	,	,	PUNCT
ejpam-5411	580	8	assume	assume	VERB
ejpam-5411	580	9	that	that	SCONJ
ejpam-5411	580	10	f	f	PROPN
ejpam-5411	580	11	∈	∈	PROPN
ejpam-5411	580	12	l1(r+,r	l1(r+,r	PROPN
ejpam-5411	580	13	)	)	PUNCT
ejpam-5411	580	14	is	be	AUX
ejpam-5411	580	15	α	α	DET
ejpam-5411	580	16	-	-	ADJ
ejpam-5411	580	17	periodic	periodic	ADJ
ejpam-5411	580	18	function	function	NOUN
ejpam-5411	580	19	with	with	ADP
ejpam-5411	580	20	period	period	NOUN
ejpam-5411	580	21	p	p	X
ejpam-5411	580	22	,	,	PUNCT
ejpam-5411	580	23	and	and	CCONJ
ejpam-5411	580	24	a(t	a(t	NOUN
ejpam-5411	580	25	)	)	PUNCT
ejpam-5411	580	26	=	=	SYM
ejpam-5411	580	27	a1	a1	PROPN
ejpam-5411	580	28	(	(	PUNCT
ejpam-5411	580	29	tα	tα	PROPN
ejpam-5411	580	30	α	α	NOUN
ejpam-5411	580	31	)	)	PUNCT
ejpam-5411	580	32	such	such	ADJ
ejpam-5411	580	33	that	that	DET
ejpam-5411	580	34	a1	a1	NOUN
ejpam-5411	580	35	∈	∈	PROPN
ejpam-5411	580	36	l1(r+	l1(r+	PROPN
ejpam-5411	580	37	)	)	PUNCT
ejpam-5411	580	38	.	.	PUNCT
ejpam-5411	581	1	then	then	ADV
ejpam-5411	581	2	(	(	PUNCT
ejpam-5411	581	3	a	a	DET
ejpam-5411	581	4	∗α	∗α	NOUN
ejpam-5411	581	5	f)−∞	f)−∞	ADV
ejpam-5411	581	6	is	be	AUX
ejpam-5411	581	7	α	α	NOUN
ejpam-5411	581	8	-	-	NOUN
ejpam-5411	581	9	periodic	periodic	NOUN
ejpam-5411	581	10	with	with	ADP
ejpam-5411	581	11	period	period	NOUN
ejpam-5411	581	12	p	p	NOUN
ejpam-5411	581	13	and	and	CCONJ
ejpam-5411	581	14	for	for	ADP
ejpam-5411	581	15	k	k	PROPN
ejpam-5411	581	16	∈	∈	PROPN
ejpam-5411	581	17	z	z	PROPN
ejpam-5411	581	18	fα((a	fα((a	PROPN
ejpam-5411	581	19	∗α	∗α	PROPN
ejpam-5411	581	20	f)−∞(t))(k	f)−∞(t))(k	PROPN
ejpam-5411	581	21	)	)	PUNCT
ejpam-5411	582	1	=	=	SYM
ejpam-5411	582	2	lα(a(t))(2ikπ	lα(a(t))(2ikπ	NOUN
ejpam-5411	582	3	α	α	NOUN
ejpam-5411	582	4	pα	pα	NOUN
ejpam-5411	582	5	)	)	PUNCT
ejpam-5411	582	6	fα(f((t))(k	fα(f((t))(k	NOUN
ejpam-5411	582	7	)	)	PUNCT
ejpam-5411	582	8	where	where	SCONJ
ejpam-5411	582	9	lα(a(t))(λ	lα(a(t))(λ	NOUN
ejpam-5411	582	10	)	)	PUNCT
ejpam-5411	582	11	is	be	AUX
ejpam-5411	582	12	the	the	DET
ejpam-5411	582	13	conformable	conformable	ADJ
ejpam-5411	582	14	laplace	laplace	NOUN
ejpam-5411	582	15	transform	transform	NOUN
ejpam-5411	582	16	of	of	ADP
ejpam-5411	582	17	a(t	a(t	NOUN
ejpam-5411	582	18	)	)	PUNCT
ejpam-5411	582	19	given	give	VERB
ejpam-5411	582	20	by	by	ADP
ejpam-5411	582	21	the	the	DET
ejpam-5411	582	22	definition	definition	NOUN
ejpam-5411	582	23	5	5	NUM
ejpam-5411	582	24	.	.	PUNCT
ejpam-5411	583	1	proof	proof	NOUN
ejpam-5411	583	2	.	.	PUNCT
ejpam-5411	584	1	let	let	VERB
ejpam-5411	584	2	0	0	NUM
ejpam-5411	584	3	<	<	X
ejpam-5411	584	4	α	α	PROPN
ejpam-5411	584	5	≤	≤	NUM
ejpam-5411	584	6	1	1	NUM
ejpam-5411	584	7	and	and	CCONJ
ejpam-5411	584	8	assume	assume	VERB
ejpam-5411	584	9	that	that	SCONJ
ejpam-5411	584	10	f	f	PROPN
ejpam-5411	584	11	∈	∈	PROPN
ejpam-5411	584	12	l1(r+,r	l1(r+,r	PROPN
ejpam-5411	584	13	)	)	PUNCT
ejpam-5411	584	14	is	be	AUX
ejpam-5411	584	15	α	α	DET
ejpam-5411	584	16	-	-	ADJ
ejpam-5411	584	17	periodic	periodic	ADJ
ejpam-5411	584	18	function	function	NOUN
ejpam-5411	584	19	with	with	ADP
ejpam-5411	584	20	period	period	NOUN
ejpam-5411	584	21	p.	p.	NOUN
ejpam-5411	584	22	for	for	ADP
ejpam-5411	584	23	t	t	PROPN
ejpam-5411	584	24	∈	∈	PROPN
ejpam-5411	585	1	[	[	X
ejpam-5411	585	2	0,+∞	0,+∞	PROPN
ejpam-5411	585	3	[	[	X
ejpam-5411	585	4	,	,	PUNCT
ejpam-5411	585	5	g(t	g(t	PROPN
ejpam-5411	585	6	)	)	PUNCT
ejpam-5411	585	7	=	=	SYM
ejpam-5411	585	8	f((αt	f((αt	NOUN
ejpam-5411	585	9	)	)	PUNCT
ejpam-5411	585	10	1	1	NUM
ejpam-5411	585	11	α	α	NOUN
ejpam-5411	585	12	)	)	PUNCT
ejpam-5411	585	13	is	be	AUX
ejpam-5411	585	14	periodic	periodic	ADJ
ejpam-5411	585	15	with	with	ADP
ejpam-5411	585	16	period	period	NOUN
ejpam-5411	585	17	t	t	NOUN
ejpam-5411	586	1	=	=	PUNCT
ejpam-5411	586	2	pα	pα	VERB
ejpam-5411	586	3	α	α	NOUN
ejpam-5411	586	4	.	.	PUNCT
ejpam-5411	587	1	by	by	ADP
ejpam-5411	587	2	theorem	theorem	NOUN
ejpam-5411	587	3	7	7	NUM
ejpam-5411	587	4	,	,	PUNCT
ejpam-5411	587	5	(	(	PUNCT
ejpam-5411	587	6	a	a	DET
ejpam-5411	587	7	∗α	∗α	PROPN
ejpam-5411	587	8	f)−∞(t	f)−∞(t	PROPN
ejpam-5411	587	9	)	)	PUNCT
ejpam-5411	587	10	is	be	AUX
ejpam-5411	587	11	α	α	DET
ejpam-5411	587	12	-	-	ADJ
ejpam-5411	587	13	periodic	periodic	ADJ
ejpam-5411	587	14	function	function	NOUN
ejpam-5411	587	15	with	with	ADP
ejpam-5411	587	16	period	period	NOUN
ejpam-5411	587	17	p	p	NOUN
ejpam-5411	587	18	,	,	PUNCT
ejpam-5411	587	19	and	and	CCONJ
ejpam-5411	587	20	we	we	PRON
ejpam-5411	587	21	showed	show	VERB
ejpam-5411	587	22	that	that	SCONJ
ejpam-5411	587	23	(	(	PUNCT
ejpam-5411	587	24	a	a	DET
ejpam-5411	587	25	∗α	∗α	NOUN
ejpam-5411	587	26	f)−∞(t	f)−∞(t	PROPN
ejpam-5411	587	27	)	)	PUNCT
ejpam-5411	588	1	=	=	SYM
ejpam-5411	588	2	f	f	PROPN
ejpam-5411	588	3	(	(	PUNCT
ejpam-5411	588	4	tα	tα	PROPN
ejpam-5411	588	5	α	α	PROPN
ejpam-5411	588	6	)	)	PUNCT
ejpam-5411	588	7	where	where	SCONJ
ejpam-5411	588	8	the	the	DET
ejpam-5411	588	9	continuous	continuous	ADJ
ejpam-5411	588	10	function	function	NOUN
ejpam-5411	588	11	f	f	PROPN
ejpam-5411	588	12	is	be	AUX
ejpam-5411	588	13	defined	define	VERB
ejpam-5411	588	14	by	by	ADP
ejpam-5411	588	15	f	f	PROPN
ejpam-5411	588	16	(	(	PUNCT
ejpam-5411	588	17	t	t	PROPN
ejpam-5411	588	18	)	)	PUNCT
ejpam-5411	588	19	=	=	SYM
ejpam-5411	589	1	∫	∫	PROPN
ejpam-5411	589	2	t	t	PROPN
ejpam-5411	589	3	−∞	−∞	ADP
ejpam-5411	589	4	a1(t−	a1(t−	NOUN
ejpam-5411	589	5	s)g(s)ds	s)g(s)ds	PROPN
ejpam-5411	589	6	.	.	PUNCT
ejpam-5411	590	1	thus	thus	ADV
ejpam-5411	590	2	,	,	PUNCT
ejpam-5411	590	3	we	we	PRON
ejpam-5411	590	4	have	have	VERB
ejpam-5411	590	5	f	f	PROPN
ejpam-5411	590	6	(	(	PUNCT
ejpam-5411	590	7	t	t	PROPN
ejpam-5411	590	8	)	)	PUNCT
ejpam-5411	590	9	=	=	SYM
ejpam-5411	591	1	∫	∫	PROPN
ejpam-5411	591	2	0	0	PUNCT
ejpam-5411	592	1	−∞	−∞	X
ejpam-5411	592	2	a1(t−	a1(t−	NOUN
ejpam-5411	592	3	s)g(s)ds	s)g(s)ds	PROPN
ejpam-5411	592	4	+	+	CCONJ
ejpam-5411	592	5	∫	∫	PROPN
ejpam-5411	592	6	t	t	PROPN
ejpam-5411	592	7	0	0	NUM
ejpam-5411	592	8	a1(t−	a1(t−	NOUN
ejpam-5411	592	9	s)g(s)ds	s)g(s)ds	PROPN
ejpam-5411	592	10	.	.	PUNCT
ejpam-5411	593	1	by	by	ADP
ejpam-5411	593	2	lemma	lemma	PROPN
ejpam-5411	593	3	3	3	NUM
ejpam-5411	593	4	,	,	PUNCT
ejpam-5411	593	5	∫	∫	PROPN
ejpam-5411	593	6	0	0	NUM
ejpam-5411	594	1	−∞	−∞	X
ejpam-5411	594	2	a1(t−	a1(t−	NOUN
ejpam-5411	594	3	s)g(s)ds	s)g(s)ds	NOUN
ejpam-5411	595	1	=	=	PROPN
ejpam-5411	596	1	+	+	ADJ
ejpam-5411	596	2	∞∑	∞∑	NUM
ejpam-5411	596	3	n=1	n=1	NUM
ejpam-5411	596	4	∫	∫	PROPN
ejpam-5411	596	5	t	t	PROPN
ejpam-5411	596	6	0	0	NUM
ejpam-5411	596	7	a1(t−	a1(t−	NOUN
ejpam-5411	596	8	u	u	NOUN
ejpam-5411	596	9	+	+	NOUN
ejpam-5411	596	10	nt	not	PART
ejpam-5411	596	11	)	)	PUNCT
ejpam-5411	596	12	g(u)du	g(u)du	X
ejpam-5411	597	1	=	=	PUNCT
ejpam-5411	598	1	+	+	NOUN
ejpam-5411	598	2	∞∑	∞∑	NUM
ejpam-5411	598	3	n=1	n=1	NUM
ejpam-5411	598	4	∫	∫	PROPN
ejpam-5411	598	5	t+nt	t+nt	VERB
ejpam-5411	598	6	t+(n−1)t	t+(n−1)t	NUM
ejpam-5411	599	1	a1(w)g(t−	a1(w)g(t−	NOUN
ejpam-5411	600	1	w)dw	w)dw	PROPN
ejpam-5411	600	2	=	=	PROPN
ejpam-5411	600	3	lim	lim	PROPN
ejpam-5411	600	4	n→+∞	n→+∞	PROPN
ejpam-5411	600	5	n∑	n∑	PROPN
ejpam-5411	600	6	n=1	n=1	PROPN
ejpam-5411	600	7	∫	∫	PROPN
ejpam-5411	600	8	t+nt	t+nt	VERB
ejpam-5411	600	9	t+(n−1)t	t+(n−1)t	NUM
ejpam-5411	601	1	a1(w)g(t−	a1(w)g(t−	NOUN
ejpam-5411	602	1	w)dw	w)dw	PROPN
ejpam-5411	602	2	=	=	PROPN
ejpam-5411	602	3	lim	lim	PROPN
ejpam-5411	602	4	n→+∞	n→+∞	PROPN
ejpam-5411	602	5	∫	∫	PROPN
