id	sid	tid	token	lemma	pos
ma-25	1	1	2021	2021	NUM
ma-25	1	2	ada	ada	PROPN
ma-25	1	3	academica	academica	PROPN
ma-25	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-25	1	5	.	.	PUNCT
ma-25	2	1	j.	j.	PROPN
ma-25	2	2	math	math	PROPN
ma-25	2	3	.	.	PUNCT
ma-25	3	1	anal	anal	ADJ
ma-25	3	2	.	.	PUNCT
ma-25	4	1	1	1	NUM
ma-25	4	2	(	(	PUNCT
ma-25	4	3	2021	2021	NUM
ma-25	4	4	)	)	PUNCT
ma-25	4	5	86	86	NUM
ma-25	4	6	-	-	SYM
ma-25	4	7	105doi	105doi	NUM
ma-25	4	8	:	:	PUNCT
ma-25	4	9	10.28924	10.28924	NUM
ma-25	4	10	/	/	SYM
ma-25	4	11	ada	ada	PROPN
ma-25	4	12	/	/	SYM
ma-25	4	13	ma.1.86	ma.1.86	NOUN
ma-25	4	14	some	some	DET
ma-25	4	15	properties	property	NOUN
ma-25	4	16	on	on	ADP
ma-25	4	17	the	the	DET
ma-25	4	18	[	[	X
ma-25	4	19	p	p	X
ma-25	4	20	,	,	PUNCT
ma-25	4	21	q]-order	q]-order	NOUN
ma-25	4	22	of	of	ADP
ma-25	4	23	meromorphic	meromorphic	ADJ
ma-25	4	24	solutions	solution	NOUN
ma-25	4	25	of	of	ADP
ma-25	4	26	homogeneous	homogeneous	ADJ
ma-25	4	27	and	and	CCONJ
ma-25	4	28	non	non	ADJ
ma-25	4	29	-	-	ADJ
ma-25	4	30	homogeneous	homogeneous	ADJ
ma-25	4	31	linear	linear	PROPN
ma-25	4	32	differential	differential	NOUN
ma-25	4	33	equations	equation	NOUN
ma-25	4	34	with	with	ADP
ma-25	4	35	meromorphic	meromorphic	ADJ
ma-25	4	36	coefficients	coefficient	NOUN
ma-25	4	37	mansouria	mansouria	PROPN
ma-25	4	38	saidani	saidani	PROPN
ma-25	4	39	,	,	PUNCT
ma-25	4	40	benharrat	benharrat	PROPN
ma-25	4	41	belaïdi∗	belaïdi∗	PROPN
ma-25	4	42	department	department	PROPN
ma-25	4	43	of	of	ADP
ma-25	4	44	mathematics	mathematics	PROPN
ma-25	4	45	,	,	PUNCT
ma-25	4	46	laboratory	laboratory	NOUN
ma-25	4	47	of	of	ADP
ma-25	4	48	pure	pure	ADJ
ma-25	4	49	and	and	CCONJ
ma-25	4	50	applied	applied	ADJ
ma-25	4	51	mathematics	mathematic	NOUN
ma-25	4	52	,	,	PUNCT
ma-25	4	53	university	university	NOUN
ma-25	4	54	of	of	ADP
ma-25	4	55	mostaganem	mostaganem	PROPN
ma-25	4	56	(	(	PUNCT
ma-25	4	57	umab	umab	NOUN
ma-25	4	58	)	)	PUNCT
ma-25	4	59	,	,	PUNCT
ma-25	5	1	b.	b.	PROPN
ma-25	5	2	p.	p.	NOUN
ma-25	5	3	227	227	NUM
ma-25	6	1	mostaganem	mostaganem	PROPN
ma-25	6	2	,	,	PUNCT
ma-25	6	3	algeria	algeria	PROPN
ma-25	6	4	saidaniman@yahoo.fr	saidaniman@yahoo.fr	PROPN
ma-25	6	5	,	,	PUNCT
ma-25	6	6	benharrat.belaidi@univ-mosta.dz	benharrat.belaidi@univ-mosta.dz	NOUN
ma-25	6	7	∗correspondence	∗correspondence	NOUN
ma-25	6	8	:	:	PUNCT
ma-25	6	9	benharrat.belaidi@univ-mosta.dz	benharrat.belaidi@univ-mosta.dz	NOUN
ma-25	6	10	abstract	abstract	NOUN
ma-25	6	11	.	.	PUNCT
ma-25	7	1	in	in	ADP
ma-25	7	2	the	the	DET
ma-25	7	3	present	present	ADJ
ma-25	7	4	paper	paper	NOUN
ma-25	7	5	,	,	PUNCT
ma-25	7	6	we	we	PRON
ma-25	7	7	investigate	investigate	VERB
ma-25	7	8	the	the	DET
ma-25	7	9	[	[	X
ma-25	7	10	p	p	X
ma-25	7	11	,	,	PUNCT
ma-25	7	12	q]-order	q]-order	NOUN
ma-25	7	13	of	of	ADP
ma-25	7	14	solutions	solution	NOUN
ma-25	7	15	of	of	ADP
ma-25	7	16	higher	high	ADJ
ma-25	7	17	order	order	NOUN
ma-25	7	18	lineardifferential	lineardifferential	ADJ
ma-25	7	19	equations	equation	NOUN
ma-25	7	20	ak	ak	PROPN
ma-25	7	21	(	(	PUNCT
ma-25	7	22	z	z	PROPN
ma-25	7	23	)	)	PUNCT
ma-25	7	24	f	f	NOUN
ma-25	7	25	(	(	PUNCT
ma-25	7	26	k	k	NOUN
ma-25	7	27	)	)	PUNCT
ma-25	7	28	+	+	CCONJ
ma-25	7	29	ak−1	ak−1	ADV
ma-25	7	30	(	(	PUNCT
ma-25	7	31	z	z	NOUN
ma-25	7	32	)	)	PUNCT
ma-25	7	33	f	f	PROPN
ma-25	7	34	(	(	PUNCT
ma-25	7	35	k−1	k−1	PROPN
ma-25	7	36	)	)	PUNCT
ma-25	7	37	+	+	PUNCT
ma-25	7	38	·	·	PUNCT
ma-25	7	39	·	·	PUNCT
ma-25	7	40	·	·	PUNCT
ma-25	8	1	+	+	NUM
ma-25	8	2	a1	a1	NOUN
ma-25	8	3	(	(	PUNCT
ma-25	8	4	z	z	NOUN
ma-25	8	5	)	)	PUNCT
ma-25	8	6	f	f	NOUN
ma-25	8	7	′	′	NUM
ma-25	9	1	+	+	CCONJ
ma-25	9	2	a0	a0	PROPN
ma-25	9	3	(	(	PUNCT
ma-25	9	4	z	z	NOUN
ma-25	9	5	)	)	PUNCT
ma-25	9	6	f	f	NOUN
ma-25	10	1	=	=	SYM
ma-25	10	2	0	0	PROPN
ma-25	10	3	and	and	CCONJ
ma-25	10	4	ak	ak	PROPN
ma-25	10	5	(	(	PUNCT
ma-25	10	6	z	z	PROPN
ma-25	10	7	)	)	PUNCT
ma-25	10	8	f	f	NOUN
ma-25	10	9	(	(	PUNCT
ma-25	10	10	k	k	NOUN
ma-25	10	11	)	)	PUNCT
ma-25	11	1	+	+	CCONJ
ma-25	11	2	ak−1	ak−1	ADV
ma-25	11	3	(	(	PUNCT
ma-25	11	4	z	z	NOUN
ma-25	11	5	)	)	PUNCT
ma-25	11	6	f	f	PROPN
ma-25	11	7	(	(	PUNCT
ma-25	11	8	k−1	k−1	PROPN
ma-25	11	9	)	)	PUNCT
ma-25	12	1	+	+	PUNCT
ma-25	12	2	·	·	PUNCT
ma-25	12	3	·	·	PUNCT
ma-25	12	4	·	·	PUNCT
ma-25	12	5	+	+	NUM
ma-25	12	6	a1	a1	NOUN
ma-25	12	7	(	(	PUNCT
ma-25	12	8	z	z	NOUN
ma-25	12	9	)	)	PUNCT
ma-25	12	10	f	f	NOUN
ma-25	12	11	′	′	NUM
ma-25	13	1	+	+	CCONJ
ma-25	13	2	a0	a0	PROPN
ma-25	13	3	(	(	PUNCT
ma-25	13	4	z	z	NOUN
ma-25	13	5	)	)	PUNCT
ma-25	13	6	f	f	NOUN
ma-25	13	7	=	=	SYM
ma-25	13	8	f	f	PROPN
ma-25	13	9	(	(	PUNCT
ma-25	13	10	z	z	NOUN
ma-25	13	11	)	)	PUNCT
ma-25	13	12	,	,	PUNCT
ma-25	13	13	where	where	SCONJ
ma-25	13	14	a0	a0	PROPN
ma-25	13	15	(	(	PUNCT
ma-25	13	16	z	z	PROPN
ma-25	13	17	)	)	PUNCT
ma-25	13	18	,	,	PUNCT
ma-25	13	19	a1	a1	NOUN
ma-25	13	20	(	(	PUNCT
ma-25	13	21	z	z	NOUN
ma-25	13	22	)	)	PUNCT
ma-25	13	23	,	,	PUNCT
ma-25	13	24	...	...	PUNCT
ma-25	13	25	,	,	PUNCT
ma-25	13	26	ak	ak	PROPN
ma-25	13	27	(	(	PUNCT
ma-25	13	28	z	z	PROPN
ma-25	13	29	)	)	PUNCT
ma-25	13	30	6≡	6≡	NUM
ma-25	13	31	0	0	NUM
ma-25	13	32	and	and	CCONJ
ma-25	13	33	f	f	PROPN
ma-25	13	34	(	(	PUNCT
ma-25	13	35	z	z	NOUN
ma-25	13	36	)	)	PUNCT
ma-25	13	37	6≡	6≡	NUM
ma-25	13	38	0	0	NUM
ma-25	13	39	are	be	AUX
ma-25	13	40	meromorphic	meromorphic	ADJ
ma-25	13	41	functions	function	NOUN
ma-25	13	42	of	of	ADP
ma-25	13	43	finite	finite	NOUN
ma-25	13	44	[	[	X
ma-25	13	45	p	p	X
ma-25	13	46	,	,	PUNCT
ma-25	13	47	q]-order.we	q]-order.we	PRON
ma-25	13	48	improve	improve	VERB
ma-25	13	49	and	and	CCONJ
ma-25	13	50	extend	extend	VERB
ma-25	13	51	some	some	DET
ma-25	13	52	results	result	NOUN
ma-25	13	53	of	of	ADP
ma-25	13	54	the	the	DET
ma-25	13	55	authors	author	NOUN
ma-25	13	56	by	by	ADP
ma-25	13	57	using	use	VERB
ma-25	13	58	the	the	DET
ma-25	13	59	concept	concept	NOUN
ma-25	13	60	[	[	X
ma-25	13	61	p	p	X
ma-25	13	62	,	,	PUNCT
ma-25	13	63	q]-order	q]-order	NOUN
ma-25	13	64	.	.	PUNCT
ma-25	14	1	1	1	X
ma-25	14	2	.	.	X
ma-25	14	3	introduction	introduction	NOUN
ma-25	14	4	and	and	CCONJ
ma-25	14	5	main	main	ADJ
ma-25	14	6	results	result	NOUN
ma-25	14	7	in	in	ADP
ma-25	14	8	this	this	DET
ma-25	14	9	paper	paper	NOUN
ma-25	14	10	,	,	PUNCT
ma-25	14	11	we	we	PRON
ma-25	14	12	assume	assume	VERB
ma-25	14	13	that	that	SCONJ
ma-25	14	14	the	the	DET
ma-25	14	15	reader	reader	NOUN
ma-25	14	16	is	be	AUX
ma-25	14	17	familiar	familiar	ADJ
ma-25	14	18	with	with	ADP
ma-25	14	19	the	the	DET
ma-25	14	20	fundamental	fundamental	ADJ
ma-25	14	21	results	result	NOUN
ma-25	14	22	and	and	CCONJ
ma-25	14	23	the	the	DET
ma-25	14	24	standardnotations	standardnotation	NOUN
ma-25	14	25	of	of	ADP
ma-25	14	26	the	the	DET
ma-25	14	27	nevanlinna	nevanlinna	NOUN
ma-25	14	28	’s	’s	PART
ma-25	14	29	value	value	NOUN
ma-25	14	30	distribution	distribution	NOUN
ma-25	14	31	theory	theory	NOUN
ma-25	14	32	of	of	ADP
ma-25	14	33	meromorphic	meromorphic	ADJ
ma-25	14	34	functions	function	NOUN
ma-25	14	35	(	(	PUNCT
ma-25	14	36	see	see	VERB
ma-25	14	37	[	[	X
ma-25	14	38	7	7	X
ma-25	14	39	]	]	PUNCT
ma-25	14	40	,	,	PUNCT
ma-25	15	1	[	[	X
ma-25	15	2	9	9	NUM
ma-25	15	3	]	]	PUNCT
ma-25	15	4	,	,	PUNCT
ma-25	15	5	[	[	X
ma-25	15	6	14	14	NUM
ma-25	15	7	]	]	PUNCT
ma-25	15	8	,	,	PUNCT
ma-25	15	9	[	[	X
ma-25	15	10	24	24	NUM
ma-25	15	11	]	]	PUNCT
ma-25	15	12	)	)	PUNCT
ma-25	15	13	.	.	PUNCT
ma-25	16	1	in	in	ADP
ma-25	16	2	addition	addition	NOUN
ma-25	16	3	,	,	PUNCT
ma-25	16	4	for	for	ADP
ma-25	16	5	any	any	DET
ma-25	16	6	integers	integer	NOUN
ma-25	16	7	p	p	NOUN
ma-25	16	8	≥	≥	NOUN
ma-25	16	9	q	q	NOUN
ma-25	16	10	≥	≥	NUM
ma-25	16	11	1	1	NUM
ma-25	16	12	and	and	CCONJ
ma-25	16	13	a	a	DET
ma-25	16	14	meromorphic	meromorphic	ADJ
ma-25	16	15	function	function	NOUN
ma-25	16	16	f	f	PROPN
ma-25	16	17	in	in	ADP
ma-25	16	18	the	the	DET
ma-25	16	19	whole	whole	ADJ
ma-25	16	20	complexplane	complexplane	NOUN
ma-25	16	21	,	,	PUNCT
ma-25	16	22	we	we	PRON
ma-25	16	23	will	will	AUX
ma-25	16	24	use	use	VERB
ma-25	16	25	ρ[p	ρ[p	NOUN
ma-25	16	26	,	,	PUNCT
ma-25	16	27	q	q	X
ma-25	16	28	]	]	X
ma-25	16	29	(	(	PUNCT
ma-25	16	30	f	f	PROPN
ma-25	16	31	)	)	PUNCT
ma-25	16	32	,	,	PUNCT
ma-25	16	33	µ[p	µ[p	ADJ
ma-25	16	34	,	,	PUNCT
ma-25	16	35	q	q	X
ma-25	16	36	]	]	X
ma-25	16	37	(	(	PUNCT
ma-25	16	38	f	f	NOUN
ma-25	16	39	)	)	PUNCT
ma-25	16	40	to	to	PART
ma-25	16	41	denote	denote	VERB
ma-25	16	42	respectively	respectively	ADV
ma-25	16	43	the	the	DET
ma-25	16	44	[	[	X
ma-25	16	45	p	p	X
ma-25	16	46	,	,	PUNCT
ma-25	16	47	q]-order	q]-order	NOUN
ma-25	16	48	and	and	CCONJ
ma-25	16	49	the	the	DET
ma-25	16	50	lower	low	ADJ
ma-25	16	51	[	[	X
ma-25	16	52	p	p	X
ma-25	16	53	,	,	PUNCT
ma-25	16	54	q]-order	q]-order	NOUN
ma-25	16	55	,	,	PUNCT
ma-25	16	56	λ[p	λ[p	PROPN
ma-25	16	57	,	,	PUNCT
ma-25	16	58	q	q	X
ma-25	16	59	]	]	X
ma-25	16	60	(	(	PUNCT
ma-25	16	61	f	f	PROPN
ma-25	16	62	−	−	PROPN
ma-25	16	63	a	a	NOUN
ma-25	16	64	)	)	PUNCT
ma-25	16	65	(	(	PUNCT
ma-25	16	66	or	or	CCONJ
ma-25	16	67	λ[p	λ[p	PROPN
ma-25	16	68	,	,	PUNCT
ma-25	16	69	q	q	X
ma-25	16	70	]	]	X
ma-25	16	71	(	(	PUNCT
ma-25	16	72	f	f	PROPN
ma-25	16	73	−	−	PROPN
ma-25	16	74	a	a	NOUN
ma-25	16	75	)	)	PUNCT
ma-25	16	76	)	)	PUNCT
ma-25	16	77	to	to	PART
ma-25	16	78	denote	denote	VERB
ma-25	16	79	the	the	DET
ma-25	16	80	[	[	X
ma-25	16	81	p	p	X
ma-25	16	82	,	,	PUNCT
ma-25	16	83	q]-convergence	q]-convergence	NOUN
ma-25	16	84	exponent	exponent	NOUN
ma-25	16	85	of	of	ADP
ma-25	16	86	the	the	DET
ma-25	16	87	sequence	sequence	NOUN
ma-25	16	88	ofdistinct	ofdistinct	PROPN
ma-25	16	89	a	a	DET
ma-25	16	90	-	-	PUNCT
ma-25	16	91	points	point	NOUN
ma-25	16	92	(	(	PUNCT
ma-25	16	93	or	or	CCONJ
ma-25	16	94	of	of	ADP
ma-25	16	95	a	a	DET
ma-25	16	96	-	-	PUNCT
ma-25	16	97	points	point	NOUN
ma-25	16	98	)	)	PUNCT
ma-25	16	99	and	and	CCONJ
ma-25	16	100	λ[p	λ[p	PROPN
ma-25	16	101	,	,	PUNCT
ma-25	16	102	q	q	X
ma-25	16	103	]	]	X
ma-25	16	104	(	(	PUNCT
ma-25	16	105	1f	1f	NOUN
ma-25	16	106	)	)	PUNCT
ma-25	16	107	to	to	PART
ma-25	16	108	denote	denote	VERB
ma-25	16	109	the	the	DET
ma-25	16	110	[	[	X
ma-25	16	111	p	p	X
ma-25	16	112	,	,	PUNCT
ma-25	16	113	q]-exponent	q]-exponent	NOUN
ma-25	16	114	of	of	ADP
ma-25	16	115	convergence	convergence	NOUN
ma-25	16	116	of	of	ADP
ma-25	16	117	thepoles	thepole	NOUN
ma-25	16	118	,	,	PUNCT
ma-25	16	119	we	we	PRON
ma-25	16	120	refer	refer	VERB
ma-25	16	121	the	the	DET
ma-25	16	122	reader	reader	NOUN
ma-25	16	123	to	to	PART
ma-25	16	124	see	see	VERB
ma-25	16	125	[	[	X
ma-25	16	126	12	12	NUM
ma-25	16	127	]	]	PUNCT
ma-25	16	128	,	,	PUNCT
ma-25	17	1	[	[	X
ma-25	17	2	15	15	NUM
ma-25	17	3	]	]	PUNCT
ma-25	17	4	,	,	PUNCT
ma-25	17	5	[	[	X
ma-25	17	6	16	16	NUM
ma-25	17	7	]	]	PUNCT
ma-25	17	8	and	and	CCONJ
ma-25	17	9	[	[	X
ma-25	17	10	25	25	NUM
ma-25	17	11	]	]	PUNCT
ma-25	17	12	.	.	PUNCT
ma-25	18	1	in	in	ADP
ma-25	18	2	particular	particular	ADJ
ma-25	18	3	for	for	ADP
ma-25	18	4	q	q	NOUN
ma-25	18	5	=	=	SYM
ma-25	18	6	1	1	NUM
ma-25	18	7	,	,	PUNCT
ma-25	18	8	ρ[p,1	ρ[p,1	VERB
ma-25	18	9	]	]	X
ma-25	18	10	(	(	PUNCT
ma-25	18	11	f	f	X
ma-25	18	12	)	)	PUNCT
ma-25	19	1	=	=	PUNCT
ma-25	19	2	ρp	ρp	INTJ
ma-25	19	3	(	(	PUNCT
ma-25	19	4	f	f	PROPN
ma-25	19	5	)	)	PUNCT
ma-25	19	6	is	be	AUX
ma-25	19	7	the	the	DET
ma-25	19	8	iterated	iterated	ADJ
ma-25	19	9	p	p	NOUN
ma-25	19	10	-	-	PUNCT
ma-25	19	11	order	order	NOUN
ma-25	19	12	,	,	PUNCT
ma-25	19	13	µ[p,1	µ[p,1	VERB
ma-25	19	14	]	]	PUNCT
ma-25	19	15	(	(	PUNCT
ma-25	19	16	f	f	X
ma-25	19	17	)	)	PUNCT
ma-25	20	1	=	=	NOUN
ma-25	20	2	µp	µp	PROPN
ma-25	20	3	(	(	PUNCT
ma-25	20	4	f	f	PROPN
ma-25	20	5	)	)	PUNCT
ma-25	20	6	is	be	AUX
ma-25	20	7	the	the	DET
ma-25	20	8	iterated	iterate	VERB
ma-25	20	9	lower	low	ADJ
ma-25	20	10	p	p	NOUN
ma-25	20	11	-	-	PUNCT
ma-25	20	12	order	order	NOUN
ma-25	20	13	,	,	PUNCT
ma-25	20	14	λ[p,1	λ[p,1	NOUN
ma-25	20	15	]	]	PUNCT
ma-25	20	16	(	(	PUNCT
ma-25	20	17	f	f	X
ma-25	20	18	−	−	PROPN
ma-25	20	19	a	a	X
ma-25	20	20	)	)	PUNCT
ma-25	20	21	=	=	SYM
ma-25	20	22	λp	λp	X
ma-25	21	1	(	(	PUNCT
ma-25	21	2	f	f	PROPN
ma-25	21	3	,	,	PUNCT
ma-25	21	4	a)(or	a)(or	PROPN
ma-25	21	5	λ[p,1	λ[p,1	NOUN
ma-25	21	6	]	]	PUNCT
ma-25	21	7	(	(	PUNCT
ma-25	21	8	f	f	X
ma-25	21	9	−	−	PROPN
ma-25	21	10	a	a	X
ma-25	21	11	)	)	PUNCT
ma-25	21	12	=	=	SYM
ma-25	22	1	λp	λp	X
ma-25	22	2	(	(	PUNCT
ma-25	22	3	f	f	PROPN
ma-25	22	4	,	,	PUNCT
ma-25	22	5	a	a	PRON
ma-25	22	6	)	)	PUNCT
ma-25	22	7	)	)	PUNCT
ma-25	22	8	is	be	AUX
ma-25	22	9	the	the	DET
ma-25	22	10	iterated	iterated	ADJ
ma-25	22	11	convergence	convergence	NOUN
ma-25	22	12	exponent	exponent	NOUN
ma-25	22	13	of	of	ADP
ma-25	22	14	the	the	DET
ma-25	22	15	sequence	sequence	NOUN
ma-25	22	16	of	of	ADP
ma-25	22	17	distinct	distinct	ADJ
ma-25	22	18	a	a	DET
ma-25	22	19	-	-	PUNCT
ma-25	22	20	points	point	NOUN
ma-25	22	21	(	(	PUNCT
ma-25	22	22	or	or	CCONJ
ma-25	22	23	of	of	ADP
ma-25	22	24	a	a	DET
ma-25	22	25	-	-	PUNCT
ma-25	22	26	points	point	NOUN
ma-25	22	27	)	)	PUNCT
ma-25	22	28	,	,	PUNCT
ma-25	22	29	λ[p,1	λ[p,1	NOUN
ma-25	22	30	]	]	PUNCT
ma-25	22	31	(	(	PUNCT
ma-25	22	32	1f	1f	NUM
ma-25	22	33	)	)	PUNCT
ma-25	23	1	=	=	PUNCT
ma-25	23	2	λp	λp	X
ma-25	23	3	(	(	PUNCT
ma-25	23	4	1	1	NUM
ma-25	23	5	f	f	NOUN
ma-25	23	6	)	)	PUNCT
ma-25	23	7	is	be	AUX
ma-25	23	8	the	the	DET
ma-25	23	9	iterated	iterated	ADJ
ma-25	23	10	exponent	exponent	NOUN
ma-25	23	11	of	of	ADP
ma-25	23	12	convergence	convergence	NOUN
ma-25	23	13	of	of	ADP
ma-25	23	14	the	the	DET
ma-25	23	15	poles	pole	NOUN
ma-25	23	16	,	,	PUNCT
ma-25	23	17	see	see	VERB
ma-25	23	18	[	[	X
ma-25	23	19	7	7	X
ma-25	23	20	]	]	PUNCT
ma-25	23	21	,	,	PUNCT
ma-25	24	1	[	[	X
ma-25	24	2	11	11	NUM
ma-25	24	3	]	]	PUNCT
ma-25	24	4	,	,	PUNCT
ma-25	24	5	[	[	X
ma-25	24	6	13	13	NUM
ma-25	24	7	]	]	PUNCT
ma-25	24	8	,	,	PUNCT
ma-25	25	1	[	[	X
ma-25	25	2	14	14	NUM
ma-25	25	3	]	]	PUNCT
ma-25	25	4	and	and	CCONJ
ma-25	25	5	[	[	X
ma-25	25	6	24	24	NUM
ma-25	25	7	]	]	PUNCT
ma-25	25	8	for	for	ADP
ma-25	25	9	notations	notation	NOUN
ma-25	25	10	and	and	CCONJ
ma-25	25	11	definitions	definition	NOUN
ma-25	25	12	.	.	PUNCT
ma-25	26	1	received	receive	VERB
ma-25	26	2	:	:	PUNCT
ma-25	26	3	6	6	NUM
ma-25	26	4	sep	sep	NOUN
ma-25	26	5	2021	2021	NUM
ma-25	26	6	.	.	PUNCT
ma-25	27	1	key	key	ADJ
ma-25	27	2	words	word	NOUN
ma-25	27	3	and	and	CCONJ
ma-25	27	4	phrases	phrase	NOUN
ma-25	27	5	.	.	PUNCT
ma-25	28	1	linear	linear	ADJ
ma-25	28	2	differential	differential	ADJ
ma-25	28	3	equations	equation	NOUN
ma-25	28	4	;	;	PUNCT
ma-25	28	5	meromorphic	meromorphic	ADJ
ma-25	28	6	functions	function	NOUN
ma-25	28	7	;	;	PUNCT
ma-25	28	8	[	[	X
ma-25	28	9	p	p	X
ma-25	28	10	,	,	PUNCT
ma-25	28	11	q]-order	q]-order	NOUN
ma-25	28	12	;	;	PUNCT
ma-25	29	1	[	[	X
ma-25	29	2	p	p	X
ma-25	29	3	,	,	PUNCT
ma-25	29	4	q]-exponent	q]-exponent	NOUN
ma-25	29	5	of	of	ADP
ma-25	29	6	conver	conver	NOUN
ma-25	29	7	-	-	PUNCT
ma-25	29	8	gence	gence	NOUN
ma-25	29	9	of	of	ADP
ma-25	29	10	zeros	zero	NOUN
ma-25	29	11	.	.	PUNCT
ma-25	30	1	86	86	NUM
ma-25	30	2	https://adac.ee	https://adac.ee	PROPN
ma-25	30	3	https://doi.org/10.28924/ada/ma.1.86	https://doi.org/10.28924/ada/ma.1.86	PROPN
ma-25	30	4	https://orcid.org/0000-0002-6635-2514	https://orcid.org/0000-0002-6635-2514	PROPN
ma-25	30	5	eur	eur	PROPN
ma-25	30	6	.	.	PUNCT
ma-25	31	1	j.	j.	PROPN
ma-25	31	2	math	math	PROPN
ma-25	31	3	.	.	PUNCT
ma-25	32	1	anal	anal	ADJ
ma-25	32	2	.	.	PUNCT
ma-25	33	1	1	1	NUM
ma-25	33	2	(	(	PUNCT
ma-25	33	3	2021	2021	NUM
ma-25	33	4	)	)	PUNCT
ma-25	33	5	87	87	NUM
ma-25	33	6	several	several	ADJ
ma-25	33	7	authors	author	NOUN
ma-25	33	8	have	have	AUX
ma-25	33	9	investigated	investigate	VERB
ma-25	33	10	the	the	DET
ma-25	33	11	growth	growth	NOUN
ma-25	33	12	of	of	ADP
ma-25	33	13	solutions	solution	NOUN
ma-25	33	14	of	of	ADP
ma-25	33	15	second	second	ADJ
ma-25	33	16	order	order	NOUN
ma-25	33	17	and	and	CCONJ
ma-25	33	18	higher	high	ADJ
ma-25	33	19	orderhomogeneous	orderhomogeneous	ADJ
ma-25	33	20	and	and	CCONJ
ma-25	33	21	non	non	ADJ
ma-25	33	22	-	-	ADJ
ma-25	33	23	homogeneous	homogeneous	ADJ
ma-25	33	24	linear	linear	PROPN
ma-25	33	25	differential	differential	NOUN
ma-25	33	26	equations	equation	NOUN
ma-25	33	27	with	with	ADP
ma-25	33	28	analytic	analytic	ADJ
ma-25	33	29	,	,	PUNCT
ma-25	33	30	entire	entire	ADJ
ma-25	33	31	or	or	CCONJ
ma-25	33	32	meromor	meromor	NOUN
ma-25	33	33	-	-	PUNCT
ma-25	33	34	phic	phic	ADJ
ma-25	33	35	coefficients	coefficient	NOUN
ma-25	33	36	,	,	PUNCT
ma-25	33	37	see	see	VERB
ma-25	33	38	(	(	PUNCT
ma-25	33	39	[	[	X
ma-25	33	40	1−3	1−3	NUM
ma-25	33	41	]	]	X
ma-25	33	42	,	,	PUNCT
ma-25	33	43	[	[	X
ma-25	33	44	6	6	NUM
ma-25	33	45	]	]	PUNCT
ma-25	33	46	,	,	PUNCT
ma-25	33	47	[	[	X
ma-25	33	48	8	8	NUM
ma-25	33	49	]	]	PUNCT
ma-25	33	50	,	,	PUNCT
ma-25	33	51	[	[	X
ma-25	33	52	11	11	NUM
ma-25	33	53	]	]	PUNCT
ma-25	33	54	,	,	PUNCT
ma-25	34	1	[	[	X
ma-25	34	2	13−16	13−16	NOUN
ma-25	34	3	]	]	X
ma-25	34	4	,	,	PUNCT
ma-25	34	5	[	[	X
ma-25	34	6	18	18	NUM
ma-25	34	7	]	]	PUNCT
ma-25	34	8	,	,	PUNCT
ma-25	34	9	[	[	X
ma-25	34	10	20−21	20−21	X
ma-25	34	11	]	]	X
ma-25	34	12	,	,	PUNCT
ma-25	34	13	[	[	X
ma-25	34	14	23	23	NUM
ma-25	34	15	]	]	PUNCT
ma-25	34	16	,	,	PUNCT
ma-25	34	17	[	[	X
ma-25	34	18	25	25	NUM
ma-25	34	19	]	]	PUNCT
ma-25	34	20	)	)	PUNCT
ma-25	34	21	.	.	PUNCT
ma-25	35	1	in	in	ADP
ma-25	35	2	the	the	DET
ma-25	35	3	recent	recent	ADJ
ma-25	35	4	years	year	NOUN
ma-25	35	5	,	,	PUNCT
ma-25	35	6	many	many	ADJ
ma-25	35	7	authors	author	NOUN
ma-25	35	8	have	have	AUX
ma-25	35	9	studied	study	VERB
ma-25	35	10	the	the	DET
ma-25	35	11	complex	complex	ADJ
ma-25	35	12	linear	linear	ADJ
ma-25	35	13	differential	differential	NOUN
ma-25	35	14	equations	equation	NOUN
ma-25	35	15	f	f	X
ma-25	35	16	(	(	PUNCT
ma-25	35	17	k	k	NOUN
ma-25	35	18	)	)	PUNCT
ma-25	35	19	+	+	CCONJ
ma-25	35	20	ak−1	ak−1	ADV
ma-25	35	21	(	(	PUNCT
ma-25	35	22	z	z	NOUN
ma-25	35	23	)	)	PUNCT
ma-25	35	24	f	f	PROPN
ma-25	35	25	(	(	PUNCT
ma-25	35	26	k−1	k−1	PROPN
ma-25	35	27	)	)	PUNCT
ma-25	36	1	+	+	PUNCT
ma-25	36	2	·	·	PUNCT
ma-25	36	3	·	·	PUNCT
ma-25	36	4	·	·	PUNCT
ma-25	36	5	+	+	NUM
ma-25	36	6	a1	a1	NOUN
ma-25	36	7	(	(	PUNCT
ma-25	36	8	z	z	NOUN
ma-25	36	9	)	)	PUNCT
ma-25	36	10	f	f	NOUN
ma-25	36	11	′	′	NUM
ma-25	37	1	+	+	CCONJ
ma-25	37	2	a0	a0	PROPN
ma-25	37	3	(	(	PUNCT
ma-25	37	4	z	z	NOUN
ma-25	37	5	)	)	PUNCT
ma-25	37	6	f	f	NOUN
ma-25	38	1	=	=	SYM
ma-25	38	2	0	0	PROPN
ma-25	38	3	,	,	PUNCT
ma-25	38	4	(	(	PUNCT
ma-25	38	5	1.1	1.1	NUM
ma-25	38	6	)	)	PUNCT
ma-25	38	7	f	f	NOUN
ma-25	38	8	(	(	PUNCT
ma-25	38	9	k	k	NOUN
ma-25	38	10	)	)	PUNCT
ma-25	39	1	+	+	CCONJ
ma-25	39	2	ak−1	ak−1	ADV
ma-25	39	3	(	(	PUNCT
ma-25	39	4	z	z	NOUN
ma-25	39	5	)	)	PUNCT
ma-25	39	6	f	f	PROPN
ma-25	39	7	(	(	PUNCT
ma-25	39	8	k−1	k−1	PROPN
ma-25	39	9	)	)	PUNCT
ma-25	40	1	+	+	PUNCT
ma-25	40	2	·	·	PUNCT
ma-25	40	3	·	·	PUNCT
ma-25	40	4	·	·	PUNCT
ma-25	40	5	+	+	NUM
ma-25	40	6	a1	a1	NOUN
ma-25	40	7	(	(	PUNCT
ma-25	40	8	z	z	NOUN
ma-25	40	9	)	)	PUNCT
ma-25	40	10	f	f	NOUN
ma-25	40	11	′	′	NUM
ma-25	41	1	+	+	CCONJ
ma-25	41	2	a0	a0	PROPN
ma-25	41	3	(	(	PUNCT
ma-25	41	4	z	z	NOUN
ma-25	41	5	)	)	PUNCT
ma-25	41	6	f	f	NOUN
ma-25	41	7	=	=	SYM
ma-25	41	8	f	f	PROPN
ma-25	41	9	(	(	PUNCT
ma-25	41	10	z	z	NOUN
ma-25	41	11	)	)	PUNCT
ma-25	41	12	,	,	PUNCT
ma-25	41	13	(	(	PUNCT
ma-25	41	14	1.2)where	1.2)where	NUM
ma-25	41	15	a0	a0	NOUN
ma-25	41	16	(	(	PUNCT
ma-25	41	17	z	z	NOUN
ma-25	41	18	)	)	PUNCT
ma-25	41	19	6≡	6≡	NUM
ma-25	41	20	0	0	NUM
ma-25	41	21	,	,	PUNCT
ma-25	41	22	a1	a1	NOUN
ma-25	41	23	(	(	PUNCT
ma-25	41	24	z	z	NOUN
ma-25	41	25	)	)	PUNCT
ma-25	41	26	,	,	PUNCT
ma-25	41	27	...	...	PUNCT
ma-25	41	28	,	,	PUNCT
ma-25	42	1	ak−1	ak−1	INTJ
ma-25	42	2	(	(	PUNCT
ma-25	42	3	z	z	NOUN
ma-25	42	4	)	)	PUNCT
ma-25	42	5	and	and	CCONJ
ma-25	42	6	f	f	PROPN
ma-25	42	7	(	(	PUNCT
ma-25	42	8	z	z	NOUN
ma-25	42	9	)	)	PUNCT
ma-25	42	10	6≡	6≡	NUM
ma-25	42	11	0	0	NUM
ma-25	42	12	are	be	AUX
ma-25	42	13	meromorphic	meromorphic	ADJ
ma-25	42	14	functions	function	NOUN
ma-25	42	15	of	of	ADP
ma-25	42	16	finite	finite	NOUN
ma-25	42	17	iterated	iterate	VERB
ma-25	42	18	p	p	NOUN
ma-25	42	19	-	-	PUNCT
ma-25	42	20	order	order	NOUN
ma-25	42	21	.	.	PUNCT
ma-25	43	1	in	in	ADP
ma-25	43	2	[	[	X
ma-25	43	3	2	2	NUM
ma-25	43	4	]	]	PUNCT
ma-25	43	5	,	,	PUNCT
ma-25	43	6	belaïdi	belaïdi	NOUN
ma-25	43	7	considered	consider	VERB
ma-25	43	8	the	the	DET
ma-25	43	9	growth	growth	NOUN
ma-25	43	10	of	of	ADP
ma-25	43	11	meromorphic	meromorphic	ADJ
ma-25	43	12	solutions	solution	NOUN
ma-25	43	13	of	of	ADP
ma-25	43	14	equations	equation	NOUN
ma-25	43	15	(	(	PUNCT
ma-25	43	16	1.1	1.1	NUM
ma-25	43	17	)	)	PUNCT
ma-25	43	18	and	and	CCONJ
ma-25	43	19	(	(	PUNCT
ma-25	43	20	1.2	1.2	NUM
ma-25	43	21	)	)	PUNCT
ma-25	43	22	with	with	ADP
ma-25	43	23	meromorphic	meromorphic	ADJ
ma-25	43	24	coefficients	coefficient	NOUN
ma-25	43	25	of	of	ADP
ma-25	43	26	finite	finite	NOUN
ma-25	43	27	iterated	iterate	VERB
ma-25	43	28	p−order	p−order	NOUN
ma-25	43	29	and	and	CCONJ
ma-25	43	30	obtained	obtain	VERB
ma-25	43	31	some	some	DET
ma-25	43	32	results	result	NOUN
ma-25	43	33	whichimprove	whichimprove	VERB
ma-25	43	34	and	and	CCONJ
ma-25	43	35	generalize	generalize	VERB
ma-25	43	36	some	some	DET
ma-25	43	37	previous	previous	ADJ
ma-25	43	38	results	result	NOUN
ma-25	43	39	.	.	PUNCT
ma-25	44	1	theorem	theorem	VERB
ma-25	44	2	a	a	PRON
ma-25	44	3	(	(	PUNCT
ma-25	44	4	[	[	X
ma-25	44	5	2	2	NUM
ma-25	44	6	]	]	PUNCT
ma-25	44	7	)	)	PUNCT
ma-25	44	8	let	let	VERB
ma-25	45	1	h	h	PRON
ma-25	45	2	⊂	⊂	PROPN
ma-25	46	1	[	[	X
ma-25	46	2	0,+∞	0,+∞	NUM
ma-25	46	3	)	)	PUNCT
ma-25	46	4	be	be	AUX
ma-25	46	5	a	a	DET
ma-25	46	6	set	set	NOUN
ma-25	46	7	with	with	ADP
ma-25	46	8	a	a	DET
ma-25	46	9	positive	positive	ADJ
ma-25	46	10	upper	upper	ADJ
ma-25	46	11	density	density	NOUN
ma-25	46	12	,	,	PUNCT
ma-25	46	13	and	and	CCONJ
ma-25	46	14	let	let	VERB
ma-25	46	15	aj	aj	PROPN
ma-25	46	16	(	(	PUNCT
ma-25	46	17	z	z	NOUN
ma-25	46	18	)	)	PUNCT
ma-25	46	19	(	(	PUNCT
ma-25	46	20	j	j	NOUN
ma-25	46	21	=	=	SYM
ma-25	46	22	0	0	NUM
ma-25	46	23	,	,	PUNCT
ma-25	46	24	1	1	NUM
ma-25	46	25	,	,	PUNCT
ma-25	46	26	...	...	PUNCT
ma-25	46	27	,	,	PUNCT
ma-25	46	28	k	k	PROPN
ma-25	47	1	−	−	NOUN
ma-25	47	2	1	1	X
ma-25	47	3	)	)	PUNCT
ma-25	47	4	be	be	AUX
ma-25	47	5	meromorphic	meromorphic	ADJ
ma-25	47	6	functions	function	NOUN
ma-25	47	7	with	with	ADP
ma-25	47	8	finite	finite	NOUN
ma-25	47	9	iterated	iterate	VERB
ma-25	47	10	p	p	NOUN
ma-25	47	11	-	-	PUNCT
ma-25	47	12	order	order	NOUN
ma-25	47	13	.	.	PUNCT
ma-25	48	1	if	if	SCONJ
ma-25	48	2	there	there	PRON
ma-25	48	3	exist	exist	VERB
ma-25	48	4	positive	positive	ADJ
ma-25	48	5	constants	constant	NOUN
ma-25	48	6	σ	σ	X
ma-25	48	7	>	>	X
ma-25	48	8	0	0	PROPN
ma-25	48	9	,	,	PUNCT
ma-25	48	10	α	α	X
ma-25	48	11	>	>	X
ma-25	48	12	0	0	NUM
ma-25	49	1	such	such	ADJ
ma-25	49	2	that	that	SCONJ
ma-25	49	3	ρ	ρ	PROPN
ma-25	49	4	=	=	SYM
ma-25	49	5	max	max	PROPN
ma-25	49	6	{	{	PUNCT
ma-25	49	7	ρp	ρp	X
ma-25	49	8	(	(	PUNCT
ma-25	49	9	aj	aj	PROPN
ma-25	49	10	)	)	PUNCT
ma-25	49	11	:	:	PUNCT
ma-25	50	1	j	j	X
ma-25	50	2	=	=	SYM
ma-25	50	3	1	1	NUM
ma-25	50	4	,	,	PUNCT
ma-25	50	5	...	...	PUNCT
ma-25	50	6	,	,	PUNCT
ma-25	50	7	k	k	PROPN
ma-25	50	8	−	−	PROPN
ma-25	50	9	1	1	NUM
ma-25	50	10	}	}	PUNCT