ejpam-5411	603	1	t+nt	t+nt	X
ejpam-5411	603	2	t	t	PROPN
ejpam-5411	603	3	a1(w)g(t−	a1(w)g(t−	NUM
ejpam-5411	604	1	w)dw	w)dw	PROPN
ejpam-5411	604	2	=	=	SYM
ejpam-5411	605	1	∫	∫	PROPN
ejpam-5411	606	1	+	+	NUM
ejpam-5411	606	2	∞	∞	PROPN
ejpam-5411	606	3	t	t	PROPN
ejpam-5411	607	1	a1(w)g(t−	a1(w)g(t−	PRON
ejpam-5411	608	1	w)dw	w)dw	PROPN
ejpam-5411	608	2	and	and	CCONJ
ejpam-5411	608	3	f	f	PROPN
ejpam-5411	608	4	(	(	PUNCT
ejpam-5411	608	5	t	t	PROPN
ejpam-5411	608	6	)	)	PUNCT
ejpam-5411	609	1	=	=	SYM
ejpam-5411	609	2	∫	∫	PROPN
ejpam-5411	610	1	+	+	NUM
ejpam-5411	610	2	∞	∞	PROPN
ejpam-5411	610	3	0	0	NUM
ejpam-5411	611	1	a1(v)g(t−	a1(v)g(t−	PROPN
ejpam-5411	611	2	v)dv	v)dv	PROPN
ejpam-5411	611	3	then	then	ADV
ejpam-5411	611	4	for	for	ADP
ejpam-5411	611	5	all	all	DET
ejpam-5411	611	6	k	k	PROPN
ejpam-5411	611	7	∈	∈	PROPN
ejpam-5411	611	8	z	z	PROPN
ejpam-5411	611	9	fα((a	fα((a	PROPN
ejpam-5411	611	10	∗α	∗α	PROPN
ejpam-5411	611	11	f)−∞(t))(k	f)−∞(t))(k	PROPN
ejpam-5411	611	12	)	)	PUNCT
ejpam-5411	612	1	=	=	SYM
ejpam-5411	612	2	α	α	PRON
ejpam-5411	612	3	pα	pα	INTJ
ejpam-5411	612	4	∫	∫	PROPN
ejpam-5411	613	1	p	p	NOUN
ejpam-5411	613	2	0	0	NUM
ejpam-5411	613	3	e	e	X
ejpam-5411	613	4	−ik	−ik	NOUN
ejpam-5411	613	5	2π	2π	NOUN
ejpam-5411	613	6	pα	pα	INTJ
ejpam-5411	613	7	tα	tα	PROPN
ejpam-5411	613	8	f	f	PROPN
ejpam-5411	614	1	(	(	PUNCT
ejpam-5411	614	2	tα	tα	PROPN
ejpam-5411	614	3	α	α	NOUN
ejpam-5411	614	4	)	)	PUNCT
ejpam-5411	614	5	tα−1dt	tα−1dt	NUM
ejpam-5411	614	6	.	.	PUNCT
ejpam-5411	615	1	t.	t.	PROPN
ejpam-5411	615	2	abdeljawad	abdeljawad	PROPN
ejpam-5411	615	3	et	et	PROPN
ejpam-5411	615	4	al	al	PROPN
ejpam-5411	615	5	.	.	PUNCT
ejpam-5411	615	6	/	/	SYM
ejpam-5411	615	7	eur	eur	PROPN
ejpam-5411	615	8	.	.	PUNCT
ejpam-5411	616	1	j.	j.	PROPN
ejpam-5411	616	2	pure	pure	PROPN
ejpam-5411	616	3	appl	appl	PROPN
ejpam-5411	616	4	.	.	PROPN
ejpam-5411	616	5	math	math	PROPN
ejpam-5411	616	6	,	,	PUNCT
ejpam-5411	616	7	17	17	NUM
ejpam-5411	616	8	(	(	PUNCT
ejpam-5411	616	9	4	4	NUM
ejpam-5411	616	10	)	)	PUNCT
ejpam-5411	616	11	(	(	PUNCT
ejpam-5411	616	12	2024	2024	NUM
ejpam-5411	616	13	)	)	PUNCT
ejpam-5411	616	14	,	,	PUNCT
ejpam-5411	616	15	2405	2405	NUM
ejpam-5411	616	16	-	-	SYM
ejpam-5411	616	17	2430	2430	NUM
ejpam-5411	616	18	2426	2426	NUM
ejpam-5411	616	19	by	by	ADP
ejpam-5411	616	20	making	make	VERB
ejpam-5411	616	21	variable	variable	ADJ
ejpam-5411	616	22	change	change	NOUN
ejpam-5411	616	23	tα	tα	ADP
ejpam-5411	616	24	α	α	NOUN
ejpam-5411	616	25	=	=	SYM
ejpam-5411	616	26	u	u	PROPN
ejpam-5411	616	27	and	and	CCONJ
ejpam-5411	616	28	u−	u−	PROPN
ejpam-5411	616	29	s	s	PART
ejpam-5411	616	30	=	=	SYM
ejpam-5411	616	31	t	t	PROPN
ejpam-5411	616	32	,	,	PUNCT
ejpam-5411	616	33	we	we	PRON
ejpam-5411	616	34	have	have	VERB
ejpam-5411	616	35	fα((a	fα((a	PROPN
ejpam-5411	616	36	∗α	∗α	PROPN
ejpam-5411	616	37	f)−∞(t))(k	f)−∞(t))(k	PROPN
ejpam-5411	616	38	)	)	PUNCT
ejpam-5411	616	39	=	=	SYM
ejpam-5411	617	1	α	α	PRON
ejpam-5411	617	2	pα	pα	INTJ
ejpam-5411	617	3	∫	∫	PROPN
ejpam-5411	617	4	pα	pα	INTJ
ejpam-5411	617	5	α	α	PROPN
ejpam-5411	617	6	0	0	PUNCT
ejpam-5411	618	1	e	e	NOUN
ejpam-5411	618	2	−ik	−ik	NOUN
ejpam-5411	618	3	2πα	2πα	NOUN
ejpam-5411	618	4	pα	pα	VERB
ejpam-5411	618	5	u	u	X
ejpam-5411	618	6	[	[	PUNCT
ejpam-5411	618	7	∫	∫	PROPN
ejpam-5411	618	8	u	u	PROPN
ejpam-5411	618	9	−∞	−∞	ADP
ejpam-5411	618	10	a((α(u−	a((α(u−	PROPN
ejpam-5411	618	11	s	s	PROPN
ejpam-5411	618	12	)	)	PUNCT
ejpam-5411	618	13	)	)	PUNCT
ejpam-5411	618	14	1	1	NUM
ejpam-5411	618	15	α	α	NOUN
ejpam-5411	618	16	)	)	PUNCT
ejpam-5411	618	17	f((αs	f((αs	NOUN
ejpam-5411	618	18	)	)	PUNCT
ejpam-5411	618	19	1	1	NUM
ejpam-5411	618	20	α	α	NOUN
ejpam-5411	618	21	)	)	PUNCT
ejpam-5411	618	22	ds]du	ds]du	PROPN
ejpam-5411	619	1	=	=	PUNCT
ejpam-5411	619	2	α	α	PRON
ejpam-5411	619	3	pα	pα	INTJ
ejpam-5411	619	4	∫	∫	PROPN
ejpam-5411	619	5	pα	pα	INTJ
ejpam-5411	619	6	α	α	PROPN
ejpam-5411	619	7	0	0	PUNCT
ejpam-5411	620	1	e	e	NOUN
ejpam-5411	620	2	−ik	−ik	NOUN
ejpam-5411	620	3	2πα	2πα	NOUN
ejpam-5411	620	4	pα	pα	VERB
ejpam-5411	620	5	u	u	X
ejpam-5411	620	6	[	[	PUNCT
ejpam-5411	620	7	∫	∫	PROPN
ejpam-5411	621	1	+	+	PROPN
ejpam-5411	621	2	∞	∞	NOUN
ejpam-5411	621	3	0	0	NUM
ejpam-5411	621	4	a((αs	a((αs	X
ejpam-5411	621	5	)	)	PUNCT
ejpam-5411	621	6	1	1	NUM
ejpam-5411	621	7	α	α	NOUN
ejpam-5411	621	8	)	)	PUNCT
ejpam-5411	621	9	f((α(u−	f((α(u−	PROPN
ejpam-5411	621	10	s	s	PROPN
ejpam-5411	621	11	)	)	PUNCT
ejpam-5411	621	12	)	)	PUNCT
ejpam-5411	621	13	1	1	NUM
ejpam-5411	621	14	α	α	NOUN
ejpam-5411	621	15	)	)	PUNCT
ejpam-5411	621	16	ds]du	ds]du	PROPN
ejpam-5411	622	1	=	=	PUNCT
ejpam-5411	623	1	[	[	PUNCT
ejpam-5411	623	2	∫	∫	PROPN
ejpam-5411	624	1	+	+	NOUN
ejpam-5411	624	2	∞	∞	NOUN
ejpam-5411	624	3	0	0	NUM
ejpam-5411	624	4	a((αs	a((αs	X
ejpam-5411	624	5	)	)	PUNCT
ejpam-5411	624	6	1	1	NUM
ejpam-5411	624	7	α	α	NOUN
ejpam-5411	624	8	)	)	PUNCT
ejpam-5411	624	9	e	e	X
ejpam-5411	624	10	2ikπ	2ikπ	NUM
ejpam-5411	624	11	α	α	PROPN
ejpam-5411	624	12	pα	pα	NOUN
ejpam-5411	624	13	s	s	X
ejpam-5411	624	14	ds	ds	NOUN
ejpam-5411	624	15	]	]	X
ejpam-5411	624	16	[	[	PUNCT
ejpam-5411	624	17	α	α	X
ejpam-5411	624	18	pα	pα	INTJ
ejpam-5411	624	19	∫	∫	PROPN
ejpam-5411	624	20	pα	pα	INTJ
ejpam-5411	624	21	α	α	PROPN
ejpam-5411	624	22	0	0	PUNCT
ejpam-5411	625	1	e	e	NOUN
ejpam-5411	625	2	−ik	−ik	NOUN
ejpam-5411	625	3	2πα	2πα	NOUN
ejpam-5411	625	4	pα	pα	VERB
ejpam-5411	625	5	t	t	NOUN
ejpam-5411	625	6	f((α(t	f((α(t	PROPN
ejpam-5411	625	7	)	)	PUNCT
ejpam-5411	625	8	)	)	PUNCT
ejpam-5411	625	9	1	1	NUM
ejpam-5411	625	10	α	α	NOUN
ejpam-5411	625	11	)	)	PUNCT
ejpam-5411	625	12	dt	dt	X
ejpam-5411	626	1	=	=	SYM
ejpam-5411	626	2	lα(a(t))(2ikπ	lα(a(t))(2ikπ	PROPN
ejpam-5411	626	3	α	α	PROPN
ejpam-5411	626	4	pα	pα	NOUN
ejpam-5411	626	5	)	)	PUNCT
ejpam-5411	626	6	fα(f(t))(k	fα(f(t))(k	PROPN
ejpam-5411	626	7	)	)	PUNCT
ejpam-5411	626	8	.	.	PUNCT
ejpam-5411	627	1	example	example	NOUN
ejpam-5411	628	1	13	13	NUM
ejpam-5411	628	2	.	.	PUNCT
ejpam-5411	629	1	let	let	VERB
ejpam-5411	629	2	f1	f1	NOUN
ejpam-5411	629	3	defined	define	VERB
ejpam-5411	629	4	by	by	ADP
ejpam-5411	629	5	example	example	NOUN
ejpam-5411	629	6	3	3	NUM
ejpam-5411	629	7	and	and	CCONJ
ejpam-5411	629	8	a(t	a(t	NOUN
ejpam-5411	629	9	)	)	PUNCT
ejpam-5411	630	1	=	=	SYM
ejpam-5411	630	2	a1	a1	PROPN
ejpam-5411	630	3	(	(	PUNCT
ejpam-5411	630	4	tα	tα	PROPN
ejpam-5411	630	5	α	α	NOUN
ejpam-5411	630	6	)	)	PUNCT
ejpam-5411	630	7	such	such	ADJ
ejpam-5411	630	8	that	that	SCONJ
ejpam-5411	630	9	a1(t	a1(t	ADV
ejpam-5411	630	10	)	)	PUNCT
ejpam-5411	630	11	=	=	SYM
ejpam-5411	630	12	e−t	e−t	NOUN
ejpam-5411	630	13	∈	∈	PROPN
ejpam-5411	630	14	l1(r+	l1(r+	PROPN
ejpam-5411	630	15	)	)	PUNCT
ejpam-5411	630	16	.	.	PUNCT
ejpam-5411	631	1	the	the	DET
ejpam-5411	631	2	function	function	NOUN
ejpam-5411	631	3	f1	f1	NOUN
ejpam-5411	631	4	is	be	AUX
ejpam-5411	631	5	α	α	NOUN
ejpam-5411	631	6	-	-	NOUN
ejpam-5411	631	7	periodic	periodic	NOUN
ejpam-5411	631	8	with	with	ADP
ejpam-5411	631	9	period	period	NOUN
ejpam-5411	631	10	(	(	PUNCT
ejpam-5411	631	11	1	1	NUM
ejpam-5411	631	12	α	α	NOUN
ejpam-5411	631	13	)	)	PUNCT
ejpam-5411	631	14	1	1	NUM
ejpam-5411	631	15	α	α	NOUN
ejpam-5411	631	16	.	.	PUNCT
ejpam-5411	632	1	let	let	VERB
ejpam-5411	632	2	k	k	PROPN
ejpam-5411	632	3	∈	∈	PROPN
ejpam-5411	632	4	z	z	PROPN
ejpam-5411	632	5	and	and	CCONJ
ejpam-5411	632	6	t	t	PROPN
ejpam-5411	632	7	∈	∈	PROPN
ejpam-5411	633	1	[	[	X
ejpam-5411	633	2	0,+∞	0,+∞	NUM
ejpam-5411	633	3	[	[	X
ejpam-5411	633	4	,	,	PUNCT
ejpam-5411	633	5	we	we	PRON
ejpam-5411	633	6	have	have	VERB
ejpam-5411	633	7	fα((a	fα((a	PROPN
ejpam-5411	633	8	∗α	∗α	PROPN
ejpam-5411	633	9	f1)−∞(t))(k	f1)−∞(t))(k	PROPN
ejpam-5411	633	10	)	)	PUNCT
ejpam-5411	633	11	=	=	SYM
ejpam-5411	633	12	f((a	f((a	NOUN
ejpam-5411	633	13	∗α	∗α	NOUN
ejpam-5411	633	14	f1)−∞((αt	f1)−∞((αt	AUX
ejpam-5411	633	15	)	)	PUNCT
ejpam-5411	633	16	1	1	NUM
ejpam-5411	633	17	α	α	NOUN
ejpam-5411	633	18	)	)	PUNCT
ejpam-5411	633	19	)	)	PUNCT
ejpam-5411	633	20	(	(	PUNCT
ejpam-5411	633	21	k	k	X
ejpam-5411	633	22	)	)	PUNCT
ejpam-5411	633	23	=	=	VERB
ejpam-5411	634	1	α2	α2	ADJ
ejpam-5411	634	2	∫	∫	PROPN
ejpam-5411	634	3	1	1	NUM
ejpam-5411	634	4	α2	α2	ADJ
ejpam-5411	634	5	0	0	NUM
ejpam-5411	634	6	e−2ikπα2tf	e−2ikπα2tf	NOUN
ejpam-5411	634	7	(	(	PUNCT
ejpam-5411	634	8	t)dt	t)dt	PROPN
ejpam-5411	634	9	=	=	PUNCT
ejpam-5411	634	10	α2	α2	PROPN
ejpam-5411	634	11	∫	∫	PROPN
ejpam-5411	635	1	1	1	NUM
ejpam-5411	635	2	α2	α2	PROPN
ejpam-5411	635	3	0	0	NUM
ejpam-5411	635	4	e−2ikπα2	e−2ikπα2	PROPN
ejpam-5411	635	5	t	t	PROPN
ejpam-5411	635	6	[	[	PUNCT
ejpam-5411	635	7	∫	∫	PROPN
ejpam-5411	635	8	0	0	NUM
ejpam-5411	636	1	−∞	−∞	X
ejpam-5411	636	2	a1(t−	a1(t−	NOUN
ejpam-5411	636	3	s)g1(s)ds	s)g1(s)ds	PROPN
ejpam-5411	636	4	+	+	CCONJ
ejpam-5411	636	5	∫	∫	PROPN
ejpam-5411	636	6	t	t	PROPN
ejpam-5411	636	7	0	0	NUM
ejpam-5411	637	1	a1(t−	a1(t−	NOUN
ejpam-5411	637	2	s)g1(s)ds]dt	s)g1(s)ds]dt	PROPN
ejpam-5411	637	3	=	=	PROPN
ejpam-5411	637	4	i1	i1	PROPN
ejpam-5411	637	5	+	+	CCONJ
ejpam-5411	637	6	i2	i2	PROPN
ejpam-5411	637	7	such	such	ADJ
ejpam-5411	637	8	that	that	DET
ejpam-5411	637	9	i1	i1	PROPN
ejpam-5411	637	10	=	=	PUNCT
ejpam-5411	637	11	α2	α2	PROPN
ejpam-5411	637	12	∫	∫	PROPN
ejpam-5411	638	1	1	1	NUM
ejpam-5411	638	2	α2	α2	PROPN
ejpam-5411	638	3	0	0	NUM
ejpam-5411	638	4	e−2ikπα2	e−2ikπα2	PROPN
ejpam-5411	638	5	t	t	PROPN
ejpam-5411	638	6	(	(	PUNCT
ejpam-5411	638	7	∫	∫	PROPN
ejpam-5411	638	8	0	0	PROPN
ejpam-5411	639	1	−∞	−∞	NOUN
ejpam-5411	639	2	a1(t−	a1(t−	NOUN
ejpam-5411	639	3	s)g1(s)ds)dt	s)g1(s)ds)dt	PROPN
ejpam-5411	639	4	and	and	CCONJ
ejpam-5411	639	5	i2	i2	PROPN
ejpam-5411	639	6	=	=	SYM
ejpam-5411	639	7	α2	α2	PROPN
ejpam-5411	639	8	∫	∫	PROPN
ejpam-5411	639	9	1	1	NUM
ejpam-5411	639	10	α2	α2	PROPN
ejpam-5411	639	11	0	0	NUM
ejpam-5411	639	12	e−2ikπα2	e−2ikπα2	PROPN
ejpam-5411	639	13	t	t	PROPN
ejpam-5411	639	14	(	(	PUNCT
ejpam-5411	639	15	∫	∫	PROPN
ejpam-5411	639	16	t	t	PROPN
ejpam-5411	639	17	0	0	NUM
ejpam-5411	639	18	a1(t−	a1(t−	NOUN
ejpam-5411	639	19	s)g1(s)ds)dt	s)g1(s)ds)dt	PROPN
ejpam-5411	639	20	.	.	PUNCT
ejpam-5411	640	1	by	by	ADP
ejpam-5411	640	2	lemma	lemma	PROPN
ejpam-5411	640	3	3	3	NUM
ejpam-5411	640	4	∫	∫	NOUN
ejpam-5411	640	5	0	0	NUM
ejpam-5411	641	1	−∞	−∞	X
ejpam-5411	641	2	a1(t−	a1(t−	NOUN
ejpam-5411	641	3	s)g(s)ds	s)g(s)ds	NOUN
ejpam-5411	641	4	=	=	PUNCT
ejpam-5411	641	5	(	(	PUNCT
ejpam-5411	641	6	e	e	PROPN
ejpam-5411	641	7	1	1	NUM
ejpam-5411	641	8	2α2	2α2	NUM
ejpam-5411	641	9	−	−	NUM
ejpam-5411	641	10	1	1	NUM
ejpam-5411	641	11	e	e	NOUN
ejpam-5411	641	12	1	1	NUM
ejpam-5411	641	13	2α2	2α2	NUM
ejpam-5411	641	14	+	+	CCONJ
ejpam-5411	641	15	1	1	NUM
ejpam-5411	641	16	)	)	PUNCT
ejpam-5411	641	17	e−t	e−t	NOUN
ejpam-5411	641	18	then	then	ADV
ejpam-5411	641	19	i1	i1	PROPN
ejpam-5411	641	20	=	=	PUNCT
ejpam-5411	642	1	α2(−2e	α2(−2e	NUM
ejpam-5411	642	2	1	1	NUM
ejpam-5411	642	3	2α2	2α2	NUM
ejpam-5411	643	1	+	+	CCONJ
ejpam-5411	643	2	e	e	X
ejpam-5411	643	3	1	1	NUM
ejpam-5411	643	4	α2	α2	ADJ
ejpam-5411	643	5	+	+	CCONJ
ejpam-5411	643	6	1)e−	1)e−	NUM
ejpam-5411	643	7	1	1	NUM
ejpam-5411	643	8	α2	α2	NOUN
ejpam-5411	643	9	2ikπα2	2ikπα2	NUM
ejpam-5411	644	1	+	+	CCONJ
ejpam-5411	644	2	1	1	NUM
ejpam-5411	644	3	and	and	CCONJ
ejpam-5411	644	4	i2	i2	PROPN
ejpam-5411	644	5	=	=	SYM
ejpam-5411	644	6	α2	α2	PROPN
ejpam-5411	644	7	∫	∫	PROPN
ejpam-5411	645	1	1	1	NUM
ejpam-5411	645	2	α2	α2	NOUN
ejpam-5411	645	3	0	0	PUNCT
ejpam-5411	646	1	e−(2ikπα2	e−(2ikπα2	PROPN
ejpam-5411	646	2	+	+	PROPN
ejpam-5411	646	3	1)t	1)t	PROPN
ejpam-5411	646	4	(	(	PUNCT
ejpam-5411	646	5	∫	∫	PROPN
ejpam-5411	646	6	t	t	PROPN
ejpam-5411	646	7	0	0	NUM
ejpam-5411	646	8	esg1(s)ds)dt	esg1(s)ds)dt	PROPN
ejpam-5411	646	9	t.	t.	NOUN
ejpam-5411	646	10	abdeljawad	abdeljawad	NOUN
ejpam-5411	646	11	et	et	PROPN
ejpam-5411	646	12	al	al	PROPN
ejpam-5411	646	13	.	.	PUNCT
ejpam-5411	646	14	/	/	SYM
ejpam-5411	646	15	eur	eur	PROPN
ejpam-5411	646	16	.	.	PUNCT
ejpam-5411	647	1	j.	j.	PROPN
ejpam-5411	647	2	pure	pure	PROPN
ejpam-5411	647	3	appl	appl	PROPN
ejpam-5411	647	4	.	.	PROPN
ejpam-5411	647	5	math	math	PROPN
ejpam-5411	647	6	,	,	PUNCT
ejpam-5411	647	7	17	17	NUM
ejpam-5411	647	8	(	(	PUNCT
ejpam-5411	647	9	4	4	NUM
ejpam-5411	647	10	)	)	PUNCT
ejpam-5411	647	11	(	(	PUNCT
ejpam-5411	647	12	2024	2024	NUM
ejpam-5411	647	13	)	)	PUNCT
ejpam-5411	647	14	,	,	PUNCT
ejpam-5411	647	15	2405	2405	NUM
ejpam-5411	647	16	-	-	SYM
ejpam-5411	647	17	2430	2430	NUM
ejpam-5411	647	18	2427	2427	NUM
ejpam-5411	647	19	=	=	PUNCT
ejpam-5411	647	20	−	−	NOUN
ejpam-5411	647	21	α2	α2	NOUN
ejpam-5411	647	22	2ikπα2	2ikπα2	NUM
ejpam-5411	648	1	+	+	CCONJ
ejpam-5411	648	2	1	1	NUM
ejpam-5411	648	3	{	{	PUNCT
ejpam-5411	648	4	e−	e−	X
ejpam-5411	648	5	1	1	NUM
ejpam-5411	648	6	α2	α2	ADJ
ejpam-5411	648	7	∫	∫	PROPN
ejpam-5411	648	8	1	1	NUM
ejpam-5411	648	9	α2	α2	PROPN
ejpam-5411	648	10	0	0	NUM
ejpam-5411	649	1	etg1(t)dt−	etg1(t)dt−	CCONJ
ejpam-5411	649	2	∫	∫	PROPN
ejpam-5411	649	3	1	1	NUM
ejpam-5411	650	1	α2	α2	PROPN
ejpam-5411	650	2	0	0	NUM
ejpam-5411	650	3	e−ik2πα2tg1(t)dt	e−ik2πα2tg1(t)dt	NOUN
ejpam-5411	650	4	}	}	PUNCT
ejpam-5411	650	5	=	=	PUNCT
ejpam-5411	650	6	−	−	PROPN
ejpam-5411	650	7	α2	α2	PROPN
ejpam-5411	650	8	2ikπα2	2ikπα2	NUM
ejpam-5411	651	1	+	+	CCONJ
ejpam-5411	651	2	1	1	NUM
ejpam-5411	651	3	{	{	PUNCT
ejpam-5411	651	4	e−	e−	PROPN
ejpam-5411	651	5	1	1	NUM
ejpam-5411	651	6	α2	α2	ADJ
ejpam-5411	651	7	[	[	PUNCT
ejpam-5411	651	8	∫	∫	PROPN
ejpam-5411	651	9	1	1	NUM
ejpam-5411	651	10	2α2	2α2	NUM
ejpam-5411	651	11	0	0	NUM
ejpam-5411	651	12	tetdt	tetdt	ADJ
ejpam-5411	652	1	+	+	CCONJ
ejpam-5411	652	2	∫	∫	PROPN
ejpam-5411	652	3	1	1	NUM
ejpam-5411	652	4	α2	α2	ADJ
ejpam-5411	652	5	1	1	NUM
ejpam-5411	652	6	2α2	2α2	NUM
ejpam-5411	652	7	(	(	PUNCT
ejpam-5411	652	8	1	1	NUM
ejpam-5411	652	9	α2	α2	ADJ
ejpam-5411	652	10	−	−	PROPN
ejpam-5411	653	1	t)etdt	t)etdt	PROPN
ejpam-5411	653	2	]	]	PUNCT
ejpam-5411	653	3	−	−	PROPN
ejpam-5411	653	4	∫	∫	PROPN
ejpam-5411	654	1	1	1	NUM
ejpam-5411	654	2	2α2	2α2	NUM
ejpam-5411	654	3	0	0	NUM
ejpam-5411	654	4	te−2ikπα2tdt−	te−2ikπα2tdt−	NUM
ejpam-5411	654	5	∫	∫	NOUN
ejpam-5411	654	6	1	1	NUM
ejpam-5411	654	7	α2	α2	ADJ
ejpam-5411	654	8	1	1	NUM
ejpam-5411	654	9	2α2	2α2	NUM
ejpam-5411	654	10	(	(	PUNCT
ejpam-5411	654	11	1	1	NUM
ejpam-5411	654	12	α2	α2	ADJ
ejpam-5411	654	13	−	−	NOUN
ejpam-5411	654	14	t)e−2ikπα2tdt	t)e−2ikπα2tdt	NOUN
ejpam-5411	654	15	}	}	PUNCT
ejpam-5411	654	16	=	=	SYM
ejpam-5411	654	17	−2π2e−	−2π2e−	PROPN
ejpam-5411	654	18	1	1	NUM
ejpam-5411	654	19	α2	α2	ADJ
ejpam-5411	654	20	α4k2	α4k2	X
ejpam-5411	654	21	+	+	CCONJ
ejpam-5411	654	22	4π4e−	4π4e−	NUM
ejpam-5411	654	23	1	1	NUM
ejpam-5411	654	24	2α2	2α2	NUM
ejpam-5411	654	25	α4k2	α4k2	X
ejpam-5411	654	26	−	−	PROPN
ejpam-5411	654	27	2k2π2α4	2k2π2α4	NUM
ejpam-5411	654	28	+	+	CCONJ
ejpam-5411	654	29	(	(	PUNCT
ejpam-5411	654	30	−1)k	−1)k	PROPN
ejpam-5411	654	31	−	−	PROPN