ma-25	50	11	<	<	X
ma-25	50	12	σ	σ	PROPN
ma-25	50	13	and	and	CCONJ
ma-25	50	14	|a0	|a0	PROPN
ma-25	50	15	(	(	PUNCT
ma-25	50	16	z	z	NOUN
ma-25	50	17	)	)	PUNCT
ma-25	50	18	|	|	ADV
ma-25	50	19	≥	≥	PRON
ma-25	50	20	expp	expp	ADJ
ma-25	50	21	(	(	PUNCT
ma-25	50	22	αrσ	αrσ	NOUN
ma-25	50	23	)	)	PUNCT
ma-25	50	24	as	as	ADP
ma-25	50	25	|z	|z	PROPN
ma-25	50	26	|	|	ADV
ma-25	50	27	=	=	SYM
ma-25	50	28	r	r	NOUN
ma-25	50	29	∈	∈	PROPN
ma-25	50	30	h	h	NOUN
ma-25	50	31	,	,	PUNCT
ma-25	50	32	r	r	NOUN
ma-25	50	33	→	→	SYM
ma-25	50	34	+	+	NOUN
ma-25	50	35	∞	∞	PROPN
ma-25	50	36	,	,	PUNCT
ma-25	50	37	then	then	ADV
ma-25	50	38	every	every	DET
ma-25	50	39	meromorphic	meromorphic	ADJ
ma-25	50	40	solution	solution	NOUN
ma-25	50	41	f	f	PROPN
ma-25	50	42	6≡	6≡	NUM
ma-25	50	43	0	0	NUM
ma-25	50	44	of	of	ADP
ma-25	50	45	equation	equation	NOUN
ma-25	50	46	(	(	PUNCT
ma-25	50	47	1.1	1.1	NUM
ma-25	50	48	)	)	PUNCT
ma-25	50	49	satisfies	satisfy	VERB
ma-25	50	50	µp	µp	PROPN
ma-25	50	51	(	(	PUNCT
ma-25	50	52	f	f	PROPN
ma-25	50	53	)	)	PUNCT
ma-25	50	54	=	=	SYM
ma-25	51	1	ρp(f	ρp(f	PROPN
ma-25	51	2	)	)	PUNCT
ma-25	52	1	=	=	PUNCT
ma-25	53	1	+	+	NUM
ma-25	53	2	∞	∞	PROPN
ma-25	53	3	,	,	PUNCT
ma-25	53	4	ρp+1(f	ρp+1(f	PROPN
ma-25	53	5	)	)	PUNCT
ma-25	53	6	≥	≥	PROPN
ma-25	53	7	σ	σ	X
ma-25	53	8	.	.	PUNCT
ma-25	54	1	furthermore	furthermore	ADV
ma-25	54	2	,	,	PUNCT
ma-25	54	3	if	if	SCONJ
ma-25	54	4	λp	λp	X
ma-25	54	5	(	(	PUNCT
ma-25	54	6	1	1	NUM
ma-25	54	7	f	f	NOUN
ma-25	54	8	)	)	PUNCT
ma-25	54	9	<	<	X
ma-25	54	10	∞	∞	PROPN
ma-25	54	11	,	,	PUNCT
ma-25	54	12	then	then	ADV
ma-25	54	13	i	i	PRON
ma-25	54	14	(	(	PUNCT
ma-25	54	15	f	f	X
ma-25	54	16	)	)	PUNCT
ma-25	55	1	=	=	PUNCT
ma-25	56	1	p	p	NOUN
ma-25	57	1	+	+	CCONJ
ma-25	57	2	1	1	NUM
ma-25	57	3	and	and	CCONJ
ma-25	57	4	σ	σ	NUM
ma-25	57	5	≤	≤	NUM
ma-25	57	6	ρp+1	ρp+1	VERB
ma-25	57	7	(	(	PUNCT
ma-25	57	8	f	f	NOUN
ma-25	57	9	)	)	PUNCT
ma-25	57	10	≤	≤	NOUN
ma-25	58	1	ρp	ρp	PRON
ma-25	58	2	(	(	PUNCT
ma-25	58	3	a0	a0	PROPN
ma-25	58	4	)	)	PUNCT
ma-25	58	5	.	.	PUNCT
ma-25	59	1	theorem	theorem	PROPN
ma-25	59	2	b	b	NOUN
ma-25	59	3	(	(	PUNCT
ma-25	59	4	[	[	X
ma-25	59	5	2	2	NUM
ma-25	59	6	]	]	PUNCT
ma-25	59	7	)	)	PUNCT
ma-25	59	8	let	let	VERB
ma-25	59	9	h	h	PRON
ma-25	59	10	⊂	⊂	PROPN
ma-25	60	1	[	[	X
ma-25	60	2	0,+∞	0,+∞	NUM
ma-25	60	3	)	)	PUNCT
ma-25	60	4	be	be	AUX
ma-25	60	5	a	a	DET
ma-25	60	6	set	set	NOUN
ma-25	60	7	with	with	ADP
ma-25	60	8	a	a	DET
ma-25	60	9	positive	positive	ADJ
ma-25	60	10	upper	upper	ADJ
ma-25	60	11	density	density	NOUN
ma-25	60	12	,	,	PUNCT
ma-25	60	13	and	and	CCONJ
ma-25	60	14	let	let	VERB
ma-25	60	15	aj	aj	PROPN
ma-25	60	16	(	(	PUNCT
ma-25	60	17	z	z	NOUN
ma-25	60	18	)	)	PUNCT
ma-25	60	19	(	(	PUNCT
ma-25	60	20	j	j	NOUN
ma-25	60	21	=	=	SYM
ma-25	60	22	0	0	NUM
ma-25	60	23	,	,	PUNCT
ma-25	60	24	1	1	NUM
ma-25	60	25	,	,	PUNCT
ma-25	60	26	...	...	PUNCT
ma-25	60	27	,	,	PUNCT
ma-25	60	28	k	k	PROPN
ma-25	61	1	−	−	PROPN
ma-25	61	2	1	1	NUM
ma-25	61	3	)	)	PUNCT
ma-25	61	4	and	and	CCONJ
ma-25	61	5	f	f	PROPN
ma-25	61	6	(	(	PUNCT
ma-25	61	7	z	z	NOUN
ma-25	61	8	)	)	PUNCT
ma-25	61	9	6≡	6≡	NUM
ma-25	61	10	0	0	NUM
ma-25	61	11	be	be	AUX
ma-25	61	12	meromorphic	meromorphic	ADJ
ma-25	61	13	functions	function	NOUN
ma-25	61	14	with	with	ADP
ma-25	61	15	finite	finite	NOUN
ma-25	61	16	iterated	iterate	VERB
ma-25	61	17	p	p	NOUN
ma-25	61	18	-	-	PUNCT
ma-25	61	19	order	order	NOUN
ma-25	61	20	.	.	PUNCT
ma-25	62	1	if	if	SCONJ
ma-25	62	2	there	there	PRON
ma-25	62	3	exist	exist	VERB
ma-25	62	4	positive	positive	ADJ
ma-25	62	5	constants	constant	NOUN
ma-25	62	6	σ	σ	X
ma-25	62	7	>	>	X
ma-25	62	8	0	0	PROPN
ma-25	62	9	,	,	PUNCT
ma-25	62	10	α	α	X
ma-25	62	11	>	>	X
ma-25	62	12	0	0	NUM
ma-25	63	1	such	such	ADJ
ma-25	63	2	that	that	SCONJ
ma-25	63	3	|a0	|a0	PROPN
ma-25	63	4	(	(	PUNCT
ma-25	63	5	z	z	NOUN
ma-25	63	6	)	)	PUNCT
ma-25	63	7	|	|	ADV
ma-25	63	8	≥	≥	PRON
ma-25	63	9	expp	expp	ADJ
ma-25	63	10	(	(	PUNCT
ma-25	63	11	αrσ	αrσ	NOUN
ma-25	63	12	)	)	PUNCT
ma-25	63	13	as	as	ADP
ma-25	63	14	|z	|z	PROPN
ma-25	63	15	|	|	ADV
ma-25	63	16	=	=	SYM
ma-25	63	17	r	r	NOUN
ma-25	63	18	∈	∈	PROPN
ma-25	63	19	h	h	NOUN
ma-25	63	20	,	,	PUNCT
ma-25	63	21	r	r	NOUN
ma-25	63	22	→	→	SYM
ma-25	63	23	+	+	NOUN
ma-25	63	24	∞	∞	PROPN
ma-25	63	25	,	,	PUNCT
ma-25	63	26	and	and	CCONJ
ma-25	63	27	ρ	ρ	PROPN
ma-25	63	28	=	=	SYM
ma-25	63	29	max	max	PROPN
ma-25	63	30	{	{	PUNCT
ma-25	63	31	ρp	ρp	X
ma-25	63	32	(	(	PUNCT
ma-25	63	33	aj	aj	PROPN
ma-25	63	34	)	)	PUNCT
ma-25	63	35	(	(	PUNCT
ma-25	63	36	j	j	NOUN
ma-25	63	37	=	=	SYM
ma-25	63	38	1	1	NUM
ma-25	63	39	,	,	PUNCT
ma-25	63	40	...	...	PUNCT
ma-25	63	41	,	,	PUNCT
ma-25	63	42	k	k	PROPN
ma-25	63	43	−	−	PROPN
ma-25	63	44	1	1	NUM
ma-25	63	45	)	)	PUNCT
ma-25	63	46	,	,	PUNCT
ma-25	63	47	ρp	ρp	PROPN
ma-25	63	48	(	(	PUNCT
ma-25	63	49	f	f	PROPN
ma-25	63	50	)	)	PUNCT
ma-25	63	51	}	}	PUNCT
ma-25	63	52	<	<	X
ma-25	63	53	σ	σ	PROPN
ma-25	63	54	,	,	PUNCT
ma-25	63	55	then	then	ADV
ma-25	63	56	every	every	DET
ma-25	63	57	meromorphic	meromorphic	ADJ
ma-25	63	58	solution	solution	NOUN
ma-25	63	59	of	of	ADP
ma-25	63	60	equation	equation	NOUN
ma-25	63	61	(	(	PUNCT
ma-25	63	62	1.2	1.2	NUM
ma-25	63	63	)	)	PUNCT
ma-25	63	64	with	with	ADP
ma-25	63	65	λp	λp	X
ma-25	64	1	(	(	PUNCT
ma-25	64	2	1	1	NUM
ma-25	64	3	f	f	NOUN
ma-25	64	4	)	)	PUNCT
ma-25	64	5	<	<	X
ma-25	64	6	σ	σ	PROPN
ma-25	64	7	satisfies	satisfie	NOUN
ma-25	64	8	λp	λp	X
ma-25	64	9	(	(	PUNCT
ma-25	64	10	f	f	PROPN
ma-25	64	11	)	)	PUNCT
ma-25	64	12	=	=	NOUN
ma-25	64	13	λp(f	λp(f	X
ma-25	64	14	)	)	PUNCT
ma-25	65	1	=	=	SYM
ma-25	65	2	ρp(f	ρp(f	PROPN
ma-25	65	3	)	)	PUNCT
ma-25	66	1	=	=	SYM
ma-25	66	2	∞	∞	PROPN
ma-25	66	3	,	,	PUNCT
ma-25	66	4	λp+1	λp+1	PROPN
ma-25	66	5	(	(	PUNCT
ma-25	66	6	f	f	X
ma-25	66	7	)	)	PUNCT
ma-25	66	8	=	=	SYM
ma-25	66	9	λp+1(f	λp+1(f	PROPN
ma-25	66	10	)	)	PUNCT
ma-25	67	1	=	=	PUNCT
ma-25	67	2	ρp+1(f	ρp+1(f	PROPN
ma-25	67	3	)	)	PUNCT
ma-25	67	4	.	.	PUNCT
ma-25	68	1	furthermore	furthermore	ADV
ma-25	68	2	,	,	PUNCT
ma-25	68	3	if	if	SCONJ
ma-25	68	4	λp	λp	X
ma-25	68	5	(	(	PUNCT
ma-25	68	6	1	1	NUM
ma-25	68	7	f	f	NOUN
ma-25	68	8	)	)	PUNCT
ma-25	68	9	<	<	X
ma-25	68	10	min	min	PROPN
ma-25	68	11	{	{	PUNCT
ma-25	68	12	µp	µp	PROPN
ma-25	68	13	(	(	PUNCT
ma-25	68	14	f	f	PROPN
ma-25	68	15	)	)	PUNCT
ma-25	68	16	,	,	PUNCT
ma-25	68	17	σ	σ	PROPN
ma-25	68	18	}	}	PUNCT
ma-25	68	19	,	,	PUNCT
ma-25	68	20	then	then	ADV
ma-25	68	21	i	i	PRON
ma-25	68	22	(	(	PUNCT
ma-25	68	23	f	f	X
ma-25	68	24	)	)	PUNCT
ma-25	68	25	=	=	PUNCT
ma-25	69	1	p	p	NOUN
ma-25	70	1	+	+	CCONJ
ma-25	70	2	1	1	NUM
ma-25	70	3	and	and	CCONJ
ma-25	70	4	λp+1	λp+1	NUM
ma-25	70	5	(	(	PUNCT
ma-25	70	6	f	f	X
ma-25	70	7	)	)	PUNCT
ma-25	70	8	=	=	SYM
ma-25	70	9	λp+1(f	λp+1(f	PROPN
ma-25	70	10	)	)	PUNCT
ma-25	71	1	=	=	PUNCT
ma-25	72	1	ρp+1	ρp+1	PROPN
ma-25	72	2	(	(	PUNCT
ma-25	72	3	f	f	NOUN
ma-25	72	4	)	)	PUNCT
ma-25	72	5	≤	≤	NOUN
ma-25	73	1	ρp	ρp	PRON
ma-25	73	2	(	(	PUNCT
ma-25	73	3	a0	a0	PROPN
ma-25	73	4	)	)	PUNCT
ma-25	73	5	.	.	PUNCT
ma-25	74	1	recently	recently	ADV
ma-25	74	2	,	,	PUNCT
ma-25	74	3	in	in	ADP
ma-25	74	4	[	[	X
ma-25	74	5	18	18	NUM
ma-25	74	6	]	]	PUNCT
ma-25	74	7	the	the	DET
ma-25	74	8	authors	author	NOUN
ma-25	74	9	have	have	AUX
ma-25	74	10	studied	study	VERB
ma-25	74	11	the	the	DET
ma-25	74	12	growth	growth	NOUN
ma-25	74	13	of	of	ADP
ma-25	74	14	solutions	solution	NOUN
ma-25	74	15	of	of	ADP
ma-25	74	16	the	the	DET
ma-25	74	17	equations	equation	NOUN
ma-25	74	18	(	(	PUNCT
ma-25	74	19	1.1	1.1	NUM
ma-25	74	20	)	)	PUNCT
ma-25	74	21	and	and	CCONJ
ma-25	74	22	(	(	PUNCT
ma-25	74	23	1.2	1.2	NUM
ma-25	74	24	)	)	PUNCT
ma-25	74	25	when	when	SCONJ
ma-25	74	26	as(z	as(z	NUM
ma-25	74	27	)	)	PUNCT
ma-25	74	28	to	to	PART
ma-25	74	29	dominate	dominate	VERB
ma-25	74	30	all	all	DET
ma-25	74	31	other	other	ADJ
ma-25	74	32	coefficients	coefficient	NOUN
ma-25	74	33	and	and	CCONJ
ma-25	74	34	they	they	PRON
ma-25	74	35	got	get	VERB
ma-25	74	36	some	some	DET
ma-25	74	37	results	result	NOUN
ma-25	74	38	about	about	ADP
ma-25	74	39	ρp+1	ρp+1	PROPN
ma-25	74	40	(	(	PUNCT
ma-25	74	41	f	f	NOUN
ma-25	74	42	)	)	PUNCT
ma-25	74	43	asfollows	asfollow	NOUN
ma-25	74	44	.	.	PUNCT
ma-25	75	1	theorem	theorem	PROPN
ma-25	75	2	c	c	NOUN
ma-25	75	3	(	(	PUNCT
ma-25	75	4	[	[	X
ma-25	75	5	18	18	NUM
ma-25	75	6	]	]	PUNCT
ma-25	75	7	)	)	PUNCT
ma-25	75	8	let	let	VERB
ma-25	75	9	h	h	PROPN
ma-25	75	10	⊂	⊂	PROPN
ma-25	75	11	(	(	PUNCT
ma-25	75	12	1,+∞	1,+∞	NUM
ma-25	75	13	)	)	PUNCT
ma-25	75	14	be	be	AUX
ma-25	75	15	a	a	DET
ma-25	75	16	set	set	NOUN
ma-25	75	17	with	with	ADP
ma-25	75	18	a	a	DET
ma-25	75	19	positive	positive	ADJ
ma-25	75	20	upper	upper	ADJ
ma-25	75	21	logarithmic	logarithmic	ADJ
ma-25	75	22	density	density	NOUN
ma-25	75	23	(	(	PUNCT
ma-25	75	24	or	or	CCONJ
ma-25	75	25	ml	ml	INTJ
ma-25	75	26	(	(	PUNCT
ma-25	75	27	h	h	NOUN
ma-25	75	28	)	)	PUNCT
ma-25	75	29	=	=	PUNCT
ma-25	76	1	+	+	NOUN
ma-25	76	2	∞	∞	NOUN
ma-25	76	3	)	)	PUNCT
ma-25	76	4	,	,	PUNCT
ma-25	76	5	and	and	CCONJ
ma-25	76	6	let	let	VERB
ma-25	76	7	aj	aj	PROPN
ma-25	76	8	(	(	PUNCT
ma-25	76	9	z	z	NOUN
ma-25	76	10	)	)	PUNCT
ma-25	76	11	(	(	PUNCT
ma-25	76	12	j	j	NOUN
ma-25	76	13	=	=	SYM
ma-25	76	14	0	0	NUM
ma-25	76	15	,	,	PUNCT
ma-25	76	16	1	1	NUM
ma-25	76	17	,	,	PUNCT
ma-25	76	18	...	...	PUNCT
ma-25	76	19	,	,	PUNCT
ma-25	77	1	k	k	PROPN
ma-25	77	2	−	−	NOUN
ma-25	77	3	1	1	X
ma-25	77	4	)	)	PUNCT
ma-25	77	5	be	be	AUX
ma-25	77	6	meromorphic	meromorphic	ADJ
ma-25	77	7	functions	function	NOUN
ma-25	77	8	with	with	ADP
ma-25	77	9	finite	finite	NOUN
ma-25	77	10	iterated	iterate	VERB
ma-25	77	11	p	p	NOUN
ma-25	77	12	-	-	PUNCT
ma-25	77	13	order	order	NOUN
ma-25	77	14	.	.	PUNCT
ma-25	78	1	eur	eur	PROPN
ma-25	78	2	.	.	PUNCT
ma-25	79	1	j.	j.	PROPN
ma-25	79	2	math	math	PROPN
ma-25	79	3	.	.	PUNCT
ma-25	80	1	anal	anal	ADJ
ma-25	80	2	.	.	PUNCT
ma-25	81	1	1	1	NUM
ma-25	81	2	(	(	PUNCT
ma-25	81	3	2021	2021	NUM
ma-25	81	4	)	)	PUNCT
ma-25	81	5	88	88	NUM
ma-25	81	6	if	if	SCONJ
ma-25	81	7	there	there	PRON
ma-25	81	8	exist	exist	VERB
ma-25	81	9	positive	positive	ADJ
ma-25	81	10	constants	constant	NOUN
ma-25	81	11	σ	σ	X
ma-25	81	12	>	>	X
ma-25	81	13	0	0	PROPN
ma-25	81	14	,	,	PUNCT
ma-25	81	15	α	α	NOUN
ma-25	81	16	>	>	X
ma-25	81	17	0	0	PUNCT
ma-25	82	1	and	and	CCONJ
ma-25	82	2	an	an	DET
ma-25	82	3	integer	integer	NOUN
ma-25	82	4	s	s	NOUN
ma-25	82	5	,	,	PUNCT
ma-25	82	6	0	0	NUM
ma-25	82	7	≤	≤	NUM
ma-25	82	8	s	s	PART
ma-25	82	9	≤	≤	NUM
ma-25	82	10	k	k	NOUN
ma-25	83	1	−	−	PROPN
ma-25	83	2	1	1	NUM
ma-25	83	3	,	,	PUNCT
ma-25	83	4	such	such	ADJ
ma-25	83	5	that	that	SCONJ
ma-25	83	6	|as	|as	PROPN
ma-25	83	7	(	(	PUNCT
ma-25	83	8	z	z	NOUN
ma-25	83	9	)	)	PUNCT
ma-25	83	10	|	|	ADV
ma-25	83	11	≥	≥	PRON
ma-25	83	12	expp	expp	ADJ
ma-25	83	13	(	(	PUNCT
ma-25	83	14	αrσ	αrσ	NOUN
ma-25	83	15	)	)	PUNCT
ma-25	83	16	as	as	ADP
ma-25	83	17	|z	|z	PROPN
ma-25	83	18	|	|	ADV
ma-25	83	19	=	=	SYM
ma-25	83	20	r	r	NOUN
ma-25	83	21	∈	∈	PROPN
ma-25	83	22	h	h	NOUN
ma-25	83	23	,	,	PUNCT
ma-25	83	24	r	r	NOUN
ma-25	83	25	→	→	SYM
ma-25	83	26	+	+	NOUN
ma-25	83	27	∞	∞	PROPN
ma-25	83	28	,	,	PUNCT
ma-25	83	29	and	and	CCONJ
ma-25	83	30	ρ	ρ	PROPN
ma-25	83	31	=	=	SYM
ma-25	83	32	max	max	PROPN
ma-25	83	33	{	{	PUNCT
ma-25	83	34	ρp	ρp	X
ma-25	83	35	(	(	PUNCT
ma-25	83	36	aj	aj	PROPN
ma-25	83	37	)	)	PUNCT
ma-25	83	38	(	(	PUNCT
ma-25	83	39	j	j	PROPN
ma-25	83	40	6=	6=	NUM
ma-25	83	41	s	s	PART
ma-25	83	42	)	)	PUNCT
ma-25	83	43	}	}	PUNCT
ma-25	83	44	<	<	X
ma-25	83	45	σ	σ	PROPN
ma-25	83	46	,	,	PUNCT
ma-25	83	47	then	then	ADV
ma-25	83	48	every	every	DET
ma-25	83	49	non	non	ADJ
ma-25	83	50	-	-	ADJ
ma-25	83	51	transcendental	transcendental	ADJ
ma-25	83	52	meromorphic	meromorphic	ADJ
ma-25	83	53	solution	solution	NOUN
ma-25	83	54	f	f	PROPN
ma-25	83	55	6≡	6≡	NUM
ma-25	83	56	0	0	NUM
ma-25	83	57	of	of	ADP
ma-25	83	58	(	(	PUNCT
ma-25	83	59	1.1	1.1	NUM
ma-25	83	60	)	)	PUNCT
ma-25	83	61	is	be	AUX
ma-25	83	62	a	a	DET
ma-25	83	63	polynomial	polynomial	NOUN
ma-25	83	64	with	with	ADP
ma-25	83	65	deg	deg	PROPN
ma-25	83	66	f	f	PROPN
ma-25	83	67	≤	≤	PROPN
ma-25	83	68	s	s	PART
ma-25	83	69	−	−	PROPN
ma-25	83	70	1	1	NUM
ma-25	83	71	and	and	CCONJ
ma-25	83	72	every	every	DET
ma-25	83	73	transcendental	transcendental	ADJ
ma-25	83	74	meromorphic	meromorphic	ADJ
ma-25	83	75	solution	solution	NOUN
ma-25	83	76	f	f	PROPN
ma-25	83	77	of	of	ADP
ma-25	83	78	(	(	PUNCT
ma-25	83	79	1.1	1.1	NUM
ma-25	83	80	)	)	PUNCT
ma-25	83	81	with	with	ADP
ma-25	83	82	λp	λp	X
ma-25	83	83	(	(	PUNCT
ma-25	83	84	1	1	NUM
ma-25	83	85	f	f	NOUN
ma-25	83	86	)	)	PUNCT
ma-25	83	87	<	<	X
ma-25	83	88	µp	µp	PROPN
ma-25	83	89	(	(	PUNCT
ma-25	83	90	f	f	NOUN
ma-25	83	91	)	)	PUNCT
ma-25	83	92	satisfies	satisfy	VERB
ma-25	83	93	i	i	PRON
ma-25	83	94	(	(	PUNCT
ma-25	83	95	f	f	PROPN
ma-25	83	96	)	)	PUNCT
ma-25	84	1	=	=	PRON
ma-25	84	2	p+	p+	VERB
ma-25	84	3	1	1	NUM
ma-25	84	4	µp	µp	NOUN
ma-25	84	5	(	(	PUNCT
ma-25	84	6	f	f	PROPN
ma-25	84	7	)	)	PUNCT
ma-25	84	8	=	=	SYM
ma-25	85	1	ρp(f	ρp(f	PROPN
ma-25	85	2	)	)	PUNCT
ma-25	86	1	=	=	PUNCT
ma-25	87	1	+	+	PUNCT
ma-25	87	2	∞	∞	NUM
ma-25	87	3	and	and	CCONJ
ma-25	87	4	σ	σ	NUM
ma-25	87	5	≤	≤	NUM
ma-25	87	6	ρp+1	ρp+1	VERB
ma-25	87	7	(	(	PUNCT
ma-25	87	8	f	f	NOUN
ma-25	87	9	)	)	PUNCT
ma-25	87	10	≤	≤	NOUN
ma-25	88	1	ρp	ρp	INTJ
ma-25	88	2	(	(	PUNCT
ma-25	88	3	as	as	ADP
ma-25	88	4	)	)	PUNCT
ma-25	88	5	.	.	PUNCT
ma-25	89	1	theorem	theorem	PROPN
ma-25	89	2	d	d	X
ma-25	89	3	(	(	PUNCT
ma-25	89	4	[	[	X
ma-25	89	5	18	18	NUM
ma-25	89	6	]	]	PUNCT
ma-25	89	7	)	)	PUNCT
ma-25	89	8	let	let	VERB
ma-25	89	9	h	h	PROPN
ma-25	89	10	⊂	⊂	PROPN
ma-25	89	11	(	(	PUNCT
ma-25	89	12	1,+∞	1,+∞	NUM
ma-25	89	13	)	)	PUNCT
ma-25	89	14	be	be	AUX
ma-25	89	15	a	a	DET
ma-25	89	16	set	set	NOUN
ma-25	89	17	with	with	ADP
ma-25	89	18	a	a	DET
ma-25	89	19	positive	positive	ADJ
ma-25	89	20	upper	upper	ADJ
ma-25	89	21	logarithmic	logarithmic	ADJ
ma-25	89	22	density	density	NOUN
ma-25	89	23	(	(	PUNCT
ma-25	89	24	or	or	CCONJ
ma-25	89	25	ml	ml	INTJ
ma-25	89	26	(	(	PUNCT
ma-25	89	27	h	h	NOUN
ma-25	89	28	)	)	PUNCT
ma-25	89	29	=	=	PUNCT
ma-25	90	1	+	+	NOUN
ma-25	90	2	∞	∞	NOUN
ma-25	90	3	)	)	PUNCT
ma-25	90	4	,	,	PUNCT
ma-25	90	5	and	and	CCONJ
ma-25	90	6	let	let	VERB
ma-25	90	7	aj	aj	PROPN
ma-25	90	8	(	(	PUNCT
ma-25	90	9	z	z	NOUN
ma-25	90	10	)	)	PUNCT
ma-25	90	11	(	(	PUNCT
ma-25	90	12	j	j	NOUN
ma-25	90	13	=	=	SYM
ma-25	90	14	0	0	NUM
ma-25	90	15	,	,	PUNCT
ma-25	90	16	1	1	NUM
ma-25	90	17	,	,	PUNCT
ma-25	90	18	...	...	PUNCT
ma-25	90	19	,	,	PUNCT
ma-25	90	20	k−1	k−1	PROPN
ma-25	90	21	)	)	PUNCT
ma-25	90	22	and	and	CCONJ
ma-25	90	23	f	f	PROPN
ma-25	90	24	(	(	PUNCT
ma-25	90	25	z	z	NOUN
ma-25	90	26	)	)	PUNCT
ma-25	90	27	6≡	6≡	NUM
ma-25	90	28	0	0	NUM
ma-25	90	29	be	be	AUX
ma-25	90	30	meromorphic	meromorphic	ADJ
ma-25	90	31	functions	function	NOUN
ma-25	90	32	with	with	ADP
ma-25	90	33	finite	finite	NOUN
ma-25	90	34	iterated	iterate	VERB
ma-25	90	35	p	p	NOUN
ma-25	90	36	-	-	PUNCT
ma-25	90	37	order	order	NOUN
ma-25	90	38	.	.	PUNCT
ma-25	91	1	if	if	SCONJ
ma-25	91	2	there	there	PRON
ma-25	91	3	exist	exist	VERB
ma-25	91	4	positive	positive	ADJ
ma-25	91	5	constants	constant	NOUN
ma-25	91	6	σ	σ	X
ma-25	91	7	>	>	X
ma-25	91	8	0	0	PROPN
ma-25	91	9	,	,	PUNCT
ma-25	91	10	α	α	NOUN
ma-25	91	11	>	>	X
ma-25	91	12	0	0	PUNCT
ma-25	92	1	and	and	CCONJ
ma-25	92	2	an	an	DET
ma-25	92	3	integer	integer	NOUN
ma-25	92	4	s	s	NOUN
ma-25	92	5	,	,	PUNCT
ma-25	92	6	0	0	NUM
ma-25	92	7	≤	≤	NUM
ma-25	92	8	s	s	PART
ma-25	92	9	≤	≤	NUM
ma-25	92	10	k	k	NOUN
ma-25	93	1	−	−	PROPN
ma-25	93	2	1	1	NUM
ma-25	93	3	,	,	PUNCT
ma-25	93	4	such	such	ADJ
ma-25	93	5	that	that	SCONJ
ma-25	93	6	|as	|as	PROPN
ma-25	93	7	(	(	PUNCT
ma-25	93	8	z	z	NOUN
ma-25	93	9	)	)	PUNCT
ma-25	93	10	|	|	ADV
ma-25	93	11	≥	≥	PRON
ma-25	93	12	expp	expp	ADJ
ma-25	93	13	(	(	PUNCT
ma-25	93	14	αrσ	αrσ	NOUN
ma-25	93	15	)	)	PUNCT
ma-25	93	16	as	as	ADP
ma-25	93	17	|z	|z	PROPN
ma-25	93	18	|	|	ADV
ma-25	93	19	=	=	SYM
ma-25	93	20	r	r	NOUN
ma-25	93	21	∈	∈	PROPN
ma-25	93	22	h	h	NOUN
ma-25	93	23	,	,	PUNCT
ma-25	93	24	r	r	NOUN
ma-25	93	25	→	→	SYM
ma-25	93	26	+	+	NOUN
ma-25	93	27	∞	∞	PROPN
ma-25	93	28	,	,	PUNCT
ma-25	93	29	and	and	CCONJ
ma-25	93	30	max	max	PROPN
ma-25	93	31	{	{	PUNCT
ma-25	93	32	ρp	ρp	X
ma-25	93	33	(	(	PUNCT
ma-25	93	34	aj	aj	PROPN
ma-25	93	35	)	)	PUNCT
ma-25	93	36	(	(	PUNCT
ma-25	93	37	j	j	PROPN
ma-25	93	38	6=	6=	NUM
ma-25	93	39	s	s	PART
ma-25	93	40	)	)	PUNCT
ma-25	93	41	,	,	PUNCT
ma-25	93	42	ρp	ρp	PROPN
ma-25	93	43	(	(	PUNCT
ma-25	93	44	f	f	PROPN
ma-25	93	45	)	)	PUNCT
ma-25	93	46	}	}	PUNCT
ma-25	93	47	<	<	X
ma-25	93	48	σ	σ	PROPN
ma-25	93	49	,	,	PUNCT
ma-25	93	50	then	then	ADV
ma-25	93	51	every	every	DET
ma-25	93	52	non	non	ADJ
ma-25	93	53	-	-	ADJ
ma-25	93	54	transcendental	transcendental	ADJ
ma-25	93	55	meromorphic	meromorphic	ADJ
ma-25	93	56	solution	solution	NOUN
ma-25	93	57	f	f	PROPN
ma-25	93	58	of	of	ADP
ma-25	93	59	(	(	PUNCT
ma-25	93	60	1.2	1.2	NUM
ma-25	93	61	)	)	PUNCT
ma-25	93	62	is	be	AUX
ma-25	93	63	a	a	DET
ma-25	93	64	polynomial	polynomial	NOUN
ma-25	93	65	with	with	ADP
ma-25	93	66	deg	deg	PROPN
ma-25	93	67	f	f	PROPN
ma-25	93	68	≤	≤	PROPN
ma-25	93	69	s	s	PART
ma-25	93	70	−	−	PROPN
ma-25	93	71	1	1	NUM
ma-25	93	72	and	and	CCONJ
ma-25	93	73	every	every	DET
ma-25	93	74	transcendental	transcendental	ADJ
ma-25	93	75	meromorphic	meromorphic	ADJ
ma-25	93	76	solution	solution	NOUN
ma-25	93	77	f	f	PROPN
ma-25	93	78	of	of	ADP
ma-25	93	79	(	(	PUNCT
ma-25	93	80	1.2	1.2	NUM
ma-25	93	81	)	)	PUNCT
ma-25	93	82	with	with	ADP
ma-25	93	83	λp	λp	X
ma-25	93	84	(	(	PUNCT
ma-25	93	85	1	1	NUM
ma-25	93	86	f	f	NOUN
ma-25	93	87	)	)	PUNCT
ma-25	93	88	<	<	X
ma-25	93	89	min	min	PROPN
ma-25	93	90	{	{	PUNCT
ma-25	93	91	σ	σ	PROPN
ma-25	93	92	,	,	PUNCT
ma-25	93	93	µp(f	µp(f	PUNCT
ma-25	93	94	)	)	PUNCT
ma-25	93	95	}	}	PUNCT
ma-25	93	96	satisfies	satisfy	VERB
ma-25	93	97	i	i	PRON
ma-25	93	98	(	(	PUNCT
ma-25	93	99	f	f	PROPN
ma-25	93	100	)	)	PUNCT
ma-25	94	1	=	=	PUNCT
ma-25	95	1	p	p	NOUN
ma-25	95	2	+	+	NOUN
ma-25	95	3	1	1	NUM
ma-25	95	4	λp	λp	PRON
ma-25	95	5	(	(	PUNCT
ma-25	95	6	f	f	PROPN
ma-25	95	7	)	)	PUNCT
ma-25	95	8	=	=	NOUN
ma-25	95	9	λp(f	λp(f	X
ma-25	95	10	)	)	PUNCT
ma-25	96	1	=	=	SYM
ma-25	96	2	ρp(f	ρp(f	PROPN
ma-25	96	3	)	)	PUNCT
ma-25	97	1	=	=	NOUN
ma-25	97	2	µp	µp	PROPN
ma-25	97	3	(	(	PUNCT
ma-25	97	4	f	f	PROPN
ma-25	97	5	)	)	PUNCT
ma-25	97	6	=	=	PUNCT
ma-25	98	1	+	+	PUNCT
ma-25	98	2	∞	∞	NUM
ma-25	98	3	and	and	CCONJ
ma-25	98	4	σ	σ	NUM
ma-25	98	5	≤	≤	NUM
ma-25	98	6	λp+1	λp+1	NOUN
ma-25	98	7	(	(	PUNCT
ma-25	98	8	f	f	X
ma-25	98	9	)	)	PUNCT
ma-25	99	1	=	=	SYM
ma-25	99	2	λp+1(f	λp+1(f	PROPN
ma-25	99	3	)	)	PUNCT
ma-25	100	1	=	=	PUNCT
ma-25	101	1	ρp+1	ρp+1	PROPN
ma-25	101	2	(	(	PUNCT
ma-25	101	3	f	f	NOUN
ma-25	101	4	)	)	PUNCT
ma-25	101	5	≤	≤	NOUN
ma-25	102	1	ρp	ρp	INTJ
ma-25	102	2	(	(	PUNCT
ma-25	102	3	as	as	ADP
ma-25	102	4	)	)	PUNCT
ma-25	102	5	.thus	.thus	ADV
ma-25	102	6	,	,	PUNCT
ma-25	102	7	the	the	DET
ma-25	102	8	following	follow	VERB
ma-25	102	9	question	question	NOUN
ma-25	102	10	arises	arise	VERB
ma-25	102	11	:	:	PUNCT
ma-25	102	12	can	can	AUX
ma-25	102	13	we	we	PRON
ma-25	102	14	have	have	VERB
ma-25	102	15	the	the	DET
ma-25	102	16	same	same	ADJ
ma-25	102	17	properties	property	NOUN
ma-25	102	18	as	as	ADP
ma-25	102	19	in	in	ADP
ma-25	102	20	theorems	theorem	NOUN
ma-25	102	21	c	c	PROPN
ma-25	102	22	andd	andd	PROPN
ma-25	102	23	for	for	ADP
ma-25	102	24	the	the	DET
ma-25	102	25	solutions	solution	NOUN
ma-25	102	26	of	of	ADP
ma-25	102	27	equations	equation	NOUN
ma-25	102	28	ak	ak	PROPN
ma-25	102	29	(	(	PUNCT
ma-25	102	30	z	z	PROPN
ma-25	102	31	)	)	PUNCT
ma-25	102	32	f	f	NOUN
ma-25	102	33	(	(	PUNCT
ma-25	102	34	k	k	NOUN
ma-25	102	35	)	)	PUNCT
ma-25	102	36	+	+	CCONJ
ma-25	102	37	ak−1	ak−1	ADV
ma-25	102	38	(	(	PUNCT
ma-25	102	39	z	z	NOUN
ma-25	102	40	)	)	PUNCT
ma-25	102	41	f	f	PROPN
ma-25	102	42	(	(	PUNCT
ma-25	102	43	k−1	k−1	PROPN
ma-25	102	44	)	)	PUNCT
ma-25	103	1	+	+	PUNCT
ma-25	103	2	·	·	PUNCT
ma-25	103	3	·	·	PUNCT
ma-25	103	4	·	·	PUNCT
ma-25	103	5	+	+	NUM
ma-25	103	6	a1	a1	NOUN
ma-25	103	7	(	(	PUNCT
ma-25	103	8	z	z	NOUN
ma-25	103	9	)	)	PUNCT
ma-25	103	10	f	f	NOUN
ma-25	103	11	′	′	NUM
ma-25	104	1	+	+	CCONJ
ma-25	104	2	a0	a0	PROPN
ma-25	104	3	(	(	PUNCT
ma-25	104	4	z	z	NOUN
ma-25	104	5	)	)	PUNCT
ma-25	104	6	f	f	NOUN
ma-25	104	7	=	=	SYM
ma-25	104	8	0	0	PROPN
ma-25	104	9	(	(	PUNCT
ma-25	104	10	1.3	1.3	NUM
ma-25	104	11	)	)	PUNCT
ma-25	104	12	and	and	CCONJ
ma-25	104	13	ak	ak	PROPN
ma-25	104	14	(	(	PUNCT
ma-25	104	15	z	z	PROPN
ma-25	104	16	)	)	PUNCT
ma-25	104	17	f	f	NOUN
ma-25	104	18	(	(	PUNCT
ma-25	104	19	k	k	NOUN
ma-25	104	20	)	)	PUNCT
ma-25	104	21	+	+	CCONJ
ma-25	104	22	ak−1	ak−1	ADV
ma-25	104	23	(	(	PUNCT
ma-25	104	24	z	z	NOUN
ma-25	104	25	)	)	PUNCT
ma-25	104	26	f	f	PROPN
ma-25	104	27	(	(	PUNCT
ma-25	104	28	k−1	k−1	PROPN
ma-25	104	29	)	)	PUNCT
ma-25	104	30	+	+	PUNCT
ma-25	104	31	·	·	PUNCT
ma-25	104	32	·	·	PUNCT
ma-25	104	33	·	·	PUNCT
ma-25	105	1	+	+	NUM
ma-25	105	2	a1	a1	NOUN
ma-25	105	3	(	(	PUNCT
ma-25	105	4	z	z	NOUN
ma-25	105	5	)	)	PUNCT
ma-25	105	6	f	f	NOUN
ma-25	105	7	′	′	NUM
ma-25	106	1	+	+	CCONJ
ma-25	106	2	a0	a0	PROPN
ma-25	106	3	(	(	PUNCT
ma-25	106	4	z	z	NOUN
ma-25	106	5	)	)	PUNCT
ma-25	106	6	f	f	NOUN
ma-25	106	7	=	=	SYM
ma-25	106	8	f	f	PROPN
ma-25	106	9	(	(	PUNCT
ma-25	106	10	z	z	NOUN
ma-25	106	11	)	)	PUNCT
ma-25	106	12	,	,	PUNCT
ma-25	106	13	(	(	PUNCT
ma-25	106	14	1.4)when	1.4)when	ADV
ma-25	106	15	the	the	DET
ma-25	106	16	coefficients	coefficient	NOUN
ma-25	106	17	aj	aj	PROPN
ma-25	106	18	(	(	PUNCT
ma-25	106	19	j	j	PROPN
ma-25	106	20	=	=	SYM
ma-25	106	21	0	0	NUM
ma-25	106	22	,	,	PUNCT
ma-25	106	23	1	1	NUM
ma-25	106	24	,	,	PUNCT
ma-25	106	25	...	...	PUNCT
ma-25	106	26	,	,	PUNCT
ma-25	106	27	k	k	X
ma-25	106	28	)	)	PUNCT
ma-25	106	29	are	be	AUX
ma-25	106	30	of	of	ADP
ma-25	106	31	[	[	X
ma-25	106	32	p	p	X
ma-25	106	33	,	,	PUNCT
ma-25	106	34	q]−order	q]−order	ADJ
ma-25	106	35	?	?	PUNCT
ma-25	107	1	in	in	ADP
ma-25	107	2	this	this	DET
ma-25	107	3	paper	paper	NOUN
ma-25	107	4	,	,	PUNCT
ma-25	107	5	we	we	PRON
ma-25	107	6	proceed	proceed	VERB
ma-25	107	7	this	this	DET
ma-25	107	8	wayand	wayand	NOUN
ma-25	107	9	we	we	PRON
ma-25	107	10	obtain	obtain	VERB
ma-25	107	11	the	the	DET
ma-25	107	12	following	follow	VERB
ma-25	107	13	results	result	NOUN
ma-25	107	14	.	.	PUNCT
ma-25	108	1	theorem	theorem	VERB
ma-25	108	2	1.1	1.1	NUM
ma-25	108	3	let	let	VERB
ma-25	108	4	h	h	PROPN
ma-25	108	5	⊂	⊂	PROPN
ma-25	108	6	(	(	PUNCT
ma-25	108	7	1,+∞	1,+∞	NUM
ma-25	108	8	)	)	PUNCT
ma-25	108	9	be	be	AUX
ma-25	108	10	a	a	DET
ma-25	108	11	set	set	NOUN
ma-25	108	12	with	with	ADP
ma-25	108	13	a	a	DET
ma-25	108	14	positive	positive	ADJ
ma-25	108	15	upper	upper	ADJ
ma-25	108	16	logarithmic	logarithmic	ADJ
ma-25	108	17	density	density	NOUN
ma-25	108	18	(	(	PUNCT
ma-25	108	19	or	or	CCONJ
ma-25	108	20	ml	ml	INTJ
ma-25	108	21	(	(	PUNCT
ma-25	108	22	h	h	NOUN
ma-25	108	23	)	)	PUNCT
ma-25	108	24	=	=	PUNCT
ma-25	109	1	+	+	NOUN
ma-25	109	2	∞	∞	NUM
ma-25	109	3	)	)	PUNCT
ma-25	109	4	and	and	CCONJ
ma-25	109	5	let	let	VERB
ma-25	109	6	aj	aj	PROPN
ma-25	109	7	(	(	PUNCT
ma-25	109	8	z	z	NOUN
ma-25	109	9	)	)	PUNCT
ma-25	109	10	(	(	PUNCT
ma-25	109	11	j	j	NOUN
ma-25	109	12	=	=	SYM
ma-25	109	13	0	0	NUM