ejpam-5411	654	32	1	1	NUM
ejpam-5411	654	33	2α2π2k2(2ikπα2	2α2π2k2(2ikπα2	NUM
ejpam-5411	654	34	+	+	CCONJ
ejpam-5411	654	35	1	1	NUM
ejpam-5411	654	36	)	)	PUNCT
ejpam-5411	654	37	.	.	PUNCT
ejpam-5411	655	1	thus	thus	ADV
ejpam-5411	655	2	fα((a	fα((a	X
ejpam-5411	655	3	∗α	∗α	PROPN
ejpam-5411	655	4	f1)−∞(t))(k	f1)−∞(t))(k	PRON
ejpam-5411	655	5	)	)	PUNCT
ejpam-5411	655	6	=	=	SYM
ejpam-5411	655	7	(	(	PUNCT
ejpam-5411	655	8	−1)k	−1)k	PROPN
ejpam-5411	655	9	−	−	PROPN
ejpam-5411	655	10	1	1	NUM
ejpam-5411	655	11	2α2π2k2(2ikπα2	2α2π2k2(2ikπα2	NUM
ejpam-5411	655	12	+	+	CCONJ
ejpam-5411	655	13	1	1	NUM
ejpam-5411	655	14	)	)	PUNCT
ejpam-5411	655	15	.	.	PUNCT
ejpam-5411	656	1	on	on	ADP
ejpam-5411	656	2	the	the	DET
ejpam-5411	656	3	other	other	ADJ
ejpam-5411	656	4	hand	hand	NOUN
ejpam-5411	656	5	,	,	PUNCT
ejpam-5411	656	6	we	we	PRON
ejpam-5411	656	7	have	have	VERB
ejpam-5411	656	8	lα(a(t))(2ikπα2	lα(a(t))(2ikπα2	NUM
ejpam-5411	656	9	)	)	PUNCT
ejpam-5411	656	10	=	=	SYM
ejpam-5411	657	1	1	1	NUM
ejpam-5411	657	2	2ikπα2	2ikπα2	NUM
ejpam-5411	657	3	+	+	CCONJ
ejpam-5411	657	4	1	1	NUM
ejpam-5411	657	5	and	and	CCONJ
ejpam-5411	657	6	by	by	ADP
ejpam-5411	657	7	example	example	NOUN
ejpam-5411	657	8	8	8	NUM
ejpam-5411	657	9	fα(f1(t))(k	fα(f1(t))(k	NOUN
ejpam-5411	657	10	)	)	PUNCT
ejpam-5411	657	11	=	=	PUNCT
ejpam-5411	658	1	(	(	PUNCT
ejpam-5411	658	2	−1)k	−1)k	PROPN
ejpam-5411	658	3	−	−	PROPN
ejpam-5411	658	4	1	1	NUM
ejpam-5411	658	5	2α2π2k2	2α2π2k2	NUM
ejpam-5411	658	6	then	then	ADV
ejpam-5411	658	7	fα((a	fα((a	PROPN
ejpam-5411	658	8	∗α	∗α	PROPN
ejpam-5411	658	9	f1)−∞(t))(k	f1)−∞(t))(k	PRON
ejpam-5411	658	10	)	)	PUNCT
ejpam-5411	658	11	=	=	SYM
ejpam-5411	658	12	lα(a(t))(2ikπα2)fα(f1((t))(k	lα(a(t))(2ikπα2)fα(f1((t))(k	PROPN
ejpam-5411	658	13	)	)	PUNCT
ejpam-5411	658	14	,	,	PUNCT
ejpam-5411	658	15	∀k	∀k	X
ejpam-5411	658	16	∈	∈	PROPN
ejpam-5411	658	17	z∗.	z∗.	PROPN
ejpam-5411	658	18	for	for	ADP
ejpam-5411	658	19	k	k	PROPN
ejpam-5411	658	20	=	=	SYM
ejpam-5411	658	21	0	0	PROPN
ejpam-5411	658	22	,	,	PUNCT
ejpam-5411	658	23	we	we	PRON
ejpam-5411	658	24	have	have	VERB
ejpam-5411	658	25	fα((a	fα((a	PROPN
ejpam-5411	658	26	∗α	∗α	PROPN
ejpam-5411	658	27	f1)−∞(t))(0	f1)−∞(t))(0	PROPN
ejpam-5411	658	28	)	)	PUNCT
ejpam-5411	658	29	=	=	SYM
ejpam-5411	658	30	i1	i1	PROPN
ejpam-5411	658	31	+	+	CCONJ
ejpam-5411	658	32	i2	i2	PROPN
ejpam-5411	658	33	such	such	ADJ
ejpam-5411	658	34	that	that	PRON
ejpam-5411	658	35	i1	i1	PROPN
ejpam-5411	658	36	=	=	PUNCT
ejpam-5411	658	37	α2	α2	PROPN
ejpam-5411	658	38	∫	∫	PROPN
ejpam-5411	659	1	1	1	NUM
ejpam-5411	659	2	α2	α2	ADJ
ejpam-5411	659	3	0	0	NUM
ejpam-5411	660	1	e−t	e−t	PROPN
ejpam-5411	660	2	(	(	PUNCT
ejpam-5411	660	3	∫	∫	PROPN
ejpam-5411	660	4	0	0	PROPN
ejpam-5411	661	1	−∞	−∞	ADP
ejpam-5411	661	2	esg1(s)ds)dt	esg1(s)ds)dt	PROPN
ejpam-5411	661	3	=	=	SYM
ejpam-5411	661	4	−α2(e−	−α2(e−	PROPN
ejpam-5411	661	5	1	1	NUM
ejpam-5411	661	6	α2	α2	ADJ
ejpam-5411	661	7	−	−	PROPN
ejpam-5411	662	1	1)(e	1)(e	NUM
ejpam-5411	662	2	1	1	NUM
ejpam-5411	662	3	2α2	2α2	NUM
ejpam-5411	662	4	−	−	NOUN
ejpam-5411	662	5	1	1	NUM
ejpam-5411	662	6	)	)	PUNCT
ejpam-5411	662	7	e	e	NOUN
ejpam-5411	662	8	1	1	NUM
ejpam-5411	662	9	2α2	2α2	NUM
ejpam-5411	662	10	+	+	CCONJ
ejpam-5411	662	11	1	1	NUM
ejpam-5411	662	12	and	and	CCONJ
ejpam-5411	662	13	i2	i2	PROPN
ejpam-5411	662	14	=	=	SYM
ejpam-5411	662	15	α2	α2	PROPN
ejpam-5411	662	16	∫	∫	PROPN
ejpam-5411	662	17	1	1	NUM
ejpam-5411	662	18	α2	α2	ADJ
ejpam-5411	662	19	0	0	NUM
ejpam-5411	663	1	e−t	e−t	PROPN
ejpam-5411	663	2	(	(	PUNCT
ejpam-5411	663	3	∫	∫	PROPN
ejpam-5411	663	4	t	t	PROPN
ejpam-5411	663	5	0	0	NUM
ejpam-5411	663	6	esg1(s)ds)dt	esg1(s)ds)dt	PROPN
ejpam-5411	663	7	=	=	SYM
ejpam-5411	663	8	α2(−e−	α2(−e−	PROPN
ejpam-5411	663	9	1	1	NUM
ejpam-5411	663	10	α2	α2	ADJ
ejpam-5411	663	11	∫	∫	PROPN
ejpam-5411	663	12	1	1	NUM
ejpam-5411	663	13	α2	α2	NOUN
ejpam-5411	663	14	0	0	NUM
ejpam-5411	663	15	etg1(t)dt	etg1(t)dt	PUNCT
ejpam-5411	664	1	+	+	CCONJ
ejpam-5411	664	2	∫	∫	PROPN
ejpam-5411	664	3	1	1	NUM
ejpam-5411	664	4	α2	α2	PROPN
ejpam-5411	664	5	0	0	NUM
ejpam-5411	664	6	g1(t)dt	g1(t)dt	NOUN
ejpam-5411	664	7	)	)	PUNCT
ejpam-5411	664	8	=	=	SYM
ejpam-5411	664	9	−α2e−	−α2e−	ADP
ejpam-5411	664	10	1	1	NUM
ejpam-5411	664	11	α2	α2	ADJ
ejpam-5411	664	12	(	(	PUNCT
ejpam-5411	664	13	−2e	−2e	PROPN
ejpam-5411	664	14	1	1	NUM
ejpam-5411	664	15	2α2	2α2	NUM
ejpam-5411	664	16	+	+	CCONJ
ejpam-5411	664	17	e	e	X
ejpam-5411	664	18	1	1	NUM
ejpam-5411	664	19	α2	α2	ADJ
ejpam-5411	664	20	+	+	CCONJ
ejpam-5411	664	21	1	1	X
ejpam-5411	664	22	)	)	PUNCT
ejpam-5411	665	1	+	+	CCONJ
ejpam-5411	665	2	1	1	NUM
ejpam-5411	665	3	4α2	4α2	NUM
ejpam-5411	665	4	.	.	PUNCT
ejpam-5411	666	1	then	then	ADV
ejpam-5411	666	2	by	by	ADP
ejpam-5411	666	3	example	example	NOUN
ejpam-5411	666	4	8	8	NUM
ejpam-5411	666	5	,	,	PUNCT
ejpam-5411	666	6	we	we	PRON
ejpam-5411	666	7	have	have	VERB
ejpam-5411	666	8	fα((a	fα((a	PROPN
ejpam-5411	666	9	∗α	∗α	PROPN
ejpam-5411	666	10	f1)−∞(t))(0	f1)−∞(t))(0	PROPN
ejpam-5411	666	11	)	)	PUNCT
ejpam-5411	666	12	=	=	SYM
ejpam-5411	667	1	1	1	NUM
ejpam-5411	667	2	4α2	4α2	NOUN
ejpam-5411	667	3	=	=	SYM
ejpam-5411	667	4	lα(a(t))(2ikπα2)fα(f1(t))(0	lα(a(t))(2ikπα2)fα(f1(t))(0	PROPN
ejpam-5411	667	5	)	)	PUNCT
ejpam-5411	667	6	.	.	PUNCT
ejpam-5411	668	1	finally	finally	ADV
ejpam-5411	668	2	fα((a	fα((a	PROPN
ejpam-5411	668	3	∗α	∗α	PROPN
ejpam-5411	668	4	f1)−∞(t))(k	f1)−∞(t))(k	NOUN
ejpam-5411	668	5	)	)	PUNCT
ejpam-5411	668	6	=	=	SYM
ejpam-5411	668	7	lα(a(t))(2ikπα2)fα(f1(t))(k	lα(a(t))(2ikπα2)fα(f1(t))(k	PROPN
ejpam-5411	668	8	)	)	PUNCT
ejpam-5411	668	9	,	,	PUNCT
ejpam-5411	668	10	∀k	∀k	X
ejpam-5411	668	11	∈	∈	PROPN
ejpam-5411	668	12	z.	z.	NOUN
ejpam-5411	668	13	references	reference	VERB
ejpam-5411	668	14	2428	2428	NUM
ejpam-5411	668	15	5	5	NUM
ejpam-5411	668	16	.	.	PUNCT
ejpam-5411	668	17	conclusion	conclusion	VERB
ejpam-5411	668	18	the	the	DET
ejpam-5411	668	19	definition	definition	NOUN
ejpam-5411	668	20	of	of	ADP
ejpam-5411	668	21	α	α	NOUN
ejpam-5411	668	22	-	-	ADJ
ejpam-5411	668	23	periodic	periodic	ADJ
ejpam-5411	668	24	function	function	NOUN
ejpam-5411	668	25	introduced	introduce	VERB
ejpam-5411	668	26	by	by	ADP
ejpam-5411	668	27	khalil	khalil	PROPN
ejpam-5411	668	28	et	et	PROPN
ejpam-5411	668	29	al	al	PROPN
ejpam-5411	669	1	[	[	X
ejpam-5411	669	2	8	8	NUM
ejpam-5411	669	3	]	]	PUNCT
ejpam-5411	669	4	has	have	AUX
ejpam-5411	669	5	been	be	AUX
ejpam-5411	669	6	investigated	investigate	VERB
ejpam-5411	669	7	.	.	PUNCT
ejpam-5411	670	1	many	many	ADJ
ejpam-5411	670	2	results	result	NOUN
ejpam-5411	670	3	and	and	CCONJ
ejpam-5411	670	4	examples	example	NOUN
ejpam-5411	670	5	related	relate	VERB
ejpam-5411	670	6	to	to	ADP
ejpam-5411	670	7	this	this	DET
ejpam-5411	670	8	definition	definition	NOUN
ejpam-5411	670	9	have	have	AUX
ejpam-5411	670	10	been	be	AUX
ejpam-5411	670	11	given	give	VERB
ejpam-5411	670	12	and	and	CCONJ
ejpam-5411	670	13	proved	prove	VERB
ejpam-5411	670	14	.	.	PUNCT
ejpam-5411	671	1	a	a	DET
ejpam-5411	671	2	new	new	ADJ
ejpam-5411	671	3	definition	definition	NOUN
ejpam-5411	671	4	of	of	ADP
ejpam-5411	671	5	conformable	conformable	ADJ
ejpam-5411	671	6	fourier	fourier	NOUN
ejpam-5411	671	7	transform	transform	NOUN
ejpam-5411	671	8	for	for	ADP
ejpam-5411	671	9	α	α	PRON
ejpam-5411	671	10	-	-	ADJ
ejpam-5411	671	11	periodic	periodic	ADJ
ejpam-5411	671	12	function	function	NOUN
ejpam-5411	671	13	has	have	AUX
ejpam-5411	671	14	been	be	AUX