ma-25	109	14	,	,	PUNCT
ma-25	109	15	1	1	NUM
ma-25	109	16	,	,	PUNCT
ma-25	109	17	...	...	PUNCT
ma-25	109	18	,	,	PUNCT
ma-25	109	19	k	k	X
ma-25	109	20	)	)	PUNCT
ma-25	109	21	with	with	ADP
ma-25	109	22	ak	ak	PROPN
ma-25	109	23	(	(	PUNCT
ma-25	109	24	z	z	PROPN
ma-25	109	25	)	)	PUNCT
ma-25	109	26	6≡	6≡	NUM
ma-25	109	27	0	0	NUM
ma-25	109	28	be	be	AUX
ma-25	109	29	meromorphic	meromorphic	ADJ
ma-25	109	30	functions	function	NOUN
ma-25	109	31	with	with	ADP
ma-25	109	32	finite	finite	NOUN
ma-25	109	33	[	[	X
ma-25	109	34	p	p	X
ma-25	109	35	,	,	PUNCT
ma-25	109	36	q]order	q]order	NOUN
ma-25	109	37	.	.	PUNCT
ma-25	110	1	if	if	SCONJ
ma-25	110	2	there	there	PRON
ma-25	110	3	exist	exist	VERB
ma-25	110	4	a	a	DET
ma-25	110	5	positive	positive	ADJ
ma-25	110	6	constant	constant	ADJ
ma-25	110	7	σ	σ	NOUN
ma-25	110	8	>	>	X
ma-25	110	9	0	0	PUNCT
ma-25	110	10	and	and	CCONJ
ma-25	110	11	an	an	DET
ma-25	110	12	integer	integer	NOUN
ma-25	110	13	s	s	NOUN
ma-25	110	14	,	,	PUNCT
ma-25	110	15	0	0	NUM
ma-25	110	16	≤	≤	NUM
ma-25	110	17	s	s	PART
ma-25	110	18	≤	≤	NUM
ma-25	110	19	k	k	NOUN
ma-25	110	20	,	,	PUNCT
ma-25	110	21	such	such	ADJ
ma-25	110	22	that	that	PRON
ma-25	110	23	for	for	ADP
ma-25	110	24	sufficiently	sufficiently	ADV
ma-25	110	25	small	small	ADJ
ma-25	110	26	ε	ε	PROPN
ma-25	110	27	>	>	X
ma-25	110	28	0	0	PROPN
ma-25	110	29	,	,	PUNCT
ma-25	110	30	we	we	PRON
ma-25	110	31	have	have	VERB
ma-25	110	32	|as	|as	NUM
ma-25	110	33	(	(	PUNCT
ma-25	110	34	z	z	NOUN
ma-25	110	35	)	)	PUNCT
ma-25	111	1	|	|	ADV
ma-25	111	2	≥	≥	NOUN
ma-25	111	3	expp+1	expp+1	PROPN
ma-25	111	4	{	{	PUNCT
ma-25	111	5	(	(	PUNCT
ma-25	111	6	σ	σ	PROPN
ma-25	111	7	−	−	PROPN
ma-25	111	8	ε	ε	PROPN
ma-25	111	9	)	)	PUNCT
ma-25	111	10	logq	logq	VERB
ma-25	111	11	r	r	NOUN
ma-25	111	12	}	}	PUNCT
ma-25	111	13	as	as	ADP
ma-25	111	14	|z	|z	PROPN
ma-25	111	15	|	|	ADV
ma-25	111	16	=	=	SYM
ma-25	111	17	r	r	NOUN
ma-25	111	18	∈	∈	PROPN
ma-25	111	19	h	h	NOUN
ma-25	111	20	,	,	PUNCT
ma-25	111	21	r	r	NOUN
ma-25	111	22	→	→	SYM
ma-25	111	23	+	+	NOUN
ma-25	111	24	∞	∞	NUM
ma-25	111	25	and	and	CCONJ
ma-25	111	26	ρ	ρ	PROPN
ma-25	111	27	=	=	SYM
ma-25	111	28	max	max	PROPN
ma-25	111	29	{	{	PUNCT
ma-25	111	30	ρ[p	ρ[p	PROPN
ma-25	111	31	,	,	PUNCT
ma-25	111	32	q	q	X
ma-25	111	33	]	]	X
ma-25	111	34	(	(	PUNCT
ma-25	111	35	aj	aj	PROPN
ma-25	111	36	)	)	PUNCT
ma-25	111	37	(	(	PUNCT
ma-25	111	38	j	j	PROPN
ma-25	111	39	6=	6=	NUM
ma-25	111	40	s	s	PART
ma-25	111	41	)	)	PUNCT
ma-25	111	42	}	}	PUNCT
ma-25	111	43	<	<	X
ma-25	111	44	σ	σ	PROPN
ma-25	111	45	,	,	PUNCT
ma-25	111	46	then	then	ADV
ma-25	111	47	every	every	DET
ma-25	111	48	non	non	ADJ
ma-25	111	49	-	-	ADJ
ma-25	111	50	transcendental	transcendental	ADJ
ma-25	111	51	meromorphic	meromorphic	ADJ
ma-25	111	52	solution	solution	NOUN
ma-25	111	53	f	f	PROPN
ma-25	111	54	6≡	6≡	NUM
ma-25	111	55	0	0	NUM
ma-25	111	56	of	of	ADP
ma-25	111	57	(	(	PUNCT
ma-25	111	58	1.3	1.3	NUM
ma-25	111	59	)	)	PUNCT
ma-25	111	60	is	be	AUX
ma-25	111	61	a	a	DET
ma-25	111	62	polynomial	polynomial	NOUN
ma-25	111	63	with	with	ADP
ma-25	111	64	deg	deg	PROPN
ma-25	111	65	f	f	PROPN
ma-25	111	66	≤	≤	ADV
ma-25	111	67	s−1	s−1	PROPN
ma-25	111	68	and	and	CCONJ
ma-25	111	69	every	every	DET
ma-25	111	70	transcendental	transcendental	ADJ
ma-25	111	71	meromorphic	meromorphic	ADJ
ma-25	111	72	solution	solution	NOUN
ma-25	111	73	f	f	PROPN
ma-25	111	74	of	of	ADP
ma-25	111	75	(	(	PUNCT
ma-25	111	76	1.3	1.3	NUM
ma-25	111	77	)	)	PUNCT
ma-25	111	78	with	with	ADP
ma-25	111	79	λ[p	λ[p	PROPN
ma-25	111	80	,	,	PUNCT
ma-25	111	81	q	q	X
ma-25	111	82	]	]	X
ma-25	111	83	(	(	PUNCT
ma-25	111	84	1	1	NUM
ma-25	111	85	f	f	NOUN
ma-25	111	86	)	)	PUNCT
ma-25	111	87	<	<	X
ma-25	111	88	µ[p	µ[p	ADJ
ma-25	111	89	,	,	PUNCT
ma-25	111	90	q	q	X
ma-25	111	91	]	]	X
ma-25	111	92	(	(	PUNCT
ma-25	111	93	f	f	NOUN
ma-25	111	94	)	)	PUNCT
ma-25	111	95	satisfies	satisfie	NOUN
ma-25	111	96	ρ[p	ρ[p	NOUN
ma-25	111	97	,	,	PUNCT
ma-25	111	98	q](f	q](f	NOUN
ma-25	111	99	)	)	PUNCT
ma-25	111	100	=	=	SYM
ma-25	111	101	µ[p	µ[p	ADJ
ma-25	111	102	,	,	PUNCT
ma-25	111	103	q	q	X
ma-25	111	104	]	]	X
ma-25	111	105	(	(	PUNCT
ma-25	111	106	f	f	X
ma-25	111	107	)	)	PUNCT
ma-25	111	108	=	=	PUNCT
ma-25	112	1	+	+	NUM
ma-25	112	2	∞	∞	PROPN
ma-25	112	3	,	,	PUNCT
ma-25	112	4	σ	σ	PROPN
ma-25	112	5	≤	≤	NUM
ma-25	112	6	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	112	7	]	]	PUNCT
ma-25	112	8	(	(	PUNCT
ma-25	112	9	f	f	PROPN
ma-25	112	10	)	)	PUNCT
ma-25	112	11	≤	≤	PROPN
ma-25	112	12	ρ[p	ρ[p	NOUN
ma-25	112	13	,	,	PUNCT
ma-25	112	14	q	q	X
ma-25	112	15	]	]	X
ma-25	112	16	(	(	PUNCT
ma-25	112	17	as	as	ADP
ma-25	112	18	)	)	PUNCT
ma-25	112	19	.	.	PUNCT
ma-25	113	1	remark	remark	VERB
ma-25	113	2	1.1	1.1	NUM
ma-25	113	3	putting	put	VERB
ma-25	113	4	ak	ak	PROPN
ma-25	113	5	(	(	PUNCT
ma-25	113	6	z	z	NOUN
ma-25	113	7	)	)	PUNCT
ma-25	113	8	≡	≡	PROPN
ma-25	113	9	1	1	NUM
ma-25	113	10	and	and	CCONJ
ma-25	113	11	q	q	AUX
ma-25	113	12	=	=	SYM
ma-25	113	13	1	1	NUM
ma-25	113	14	in	in	ADP
ma-25	113	15	theorem	theorem	ADJ
ma-25	113	16	1.1	1.1	NUM
ma-25	113	17	,	,	PUNCT
ma-25	113	18	we	we	PRON
ma-25	113	19	obtain	obtain	AUX
ma-25	113	20	theorem	theorem	ADJ
ma-25	113	21	c.	c.	PROPN
ma-25	113	22	eur	eur	PROPN
ma-25	113	23	.	.	PUNCT
ma-25	114	1	j.	j.	PROPN
ma-25	114	2	math	math	PROPN
ma-25	114	3	.	.	PUNCT
ma-25	115	1	anal	anal	ADJ
ma-25	115	2	.	.	PUNCT
ma-25	116	1	1	1	NUM
ma-25	116	2	(	(	PUNCT
ma-25	116	3	2021	2021	NUM
ma-25	116	4	)	)	PUNCT
ma-25	116	5	89	89	NUM
ma-25	116	6	corollary	corollary	NOUN
ma-25	116	7	1.1	1.1	NUM
ma-25	116	8	under	under	ADP
ma-25	116	9	the	the	DET
ma-25	116	10	hypotheses	hypothesis	NOUN
ma-25	116	11	of	of	ADP
ma-25	116	12	theorem	theorem	NOUN
ma-25	116	13	1.1	1.1	NUM
ma-25	116	14	,	,	PUNCT
ma-25	116	15	suppose	suppose	VERB
ma-25	116	16	further	far	ADV
ma-25	116	17	that	that	SCONJ
ma-25	116	18	ϕ	ϕ	PROPN
ma-25	116	19	is	be	AUX
ma-25	116	20	a	a	DET
ma-25	116	21	transcendental	transcendental	ADJ
ma-25	116	22	meromorphic	meromorphic	ADJ
ma-25	116	23	function	function	NOUN
ma-25	116	24	satisfying	satisfy	VERB
ma-25	116	25	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	116	26	]	]	PUNCT
ma-25	116	27	(	(	PUNCT
ma-25	116	28	ϕ	ϕ	NOUN
ma-25	116	29	)	)	PUNCT
ma-25	116	30	<	<	X
ma-25	116	31	σ	σ	PROPN
ma-25	116	32	.	.	PUNCT
ma-25	117	1	then	then	ADV
ma-25	117	2	,	,	PUNCT
ma-25	117	3	every	every	DET
ma-25	117	4	transcendental	transcendental	ADJ
ma-25	117	5	meromorphic	meromorphic	ADJ
ma-25	117	6	solution	solution	NOUN
ma-25	117	7	f	f	PROPN
ma-25	117	8	of	of	ADP
ma-25	117	9	equation	equation	NOUN
ma-25	117	10	(	(	PUNCT
ma-25	117	11	1.3	1.3	NUM
ma-25	117	12	)	)	PUNCT
ma-25	117	13	with	with	ADP
ma-25	117	14	λ[p	λ[p	PROPN
ma-25	117	15	,	,	PUNCT
ma-25	117	16	q	q	X
ma-25	117	17	]	]	X
ma-25	117	18	(	(	PUNCT
ma-25	117	19	1	1	NUM
ma-25	117	20	f	f	NOUN
ma-25	117	21	)	)	PUNCT
ma-25	117	22	<	<	X
ma-25	117	23	µ[p	µ[p	ADJ
ma-25	117	24	,	,	PUNCT
ma-25	117	25	q	q	X
ma-25	117	26	]	]	X
ma-25	117	27	(	(	PUNCT
ma-25	117	28	f	f	NOUN
ma-25	117	29	)	)	PUNCT
ma-25	117	30	satisfies	satisfy	VERB
ma-25	117	31	σ	σ	NOUN
ma-25	117	32	≤	≤	PROPN
ma-25	117	33	λ[p+1,q	λ[p+1,q	NOUN
ma-25	117	34	]	]	X
ma-25	117	35	(	(	PUNCT
ma-25	117	36	f	f	PROPN
ma-25	117	37	−	−	PROPN
ma-25	117	38	ϕ	ϕ	PROPN
ma-25	117	39	)	)	PUNCT
ma-25	117	40	=	=	SYM
ma-25	117	41	λ[p+1,q	λ[p+1,q	PROPN
ma-25	117	42	]	]	X
ma-25	117	43	(	(	PUNCT
ma-25	117	44	f	f	PROPN
ma-25	117	45	−	−	PROPN
ma-25	117	46	ϕ	ϕ	PROPN
ma-25	117	47	)	)	PUNCT
ma-25	117	48	=	=	SYM
ma-25	117	49	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	117	50	]	]	PUNCT
ma-25	117	51	(	(	PUNCT
ma-25	117	52	f	f	X
ma-25	117	53	−	−	PROPN
ma-25	117	54	ϕ	ϕ	PROPN
ma-25	117	55	)	)	PUNCT
ma-25	117	56	=	=	SYM
ma-25	117	57	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	117	58	]	]	PUNCT
ma-25	117	59	(	(	PUNCT
ma-25	117	60	f	f	PROPN
ma-25	117	61	)	)	PUNCT
ma-25	117	62	≤	≤	PROPN
ma-25	117	63	ρ[p	ρ[p	NOUN
ma-25	117	64	,	,	PUNCT
ma-25	117	65	q	q	X
ma-25	117	66	]	]	X
ma-25	117	67	(	(	PUNCT
ma-25	117	68	as	as	ADP
ma-25	117	69	)	)	PUNCT
ma-25	117	70	.	.	PUNCT
ma-25	118	1	considering	consider	VERB
ma-25	118	2	the	the	DET
ma-25	118	3	non	non	ADJ
ma-25	118	4	-	-	ADJ
ma-25	118	5	homogeneous	homogeneous	ADJ
ma-25	118	6	linear	linear	PROPN
ma-25	118	7	differential	differential	NOUN
ma-25	118	8	equation	equation	NOUN
ma-25	118	9	(	(	PUNCT
ma-25	118	10	1.4	1.4	NUM
ma-25	118	11	)	)	PUNCT
ma-25	118	12	,	,	PUNCT
ma-25	118	13	we	we	PRON
ma-25	118	14	obtain	obtain	VERB
ma-25	118	15	the	the	DET
ma-25	118	16	followingresults	followingresult	NOUN
ma-25	118	17	.	.	PUNCT
ma-25	119	1	theorem	theorem	VERB
ma-25	119	2	1.2	1.2	NUM
ma-25	119	3	let	let	VERB
ma-25	119	4	h	h	PROPN
ma-25	119	5	⊂	⊂	PROPN
ma-25	119	6	(	(	PUNCT
ma-25	119	7	1,+∞	1,+∞	NUM
ma-25	119	8	)	)	PUNCT
ma-25	119	9	be	be	AUX
ma-25	119	10	a	a	DET
ma-25	119	11	set	set	NOUN
ma-25	119	12	with	with	ADP
ma-25	119	13	a	a	DET
ma-25	119	14	positive	positive	ADJ
ma-25	119	15	upper	upper	ADJ
ma-25	119	16	logarithmic	logarithmic	ADJ
ma-25	119	17	density	density	NOUN
ma-25	119	18	(	(	PUNCT
ma-25	119	19	or	or	CCONJ
ma-25	119	20	ml	ml	INTJ
ma-25	119	21	(	(	PUNCT
ma-25	119	22	h	h	NOUN
ma-25	119	23	)	)	PUNCT
ma-25	119	24	=	=	PUNCT
ma-25	120	1	+	+	NOUN
ma-25	120	2	∞	∞	NOUN
ma-25	120	3	)	)	PUNCT
ma-25	120	4	,	,	PUNCT
ma-25	120	5	and	and	CCONJ
ma-25	120	6	let	let	VERB
ma-25	120	7	aj	aj	PROPN
ma-25	120	8	(	(	PUNCT
ma-25	120	9	z	z	NOUN
ma-25	120	10	)	)	PUNCT
ma-25	120	11	(	(	PUNCT
ma-25	120	12	j	j	NOUN
ma-25	120	13	=	=	SYM
ma-25	120	14	0	0	NUM
ma-25	120	15	,	,	PUNCT
ma-25	120	16	1	1	NUM
ma-25	120	17	,	,	PUNCT
ma-25	120	18	...	...	PUNCT
ma-25	120	19	,	,	PUNCT
ma-25	120	20	k	k	X
ma-25	120	21	)	)	PUNCT
ma-25	120	22	with	with	ADP
ma-25	120	23	ak	ak	PROPN
ma-25	120	24	(	(	PUNCT
ma-25	120	25	z	z	PROPN
ma-25	120	26	)	)	PUNCT
ma-25	120	27	6≡	6≡	NUM
ma-25	120	28	0	0	NUM
ma-25	120	29	and	and	CCONJ
ma-25	120	30	f	f	PROPN
ma-25	120	31	(	(	PUNCT
ma-25	120	32	z	z	NOUN
ma-25	120	33	)	)	PUNCT
ma-25	120	34	6≡	6≡	NUM
ma-25	120	35	0	0	NUM
ma-25	120	36	be	be	AUX
ma-25	120	37	meromorphic	meromorphic	ADJ
ma-25	120	38	functions	function	NOUN
ma-25	120	39	with	with	ADP
ma-25	120	40	finite	finite	NOUN
ma-25	120	41	[	[	X
ma-25	120	42	p	p	X
ma-25	120	43	,	,	PUNCT
ma-25	120	44	q]-order	q]-order	NOUN
ma-25	120	45	.	.	PUNCT
ma-25	121	1	if	if	SCONJ
ma-25	121	2	there	there	PRON
ma-25	121	3	exist	exist	VERB
ma-25	121	4	a	a	DET
ma-25	121	5	positive	positive	ADJ
ma-25	121	6	constant	constant	ADJ
ma-25	121	7	σ	σ	NOUN
ma-25	121	8	>	>	X
ma-25	121	9	0	0	PUNCT
ma-25	121	10	and	and	CCONJ
ma-25	121	11	an	an	DET
ma-25	121	12	integer	integer	NOUN
ma-25	121	13	s	s	NOUN
ma-25	121	14	,	,	PUNCT
ma-25	121	15	0	0	NUM
ma-25	121	16	≤	≤	NUM
ma-25	121	17	s	s	PART
ma-25	121	18	≤	≤	NUM
ma-25	121	19	k	k	NOUN
ma-25	121	20	,	,	PUNCT
ma-25	121	21	such	such	ADJ
ma-25	121	22	that	that	PRON
ma-25	121	23	for	for	ADP
ma-25	121	24	sufficiently	sufficiently	ADV
ma-25	121	25	small	small	ADJ
ma-25	121	26	ε	ε	PROPN
ma-25	121	27	>	>	X
ma-25	121	28	0	0	PROPN
ma-25	121	29	,	,	PUNCT
ma-25	121	30	we	we	PRON
ma-25	121	31	have	have	VERB
ma-25	121	32	|as	|as	NUM
ma-25	121	33	(	(	PUNCT
ma-25	121	34	z	z	NOUN
ma-25	121	35	)	)	PUNCT
ma-25	122	1	|	|	ADV
ma-25	122	2	≥	≥	NOUN
ma-25	122	3	expp+1	expp+1	PROPN
ma-25	122	4	{	{	PUNCT
ma-25	122	5	(	(	PUNCT
ma-25	122	6	σ	σ	PROPN
ma-25	122	7	−	−	PROPN
ma-25	122	8	ε	ε	PROPN
ma-25	122	9	)	)	PUNCT
ma-25	122	10	logq	logq	VERB
ma-25	122	11	r	r	NOUN
ma-25	122	12	}	}	PUNCT
ma-25	122	13	as	as	ADP
ma-25	122	14	|z	|z	PROPN
ma-25	122	15	|	|	ADV
ma-25	122	16	=	=	SYM
ma-25	122	17	r	r	NOUN
ma-25	122	18	∈	∈	PROPN
ma-25	122	19	h	h	NOUN
ma-25	122	20	,	,	PUNCT
ma-25	122	21	r	r	NOUN
ma-25	122	22	→	→	SYM
ma-25	122	23	+	+	NOUN
ma-25	122	24	∞	∞	NUM
ma-25	122	25	and	and	CCONJ
ma-25	122	26	max	max	PROPN
ma-25	122	27	{	{	PUNCT
ma-25	122	28	ρ[p	ρ[p	PROPN
ma-25	122	29	,	,	PUNCT
ma-25	122	30	q	q	X
ma-25	122	31	]	]	X
ma-25	122	32	(	(	PUNCT
ma-25	122	33	aj	aj	PROPN
ma-25	122	34	)	)	PUNCT
ma-25	122	35	(	(	PUNCT
ma-25	122	36	j	j	PROPN
ma-25	122	37	6=	6=	NUM
ma-25	122	38	s	s	PROPN
ma-25	122	39	)	)	PUNCT
ma-25	122	40	,	,	PUNCT
ma-25	122	41	ρ[p	ρ[p	PROPN
ma-25	122	42	,	,	PUNCT
ma-25	122	43	q	q	X
ma-25	122	44	]	]	X
ma-25	122	45	(	(	PUNCT
ma-25	122	46	f	f	PROPN
ma-25	122	47	)	)	PUNCT
ma-25	122	48	}	}	PUNCT
ma-25	122	49	<	<	X
ma-25	122	50	σ	σ	PROPN
ma-25	122	51	,	,	PUNCT
ma-25	122	52	then	then	ADV
ma-25	122	53	every	every	DET
ma-25	122	54	non	non	ADJ
ma-25	122	55	-	-	ADJ
ma-25	122	56	transcendental	transcendental	ADJ
ma-25	122	57	meromorphic	meromorphic	ADJ
ma-25	122	58	solution	solution	NOUN
ma-25	122	59	f	f	PROPN
ma-25	122	60	of	of	ADP
ma-25	122	61	(	(	PUNCT
ma-25	122	62	1.4	1.4	NUM
ma-25	122	63	)	)	PUNCT
ma-25	122	64	is	be	AUX
ma-25	122	65	a	a	DET
ma-25	122	66	polynomial	polynomial	NOUN
ma-25	122	67	with	with	ADP
ma-25	122	68	deg	deg	PROPN
ma-25	122	69	f	f	PROPN
ma-25	122	70	≤	≤	PROPN
ma-25	122	71	s	s	PART
ma-25	122	72	−	−	PROPN
ma-25	122	73	1	1	NUM
ma-25	122	74	and	and	CCONJ
ma-25	122	75	every	every	DET
ma-25	122	76	transcendental	transcendental	ADJ
ma-25	122	77	meromorphic	meromorphic	ADJ
ma-25	122	78	solution	solution	NOUN
ma-25	122	79	f	f	PROPN
ma-25	122	80	of	of	ADP
ma-25	122	81	(	(	PUNCT
ma-25	122	82	1.4	1.4	NUM
ma-25	122	83	)	)	PUNCT
ma-25	122	84	with	with	ADP
ma-25	122	85	λ[p	λ[p	PROPN
ma-25	122	86	,	,	PUNCT
ma-25	122	87	q	q	X
ma-25	122	88	]	]	X
ma-25	122	89	(	(	PUNCT
ma-25	122	90	1	1	NUM
ma-25	122	91	f	f	NOUN
ma-25	122	92	)	)	PUNCT
ma-25	122	93	<	<	X
ma-25	122	94	min	min	PROPN
ma-25	122	95	{	{	PUNCT
ma-25	122	96	σ	σ	PROPN
ma-25	122	97	,	,	PUNCT
ma-25	122	98	µ[p	µ[p	ADJ
ma-25	122	99	,	,	PUNCT
ma-25	122	100	q](f	q](f	NOUN
ma-25	122	101	)	)	PUNCT
ma-25	122	102	}	}	PUNCT
ma-25	122	103	satisfies	satisfy	VERB
ma-25	122	104	λ[p	λ[p	NOUN
ma-25	122	105	,	,	PUNCT
ma-25	122	106	q	q	X
ma-25	122	107	]	]	X
ma-25	122	108	(	(	PUNCT
ma-25	122	109	f	f	X
ma-25	122	110	)	)	PUNCT
ma-25	123	1	=	=	SYM
ma-25	123	2	λ[p	λ[p	ADJ
ma-25	123	3	,	,	PUNCT
ma-25	123	4	q](f	q](f	NOUN
ma-25	123	5	)	)	PUNCT
ma-25	123	6	=	=	SYM
ma-25	123	7	ρ[p	ρ[p	NOUN
ma-25	123	8	,	,	PUNCT
ma-25	123	9	q](f	q](f	NOUN
ma-25	123	10	)	)	PUNCT
ma-25	123	11	=	=	SYM
ma-25	123	12	µ[p	µ[p	ADJ
ma-25	123	13	,	,	PUNCT
ma-25	123	14	q	q	X
ma-25	123	15	]	]	X
ma-25	123	16	(	(	PUNCT
ma-25	123	17	f	f	X
ma-25	123	18	)	)	PUNCT
ma-25	123	19	=	=	PUNCT
ma-25	124	1	+	+	PUNCT
ma-25	124	2	∞	∞	NUM
ma-25	124	3	and	and	CCONJ
ma-25	124	4	σ	σ	PROPN
ma-25	124	5	≤	≤	PROPN
ma-25	124	6	λ[p+1,q	λ[p+1,q	PROPN
ma-25	124	7	]	]	X
ma-25	124	8	(	(	PUNCT
ma-25	124	9	f	f	X
ma-25	124	10	)	)	PUNCT
ma-25	124	11	=	=	SYM
ma-25	124	12	λ[p+1,q](f	λ[p+1,q](f	X
ma-25	124	13	)	)	PUNCT
ma-25	124	14	=	=	SYM
ma-25	124	15	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	124	16	]	]	PUNCT
ma-25	124	17	(	(	PUNCT
ma-25	124	18	f	f	PROPN
ma-25	124	19	)	)	PUNCT
ma-25	124	20	≤	≤	PROPN
ma-25	124	21	ρ[p	ρ[p	NOUN
ma-25	124	22	,	,	PUNCT
ma-25	124	23	q	q	X
ma-25	124	24	]	]	X
ma-25	124	25	(	(	PUNCT
ma-25	124	26	as	as	ADP
ma-25	124	27	)	)	PUNCT
ma-25	124	28	.	.	PUNCT
ma-25	125	1	remark	remark	VERB
ma-25	125	2	1.2	1.2	NUM
ma-25	125	3	putting	put	VERB
ma-25	125	4	ak	ak	PROPN
ma-25	125	5	(	(	PUNCT
ma-25	125	6	z	z	NOUN
ma-25	125	7	)	)	PUNCT
ma-25	125	8	≡	≡	PROPN
ma-25	125	9	1	1	NUM
ma-25	125	10	and	and	CCONJ
ma-25	125	11	q	q	NOUN
ma-25	126	1	=	=	SYM
ma-25	126	2	1	1	NUM
ma-25	126	3	in	in	ADP
ma-25	126	4	theorem	theorem	ADJ
ma-25	126	5	1.2	1.2	NUM
ma-25	126	6	,	,	PUNCT
ma-25	126	7	we	we	PRON
ma-25	126	8	obtain	obtain	VERB
ma-25	126	9	theorem	theorem	ADJ
ma-25	126	10	d.	d.	PROPN
ma-25	126	11	corollary	corollary	PROPN
ma-25	126	12	1.2	1.2	NUM
ma-25	126	13	let	let	VERB
ma-25	126	14	aj	aj	PROPN
ma-25	126	15	(	(	PUNCT
ma-25	126	16	z	z	NOUN
ma-25	126	17	)	)	PUNCT
ma-25	126	18	(	(	PUNCT
ma-25	126	19	j	j	NOUN
ma-25	126	20	=	=	SYM
ma-25	126	21	0	0	NUM
ma-25	126	22	,	,	PUNCT
ma-25	126	23	1	1	NUM
ma-25	126	24	,	,	PUNCT
ma-25	126	25	...	...	PUNCT
ma-25	126	26	,	,	PUNCT
ma-25	126	27	k	k	PROPN
ma-25	126	28	)	)	PUNCT
ma-25	126	29	,	,	PUNCT
ma-25	126	30	f	f	PROPN
ma-25	126	31	(	(	PUNCT
ma-25	126	32	z	z	NOUN
ma-25	126	33	)	)	PUNCT
ma-25	126	34	,	,	PUNCT
ma-25	126	35	h	h	NOUN
ma-25	126	36	satisfy	satisfy	VERB
ma-25	126	37	all	all	DET
ma-25	126	38	the	the	DET
ma-25	126	39	hypotheses	hypothesis	NOUN
ma-25	126	40	of	of	ADP
ma-25	126	41	theorem	theorem	ADJ
ma-25	126	42	1.2	1.2	NUM
ma-25	126	43	,	,	PUNCT
ma-25	126	44	and	and	CCONJ
ma-25	126	45	let	let	VERB
ma-25	126	46	ϕ	ϕ	NOUN
ma-25	126	47	be	be	AUX
ma-25	126	48	a	a	DET
ma-25	126	49	transcendental	transcendental	ADJ
ma-25	126	50	meromorphic	meromorphic	ADJ
ma-25	126	51	function	function	NOUN
ma-25	126	52	satisfying	satisfy	VERB
ma-25	126	53	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	126	54	]	]	PUNCT
ma-25	126	55	(	(	PUNCT
ma-25	126	56	ϕ	ϕ	NOUN
ma-25	126	57	)	)	PUNCT
ma-25	126	58	<	<	X
ma-25	126	59	σ	σ	PROPN
ma-25	126	60	.	.	PUNCT
ma-25	127	1	then	then	ADV
ma-25	127	2	,	,	PUNCT
ma-25	127	3	every	every	DET
ma-25	127	4	transcendental	transcendental	ADJ
ma-25	127	5	meromorphic	meromorphic	ADJ
ma-25	127	6	solution	solution	NOUN
ma-25	127	7	f	f	PROPN
ma-25	127	8	with	with	ADP
ma-25	127	9	λ[p	λ[p	PROPN
ma-25	127	10	,	,	PUNCT
ma-25	127	11	q	q	X
ma-25	127	12	]	]	X
ma-25	127	13	(	(	PUNCT
ma-25	127	14	1	1	NUM
ma-25	127	15	f	f	NOUN
ma-25	127	16	)	)	PUNCT
ma-25	127	17	<	<	X
ma-25	127	18	min{σ	min{σ	PROPN
ma-25	127	19	,	,	PUNCT
ma-25	127	20	µ[p	µ[p	ADJ
ma-25	127	21	,	,	PUNCT
ma-25	127	22	q	q	X
ma-25	127	23	]	]	X
ma-25	127	24	(	(	PUNCT
ma-25	127	25	f	f	NOUN
ma-25	127	26	)	)	PUNCT
ma-25	127	27	}	}	PUNCT
ma-25	127	28	of	of	ADP
ma-25	127	29	equation	equation	NOUN
ma-25	127	30	(	(	PUNCT
ma-25	127	31	1.4	1.4	NUM
ma-25	127	32	)	)	PUNCT
ma-25	127	33	satisfies	satisfy	VERB
ma-25	127	34	σ	σ	NOUN
ma-25	127	35	≤	≤	PROPN
ma-25	127	36	λ[p+1,q	λ[p+1,q	NOUN
ma-25	127	37	]	]	X
ma-25	127	38	(	(	PUNCT
ma-25	127	39	f	f	PROPN
ma-25	127	40	−	−	PROPN
ma-25	127	41	ϕ	ϕ	PROPN
ma-25	127	42	)	)	PUNCT
ma-25	127	43	=	=	SYM
ma-25	127	44	λ[p+1,q	λ[p+1,q	PROPN
ma-25	127	45	]	]	X
ma-25	127	46	(	(	PUNCT
ma-25	127	47	f	f	PROPN
ma-25	127	48	−	−	PROPN
ma-25	127	49	ϕ	ϕ	PROPN
ma-25	127	50	)	)	PUNCT
ma-25	127	51	=	=	SYM
ma-25	127	52	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	127	53	]	]	PUNCT
ma-25	127	54	(	(	PUNCT
ma-25	127	55	f	f	X
ma-25	127	56	−	−	PROPN
ma-25	127	57	ϕ	ϕ	PROPN
ma-25	127	58	)	)	PUNCT
ma-25	127	59	≤	≤	NOUN
ma-25	127	60	ρ[p	ρ[p	NOUN
ma-25	127	61	,	,	PUNCT
ma-25	127	62	q	q	X
ma-25	127	63	]	]	X
ma-25	127	64	(	(	PUNCT
ma-25	127	65	as	as	ADP
ma-25	127	66	)	)	PUNCT
ma-25	127	67	.	.	PUNCT
ma-25	128	1	remark	remark	VERB
ma-25	128	2	1.3	1.3	NUM
ma-25	128	3	in	in	ADP
ma-25	128	4	[	[	X
ma-25	128	5	17	17	NUM
ma-25	128	6	,	,	PUNCT
ma-25	128	7	19	19	NUM
ma-25	128	8	]	]	PUNCT
ma-25	128	9	,	,	PUNCT
ma-25	128	10	the	the	DET
ma-25	128	11	authors	author	NOUN
ma-25	128	12	have	have	AUX
ma-25	128	13	studied	study	VERB
ma-25	128	14	the	the	DET
ma-25	128	15	growth	growth	NOUN
ma-25	128	16	and	and	CCONJ
ma-25	128	17	the	the	DET
ma-25	128	18	oscillation	oscillation	NOUN
ma-25	128	19	of	of	ADP
ma-25	128	20	solutionsof	solutionsof	NOUN
ma-25	128	21	equations	equation	NOUN
ma-25	128	22	(	(	PUNCT
ma-25	128	23	1.3	1.3	NUM
ma-25	128	24	)	)	PUNCT
ma-25	128	25	and	and	CCONJ
ma-25	128	26	(	(	PUNCT
ma-25	128	27	1.4	1.4	NUM
ma-25	128	28	)	)	PUNCT
ma-25	128	29	when	when	SCONJ
ma-25	128	30	the	the	DET
ma-25	128	31	coefficients	coefficient	NOUN
ma-25	128	32	aj	aj	PROPN
ma-25	128	33	(	(	PUNCT
ma-25	128	34	z	z	NOUN
ma-25	128	35	)	)	PUNCT
ma-25	128	36	(	(	PUNCT
ma-25	128	37	j	j	NOUN
ma-25	128	38	=	=	SYM
ma-25	128	39	0	0	NUM
ma-25	128	40	,	,	PUNCT
ma-25	128	41	1	1	NUM
ma-25	128	42	,	,	PUNCT
ma-25	128	43	...	...	PUNCT
ma-25	128	44	,	,	PUNCT
ma-25	128	45	k	k	X
ma-25	128	46	)	)	PUNCT
ma-25	128	47	and	and	CCONJ
ma-25	128	48	f	f	PROPN
ma-25	128	49	(	(	PUNCT
ma-25	128	50	z	z	NOUN
ma-25	128	51	)	)	PUNCT
ma-25	128	52	are	be	AUX
ma-25	128	53	entirefunctions	entirefunction	NOUN
ma-25	128	54	of	of	ADP
ma-25	128	55	iterated	iterated	ADJ
ma-25	128	56	p	p	NOUN
ma-25	128	57	-	-	PUNCT
ma-25	128	58	order	order	NOUN
ma-25	128	59	or	or	CCONJ
ma-25	128	60	of	of	ADP
ma-25	128	61	[	[	X
ma-25	128	62	p	p	X
ma-25	128	63	,	,	PUNCT
ma-25	128	64	q]-order	q]-order	NOUN
ma-25	128	65	.	.	PUNCT
ma-25	129	1	however	however	ADV
ma-25	129	2	,	,	PUNCT
ma-25	129	3	in	in	ADP
ma-25	129	4	the	the	DET
ma-25	129	5	present	present	ADJ
ma-25	129	6	paper	paper	NOUN
ma-25	129	7	the	the	DET
ma-25	129	8	coefficients	coefficient	NOUN
ma-25	129	9	aj	aj	PROPN
ma-25	129	10	(	(	PUNCT
ma-25	129	11	z	z	NOUN
ma-25	129	12	)	)	PUNCT
ma-25	129	13	(	(	PUNCT
ma-25	129	14	j	j	NOUN
ma-25	129	15	=	=	SYM
ma-25	129	16	0	0	NUM
ma-25	129	17	,	,	PUNCT
ma-25	129	18	1	1	NUM
ma-25	129	19	,	,	PUNCT
ma-25	129	20	...	...	PUNCT
ma-25	129	21	,	,	PUNCT
ma-25	129	22	k	k	X
ma-25	129	23	)	)	PUNCT
ma-25	129	24	and	and	CCONJ
ma-25	129	25	f	f	PROPN
ma-25	129	26	(	(	PUNCT
ma-25	129	27	z	z	NOUN
ma-25	129	28	)	)	PUNCT
ma-25	129	29	are	be	AUX
ma-25	129	30	meromorphic	meromorphic	ADJ
ma-25	129	31	functions	function	NOUN
ma-25	129	32	with	with	ADP
ma-25	129	33	reduction	reduction	NOUN
ma-25	129	34	of	of	ADP
ma-25	129	35	the	the	DET
ma-25	129	36	hypotheses	hypothesis	NOUN
ma-25	129	37	in	in	ADP
ma-25	129	38	theorems1.1	theorems1.1	NOUN
ma-25	129	39	and	and	CCONJ
ma-25	129	40	1.2	1.2	NUM
ma-25	129	41	.	.	PUNCT
ma-25	130	1	so	so	ADV
ma-25	130	2	,	,	PUNCT
ma-25	130	3	this	this	DET
ma-25	130	4	article	article	NOUN
ma-25	130	5	may	may	AUX
ma-25	130	6	be	be	AUX
ma-25	130	7	understood	understand	VERB
ma-25	130	8	as	as	ADP
ma-25	130	9	an	an	DET
ma-25	130	10	extension	extension	NOUN
ma-25	130	11	and	and	CCONJ
ma-25	130	12	an	an	DET
ma-25	130	13	improvement	improvement	NOUN
ma-25	130	14	of	of	ADP
ma-25	130	15	[	[	X
ma-25	130	16	17	17	NUM
ma-25	130	17	,	,	PUNCT
ma-25	130	18	19	19	NUM
ma-25	130	19	]	]	PUNCT
ma-25	130	20	.	.	PUNCT
ma-25	131	1	eur	eur	PROPN
ma-25	131	2	.	.	PUNCT
ma-25	132	1	j.	j.	PROPN
ma-25	132	2	math	math	PROPN
ma-25	132	3	.	.	PUNCT
ma-25	133	1	anal	anal	ADJ
ma-25	133	2	.	.	PUNCT
ma-25	134	1	1	1	NUM
ma-25	134	2	(	(	PUNCT
ma-25	134	3	2021	2021	NUM
ma-25	134	4	)	)	PUNCT
ma-25	135	1	902	902	NUM
ma-25	135	2	.	.	PUNCT
ma-25	136	1	some	some	DET
ma-25	136	2	auxiliary	auxiliary	NOUN
ma-25	136	3	lemmas	lemma	VERB
ma-25	136	4	in	in	ADP
ma-25	136	5	order	order	NOUN
ma-25	136	6	to	to	PART
ma-25	136	7	prove	prove	VERB
ma-25	136	8	our	our	PRON
ma-25	136	9	theorems	theorem	NOUN
ma-25	136	10	,	,	PUNCT
ma-25	136	11	we	we	PRON
ma-25	136	12	need	need	VERB
ma-25	136	13	the	the	DET
ma-25	136	14	following	follow	VERB
ma-25	136	15	definition	definition	NOUN
ma-25	136	16	,	,	PUNCT
ma-25	136	17	proposition	proposition	NOUN
ma-25	136	18	and	and	CCONJ
ma-25	136	19	lemmas	lemmas	PROPN
ma-25	136	20	.	.	PUNCT
ma-25	137	1	thelebesgue	thelebesgue	PROPN
ma-25	137	2	linear	linear	ADJ
ma-25	137	3	measure	measure	NOUN
ma-25	137	4	of	of	ADP
ma-25	137	5	a	a	DET
ma-25	137	6	set	set	NOUN
ma-25	137	7	e	e	X
ma-25	137	8	⊂	⊂	PROPN
ma-25	138	1	[	[	X
ma-25	138	2	0,+∞	0,+∞	NUM
ma-25	138	3	)	)	PUNCT
ma-25	138	4	is	be	AUX
ma-25	138	5	m	m	PROPN
ma-25	138	6	(	(	PUNCT
ma-25	138	7	e	e	NOUN
ma-25	138	8	)	)	PUNCT
ma-25	138	9	=	=	SYM
ma-25	139	1	∫	∫	PROPN
ma-25	139	2	e	e	X
ma-25	139	3	dt	dt	PROPN
ma-25	139	4	,	,	PUNCT
ma-25	139	5	and	and	CCONJ
ma-25	139	6	the	the	DET
ma-25	139	7	logarithmic	logarithmic	ADJ
ma-25	139	8	measure	measure	NOUN
ma-25	139	9	of	of	ADP
ma-25	139	10	a	a	DET
ma-25	139	11	set	set	NOUN
ma-25	139	12	f	f	PROPN
ma-25	139	13	⊂	⊂	PROPN
ma-25	140	1	[	[	X
ma-25	140	2	1,+∞	1,+∞	NUM
ma-25	140	3	)	)	PUNCT
ma-25	140	4	is	be	AUX
ma-25	140	5	ml	ml	NOUN
ma-25	140	6	(	(	PUNCT
ma-25	140	7	f	f	X
ma-25	140	8	)	)	PUNCT
ma-25	141	1	=	=	PUNCT
ma-25	142	1	∫	∫	PROPN
ma-25	143	1	f	f	PROPN
ma-25	143	2	dt	dt	PROPN
ma-25	143	3	t	t	PROPN
ma-25	143	4	.	.	PUNCT
ma-25	144	1	the	the	DET
ma-25	144	2	upper	upper	ADJ
ma-25	144	3	density	density	NOUN
ma-25	144	4	of	of	ADP
ma-25	144	5	e	e	PROPN
ma-25	144	6	⊂	⊂	PROPN
ma-25	144	7	[	[	X
ma-25	144	8	0,+∞	0,+∞	NUM
ma-25	144	9	)	)	PUNCT
ma-25	144	10	is	be	AUX
ma-25	144	11	given	give	VERB
ma-25	144	12	by	by	ADP
ma-25	144	13	dens	den	NOUN
ma-25	144	14	(	(	PUNCT
ma-25	144	15	e	e	NOUN
ma-25	144	16	)	)	PUNCT
ma-25	144	17	=	=	SYM
ma-25	144	18	lim	lim	PROPN
ma-25	144	19	sup	sup	PROPN
ma-25	144	20	r→∞	r→∞	PRON
ma-25	144	21	m	m	VERB
ma-25	144	22	(	(	PUNCT
ma-25	144	23	e	e	X
ma-25	144	24	∩	∩	X
ma-25	144	25	[	[	X
ma-25	144	26	0	0	NUM
ma-25	144	27	,	,	PUNCT
ma-25	144	28	r	r	NOUN
ma-25	144	29	]	]	PUNCT
ma-25	144	30	)	)	PUNCT
ma-25	144	31	rand	rand	NOUN
ma-25	144	32	the	the	DET
ma-25	144	33	upper	upper	ADJ
ma-25	144	34	logarithmic	logarithmic	ADJ
ma-25	144	35	density	density	NOUN
ma-25	144	36	of	of	ADP
ma-25	144	37	the	the	DET
ma-25	144	38	set	set	NOUN
ma-25	144	39	f	f	PROPN
ma-25	144	40	⊂	⊂	PROPN
ma-25	145	1	[	[	X
ma-25	145	2	1,+∞	1,+∞	NUM
ma-25	145	3	)	)	PUNCT
ma-25	145	4	is	be	AUX
ma-25	145	5	defined	define	VERB
ma-25	145	6	by	by	ADP
ma-25	145	7	log	log	NOUN
ma-25	145	8	dens	den	NOUN
ma-25	145	9	(	(	PUNCT
ma-25	145	10	f	f	X
ma-25	145	11	)	)	PUNCT
ma-25	146	1	=	=	SYM
ma-25	146	2	lim	lim	PROPN
ma-25	146	3	sup	sup	PROPN
ma-25	146	4	r−→+∞	r−→+∞	PROPN
ma-25	146	5	ml	ml	PROPN
ma-25	146	6	(	(	PUNCT
ma-25	146	7	f	f	PROPN
ma-25	146	8	∩	∩	PROPN
ma-25	146	9	[	[	X
ma-25	146	10	1	1	NUM
ma-25	146	11	,	,	PUNCT
ma-25	146	12	r	r	NOUN
ma-25	146	13	]	]	PUNCT
ma-25	146	14	)	)	PUNCT
ma-25	146	15	log	log	PROPN
ma-25	146	16	r	r	NOUN
ma-25	146	17	.	.	PUNCT
ma-25	147	1	proposition	proposition	NOUN
ma-25	147	2	2.1	2.1	NUM
ma-25	147	3	(	(	PUNCT
ma-25	147	4	[	[	X
ma-25	147	5	2	2	NUM
ma-25	147	6	]	]	PUNCT
ma-25	147	7	)	)	PUNCT
ma-25	147	8	for	for	ADP
ma-25	147	9	all	all	DET
ma-25	147	10	h	h	NOUN
ma-25	147	11	⊂	⊂	PROPN
ma-25	147	12	(	(	PUNCT
ma-25	147	13	1,+∞	1,+∞	NUM
ma-25	147	14	)	)	PUNCT
ma-25	147	15	the	the	DET
ma-25	147	16	following	follow	VERB
ma-25	147	17	statements	statement	NOUN
ma-25	147	18	hold	hold	VERB
ma-25	147	19	:	:	PUNCT
ma-25	147	20	(	(	PUNCT
ma-25	147	21	i	i	NOUN
ma-25	147	22	)	)	PUNCT
ma-25	147	23	if	if	SCONJ
ma-25	147	24	ml	ml	INTJ
ma-25	147	25	(	(	PUNCT
ma-25	147	26	h	h	NOUN
ma-25	147	27	)	)	PUNCT
ma-25	147	28	=	=	PUNCT
ma-25	148	1	+	+	NUM
ma-25	148	2	∞	∞	PROPN
ma-25	148	3	,	,	PUNCT
ma-25	148	4	then	then	ADV
ma-25	148	5	m	m	PROPN
ma-25	148	6	(	(	PUNCT
ma-25	148	7	h	h	NOUN
ma-25	148	8	)	)	PUNCT
ma-25	148	9	=	=	PUNCT
ma-25	149	1	+	+	NOUN
ma-25	149	2	∞	∞	NUM
ma-25	149	3	;	;	PUNCT
ma-25	149	4	(	(	PUNCT
ma-25	149	5	ii	ii	NOUN
ma-25	149	6	)	)	PUNCT
ma-25	149	7	if	if	SCONJ
ma-25	149	8	dens	den	NOUN
ma-25	149	9	(	(	PUNCT
ma-25	149	10	h	h	NOUN
ma-25	149	11	)	)	PUNCT
ma-25	149	12	>	>	X
ma-25	149	13	0	0	NUM
ma-25	149	14	,	,	PUNCT
ma-25	149	15	then	then	ADV
ma-25	149	16	m	m	PRON
ma-25	149	17	(	(	PUNCT
ma-25	149	18	h	h	NOUN
ma-25	149	19	)	)	PUNCT
ma-25	149	20	=	=	PUNCT
ma-25	150	1	+	+	NOUN
ma-25	150	2	∞	∞	NUM
ma-25	150	3	;	;	PUNCT
ma-25	150	4	(	(	PUNCT
ma-25	150	5	iii	iii	X
ma-25	150	6	)	)	PUNCT
ma-25	150	7	if	if	SCONJ
ma-25	150	8	log	log	VERB
ma-25	150	9	dens	dens	ADP
ma-25	150	10	(	(	PUNCT
ma-25	150	11	h	h	NOUN
ma-25	150	12	)	)	PUNCT
ma-25	150	13	>	>	X
ma-25	150	14	0	0	NUM
ma-25	150	15	,	,	PUNCT
ma-25	150	16	then	then	ADV
ma-25	150	17	ml	ml	INTJ
ma-25	150	18	(	(	PUNCT
ma-25	150	19	h	h	NOUN
ma-25	150	20	)	)	PUNCT
ma-25	150	21	=	=	PUNCT
ma-25	151	1	+	+	NUM
ma-25	151	2	∞.	∞.	PROPN
ma-25	151	3	lemma	lemma	PROPN
ma-25	151	4	2.1	2.1	NUM
ma-25	151	5	(	(	PUNCT
ma-25	151	6	[	[	X
ma-25	151	7	5	5	NUM
ma-25	151	8	]	]	PUNCT
ma-25	151	9	)	)	PUNCT
ma-25	151	10	let	let	VERB
ma-25	151	11	f	f	PRON
ma-25	151	12	be	be	AUX
ma-25	151	13	a	a	DET
ma-25	151	14	transcendental	transcendental	ADJ
ma-25	151	15	meromorphic	meromorphic	ADJ
ma-25	151	16	function	function	NOUN
ma-25	151	17	in	in	ADP
ma-25	151	18	the	the	DET
ma-25	151	19	plane	plane	NOUN
ma-25	151	20	,	,	PUNCT
ma-25	151	21	and	and	CCONJ
ma-25	151	22	let	let	VERB
ma-25	151	23	α	α	PRON
ma-25	151	24	>	>	X
ma-25	151	25	1	1	NUM
ma-25	151	26	be	be	AUX
ma-25	151	27	a	a	DET
ma-25	151	28	given	give	VERB
ma-25	151	29	constant	constant	NOUN
ma-25	151	30	.	.	PUNCT
ma-25	152	1	then	then	ADV
ma-25	152	2	,	,	PUNCT
ma-25	152	3	there	there	PRON
ma-25	152	4	exist	exist	VERB
ma-25	152	5	a	a	DET
ma-25	152	6	set	set	NOUN
ma-25	152	7	e1	e1	NOUN
ma-25	152	8	⊂	⊂	PROPN
ma-25	152	9	(	(	PUNCT
ma-25	152	10	1,+∞	1,+∞	NUM
ma-25	152	11	)	)	PUNCT
ma-25	152	12	that	that	PRON
ma-25	152	13	has	have	VERB
ma-25	152	14	a	a	DET
ma-25	152	15	finite	finite	ADJ
ma-25	152	16	logarithmic	logarithmic	ADJ
ma-25	152	17	measure	measure	NOUN
ma-25	152	18	,	,	PUNCT
ma-25	152	19	and	and	CCONJ
ma-25	152	20	a	a	DET
ma-25	152	21	constant	constant	ADJ
ma-25	152	22	b	b	NOUN
ma-25	152	23	>	>	X
ma-25	152	24	0	0	PUNCT
ma-25	153	1	depending	depend	VERB
ma-25	153	2	only	only	ADV
ma-25	153	3	on	on	ADP
ma-25	153	4	α	α	PROPN
ma-25	153	5	and	and	CCONJ
ma-25	153	6	(	(	PUNCT
ma-25	153	7	i	i	PROPN
ma-25	153	8	,	,	PUNCT
ma-25	153	9	j	j	PROPN
ma-25	153	10	)	)	PUNCT
ma-25	153	11	(	(	PUNCT
ma-25	153	12	(	(	PUNCT
ma-25	153	13	i	i	INTJ
ma-25	153	14	,	,	PUNCT
ma-25	153	15	j	j	PROPN
ma-25	153	16	)	)	PUNCT
ma-25	153	17	positive	positive	ADJ
ma-25	153	18	integers	integer	NOUN
ma-25	153	19	with	with	ADP
ma-25	153	20	i	i	PROPN
ma-25	153	21	>	>	X
ma-25	153	22	j	j	PROPN
ma-25	153	23	)	)	PUNCT
ma-25	153	24	such	such	ADJ
ma-25	153	25	that	that	PRON
ma-25	153	26	for	for	ADP
ma-25	153	27	all	all	DET
ma-25	153	28	z	z	NOUN
ma-25	153	29	with	with	ADP
ma-25	153	30	|z	|z	PROPN
ma-25	154	1	|	|	ADV
ma-25	155	1	=	=	SYM
ma-25	155	2	r	r	NOUN
ma-25	155	3	6∈	6∈	NOUN
ma-25	156	1	[	[	X
ma-25	156	2	0	0	NUM
ma-25	156	3	,	,	PUNCT
ma-25	156	4	1	1	NUM
ma-25	156	5	]	]	PUNCT
ma-25	156	6	∪	∪	NOUN
ma-25	156	7	e1	e1	NOUN
ma-25	156	8	,	,	PUNCT
ma-25	156	9	we	we	PRON
ma-25	156	10	have∣∣∣∣∣	have∣∣∣∣∣	PROPN
ma-25	156	11	f	f	PROPN
ma-25	156	12	(	(	PUNCT
ma-25	156	13	i)(z	i)(z	PROPN
ma-25	156	14	)	)	PUNCT
ma-25	156	15	f	f	NOUN
ma-25	156	16	(	(	PUNCT
ma-25	156	17	j)(z	j)(z	PROPN
ma-25	156	18	)	)	PUNCT
ma-25	156	19	∣∣∣∣∣	∣∣∣∣∣	ADP
ma-25	157	1	≤	≤	PROPN
ma-25	157	2	b	b	X
ma-25	157	3	(	(	PUNCT
ma-25	157	4	t	t	PROPN
ma-25	157	5	(	(	PUNCT
ma-25	157	6	αr	αr	PROPN
ma-25	157	7	,	,	PUNCT
ma-25	157	8	f	f	PROPN
ma-25	157	9	)	)	PUNCT
ma-25	157	10	r	r	NOUN
ma-25	157	11	(	(	PUNCT
ma-25	157	12	logα	logα	NOUN
ma-25	157	13	r	r	NOUN
ma-25	157	14	)	)	PUNCT
ma-25	157	15	logt	logt	NOUN
ma-25	157	16	(	(	PUNCT
ma-25	157	17	αr	αr	PROPN
ma-25	157	18	,	,	PUNCT
ma-25	157	19	f	f	PROPN
ma-25	157	20	)	)	PUNCT
ma-25	157	21	)	)	PUNCT
ma-25	157	22	i−j	i−j	PROPN
ma-25	157	23	.	.	PUNCT
ma-25	158	1	lemma	lemma	PROPN
ma-25	158	2	2.2	2.2	NUM
ma-25	158	3	(	(	PUNCT
ma-25	158	4	[	[	X
ma-25	158	5	4	4	NUM
ma-25	158	6	]	]	PUNCT
ma-25	158	7	)	)	PUNCT
ma-25	158	8	let	let	VERB
ma-25	158	9	p	p	PRON
ma-25	158	10	≥	≥	PRON
ma-25	158	11	q	q	NOUN
ma-25	158	12	≥	≥	NUM
ma-25	158	13	1	1	NUM
ma-25	158	14	be	be	AUX
ma-25	158	15	integers	integer	NOUN
ma-25	158	16	and	and	CCONJ
ma-25	158	17	g	g	NOUN
ma-25	158	18	be	be	AUX
ma-25	158	19	an	an	DET
ma-25	158	20	entire	entire	ADJ
ma-25	158	21	function	function	NOUN
ma-25	158	22	such	such	ADJ
ma-25	158	23	that	that	SCONJ
ma-25	158	24	ρ[p	ρ[p	NOUN
ma-25	158	25	,	,	PUNCT
ma-25	158	26	q	q	X
ma-25	158	27	]	]	X
ma-25	158	28	(	(	PUNCT
ma-25	158	29	g	g	NOUN
ma-25	158	30	)	)	PUNCT
ma-25	158	31	<	<	X
ma-25	159	1	+	+	PUNCT
ma-25	159	2	∞.	∞.	PROPN
ma-25	159	3	then	then	ADV
ma-25	159	4	,	,	PUNCT
ma-25	159	5	there	there	PRON
ma-25	159	6	exist	exist	VERB
ma-25	159	7	entire	entire	ADJ
ma-25	159	8	functions	function	NOUN
ma-25	159	9	u(z	u(z	NOUN
ma-25	159	10	)	)	PUNCT
ma-25	159	11	and	and	CCONJ
ma-25	159	12	v(z	v(z	NOUN
ma-25	159	13	)	)	PUNCT
ma-25	159	14	such	such	ADJ
ma-25	159	15	that	that	SCONJ
ma-25	159	16	g	g	PROPN
ma-25	159	17	(	(	PUNCT
ma-25	159	18	z	z	NOUN
ma-25	159	19	)	)	PUNCT
ma-25	159	20	=	=	SYM
ma-25	159	21	u(z)ev(z	u(z)ev(z	NOUN
ma-25	159	22	)	)	PUNCT
ma-25	159	23	,	,	PUNCT
ma-25	159	24	ρ[p	ρ[p	PROPN
ma-25	159	25	,	,	PUNCT
ma-25	159	26	q	q	X
ma-25	159	27	]	]	X
ma-25	159	28	(	(	PUNCT
ma-25	159	29	g	g	NOUN
ma-25	159	30	)	)	PUNCT
ma-25	160	1	=	=	SYM
ma-25	160	2	max	max	PROPN
ma-25	160	3	{	{	PUNCT
ma-25	160	4	ρ[p	ρ[p	PROPN
ma-25	160	5	,	,	PUNCT
ma-25	160	6	q	q	X
ma-25	160	7	]	]	X
ma-25	160	8	(	(	PUNCT
ma-25	160	9	u	u	NOUN
ma-25	160	10	)	)	PUNCT
ma-25	160	11	,	,	PUNCT
ma-25	160	12	ρ[p	ρ[p	PROPN
ma-25	160	13	,	,	PUNCT
ma-25	160	14	q	q	X
ma-25	160	15	]	]	X
ma-25	160	16	(	(	PUNCT
ma-25	160	17	ev(z	ev(z	X
ma-25	160	18	)	)	PUNCT
ma-25	160	19	)	)	PUNCT
ma-25	160	20	}	}	PUNCT
ma-25	160	21	and	and	CCONJ
ma-25	160	22	ρ[p	ρ[p	NOUN
ma-25	160	23	,	,	PUNCT
ma-25	160	24	q	q	X
ma-25	160	25	]	]	X
ma-25	160	26	(	(	PUNCT
ma-25	160	27	u	u	NOUN
ma-25	160	28	)	)	PUNCT
ma-25	160	29	=	=	SYM
ma-25	161	1	lim	lim	PROPN
ma-25	161	2	sup	sup	VERB
ma-25	161	3	r→+∞	r→+∞	PROPN
ma-25	161	4	logp	logp	NOUN
ma-25	161	5	n	n	CCONJ
ma-25	161	6	(	(	PUNCT
ma-25	161	7	r	r	NOUN
ma-25	161	8	,	,	PUNCT
ma-25	161	9	1	1	NUM
ma-25	161	10	g	g	NOUN
ma-25	161	11	)	)	PUNCT
ma-25	161	12	logq	logq	ADJ
ma-25	161	13	r	r	NOUN
ma-25	161	14	.	.	PUNCT
ma-25	162	1	moreover	moreover	ADV
ma-25	162	2	,	,	PUNCT
ma-25	162	3	for	for	ADP
ma-25	162	4	any	any	DET
ma-25	162	5	given	give	VERB
ma-25	162	6	ε	ε	PROPN
ma-25	162	7	>	>	X
ma-25	162	8	0	0	PROPN
ma-25	162	9	,	,	PUNCT
ma-25	162	10	we	we	PRON
ma-25	162	11	have	have	VERB
ma-25	162	12	|u(z)|	|u(z)|	VERB
ma-25	162	13	≥	≥	NUM
ma-25	162	14	exp	exp	NOUN
ma-25	162	15	{	{	PUNCT
ma-25	162	16	−	−	PUNCT
ma-25	162	17	expp	expp	ADJ
ma-25	162	18	{	{	PUNCT
ma-25	162	19	(	(	PUNCT
ma-25	162	20	ρ[p	ρ[p	NOUN
ma-25	162	21	,	,	PUNCT
ma-25	162	22	q	q	X
ma-25	162	23	]	]	X
ma-25	162	24	(	(	PUNCT
ma-25	162	25	u	u	NOUN
ma-25	162	26	)	)	PUNCT
ma-25	163	1	+	+	NUM
ma-25	163	2	ε	ε	PROPN
ma-25	163	3	)	)	PUNCT
ma-25	163	4	logq	logq	VERB
ma-25	163	5	r	r	NOUN
ma-25	163	6	}	}	PUNCT
ma-25	163	7	}	}	PUNCT
ma-25	163	8	(	(	PUNCT
ma-25	163	9	r	r	NOUN
ma-25	163	10	/∈	/∈	SYM
ma-25	163	11	e2	e2	PROPN
ma-25	163	12	)	)	PUNCT
ma-25	163	13	,	,	PUNCT
ma-25	163	14	where	where	SCONJ
ma-25	163	15	e2	e2	PROPN
ma-25	163	16	⊂	⊂	PROPN
ma-25	163	17	(	(	PUNCT
ma-25	163	18	1,+∞	1,+∞	NUM
ma-25	163	19	)	)	PUNCT
ma-25	163	20	is	be	AUX
ma-25	163	21	a	a	DET
ma-25	163	22	set	set	NOUN
ma-25	163	23	of	of	ADP
ma-25	163	24	r	r	NOUN
ma-25	163	25	of	of	ADP
ma-25	163	26	finite	finite	ADJ
ma-25	163	27	linear	linear	PROPN
ma-25	163	28	measure	measure	NOUN
ma-25	163	29	.	.	PUNCT
ma-25	164	1	eur	eur	PROPN
ma-25	164	2	.	.	PUNCT
ma-25	165	1	j.	j.	PROPN
ma-25	165	2	math	math	PROPN
ma-25	165	3	.	.	PUNCT
ma-25	166	1	anal	anal	ADJ
ma-25	166	2	.	.	PUNCT
ma-25	167	1	1	1	NUM
ma-25	167	2	(	(	PUNCT
ma-25	167	3	2021	2021	NUM
ma-25	167	4	)	)	PUNCT
ma-25	167	5	91	91	NUM
ma-25	167	6	lemma	lemma	PROPN
ma-25	167	7	2.3	2.3	NUM
ma-25	167	8	let	let	VERB
ma-25	167	9	p	p	PRON
ma-25	167	10	≥	≥	PRON
ma-25	167	11	q	q	NOUN
ma-25	167	12	≥	≥	NUM
ma-25	167	13	1	1	NUM
ma-25	167	14	be	be	AUX
ma-25	167	15	integers	integer	NOUN
ma-25	167	16	.	.	PUNCT
ma-25	167	17	suppose	suppose	VERB
ma-25	167	18	that	that	SCONJ
ma-25	167	19	f	f	PROPN
ma-25	167	20	is	be	AUX
ma-25	167	21	a	a	DET
ma-25	167	22	meromorphic	meromorphic	ADJ
ma-25	167	23	function	function	NOUN
ma-25	167	24	such	such	ADJ
ma-25	167	25	that	that	SCONJ
ma-25	167	26	ρ[p	ρ[p	NOUN
ma-25	167	27	,	,	PUNCT
ma-25	167	28	q	q	X
ma-25	167	29	]	]	X
ma-25	167	30	(	(	PUNCT
ma-25	167	31	f	f	X
ma-25	167	32	)	)	PUNCT
ma-25	167	33	<	<	X
ma-25	168	1	+	+	PUNCT
ma-25	168	2	∞.	∞.	PROPN
ma-25	168	3	then	then	ADV
ma-25	168	4	,	,	PUNCT
ma-25	168	5	there	there	PRON
ma-25	168	6	exist	exist	VERB
ma-25	168	7	entire	entire	ADJ
ma-25	168	8	functions	function	NOUN
ma-25	168	9	u1	u1	NOUN
ma-25	168	10	(	(	PUNCT
ma-25	168	11	z	z	NOUN
ma-25	168	12	)	)	PUNCT
ma-25	168	13	,	,	PUNCT
ma-25	168	14	u2	u2	PROPN
ma-25	168	15	(	(	PUNCT
ma-25	168	16	z	z	NOUN
ma-25	168	17	)	)	PUNCT
ma-25	168	18	and	and	CCONJ
ma-25	168	19	v	v	X
ma-25	168	20	(	(	PUNCT
ma-25	168	21	z	z	NOUN
ma-25	168	22	)	)	PUNCT
ma-25	168	23	such	such	ADJ
ma-25	168	24	that	that	SCONJ
ma-25	168	25	f	f	PROPN
ma-25	168	26	(	(	PUNCT
ma-25	168	27	z	z	NOUN
ma-25	168	28	)	)	PUNCT
ma-25	168	29	=	=	SYM
ma-25	168	30	u1	u1	NOUN
ma-25	168	31	(	(	PUNCT
ma-25	168	32	z	z	NOUN
ma-25	168	33	)	)	PUNCT
ma-25	168	34	ev(z	ev(z	X
ma-25	168	35	)	)	PUNCT
ma-25	168	36	u2	u2	NOUN
ma-25	168	37	(	(	PUNCT
ma-25	168	38	z	z	NOUN
ma-25	168	39	)	)	PUNCT
ma-25	168	40	(	(	PUNCT
ma-25	168	41	2.1	2.1	NUM
ma-25	168	42	)	)	PUNCT
ma-25	168	43	and	and	CCONJ
ma-25	168	44	ρ[p	ρ[p	NOUN
ma-25	168	45	,	,	PUNCT
ma-25	168	46	q](f	q](f	NOUN
ma-25	168	47	)	)	PUNCT
ma-25	168	48	=	=	SYM
ma-25	168	49	max	max	PROPN
ma-25	168	50	{	{	PUNCT
ma-25	168	51	ρ[p	ρ[p	PROPN
ma-25	168	52	,	,	PUNCT
ma-25	168	53	q](u1	q](u1	NOUN
ma-25	168	54	)	)	PUNCT
ma-25	168	55	,	,	PUNCT
ma-25	168	56	ρ[p	ρ[p	PROPN
ma-25	168	57	,	,	PUNCT
ma-25	168	58	q](u2	q](u2	PROPN
ma-25	168	59	)	)	PUNCT
ma-25	168	60	,	,	PUNCT
ma-25	168	61	ρ[p	ρ[p	PROPN
ma-25	168	62	,	,	PUNCT
ma-25	168	63	q](e	q](e	ADJ
ma-25	168	64	v(z	v(z	NOUN
ma-25	168	65	)	)	PUNCT
ma-25	168	66	)	)	PUNCT
ma-25	168	67	}	}	PUNCT
ma-25	168	68	.	.	PUNCT
ma-25	169	1	(	(	PUNCT
ma-25	169	2	2.2	2.2	NUM
ma-25	169	3	)	)	PUNCT
ma-25	169	4	moreover	moreover	ADV
ma-25	169	5	,	,	PUNCT
ma-25	169	6	for	for	ADP
ma-25	169	7	any	any	DET
ma-25	169	8	given	give	VERB
ma-25	169	9	ε	ε	PROPN
ma-25	169	10	>	>	X
ma-25	169	11	0	0	PROPN
ma-25	169	12	,	,	PUNCT
ma-25	169	13	we	we	PRON
ma-25	169	14	have	have	VERB
ma-25	169	15	exp	exp	NOUN
ma-25	169	16	{	{	PUNCT
ma-25	169	17	−	−	PUNCT
ma-25	169	18	expp	expp	ADJ
ma-25	169	19	{	{	PUNCT
ma-25	169	20	(	(	PUNCT
ma-25	169	21	ρ(p	ρ(p	PROPN
ma-25	169	22	,	,	PUNCT
ma-25	169	23	q	q	NOUN
ma-25	169	24	)	)	PUNCT
ma-25	169	25	(	(	PUNCT
ma-25	169	26	f	f	PROPN
ma-25	169	27	)	)	PUNCT
ma-25	170	1	+	+	CCONJ
ma-25	170	2	ε	ε	PROPN
ma-25	170	3	)	)	PUNCT
ma-25	170	4	logq	logq	VERB
ma-25	170	5	r	r	NOUN
ma-25	170	6	}	}	PUNCT
ma-25	170	7	}	}	PUNCT
ma-25	170	8	≤	≤	NOUN
ma-25	170	9	|f	|f	PROPN
ma-25	171	1	(	(	PUNCT
ma-25	171	2	z)|	z)|	ADP
ma-25	171	3	≤	≤	NUM
ma-25	171	4	expp+1	expp+1	PROPN
ma-25	171	5	{	{	PUNCT
ma-25	171	6	(	(	PUNCT
ma-25	171	7	ρ(p	ρ(p	PROPN
ma-25	171	8	,	,	PUNCT
ma-25	171	9	q	q	NOUN
ma-25	171	10	)	)	PUNCT
ma-25	171	11	(	(	PUNCT
ma-25	171	12	f	f	PROPN
ma-25	171	13	)	)	PUNCT
ma-25	171	14	+	+	CCONJ
ma-25	171	15	ε	ε	PROPN
ma-25	171	16	)	)	PUNCT
ma-25	171	17	logq	logq	VERB
ma-25	171	18	r	r	NOUN
ma-25	171	19	}	}	PUNCT
ma-25	171	20	(	(	PUNCT
ma-25	171	21	r	r	NOUN
ma-25	171	22	/∈	/∈	SYM
ma-25	171	23	e3	e3	NOUN
ma-25	171	24	)	)	PUNCT
ma-25	171	25	,	,	PUNCT
ma-25	171	26	(	(	PUNCT
ma-25	171	27	2.3	2.3	NUM
ma-25	171	28	)	)	PUNCT
ma-25	171	29	where	where	SCONJ
ma-25	171	30	e3	e3	PROPN
ma-25	171	31	⊂	⊂	X
ma-25	171	32	(	(	PUNCT
ma-25	171	33	1,+∞	1,+∞	NUM
ma-25	171	34	)	)	PUNCT
ma-25	171	35	is	be	AUX
ma-25	171	36	a	a	DET
ma-25	171	37	set	set	NOUN
ma-25	171	38	of	of	ADP
ma-25	171	39	r	r	NOUN
ma-25	171	40	of	of	ADP
ma-25	171	41	finite	finite	ADJ
ma-25	171	42	linear	linear	PROPN
ma-25	171	43	measure	measure	NOUN
ma-25	171	44	.	.	PUNCT
ma-25	172	1	proof	proof	NOUN
ma-25	172	2	.	.	PUNCT
ma-25	173	1	when	when	SCONJ
ma-25	173	2	p	p	PRON
ma-25	173	3	≥	≥	X
ma-25	173	4	q	q	X
ma-25	173	5	=	=	NOUN
ma-25	173	6	1	1	NUM
ma-25	173	7	,	,	PUNCT
ma-25	173	8	the	the	DET
ma-25	173	9	lemma	lemma	PROPN
ma-25	173	10	is	be	AUX
ma-25	173	11	due	due	ADJ
ma-25	173	12	to	to	ADP
ma-25	173	13	tu	tu	PROPN
ma-25	173	14	and	and	CCONJ
ma-25	173	15	long	long	ADJ
ma-25	173	16	[	[	X
ma-25	173	17	21	21	NUM
ma-25	173	18	]	]	PUNCT
ma-25	173	19	.	.	PUNCT
ma-25	174	1	thus	thus	ADV
ma-25	174	2	,	,	PUNCT
ma-25	174	3	we	we	PRON
ma-25	174	4	assume	assume	VERB
ma-25	174	5	that	that	SCONJ
ma-25	174	6	p	p	X
ma-25	174	7	>	>	X
ma-25	174	8	q	q	PUNCT
ma-25	174	9	>	>	PUNCT
ma-25	174	10	1	1	NUM
ma-25	174	11	or	or	CCONJ
ma-25	174	12	p	p	NOUN
ma-25	174	13	=	=	ADJ
ma-25	174	14	q	q	X
ma-25	174	15	>	>	X
ma-25	174	16	1	1	X
ma-25	174	17	.	.	PUNCT
ma-25	175	1	by	by	ADP
ma-25	175	2	hadamard	hadamard	ADJ
ma-25	175	3	factorization	factorization	NOUN
ma-25	175	4	theorem	theorem	NOUN
ma-25	175	5	,	,	PUNCT
ma-25	175	6	we	we	PRON
ma-25	175	7	can	can	AUX
ma-25	175	8	write	write	VERB
ma-25	175	9	f	f	PROPN
ma-25	175	10	as	as	ADP
ma-25	175	11	f	f	PROPN
ma-25	175	12	(	(	PUNCT
ma-25	175	13	z	z	NOUN
ma-25	175	14	)	)	PUNCT
ma-25	175	15	=	=	SYM
ma-25	175	16	g(z	g(z	ADJ
ma-25	175	17	)	)	PUNCT
ma-25	175	18	d(z	d(z	PROPN
ma-25	175	19	)	)	PUNCT
ma-25	175	20	,	,	PUNCT
ma-25	175	21	where	where	SCONJ
ma-25	175	22	g	g	PROPN
ma-25	175	23	(	(	PUNCT
ma-25	175	24	z	z	NOUN
ma-25	175	25	)	)	PUNCT
ma-25	175	26	and	and	CCONJ
ma-25	175	27	d	d	X
ma-25	175	28	(	(	PUNCT
ma-25	175	29	z	z	NOUN
ma-25	175	30	)	)	PUNCT
ma-25	175	31	are	be	AUX
ma-25	175	32	entire	entire	ADJ
ma-25	175	33	functions	function	NOUN
ma-25	175	34	satisfying	satisfy	VERB
ma-25	175	35	µ[p	µ[p	ADJ
ma-25	175	36	,	,	PUNCT
ma-25	175	37	q	q	X
ma-25	175	38	]	]	X
ma-25	175	39	(	(	PUNCT
ma-25	175	40	g	g	NOUN
ma-25	175	41	)	)	PUNCT
ma-25	175	42	=	=	SYM
ma-25	175	43	µ[p	µ[p	ADJ
ma-25	175	44	,	,	PUNCT
ma-25	175	45	q	q	X
ma-25	175	46	]	]	X
ma-25	175	47	(	(	PUNCT
ma-25	175	48	f	f	X
ma-25	175	49	)	)	PUNCT
ma-25	175	50	=	=	SYM
ma-25	175	51	µ	µ	PRON
ma-25	175	52	≤	≤	NUM
ma-25	175	53	ρ[p	ρ[p	NUM
ma-25	175	54	,	,	PUNCT
ma-25	175	55	q	q	X
ma-25	175	56	]	]	X
ma-25	175	57	(	(	PUNCT
ma-25	175	58	f	f	X
ma-25	175	59	)	)	PUNCT
ma-25	175	60	=	=	PUNCT
ma-25	176	1	ρ[p	ρ[p	PROPN
ma-25	176	2	,	,	PUNCT
ma-25	176	3	q	q	X
ma-25	176	4	]	]	X
ma-25	176	5	(	(	PUNCT
ma-25	176	6	g	g	NOUN
ma-25	176	7	)	)	PUNCT
ma-25	176	8	<	<	X
ma-25	177	1	+	+	X
ma-25	177	2	∞	∞	PROPN
ma-25	177	3	and	and	CCONJ
ma-25	177	4	λ[p	λ[p	PROPN
ma-25	177	5	,	,	PUNCT
ma-25	177	6	q	q	X
ma-25	177	7	]	]	X
ma-25	177	8	(	(	PUNCT
ma-25	177	9	d	d	NOUN
ma-25	177	10	)	)	PUNCT
ma-25	177	11	=	=	SYM
ma-25	177	12	ρ[p	ρ[p	PROPN
ma-25	177	13	,	,	PUNCT
ma-25	177	14	q	q	X
ma-25	177	15	]	]	X
ma-25	177	16	(	(	PUNCT
ma-25	177	17	d	d	NOUN
ma-25	177	18	)	)	PUNCT
ma-25	177	19	=	=	SYM
ma-25	177	20	λ[p	λ[p	NUM
ma-25	177	21	,	,	PUNCT
ma-25	177	22	q	q	X
ma-25	177	23	]	]	X
ma-25	177	24	(	(	PUNCT
ma-25	177	25	1	1	NUM
ma-25	177	26	f	f	NOUN
ma-25	177	27	)	)	PUNCT
ma-25	177	28	<	<	X
ma-25	177	29	µ.	µ.	PROPN
ma-25	177	30	by	by	ADP
ma-25	177	31	lemma	lemma	PROPN
ma-25	177	32	2.2	2.2	NUM
ma-25	177	33	,	,	PUNCT
ma-25	177	34	there	there	PRON
ma-25	177	35	exist	exist	VERB
ma-25	177	36	entire	entire	ADJ
ma-25	177	37	functions	function	NOUN
ma-25	177	38	u(z	u(z	NOUN
ma-25	177	39	)	)	PUNCT
ma-25	177	40	and	and	CCONJ
ma-25	177	41	v(z	v(z	NOUN
ma-25	177	42	)	)	PUNCT
ma-25	177	43	such	such	ADJ
ma-25	177	44	that	that	SCONJ
ma-25	177	45	g	g	PROPN
ma-25	177	46	(	(	PUNCT
ma-25	177	47	z	z	NOUN
ma-25	177	48	)	)	PUNCT
ma-25	177	49	=	=	SYM
ma-25	177	50	u(z)ev(z	u(z)ev(z	NOUN
ma-25	177	51	)	)	PUNCT
ma-25	177	52	,	,	PUNCT
ma-25	177	53	ρ[p	ρ[p	PROPN
ma-25	177	54	,	,	PUNCT
ma-25	177	55	q	q	X
ma-25	177	56	]	]	X
ma-25	177	57	(	(	PUNCT
ma-25	177	58	g	g	NOUN
ma-25	177	59	)	)	PUNCT
ma-25	178	1	=	=	SYM
ma-25	178	2	max	max	PROPN
ma-25	178	3	{	{	PUNCT
ma-25	178	4	ρ[p	ρ[p	PROPN
ma-25	178	5	,	,	PUNCT
ma-25	178	6	q	q	X
ma-25	178	7	]	]	X
ma-25	178	8	(	(	PUNCT
ma-25	178	9	u	u	NOUN
ma-25	178	10	)	)	PUNCT
ma-25	178	11	,	,	PUNCT
ma-25	178	12	ρ[p	ρ[p	PROPN
ma-25	178	13	,	,	PUNCT
ma-25	178	14	q	q	X
ma-25	178	15	]	]	X
ma-25	178	16	(	(	PUNCT
ma-25	178	17	ev(z	ev(z	X
ma-25	178	18	)	)	PUNCT
ma-25	178	19	)	)	PUNCT
ma-25	178	20	}	}	PUNCT
ma-25	178	21	.	.	PUNCT
ma-25	179	1	so	so	ADV
ma-25	179	2	,	,	PUNCT
ma-25	179	3	there	there	PRON
ma-25	179	4	exist	exist	VERB
ma-25	179	5	entire	entire	ADJ
ma-25	179	6	functions	function	NOUN
ma-25	179	7	u(z	u(z	NOUN
ma-25	179	8	)	)	PUNCT
ma-25	179	9	,	,	PUNCT
ma-25	179	10	v(z	v(z	NOUN
ma-25	179	11	)	)	PUNCT
ma-25	179	12	and	and	CCONJ
ma-25	179	13	d	d	X
ma-25	179	14	(	(	PUNCT
ma-25	179	15	z	z	NOUN
ma-25	179	16	)	)	PUNCT
ma-25	179	17	such	such	ADJ
ma-25	179	18	that	that	SCONJ
ma-25	179	19	f	f	PROPN
ma-25	179	20	(	(	PUNCT
ma-25	179	21	z	z	NOUN
ma-25	179	22	)	)	PUNCT
ma-25	179	23	=	=	SYM
ma-25	179	24	u(z)ev(z	u(z)ev(z	NOUN
ma-25	179	25	)	)	PUNCT
ma-25	179	26	d	d	NOUN
ma-25	179	27	(	(	PUNCT
ma-25	179	28	z)and	z)and	PROPN
ma-25	179	29	ρ[p	ρ[p	NUM
ma-25	179	30	,	,	PUNCT
ma-25	179	31	q](f	q](f	NOUN
ma-25	179	32	)	)	PUNCT
ma-25	179	33	=	=	SYM
ma-25	179	34	max	max	PROPN
ma-25	179	35	{	{	PUNCT
ma-25	179	36	ρ[p	ρ[p	PROPN
ma-25	179	37	,	,	PUNCT
ma-25	179	38	q	q	X
ma-25	179	39	]	]	X
ma-25	179	40	(	(	PUNCT
ma-25	179	41	u	u	NOUN
ma-25	179	42	)	)	PUNCT
ma-25	179	43	,	,	PUNCT
ma-25	179	44	ρ[p	ρ[p	NOUN
ma-25	179	45	,	,	PUNCT
ma-25	179	46	q](d	q](d	NOUN
ma-25	179	47	)	)	PUNCT
ma-25	179	48	,	,	PUNCT
ma-25	179	49	ρ[p	ρ[p	PROPN
ma-25	179	50	,	,	PUNCT
ma-25	179	51	q	q	X
ma-25	179	52	]	]	X
ma-25	179	53	(	(	PUNCT
ma-25	179	54	ev(z	ev(z	X
ma-25	179	55	)	)	PUNCT
ma-25	179	56	)	)	PUNCT
ma-25	179	57	}	}	PUNCT
ma-25	179	58	.	.	PUNCT
ma-25	180	1	thus	thus	ADV
ma-25	180	2	(	(	PUNCT
ma-25	180	3	2.1	2.1	NUM
ma-25	180	4	)	)	PUNCT
ma-25	180	5	and	and	CCONJ
ma-25	180	6	(	(	PUNCT
ma-25	180	7	2.2	2.2	NUM
ma-25	180	8	)	)	PUNCT
ma-25	180	9	hold	hold	NOUN
ma-25	180	10	.	.	PUNCT
ma-25	181	1	set	set	VERB
ma-25	181	2	f	f	PROPN
ma-25	181	3	(	(	PUNCT
ma-25	181	4	z	z	NOUN
ma-25	181	5	)	)	PUNCT
ma-25	181	6	=	=	SYM
ma-25	182	1	u1(z)e	u1(z)e	ADJ
ma-25	182	2	v(z	v(z	PROPN
ma-25	182	3	)	)	PUNCT
ma-25	182	4	u2(z	u2(z	SYM
ma-25	182	5	)	)	PUNCT
ma-25	182	6	,	,	PUNCT
ma-25	182	7	where	where	SCONJ
ma-25	182	8	u1	u1	NOUN
ma-25	182	9	(	(	PUNCT
ma-25	182	10	z	z	NOUN
ma-25	182	11	)	)	PUNCT
ma-25	182	12	,	,	PUNCT
ma-25	182	13	u2	u2	PROPN
ma-25	182	14	(	(	PUNCT
ma-25	182	15	z	z	NOUN
ma-25	182	16	)	)	PUNCT
ma-25	182	17	are	be	AUX
ma-25	182	18	the	the	DET
ma-25	182	19	canonical	canonical	ADJ
ma-25	182	20	productsformed	productsformed	NOUN
ma-25	182	21	with	with	ADP
ma-25	182	22	the	the	DET
ma-25	182	23	zeros	zero	NOUN
ma-25	182	24	and	and	CCONJ
ma-25	182	25	poles	pole	NOUN
ma-25	182	26	of	of	ADP
ma-25	182	27	f	f	PROPN
ma-25	182	28	respectively	respectively	ADV
ma-25	182	29	.	.	PUNCT
ma-25	183	1	by	by	ADP
ma-25	183	2	the	the	DET
ma-25	183	3	definition	definition	NOUN
ma-25	183	4	of	of	ADP