ejpam-5411	671	15	given	give	VERB
ejpam-5411	671	16	.	.	PUNCT
ejpam-5411	672	1	a	a	DET
ejpam-5411	672	2	relationship	relationship	NOUN
ejpam-5411	672	3	between	between	ADP
ejpam-5411	672	4	the	the	DET
ejpam-5411	672	5	conformable	conformable	ADJ
ejpam-5411	672	6	fourier	fourier	NOUN
ejpam-5411	672	7	transform	transform	NOUN
ejpam-5411	672	8	and	and	CCONJ
ejpam-5411	672	9	the	the	DET
ejpam-5411	672	10	classical	classical	ADJ
ejpam-5411	672	11	fourier	fourier	NOUN
ejpam-5411	672	12	transform	transform	NOUN
ejpam-5411	672	13	have	have	AUX
ejpam-5411	672	14	been	be	AUX
ejpam-5411	672	15	established	establish	VERB
ejpam-5411	672	16	.	.	PUNCT
ejpam-5411	673	1	many	many	ADJ
ejpam-5411	673	2	results	result	NOUN
ejpam-5411	673	3	relating	relate	VERB
ejpam-5411	673	4	to	to	ADP
ejpam-5411	673	5	the	the	DET
ejpam-5411	673	6	classical	classical	ADJ
ejpam-5411	673	7	fourier	fourier	NOUN
ejpam-5411	673	8	case	case	NOUN
ejpam-5411	673	9	have	have	AUX
ejpam-5411	673	10	been	be	AUX
ejpam-5411	673	11	obtained	obtain	VERB
ejpam-5411	673	12	and	and	CCONJ
ejpam-5411	673	13	demonstrated	demonstrate	VERB
ejpam-5411	673	14	in	in	ADP
ejpam-5411	673	15	the	the	DET
ejpam-5411	673	16	conformable	conformable	ADJ
ejpam-5411	673	17	fourier	fourier	NOUN
ejpam-5411	673	18	case	case	NOUN
ejpam-5411	673	19	.	.	PUNCT
ejpam-5411	674	1	many	many	ADJ
ejpam-5411	674	2	examples	example	NOUN
ejpam-5411	674	3	have	have	AUX
ejpam-5411	674	4	been	be	AUX
ejpam-5411	674	5	constructed	construct	VERB
ejpam-5411	674	6	to	to	PART
ejpam-5411	674	7	illustrate	illustrate	VERB
ejpam-5411	674	8	these	these	DET
ejpam-5411	674	9	results	result	NOUN
ejpam-5411	674	10	.	.	PUNCT
ejpam-5411	675	1	our	our	PRON
ejpam-5411	675	2	interest	interest	NOUN
ejpam-5411	675	3	for	for	ADP
ejpam-5411	675	4	future	future	ADJ
ejpam-5411	675	5	work	work	NOUN
ejpam-5411	675	6	is	be	AUX
ejpam-5411	675	7	to	to	PART
ejpam-5411	675	8	apply	apply	VERB
ejpam-5411	675	9	this	this	DET
ejpam-5411	675	10	results	result	NOUN
ejpam-5411	675	11	to	to	PART
ejpam-5411	675	12	solve	solve	VERB
ejpam-5411	675	13	some	some	DET
ejpam-5411	675	14	conformable	conformable	ADJ
ejpam-5411	675	15	partial	partial	ADJ
ejpam-5411	675	16	differential	differential	NOUN
ejpam-5411	675	17	equations	equation	NOUN
ejpam-5411	675	18	,	,	PUNCT
ejpam-5411	675	19	conformable	conformable	ADJ
ejpam-5411	675	20	ordinary	ordinary	ADJ
ejpam-5411	675	21	differential	differential	ADJ
ejpam-5411	675	22	equations	equation	NOUN
ejpam-5411	675	23	,	,	PUNCT
ejpam-5411	675	24	conformable	conformable	ADJ
ejpam-5411	675	25	integro	integro	ADJ
ejpam-5411	675	26	-	-	PUNCT
ejpam-5411	675	27	differential	differential	NOUN
ejpam-5411	675	28	equations	equation	NOUN
ejpam-5411	675	29	and	and	CCONJ
ejpam-5411	675	30	conformable	conformable	ADJ
ejpam-5411	675	31	cauchy	cauchy	NOUN
ejpam-5411	675	32	problems	problem	NOUN
ejpam-5411	675	33	.	.	PUNCT
ejpam-5411	676	1	also	also	ADV
ejpam-5411	676	2	,	,	PUNCT
ejpam-5411	676	3	it	it	PRON
ejpam-5411	676	4	may	may	AUX
ejpam-5411	676	5	be	be	AUX
ejpam-5411	676	6	of	of	ADP
ejpam-5411	676	7	interest	interest	NOUN
ejpam-5411	676	8	to	to	PART
ejpam-5411	676	9	investigate	investigate	VERB
ejpam-5411	676	10	several	several	ADJ
ejpam-5411	676	11	modifications	modification	NOUN
ejpam-5411	676	12	of	of	ADP
ejpam-5411	676	13	the	the	DET
ejpam-5411	676	14	introduced	introduce	VERB
ejpam-5411	676	15	conformable	conformable	ADJ
ejpam-5411	676	16	fourier	fourier	NOUN
ejpam-5411	676	17	transform	transform	NOUN
ejpam-5411	676	18	in	in	ADP
ejpam-5411	676	19	this	this	DET
ejpam-5411	676	20	article	article	NOUN
ejpam-5411	676	21	to	to	PART
ejpam-5411	676	22	serve	serve	VERB
ejpam-5411	676	23	other	other	ADJ
ejpam-5411	676	24	modifications	modification	NOUN
ejpam-5411	676	25	of	of	ADP
ejpam-5411	676	26	conformable	conformable	ADJ
ejpam-5411	676	27	derivatives	derivative	NOUN
ejpam-5411	676	28	,	,	PUNCT
ejpam-5411	676	29	such	such	ADJ
ejpam-5411	676	30	as	as	ADP
ejpam-5411	676	31	m−truncated	m−truncate	VERB
ejpam-5411	676	32	fractional	fractional	ADJ
ejpam-5411	676	33	derivatives	derivative	NOUN
ejpam-5411	676	34	.	.	PUNCT
ejpam-5411	677	1	acknowledgements	acknowledgement	NOUN
ejpam-5411	677	2	the	the	DET
ejpam-5411	677	3	authors	author	NOUN
ejpam-5411	677	4	t.	t.	PROPN
ejpam-5411	677	5	abdeljawad	abdeljawad	PROPN
ejpam-5411	677	6	and	and	CCONJ
ejpam-5411	677	7	b.	b.	PROPN
ejpam-5411	677	8	abdalla	abdalla	PROPN
ejpam-5411	677	9	would	would	AUX
ejpam-5411	677	10	like	like	VERB
ejpam-5411	677	11	to	to	PART
ejpam-5411	677	12	thank	thank	VERB
ejpam-5411	677	13	prince	prince	PROPN
ejpam-5411	677	14	sultan	sultan	PROPN
ejpam-5411	677	15	university	university	PROPN
ejpam-5411	677	16	for	for	ADP
ejpam-5411	677	17	the	the	DET
ejpam-5411	677	18	support	support	NOUN
ejpam-5411	677	19	through	through	ADP
ejpam-5411	677	20	tas	tas	PROPN
ejpam-5411	677	21	research	research	NOUN
ejpam-5411	677	22	lab	lab	NOUN
ejpam-5411	677	23	.	.	PUNCT
ejpam-5411	678	1	references	reference	NOUN
ejpam-5411	678	2	[	[	X
ejpam-5411	678	3	1	1	NUM
ejpam-5411	678	4	]	]	PUNCT
ejpam-5411	678	5	t	t	NOUN
ejpam-5411	678	6	abdeljawad	abdeljawad	NOUN
ejpam-5411	678	7	.	.	PUNCT
ejpam-5411	679	1	on	on	ADP
ejpam-5411	679	2	conformable	conformable	ADJ
ejpam-5411	679	3	fractional	fractional	ADJ
ejpam-5411	679	4	calculus	calculus	NOUN
ejpam-5411	679	5	.	.	PUNCT
ejpam-5411	680	1	journal	journal	PROPN
ejpam-5411	680	2	of	of	ADP
ejpam-5411	680	3	computational	computational	ADJ
ejpam-5411	680	4	and	and	CCONJ
ejpam-5411	680	5	applied	applied	ADJ
ejpam-5411	680	6	mathematics	mathematic	NOUN
ejpam-5411	680	7	,	,	PUNCT
ejpam-5411	680	8	279:57–66	279:57–66	NUM
ejpam-5411	680	9	,	,	PUNCT
ejpam-5411	680	10	2015	2015	NUM
ejpam-5411	680	11	.	.	PUNCT
ejpam-5411	681	1	[	[	X
ejpam-5411	681	2	2	2	NUM
ejpam-5411	681	3	]	]	PUNCT
ejpam-5411	681	4	anderson	anderson	PROPN
ejpam-5411	681	5	and	and	CCONJ
ejpam-5411	681	6	ulness	ulness	PROPN
ejpam-5411	681	7	.	.	PUNCT
ejpam-5411	682	1	properties	property	NOUN
ejpam-5411	682	2	of	of	ADP
ejpam-5411	682	3	the	the	DET
ejpam-5411	682	4	katugampola	katugampola	ADJ
ejpam-5411	682	5	fractional	fractional	PROPN
ejpam-5411	682	6	derivative	derivative	NOUN
ejpam-5411	682	7	with	with	ADP
ejpam-5411	682	8	potential	potential	ADJ
ejpam-5411	682	9	application	application	NOUN
ejpam-5411	682	10	in	in	ADP
ejpam-5411	682	11	quantum	quantum	ADJ
ejpam-5411	682	12	mechanics	mechanic	NOUN
ejpam-5411	682	13	.	.	PUNCT
ejpam-5411	683	1	journal	journal	PROPN
ejpam-5411	683	2	of	of	ADP
ejpam-5411	683	3	mathematical	mathematical	ADJ
ejpam-5411	683	4	physic	physic	NOUN
ejpam-5411	683	5	,	,	PUNCT
ejpam-5411	683	6	56(6	56(6	NUM
ejpam-5411	683	7	)	)	PUNCT
ejpam-5411	683	8	,	,	PUNCT
ejpam-5411	683	9	2015	2015	NUM
ejpam-5411	683	10	.	.	PUNCT
ejpam-5411	684	1	[	[	X
ejpam-5411	684	2	3	3	NUM
ejpam-5411	684	3	]	]	X
ejpam-5411	684	4	ayata	ayata	NOUN
ejpam-5411	684	5	and	and	CCONJ
ejpam-5411	684	6	ozkan	ozkan	NOUN
ejpam-5411	684	7	.	.	PUNCT
ejpam-5411	685	1	a	a	DET
ejpam-5411	685	2	new	new	ADJ
ejpam-5411	685	3	application	application	NOUN
ejpam-5411	685	4	of	of	ADP
ejpam-5411	685	5	conformable	conformable	ADJ
ejpam-5411	685	6	laplace	laplace	NOUN
ejpam-5411	685	7	decomposition	decomposition	NOUN
ejpam-5411	685	8	method	method	NOUN
ejpam-5411	685	9	for	for	ADP
ejpam-5411	685	10	fractional	fractional	ADJ
ejpam-5411	685	11	newell	newell	PROPN
ejpam-5411	685	12	-	-	PUNCT
ejpam-5411	685	13	whitehead	whitehead	PROPN