ma-25	183	5	[	[	X
ma-25	183	6	p	p	X
ma-25	183	7	,	,	PUNCT
ma-25	183	8	q]-order	q]-order	NOUN
ma-25	183	9	,	,	PUNCT
ma-25	183	10	for	for	ADP
ma-25	183	11	sufficientlylarge	sufficientlylarge	NOUN
ma-25	183	12	r	r	NOUN
ma-25	183	13	and	and	CCONJ
ma-25	183	14	any	any	DET
ma-25	183	15	given	give	VERB
ma-25	183	16	ε	ε	PROPN
ma-25	183	17	>	>	X
ma-25	183	18	0	0	PROPN
ma-25	183	19	,	,	PUNCT
ma-25	183	20	we	we	PRON
ma-25	183	21	have	have	VERB
ma-25	183	22	|u1	|u1	NOUN
ma-25	183	23	(	(	PUNCT
ma-25	183	24	z)|	z)|	NOUN
ma-25	183	25	≤	≤	NUM
ma-25	183	26	expp+1	expp+1	PROPN
ma-25	183	27	{	{	PUNCT
ma-25	183	28	(	(	PUNCT
ma-25	183	29	ρ[p	ρ[p	NOUN
ma-25	183	30	,	,	PUNCT
ma-25	183	31	q	q	X
ma-25	183	32	]	]	X
ma-25	183	33	(	(	PUNCT
ma-25	183	34	u1	u1	NOUN
ma-25	183	35	)	)	PUNCT
ma-25	184	1	+	+	CCONJ
ma-25	184	2	ε	ε	PROPN
ma-25	184	3	3	3	NUM
ma-25	184	4	)	)	PUNCT
ma-25	184	5	logq	logq	VERB
ma-25	184	6	r	r	NOUN
ma-25	184	7	}	}	PUNCT
ma-25	184	8	,	,	PUNCT
ma-25	184	9	|u2	|u2	NOUN
ma-25	184	10	(	(	PUNCT
ma-25	184	11	z)|	z)|	NOUN
ma-25	184	12	≤	≤	NUM
ma-25	184	13	expp+1	expp+1	PROPN
ma-25	184	14	{	{	PUNCT
ma-25	184	15	(	(	PUNCT
ma-25	184	16	ρ[p	ρ[p	NOUN
ma-25	184	17	,	,	PUNCT
ma-25	184	18	q	q	X
ma-25	184	19	]	]	X
ma-25	184	20	(	(	PUNCT
ma-25	184	21	u2	u2	NOUN
ma-25	184	22	)	)	PUNCT
ma-25	185	1	+	+	CCONJ
ma-25	185	2	ε	ε	PROPN
ma-25	185	3	3	3	NUM
ma-25	185	4	)	)	PUNCT
ma-25	185	5	logq	logq	VERB
ma-25	185	6	r	r	NOUN
ma-25	185	7	}	}	PUNCT
ma-25	185	8	.	.	PUNCT
ma-25	186	1	(	(	PUNCT
ma-25	186	2	2.4	2.4	NUM
ma-25	186	3	)	)	PUNCT
ma-25	186	4	since	since	SCONJ
ma-25	186	5	max	max	PROPN
ma-25	186	6	{	{	PUNCT
ma-25	186	7	ρ[p	ρ[p	PROPN
ma-25	186	8	,	,	PUNCT
ma-25	186	9	q](u1	q](u1	NOUN
ma-25	186	10	)	)	PUNCT
ma-25	186	11	,	,	PUNCT
ma-25	186	12	ρ[p	ρ[p	PROPN
ma-25	186	13	,	,	PUNCT
ma-25	186	14	q](u2	q](u2	PROPN
ma-25	186	15	)	)	PUNCT
ma-25	186	16	,	,	PUNCT
ma-25	186	17	ρ[p	ρ[p	PROPN
ma-25	186	18	,	,	PUNCT
ma-25	186	19	q](e	q](e	ADJ
ma-25	186	20	v(z	v(z	NOUN
ma-25	186	21	)	)	PUNCT
ma-25	186	22	)	)	PUNCT
ma-25	186	23	}	}	PUNCT
ma-25	187	1	=	=	PUNCT
ma-25	187	2	ρ[p	ρ[p	NOUN
ma-25	187	3	,	,	PUNCT
ma-25	187	4	q](f	q](f	NOUN
ma-25	187	5	)	)	PUNCT
ma-25	187	6	,	,	PUNCT
ma-25	187	7	then	then	ADV
ma-25	187	8	we	we	PRON
ma-25	187	9	obtain	obtain	VERB
ma-25	187	10	|u1	|u1	NOUN
ma-25	187	11	(	(	PUNCT
ma-25	187	12	z)|	z)|	ADP
ma-25	187	13	≤	≤	NUM
ma-25	187	14	expp+1	expp+1	PROPN
ma-25	187	15	{	{	PUNCT
ma-25	187	16	(	(	PUNCT
ma-25	187	17	ρ[p	ρ[p	NOUN
ma-25	187	18	,	,	PUNCT
ma-25	187	19	q	q	X
ma-25	187	20	]	]	X
ma-25	187	21	(	(	PUNCT
ma-25	187	22	f	f	PROPN
ma-25	187	23	)	)	PUNCT
ma-25	188	1	+	+	CCONJ
ma-25	188	2	ε	ε	PROPN
ma-25	188	3	3	3	NUM
ma-25	188	4	)	)	PUNCT
ma-25	188	5	logq	logq	VERB
ma-25	188	6	r	r	NOUN
ma-25	188	7	}	}	PUNCT
ma-25	188	8	,	,	PUNCT
ma-25	188	9	(	(	PUNCT
ma-25	188	10	2.5	2.5	NUM
ma-25	188	11	)	)	PUNCT
ma-25	188	12	|u2	|u2	NOUN
ma-25	188	13	(	(	PUNCT
ma-25	188	14	z)|	z)|	ADP
ma-25	188	15	≤	≤	NUM
ma-25	188	16	expp+1	expp+1	PROPN
ma-25	188	17	{	{	PUNCT
ma-25	188	18	(	(	PUNCT
ma-25	188	19	ρ[p	ρ[p	NOUN
ma-25	188	20	,	,	PUNCT
ma-25	188	21	q	q	X
ma-25	188	22	]	]	X
ma-25	188	23	(	(	PUNCT
ma-25	188	24	f	f	PROPN
ma-25	188	25	)	)	PUNCT
ma-25	189	1	+	+	CCONJ
ma-25	189	2	ε	ε	PROPN
ma-25	189	3	3	3	NUM
ma-25	189	4	)	)	PUNCT
ma-25	189	5	logq	logq	VERB
ma-25	189	6	r	r	NOUN
ma-25	189	7	}	}	PUNCT
ma-25	189	8	,	,	PUNCT
ma-25	189	9	(	(	PUNCT
ma-25	189	10	2.6	2.6	NUM
ma-25	189	11	)	)	PUNCT
ma-25	189	12	eur	eur	PROPN
ma-25	189	13	.	.	PUNCT
ma-25	190	1	j.	j.	PROPN
ma-25	190	2	math	math	PROPN
ma-25	190	3	.	.	PUNCT
ma-25	191	1	anal	anal	ADJ
ma-25	191	2	.	.	PUNCT
ma-25	192	1	1	1	NUM
ma-25	192	2	(	(	PUNCT
ma-25	192	3	2021	2021	NUM
ma-25	192	4	)	)	PUNCT
ma-25	192	5	92∣∣∣ev(z)∣∣∣	92∣∣∣ev(z)∣∣∣	NOUN
ma-25	192	6	≤	≤	NUM
ma-25	192	7	expp+1	expp+1	PROPN
ma-25	192	8	{	{	PUNCT
ma-25	192	9	(	(	PUNCT
ma-25	192	10	ρ[p	ρ[p	NOUN
ma-25	192	11	,	,	PUNCT
ma-25	192	12	q	q	X
ma-25	192	13	]	]	X
ma-25	192	14	(	(	PUNCT
ma-25	192	15	f	f	PROPN
ma-25	192	16	)	)	PUNCT
ma-25	193	1	+	+	CCONJ
ma-25	193	2	ε	ε	PROPN
ma-25	193	3	3	3	NUM
ma-25	193	4	)	)	PUNCT
ma-25	193	5	logq	logq	VERB
ma-25	193	6	r	r	NOUN
ma-25	193	7	}	}	PUNCT
ma-25	193	8	.	.	PUNCT
ma-25	194	1	(	(	PUNCT
ma-25	194	2	2.7	2.7	NUM
ma-25	194	3	)	)	PUNCT
ma-25	194	4	by	by	ADP
ma-25	194	5	lemma	lemma	PROPN
ma-25	194	6	2.2	2.2	NUM
ma-25	194	7	,	,	PUNCT
ma-25	194	8	there	there	PRON
ma-25	194	9	exists	exist	VERB
ma-25	194	10	a	a	DET
ma-25	194	11	set	set	NOUN
ma-25	194	12	e3	e3	NOUN
ma-25	194	13	⊂	⊂	X
ma-25	194	14	(	(	PUNCT
ma-25	194	15	1,+∞	1,+∞	NUM
ma-25	194	16	)	)	PUNCT
ma-25	194	17	of	of	ADP
ma-25	194	18	r	r	NOUN
ma-25	194	19	with	with	ADP
ma-25	194	20	a	a	DET
ma-25	194	21	finite	finite	ADJ
ma-25	194	22	linear	linear	ADJ
ma-25	194	23	measure	measure	NOUN
ma-25	194	24	such	such	ADJ
ma-25	194	25	that	that	PRON
ma-25	194	26	for	for	ADP
ma-25	194	27	anygiven	anygiven	ADJ
ma-25	194	28	ε	ε	PROPN
ma-25	194	29	>	>	X
ma-25	194	30	0	0	PROPN
ma-25	194	31	,	,	PUNCT
ma-25	194	32	we	we	PRON
ma-25	194	33	have	have	VERB
ma-25	194	34	|u1	|u1	PRON
ma-25	194	35	(	(	PUNCT
ma-25	194	36	z)|	z)|	X
ma-25	194	37	≥	≥	NOUN
ma-25	194	38	exp	exp	NOUN
ma-25	194	39	{	{	PUNCT
ma-25	194	40	−	−	PUNCT
ma-25	194	41	expp	expp	ADJ
ma-25	194	42	{	{	PUNCT
ma-25	194	43	(	(	PUNCT
ma-25	194	44	ρ[p	ρ[p	NOUN
ma-25	194	45	,	,	PUNCT
ma-25	194	46	q	q	X
ma-25	194	47	]	]	X
ma-25	194	48	(	(	PUNCT
ma-25	194	49	u1	u1	NOUN
ma-25	194	50	)	)	PUNCT
ma-25	195	1	+	+	CCONJ
ma-25	195	2	ε	ε	PROPN
ma-25	195	3	3	3	NUM
ma-25	195	4	)	)	PUNCT
ma-25	195	5	logq	logq	VERB
ma-25	195	6	r	r	NOUN
ma-25	195	7	}	}	PUNCT
ma-25	195	8	}	}	PUNCT
ma-25	195	9	≥	≥	NOUN
ma-25	195	10	exp	exp	NOUN
ma-25	195	11	{	{	PUNCT
ma-25	195	12	−	−	PUNCT
ma-25	195	13	expp	expp	ADJ
ma-25	195	14	{	{	PUNCT
ma-25	195	15	(	(	PUNCT
ma-25	195	16	ρ[p	ρ[p	NOUN
ma-25	195	17	,	,	PUNCT
ma-25	195	18	q	q	X
ma-25	195	19	]	]	X
ma-25	195	20	(	(	PUNCT
ma-25	195	21	f	f	PROPN
ma-25	195	22	)	)	PUNCT
ma-25	196	1	+	+	CCONJ
ma-25	196	2	ε	ε	PROPN
ma-25	196	3	3	3	NUM
ma-25	196	4	)	)	PUNCT
ma-25	196	5	logq	logq	VERB
ma-25	196	6	r	r	NOUN
ma-25	196	7	}	}	PUNCT
ma-25	196	8	}	}	PUNCT
ma-25	196	9	,	,	PUNCT
ma-25	196	10	(	(	PUNCT
ma-25	196	11	r	r	NOUN
ma-25	196	12	/∈	/∈	SYM
ma-25	196	13	e3	e3	NOUN
ma-25	196	14	)	)	PUNCT
ma-25	196	15	,	,	PUNCT
ma-25	196	16	(	(	PUNCT
ma-25	196	17	2.8	2.8	NUM
ma-25	196	18	)	)	PUNCT
ma-25	196	19	|u2	|u2	NOUN
ma-25	197	1	(	(	PUNCT
ma-25	197	2	z)|	z)|	X
ma-25	197	3	≥	≥	NOUN
ma-25	197	4	exp	exp	NOUN
ma-25	197	5	{	{	PUNCT
ma-25	197	6	−	−	PUNCT
ma-25	197	7	expp	expp	ADJ
ma-25	197	8	{	{	PUNCT
ma-25	197	9	(	(	PUNCT
ma-25	197	10	ρ[p	ρ[p	NOUN
ma-25	197	11	,	,	PUNCT
ma-25	197	12	q	q	X
ma-25	197	13	]	]	X
ma-25	197	14	(	(	PUNCT
ma-25	197	15	u2	u2	NOUN
ma-25	197	16	)	)	PUNCT
ma-25	197	17	+	+	CCONJ
ma-25	197	18	ε	ε	PROPN
ma-25	197	19	3	3	NUM
ma-25	197	20	)	)	PUNCT
ma-25	197	21	logq	logq	VERB
ma-25	197	22	r	r	NOUN
ma-25	197	23	}	}	PUNCT
ma-25	197	24	}	}	PUNCT
ma-25	197	25	≥	≥	NOUN
ma-25	197	26	exp	exp	NOUN
ma-25	197	27	{	{	PUNCT
ma-25	197	28	−	−	PUNCT
ma-25	197	29	expp	expp	ADJ
ma-25	197	30	{	{	PUNCT
ma-25	197	31	(	(	PUNCT
ma-25	197	32	ρ[p	ρ[p	NOUN
ma-25	197	33	,	,	PUNCT
ma-25	197	34	q	q	X
ma-25	197	35	]	]	X
ma-25	197	36	(	(	PUNCT
ma-25	197	37	f	f	PROPN
ma-25	197	38	)	)	PUNCT
ma-25	198	1	+	+	CCONJ
ma-25	198	2	ε	ε	PROPN
ma-25	198	3	3	3	NUM
ma-25	198	4	)	)	PUNCT
ma-25	198	5	logq	logq	VERB
ma-25	198	6	r	r	NOUN
ma-25	198	7	}	}	PUNCT
ma-25	198	8	}	}	PUNCT
ma-25	198	9	,	,	PUNCT
ma-25	198	10	(	(	PUNCT
ma-25	198	11	r	r	NOUN
ma-25	198	12	/∈	/∈	SYM
ma-25	198	13	e3	e3	NOUN
ma-25	198	14	)	)	PUNCT
ma-25	198	15	.	.	PUNCT
ma-25	199	1	(	(	PUNCT
ma-25	199	2	2.9	2.9	NUM
ma-25	199	3	)	)	PUNCT
ma-25	199	4	then	then	ADV
ma-25	199	5	,	,	PUNCT
ma-25	199	6	by	by	ADP
ma-25	199	7	using	use	VERB
ma-25	199	8	(	(	PUNCT
ma-25	199	9	2.5	2.5	NUM
ma-25	199	10	)	)	PUNCT
ma-25	199	11	,	,	PUNCT
ma-25	199	12	(	(	PUNCT
ma-25	199	13	2.7	2.7	NUM
ma-25	199	14	)	)	PUNCT
ma-25	199	15	and	and	CCONJ
ma-25	199	16	(	(	PUNCT
ma-25	199	17	2.9	2.9	NUM
ma-25	199	18	)	)	PUNCT
ma-25	199	19	,	,	PUNCT
ma-25	199	20	we	we	PRON
ma-25	199	21	obtain	obtain	VERB
ma-25	199	22	for	for	ADP
ma-25	199	23	sufficiently	sufficiently	ADV
ma-25	199	24	large	large	ADJ
ma-25	199	25	r	r	NOUN
ma-25	199	26	/∈	/∈	PUNCT
ma-25	199	27	e3	e3	NOUN
ma-25	199	28	and	and	CCONJ
ma-25	199	29	any	any	DET
ma-25	199	30	given	give	VERB
ma-25	199	31	ε	ε	PROPN
ma-25	199	32	>	>	X
ma-25	199	33	0	0	NUM
ma-25	200	1	|f	|f	PROPN
ma-25	201	1	(	(	PUNCT
ma-25	201	2	z)|	z)|	NOUN
ma-25	201	3	=	=	SYM
ma-25	201	4	|u1	|u1	PROPN
ma-25	201	5	(	(	PUNCT
ma-25	201	6	z)|	z)|	PROPN
ma-25	201	7	∣∣ev(z)∣∣	∣∣ev(z)∣∣	ADJ
ma-25	201	8	|u2	|u2	NOUN
ma-25	201	9	(	(	PUNCT
ma-25	201	10	z)|	z)|	NOUN
ma-25	201	11	≤	≤	NUM
ma-25	201	12	expp+1	expp+1	PROPN
ma-25	201	13	{	{	PUNCT
ma-25	201	14	(	(	PUNCT
ma-25	201	15	ρ[p	ρ[p	NOUN
ma-25	201	16	,	,	PUNCT
ma-25	201	17	q	q	X
ma-25	201	18	]	]	X
ma-25	201	19	(	(	PUNCT
ma-25	201	20	f	f	PROPN
ma-25	201	21	)	)	PUNCT
ma-25	202	1	+	+	CCONJ
ma-25	202	2	ε	ε	PROPN
ma-25	202	3	3	3	NUM
ma-25	202	4	)	)	PUNCT
ma-25	202	5	logq	logq	VERB
ma-25	202	6	r	r	NOUN
ma-25	202	7	}	}	PUNCT
ma-25	202	8	expp+1	expp+1	NOUN
ma-25	202	9	{	{	PUNCT
ma-25	202	10	(	(	PUNCT
ma-25	202	11	ρ[p	ρ[p	NOUN
ma-25	202	12	,	,	PUNCT
ma-25	202	13	q	q	X
ma-25	202	14	]	]	X
ma-25	202	15	(	(	PUNCT
ma-25	202	16	f	f	PROPN
ma-25	202	17	)	)	PUNCT
ma-25	203	1	+	+	CCONJ
ma-25	203	2	ε	ε	PROPN
ma-25	203	3	3	3	NUM
ma-25	203	4	)	)	PUNCT
ma-25	203	5	logq	logq	VERB
ma-25	203	6	r	r	NOUN
ma-25	203	7	}	}	PUNCT
ma-25	203	8	exp	exp	NOUN
ma-25	203	9	{	{	PUNCT
ma-25	203	10	−	−	PUNCT
ma-25	203	11	expp	expp	ADJ
ma-25	203	12	{	{	PUNCT
ma-25	203	13	(	(	PUNCT
ma-25	203	14	ρ[p	ρ[p	NOUN
ma-25	203	15	,	,	PUNCT
ma-25	203	16	q	q	X
ma-25	203	17	]	]	X
ma-25	203	18	(	(	PUNCT
ma-25	203	19	f	f	PROPN
ma-25	203	20	)	)	PUNCT
ma-25	204	1	+	+	CCONJ
ma-25	204	2	ε	ε	PROPN
ma-25	204	3	3	3	NUM
ma-25	204	4	)	)	PUNCT
ma-25	204	5	logq	logq	VERB
ma-25	204	6	r	r	NOUN
ma-25	204	7	}	}	PUNCT
ma-25	204	8	}	}	PUNCT
ma-25	204	9	≤	≤	NUM
ma-25	204	10	expp+1	expp+1	NOUN
ma-25	204	11	{	{	PUNCT
ma-25	204	12	(	(	PUNCT
ma-25	204	13	ρ[p	ρ[p	NOUN
ma-25	204	14	,	,	PUNCT
ma-25	204	15	q	q	X
ma-25	204	16	]	]	X
ma-25	204	17	(	(	PUNCT
ma-25	204	18	f	f	PROPN
ma-25	204	19	)	)	PUNCT
ma-25	205	1	+	+	CCONJ
ma-25	205	2	ε	ε	PROPN
ma-25	205	3	)	)	PUNCT
ma-25	205	4	logq	logq	VERB
ma-25	205	5	r	r	NOUN
ma-25	205	6	}	}	PUNCT
ma-25	205	7	.	.	PUNCT
ma-25	206	1	(	(	PUNCT
ma-25	206	2	2.10	2.10	NUM
ma-25	206	3	)	)	PUNCT
ma-25	206	4	on	on	ADP
ma-25	206	5	the	the	DET
ma-25	206	6	other	other	ADJ
ma-25	206	7	hand	hand	NOUN
ma-25	206	8	,	,	PUNCT
ma-25	206	9	we	we	PRON
ma-25	206	10	have	have	VERB
ma-25	206	11	ρ[p−1,q	ρ[p−1,q	NOUN
ma-25	206	12	]	]	X
ma-25	206	13	(	(	PUNCT
ma-25	206	14	v	v	NOUN
ma-25	206	15	)	)	PUNCT
ma-25	206	16	=	=	PUNCT
ma-25	206	17	ρ[p	ρ[p	PROPN
ma-25	206	18	,	,	PUNCT
ma-25	206	19	q	q	X
ma-25	206	20	]	]	X
ma-25	206	21	(	(	PUNCT
ma-25	206	22	ev(z	ev(z	X
ma-25	206	23	)	)	PUNCT
ma-25	206	24	)	)	PUNCT
ma-25	207	1	≤	≤	NOUN
ma-25	207	2	ρ[p	ρ[p	NOUN
ma-25	207	3	,	,	PUNCT
ma-25	207	4	q	q	X
ma-25	207	5	]	]	X
ma-25	207	6	(	(	PUNCT
ma-25	207	7	f	f	PROPN
ma-25	207	8	)	)	PUNCT
ma-25	207	9	and	and	CCONJ
ma-25	207	10	∣∣ev(z)∣∣	∣∣ev(z)∣∣	X
ma-25	207	11	≥	≥	NOUN
ma-25	207	12	e−|v(z)|	e−|v(z)|	VERB
ma-25	207	13	.	.	PUNCT
ma-25	208	1	makinguse	makinguse	NOUN
ma-25	208	2	of	of	ADP
ma-25	208	3	the	the	DET
ma-25	208	4	definition	definition	NOUN
ma-25	208	5	of	of	ADP
ma-25	208	6	[	[	X
ma-25	208	7	p	p	X
ma-25	208	8	,	,	PUNCT
ma-25	208	9	q]-order	q]-order	NOUN
ma-25	208	10	,	,	PUNCT
ma-25	208	11	we	we	PRON
ma-25	208	12	obtain	obtain	VERB
ma-25	208	13	|v	|v	X
ma-25	208	14	(	(	PUNCT
ma-25	208	15	z)|	z)|	ADP
ma-25	208	16	≤	≤	PROPN
ma-25	208	17	m(r	m(r	PROPN
ma-25	208	18	,	,	PUNCT
ma-25	208	19	v	v	NOUN
ma-25	208	20	)	)	PUNCT
ma-25	208	21	≤	≤	X
ma-25	208	22	expp	expp	ADJ
ma-25	208	23	{	{	PUNCT
ma-25	208	24	(	(	PUNCT
ma-25	208	25	ρ(p−1,q	ρ(p−1,q	NOUN
ma-25	208	26	)	)	PUNCT
ma-25	208	27	(	(	PUNCT
ma-25	208	28	v	v	NOUN
ma-25	208	29	)	)	PUNCT
ma-25	208	30	+	+	CCONJ
ma-25	208	31	ε	ε	PROPN
ma-25	208	32	3	3	NUM
ma-25	208	33	)	)	PUNCT
ma-25	208	34	logq	logq	VERB
ma-25	208	35	r	r	NOUN
ma-25	208	36	}	}	PUNCT
ma-25	208	37	≤	≤	X
ma-25	208	38	expp	expp	ADJ
ma-25	208	39	{	{	PUNCT
ma-25	208	40	(	(	PUNCT
ma-25	208	41	ρ[p	ρ[p	NOUN
ma-25	208	42	,	,	PUNCT
ma-25	208	43	q	q	X
ma-25	208	44	]	]	X
ma-25	208	45	(	(	PUNCT
ma-25	208	46	f	f	PROPN
ma-25	208	47	)	)	PUNCT
ma-25	209	1	+	+	CCONJ
ma-25	209	2	ε	ε	PROPN
ma-25	209	3	3	3	NUM
ma-25	209	4	)	)	PUNCT
ma-25	209	5	logq	logq	VERB
ma-25	209	6	r	r	NOUN
ma-25	209	7	}	}	PUNCT
ma-25	209	8	.	.	PUNCT
ma-25	210	1	then	then	ADV
ma-25	210	2	,	,	PUNCT
ma-25	210	3	for	for	ADP
ma-25	210	4	sufficiently	sufficiently	ADV
ma-25	210	5	large	large	ADJ
ma-25	210	6	r	r	NOUN
ma-25	210	7	and	and	CCONJ
ma-25	210	8	any	any	DET
ma-25	210	9	given	give	VERB
ma-25	210	10	ε	ε	PROPN
ma-25	210	11	>	>	X
ma-25	210	12	0	0	PROPN
ma-25	210	13	,	,	PUNCT
ma-25	210	14	we	we	PRON
ma-25	210	15	have∣∣∣ev(z)∣∣∣	have∣∣∣ev(z)∣∣∣	VERB
ma-25	210	16	≥	≥	NUM
ma-25	210	17	e−|v(z)|	e−|v(z)|	PROPN
ma-25	210	18	≥	≥	PROPN
ma-25	210	19	exp	exp	NOUN
ma-25	210	20	{	{	PUNCT
ma-25	210	21	−	−	PUNCT
ma-25	210	22	expp	expp	ADJ
ma-25	210	23	{	{	PUNCT
ma-25	210	24	(	(	PUNCT
ma-25	210	25	ρ[p	ρ[p	NOUN
ma-25	210	26	,	,	PUNCT
ma-25	210	27	q	q	X
ma-25	210	28	]	]	X
ma-25	210	29	(	(	PUNCT
ma-25	210	30	f	f	PROPN
ma-25	210	31	)	)	PUNCT
ma-25	211	1	+	+	CCONJ
ma-25	211	2	ε	ε	PROPN
ma-25	211	3	3	3	NUM
ma-25	211	4	)	)	PUNCT
ma-25	211	5	logq	logq	VERB
ma-25	211	6	r	r	NOUN
ma-25	211	7	}	}	PUNCT
ma-25	211	8	}	}	PUNCT
ma-25	211	9	.	.	PUNCT
ma-25	212	1	(	(	PUNCT
ma-25	212	2	2.11	2.11	NUM
ma-25	212	3	)	)	PUNCT
ma-25	212	4	by	by	ADP
ma-25	212	5	(	(	PUNCT
ma-25	212	6	2.6	2.6	NUM
ma-25	212	7	)	)	PUNCT
ma-25	212	8	,	,	PUNCT
ma-25	212	9	(	(	PUNCT
ma-25	212	10	2.8	2.8	NUM
ma-25	212	11	)	)	PUNCT
ma-25	212	12	and	and	CCONJ
ma-25	212	13	(	(	PUNCT
ma-25	212	14	2.11	2.11	NUM
ma-25	212	15	)	)	PUNCT
ma-25	212	16	,	,	PUNCT
ma-25	212	17	we	we	PRON
ma-25	212	18	can	can	AUX
ma-25	212	19	easily	easily	ADV
ma-25	212	20	obtain	obtain	VERB
ma-25	212	21	|f	|f	PROPN
ma-25	213	1	(	(	PUNCT
ma-25	213	2	z)|	z)|	NOUN
ma-25	213	3	=	=	SYM
ma-25	213	4	|u1	|u1	PROPN
ma-25	213	5	(	(	PUNCT
ma-25	213	6	z)|	z)|	PROPN
ma-25	213	7	∣∣ev(z)∣∣	∣∣ev(z)∣∣	ADJ
ma-25	213	8	|u2	|u2	NOUN
ma-25	213	9	(	(	PUNCT
ma-25	213	10	z)|	z)|	X
ma-25	213	11	≥	≥	NOUN
ma-25	213	12	exp	exp	NOUN
ma-25	213	13	{	{	PUNCT
ma-25	213	14	−	−	PUNCT
ma-25	213	15	expp	expp	ADJ
ma-25	213	16	{	{	PUNCT
ma-25	213	17	(	(	PUNCT
ma-25	213	18	ρ[p	ρ[p	NOUN
ma-25	213	19	,	,	PUNCT
ma-25	213	20	q	q	X
ma-25	213	21	]	]	X
ma-25	213	22	(	(	PUNCT
ma-25	213	23	f	f	PROPN
ma-25	213	24	)	)	PUNCT
ma-25	214	1	+	+	CCONJ
ma-25	214	2	ε	ε	PROPN
ma-25	214	3	3	3	NUM
ma-25	214	4	)	)	PUNCT
ma-25	214	5	logq	logq	VERB
ma-25	214	6	r	r	NOUN
ma-25	214	7	}	}	PUNCT
ma-25	214	8	}	}	PUNCT
ma-25	214	9	exp	exp	NOUN
ma-25	214	10	{	{	PUNCT
ma-25	214	11	−	−	PUNCT
ma-25	214	12	expp	expp	ADJ
ma-25	214	13	{	{	PUNCT
ma-25	214	14	(	(	PUNCT
ma-25	214	15	ρ[p	ρ[p	NOUN
ma-25	214	16	,	,	PUNCT
ma-25	214	17	q	q	X
ma-25	214	18	]	]	X
ma-25	214	19	(	(	PUNCT
ma-25	214	20	f	f	PROPN
ma-25	214	21	)	)	PUNCT
ma-25	215	1	+	+	CCONJ
ma-25	215	2	ε	ε	PROPN
ma-25	215	3	3	3	NUM
ma-25	215	4	)	)	PUNCT
ma-25	215	5	logq	logq	VERB
ma-25	215	6	r	r	NOUN
ma-25	215	7	}	}	PUNCT
ma-25	215	8	}	}	PUNCT
ma-25	215	9	expp+1	expp+1	NOUN
ma-25	215	10	{	{	PUNCT
ma-25	215	11	(	(	PUNCT
ma-25	215	12	ρ[p	ρ[p	NOUN
ma-25	215	13	,	,	PUNCT
ma-25	215	14	q	q	X
ma-25	215	15	]	]	X
ma-25	215	16	(	(	PUNCT
ma-25	215	17	f	f	PROPN
ma-25	215	18	)	)	PUNCT
ma-25	216	1	+	+	CCONJ
ma-25	216	2	ε	ε	PROPN
ma-25	216	3	3	3	NUM
ma-25	216	4	)	)	PUNCT
ma-25	216	5	logq	logq	VERB
ma-25	216	6	r	r	NOUN
ma-25	216	7	}	}	PUNCT
ma-25	216	8	.	.	PUNCT
ma-25	217	1	=	=	PRON
ma-25	217	2	exp	exp	NOUN
ma-25	217	3	{	{	PUNCT
ma-25	217	4	−3	−3	NOUN
ma-25	217	5	expp	expp	ADJ
ma-25	217	6	{	{	PUNCT
ma-25	217	7	(	(	PUNCT
ma-25	217	8	ρ[p	ρ[p	NOUN
ma-25	217	9	,	,	PUNCT
ma-25	217	10	q	q	X
ma-25	217	11	]	]	X
ma-25	217	12	(	(	PUNCT
ma-25	217	13	f	f	PROPN
ma-25	217	14	)	)	PUNCT
ma-25	217	15	+	+	CCONJ
ma-25	217	16	ε	ε	PROPN
ma-25	217	17	3	3	NUM
ma-25	217	18	)	)	PUNCT
ma-25	217	19	logq	logq	VERB
ma-25	217	20	r	r	NOUN
ma-25	217	21	}	}	PUNCT
ma-25	217	22	}	}	PUNCT
ma-25	217	23	≥	≥	NOUN
ma-25	217	24	exp	exp	NOUN
ma-25	217	25	{	{	PUNCT
ma-25	217	26	−	−	PUNCT
ma-25	217	27	expp	expp	ADJ
ma-25	217	28	{	{	PUNCT
ma-25	217	29	(	(	PUNCT
ma-25	217	30	ρ[p	ρ[p	NOUN
ma-25	217	31	,	,	PUNCT
ma-25	217	32	q	q	X
ma-25	217	33	]	]	X
ma-25	217	34	(	(	PUNCT
ma-25	217	35	f	f	PROPN
ma-25	217	36	)	)	PUNCT
ma-25	218	1	+	+	CCONJ
ma-25	218	2	ε	ε	PROPN
ma-25	218	3	)	)	PUNCT
ma-25	218	4	logq	logq	VERB
ma-25	218	5	r	r	NOUN
ma-25	218	6	}	}	PUNCT
ma-25	218	7	}	}	PUNCT
ma-25	218	8	.	.	PUNCT
ma-25	219	1	thus	thus	ADV
ma-25	219	2	,	,	PUNCT
ma-25	219	3	we	we	PRON
ma-25	219	4	complete	complete	VERB
ma-25	219	5	the	the	DET
ma-25	219	6	proof	proof	NOUN
ma-25	219	7	of	of	ADP
ma-25	219	8	lemma	lemma	PROPN
ma-25	219	9	2.3	2.3	NUM
ma-25	219	10	.	.	PUNCT
ma-25	220	1	lemma	lemma	PROPN
ma-25	220	2	2.4	2.4	NUM
ma-25	220	3	under	under	ADP
ma-25	220	4	the	the	DET
ma-25	220	5	assumptions	assumption	NOUN
ma-25	220	6	of	of	ADP
ma-25	220	7	theorem	theorem	ADJ
ma-25	220	8	1.1	1.1	NUM
ma-25	220	9	or	or	CCONJ
ma-25	220	10	theorem	theorem	VERB
ma-25	220	11	1.2	1.2	NUM
ma-25	220	12	,	,	PUNCT
ma-25	220	13	we	we	PRON
ma-25	220	14	have	have	VERB
ma-25	220	15	ρ[p	ρ[p	NOUN
ma-25	220	16	,	,	PUNCT
ma-25	220	17	q	q	X
ma-25	220	18	]	]	X
ma-25	220	19	(	(	PUNCT
ma-25	220	20	as	as	ADP
ma-25	220	21	)	)	PUNCT
ma-25	220	22	=	=	SYM
ma-25	220	23	β	β	X
ma-25	220	24	≥	≥	PROPN
ma-25	220	25	σ	σ	PROPN
ma-25	220	26	.	.	PUNCT
ma-25	220	27	eur	eur	PROPN
ma-25	220	28	.	.	PUNCT
ma-25	221	1	j.	j.	PROPN
ma-25	221	2	math	math	PROPN
ma-25	221	3	.	.	PUNCT
ma-25	222	1	anal	anal	ADJ
ma-25	222	2	.	.	PUNCT
ma-25	223	1	1	1	NUM
ma-25	223	2	(	(	PUNCT
ma-25	223	3	2021	2021	NUM
ma-25	223	4	)	)	PUNCT
ma-25	223	5	93	93	NUM
ma-25	223	6	proof	proof	NOUN
ma-25	223	7	.	.	PUNCT
ma-25	224	1	assume	assume	VERB
ma-25	224	2	that	that	SCONJ
ma-25	224	3	ρ[p	ρ[p	NOUN
ma-25	224	4	,	,	PUNCT
ma-25	224	5	q	q	X
ma-25	224	6	]	]	X
ma-25	224	7	(	(	PUNCT
ma-25	224	8	as	as	ADP
ma-25	224	9	)	)	PUNCT
ma-25	224	10	=	=	PUNCT
ma-25	224	11	β	β	X
ma-25	224	12	<	<	X
ma-25	224	13	σ	σ	PROPN
ma-25	224	14	.	.	PUNCT
ma-25	225	1	according	accord	VERB
ma-25	225	2	to	to	ADP
ma-25	225	3	the	the	DET
ma-25	225	4	hypotheses	hypothesis	NOUN
ma-25	225	5	of	of	ADP
ma-25	225	6	theorems	theorem	NOUN
ma-25	225	7	1.1	1.1	NUM
ma-25	225	8	or	or	CCONJ
ma-25	225	9	1.2	1.2	NUM
ma-25	225	10	,	,	PUNCT
ma-25	225	11	thereexists	thereexist	NOUN
ma-25	225	12	a	a	DET
ma-25	225	13	positive	positive	ADJ
ma-25	225	14	constant	constant	ADJ
ma-25	225	15	σ	σ	NOUN
ma-25	225	16	>	>	X
ma-25	225	17	0	0	NUM
ma-25	225	18	such	such	ADJ
ma-25	225	19	that	that	PRON
ma-25	225	20	for	for	ADP
ma-25	225	21	sufficiently	sufficiently	ADV
ma-25	225	22	small	small	ADJ
ma-25	225	23	ε	ε	PROPN
ma-25	225	24	>	>	X
ma-25	225	25	0	0	PROPN
ma-25	225	26	,	,	PUNCT
ma-25	225	27	we	we	PRON
ma-25	225	28	have	have	VERB
ma-25	225	29	|as	|as	NUM
ma-25	225	30	(	(	PUNCT
ma-25	225	31	z	z	NOUN
ma-25	225	32	)	)	PUNCT
ma-25	225	33	|	|	ADV
ma-25	225	34	≥	≥	NOUN
ma-25	225	35	expp+1	expp+1	PROPN
ma-25	225	36	{	{	PUNCT
ma-25	225	37	(	(	PUNCT
ma-25	225	38	σ	σ	PROPN
ma-25	225	39	−	−	PROPN
ma-25	225	40	ε	ε	PROPN
ma-25	225	41	)	)	PUNCT
ma-25	225	42	logq	logq	VERB
ma-25	225	43	r	r	NOUN
ma-25	225	44	}	}	PUNCT
ma-25	225	45	(	(	PUNCT
ma-25	225	46	2.12	2.12	NUM
ma-25	225	47	)	)	PUNCT
ma-25	225	48	as	as	ADP
ma-25	225	49	|z	|z	PROPN
ma-25	225	50	|	|	ADV
ma-25	225	51	=	=	SYM
ma-25	225	52	r	r	NOUN
ma-25	225	53	∈	∈	PROPN
ma-25	225	54	h	h	NOUN
ma-25	225	55	,	,	PUNCT
ma-25	225	56	r	r	NOUN
ma-25	225	57	→	→	SYM
ma-25	225	58	+	+	NOUN
ma-25	225	59	∞	∞	PROPN
ma-25	225	60	,	,	PUNCT
ma-25	225	61	where	where	SCONJ
ma-25	225	62	h	h	PROPN
ma-25	225	63	⊂	⊂	PROPN
ma-25	225	64	(	(	PUNCT
ma-25	225	65	1,+∞	1,+∞	NUM
ma-25	225	66	)	)	PUNCT
ma-25	225	67	is	be	AUX
ma-25	225	68	a	a	DET
ma-25	225	69	set	set	NOUN
ma-25	225	70	with	with	ADP
ma-25	225	71	a	a	DET
ma-25	225	72	positive	positive	ADJ
ma-25	225	73	upper	upper	ADJ
ma-25	225	74	logarithmic	logarithmic	ADJ
ma-25	225	75	density	density	NOUN
ma-25	225	76	(	(	PUNCT
ma-25	225	77	by	by	ADP
ma-25	225	78	proposition	proposition	NOUN
ma-25	225	79	2.1	2.1	NUM
ma-25	225	80	,	,	PUNCT
ma-25	225	81	we	we	PRON
ma-25	225	82	have	have	VERB
ma-25	225	83	ml	ml	NOUN
ma-25	225	84	(	(	PUNCT
ma-25	225	85	h	h	NOUN
ma-25	225	86	)	)	PUNCT
ma-25	225	87	=	=	PUNCT
ma-25	226	1	+	+	NOUN
ma-25	226	2	∞	∞	NOUN
ma-25	226	3	)	)	PUNCT
ma-25	226	4	.	.	PUNCT
ma-25	227	1	by	by	ADP
ma-25	227	2	lemma	lemma	PROPN
ma-25	227	3	2.3	2.3	NUM
ma-25	227	4	,	,	PUNCT
ma-25	227	5	we	we	PRON
ma-25	227	6	can	can	AUX
ma-25	227	7	find	find	VERB
ma-25	227	8	a	a	DET
ma-25	227	9	set	set	NOUN
ma-25	227	10	e3	e3	NOUN
ma-25	227	11	⊂	⊂	X
ma-25	227	12	(	(	PUNCT
ma-25	227	13	1,+∞	1,+∞	NUM
ma-25	227	14	)	)	PUNCT
ma-25	227	15	thathas	thathas	PROPN
ma-25	227	16	finite	finite	PROPN
ma-25	227	17	linear	linear	PROPN
ma-25	227	18	measure	measure	NOUN
ma-25	227	19	(	(	PUNCT
ma-25	227	20	and	and	CCONJ
ma-25	227	21	so	so	ADV
ma-25	227	22	of	of	ADP
ma-25	227	23	finite	finite	ADJ
ma-25	227	24	logarithmic	logarithmic	ADJ
ma-25	227	25	measure	measure	NOUN
ma-25	227	26	)	)	PUNCT
ma-25	227	27	such	such	ADJ
ma-25	227	28	that	that	SCONJ
ma-25	227	29	when	when	SCONJ
ma-25	227	30	|z	|z	PROPN
ma-25	227	31	|	|	ADV
ma-25	227	32	=	=	NOUN
ma-25	227	33	r	r	NOUN
ma-25	227	34	/∈	/∈	NOUN
ma-25	227	35	e3	e3	NOUN
ma-25	227	36	,	,	PUNCT
ma-25	227	37	wehave	wehave	NOUN
ma-25	227	38	for	for	ADP
ma-25	227	39	any	any	DET
ma-25	227	40	given	give	VERB
ma-25	227	41	ε	ε	PROPN
ma-25	227	42	(	(	PUNCT
ma-25	227	43	0	0	PUNCT
ma-25	227	44	<	<	X
ma-25	227	45	2ε	2ε	PROPN
ma-25	227	46	<	<	X
ma-25	227	47	σ	σ	X
ma-25	227	48	−	−	PROPN
ma-25	227	49	β	β	X
ma-25	227	50	)	)	PUNCT
ma-25	227	51	|as	|as	PROPN
ma-25	227	52	(	(	PUNCT
ma-25	227	53	z	z	NOUN
ma-25	227	54	)	)	PUNCT
ma-25	227	55	|	|	ADV
ma-25	227	56	≤	≤	NUM
ma-25	227	57	expp+1	expp+1	NOUN
ma-25	227	58	{	{	PUNCT
ma-25	227	59	(	(	PUNCT
ma-25	227	60	β	β	X
ma-25	227	61	+	+	CCONJ
ma-25	227	62	ε	ε	PROPN
ma-25	227	63	)	)	PUNCT
ma-25	227	64	logq	logq	VERB
ma-25	227	65	r	r	NOUN
ma-25	227	66	}	}	PUNCT
ma-25	227	67	.	.	PUNCT
ma-25	228	1	(	(	PUNCT
ma-25	228	2	2.13	2.13	NUM
ma-25	228	3	)	)	PUNCT
ma-25	228	4	by	by	ADP
ma-25	228	5	(	(	PUNCT
ma-25	228	6	2.12	2.12	NUM
ma-25	228	7	)	)	PUNCT
ma-25	228	8	and	and	CCONJ
ma-25	228	9	(	(	PUNCT
ma-25	228	10	2.13	2.13	NUM
ma-25	228	11	)	)	PUNCT
ma-25	228	12	,	,	PUNCT
ma-25	228	13	we	we	PRON
ma-25	228	14	obtain	obtain	VERB
ma-25	228	15	for	for	ADP
ma-25	228	16	|z	|z	PROPN
ma-25	229	1	|	|	ADV
ma-25	229	2	=	=	SYM
ma-25	229	3	r	r	NOUN
ma-25	229	4	∈	∈	NOUN
ma-25	229	5	h	h	NOUN
ma-25	229	6	r	r	NOUN
ma-25	229	7	e3	e3	NOUN
ma-25	229	8	,	,	PUNCT
ma-25	229	9	r	r	NOUN
ma-25	229	10	→	→	SYM
ma-25	229	11	+	+	ADJ
ma-25	229	12	∞	∞	PROPN
ma-25	229	13	expp+1	expp+1	NOUN
ma-25	229	14	{	{	PUNCT
ma-25	229	15	(	(	PUNCT
ma-25	229	16	σ	σ	PROPN
ma-25	229	17	−	−	PROPN
ma-25	229	18	ε	ε	PROPN
ma-25	229	19	)	)	PUNCT
ma-25	229	20	logq	logq	VERB
ma-25	229	21	r	r	NOUN
ma-25	229	22	}	}	PUNCT
ma-25	229	23	≤	≤	X
ma-25	229	24	|as	|as	ADP
ma-25	229	25	(	(	PUNCT
ma-25	229	26	z	z	NOUN
ma-25	229	27	)	)	PUNCT
ma-25	229	28	|	|	ADV
ma-25	229	29	≤	≤	NUM
ma-25	229	30	expp+1	expp+1	NOUN
ma-25	229	31	{	{	PUNCT
ma-25	229	32	(	(	PUNCT
ma-25	229	33	β	β	X
ma-25	229	34	+	+	CCONJ
ma-25	229	35	ε	ε	PROPN
ma-25	229	36	)	)	PUNCT
ma-25	229	37	logq	logq	VERB
ma-25	229	38	r	r	NOUN
ma-25	229	39	}	}	PUNCT
ma-25	229	40	and	and	CCONJ
ma-25	229	41	by	by	ADP
ma-25	229	42	ε	ε	PROPN
ma-25	229	43	(	(	PUNCT
ma-25	229	44	0	0	PUNCT
ma-25	229	45	<	<	X
ma-25	229	46	2ε	2ε	PROPN
ma-25	229	47	<	<	X
ma-25	229	48	σ	σ	X
ma-25	230	1	−	−	PROPN
ma-25	230	2	β	β	X
ma-25	230	3	)	)	PUNCT
ma-25	230	4	this	this	PRON
ma-25	230	5	is	be	AUX
ma-25	230	6	a	a	DET
ma-25	230	7	contradiction	contradiction	NOUN
ma-25	230	8	.	.	PUNCT
ma-25	231	1	hence	hence	ADV
ma-25	231	2	ρ[p	ρ[p	NUM
ma-25	231	3	,	,	PUNCT
ma-25	231	4	q	q	X
ma-25	231	5	]	]	X
ma-25	231	6	(	(	PUNCT
ma-25	231	7	as	as	ADP
ma-25	231	8	)	)	PUNCT
ma-25	231	9	=	=	SYM
ma-25	231	10	β	β	X
ma-25	231	11	≥	≥	PROPN
ma-25	231	12	σ	σ	PROPN
ma-25	231	13	.	.	PUNCT
ma-25	232	1	lemma	lemma	PROPN
ma-25	232	2	2.5	2.5	PROPN
ma-25	232	3	(	(	PUNCT
ma-25	232	4	wiman	wiman	PROPN
ma-25	232	5	-	-	PUNCT
ma-25	232	6	valiron	valiron	PROPN
ma-25	232	7	,	,	PUNCT
ma-25	232	8	[	[	X
ma-25	232	9	10	10	NUM
ma-25	232	10	]	]	PUNCT
ma-25	232	11	,	,	PUNCT
ma-25	232	12	[	[	X
ma-25	232	13	22	22	NUM
ma-25	232	14	]	]	PUNCT
ma-25	232	15	)	)	PUNCT
ma-25	232	16	let	let	VERB
ma-25	232	17	f	f	PRON
ma-25	232	18	be	be	AUX
ma-25	232	19	a	a	DET
ma-25	232	20	transcendental	transcendental	ADJ
ma-25	232	21	entire	entire	ADJ
ma-25	232	22	function	function	NOUN
ma-25	232	23	,	,	PUNCT
ma-25	232	24	and	and	CCONJ
ma-25	232	25	let	let	VERB
ma-25	232	26	z	z	PRON
ma-25	232	27	be	be	AUX
ma-25	232	28	a	a	DET
ma-25	232	29	point	point	NOUN
ma-25	232	30	with	with	ADP
ma-25	232	31	|z	|z	PROPN
ma-25	233	1	|	|	ADV
ma-25	233	2	=	=	NOUN
ma-25	233	3	r	r	NOUN
ma-25	233	4	at	at	ADP
ma-25	233	5	which	which	PRON
ma-25	233	6	|f	|f	PROPN
ma-25	233	7	(	(	PUNCT
ma-25	233	8	z)|	z)|	NOUN
ma-25	233	9	=	=	PRON
ma-25	233	10	m	m	PROPN
ma-25	233	11	(	(	PUNCT
ma-25	233	12	r	r	NOUN
ma-25	233	13	,	,	PUNCT
ma-25	233	14	f	f	PROPN
ma-25	233	15	)	)	PUNCT
ma-25	233	16	.	.	PUNCT
ma-25	234	1	then	then	ADV
ma-25	234	2	the	the	DET
ma-25	234	3	estimation	estimation	NOUN
ma-25	234	4	f	f	X
ma-25	234	5	(	(	PUNCT
ma-25	234	6	j	j	PROPN
ma-25	234	7	)	)	PUNCT
ma-25	234	8	(	(	PUNCT
ma-25	234	9	z	z	X
ma-25	234	10	)	)	PUNCT
ma-25	234	11	f	f	NOUN
ma-25	234	12	(	(	PUNCT
ma-25	234	13	z	z	NOUN
ma-25	234	14	)	)	PUNCT
ma-25	234	15	=	=	SYM
ma-25	234	16	(	(	PUNCT
ma-25	234	17	νf	νf	X
ma-25	234	18	(	(	PUNCT
ma-25	234	19	r	r	NOUN
ma-25	234	20	)	)	PUNCT
ma-25	234	21	z	z	NOUN
ma-25	234	22	)	)	PUNCT
ma-25	234	23	j	j	NOUN
ma-25	234	24	(	(	PUNCT
ma-25	234	25	1	1	NUM
ma-25	234	26	+	+	NUM
ma-25	234	27	o	o	NOUN
ma-25	234	28	(	(	PUNCT
ma-25	234	29	1	1	NUM
ma-25	234	30	)	)	PUNCT
ma-25	234	31	)	)	PUNCT
ma-25	234	32	(	(	PUNCT
ma-25	234	33	j	j	PROPN
ma-25	234	34	≥	≥	NUM
ma-25	234	35	1	1	NUM
ma-25	234	36	is	be	AUX
ma-25	234	37	an	an	DET
ma-25	234	38	integer	integer	NOUN
ma-25	234	39	)	)	PUNCT
ma-25	234	40	holds	hold	VERB
ma-25	234	41	for	for	ADP
ma-25	234	42	all	all	PRON
ma-25	234	43	|z	|z	NOUN
ma-25	235	1	|	|	ADV
ma-25	235	2	outside	outside	ADP
ma-25	235	3	a	a	DET
ma-25	235	4	set	set	ADJ
ma-25	235	5	e4	e4	PROPN
ma-25	235	6	of	of	ADP
ma-25	235	7	r	r	NOUN
ma-25	235	8	of	of	ADP
ma-25	235	9	finite	finite	ADJ
ma-25	235	10	logarithmic	logarithmic	ADJ
ma-25	235	11	measure	measure	NOUN
ma-25	235	12	,	,	PUNCT
ma-25	235	13	where	where	SCONJ
ma-25	235	14	νf	νf	NOUN
ma-25	235	15	(	(	PUNCT
ma-25	235	16	r	r	NOUN
ma-25	235	17	)	)	PUNCT
ma-25	235	18	is	be	AUX
ma-25	235	19	the	the	DET
ma-25	235	20	central	central	ADJ
ma-25	235	21	index	index	NOUN
ma-25	235	22	of	of	ADP
ma-25	235	23	f	f	PROPN
ma-25	235	24	.	.	PUNCT
ma-25	236	1	lemma	lemma	PROPN
ma-25	236	2	2.6	2.6	NUM
ma-25	236	3	(	(	PUNCT
ma-25	236	4	[	[	X
ma-25	236	5	12	12	NUM
ma-25	236	6	]	]	PUNCT
ma-25	236	7	)	)	PUNCT
ma-25	236	8	let	let	VERB
ma-25	236	9	f	f	PRON
ma-25	236	10	be	be	AUX
ma-25	236	11	an	an	DET
ma-25	236	12	entire	entire	ADJ
ma-25	236	13	function	function	NOUN
ma-25	236	14	of	of	ADP
ma-25	236	15	[	[	X
ma-25	236	16	p	p	X
ma-25	236	17	,	,	PUNCT
ma-25	236	18	q]-order	q]-order	NOUN
ma-25	236	19	and	and	CCONJ
ma-25	236	20	let	let	VERB
ma-25	236	21	νf	νf	NOUN
ma-25	236	22	(	(	PUNCT
ma-25	236	23	r	r	NOUN
ma-25	236	24	)	)	PUNCT
ma-25	236	25	be	be	AUX
ma-25	236	26	the	the	DET
ma-25	236	27	central	central	ADJ
ma-25	236	28	index	index	NOUN
ma-25	236	29	of	of	ADP
ma-25	236	30	f	f	PROPN
ma-25	236	31	.	.	PUNCT
ma-25	237	1	then	then	ADV
ma-25	237	2	ρ[p	ρ[p	NOUN
ma-25	237	3	,	,	PUNCT
ma-25	237	4	q	q	X
ma-25	237	5	]	]	X
ma-25	237	6	(	(	PUNCT
ma-25	237	7	f	f	X
ma-25	237	8	)	)	PUNCT
ma-25	238	1	=	=	SYM
ma-25	238	2	lim	lim	PROPN
ma-25	238	3	sup	sup	NOUN
ma-25	238	4	r→+∞	r→+∞	PROPN
ma-25	238	5	logp	logp	NOUN
ma-25	238	6	νf	νf	NOUN
ma-25	238	7	(	(	PUNCT
ma-25	238	8	r	r	NOUN
ma-25	238	9	)	)	PUNCT
ma-25	238	10	logq	logq	ADJ
ma-25	238	11	r	r	NOUN
ma-25	238	12	,	,	PUNCT
ma-25	238	13	µ[p	µ[p	ADJ
ma-25	238	14	,	,	PUNCT
ma-25	238	15	q	q	X
ma-25	238	16	]	]	X
ma-25	238	17	(	(	PUNCT
ma-25	238	18	f	f	X
ma-25	238	19	)	)	PUNCT
ma-25	239	1	=	=	PROPN
ma-25	239	2	lim	lim	PROPN
ma-25	239	3	inf	inf	PROPN
ma-25	239	4	r→+∞	r→+∞	PROPN
ma-25	239	5	logp	logp	NOUN
ma-25	239	6	νf	νf	NOUN
ma-25	239	7	(	(	PUNCT
ma-25	239	8	r	r	NOUN
ma-25	239	9	)	)	PUNCT
ma-25	239	10	logq	logq	ADJ
ma-25	239	11	r	r	NOUN
ma-25	239	12	.	.	PUNCT
ma-25	240	1	the	the	DET
ma-25	240	2	following	follow	VERB
ma-25	240	3	two	two	NUM
ma-25	240	4	lemmas	lemma	NOUN
ma-25	240	5	were	be	AUX
ma-25	240	6	given	give	VERB
ma-25	240	7	in	in	ADP
ma-25	240	8	[	[	NOUN
ma-25	240	9	4	4	X
ma-25	240	10	]	]	PUNCT
ma-25	240	11	without	without	ADP
ma-25	240	12	proof	proof	NOUN
ma-25	240	13	,	,	PUNCT
ma-25	240	14	so	so	ADV
ma-25	240	15	for	for	ADP
ma-25	240	16	the	the	DET
ma-25	240	17	convenience	convenience	NOUN
ma-25	240	18	of	of	ADP
ma-25	240	19	the	the	DET
ma-25	240	20	reader	reader	NOUN
ma-25	240	21	,	,	PUNCT
ma-25	240	22	we	we	PRON
ma-25	240	23	prove	prove	VERB
ma-25	240	24	them	they	PRON
ma-25	240	25	.	.	PUNCT
ma-25	241	1	lemma	lemma	PROPN
ma-25	241	2	2.7	2.7	NUM
ma-25	241	3	let	let	VERB
ma-25	241	4	f	f	PROPN
ma-25	241	5	(	(	PUNCT
ma-25	241	6	z	z	NOUN
ma-25	241	7	)	)	PUNCT
ma-25	241	8	=	=	SYM
ma-25	241	9	g(z	g(z	ADJ
ma-25	241	10	)	)	PUNCT
ma-25	241	11	d(z	d(z	PROPN
ma-25	241	12	)	)	PUNCT
ma-25	241	13	be	be	AUX
ma-25	241	14	a	a	DET
ma-25	241	15	meromorphic	meromorphic	ADJ
ma-25	241	16	function	function	NOUN
ma-25	241	17	,	,	PUNCT
ma-25	241	18	where	where	SCONJ
ma-25	241	19	g	g	PROPN
ma-25	241	20	(	(	PUNCT
ma-25	241	21	z	z	PROPN
ma-25	241	22	)	)	PUNCT
ma-25	241	23	,	,	PUNCT
ma-25	241	24	d	d	X
ma-25	241	25	(	(	PUNCT
ma-25	241	26	z	z	NOUN
ma-25	241	27	)	)	PUNCT
ma-25	241	28	are	be	AUX
ma-25	241	29	entire	entire	ADJ
ma-25	241	30	functions	function	NOUN
ma-25	241	31	satisfying	satisfy	VERB
ma-25	241	32	µ[p	µ[p	ADJ
ma-25	241	33	,	,	PUNCT
ma-25	241	34	q	q	X
ma-25	241	35	]	]	X
ma-25	241	36	(	(	PUNCT
ma-25	241	37	g	g	NOUN
ma-25	241	38	)	)	PUNCT
ma-25	241	39	=	=	SYM
ma-25	241	40	µ[p	µ[p	ADJ
ma-25	241	41	,	,	PUNCT
ma-25	241	42	q	q	X
ma-25	241	43	]	]	X
ma-25	241	44	(	(	PUNCT
ma-25	241	45	f	f	X
ma-25	241	46	)	)	PUNCT
ma-25	241	47	=	=	SYM
ma-25	241	48	µ	µ	PRON
ma-25	241	49	≤	≤	NUM
ma-25	241	50	ρ[p	ρ[p	NUM
ma-25	241	51	,	,	PUNCT
ma-25	241	52	q	q	X
ma-25	241	53	]	]	X
ma-25	241	54	(	(	PUNCT
ma-25	241	55	f	f	X
ma-25	241	56	)	)	PUNCT
ma-25	241	57	=	=	PUNCT
ma-25	242	1	ρ[p	ρ[p	PROPN
ma-25	242	2	,	,	PUNCT
ma-25	242	3	q	q	X
ma-25	242	4	]	]	X
ma-25	242	5	(	(	PUNCT
ma-25	242	6	g	g	NOUN
ma-25	242	7	)	)	PUNCT
ma-25	242	8	≤	≤	NOUN
ma-25	243	1	+	+	PUNCT
ma-25	243	2	∞	∞	PROPN
ma-25	243	3	and	and	CCONJ
ma-25	243	4	λ[p	λ[p	PROPN
ma-25	243	5	,	,	PUNCT
ma-25	243	6	q	q	X
ma-25	243	7	]	]	X
ma-25	243	8	(	(	PUNCT
ma-25	243	9	d	d	NOUN
ma-25	243	10	)	)	PUNCT
ma-25	243	11	=	=	SYM
ma-25	243	12	ρ[p	ρ[p	PROPN
ma-25	243	13	,	,	PUNCT
ma-25	243	14	q	q	X
ma-25	243	15	]	]	X
ma-25	243	16	(	(	PUNCT
ma-25	243	17	d	d	NOUN
ma-25	243	18	)	)	PUNCT
ma-25	243	19	=	=	NOUN
ma-25	243	20	β	β	X
ma-25	243	21	=	=	SYM
ma-25	243	22	λ[p	λ[p	NUM
ma-25	243	23	,	,	PUNCT
ma-25	243	24	q	q	X
ma-25	243	25	]	]	X
ma-25	243	26	(	(	PUNCT
ma-25	243	27	1	1	NUM
ma-25	243	28	f	f	NOUN
ma-25	243	29	)	)	PUNCT
ma-25	243	30	<	<	X
ma-25	243	31	µ.	µ.	NOUN
ma-25	243	32	then	then	ADV
ma-25	243	33	,	,	PUNCT
ma-25	243	34	there	there	PRON
ma-25	243	35	exists	exist	VERB
ma-25	243	36	a	a	DET
ma-25	243	37	set	set	NOUN
ma-25	243	38	e5	e5	PROPN
ma-25	243	39	⊂	⊂	PROPN
ma-25	243	40	(	(	PUNCT
ma-25	243	41	1,+∞	1,+∞	NUM
ma-25	243	42	)	)	PUNCT
ma-25	243	43	of	of	ADP
ma-25	243	44	finite	finite	ADJ
ma-25	243	45	logarithmic	logarithmic	ADJ
ma-25	243	46	measure	measure	NOUN
ma-25	243	47	such	such	ADJ
ma-25	243	48	that	that	PRON
ma-25	243	49	for	for	ADP
ma-25	243	50	all	all	PRON
ma-25	243	51	|z	|z	NOUN
ma-25	244	1	|	|	NOUN
ma-25	244	2	=	=	NOUN
ma-25	244	3	r	r	NOUN
ma-25	244	4	/∈	/∈	PUNCT
ma-25	245	1	[	[	X
ma-25	245	2	0	0	NUM
ma-25	245	3	,	,	PUNCT
ma-25	245	4	1	1	NUM
ma-25	245	5	]	]	PUNCT
ma-25	245	6	∪	∪	X
ma-25	245	7	e5	e5	PROPN
ma-25	245	8	and	and	CCONJ
ma-25	245	9	|g	|g	NOUN
ma-25	245	10	(	(	PUNCT
ma-25	245	11	z	z	NOUN
ma-25	245	12	)	)	PUNCT
ma-25	246	1	|	|	ADV
ma-25	246	2	=	=	SYM
ma-25	246	3	m	m	PROPN
ma-25	246	4	(	(	PUNCT
ma-25	246	5	r	r	NOUN
ma-25	246	6	,	,	PUNCT
ma-25	246	7	g	g	NOUN
ma-25	246	8	)	)	PUNCT
ma-25	246	9	,	,	PUNCT
ma-25	246	10	we	we	PRON
ma-25	246	11	have	have	VERB
ma-25	246	12	f	f	PROPN
ma-25	246	13	(	(	PUNCT
ma-25	246	14	n	n	CCONJ
ma-25	246	15	)	)	PUNCT
ma-25	246	16	(	(	PUNCT
ma-25	246	17	z	z	X
ma-25	246	18	)	)	PUNCT
ma-25	246	19	f	f	NOUN
ma-25	246	20	(	(	PUNCT
ma-25	246	21	z	z	NOUN
ma-25	246	22	)	)	PUNCT
ma-25	246	23	=	=	SYM
ma-25	247	1	(	(	PUNCT
ma-25	247	2	νg	νg	X
ma-25	247	3	(	(	PUNCT
ma-25	247	4	r	r	NOUN
ma-25	247	5	)	)	PUNCT
ma-25	247	6	z	z	NOUN
ma-25	247	7	)	)	PUNCT
ma-25	247	8	n	n	CCONJ
ma-25	247	9	(	(	PUNCT
ma-25	247	10	1	1	NUM
ma-25	248	1	+	+	NUM
ma-25	248	2	o	o	NOUN
ma-25	248	3	(	(	PUNCT
ma-25	248	4	1	1	NUM
ma-25	248	5	)	)	PUNCT
ma-25	248	6	)	)	PUNCT
ma-25	248	7	,	,	PUNCT
ma-25	248	8	n	n	PROPN
ma-25	248	9	∈	∈	PROPN
ma-25	248	10	n	n	CCONJ
ma-25	248	11	,	,	PUNCT
ma-25	248	12	where	where	SCONJ
ma-25	248	13	νg	νg	NOUN
ma-25	248	14	(	(	PUNCT
ma-25	248	15	r	r	NOUN
ma-25	248	16	)	)	PUNCT
ma-25	248	17	denote	denote	VERB
ma-25	248	18	the	the	DET
ma-25	248	19	central	central	ADJ
ma-25	248	20	index	index	NOUN
ma-25	248	21	of	of	ADP
ma-25	248	22	g.	g.	PROPN
ma-25	248	23	eur	eur	PROPN
ma-25	248	24	.	.	PUNCT
ma-25	249	1	j.	j.	PROPN
ma-25	249	2	math	math	PROPN
ma-25	249	3	.	.	PUNCT
ma-25	250	1	anal	anal	ADJ
ma-25	250	2	.	.	PUNCT
ma-25	251	1	1	1	NUM
ma-25	251	2	(	(	PUNCT
ma-25	251	3	2021	2021	NUM
ma-25	251	4	)	)	PUNCT
ma-25	252	1	94	94	NUM
ma-25	252	2	proof	proof	NOUN
ma-25	252	3	.	.	PUNCT
ma-25	253	1	by	by	ADP
ma-25	253	2	mathematical	mathematical	ADJ
ma-25	253	3	induction	induction	NOUN
ma-25	253	4	,	,	PUNCT
ma-25	253	5	we	we	PRON
ma-25	253	6	obtain	obtain	VERB
ma-25	253	7	f	f	PROPN
ma-25	253	8	(	(	PUNCT
ma-25	253	9	n	n	CCONJ
ma-25	253	10	)	)	PUNCT
ma-25	253	11	=	=	SYM
ma-25	253	12	g(n	g(n	PROPN
ma-25	253	13	)	)	PUNCT
ma-25	254	1	d	d	PROPN
ma-25	254	2	+	+	CCONJ
ma-25	254	3	n−1∑	n−1∑	NUM
ma-25	254	4	j=0	j=0	PROPN
ma-25	254	5	g(j	g(j	PROPN
ma-25	254	6	)	)	PUNCT
ma-25	255	1	d	d	X
ma-25	255	2	∑	∑	PROPN
ma-25	255	3	(	(	PUNCT
ma-25	255	4	j1	j1	PROPN
ma-25	255	5	...	...	SYM
ma-25	255	6	jn	jn	PROPN
ma-25	255	7	)	)	PUNCT
ma-25	255	8	cj	cj	PROPN
ma-25	255	9	j1	j1	PROPN
ma-25	255	10	...	...	PUNCT
ma-25	255	11	jn	jn	PROPN
ma-25	256	1	(	(	PUNCT
ma-25	256	2	d	d	NOUN
ma-25	256	3	′	′	NUM
ma-25	257	1	d	d	NOUN
ma-25	257	2	)	)	PUNCT
ma-25	257	3	j1	j1	PROPN
ma-25	257	4	×	×	NOUN
ma-25	257	5	·	·	PUNCT
ma-25	257	6	·	·	PUNCT
ma-25	257	7	·	·	PUNCT
ma-25	258	1	×	×	NOUN
ma-25	258	2	(	(	PUNCT
ma-25	258	3	d	d	X
ma-25	258	4	(	(	PUNCT
ma-25	258	5	n	n	CCONJ
ma-25	258	6	)	)	PUNCT
ma-25	258	7	d	d	NOUN
ma-25	258	8	)	)	PUNCT
ma-25	258	9	jn	jn	PROPN
ma-25	258	10	,	,	PUNCT
ma-25	258	11	(	(	PUNCT
ma-25	258	12	2.14	2.14	NUM
ma-25	258	13	)	)	PUNCT
ma-25	258	14	where	where	SCONJ
ma-25	258	15	cj	cj	PROPN
ma-25	258	16	j1	j1	PROPN
ma-25	258	17	...	...	PUNCT
ma-25	258	18	jn	jn	PROPN
ma-25	258	19	are	be	AUX
ma-25	258	20	constants	constant	NOUN
ma-25	258	21	and	and	CCONJ
ma-25	258	22	j	j	PROPN
ma-25	258	23	+	+	CCONJ
ma-25	258	24	j1	j1	PROPN
ma-25	258	25	+	+	CCONJ
ma-25	258	26	2j2	2j2	NUM
ma-25	258	27	+	+	CCONJ
ma-25	258	28	·	·	PUNCT
ma-25	258	29	·	·	PUNCT
ma-25	258	30	·	·	PUNCT
ma-25	258	31	+	+	NUM
ma-25	258	32	njn	njn	NOUN
ma-25	258	33	=	=	NOUN
ma-25	258	34	n.	n.	NOUN
ma-25	259	1	hence	hence	ADV
ma-25	259	2	f	f	PROPN
ma-25	259	3	(	(	PUNCT
ma-25	259	4	n	n	CCONJ
ma-25	259	5	)	)	PUNCT
ma-25	259	6	f	f	PROPN
ma-25	259	7	=	=	SYM
ma-25	259	8	g(n	g(n	PROPN
ma-25	259	9	)	)	PUNCT
ma-25	259	10	g	g	PROPN
ma-25	259	11	+	+	NUM
ma-25	259	12	n−1∑	n−1∑	PROPN
ma-25	259	13	j=0	j=0	PROPN
ma-25	259	14	g(j	g(j	PROPN
ma-25	259	15	)	)	PUNCT
ma-25	260	1	g	g	PROPN
ma-25	260	2	∑	∑	PROPN
ma-25	260	3	(	(	PUNCT
ma-25	260	4	j1	j1	PROPN
ma-25	260	5	...	...	SYM
ma-25	260	6	jn	jn	PROPN
ma-25	260	7	)	)	PUNCT
ma-25	260	8	cj	cj	PROPN
ma-25	260	9	j1	j1	PROPN
ma-25	260	10	...	...	PUNCT
ma-25	260	11	jn	jn	PROPN
ma-25	261	1	(	(	PUNCT
ma-25	261	2	d	d	NOUN
ma-25	261	3	′	′	NUM
ma-25	262	1	d	d	NOUN
ma-25	262	2	)	)	PUNCT
ma-25	262	3	j1	j1	PROPN
ma-25	262	4	×	×	NOUN
ma-25	262	5	·	·	PUNCT
ma-25	262	6	·	·	PUNCT
ma-25	262	7	·	·	PUNCT
ma-25	263	1	×	×	NOUN
ma-25	263	2	(	(	PUNCT
ma-25	263	3	d	d	X
ma-25	263	4	(	(	PUNCT
ma-25	263	5	n	n	CCONJ
ma-25	263	6	)	)	PUNCT
ma-25	263	7	d	d	NOUN
ma-25	263	8	)	)	PUNCT
ma-25	263	9	jn	jn	PROPN
ma-25	263	10	.	.	PUNCT
ma-25	264	1	(	(	PUNCT
ma-25	264	2	2.15	2.15	NUM
ma-25	264	3	)	)	PUNCT
ma-25	264	4	from	from	ADP
ma-25	264	5	lemma	lemma	PROPN
ma-25	264	6	2.5	2.5	NUM
ma-25	264	7	,	,	PUNCT
ma-25	264	8	there	there	PRON
ma-25	264	9	exists	exist	VERB
ma-25	264	10	a	a	DET
ma-25	264	11	set	set	NOUN
ma-25	264	12	e4	e4	PROPN
ma-25	264	13	⊂	⊂	PROPN
ma-25	264	14	(	(	PUNCT
ma-25	264	15	1,+∞	1,+∞	NUM
ma-25	264	16	)	)	PUNCT
ma-25	264	17	with	with	ADP
ma-25	264	18	finite	finite	ADJ
ma-25	264	19	logarithmic	logarithmic	ADJ
ma-25	264	20	measure	measure	NOUN
ma-25	264	21	such	such	ADJ
ma-25	264	22	that	that	PRON
ma-25	264	23	for	for	ADP
ma-25	264	24	apoint	apoint	NOUN
ma-25	264	25	z	z	NOUN
ma-25	264	26	satisfying	satisfy	VERB
ma-25	264	27	|z	|z	PROPN
ma-25	265	1	|	|	ADV
ma-25	265	2	=	=	SYM
ma-25	265	3	r	r	NOUN
ma-25	265	4	/∈	/∈	PUNCT
ma-25	265	5	e4	e4	PROPN
ma-25	265	6	and	and	CCONJ
ma-25	265	7	|g	|g	NOUN
ma-25	265	8	(	(	PUNCT
ma-25	265	9	z)|	z)|	X
ma-25	265	10	=	=	PRON
ma-25	265	11	m	m	PROPN
ma-25	265	12	(	(	PUNCT
ma-25	265	13	r	r	NOUN
ma-25	265	14	,	,	PUNCT
ma-25	265	15	g	g	NOUN
ma-25	265	16	)	)	PUNCT
ma-25	265	17	,	,	PUNCT
ma-25	265	18	we	we	PRON
ma-25	265	19	have	have	VERB
ma-25	265	20	g(j)(z	g(j)(z	VERB
ma-25	265	21	)	)	PUNCT
ma-25	265	22	g(z	g(z	PROPN
ma-25	265	23	)	)	PUNCT
ma-25	266	1	=	=	PRON
ma-25	266	2	(	(	PUNCT
ma-25	266	3	νg	νg	X
ma-25	266	4	(	(	PUNCT
ma-25	266	5	r	r	NOUN
ma-25	266	6	)	)	PUNCT
ma-25	266	7	z	z	NOUN
ma-25	266	8	)	)	PUNCT
ma-25	266	9	j	j	NOUN
ma-25	266	10	(	(	PUNCT
ma-25	266	11	1	1	NUM
ma-25	266	12	+	+	NUM
ma-25	266	13	o	o	NOUN
ma-25	266	14	(	(	PUNCT
ma-25	266	15	1	1	NUM
ma-25	266	16	)	)	PUNCT
ma-25	266	17	)	)	PUNCT
ma-25	267	1	(	(	PUNCT
ma-25	267	2	j	j	NOUN
ma-25	267	3	=	=	SYM
ma-25	267	4	1	1	NUM
ma-25	267	5	,	,	PUNCT
ma-25	267	6	2	2	NUM
ma-25	267	7	,	,	PUNCT
ma-25	267	8	...	...	PUNCT
ma-25	267	9	,	,	PUNCT
ma-25	267	10	n	n	CCONJ
ma-25	267	11	)	)	PUNCT
ma-25	267	12	,	,	PUNCT
ma-25	267	13	(	(	PUNCT
ma-25	267	14	2.16	2.16	NUM
ma-25	267	15	)	)	PUNCT
ma-25	268	1	where	where	SCONJ
ma-25	268	2	νg	νg	NOUN
ma-25	268	3	(	(	PUNCT
ma-25	268	4	r	r	NOUN
ma-25	268	5	)	)	PUNCT
ma-25	268	6	is	be	AUX
ma-25	268	7	the	the	DET
ma-25	268	8	central	central	ADJ
ma-25	268	9	index	index	NOUN
ma-25	268	10	of	of	ADP
ma-25	268	11	g.	g.	PROPN
ma-25	268	12	substituting	substitute	VERB
ma-25	268	13	(	(	PUNCT
ma-25	268	14	2.16	2.16	NUM
ma-25	268	15	)	)	PUNCT
ma-25	268	16	into	into	ADP
ma-25	268	17	(	(	PUNCT
ma-25	268	18	2.15	2.15	NUM
ma-25	268	19	)	)	PUNCT
ma-25	268	20	yields	yield	NOUN
ma-25	268	21	f	f	PROPN
ma-25	268	22	(	(	PUNCT
ma-25	268	23	n	n	CCONJ
ma-25	268	24	)	)	PUNCT
ma-25	268	25	(	(	PUNCT
ma-25	268	26	z	z	X
ma-25	268	27	)	)	PUNCT
ma-25	268	28	f	f	NOUN
ma-25	268	29	(	(	PUNCT
ma-25	268	30	z	z	NOUN
ma-25	268	31	)	)	PUNCT
ma-25	268	32	=	=	SYM
ma-25	268	33	(	(	PUNCT
ma-25	268	34	νg	νg	X
ma-25	268	35	(	(	PUNCT
ma-25	268	36	r	r	NOUN
ma-25	268	37	)	)	PUNCT
ma-25	268	38	z	z	NOUN
ma-25	268	39	)	)	PUNCT
ma-25	268	40	n	n	CCONJ
ma-25	269	1	[	[	X
ma-25	269	2	(	(	PUNCT
ma-25	269	3	1	1	NUM
ma-25	269	4	+	+	NUM
ma-25	269	5	o	o	NOUN
ma-25	269	6	(	(	PUNCT
ma-25	269	7	1	1	NUM
ma-25	269	8	)	)	PUNCT
ma-25	269	9	)	)	PUNCT
ma-25	270	1	+	+	CCONJ
ma-25	271	1	n−1∑	n−1∑	PROPN
ma-25	271	2	j=0	j=0	PROPN
ma-25	271	3	(	(	PUNCT
ma-25	271	4	νg	νg	X
ma-25	271	5	(	(	PUNCT
ma-25	271	6	r	r	NOUN
ma-25	271	7	)	)	PUNCT
ma-25	271	8	z	z	NOUN
ma-25	271	9	)	)	PUNCT
ma-25	271	10	j−n	j−n	PROPN
ma-25	271	11	(	(	PUNCT
ma-25	271	12	1	1	NUM
ma-25	271	13	+	+	NUM
ma-25	271	14	o	o	NOUN
ma-25	271	15	(	(	PUNCT
ma-25	271	16	1	1	NUM
ma-25	271	17	)	)	PUNCT
ma-25	271	18	)	)	PUNCT
ma-25	271	19	∑	∑	PROPN
ma-25	271	20	(	(	PUNCT
ma-25	271	21	j1	j1	PROPN
ma-25	271	22	...	...	SYM
ma-25	271	23	jn	jn	PROPN
ma-25	271	24	)	)	PUNCT
ma-25	271	25	cj	cj	PROPN
ma-25	271	26	j1	j1	PROPN
ma-25	271	27	...	...	PUNCT
ma-25	271	28	jn	jn	PROPN
ma-25	271	29	(	(	PUNCT
ma-25	271	30	d	d	NOUN
ma-25	271	31	′	′	NUM
ma-25	271	32	d	d	NOUN
ma-25	271	33	)	)	PUNCT
ma-25	271	34	j1	j1	PROPN
ma-25	271	35	×	×	NOUN
ma-25	271	36	·	·	PUNCT
ma-25	271	37	·	·	PUNCT
ma-25	271	38	·	·	PUNCT
ma-25	271	39	×	×	NOUN
ma-25	271	40	(	(	PUNCT
ma-25	271	41	d	d	X
ma-25	271	42	(	(	PUNCT
ma-25	271	43	n	n	CCONJ
ma-25	271	44	)	)	PUNCT
ma-25	271	45	d	d	NOUN
ma-25	271	46	)	)	PUNCT
ma-25	271	47	jn	jn	PROPN
ma-25	271	48	.	.	PUNCT
ma-25	272	1	(	(	PUNCT
ma-25	272	2	2.17	2.17	NUM
ma-25	272	3	)	)	PUNCT
ma-25	272	4	since	since	SCONJ
ma-25	272	5	ρ[p	ρ[p	NOUN
ma-25	272	6	,	,	PUNCT
ma-25	272	7	q	q	X
ma-25	272	8	]	]	X
ma-25	272	9	(	(	PUNCT
ma-25	272	10	d	d	NOUN
ma-25	272	11	)	)	PUNCT
ma-25	272	12	=	=	SYM
ma-25	272	13	β	β	X
ma-25	272	14	<	<	X
ma-25	272	15	µ	µ	X
ma-25	272	16	,	,	PUNCT
ma-25	272	17	then	then	ADV
ma-25	272	18	for	for	ADP
ma-25	272	19	any	any	DET
ma-25	272	20	given	give	VERB
ma-25	272	21	ε	ε	PROPN
ma-25	272	22	(	(	PUNCT
ma-25	272	23	0	0	PUNCT
ma-25	272	24	<	<	X
ma-25	272	25	2ε	2ε	NOUN
ma-25	272	26	<	<	X
ma-25	272	27	µ−	µ−	PROPN
ma-25	272	28	β	β	NOUN
ma-25	272	29	)	)	PUNCT
ma-25	272	30	and	and	CCONJ
ma-25	272	31	sufficiently	sufficiently	ADV
ma-25	272	32	large	large	ADJ
ma-25	272	33	r	r	NOUN
ma-25	272	34	,	,	PUNCT
ma-25	272	35	we	we	PRON
ma-25	272	36	have	have	VERB
ma-25	272	37	t	t	NOUN
ma-25	272	38	(	(	PUNCT
ma-25	272	39	r	r	NOUN
ma-25	272	40	,	,	PUNCT
ma-25	272	41	d	d	NOUN
ma-25	272	42	)	)	PUNCT
ma-25	272	43	≤	≤	X
ma-25	272	44	expp	expp	ADJ
ma-25	272	45	{	{	PUNCT
ma-25	272	46	(	(	PUNCT
ma-25	272	47	β	β	X
ma-25	272	48	+	+	X
ma-25	272	49	ε	ε	PROPN
ma-25	272	50	2	2	NUM
ma-25	272	51	)	)	PUNCT
ma-25	272	52	logq	logq	VERB
ma-25	272	53	r	r	NOUN
ma-25	272	54	}	}	PUNCT
ma-25	272	55	by	by	ADP
ma-25	272	56	using	use	VERB
ma-25	272	57	lemma	lemma	PROPN
ma-25	272	58	2.1	2.1	NUM
ma-25	272	59	,	,	PUNCT
ma-25	272	60	for	for	ADP
ma-25	272	61	α	α	NOUN
ma-25	272	62	=	=	SYM
ma-25	272	63	2	2	NUM
ma-25	272	64	,	,	PUNCT
ma-25	272	65	there	there	PRON
ma-25	272	66	exist	exist	VERB
ma-25	272	67	a	a	DET
ma-25	272	68	set	set	NOUN
ma-25	272	69	e1	e1	NOUN
ma-25	272	70	⊂	⊂	PROPN
ma-25	272	71	(	(	PUNCT
ma-25	272	72	1,+∞	1,+∞	NUM
ma-25	272	73	)	)	PUNCT
ma-25	272	74	with	with	ADP
ma-25	272	75	ml(e1	ml(e1	NOUN
ma-25	272	76	)	)	PUNCT
ma-25	273	1	<	<	X
ma-25	273	2	∞	∞	NUM
ma-25	273	3	and	and	CCONJ
ma-25	273	4	a	a	DET
ma-25	273	5	constant	constant	ADJ
ma-25	273	6	b	b	NOUN
ma-25	273	7	>	>	X
ma-25	273	8	0	0	NUM
ma-25	273	9	,	,	PUNCT
ma-25	273	10	such	such	ADJ
ma-25	273	11	that	that	SCONJ
ma-25	273	12	for	for	ADP
ma-25	273	13	all	all	DET
ma-25	273	14	z	z	NOUN
ma-25	273	15	satisfying	satisfy	VERB
ma-25	273	16	|z	|z	PROPN
ma-25	274	1	|	|	ADV
ma-25	274	2	=	=	NOUN
ma-25	274	3	r	r	NOUN
ma-25	274	4	/∈	/∈	PUNCT
ma-25	275	1	[	[	X
ma-25	275	2	0	0	NUM
ma-25	275	3	,	,	PUNCT
ma-25	275	4	1	1	NUM
ma-25	275	5	]	]	PUNCT
ma-25	275	6	∪	∪	NOUN
ma-25	275	7	e1	e1	NOUN
ma-25	275	8	,	,	PUNCT
ma-25	275	9	we	we	PRON
ma-25	275	10	have∣∣∣∣∣d	have∣∣∣∣∣d	VERB
ma-25	275	11	(	(	PUNCT
ma-25	275	12	m	m	NOUN
ma-25	275	13	)	)	PUNCT
ma-25	275	14	(	(	PUNCT
ma-25	275	15	z	z	X
ma-25	275	16	)	)	PUNCT
ma-25	275	17	d	d	NOUN
ma-25	275	18	(	(	PUNCT
ma-25	275	19	z	z	NOUN
ma-25	275	20	)	)	PUNCT
ma-25	275	21	∣∣∣∣∣	∣∣∣∣∣	ADP
ma-25	276	1	≤	≤	PROPN
ma-25	276	2	b	b	X
ma-25	277	1	[	[	X
ma-25	277	2	t	t	X
ma-25	277	3	(	(	PUNCT
ma-25	277	4	2r	2r	NUM
ma-25	277	5	,	,	PUNCT
ma-25	277	6	d)]m+1	d)]m+1	PROPN
ma-25	277	7	≤	≤	PROPN
ma-25	277	8	b	b	PROPN
ma-25	277	9	[	[	PUNCT
ma-25	277	10	expp	expp	ADJ
ma-25	277	11	{	{	PUNCT
ma-25	277	12	(	(	PUNCT
ma-25	277	13	β	β	X
ma-25	277	14	+	+	X
ma-25	277	15	ε	ε	PROPN
ma-25	277	16	2	2	NUM
ma-25	277	17	)	)	PUNCT
ma-25	277	18	logq	logq	NOUN
ma-25	277	19	(	(	PUNCT
ma-25	277	20	2r	2r	NUM
ma-25	277	21	)	)	PUNCT
ma-25	277	22	}	}	PUNCT
ma-25	277	23	]	]	PUNCT
ma-25	277	24	m+1	m+1	X
ma-25	277	25	≤	≤	NUM
ma-25	277	26	expp	expp	ADJ
ma-25	277	27	{	{	PUNCT
ma-25	277	28	(	(	PUNCT
ma-25	277	29	β	β	X
ma-25	277	30	+	+	CCONJ
ma-25	277	31	ε	ε	PROPN
ma-25	277	32	)	)	PUNCT
ma-25	277	33	logq	logq	VERB
ma-25	277	34	r	r	NOUN
ma-25	277	35	}	}	PUNCT
ma-25	277	36	m	m	PROPN
ma-25	277	37	,	,	PUNCT
ma-25	277	38	m	m	VERB
ma-25	277	39	=	=	NOUN
ma-25	277	40	1	1	NUM
ma-25	277	41	,	,	PUNCT
ma-25	277	42	2	2	NUM
ma-25	277	43	,	,	PUNCT
ma-25	277	44	...	...	PUNCT
ma-25	277	45	,	,	PUNCT
ma-25	277	46	n.	n.	PROPN
ma-25	277	47	(	(	PUNCT
ma-25	277	48	2.18)by	2.18)by	NUM
ma-25	277	49	lemma	lemma	PROPN
ma-25	277	50	2.6	2.6	NUM
ma-25	277	51	and	and	CCONJ
ma-25	277	52	µ[p	µ[p	ADJ
ma-25	277	53	,	,	PUNCT
ma-25	277	54	q	q	X
ma-25	277	55	]	]	X
ma-25	277	56	(	(	PUNCT
ma-25	277	57	g	g	NOUN
ma-25	277	58	)	)	PUNCT
ma-25	277	59	=	=	SYM
ma-25	277	60	µ[p	µ[p	ADJ
ma-25	277	61	,	,	PUNCT
ma-25	277	62	q	q	X
ma-25	277	63	]	]	X
ma-25	277	64	(	(	PUNCT
ma-25	277	65	f	f	X
ma-25	277	66	)	)	PUNCT
ma-25	277	67	=	=	SYM
ma-25	277	68	µ	µ	X
ma-25	277	69	,	,	PUNCT
ma-25	277	70	it	it	PRON
ma-25	277	71	follows	follow	VERB
ma-25	277	72	that	that	DET
ma-25	277	73	νg	νg	NOUN
ma-25	277	74	(	(	PUNCT