ejpam-5411	685	14	-	-	PUNCT
ejpam-5411	685	15	segel	segel	NOUN
ejpam-5411	685	16	equation	equation	NOUN
ejpam-5411	685	17	.	.	PUNCT
ejpam-5411	686	1	aims	aim	VERB
ejpam-5411	686	2	mathematics	mathematic	NOUN
ejpam-5411	686	3	,	,	PUNCT
ejpam-5411	686	4	5(3):7402–7412	5(3):7402–7412	NUM
ejpam-5411	686	5	,	,	PUNCT
ejpam-5411	686	6	2020	2020	NUM
ejpam-5411	686	7	.	.	PUNCT
ejpam-5411	687	1	[	[	X
ejpam-5411	687	2	4	4	NUM
ejpam-5411	687	3	]	]	X
ejpam-5411	687	4	bahloul	bahloul	NOUN
ejpam-5411	687	5	.	.	PUNCT
ejpam-5411	687	6	existence	existence	NOUN
ejpam-5411	687	7	and	and	CCONJ
ejpam-5411	687	8	uniqueness	uniqueness	NOUN
ejpam-5411	687	9	of	of	ADP
ejpam-5411	687	10	solutions	solution	NOUN
ejpam-5411	687	11	of	of	ADP
ejpam-5411	687	12	the	the	DET
ejpam-5411	687	13	fractional	fractional	ADJ
ejpam-5411	687	14	integro	integro	ADJ
ejpam-5411	687	15	-	-	PUNCT
ejpam-5411	687	16	differential	differential	NOUN
ejpam-5411	687	17	equations	equation	NOUN
ejpam-5411	687	18	in	in	ADP
ejpam-5411	687	19	vector	vector	NOUN
ejpam-5411	687	20	-	-	PUNCT
ejpam-5411	687	21	valued	value	VERB
ejpam-5411	687	22	functional	functional	ADJ
ejpam-5411	687	23	space	space	NOUN
ejpam-5411	687	24	.	.	PUNCT
ejpam-5411	688	1	archivum	archivum	PROPN
ejpam-5411	688	2	mathematicuml	mathematicuml	PROPN
ejpam-5411	688	3	,	,	PUNCT
ejpam-5411	688	4	55:97–108	55:97–108	NUM
ejpam-5411	688	5	,	,	PUNCT
ejpam-5411	688	6	2019	2019	NUM
ejpam-5411	688	7	.	.	PUNCT
ejpam-5411	689	1	[	[	X
ejpam-5411	689	2	5	5	NUM
ejpam-5411	689	3	]	]	SYM
ejpam-5411	689	4	ding	ding	NOUN
ejpam-5411	689	5	and	and	CCONJ
ejpam-5411	689	6	wang	wang	PROPN
ejpam-5411	689	7	.	.	PUNCT
ejpam-5411	690	1	conformable	conformable	ADJ
ejpam-5411	690	2	linear	linear	NOUN
ejpam-5411	690	3	and	and	CCONJ
ejpam-5411	690	4	nonlinear	nonlinear	ADJ
ejpam-5411	690	5	non	non	ADJ
ejpam-5411	690	6	-	-	ADJ
ejpam-5411	690	7	instantaneous	instantaneous	ADJ
ejpam-5411	690	8	impulsive	impulsive	ADJ
ejpam-5411	690	9	differential	differential	ADJ
ejpam-5411	690	10	equations	equation	NOUN
ejpam-5411	690	11	.	.	PUNCT
ejpam-5411	691	1	electronic	electronic	ADJ
ejpam-5411	691	2	journal	journal	PROPN
ejpam-5411	691	3	differential	differential	NOUN
ejpam-5411	691	4	equations	equation	NOUN
ejpam-5411	691	5	,	,	PUNCT
ejpam-5411	691	6	2020:1–19	2020:1–19	NUM
ejpam-5411	691	7	,	,	PUNCT
ejpam-5411	691	8	2020	2020	NUM
ejpam-5411	691	9	.	.	PUNCT
ejpam-5411	692	1	[	[	X
ejpam-5411	692	2	6	6	NUM
ejpam-5411	692	3	]	]	PUNCT
ejpam-5411	692	4	a.	a.	PROPN
ejpam-5411	692	5	el	el	PROPN
ejpam-5411	692	6	-	-	PUNCT
ejpam-5411	692	7	ajou	ajou	ADJ
ejpam-5411	692	8	.	.	PUNCT
ejpam-5411	693	1	a	a	DET
ejpam-5411	693	2	modification	modification	NOUN
ejpam-5411	693	3	to	to	ADP
ejpam-5411	693	4	the	the	DET
ejpam-5411	693	5	conformable	conformable	ADJ
ejpam-5411	693	6	fractional	fractional	ADJ
ejpam-5411	693	7	calculus	calculus	NOUN
ejpam-5411	693	8	with	with	ADP
ejpam-5411	693	9	some	some	DET
ejpam-5411	693	10	applications	application	NOUN
ejpam-5411	693	11	.	.	PUNCT
ejpam-5411	694	1	alexandria	alexandria	PROPN
ejpam-5411	694	2	engineering	engineering	PROPN
ejpam-5411	694	3	journal	journal	PROPN
ejpam-5411	694	4	,	,	PUNCT
ejpam-5411	694	5	59:2239–2249	59:2239–2249	NUM
ejpam-5411	694	6	,	,	PUNCT
ejpam-5411	694	7	2020	2020	NUM
ejpam-5411	694	8	.	.	PUNCT
ejpam-5411	695	1	[	[	X
ejpam-5411	695	2	7	7	X
ejpam-5411	695	3	]	]	X
ejpam-5411	695	4	keyantuo	keyantuo	NOUN
ejpam-5411	695	5	and	and	CCONJ
ejpam-5411	695	6	lizama	lizama	NOUN
ejpam-5411	695	7	.	.	PUNCT
ejpam-5411	696	1	fourier	fouri	ADJ
ejpam-5411	696	2	multipliers	multiplier	NOUN
ejpam-5411	696	3	and	and	CCONJ
ejpam-5411	696	4	integro	integro	ADJ
ejpam-5411	696	5	-	-	PUNCT
ejpam-5411	696	6	differential	differential	NOUN
ejpam-5411	696	7	equations	equation	NOUN
ejpam-5411	696	8	in	in	ADP
ejpam-5411	696	9	banach	banach	NOUN
ejpam-5411	696	10	spaces	space	NOUN
ejpam-5411	696	11	.	.	PUNCT
ejpam-5411	697	1	journal	journal	NOUN
ejpam-5411	697	2	of	of	ADP
ejpam-5411	697	3	the	the	DET
ejpam-5411	697	4	london	london	PROPN
ejpam-5411	697	5	mathematical	mathematical	ADJ
ejpam-5411	697	6	society	society	NOUN
ejpam-5411	697	7	,	,	PUNCT
ejpam-5411	697	8	69(3):737–750	69(3):737–750	PROPN
ejpam-5411	697	9	,	,	PUNCT
ejpam-5411	697	10	2004	2004	NUM
ejpam-5411	697	11	.	.	PUNCT
ejpam-5411	698	1	references	reference	NOUN
ejpam-5411	698	2	2429	2429	NUM
ejpam-5411	699	1	[	[	X
ejpam-5411	699	2	8	8	NUM
ejpam-5411	699	3	]	]	X
ejpam-5411	699	4	khalil	khalil	PROPN
ejpam-5411	699	5	and	and	CCONJ
ejpam-5411	699	6	abu	abu	PROPN
ejpam-5411	699	7	hammad	hammad	PROPN
ejpam-5411	699	8	.	.	PUNCT
ejpam-5411	700	1	fractional	fractional	ADJ
ejpam-5411	700	2	fourier	fourier	PROPN
ejpam-5411	700	3	series	series	NOUN
ejpam-5411	700	4	with	with	ADP
ejpam-5411	700	5	applications	application	NOUN
ejpam-5411	700	6	.	.	PUNCT
ejpam-5411	701	1	american	american	ADJ
ejpam-5411	701	2	journal	journal	PROPN
ejpam-5411	701	3	of	of	ADP
ejpam-5411	701	4	computational	computational	ADJ
ejpam-5411	701	5	and	and	CCONJ
ejpam-5411	701	6	applied	applied	ADJ
ejpam-5411	701	7	mathematics	mathematic	NOUN
ejpam-5411	701	8	,	,	PUNCT
ejpam-5411	701	9	4(6):187–191	4(6):187–191	NUM
ejpam-5411	701	10	,	,	PUNCT
ejpam-5411	701	11	2014	2014	NUM
ejpam-5411	701	12	.	.	PUNCT
ejpam-5411	702	1	[	[	X
ejpam-5411	702	2	9	9	NUM
ejpam-5411	702	3	]	]	X
ejpam-5411	702	4	yousef	yousef	PROPN
ejpam-5411	702	5	khalil	khalil	PROPN
ejpam-5411	702	6	,	,	PUNCT
ejpam-5411	702	7	al	al	PROPN
ejpam-5411	702	8	horani	horani	PROPN
ejpam-5411	702	9	and	and	CCONJ
ejpam-5411	702	10	sababheh	sababheh	VERB
ejpam-5411	702	11	.	.	PUNCT
ejpam-5411	703	1	a	a	DET
ejpam-5411	703	2	new	new	ADJ
ejpam-5411	703	3	definition	definition	NOUN
ejpam-5411	703	4	of	of	ADP
ejpam-5411	703	5	fractional	fractional	ADJ
ejpam-5411	703	6	derivative	derivative	NOUN
ejpam-5411	703	7	.	.	PUNCT
ejpam-5411	704	1	journal	journal	PROPN
ejpam-5411	704	2	of	of	ADP
ejpam-5411	704	3	computational	computational	ADJ
ejpam-5411	704	4	and	and	CCONJ
ejpam-5411	704	5	applied	applied	ADJ
ejpam-5411	704	6	mathematics	mathematic	NOUN
ejpam-5411	704	7	,	,	PUNCT
ejpam-5411	704	8	264:65–70	264:65–70	NUM
ejpam-5411	704	9	,	,	PUNCT
ejpam-5411	704	10	2014	2014	NUM
ejpam-5411	704	11	.	.	PUNCT
ejpam-5411	705	1	[	[	X
ejpam-5411	705	2	10	10	NUM
ejpam-5411	705	3	]	]	X
ejpam-5411	705	4	lorente	lorente	NOUN
ejpam-5411	705	5	khitab	khitab	NOUN
ejpam-5411	705	6	and	and	CCONJ
ejpam-5411	705	7	ollivier	ollivi	ADJ
ejpam-5411	705	8	.	.	PUNCT
ejpam-5411	706	1	predictive	predictive	ADJ
ejpam-5411	706	2	model	model	NOUN
ejpam-5411	706	3	for	for	ADP
ejpam-5411	706	4	chloride	chloride	ADJ
ejpam-5411	706	5	penetration	penetration	NOUN
ejpam-5411	706	6	through	through	ADP
ejpam-5411	706	7	concrete	concrete	NOUN
ejpam-5411	706	8	.	.	PUNCT
ejpam-5411	707	1	magazine	magazine	NOUN
ejpam-5411	707	2	of	of	ADP
ejpam-5411	707	3	concrete	concrete	ADJ
ejpam-5411	707	4	research	research	NOUN
ejpam-5411	707	5	,	,	PUNCT
ejpam-5411	707	6	57(9):511–520	57(9):511–520	PROPN
ejpam-5411	707	7	,	,	PUNCT
ejpam-5411	707	8	2005	2005	NUM
ejpam-5411	707	9	.	.	PUNCT
ejpam-5411	708	1	[	[	X
ejpam-5411	708	2	11	11	NUM
ejpam-5411	708	3	]	]	X
ejpam-5411	708	4	srivastava	srivastava	PROPN
ejpam-5411	708	5	kilbas	kilbas	PROPN
ejpam-5411	708	6	and	and	CCONJ
ejpam-5411	708	7	trujillo	trujillo	PROPN
ejpam-5411	708	8	.	.	PUNCT
ejpam-5411	708	9	theory	theory	NOUN
ejpam-5411	708	10	and	and	CCONJ
ejpam-5411	708	11	application	application	NOUN
ejpam-5411	708	12	of	of	ADP
ejpam-5411	708	13	fractional	fractional	ADJ
ejpam-5411	708	14	differential	differential	ADJ
ejpam-5411	708	15	equations	equation	NOUN
ejpam-5411	708	16	.	.	PUNCT
ejpam-5411	709	1	north	north	NOUN
ejpam-5411	709	2	holland	holland	PROPN
ejpam-5411	709	3	mathematics	mathematics	PROPN
ejpam-5411	709	4	studies	study	NOUN
ejpam-5411	709	5	,	,	PUNCT
ejpam-5411	709	6	2006	2006	NUM
ejpam-5411	709	7	.	.	PUNCT
ejpam-5411	710	1	[	[	X
ejpam-5411	710	2	12	12	NUM
ejpam-5411	710	3	]	]	X
ejpam-5411	710	4	wang	wang	PROPN
ejpam-5411	710	5	j	j	PROPN
ejpam-5411	710	6	r	r	PROPN
ejpam-5411	710	7	li	li	PROPN
ejpam-5411	710	8	m	m	VERB
ejpam-5411	710	9	m	m	VERB
ejpam-5411	710	10	and	and	CCONJ
ejpam-5411	710	11	o’regan	o’regan	PROPN
ejpam-5411	710	12	d.	d.	PROPN
ejpam-5411	710	13	existence	existence	PROPN
ejpam-5411	710	14	and	and	CCONJ
ejpam-5411	710	15	ulam	ulam	PROPN
ejpam-5411	710	16	’s	’s	PART
ejpam-5411	710	17	stability	stability	NOUN
ejpam-5411	710	18	for	for	ADP
ejpam-5411	710	19	conformable	conformable	ADJ
ejpam-5411	710	20	fractional	fractional	ADJ
ejpam-5411	710	21	differential	differential	ADJ
ejpam-5411	710	22	equations	equation	NOUN
ejpam-5411	710	23	with	with	ADP
ejpam-5411	710	24	constant	constant	ADJ
ejpam-5411	710	25	coefficients	coefficient	NOUN
ejpam-5411	710	26	.	.	PUNCT
ejpam-5411	711	1	bulletin	bulletin	NOUN
ejpam-5411	711	2	of	of	ADP
ejpam-5411	711	3	the	the	DET
ejpam-5411	711	4	malaysian	malaysian	PROPN
ejpam-5411	711	5	mathematical	mathematical	PROPN
ejpam-5411	711	6	sciences	sciences	PROPN
ejpam-5411	711	7	society	society	NOUN
ejpam-5411	711	8	,	,	PUNCT
ejpam-5411	711	9	42:1791–1812	42:1791–1812	NOUN
ejpam-5411	711	10	,	,	PUNCT
ejpam-5411	711	11	2017	2017	NUM
ejpam-5411	711	12	.	.	PUNCT
ejpam-5411	712	1	[	[	X
ejpam-5411	712	2	13	13	NUM
ejpam-5411	712	3	]	]	PUNCT
ejpam-5411	712	4	eslami	eslami	NOUN
ejpam-5411	712	5	mohammadnezhad	mohammadnezhad	VERB
ejpam-5411	712	6	and	and	CCONJ
ejpam-5411	712	7	rezazadeh	rezazadeh	PROPN
ejpam-5411	712	8	.	.	PUNCT
ejpam-5411	713	1	stability	stability	NOUN
ejpam-5411	713	2	analysis	analysis	NOUN
ejpam-5411	713	3	of	of	ADP
ejpam-5411	713	4	linear	linear	ADJ
ejpam-5411	713	5	conformable	conformable	ADJ
ejpam-5411	713	6	fractional	fractional	ADJ
ejpam-5411	713	7	differential	differential	NOUN
ejpam-5411	713	8	equations	equation	NOUN
ejpam-5411	713	9	system	system	NOUN
ejpam-5411	713	10	with	with	ADP
ejpam-5411	713	11	time	time	NOUN
ejpam-5411	713	12	delays	delay	NOUN
ejpam-5411	713	13	.	.	PUNCT
ejpam-5411	714	1	boletim	boletim	PROPN
ejpam-5411	714	2	da	da	PROPN
ejpam-5411	714	3	sociedade	sociedade	PROPN
ejpam-5411	714	4	paranaense	paranaense	PROPN
ejpam-5411	714	5	de	de	PROPN
ejpam-5411	714	6	matematica	matematica	PROPN
ejpam-5411	714	7	,	,	PUNCT
ejpam-5411	714	8	38(6):159–171	38(6):159–171	PROPN
ejpam-5411	714	9	,	,	PUNCT
ejpam-5411	714	10	2020	2020	NUM
ejpam-5411	714	11	.	.	PUNCT
ejpam-5411	715	1	[	[	X
ejpam-5411	715	2	14	14	NUM
ejpam-5411	715	3	]	]	SYM
ejpam-5411	715	4	podlubny	podlubny	NOUN
ejpam-5411	715	5	.	.	PUNCT
ejpam-5411	716	1	fractional	fractional	ADJ
ejpam-5411	716	2	differential	differential	ADJ
ejpam-5411	716	3	equations	equation	NOUN
ejpam-5411	716	4	,	,	PUNCT
ejpam-5411	716	5	academic	academic	ADJ
ejpam-5411	716	6	press	press	NOUN
ejpam-5411	716	7	.	.	PUNCT
ejpam-5411	717	1	san	san	PROPN
ejpam-5411	717	2	diego	diego	PROPN
ejpam-5411	717	3	ca	ca	PROPN
ejpam-5411	717	4	,	,	PUNCT
ejpam-5411	717	5	1999	1999	NUM
ejpam-5411	717	6	.	.	PUNCT
ejpam-5411	718	1	[	[	X
ejpam-5411	718	2	15	15	X
ejpam-5411	718	3	]	]	X
ejpam-5411	718	4	riaz	riaz	PROPN
ejpam-5411	718	5	ur	ur	PROPN
ejpam-5411	718	6	rahman	rahman	PROPN
ejpam-5411	718	7	,	,	PUNCT
ejpam-5411	718	8	amal	amal	PROPN
ejpam-5411	718	9	f	f	PROPN
ejpam-5411	718	10	al	al	PROPN
ejpam-5411	718	11	-	-	PUNCT
ejpam-5411	718	12	maaitah	maaitah	PROPN
ejpam-5411	718	13	,	,	PUNCT
ejpam-5411	718	14	maysoon	maysoon	NOUN
ejpam-5411	718	15	qousini	qousini	PROPN
ejpam-5411	718	16	,	,	PUNCT
ejpam-5411	718	17	emad	emad	PROPN
ejpam-5411	718	18	ahmad	ahmad	PROPN
ejpam-5411	718	19	az	az	PROPN
ejpam-5411	718	20	-	-	PROPN
ejpam-5411	718	21	zo’bi	zo’bi	PROPN
ejpam-5411	718	22	,	,	PUNCT
ejpam-5411	718	23	sayed	say	VERB
ejpam-5411	718	24	m.	m.	PROPN
ejpam-5411	718	25	eldin	eldin	PROPN
ejpam-5411	718	26	,	,	PUNCT
ejpam-5411	718	27	and	and	CCONJ
ejpam-5411	718	28	muhammad	muhammad	PROPN
ejpam-5411	718	29	abuzar	abuzar	PROPN
ejpam-5411	718	30	.	.	PUNCT
ejpam-5411	719	1	new	new	ADJ
ejpam-5411	719	2	soliton	soliton	NOUN
ejpam-5411	719	3	solutions	solution	NOUN
ejpam-5411	719	4	and	and	CCONJ
ejpam-5411	719	5	modulation	modulation	NOUN
ejpam-5411	719	6	instability	instability	NOUN
ejpam-5411	719	7	analysis	analysis	NOUN
ejpam-5411	719	8	of	of	ADP
ejpam-5411	719	9	fractional	fractional	ADJ
ejpam-5411	719	10	huxley	huxley	PROPN
ejpam-5411	719	11	equation	equation	NOUN
ejpam-5411	719	12	.	.	PUNCT
ejpam-5411	720	1	results	result	NOUN
ejpam-5411	720	2	in	in	ADP
ejpam-5411	720	3	physics	physics	NOUN
ejpam-5411	720	4	,	,	PUNCT
ejpam-5411	720	5	44:106163	44:106163	NUM
ejpam-5411	720	6	,	,	PUNCT
ejpam-5411	720	7	2023	2023	NUM
ejpam-5411	720	8	.	.	PUNCT
ejpam-5411	721	1	[	[	X
ejpam-5411	721	2	16	16	NUM
ejpam-5411	721	3	]	]	X
ejpam-5411	721	4	ma	ma	PROPN
ejpam-5411	721	5	lgorzata	lgorzata	PROPN
ejpam-5411	721	6	guzowska	guzowska	PROPN
ejpam-5411	721	7	ricardo	ricardo	PROPN
ejpam-5411	721	8	almeida	almeida	PROPN
ejpam-5411	721	9	and	and	CCONJ
ejpam-5411	721	10	tatiana	tatiana	PROPN
ejpam-5411	721	11	odzijewicz	odzijewicz	PROPN
ejpam-5411	721	12	.	.	PUNCT
ejpam-5411	722	1	a	a	DET
ejpam-5411	722	2	remark	remark	NOUN
ejpam-5411	722	3	on	on	ADP
ejpam-5411	722	4	local	local	ADJ
ejpam-5411	722	5	fractional	fractional	ADJ
ejpam-5411	722	6	calculus	calculus	NOUN
ejpam-5411	722	7	and	and	CCONJ
ejpam-5411	722	8	ordinary	ordinary	ADJ
ejpam-5411	722	9	derivatives	derivative	NOUN
ejpam-5411	722	10	.	.	PUNCT
ejpam-5411	723	1	open	open	ADJ
ejpam-5411	723	2	mathematics	mathematic	NOUN
ejpam-5411	723	3	,	,	PUNCT
ejpam-5411	723	4	14(1):1122–1124	14(1):1122–1124	NUM
ejpam-5411	723	5	,	,	PUNCT
ejpam-5411	723	6	2016	2016	NUM
ejpam-5411	723	7	.	.	PUNCT
ejpam-5411	724	1	[	[	X
ejpam-5411	724	2	17	17	NUM
ejpam-5411	724	3	]	]	PUNCT
ejpam-5411	724	4	kilbas	kilbas	PROPN
ejpam-5411	724	5	samko	samko	NOUN
ejpam-5411	724	6	and	and	CCONJ
ejpam-5411	724	7	marichev	marichev	ADJ
ejpam-5411	724	8	.	.	PUNCT
ejpam-5411	724	9	fractional	fractional	ADJ
ejpam-5411	724	10	integrals	integral	NOUN
ejpam-5411	724	11	and	and	CCONJ
ejpam-5411	724	12	derivatives	derivative	NOUN
ejpam-5411	724	13	:	:	PUNCT
ejpam-5411	724	14	theory	theory	NOUN
ejpam-5411	724	15	and	and	CCONJ
ejpam-5411	724	16	applications	application	NOUN
ejpam-5411	724	17	.	.	PUNCT
ejpam-5411	725	1	gordon	gordon	PROPN
ejpam-5411	725	2	and	and	CCONJ
ejpam-5411	725	3	breach	breach	NOUN
ejpam-5411	725	4	,	,	PUNCT
ejpam-5411	725	5	yverdon	yverdon	PROPN
ejpam-5411	725	6	,	,	PUNCT
ejpam-5411	725	7	1993	1993	NUM
ejpam-5411	725	8	.	.	PUNCT
ejpam-5411	726	1	[	[	X
ejpam-5411	726	2	18	18	NUM
ejpam-5411	726	3	]	]	X
ejpam-5411	726	4	makhlouf	makhlouf	ADJ
ejpam-5411	726	5	souahi	souahi	NOUN
ejpam-5411	726	6	and	and	CCONJ
ejpam-5411	726	7	hammami	hammami	NOUN
ejpam-5411	726	8	.	.	PUNCT
ejpam-5411	727	1	stability	stability	NOUN
ejpam-5411	727	2	analysis	analysis	NOUN