ma-25	277	75	r	r	NOUN
ma-25	277	76	)	)	PUNCT
ma-25	277	77	>	>	X
ma-25	277	78	expp	expp	ADJ
ma-25	277	79	{	{	PUNCT
ma-25	277	80	(	(	PUNCT
ma-25	277	81	µ−	µ−	PROPN
ma-25	277	82	ε	ε	PROPN
ma-25	277	83	)	)	PUNCT
ma-25	277	84	logq	logq	VERB
ma-25	277	85	r	r	NOUN
ma-25	277	86	}	}	PUNCT
ma-25	277	87	for	for	ADP
ma-25	277	88	sufficiently	sufficiently	ADV
ma-25	277	89	large	large	ADJ
ma-25	277	90	r	r	NOUN
ma-25	277	91	.	.	PUNCT
ma-25	278	1	thus	thus	ADV
ma-25	278	2	,	,	PUNCT
ma-25	278	3	by	by	ADP
ma-25	278	4	using	use	VERB
ma-25	278	5	j1	j1	PROPN
ma-25	278	6	+	+	CCONJ
ma-25	278	7	2j2	2j2	NUM
ma-25	278	8	+	+	CCONJ
ma-25	278	9	·	·	PUNCT
ma-25	278	10	·	·	PUNCT
ma-25	278	11	·	·	PUNCT
ma-25	278	12	+	+	NUM
ma-25	278	13	njn	njn	NOUN
ma-25	278	14	=	=	SYM
ma-25	278	15	n	n	CCONJ
ma-25	278	16	−	−	PROPN
ma-25	278	17	j	j	NOUN
ma-25	278	18	,	,	PUNCT
ma-25	278	19	we	we	PRON
ma-25	278	20	obtain∣∣∣∣∣∣	obtain∣∣∣∣∣∣	VERB
ma-25	278	21	(	(	PUNCT
ma-25	278	22	νg	νg	X
ma-25	278	23	(	(	PUNCT
ma-25	278	24	r	r	NOUN
ma-25	278	25	)	)	PUNCT
ma-25	278	26	z	z	NOUN
ma-25	278	27	)	)	PUNCT
ma-25	278	28	j−n	j−n	PROPN
ma-25	279	1	(	(	PUNCT
ma-25	279	2	d	d	NOUN
ma-25	279	3	′	′	NUM
ma-25	280	1	d	d	NOUN
ma-25	280	2	)	)	PUNCT
ma-25	280	3	j1	j1	PROPN
ma-25	280	4	×	×	NOUN
ma-25	280	5	·	·	PUNCT
ma-25	280	6	·	·	PUNCT
ma-25	280	7	·	·	PUNCT
ma-25	281	1	×	×	NOUN
ma-25	281	2	(	(	PUNCT
ma-25	281	3	d	d	X
ma-25	281	4	(	(	PUNCT
ma-25	281	5	n	n	CCONJ
ma-25	281	6	)	)	PUNCT
ma-25	281	7	d	d	NOUN
ma-25	281	8	)	)	PUNCT
ma-25	281	9	jn	jn	PROPN
ma-25	281	10	∣∣∣∣∣∣	∣∣∣∣∣∣	ADP
ma-25	281	11	≤	≤	PROPN
ma-25	281	12	[	[	PUNCT
ma-25	281	13	expp	expp	ADJ
ma-25	281	14	{	{	PUNCT
ma-25	281	15	(	(	PUNCT
ma-25	281	16	µ−	µ−	PROPN
ma-25	281	17	ε	ε	PROPN
ma-25	281	18	)	)	PUNCT
ma-25	281	19	logq	logq	VERB
ma-25	281	20	r	r	NOUN
ma-25	281	21	}	}	PUNCT
ma-25	281	22	r	r	NOUN
ma-25	281	23	]	]	PUNCT
ma-25	281	24	j−n	j−n	ADJ
ma-25	281	25	×	×	NOUN
ma-25	281	26	[	[	PUNCT
ma-25	281	27	expp	expp	ADJ
ma-25	281	28	{	{	PUNCT
ma-25	281	29	(	(	PUNCT
ma-25	281	30	β	β	X
ma-25	281	31	+	+	CCONJ
ma-25	281	32	ε	ε	PROPN
ma-25	281	33	)	)	PUNCT
ma-25	281	34	logq	logq	VERB
ma-25	281	35	r	r	NOUN
ma-25	281	36	}	}	PUNCT
ma-25	281	37	]	]	PUNCT
ma-25	281	38	n−j	n−j	X
ma-25	281	39	=	=	PUNCT
ma-25	282	1	[	[	PUNCT
ma-25	282	2	r	r	X
ma-25	282	3	expp	expp	ADJ
ma-25	282	4	{	{	PUNCT
ma-25	282	5	(	(	PUNCT
ma-25	282	6	β	β	X
ma-25	282	7	+	+	CCONJ
ma-25	282	8	ε	ε	PROPN
ma-25	282	9	)	)	PUNCT
ma-25	282	10	logq	logq	ADJ
ma-25	282	11	r	r	NOUN
ma-25	282	12	}	}	PUNCT
ma-25	282	13	expp	expp	ADJ
ma-25	282	14	{	{	PUNCT
ma-25	282	15	(	(	PUNCT
ma-25	282	16	µ−	µ−	PROPN
ma-25	282	17	ε	ε	PROPN
ma-25	282	18	)	)	PUNCT
ma-25	282	19	logq	logq	VERB
ma-25	282	20	r	r	NOUN
ma-25	282	21	}	}	PUNCT
ma-25	282	22	]	]	PUNCT
ma-25	282	23	n−j	n−j	X
ma-25	282	24	→	→	X
ma-25	282	25	0	0	NUM
ma-25	282	26	(	(	PUNCT
ma-25	282	27	2.19	2.19	NUM
ma-25	282	28	)	)	PUNCT
ma-25	282	29	as	as	ADP
ma-25	282	30	r	r	NOUN
ma-25	282	31	→	→	SYM
ma-25	282	32	+	+	NOUN
ma-25	282	33	∞	∞	PROPN
ma-25	282	34	,	,	PUNCT
ma-25	282	35	where	where	SCONJ
ma-25	282	36	|z	|z	PROPN
ma-25	283	1	|	|	ADV
ma-25	283	2	=	=	NOUN
ma-25	283	3	r	r	NOUN
ma-25	283	4	/∈	/∈	PUNCT
ma-25	284	1	[	[	X
ma-25	284	2	0	0	NUM
ma-25	284	3	,	,	PUNCT
ma-25	284	4	1	1	NUM
ma-25	284	5	]	]	PUNCT
ma-25	284	6	∪	∪	X
ma-25	284	7	e5	e5	PROPN
ma-25	284	8	,	,	PUNCT
ma-25	284	9	e5	e5	PROPN
ma-25	284	10	=	=	PROPN
ma-25	284	11	e1	e1	PROPN
ma-25	284	12	∪	∪	ADJ
ma-25	284	13	e4	e4	PROPN
ma-25	284	14	and	and	CCONJ
ma-25	284	15	|g	|g	NOUN
ma-25	284	16	(	(	PUNCT
ma-25	284	17	z)|	z)|	X
ma-25	284	18	=	=	PRON
ma-25	284	19	m	m	PROPN
ma-25	284	20	(	(	PUNCT
ma-25	284	21	r	r	NOUN
ma-25	284	22	,	,	PUNCT
ma-25	284	23	g	g	NOUN
ma-25	284	24	)	)	PUNCT
ma-25	284	25	.	.	PUNCT
ma-25	285	1	from	from	ADP
ma-25	285	2	(	(	PUNCT
ma-25	285	3	2.17	2.17	NUM
ma-25	285	4	)	)	PUNCT
ma-25	285	5	and	and	CCONJ
ma-25	285	6	(	(	PUNCT
ma-25	285	7	2.19	2.19	NUM
ma-25	285	8	)	)	PUNCT
ma-25	285	9	,	,	PUNCT
ma-25	285	10	we	we	PRON
ma-25	285	11	obtain	obtain	VERB
ma-25	285	12	our	our	PRON
ma-25	285	13	assertion	assertion	NOUN
ma-25	285	14	.	.	PUNCT
ma-25	286	1	eur	eur	PROPN
ma-25	286	2	.	.	PUNCT
ma-25	287	1	j.	j.	PROPN
ma-25	287	2	math	math	PROPN
ma-25	287	3	.	.	PUNCT
ma-25	288	1	anal	anal	ADJ
ma-25	288	2	.	.	PUNCT
ma-25	289	1	1	1	NUM
ma-25	289	2	(	(	PUNCT
ma-25	289	3	2021	2021	NUM
ma-25	289	4	)	)	PUNCT
ma-25	289	5	95	95	NUM
ma-25	289	6	lemma	lemma	PROPN
ma-25	289	7	2.8	2.8	NUM
ma-25	289	8	let	let	VERB
ma-25	289	9	f	f	PROPN
ma-25	289	10	(	(	PUNCT
ma-25	289	11	z	z	NOUN
ma-25	289	12	)	)	PUNCT
ma-25	289	13	=	=	SYM
ma-25	289	14	g(z	g(z	ADJ
ma-25	289	15	)	)	PUNCT
ma-25	289	16	d(z	d(z	PROPN
ma-25	289	17	)	)	PUNCT
ma-25	289	18	be	be	AUX
ma-25	289	19	a	a	DET
ma-25	289	20	meromorphic	meromorphic	ADJ
ma-25	289	21	function	function	NOUN
ma-25	289	22	,	,	PUNCT
ma-25	289	23	where	where	SCONJ
ma-25	289	24	g	g	PROPN
ma-25	289	25	(	(	PUNCT
ma-25	289	26	z	z	NOUN
ma-25	289	27	)	)	PUNCT
ma-25	289	28	,	,	PUNCT
ma-25	289	29	d	d	X
ma-25	289	30	(	(	PUNCT
ma-25	289	31	z	z	NOUN
ma-25	289	32	)	)	PUNCT
ma-25	289	33	are	be	AUX
ma-25	289	34	entire	entire	ADJ
ma-25	289	35	functions	function	NOUN
ma-25	289	36	satisfying	satisfy	VERB
ma-25	289	37	µ[p	µ[p	ADJ
ma-25	289	38	,	,	PUNCT
ma-25	289	39	q	q	X
ma-25	289	40	]	]	X
ma-25	289	41	(	(	PUNCT
ma-25	289	42	g	g	NOUN
ma-25	289	43	)	)	PUNCT
ma-25	289	44	=	=	SYM
ma-25	289	45	µ[p	µ[p	ADJ
ma-25	289	46	,	,	PUNCT
ma-25	289	47	q	q	X
ma-25	289	48	]	]	X
ma-25	289	49	(	(	PUNCT
ma-25	289	50	f	f	X
ma-25	289	51	)	)	PUNCT
ma-25	289	52	=	=	SYM
ma-25	289	53	µ	µ	PRON
ma-25	289	54	≤	≤	NUM
ma-25	289	55	ρ[p	ρ[p	NUM
ma-25	289	56	,	,	PUNCT
ma-25	289	57	q	q	X
ma-25	289	58	]	]	X
ma-25	289	59	(	(	PUNCT
ma-25	289	60	f	f	X
ma-25	289	61	)	)	PUNCT
ma-25	289	62	=	=	PUNCT
ma-25	289	63	ρ[p	ρ[p	PROPN
ma-25	289	64	,	,	PUNCT
ma-25	289	65	q	q	X
ma-25	289	66	]	]	X
ma-25	289	67	(	(	PUNCT
ma-25	289	68	g	g	NOUN
ma-25	289	69	)	)	PUNCT
ma-25	289	70	≤	≤	NOUN
ma-25	290	1	+	+	PUNCT
ma-25	290	2	∞	∞	PROPN
ma-25	290	3	and	and	CCONJ
ma-25	290	4	λ[p	λ[p	PROPN
ma-25	290	5	,	,	PUNCT
ma-25	290	6	q	q	X
ma-25	290	7	]	]	X
ma-25	290	8	(	(	PUNCT
ma-25	290	9	d	d	NOUN
ma-25	290	10	)	)	PUNCT
ma-25	290	11	=	=	SYM
ma-25	290	12	ρ[p	ρ[p	PROPN
ma-25	290	13	,	,	PUNCT
ma-25	290	14	q	q	X
ma-25	290	15	]	]	X
ma-25	290	16	(	(	PUNCT
ma-25	290	17	d	d	NOUN
ma-25	290	18	)	)	PUNCT
ma-25	291	1	=	=	SYM
ma-25	291	2	λ[p	λ[p	NUM
ma-25	291	3	,	,	PUNCT
ma-25	291	4	q	q	X
ma-25	291	5	]	]	X
ma-25	291	6	(	(	PUNCT
ma-25	291	7	1	1	NUM
ma-25	291	8	f	f	NOUN
ma-25	291	9	)	)	PUNCT
ma-25	291	10	<	<	X
ma-25	291	11	µ.	µ.	NOUN
ma-25	291	12	then	then	ADV
ma-25	291	13	,	,	PUNCT
ma-25	291	14	there	there	PRON
ma-25	291	15	exists	exist	VERB
ma-25	291	16	a	a	DET
ma-25	291	17	set	set	NOUN
ma-25	291	18	e6	e6	PROPN
ma-25	291	19	⊂	⊂	PROPN
ma-25	291	20	(	(	PUNCT
ma-25	291	21	1,+∞	1,+∞	NUM
ma-25	291	22	)	)	PUNCT
ma-25	291	23	of	of	ADP
ma-25	291	24	finite	finite	ADJ
ma-25	291	25	logarithmic	logarithmic	ADJ
ma-25	291	26	measure	measure	NOUN
ma-25	291	27	such	such	ADJ
ma-25	291	28	that	that	PRON
ma-25	291	29	for	for	ADP
ma-25	291	30	all	all	PRON
ma-25	291	31	|z	|z	NOUN
ma-25	292	1	|	|	NOUN
ma-25	292	2	=	=	NOUN
ma-25	292	3	r	r	NOUN
ma-25	292	4	/∈	/∈	PUNCT
ma-25	293	1	[	[	X
ma-25	293	2	0	0	NUM
ma-25	293	3	,	,	PUNCT
ma-25	293	4	1	1	NUM
ma-25	293	5	]	]	PUNCT
ma-25	293	6	∪	∪	NOUN
ma-25	293	7	e6	e6	NOUN
ma-25	293	8	and	and	CCONJ
ma-25	293	9	|g	|g	NOUN
ma-25	293	10	(	(	PUNCT
ma-25	293	11	z	z	NOUN
ma-25	293	12	)	)	PUNCT
ma-25	294	1	|	|	ADV
ma-25	294	2	=	=	SYM
ma-25	294	3	m	m	PROPN
ma-25	294	4	(	(	PUNCT
ma-25	294	5	r	r	NOUN
ma-25	294	6	,	,	PUNCT
ma-25	294	7	g	g	NOUN
ma-25	294	8	)	)	PUNCT
ma-25	294	9	,	,	PUNCT
ma-25	294	10	we	we	PRON
ma-25	294	11	have∣∣∣∣	have∣∣∣∣	PROPN
ma-25	294	12	f	f	PROPN
ma-25	294	13	(	(	PUNCT
ma-25	294	14	z	z	PROPN
ma-25	294	15	)	)	PUNCT
ma-25	294	16	f	f	NOUN
ma-25	294	17	(	(	PUNCT
ma-25	294	18	s	s	NOUN
ma-25	294	19	)	)	PUNCT
ma-25	294	20	(	(	PUNCT
ma-25	294	21	z	z	NOUN
ma-25	294	22	)	)	PUNCT
ma-25	294	23	∣∣∣∣	∣∣∣∣	NOUN
ma-25	294	24	≤	≤	ADJ
ma-25	294	25	r2s	r2s	NOUN
ma-25	294	26	,	,	PUNCT
ma-25	294	27	(	(	PUNCT
ma-25	294	28	s	s	NOUN
ma-25	294	29	∈	∈	PROPN
ma-25	294	30	n	n	CCONJ
ma-25	294	31	)	)	PUNCT
ma-25	294	32	.	.	PUNCT
ma-25	295	1	proof	proof	NOUN
ma-25	295	2	.	.	PUNCT
ma-25	296	1	by	by	ADP
ma-25	296	2	lemma	lemma	PROPN
ma-25	296	3	2.7	2.7	NUM
ma-25	296	4	,	,	PUNCT
ma-25	296	5	there	there	PRON
ma-25	296	6	exists	exist	VERB
ma-25	296	7	a	a	DET
ma-25	296	8	set	set	ADJ
ma-25	296	9	e5	e5	PROPN
ma-25	296	10	of	of	ADP
ma-25	296	11	finite	finite	PROPN
ma-25	296	12	logarithmic	logarithmic	ADJ
ma-25	296	13	measure	measure	NOUN
ma-25	296	14	such	such	ADJ
ma-25	296	15	that	that	SCONJ
ma-25	296	16	the	the	DET
ma-25	296	17	estimation	estimation	NOUN
ma-25	296	18	f	f	X
ma-25	296	19	(	(	PUNCT
ma-25	296	20	s)(z	s)(z	PROPN
ma-25	296	21	)	)	PUNCT
ma-25	296	22	f	f	NOUN
ma-25	296	23	(	(	PUNCT
ma-25	296	24	z	z	NOUN
ma-25	296	25	)	)	PUNCT
ma-25	296	26	=	=	SYM
ma-25	296	27	(	(	PUNCT
ma-25	296	28	νg	νg	X
ma-25	296	29	(	(	PUNCT
ma-25	296	30	r	r	NOUN
ma-25	296	31	)	)	PUNCT
ma-25	296	32	z	z	NOUN
ma-25	296	33	)	)	PUNCT
ma-25	296	34	s	s	PART
ma-25	296	35	(	(	PUNCT
ma-25	296	36	1	1	NUM
ma-25	296	37	+	+	NUM
ma-25	296	38	o	o	NOUN
ma-25	296	39	(	(	PUNCT
ma-25	296	40	1	1	NUM
ma-25	296	41	)	)	PUNCT
ma-25	296	42	)	)	PUNCT
ma-25	296	43	(	(	PUNCT
ma-25	296	44	s	s	X
ma-25	296	45	≥	≥	NOUN
ma-25	296	46	1	1	NUM
ma-25	296	47	is	be	AUX
ma-25	296	48	an	an	DET
ma-25	296	49	integer	integer	NOUN
ma-25	296	50	)	)	PUNCT
ma-25	296	51	(	(	PUNCT
ma-25	296	52	2.20	2.20	NUM
ma-25	296	53	)	)	PUNCT
ma-25	296	54	holds	hold	VERB
ma-25	296	55	for	for	ADP
ma-25	296	56	all	all	PRON
ma-25	296	57	|z	|z	NOUN
ma-25	297	1	|	|	NOUN
ma-25	297	2	=	=	NOUN
ma-25	297	3	r	r	NOUN
ma-25	297	4	/∈	/∈	PUNCT
ma-25	298	1	[	[	X
ma-25	298	2	0	0	NUM
ma-25	298	3	,	,	PUNCT
ma-25	298	4	1]∪e5	1]∪e5	NUM
ma-25	298	5	and	and	CCONJ
ma-25	298	6	|g	|g	PROPN
ma-25	298	7	(	(	PUNCT
ma-25	298	8	z)|	z)|	X
ma-25	298	9	=	=	PRON
ma-25	298	10	m	m	PROPN
ma-25	298	11	(	(	PUNCT
ma-25	298	12	r	r	NOUN
ma-25	298	13	,	,	PUNCT
ma-25	298	14	g	g	NOUN
ma-25	298	15	)	)	PUNCT
ma-25	298	16	,	,	PUNCT
ma-25	298	17	where	where	SCONJ
ma-25	298	18	νg	νg	NOUN
ma-25	298	19	(	(	PUNCT
ma-25	298	20	r	r	NOUN
ma-25	298	21	)	)	PUNCT
ma-25	298	22	is	be	AUX
ma-25	298	23	the	the	DET
ma-25	298	24	central	central	ADJ
ma-25	298	25	index	index	NOUN
ma-25	298	26	of	of	ADP
ma-25	298	27	g.	g.	PROPN
ma-25	298	28	onthe	onthe	PROPN
ma-25	298	29	other	other	ADJ
ma-25	298	30	hand	hand	NOUN
ma-25	298	31	,	,	PUNCT
ma-25	298	32	by	by	ADP
ma-25	298	33	lemma	lemma	PROPN
ma-25	298	34	2.6	2.6	NUM
ma-25	298	35	,	,	PUNCT
ma-25	298	36	for	for	ADP
ma-25	298	37	any	any	DET
ma-25	298	38	given	give	VERB
ma-25	298	39	ε	ε	PROPN
ma-25	298	40	(	(	PUNCT
ma-25	298	41	0	0	X
ma-25	298	42	<	<	X
ma-25	298	43	ε	ε	X
ma-25	298	44	<	<	X
ma-25	298	45	1	1	NUM
ma-25	298	46	)	)	PUNCT
ma-25	298	47	,	,	PUNCT
ma-25	298	48	there	there	PRON
ma-25	298	49	exists	exist	VERB
ma-25	298	50	r	r	NOUN
ma-25	298	51	>	>	X
ma-25	298	52	1	1	NUM
ma-25	298	53	such	such	ADJ
ma-25	298	54	that	that	PRON
ma-25	298	55	for	for	ADP
ma-25	298	56	all	all	DET
ma-25	298	57	r	r	NOUN
ma-25	298	58	>	>	X
ma-25	298	59	r	r	NOUN
ma-25	298	60	,	,	PUNCT
ma-25	298	61	we	we	PRON
ma-25	298	62	have	have	VERB
ma-25	298	63	νg	νg	NOUN
ma-25	298	64	(	(	PUNCT
ma-25	298	65	r	r	NOUN
ma-25	298	66	)	)	PUNCT
ma-25	298	67	>	>	X
ma-25	298	68	expp	expp	ADJ
ma-25	298	69	{	{	PUNCT
ma-25	298	70	(	(	PUNCT
ma-25	298	71	µ−	µ−	PROPN
ma-25	298	72	ε	ε	PROPN
ma-25	298	73	)	)	PUNCT
ma-25	298	74	logq	logq	NOUN
ma-25	298	75	(	(	PUNCT
ma-25	298	76	r	r	NOUN
ma-25	298	77	)	)	PUNCT
ma-25	298	78	}	}	PUNCT
ma-25	298	79	.	.	PUNCT
ma-25	299	1	(	(	PUNCT
ma-25	299	2	2.21	2.21	NUM
ma-25	299	3	)	)	PUNCT
ma-25	299	4	if	if	SCONJ
ma-25	299	5	µ	µ	X
ma-25	299	6	=	=	SYM
ma-25	299	7	+	+	NOUN
ma-25	299	8	∞	∞	PROPN
ma-25	299	9	,	,	PUNCT
ma-25	299	10	then	then	ADV
ma-25	299	11	µ−	µ−	PROPN
ma-25	299	12	ε	ε	PROPN
ma-25	299	13	can	can	AUX
ma-25	299	14	be	be	AUX
ma-25	299	15	replaced	replace	VERB
ma-25	299	16	by	by	ADP
ma-25	299	17	a	a	DET
ma-25	299	18	large	large	ADJ
ma-25	299	19	enough	enough	ADJ
ma-25	299	20	real	real	ADJ
ma-25	299	21	number	number	NOUN
ma-25	299	22	m	m	PROPN
ma-25	299	23	.	.	PUNCT
ma-25	300	1	set	set	VERB
ma-25	300	2	e6	e6	PROPN
ma-25	301	1	=	=	PUNCT
ma-25	302	1	[	[	X
ma-25	302	2	1	1	NUM
ma-25	302	3	,	,	PUNCT
ma-25	302	4	r	r	NOUN
ma-25	302	5	]	]	PUNCT
ma-25	302	6	∪	∪	X
ma-25	302	7	e5	e5	PROPN
ma-25	302	8	,	,	PUNCT
ma-25	302	9	lm	lm	PROPN
ma-25	302	10	(	(	PUNCT
ma-25	302	11	e6	e6	PROPN
ma-25	302	12	)	)	PUNCT
ma-25	302	13	<	<	X
ma-25	303	1	+	+	PUNCT
ma-25	303	2	∞.	∞.	PROPN
ma-25	303	3	hence	hence	ADV
ma-25	303	4	from	from	ADP
ma-25	303	5	(	(	PUNCT
ma-25	303	6	2.20	2.20	NUM
ma-25	303	7	)	)	PUNCT
ma-25	303	8	and	and	CCONJ
ma-25	303	9	(	(	PUNCT
ma-25	303	10	2.21	2.21	NUM
ma-25	303	11	)	)	PUNCT
ma-25	303	12	,	,	PUNCT
ma-25	303	13	we	we	PRON
ma-25	303	14	obtain∣∣∣∣	obtain∣∣∣∣	ADJ
ma-25	303	15	f	f	PROPN
ma-25	303	16	(	(	PUNCT
ma-25	303	17	z	z	NOUN
ma-25	303	18	)	)	PUNCT
ma-25	303	19	f	f	NOUN
ma-25	303	20	(	(	PUNCT
ma-25	303	21	s	s	NOUN
ma-25	303	22	)	)	PUNCT
ma-25	303	23	(	(	PUNCT
ma-25	303	24	z	z	NOUN
ma-25	303	25	)	)	PUNCT
ma-25	303	26	∣∣∣∣	∣∣∣∣	NOUN
ma-25	303	27	=	=	SYM
ma-25	303	28	∣∣∣∣	∣∣∣∣	PROPN
ma-25	303	29	z	z	PROPN
ma-25	303	30	νg	νg	NOUN
ma-25	303	31	(	(	PUNCT
ma-25	303	32	r	r	NOUN
ma-25	303	33	)	)	PUNCT
ma-25	303	34	∣∣∣∣s	∣∣∣∣s	ADJ
ma-25	303	35	1	1	NUM
ma-25	303	36	|1	|1	PRON
ma-25	303	37	+	+	NUM
ma-25	303	38	o	o	X
ma-25	303	39	(	(	PUNCT
ma-25	303	40	1)|	1)|	NUM
ma-25	303	41	≤	≤	NOUN
ma-25	303	42	r	r	NOUN
ma-25	303	43	s	s	X
ma-25	303	44	(	(	PUNCT
ma-25	303	45	expp	expp	ADJ
ma-25	303	46	{	{	PUNCT
ma-25	303	47	(	(	PUNCT
ma-25	303	48	µ−	µ−	PROPN
ma-25	303	49	ε	ε	PROPN
ma-25	303	50	)	)	PUNCT
ma-25	303	51	logq	logq	NOUN
ma-25	303	52	(	(	PUNCT
ma-25	303	53	r	r	NOUN
ma-25	303	54	)	)	PUNCT
ma-25	303	55	}	}	PUNCT
ma-25	303	56	)	)	PUNCT
ma-25	303	57	s	s	VERB
ma-25	303	58	≤	≤	ADJ
ma-25	303	59	r2s	r2s	X
ma-25	303	60	,	,	PUNCT
ma-25	303	61	where	where	SCONJ
ma-25	303	62	|z	|z	PROPN
ma-25	304	1	|	|	ADV
ma-25	304	2	=	=	NOUN
ma-25	304	3	r	r	NOUN
ma-25	304	4	/∈	/∈	PUNCT
ma-25	305	1	[	[	X
ma-25	305	2	0	0	NUM
ma-25	305	3	,	,	PUNCT
ma-25	305	4	1	1	NUM
ma-25	305	5	]	]	PUNCT
ma-25	305	6	∪	∪	X
ma-25	305	7	e6	e6	NOUN
ma-25	305	8	,	,	PUNCT
ma-25	305	9	r	r	NOUN
ma-25	305	10	→	→	SYM
ma-25	305	11	+	+	NOUN
ma-25	305	12	∞	∞	NUM
ma-25	305	13	and	and	CCONJ
ma-25	305	14	|g	|g	NOUN
ma-25	305	15	(	(	PUNCT
ma-25	305	16	z)|	z)|	X
ma-25	305	17	=	=	PRON
ma-25	305	18	m	m	PROPN
ma-25	305	19	(	(	PUNCT
ma-25	305	20	r	r	NOUN
ma-25	305	21	,	,	PUNCT
ma-25	305	22	g	g	NOUN
ma-25	305	23	)	)	PUNCT
ma-25	305	24	.	.	PUNCT
ma-25	306	1	lemma	lemma	PROPN
ma-25	306	2	2.9	2.9	NUM
ma-25	306	3	(	(	PUNCT
ma-25	306	4	[	[	X
ma-25	306	5	6	6	NUM
ma-25	306	6	]	]	PUNCT
ma-25	306	7	)	)	PUNCT
ma-25	306	8	let	let	VERB
ma-25	306	9	ϕ	ϕ	NOUN
ma-25	306	10	:	:	PUNCT
ma-25	307	1	[	[	X
ma-25	307	2	0,+∞)→	0,+∞)→	NUM
ma-25	307	3	r	r	NOUN
ma-25	307	4	and	and	CCONJ
ma-25	307	5	ψ	ψ	X
ma-25	307	6	:	:	PUNCT
ma-25	308	1	[	[	X
ma-25	308	2	0,+∞)→	0,+∞)→	NUM
ma-25	308	3	r	r	NOUN
ma-25	308	4	be	be	VERB
ma-25	308	5	monotone	monotone	ADJ
ma-25	308	6	nondecreasing	nondecrease	VERB
ma-25	308	7	functions	function	NOUN
ma-25	308	8	such	such	ADJ
ma-25	308	9	that	that	SCONJ
ma-25	308	10	ϕ(r	ϕ(r	PROPN
ma-25	308	11	)	)	PUNCT
ma-25	308	12	≤	≤	NUM
ma-25	308	13	ψ(r	ψ(r	NOUN
ma-25	308	14	)	)	PUNCT
ma-25	308	15	for	for	ADP
ma-25	308	16	all	all	DET
ma-25	308	17	r	r	NOUN
ma-25	308	18	/∈	/∈	PUNCT
ma-25	309	1	(	(	PUNCT
ma-25	309	2	e7	e7	PROPN
ma-25	309	3	∪	∪	ADP
ma-25	309	4	[	[	X
ma-25	309	5	0	0	NUM
ma-25	309	6	,	,	PUNCT
ma-25	309	7	1	1	NUM
ma-25	309	8	]	]	PUNCT
ma-25	309	9	)	)	PUNCT
ma-25	309	10	,	,	PUNCT
ma-25	309	11	where	where	SCONJ
ma-25	309	12	e7	e7	PROPN
ma-25	309	13	is	be	AUX
ma-25	309	14	a	a	DET
ma-25	309	15	set	set	NOUN
ma-25	309	16	of	of	ADP
ma-25	309	17	finite	finite	ADJ
ma-25	309	18	logarithmic	logarithmic	ADJ
ma-25	309	19	measure	measure	NOUN
ma-25	309	20	.	.	PUNCT
ma-25	310	1	let	let	VERB
ma-25	310	2	α	α	PRON
ma-25	310	3	>	>	X
ma-25	310	4	1	1	NUM
ma-25	310	5	be	be	AUX
ma-25	310	6	a	a	DET
ma-25	310	7	given	give	VERB
ma-25	310	8	constant	constant	NOUN
ma-25	310	9	.	.	PUNCT
ma-25	311	1	then	then	ADV
ma-25	311	2	,	,	PUNCT
ma-25	311	3	there	there	PRON
ma-25	311	4	exists	exist	VERB
ma-25	311	5	an	an	DET
ma-25	311	6	r1	r1	PROPN
ma-25	311	7	=	=	SYM
ma-25	311	8	r1(α	r1(α	PROPN
ma-25	311	9	)	)	PUNCT
ma-25	311	10	>	>	X
ma-25	311	11	0	0	NUM
ma-25	312	1	such	such	ADJ
ma-25	312	2	that	that	SCONJ
ma-25	312	3	ϕ(r	ϕ(r	PROPN
ma-25	312	4	)	)	PUNCT
ma-25	312	5	≤	≤	PROPN
ma-25	312	6	ψ(αr	ψ(αr	ADV
ma-25	312	7	)	)	PUNCT
ma-25	312	8	for	for	ADP
ma-25	312	9	all	all	DET
ma-25	312	10	r	r	NOUN
ma-25	312	11	>	>	X
ma-25	312	12	r1	r1	PROPN
ma-25	312	13	.	.	PUNCT
ma-25	313	1	lemma	lemma	PROPN
ma-25	313	2	2.10	2.10	NUM
ma-25	313	3	(	(	PUNCT
ma-25	313	4	[	[	X
ma-25	313	5	19	19	NUM
ma-25	313	6	]	]	PUNCT
ma-25	313	7	)	)	PUNCT
ma-25	313	8	let	let	VERB
ma-25	313	9	f	f	PROPN
ma-25	313	10	(	(	PUNCT
ma-25	313	11	z	z	NOUN
ma-25	313	12	)	)	PUNCT
ma-25	313	13	=	=	SYM
ma-25	313	14	g(z	g(z	ADJ
ma-25	313	15	)	)	PUNCT
ma-25	313	16	d(z	d(z	PROPN
ma-25	313	17	)	)	PUNCT
ma-25	313	18	be	be	AUX
ma-25	313	19	a	a	DET
ma-25	313	20	meromorphic	meromorphic	ADJ
ma-25	313	21	function	function	NOUN
ma-25	313	22	,	,	PUNCT
ma-25	313	23	where	where	SCONJ
ma-25	313	24	g	g	PROPN
ma-25	313	25	(	(	PUNCT
ma-25	313	26	z	z	NOUN
ma-25	313	27	)	)	PUNCT
ma-25	313	28	,	,	PUNCT
ma-25	313	29	d	d	X
ma-25	313	30	(	(	PUNCT
ma-25	313	31	z	z	NOUN
ma-25	313	32	)	)	PUNCT
ma-25	313	33	are	be	AUX
ma-25	313	34	entire	entire	ADJ
ma-25	313	35	functions	function	NOUN
ma-25	313	36	.	.	PUNCT
ma-25	314	1	if	if	SCONJ
ma-25	314	2	0	0	NUM
ma-25	314	3	≤	≤	NUM
ma-25	314	4	ρ[p	ρ[p	NOUN
ma-25	314	5	,	,	PUNCT
ma-25	314	6	q	q	X
ma-25	314	7	]	]	X
ma-25	314	8	(	(	PUNCT
ma-25	314	9	d	d	X
ma-25	314	10	)	)	PUNCT
ma-25	314	11	<	<	X
ma-25	314	12	µ[p	µ[p	ADJ
ma-25	314	13	,	,	PUNCT
ma-25	314	14	q	q	X
ma-25	314	15	]	]	X
ma-25	314	16	(	(	PUNCT
ma-25	314	17	f	f	PROPN
ma-25	314	18	)	)	PUNCT
ma-25	314	19	,	,	PUNCT
ma-25	314	20	then	then	ADV
ma-25	314	21	µ[p	µ[p	VERB
ma-25	314	22	,	,	PUNCT
ma-25	314	23	q	q	X
ma-25	314	24	]	]	X
ma-25	314	25	(	(	PUNCT
ma-25	314	26	g	g	NOUN
ma-25	314	27	)	)	PUNCT
ma-25	314	28	=	=	SYM
ma-25	314	29	µ[p	µ[p	ADJ
ma-25	314	30	,	,	PUNCT
ma-25	314	31	q	q	X
ma-25	314	32	]	]	X
ma-25	314	33	(	(	PUNCT
ma-25	314	34	f	f	PROPN
ma-25	314	35	)	)	PUNCT
ma-25	314	36	and	and	CCONJ
ma-25	314	37	ρ[p	ρ[p	NOUN
ma-25	314	38	,	,	PUNCT
ma-25	314	39	q	q	X
ma-25	314	40	]	]	X
ma-25	314	41	(	(	PUNCT
ma-25	314	42	g	g	NOUN
ma-25	314	43	)	)	PUNCT
ma-25	314	44	=	=	PUNCT
ma-25	314	45	ρ[p	ρ[p	PROPN
ma-25	314	46	,	,	PUNCT
ma-25	314	47	q	q	X
ma-25	314	48	]	]	X
ma-25	314	49	(	(	PUNCT
ma-25	314	50	f	f	PROPN
ma-25	314	51	)	)	PUNCT
ma-25	314	52	.	.	PUNCT
ma-25	315	1	moreover	moreover	ADV
ma-25	315	2	,	,	PUNCT
ma-25	315	3	if	if	SCONJ
ma-25	315	4	ρ[p	ρ[p	NOUN
ma-25	315	5	,	,	PUNCT
ma-25	315	6	q	q	X
ma-25	315	7	]	]	X
ma-25	315	8	(	(	PUNCT
ma-25	315	9	f	f	X
ma-25	315	10	)	)	PUNCT
ma-25	315	11	=	=	PUNCT
ma-25	316	1	+	+	NUM
ma-25	316	2	∞	∞	PROPN
ma-25	316	3	,	,	PUNCT
ma-25	316	4	then	then	ADV
ma-25	316	5	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	316	6	]	]	PUNCT
ma-25	316	7	(	(	PUNCT
ma-25	316	8	g	g	NOUN
ma-25	316	9	)	)	PUNCT
ma-25	316	10	=	=	SYM
ma-25	316	11	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	316	12	]	]	PUNCT
ma-25	316	13	(	(	PUNCT
ma-25	316	14	f	f	PROPN
ma-25	316	15	)	)	PUNCT
ma-25	316	16	.	.	PUNCT
ma-25	317	1	lemma	lemma	PROPN
ma-25	317	2	2.11	2.11	NUM
ma-25	317	3	assume	assume	VERB
ma-25	317	4	that	that	SCONJ
ma-25	317	5	k	k	PROPN
ma-25	317	6	≥	≥	NUM
ma-25	317	7	2	2	NUM
ma-25	317	8	and	and	CCONJ
ma-25	317	9	a0	a0	PROPN
ma-25	317	10	,	,	PUNCT
ma-25	317	11	a1	a1	PROPN
ma-25	317	12	,	,	PUNCT
ma-25	317	13	...	...	PUNCT
ma-25	317	14	,	,	PUNCT
ma-25	317	15	ak	ak	PROPN
ma-25	317	16	6≡	6≡	NUM
ma-25	317	17	0	0	NUM
ma-25	317	18	,	,	PUNCT
ma-25	317	19	f	f	PROPN
ma-25	317	20	are	be	AUX
ma-25	317	21	meromorphic	meromorphic	ADJ
ma-25	317	22	functions	function	NOUN
ma-25	317	23	.	.	PUNCT
ma-25	318	1	let	let	VERB
ma-25	318	2	ρ	ρ	PROPN
ma-25	318	3	=	=	SYM
ma-25	318	4	max	max	PROPN
ma-25	318	5	{	{	PUNCT
ma-25	318	6	ρ[p	ρ[p	PROPN
ma-25	318	7	,	,	PUNCT
ma-25	318	8	q	q	X
ma-25	318	9	]	]	X
ma-25	318	10	(	(	PUNCT
ma-25	318	11	aj	aj	PROPN
ma-25	318	12	)	)	PUNCT
ma-25	318	13	(	(	PUNCT
ma-25	318	14	j	j	NOUN
ma-25	318	15	=	=	SYM
ma-25	318	16	0	0	NUM
ma-25	318	17	,	,	PUNCT
ma-25	318	18	1	1	NUM
ma-25	318	19	,	,	PUNCT
ma-25	318	20	...	...	PUNCT
ma-25	318	21	,	,	PUNCT
ma-25	318	22	k	k	NOUN
ma-25	318	23	)	)	PUNCT
ma-25	318	24	,	,	PUNCT
ma-25	318	25	ρ[p	ρ[p	PROPN
ma-25	318	26	,	,	PUNCT
ma-25	318	27	q	q	X
ma-25	318	28	]	]	X
ma-25	318	29	(	(	PUNCT
ma-25	318	30	f	f	PROPN
ma-25	318	31	)	)	PUNCT
ma-25	318	32	}	}	PUNCT
ma-25	318	33	<	<	X
ma-25	318	34	∞	∞	PROPN
ma-25	318	35	and	and	CCONJ
ma-25	318	36	let	let	VERB
ma-25	318	37	f	f	PRON
ma-25	318	38	be	be	AUX
ma-25	318	39	a	a	DET
ma-25	318	40	meromorphic	meromorphic	ADJ
ma-25	318	41	solution	solution	NOUN
ma-25	318	42	of	of	ADP
ma-25	318	43	infinite	infinite	ADJ
ma-25	318	44	[	[	X
ma-25	318	45	p	p	NOUN
ma-25	318	46	,	,	PUNCT
ma-25	318	47	q]-order	q]-order	NOUN
ma-25	318	48	of	of	ADP
ma-25	318	49	equation	equation	NOUN
ma-25	318	50	(	(	PUNCT
ma-25	318	51	1.4	1.4	NUM
ma-25	318	52	)	)	PUNCT
ma-25	318	53	with	with	ADP
ma-25	318	54	λ[p	λ[p	PROPN
ma-25	318	55	,	,	PUNCT
ma-25	318	56	q	q	X
ma-25	318	57	]	]	X
ma-25	318	58	(	(	PUNCT
ma-25	318	59	1	1	NUM
ma-25	318	60	f	f	NOUN
ma-25	318	61	)	)	PUNCT
ma-25	318	62	<	<	X
ma-25	318	63	µ[p	µ[p	ADJ
ma-25	318	64	,	,	PUNCT
ma-25	318	65	q	q	X
ma-25	318	66	]	]	X
ma-25	318	67	(	(	PUNCT
ma-25	318	68	f	f	PROPN
ma-25	318	69	)	)	PUNCT
ma-25	318	70	.	.	PUNCT
ma-25	319	1	then	then	ADV
ma-25	319	2	,	,	PUNCT
ma-25	319	3	ρ[p+1,q](f	ρ[p+1,q](f	PROPN
ma-25	319	4	)	)	PUNCT
ma-25	319	5	≤	≤	NOUN
ma-25	319	6	ρ	ρ	NOUN
ma-25	319	7	.	.	PUNCT
ma-25	320	1	proof	proof	NOUN
ma-25	320	2	.	.	PUNCT
ma-25	321	1	let	let	VERB
ma-25	321	2	f	f	PRON
ma-25	321	3	be	be	AUX
ma-25	321	4	a	a	DET
ma-25	321	5	meromorphic	meromorphic	ADJ
ma-25	321	6	solution	solution	NOUN
ma-25	321	7	of	of	ADP
ma-25	321	8	infinite	infinite	ADJ
ma-25	321	9	[	[	X
ma-25	321	10	p	p	NOUN
ma-25	321	11	,	,	PUNCT
ma-25	321	12	q]-order	q]-order	NOUN
ma-25	321	13	of	of	ADP
ma-25	321	14	equation	equation	NOUN
ma-25	321	15	(	(	PUNCT
ma-25	321	16	1.4	1.4	NUM
ma-25	321	17	)	)	PUNCT
ma-25	321	18	with	with	ADP
ma-25	321	19	λ[p	λ[p	PROPN
ma-25	321	20	,	,	PUNCT
ma-25	321	21	q	q	X
ma-25	321	22	]	]	X
ma-25	321	23	(	(	PUNCT
ma-25	321	24	1f	1f	NUM
ma-25	321	25	)	)	PUNCT
ma-25	321	26	<	<	X
ma-25	321	27	µ[p	µ[p	ADJ
ma-25	321	28	,	,	PUNCT
ma-25	321	29	q	q	X
ma-25	321	30	]	]	X
ma-25	321	31	(	(	PUNCT
ma-25	321	32	f	f	PROPN
ma-25	321	33	)	)	PUNCT
ma-25	321	34	.	.	PUNCT
ma-25	322	1	so	so	ADV
ma-25	322	2	,	,	PUNCT
ma-25	322	3	we	we	PRON