ejpam-5411	727	3	of	of	ADP
ejpam-5411	727	4	conformable	conformable	ADJ
ejpam-5411	727	5	fractional	fractional	ADJ
ejpam-5411	727	6	-	-	PUNCT
ejpam-5411	727	7	order	order	NOUN
ejpam-5411	727	8	nonlinear	nonlinear	ADJ
ejpam-5411	727	9	systems	system	NOUN
ejpam-5411	727	10	.	.	PUNCT
ejpam-5411	728	1	indagationes	indagatione	NOUN
ejpam-5411	728	2	mathematicae	mathematicae	PROPN
ejpam-5411	728	3	,	,	PUNCT
ejpam-5411	728	4	28:1265–1274	28:1265–1274	NUM
ejpam-5411	728	5	,	,	PUNCT
ejpam-5411	728	6	2017	2017	NUM
ejpam-5411	728	7	.	.	PUNCT
ejpam-5411	729	1	[	[	X
ejpam-5411	729	2	19	19	NUM
ejpam-5411	729	3	]	]	SYM
ejpam-5411	729	4	vanterler	vanterler	NOUN
ejpam-5411	729	5	da	da	PROPN
ejpam-5411	729	6	c.	c.	PROPN
ejpam-5411	729	7	sousa	sousa	PROPN
ejpam-5411	729	8	and	and	CCONJ
ejpam-5411	729	9	e.	e.	PROPN
ejpam-5411	729	10	capelas	capelas	PROPN
ejpam-5411	729	11	de	de	PROPN
ejpam-5411	729	12	oliveira	oliveira	PROPN
ejpam-5411	729	13	.	.	PUNCT
ejpam-5411	730	1	a	a	DET
ejpam-5411	730	2	new	new	ADJ
ejpam-5411	730	3	truncated	truncated	ADJ
ejpam-5411	730	4	m	m	ADJ
ejpam-5411	730	5	-	-	ADJ
ejpam-5411	730	6	fractional	fractional	ADJ
ejpam-5411	730	7	derivative	derivative	ADJ
ejpam-5411	730	8	type	type	NOUN
ejpam-5411	730	9	unifying	unify	VERB
ejpam-5411	730	10	some	some	DET
ejpam-5411	730	11	fractional	fractional	ADJ
ejpam-5411	730	12	derivative	derivative	ADJ
ejpam-5411	730	13	types	type	NOUN
ejpam-5411	730	14	with	with	ADP
ejpam-5411	730	15	classical	classical	ADJ
ejpam-5411	730	16	properties	property	NOUN
ejpam-5411	730	17	.	.	PUNCT
ejpam-5411	731	1	international	international	ADJ
ejpam-5411	731	2	journal	journal	NOUN
ejpam-5411	731	3	of	of	ADP
ejpam-5411	731	4	analysis	analysis	NOUN
ejpam-5411	731	5	and	and	CCONJ
ejpam-5411	731	6	applications	application	NOUN
ejpam-5411	731	7	,	,	PUNCT
ejpam-5411	731	8	16(1):83–96	16(1):83–96	NUM
ejpam-5411	731	9	,	,	PUNCT
ejpam-5411	731	10	2018	2018	NUM
ejpam-5411	731	11	.	.	PUNCT
ejpam-5411	732	1	[	[	X
ejpam-5411	732	2	20	20	NUM
ejpam-5411	732	3	]	]	X
ejpam-5411	732	4	thomas	thomas	PROPN
ejpam-5411	732	5	and	and	CCONJ
ejpam-5411	732	6	bamforth	bamforth	NOUN
ejpam-5411	732	7	.	.	PUNCT
ejpam-5411	733	1	modelling	model	VERB
ejpam-5411	733	2	chloride	chloride	NOUN
ejpam-5411	733	3	diffusion	diffusion	NOUN
ejpam-5411	733	4	in	in	ADP
ejpam-5411	733	5	concrete	concrete	ADJ
ejpam-5411	733	6	:	:	PUNCT
ejpam-5411	733	7	effect	effect	NOUN
ejpam-5411	733	8	of	of	ADP
ejpam-5411	733	9	fly	fly	VERB
ejpam-5411	733	10	ash	ash	NOUN
ejpam-5411	733	11	and	and	CCONJ
ejpam-5411	733	12	slag	slag	NOUN
ejpam-5411	733	13	.	.	PUNCT
ejpam-5411	734	1	cement	cement	NOUN
ejpam-5411	734	2	and	and	CCONJ
ejpam-5411	734	3	concrete	concrete	ADJ
ejpam-5411	734	4	research	research	NOUN
ejpam-5411	734	5	,	,	PUNCT
ejpam-5411	734	6	29(4):487–495	29(4):487–495	NUM
ejpam-5411	734	7	,	,	PUNCT
ejpam-5411	734	8	1999	1999	NUM
ejpam-5411	734	9	.	.	PUNCT
ejpam-5411	735	1	[	[	X
ejpam-5411	735	2	21	21	NUM
ejpam-5411	735	3	]	]	X
ejpam-5411	735	4	yang	yang	PROPN
ejpam-5411	735	5	.	.	PUNCT
ejpam-5411	736	1	advanced	advanced	ADJ
ejpam-5411	736	2	local	local	ADJ
ejpam-5411	736	3	fractional	fractional	ADJ
ejpam-5411	736	4	calculus	calculus	NOUN
ejpam-5411	736	5	and	and	CCONJ
ejpam-5411	736	6	its	its	PRON
ejpam-5411	736	7	applications	application	NOUN
ejpam-5411	736	8	.	.	PUNCT
ejpam-5411	737	1	world	world	NOUN
ejpam-5411	737	2	science	science	PROPN
ejpam-5411	737	3	publisher	publisher	NOUN
ejpam-5411	737	4	,	,	PUNCT
ejpam-5411	737	5	new	new	PROPN
ejpam-5411	737	6	york	york	PROPN
ejpam-5411	737	7	,	,	PUNCT
ejpam-5411	737	8	ny	ny	PROPN
ejpam-5411	737	9	,	,	PUNCT
ejpam-5411	737	10	usa	usa	PROPN
ejpam-5411	737	11	,	,	PUNCT
ejpam-5411	737	12	2012	2012	NUM
ejpam-5411	737	13	.	.	PUNCT
ejpam-5411	738	1	[	[	X
ejpam-5411	738	2	22	22	NUM
ejpam-5411	738	3	]	]	X
ejpam-5411	738	4	fatima	fatima	PROPN
ejpam-5411	738	5	alrawajeh	alrawajeh	PROPN
ejpam-5411	738	6	zeyade	zeyade	PROPN
ejpam-5411	738	7	al	al	PROPN
ejpam-5411	738	8	-	-	PROPN
ejpam-5411	738	9	zhouri	zhouri	PROPN
ejpam-5411	738	10	,	,	PUNCT
ejpam-5411	738	11	nouf	nouf	PROPN
ejpam-5411	738	12	al	al	PROPN
ejpam-5411	738	13	-	-	PUNCT
ejpam-5411	738	14	mutai	mutai	PROPN
ejpam-5411	738	15	,	,	PUNCT
ejpam-5411	738	16	and	and	CCONJ
ejpam-5411	738	17	raed	raed	PROPN
ejpam-5411	738	18	alkhasawnh	alkhasawnh	PROPN
ejpam-5411	738	19	.	.	PUNCT
ejpam-5411	739	1	new	new	ADJ
ejpam-5411	739	2	results	result	NOUN
ejpam-5411	739	3	on	on	ADP
ejpam-5411	739	4	conformable	conformable	ADJ
ejpam-5411	739	5	fractional	fractional	ADJ
ejpam-5411	739	6	sumudo	sumudo	NOUN
ejpam-5411	739	7	transform	transform	NOUN
ejpam-5411	739	8	.	.	PUNCT
ejpam-5411	740	1	theories	theory	NOUN
ejpam-5411	740	2	and	and	CCONJ
ejpam-5411	740	3	applications	application	NOUN
ejpam-5411	740	4	,	,	PUNCT
ejpam-5411	740	5	international	international	ADJ
ejpam-5411	740	6	journal	journal	NOUN
ejpam-5411	740	7	of	of	ADP
ejpam-5411	740	8	analysis	analysis	NOUN
ejpam-5411	740	9	and	and	CCONJ
ejpam-5411	740	10	applications	application	NOUN
ejpam-5411	740	11	,	,	PUNCT
ejpam-5411	740	12	17(6):1019–1033	17(6):1019–1033	NUM
ejpam-5411	740	13	,	,	PUNCT
ejpam-5411	740	14	2019	2019	NUM
ejpam-5411	740	15	.	.	PUNCT
ejpam-5411	741	1	references	reference	NOUN
ejpam-5411	741	2	2430	2430	NUM
ejpam-5411	742	1	[	[	X
ejpam-5411	742	2	23	23	NUM
ejpam-5411	742	3	]	]	PUNCT
ejpam-5411	742	4	pan	pan	PROPN
ejpam-5411	742	5	zhao	zhao	PROPN
ejpam-5411	742	6	and	and	CCONJ
ejpam-5411	742	7	luo	luo	PROPN
ejpam-5411	742	8	.	.	PUNCT
ejpam-5411	743	1	a	a	DET
ejpam-5411	743	2	new	new	ADJ
ejpam-5411	743	3	framework	framework	NOUN
ejpam-5411	743	4	for	for	ADP
ejpam-5411	743	5	multivariate	multivariate	NOUN
ejpam-5411	743	6	general	general	ADJ
ejpam-5411	743	7	conformable	conformable	ADJ
ejpam-5411	743	8	fractional	fractional	ADJ
ejpam-5411	743	9	calculus	calculus	NOUN
ejpam-5411	743	10	and	and	CCONJ
ejpam-5411	743	11	potential	potential	ADJ
ejpam-5411	743	12	applications	application	NOUN
ejpam-5411	743	13	.	.	PUNCT
ejpam-5411	744	1	physica	physica	PROPN
ejpam-5411	744	2	a	a	DET
ejpam-5411	744	3	:	:	PUNCT
ejpam-5411	744	4	statistical	statistical	ADJ
ejpam-5411	744	5	mechanics	mechanic	NOUN
ejpam-5411	744	6	and	and	CCONJ
ejpam-5411	744	7	its	its	PRON
ejpam-5411	744	8	applications	application	NOUN
ejpam-5411	744	9	,	,	PUNCT
ejpam-5411	744	10	510(4):271–280	510(4):271–280	NUM
ejpam-5411	744	11	,	,	PUNCT
ejpam-5411	744	12	2018	2018	NUM
ejpam-5411	744	13	.	.	PUNCT
ejpam-5411	745	1	[	[	X
ejpam-5411	745	2	24	24	NUM
ejpam-5411	745	3	]	]	X
ejpam-5411	745	4	yang	yang	PROPN
ejpam-5411	745	5	zhou	zhou	PROPN
ejpam-5411	745	6	and	and	CCONJ
ejpam-5411	745	7	zhang	zhang	PROPN
ejpam-5411	745	8	.	.	PUNCT
ejpam-5411	746	1	conformable	conformable	ADJ
ejpam-5411	746	2	derivative	derivative	ADJ
ejpam-5411	746	3	approach	approach	NOUN
ejpam-5411	746	4	to	to	ADP
ejpam-5411	746	5	anomalous	anomalous	ADJ
ejpam-5411	746	6	diffusion	diffusion	NOUN
ejpam-5411	746	7	.	.	PUNCT
ejpam-5411	747	1	physica	physica	PROPN
ejpam-5411	747	2	a	a	DET
ejpam-5411	747	3	:	:	PUNCT
ejpam-5411	747	4	statistical	statistical	ADJ
ejpam-5411	747	5	mechanics	mechanic	NOUN
ejpam-5411	747	6	and	and	CCONJ
ejpam-5411	747	7	its	its	PRON
ejpam-5411	747	8	applications	application	NOUN
ejpam-5411	747	9	,	,	PUNCT
ejpam-5411	747	10	491:1001–1013	491:1001–1013	NOUN
ejpam-5411	747	11	,	,	PUNCT
ejpam-5411	747	12	2018	2018	NUM
ejpam-5411	747	13	.	.	PUNCT