ma-25	322	4	can	can	AUX
ma-25	322	5	use	use	VERB
ma-25	322	6	hadamard	hadamard	ADJ
ma-25	322	7	factorization	factorization	NOUN
ma-25	322	8	theorem	theorem	NOUN
ma-25	322	9	and	and	CCONJ
ma-25	322	10	write	write	VERB
ma-25	322	11	f	f	PROPN
ma-25	322	12	as	as	ADP
ma-25	322	13	f	f	PROPN
ma-25	322	14	(	(	PUNCT
ma-25	322	15	z	z	NOUN
ma-25	322	16	)	)	PUNCT
ma-25	322	17	=	=	SYM
ma-25	322	18	g(z	g(z	ADJ
ma-25	322	19	)	)	PUNCT
ma-25	322	20	d(z	d(z	PROPN
ma-25	322	21	)	)	PUNCT
ma-25	322	22	,	,	PUNCT
ma-25	322	23	where	where	SCONJ
ma-25	322	24	g(z)and	g(z)and	PROPN
ma-25	322	25	d(z	d(z	PROPN
ma-25	322	26	)	)	PUNCT
ma-25	322	27	are	be	AUX
ma-25	322	28	entire	entire	ADJ
ma-25	322	29	functions	function	NOUN
ma-25	322	30	satisfying	satisfy	VERB
ma-25	322	31	µ[p	µ[p	ADJ
ma-25	322	32	,	,	PUNCT
ma-25	322	33	q	q	X
ma-25	322	34	]	]	X
ma-25	322	35	(	(	PUNCT
ma-25	322	36	g	g	NOUN
ma-25	322	37	)	)	PUNCT
ma-25	322	38	=	=	SYM
ma-25	322	39	µ[p	µ[p	ADJ
ma-25	322	40	,	,	PUNCT
ma-25	322	41	q	q	X
ma-25	322	42	]	]	X
ma-25	322	43	(	(	PUNCT
ma-25	322	44	f	f	X
ma-25	322	45	)	)	PUNCT
ma-25	322	46	=	=	SYM
ma-25	322	47	µ	µ	PRON
ma-25	322	48	≤	≤	NUM
ma-25	322	49	ρ[p	ρ[p	NUM
ma-25	322	50	,	,	PUNCT
ma-25	322	51	q	q	X
ma-25	322	52	]	]	X
ma-25	322	53	(	(	PUNCT
ma-25	322	54	f	f	X
ma-25	322	55	)	)	PUNCT
ma-25	322	56	=	=	PUNCT
ma-25	322	57	ρ[p	ρ[p	PROPN
ma-25	322	58	,	,	PUNCT
ma-25	322	59	q	q	X
ma-25	322	60	]	]	X
ma-25	322	61	(	(	PUNCT
ma-25	322	62	g	g	NOUN
ma-25	322	63	)	)	PUNCT
ma-25	322	64	≤	≤	NOUN
ma-25	323	1	+	+	PROPN
ma-25	323	2	∞	∞	PROPN
ma-25	323	3	eur	eur	NOUN
ma-25	323	4	.	.	PUNCT
ma-25	324	1	j.	j.	PROPN
ma-25	324	2	math	math	PROPN
ma-25	324	3	.	.	PUNCT
ma-25	325	1	anal	anal	ADJ
ma-25	325	2	.	.	PUNCT
ma-25	326	1	1	1	NUM
ma-25	326	2	(	(	PUNCT
ma-25	326	3	2021	2021	NUM
ma-25	326	4	)	)	PUNCT
ma-25	327	1	96and	96and	NOUN
ma-25	327	2	λ[p	λ[p	NUM
ma-25	327	3	,	,	PUNCT
ma-25	327	4	q	q	X
ma-25	327	5	]	]	X
ma-25	327	6	(	(	PUNCT
ma-25	327	7	d	d	NOUN
ma-25	327	8	)	)	PUNCT
ma-25	327	9	=	=	SYM
ma-25	327	10	ρ[p	ρ[p	PROPN
ma-25	327	11	,	,	PUNCT
ma-25	327	12	q	q	X
ma-25	327	13	]	]	X
ma-25	327	14	(	(	PUNCT
ma-25	327	15	d	d	NOUN
ma-25	327	16	)	)	PUNCT
ma-25	327	17	=	=	SYM
ma-25	327	18	λ[p	λ[p	NUM
ma-25	327	19	,	,	PUNCT
ma-25	327	20	q	q	X
ma-25	327	21	]	]	X
ma-25	327	22	(	(	PUNCT
ma-25	327	23	1	1	NUM
ma-25	327	24	f	f	NOUN
ma-25	327	25	)	)	PUNCT
ma-25	327	26	<	<	X
ma-25	327	27	µ.	µ.	PROPN
ma-25	327	28	by	by	ADP
ma-25	327	29	lemma	lemma	PROPN
ma-25	327	30	2.3	2.3	NUM
ma-25	327	31	,	,	PUNCT
ma-25	327	32	there	there	PRON
ma-25	327	33	exists	exist	VERB
ma-25	327	34	a	a	DET
ma-25	327	35	set	set	NOUN
ma-25	327	36	e3	e3	NOUN
ma-25	327	37	⊂	⊂	X
ma-25	327	38	(	(	PUNCT
ma-25	327	39	1,+∞)of	1,+∞)of	NUM
ma-25	327	40	r	r	NOUN
ma-25	327	41	with	with	ADP
ma-25	327	42	a	a	DET
ma-25	327	43	finite	finite	ADJ
ma-25	327	44	linear	linear	ADJ
ma-25	327	45	measure	measure	NOUN
ma-25	327	46	such	such	ADJ
ma-25	327	47	that	that	PRON
ma-25	327	48	for	for	ADP
ma-25	327	49	all	all	PRON
ma-25	327	50	|z	|z	NOUN
ma-25	328	1	|	|	NOUN
ma-25	329	1	=	=	SYM
ma-25	330	1	r	r	NOUN
ma-25	330	2	/∈	/∈	PUNCT
ma-25	330	3	e3	e3	NOUN
ma-25	330	4	and	and	CCONJ
ma-25	330	5	any	any	DET
ma-25	330	6	given	give	VERB
ma-25	330	7	ε	ε	PROPN
ma-25	330	8	(	(	PUNCT
ma-25	330	9	0	0	PUNCT
ma-25	330	10	<	<	X
ma-25	330	11	2ε	2ε	PROPN
ma-25	330	12	<	<	X
ma-25	330	13	µ[p	µ[p	PROPN
ma-25	330	14	,	,	PUNCT
ma-25	330	15	q	q	X
ma-25	330	16	]	]	X
ma-25	330	17	(	(	PUNCT
ma-25	330	18	f	f	NOUN
ma-25	330	19	)	)	PUNCT
ma-25	330	20	−	−	PROPN
ma-25	330	21	ρ[p	ρ[p	NOUN
ma-25	330	22	,	,	PUNCT
ma-25	330	23	q	q	X
ma-25	330	24	]	]	X
ma-25	330	25	(	(	PUNCT
ma-25	330	26	d	d	NOUN
ma-25	330	27	)	)	PUNCT
ma-25	330	28	)	)	PUNCT
ma-25	330	29	,	,	PUNCT
ma-25	330	30	we	we	PRON
ma-25	330	31	have	have	VERB
ma-25	330	32	|aj	|aj	NUM
ma-25	330	33	(	(	PUNCT
ma-25	330	34	z	z	NOUN
ma-25	330	35	)	)	PUNCT
ma-25	330	36	|	|	ADV
ma-25	330	37	≤	≤	NUM
ma-25	330	38	expp+1	expp+1	NOUN
ma-25	330	39	{	{	PUNCT
ma-25	330	40	(	(	PUNCT
ma-25	330	41	ρ(p	ρ(p	PROPN
ma-25	330	42	,	,	PUNCT
ma-25	330	43	q	q	NOUN
ma-25	330	44	)	)	PUNCT
ma-25	330	45	(	(	PUNCT
ma-25	330	46	aj	aj	PROPN
ma-25	330	47	)	)	PUNCT
ma-25	331	1	+	+	CCONJ
ma-25	331	2	ε	ε	PROPN
ma-25	331	3	)	)	PUNCT
ma-25	331	4	logq	logq	VERB
ma-25	331	5	r	r	NOUN
ma-25	331	6	}	}	PUNCT
ma-25	331	7	≤	≤	NUM
ma-25	331	8	expp+1	expp+1	NOUN
ma-25	331	9	{	{	PUNCT
ma-25	331	10	(	(	PUNCT
ma-25	331	11	ρ+	ρ+	NUM
ma-25	331	12	ε	ε	NOUN
ma-25	331	13	)	)	PUNCT
ma-25	331	14	logq	logq	VERB
ma-25	331	15	r	r	NOUN
ma-25	331	16	}	}	PUNCT
ma-25	331	17	,	,	PUNCT
ma-25	331	18	j	j	PROPN
ma-25	331	19	=	=	SYM
ma-25	331	20	0	0	NUM
ma-25	331	21	,	,	PUNCT
ma-25	331	22	1	1	NUM
ma-25	331	23	,	,	PUNCT
ma-25	331	24	...	...	PUNCT
ma-25	331	25	,	,	PUNCT
ma-25	331	26	k	k	PROPN
ma-25	332	1	−	−	PROPN
ma-25	332	2	1	1	NUM
ma-25	332	3	,	,	PUNCT
ma-25	332	4	(	(	PUNCT
ma-25	332	5	2.22	2.22	NUM
ma-25	332	6	)	)	PUNCT
ma-25	332	7	|ak	|ak	X
ma-25	332	8	(	(	PUNCT
ma-25	332	9	z	z	NOUN
ma-25	332	10	)	)	PUNCT
ma-25	332	11	|	|	ADV
ma-25	332	12	≥	≥	NOUN
ma-25	332	13	exp	exp	NOUN
ma-25	332	14	{	{	PUNCT
ma-25	332	15	−	−	PUNCT
ma-25	332	16	expp	expp	ADJ
ma-25	332	17	{	{	PUNCT
ma-25	332	18	(	(	PUNCT
ma-25	332	19	ρ(p	ρ(p	PROPN
ma-25	332	20	,	,	PUNCT
ma-25	332	21	q	q	NOUN
ma-25	332	22	)	)	PUNCT
ma-25	332	23	(	(	PUNCT
ma-25	332	24	ak	ak	PROPN
ma-25	332	25	)	)	PUNCT
ma-25	332	26	+	+	CCONJ
ma-25	332	27	ε	ε	PROPN
ma-25	332	28	)	)	PUNCT
ma-25	332	29	logq	logq	VERB
ma-25	332	30	r	r	NOUN
ma-25	332	31	}	}	PUNCT
ma-25	332	32	}	}	PUNCT
ma-25	332	33	≥	≥	NOUN
ma-25	332	34	exp	exp	NOUN
ma-25	332	35	{	{	PUNCT
ma-25	332	36	−	−	PUNCT
ma-25	332	37	expp	expp	ADJ
ma-25	332	38	{	{	PUNCT
ma-25	332	39	(	(	PUNCT
ma-25	332	40	ρ+	ρ+	NUM
ma-25	332	41	ε	ε	NOUN
ma-25	332	42	)	)	PUNCT
ma-25	332	43	logq	logq	VERB
ma-25	332	44	r	r	NOUN
ma-25	332	45	}	}	PUNCT
ma-25	332	46	}	}	PUNCT
ma-25	332	47	(	(	PUNCT
ma-25	332	48	2.23)and	2.23)and	NUM
ma-25	332	49	|f	|f	PROPN
ma-25	333	1	(	(	PUNCT
ma-25	333	2	z)|	z)|	ADP
ma-25	333	3	≤	≤	NUM
ma-25	333	4	expp+1	expp+1	PROPN
ma-25	333	5	{	{	PUNCT
ma-25	333	6	(	(	PUNCT
ma-25	333	7	ρ(p	ρ(p	PROPN
ma-25	333	8	,	,	PUNCT
ma-25	333	9	q	q	NOUN
ma-25	333	10	)	)	PUNCT
ma-25	333	11	(	(	PUNCT
ma-25	333	12	f	f	PROPN
ma-25	333	13	)	)	PUNCT
ma-25	333	14	+	+	CCONJ
ma-25	333	15	ε	ε	PROPN
ma-25	333	16	)	)	PUNCT
ma-25	333	17	logq	logq	VERB
ma-25	333	18	r	r	NOUN
ma-25	333	19	}	}	PUNCT
ma-25	333	20	≤	≤	NUM
ma-25	333	21	expp+1	expp+1	NOUN
ma-25	333	22	{	{	PUNCT
ma-25	333	23	(	(	PUNCT
ma-25	333	24	ρ+	ρ+	NUM
ma-25	333	25	ε	ε	NOUN
ma-25	333	26	)	)	PUNCT
ma-25	333	27	logq	logq	VERB
ma-25	333	28	r	r	NOUN
ma-25	333	29	}	}	PUNCT
ma-25	333	30	.	.	PUNCT
ma-25	334	1	(	(	PUNCT
ma-25	334	2	2.24)by	2.24)by	NUM
ma-25	334	3	(	(	PUNCT
ma-25	334	4	2.24	2.24	NUM
ma-25	334	5	)	)	PUNCT
ma-25	334	6	,	,	PUNCT
ma-25	334	7	for	for	ADP
ma-25	334	8	all	all	DET
ma-25	334	9	z	z	NOUN
ma-25	334	10	satisfying	satisfy	VERB
ma-25	334	11	|z	|z	PROPN
ma-25	335	1	|	|	ADV
ma-25	335	2	=	=	NOUN
ma-25	335	3	r	r	NOUN
ma-25	335	4	/∈	/∈	PUNCT
ma-25	335	5	e3	e3	NOUN
ma-25	335	6	at	at	ADP
ma-25	335	7	which	which	PRON
ma-25	335	8	|g(z)|	|g(z)|	PROPN
ma-25	335	9	=	=	SYM
ma-25	335	10	m(r	m(r	PROPN
ma-25	335	11	,	,	PUNCT
ma-25	335	12	g	g	NOUN
ma-25	335	13	)	)	PUNCT
ma-25	335	14	and	and	CCONJ
ma-25	335	15	any	any	DET
ma-25	335	16	given	give	VERB
ma-25	335	17	ε	ε	PROPN
ma-25	335	18	(	(	PUNCT
ma-25	335	19	0	0	PUNCT
ma-25	335	20	<	<	X
ma-25	335	21	2ε	2ε	PROPN
ma-25	335	22	<	<	X
ma-25	335	23	µ[p	µ[p	PROPN
ma-25	335	24	,	,	PUNCT
ma-25	335	25	q	q	X
ma-25	335	26	]	]	X
ma-25	335	27	(	(	PUNCT
ma-25	335	28	f	f	NOUN
ma-25	335	29	)	)	PUNCT
ma-25	335	30	−	−	PROPN
ma-25	335	31	ρ[p	ρ[p	NOUN
ma-25	335	32	,	,	PUNCT
ma-25	335	33	q	q	X
ma-25	335	34	]	]	X
ma-25	335	35	(	(	PUNCT
ma-25	335	36	d	d	NOUN
ma-25	335	37	)	)	PUNCT
ma-25	335	38	)	)	PUNCT
ma-25	335	39	,	,	PUNCT
ma-25	335	40	we	we	PRON
ma-25	335	41	obtain∣∣∣∣f	obtain∣∣∣∣f	VERB
ma-25	335	42	(	(	PUNCT
ma-25	335	43	z	z	NOUN
ma-25	335	44	)	)	PUNCT
ma-25	335	45	f	f	NOUN
ma-25	335	46	(	(	PUNCT
ma-25	335	47	z	z	NOUN
ma-25	335	48	)	)	PUNCT
ma-25	335	49	∣∣∣∣	∣∣∣∣	PROPN
ma-25	335	50	=	=	SYM
ma-25	335	51	|f	|f	PROPN
ma-25	335	52	(	(	PUNCT
ma-25	335	53	z)|	z)|	INTJ
ma-25	335	54	|g(z)|	|g(z)|	PROPN
ma-25	335	55	|d	|d	NOUN
ma-25	335	56	(	(	PUNCT
ma-25	335	57	z)|	z)|	ADP
ma-25	335	58	≤	≤	NUM
ma-25	335	59	expp+1	expp+1	NOUN
ma-25	335	60	{	{	PUNCT
ma-25	335	61	(	(	PUNCT
ma-25	335	62	ρ[p	ρ[p	NOUN
ma-25	335	63	,	,	PUNCT
ma-25	335	64	q	q	X
ma-25	335	65	]	]	X
ma-25	335	66	(	(	PUNCT
ma-25	335	67	d	d	NOUN
ma-25	335	68	)	)	PUNCT
ma-25	336	1	+	+	CCONJ
ma-25	336	2	ε	ε	PROPN
ma-25	336	3	)	)	PUNCT
ma-25	336	4	logq	logq	VERB
ma-25	336	5	r	r	NOUN
ma-25	336	6	}	}	PUNCT
ma-25	336	7	expp+1	expp+1	NOUN
ma-25	336	8	{	{	PUNCT
ma-25	336	9	(	(	PUNCT
ma-25	336	10	ρ+	ρ+	NUM
ma-25	336	11	ε	ε	NOUN
ma-25	336	12	)	)	PUNCT
ma-25	336	13	logq	logq	VERB
ma-25	336	14	r	r	NOUN
ma-25	336	15	}	}	PUNCT
ma-25	336	16	expp+1	expp+1	NOUN
ma-25	336	17	{	{	PUNCT
ma-25	336	18	(	(	PUNCT
ma-25	336	19	µ[p	µ[p	ADJ
ma-25	336	20	,	,	PUNCT
ma-25	336	21	q	q	X
ma-25	336	22	]	]	X
ma-25	336	23	(	(	PUNCT
ma-25	336	24	f	f	NOUN
ma-25	336	25	)	)	PUNCT
ma-25	336	26	−	−	ADP
ma-25	336	27	ε	ε	PROPN
ma-25	336	28	)	)	PUNCT
ma-25	336	29	logq	logq	VERB
ma-25	336	30	r	r	NOUN
ma-25	336	31	}	}	PUNCT
ma-25	336	32	≤	≤	NUM
ma-25	336	33	expp+1	expp+1	NOUN
ma-25	336	34	{	{	PUNCT
ma-25	336	35	(	(	PUNCT
ma-25	336	36	ρ+	ρ+	NUM
ma-25	336	37	ε	ε	NOUN
ma-25	336	38	)	)	PUNCT
ma-25	336	39	logq	logq	VERB
ma-25	336	40	r	r	NOUN
ma-25	336	41	}	}	PUNCT
ma-25	336	42	.	.	PUNCT
ma-25	337	1	(	(	PUNCT
ma-25	337	2	2.25)by	2.25)by	NUM
ma-25	337	3	lemma	lemma	PROPN
ma-25	337	4	2.7	2.7	NUM
ma-25	337	5	,	,	PUNCT
ma-25	337	6	there	there	PRON
ma-25	337	7	exists	exist	VERB
ma-25	337	8	a	a	DET
ma-25	337	9	set	set	NOUN
ma-25	337	10	e5	e5	PROPN
ma-25	337	11	⊂	⊂	PROPN
ma-25	337	12	(	(	PUNCT
ma-25	337	13	1,+∞	1,+∞	NUM
ma-25	337	14	)	)	PUNCT
ma-25	337	15	of	of	ADP
ma-25	337	16	finite	finite	ADJ
ma-25	337	17	logarithmic	logarithmic	ADJ
ma-25	337	18	measure	measure	NOUN
ma-25	337	19	such	such	ADJ
ma-25	337	20	that	that	PRON
ma-25	337	21	for	for	ADP
ma-25	337	22	all	all	PRON
ma-25	337	23	|z	|z	NOUN
ma-25	338	1	|	|	NOUN
ma-25	338	2	=	=	NOUN
ma-25	338	3	r	r	NOUN
ma-25	338	4	/∈	/∈	PUNCT
ma-25	339	1	[	[	X
ma-25	339	2	0	0	NUM
ma-25	339	3	,	,	PUNCT
ma-25	339	4	1	1	NUM
ma-25	339	5	]	]	PUNCT
ma-25	339	6	∪	∪	X
ma-25	339	7	e5	e5	PROPN
ma-25	339	8	and	and	CCONJ
ma-25	339	9	|g	|g	NOUN
ma-25	339	10	(	(	PUNCT
ma-25	339	11	z	z	NOUN
ma-25	339	12	)	)	PUNCT
ma-25	340	1	|	|	ADV
ma-25	340	2	=	=	SYM
ma-25	340	3	m	m	PROPN
ma-25	340	4	(	(	PUNCT
ma-25	340	5	r	r	NOUN
ma-25	340	6	,	,	PUNCT
ma-25	340	7	g	g	NOUN
ma-25	340	8	)	)	PUNCT
ma-25	340	9	,	,	PUNCT
ma-25	340	10	we	we	PRON
ma-25	340	11	have	have	VERB
ma-25	340	12	f	f	PROPN
ma-25	340	13	(	(	PUNCT
ma-25	340	14	j	j	PROPN
ma-25	340	15	)	)	PUNCT
ma-25	340	16	(	(	PUNCT
ma-25	340	17	z	z	X
ma-25	340	18	)	)	PUNCT
ma-25	340	19	f	f	NOUN
ma-25	340	20	(	(	PUNCT
ma-25	340	21	z	z	NOUN
ma-25	340	22	)	)	PUNCT
ma-25	340	23	=	=	SYM
ma-25	340	24	(	(	PUNCT
ma-25	340	25	νg	νg	X
ma-25	340	26	(	(	PUNCT
ma-25	340	27	r	r	NOUN
ma-25	340	28	)	)	PUNCT
ma-25	340	29	z	z	NOUN
ma-25	340	30	)	)	PUNCT
ma-25	340	31	j	j	NOUN
ma-25	340	32	(	(	PUNCT
ma-25	340	33	1	1	NUM
ma-25	341	1	+	+	NUM
ma-25	341	2	o	o	NOUN
ma-25	341	3	(	(	PUNCT
ma-25	341	4	1	1	NUM
ma-25	341	5	)	)	PUNCT
ma-25	341	6	)	)	PUNCT
ma-25	341	7	,	,	PUNCT
ma-25	341	8	j	j	PROPN
ma-25	341	9	=	=	SYM
ma-25	341	10	1	1	NUM
ma-25	341	11	,	,	PUNCT
ma-25	341	12	...	...	PUNCT
ma-25	341	13	,	,	PUNCT
ma-25	341	14	k.	k.	PROPN
ma-25	341	15	(	(	PUNCT
ma-25	341	16	2.26	2.26	NUM
ma-25	341	17	)	)	PUNCT
ma-25	341	18	we	we	PRON
ma-25	341	19	can	can	AUX
ma-25	341	20	rewrite	rewrite	VERB
ma-25	341	21	(	(	PUNCT
ma-25	341	22	1.4	1.4	NUM
ma-25	341	23	)	)	PUNCT
ma-25	341	24	as∣∣∣∣∣	as∣∣∣∣∣	ADP
ma-25	341	25	f	f	PROPN
ma-25	341	26	(	(	PUNCT
ma-25	341	27	k	k	NOUN
ma-25	341	28	)	)	PUNCT
ma-25	341	29	(	(	PUNCT
ma-25	341	30	z	z	X
ma-25	341	31	)	)	PUNCT
ma-25	341	32	f	f	NOUN
ma-25	341	33	(	(	PUNCT
ma-25	341	34	z	z	NOUN
ma-25	341	35	)	)	PUNCT
ma-25	341	36	∣∣∣∣∣	∣∣∣∣∣	ADP
ma-25	342	1	≤	≤	ADV
ma-25	342	2	1	1	NUM
ma-25	342	3	|ak	|ak	NOUN
ma-25	342	4	(	(	PUNCT
ma-25	342	5	z	z	NOUN
ma-25	342	6	)	)	PUNCT
ma-25	343	1	|	|	ADV
ma-25	343	2	|a0	|a0	NUM
ma-25	343	3	(	(	PUNCT
ma-25	343	4	z	z	NOUN
ma-25	343	5	)	)	PUNCT
ma-25	343	6	|+	|+	NOUN
ma-25	344	1	∣∣∣∣f	∣∣∣∣f	NOUN
ma-25	345	1	(	(	PUNCT
ma-25	345	2	z	z	X
ma-25	345	3	)	)	PUNCT
ma-25	345	4	f	f	NOUN
ma-25	345	5	(	(	PUNCT
ma-25	345	6	z	z	NOUN
ma-25	345	7	)	)	PUNCT
ma-25	345	8	∣∣∣∣+	∣∣∣∣+	NOUN
ma-25	345	9	k−1∑	k−1∑	PROPN
ma-25	345	10	j=1	j=1	PROPN
ma-25	345	11	|aj	|aj	PUNCT
ma-25	345	12	(	(	PUNCT
ma-25	345	13	z	z	NOUN
ma-25	345	14	)	)	PUNCT
ma-25	345	15	|	|	ADV
ma-25	345	16	∣∣∣∣∣	∣∣∣∣∣	PROPN
ma-25	346	1	f	f	PROPN
ma-25	346	2	(	(	PUNCT
ma-25	346	3	j	j	PROPN
ma-25	346	4	)	)	PUNCT
ma-25	346	5	(	(	PUNCT
ma-25	346	6	z	z	X
ma-25	346	7	)	)	PUNCT
ma-25	346	8	f	f	NOUN
ma-25	346	9	(	(	PUNCT
ma-25	346	10	z	z	NOUN
ma-25	346	11	)	)	PUNCT
ma-25	346	12	∣∣∣∣∣	∣∣∣∣∣	PUNCT
ma-25	347	1			PROPN
ma-25	347	2	.	.	PUNCT
ma-25	348	1	(	(	PUNCT
ma-25	348	2	2.27	2.27	NUM
ma-25	348	3	)	)	PUNCT
ma-25	348	4	by	by	ADP
ma-25	348	5	substituting	substitute	VERB
ma-25	348	6	(	(	PUNCT
ma-25	348	7	2.22	2.22	NUM
ma-25	348	8	)	)	PUNCT
ma-25	348	9	,	,	PUNCT
ma-25	348	10	(	(	PUNCT
ma-25	348	11	2.23	2.23	NUM
ma-25	348	12	)	)	PUNCT
ma-25	348	13	,	,	PUNCT
ma-25	348	14	(	(	PUNCT
ma-25	348	15	2.25	2.25	NUM
ma-25	348	16	)	)	PUNCT
ma-25	348	17	and	and	CCONJ
ma-25	348	18	(	(	PUNCT
ma-25	348	19	2.26	2.26	NUM
ma-25	348	20	)	)	PUNCT
ma-25	348	21	into	into	ADP
ma-25	348	22	(	(	PUNCT
ma-25	348	23	2.27	2.27	NUM
ma-25	348	24	)	)	PUNCT
ma-25	348	25	,	,	PUNCT
ma-25	348	26	we	we	PRON
ma-25	348	27	obtain∣∣∣∣νg	obtain∣∣∣∣νg	PROPN
ma-25	348	28	(	(	PUNCT
ma-25	348	29	r	r	NOUN
ma-25	348	30	)	)	PUNCT
ma-25	348	31	z	z	NOUN
ma-25	348	32	∣∣∣∣k	∣∣∣∣k	NOUN
ma-25	348	33	|1	|1	PRON
ma-25	349	1	+	+	NUM
ma-25	349	2	o	o	AUX
ma-25	349	3	(	(	PUNCT
ma-25	349	4	1)|	1)|	NUM
ma-25	349	5	≤	≤	NUM
ma-25	349	6	1	1	NUM
ma-25	349	7	exp	exp	NOUN
ma-25	349	8	{	{	PUNCT
ma-25	349	9	−	−	PUNCT
ma-25	349	10	expp	expp	ADJ
ma-25	349	11	{	{	PUNCT
ma-25	349	12	(	(	PUNCT
ma-25	349	13	ρ+	ρ+	NUM
ma-25	349	14	ε	ε	NOUN
ma-25	349	15	)	)	PUNCT
ma-25	349	16	logq	logq	VERB
ma-25	349	17	r	r	NOUN
ma-25	349	18	}	}	PUNCT
ma-25	349	19	}	}	PUNCT
ma-25	349	20	×1	×1	PROPN
ma-25	349	21	+	+	CCONJ
ma-25	349	22	k−1∑	k−1∑	PROPN
ma-25	349	23	j=1	j=1	PROPN
ma-25	349	24	∣∣∣∣νg	∣∣∣∣νg	NOUN
ma-25	349	25	(	(	PUNCT
ma-25	349	26	r	r	NOUN
ma-25	349	27	)	)	PUNCT
ma-25	349	28	z	z	NOUN
ma-25	349	29	∣∣∣∣j	∣∣∣∣j	NOUN
ma-25	349	30	|1	|1	PRON
ma-25	350	1	+	+	X
ma-25	351	1	o	o	X
ma-25	351	2	(	(	PUNCT
ma-25	351	3	1)|	1)|	NUM
ma-25	351	4			PROPN
ma-25	351	5	expp+1	expp+1	NOUN
ma-25	351	6	{	{	PUNCT
ma-25	351	7	(	(	PUNCT
ma-25	351	8	ρ+	ρ+	NUM
ma-25	351	9	ε	ε	NOUN
ma-25	351	10	)	)	PUNCT
ma-25	351	11	logq	logq	VERB
ma-25	351	12	r	r	NOUN
ma-25	351	13	}	}	PUNCT
ma-25	351	14	+	+	CCONJ
ma-25	351	15	expp+1	expp+1	NOUN
ma-25	351	16	{	{	PUNCT
ma-25	351	17	(	(	PUNCT
ma-25	351	18	ρ+	ρ+	NUM
ma-25	351	19	ε	ε	NOUN
ma-25	351	20	)	)	PUNCT
ma-25	351	21	logq	logq	VERB
ma-25	351	22	r	r	NOUN
ma-25	351	23	}	}	PUNCT
ma-25	351	24	)	)	PUNCT
ma-25	351	25	=	=	SYM
ma-25	351	26	2	2	PROPN
ma-25	352	1	+	+	CCONJ
ma-25	352	2	k−1∑	k−1∑	PROPN
ma-25	352	3	j=1	j=1	PROPN
ma-25	352	4	∣∣∣∣νg	∣∣∣∣νg	NOUN
ma-25	352	5	(	(	PUNCT
ma-25	352	6	r	r	NOUN
ma-25	352	7	)	)	PUNCT
ma-25	352	8	z	z	NOUN
ma-25	352	9	∣∣∣∣j	∣∣∣∣j	NOUN
ma-25	352	10	|1	|1	PRON
ma-25	353	1	+	+	X
ma-25	353	2	o	o	X
ma-25	353	3	(	(	PUNCT
ma-25	353	4	1)|	1)|	NUM
ma-25	353	5			NOUN
ma-25	353	6	exp	exp	NOUN
ma-25	353	7	{	{	PUNCT
ma-25	353	8	2	2	NUM
ma-25	353	9	expp	expp	ADJ
ma-25	353	10	{	{	PUNCT
ma-25	353	11	(	(	PUNCT
ma-25	353	12	ρ+	ρ+	NUM
ma-25	353	13	ε	ε	NOUN
ma-25	353	14	)	)	PUNCT
ma-25	353	15	logq	logq	VERB
ma-25	353	16	r	r	NOUN
ma-25	353	17	}	}	PUNCT
ma-25	353	18	}	}	PUNCT
ma-25	353	19	.	.	PUNCT
ma-25	354	1	hence	hence	ADV
ma-25	354	2	|νg	|νg	PROPN
ma-25	354	3	(	(	PUNCT
ma-25	354	4	r)|	r)|	VERB
ma-25	354	5	|1	|1	X
ma-25	355	1	+	+	NUM
ma-25	355	2	o	o	X
ma-25	355	3	(	(	PUNCT
ma-25	355	4	1)|	1)|	NUM
ma-25	355	5	≤	≤	NOUN
ma-25	355	6	(	(	PUNCT
ma-25	355	7	k	k	NOUN
ma-25	355	8	+	+	PROPN
ma-25	355	9	1	1	X
ma-25	355	10	)	)	PUNCT
ma-25	355	11	r	r	NOUN
ma-25	355	12	|1	|1	NUM
ma-25	356	1	+	+	NUM
ma-25	356	2	o	o	X
ma-25	356	3	(	(	PUNCT
ma-25	356	4	1)|	1)|	NUM
ma-25	356	5	exp	exp	NOUN
ma-25	356	6	{	{	PUNCT
ma-25	356	7	2	2	NUM
ma-25	356	8	expp	expp	ADJ
ma-25	356	9	{	{	PUNCT
ma-25	356	10	(	(	PUNCT
ma-25	356	11	ρ+	ρ+	NUM
ma-25	356	12	ε	ε	NOUN
ma-25	356	13	)	)	PUNCT
ma-25	356	14	logq	logq	VERB
ma-25	356	15	r	r	NOUN
ma-25	356	16	}	}	PUNCT
ma-25	356	17	}	}	PUNCT
ma-25	356	18	(	(	PUNCT
ma-25	356	19	2.28	2.28	NUM
ma-25	356	20	)	)	PUNCT
ma-25	356	21	eur	eur	PROPN
ma-25	356	22	.	.	PUNCT
ma-25	357	1	j.	j.	PROPN
ma-25	357	2	math	math	PROPN
ma-25	357	3	.	.	PUNCT
ma-25	358	1	anal	anal	ADJ
ma-25	358	2	.	.	PUNCT
ma-25	359	1	1	1	NUM
ma-25	359	2	(	(	PUNCT
ma-25	359	3	2021	2021	NUM
ma-25	359	4	)	)	PUNCT
ma-25	359	5	97holds	97hold	NOUN
ma-25	359	6	for	for	ADP
ma-25	359	7	all	all	DET
ma-25	359	8	z	z	NOUN
ma-25	359	9	satisfying	satisfy	VERB
ma-25	359	10	|z	|z	PROPN
ma-25	360	1	|	|	ADV
ma-25	360	2	=	=	NOUN
ma-25	360	3	r	r	NOUN
ma-25	360	4	/∈	/∈	PUNCT
ma-25	361	1	[	[	X
ma-25	361	2	0	0	NUM
ma-25	361	3	,	,	PUNCT
ma-25	361	4	1	1	NUM
ma-25	361	5	]	]	PUNCT
ma-25	361	6	∪	∪	ADP
ma-25	361	7	e3	e3	PROPN
ma-25	361	8	∪	∪	X
ma-25	361	9	e5	e5	PROPN
ma-25	361	10	and	and	CCONJ
ma-25	361	11	|g	|g	NOUN
ma-25	361	12	(	(	PUNCT
ma-25	361	13	z	z	NOUN
ma-25	361	14	)	)	PUNCT
ma-25	362	1	|	|	ADV
ma-25	362	2	=	=	SYM
ma-25	362	3	m	m	PROPN
ma-25	362	4	(	(	PUNCT
ma-25	362	5	r	r	NOUN
ma-25	362	6	,	,	PUNCT
ma-25	362	7	g	g	NOUN
ma-25	362	8	)	)	PUNCT
ma-25	362	9	,	,	PUNCT
ma-25	362	10	r	r	NOUN
ma-25	362	11	→	→	SYM
ma-25	362	12	+	+	ADJ
ma-25	362	13	∞.	∞.	PROPN
ma-25	362	14	by	by	ADP
ma-25	362	15	(	(	PUNCT
ma-25	362	16	2.28),we	2.28),we	PROPN
ma-25	362	17	get	get	VERB
ma-25	362	18	lim	lim	PROPN
ma-25	362	19	sup	sup	PROPN
ma-25	362	20	r→+∞	r→+∞	PROPN
ma-25	362	21	logp+1	logp+1	PROPN
ma-25	362	22	νg	νg	NOUN
ma-25	362	23	(	(	PUNCT
ma-25	362	24	r	r	NOUN
ma-25	362	25	)	)	PUNCT
ma-25	362	26	logq	logq	ADJ
ma-25	362	27	r	r	NOUN
ma-25	362	28	≤	≤	NOUN
ma-25	362	29	ρ+	ρ+	NUM
ma-25	362	30	ε	ε	PROPN
ma-25	362	31	.	.	PUNCT
ma-25	363	1	(	(	PUNCT
ma-25	363	2	2.29	2.29	NUM
ma-25	363	3	)	)	PUNCT
ma-25	363	4	since	since	SCONJ
ma-25	363	5	ε	ε	PROPN
ma-25	363	6	>	>	X
ma-25	363	7	0	0	NUM
ma-25	363	8	is	be	AUX
ma-25	363	9	arbitrary	arbitrary	ADJ
ma-25	363	10	,	,	PUNCT
ma-25	363	11	by	by	ADP
ma-25	363	12	(	(	PUNCT
ma-25	363	13	2.29	2.29	NUM
ma-25	363	14	)	)	PUNCT
ma-25	363	15	and	and	CCONJ
ma-25	363	16	lemma	lemma	PROPN
ma-25	363	17	2.6	2.6	NUM
ma-25	363	18	,	,	PUNCT
ma-25	363	19	we	we	PRON
ma-25	363	20	obtain	obtain	VERB
ma-25	363	21	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	363	22	]	]	PUNCT
ma-25	363	23	(	(	PUNCT
ma-25	363	24	g	g	NOUN
ma-25	363	25	)	)	PUNCT
ma-25	363	26	≤	≤	NOUN
ma-25	363	27	ρ	ρ	NOUN
ma-25	363	28	.	.	PUNCT
ma-25	364	1	since	since	SCONJ
ma-25	364	2	ρ[p	ρ[p	NOUN
ma-25	364	3	,	,	PUNCT
ma-25	364	4	q	q	X
ma-25	364	5	]	]	X
ma-25	364	6	(	(	PUNCT
ma-25	364	7	d	d	X
ma-25	364	8	)	)	PUNCT
ma-25	364	9	<	<	X
ma-25	364	10	µ[p	µ[p	ADJ
ma-25	364	11	,	,	PUNCT
ma-25	364	12	q	q	X
ma-25	364	13	]	]	X
ma-25	364	14	(	(	PUNCT
ma-25	364	15	f	f	PROPN
ma-25	364	16	)	)	PUNCT
ma-25	364	17	,	,	PUNCT
ma-25	364	18	so	so	CCONJ
ma-25	364	19	by	by	ADP
ma-25	364	20	lemma	lemma	PROPN
ma-25	364	21	2.10	2.10	NUM
ma-25	364	22	,	,	PUNCT
ma-25	364	23	we	we	PRON
ma-25	364	24	have	have	VERB
ma-25	364	25	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	364	26	]	]	PUNCT
ma-25	364	27	(	(	PUNCT
ma-25	364	28	g	g	NOUN
ma-25	364	29	)	)	PUNCT
ma-25	364	30	=	=	SYM
ma-25	364	31	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	364	32	]	]	PUNCT
ma-25	364	33	(	(	PUNCT
ma-25	364	34	f	f	PROPN
ma-25	364	35	)	)	PUNCT
ma-25	364	36	.	.	PUNCT
ma-25	365	1	thus	thus	ADV
ma-25	365	2	,	,	PUNCT
ma-25	365	3	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	365	4	]	]	PUNCT
ma-25	365	5	(	(	PUNCT
ma-25	365	6	f	f	PROPN
ma-25	365	7	)	)	PUNCT
ma-25	365	8	≤	≤	PROPN
ma-25	365	9	ρ	ρ	PROPN
ma-25	365	10	.	.	PUNCT
ma-25	366	1	therefore	therefore	ADV
ma-25	366	2	,	,	PUNCT
ma-25	366	3	lemma	lemma	PROPN
ma-25	366	4	2.11	2.11	NUM
ma-25	366	5	is	be	AUX
ma-25	366	6	proved	prove	VERB
ma-25	366	7	.	.	PUNCT
ma-25	367	1	lemma	lemma	PROPN
ma-25	367	2	2.12	2.12	NUM
ma-25	367	3	(	(	PUNCT
ma-25	367	4	[	[	X
ma-25	367	5	19	19	NUM
ma-25	367	6	]	]	PUNCT
ma-25	367	7	)	)	PUNCT
ma-25	367	8	let	let	VERB
ma-25	367	9	aj	aj	PROPN
ma-25	367	10	(	(	PUNCT
ma-25	367	11	z	z	NOUN
ma-25	367	12	)	)	PUNCT
ma-25	367	13	(	(	PUNCT
ma-25	367	14	j	j	NOUN
ma-25	367	15	=	=	SYM
ma-25	367	16	0	0	NUM
ma-25	367	17	,	,	PUNCT
ma-25	367	18	1	1	NUM
ma-25	367	19	,	,	PUNCT
ma-25	367	20	...	...	PUNCT
ma-25	367	21	,	,	PUNCT
ma-25	367	22	k	k	PROPN
ma-25	367	23	)	)	PUNCT
ma-25	367	24	,	,	PUNCT
ma-25	367	25	ak	ak	PROPN
ma-25	367	26	(	(	PUNCT
ma-25	367	27	z	z	NOUN
ma-25	367	28	)	)	PUNCT
ma-25	367	29	(	(	PUNCT
ma-25	367	30	6≡	6≡	NUM
ma-25	367	31	0	0	NUM
ma-25	367	32	)	)	PUNCT
ma-25	367	33	,	,	PUNCT
ma-25	367	34	f	f	PROPN
ma-25	367	35	(	(	PUNCT
ma-25	367	36	z	z	NOUN
ma-25	367	37	)	)	PUNCT
ma-25	367	38	(	(	PUNCT
ma-25	367	39	6≡	6≡	NUM
ma-25	367	40	0	0	X
ma-25	367	41	)	)	PUNCT
ma-25	367	42	be	be	AUX
ma-25	367	43	meromorphic	meromorphic	ADJ
ma-25	367	44	functions	function	NOUN
ma-25	367	45	and	and	CCONJ
ma-25	367	46	let	let	VERB
ma-25	367	47	f	f	PRON
ma-25	367	48	be	be	AUX
ma-25	367	49	a	a	DET
ma-25	367	50	meromorphic	meromorphic	ADJ
ma-25	367	51	solution	solution	NOUN
ma-25	367	52	of	of	ADP
ma-25	367	53	(	(	PUNCT
ma-25	367	54	1.4	1.4	NUM
ma-25	367	55	)	)	PUNCT
ma-25	367	56	of	of	ADP
ma-25	367	57	infinite	infinite	ADJ
ma-25	367	58	[	[	X
ma-25	367	59	p	p	X
ma-25	367	60	,	,	PUNCT
ma-25	367	61	q]-order	q]-order	NOUN
ma-25	367	62	satisfying	satisfy	VERB
ma-25	367	63	the	the	DET
ma-25	367	64	following	follow	VERB
ma-25	367	65	condition	condition	NOUN
ma-25	367	66	b	b	NOUN
ma-25	367	67	=	=	SYM
ma-25	367	68	max	max	PROPN
ma-25	367	69	{	{	PUNCT
ma-25	367	70	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	367	71	]	]	PUNCT
ma-25	367	72	(	(	PUNCT
ma-25	367	73	f	f	PROPN
ma-25	367	74	)	)	PUNCT
ma-25	367	75	,	,	PUNCT
ma-25	367	76	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	367	77	]	]	PUNCT
ma-25	367	78	(	(	PUNCT
ma-25	367	79	aj	aj	PROPN
ma-25	367	80	)	)	PUNCT
ma-25	367	81	(	(	PUNCT
ma-25	367	82	j	j	NOUN
ma-25	367	83	=	=	SYM
ma-25	367	84	0	0	NUM
ma-25	367	85	,	,	PUNCT
ma-25	367	86	1	1	NUM
ma-25	367	87	,	,	PUNCT
ma-25	367	88	...	...	PUNCT
ma-25	367	89	,	,	PUNCT
ma-25	367	90	k	k	NOUN
ma-25	367	91	)	)	PUNCT
ma-25	367	92	}	}	PUNCT
ma-25	367	93	<	<	X
ma-25	367	94	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	367	95	]	]	PUNCT
ma-25	367	96	(	(	PUNCT
ma-25	367	97	f	f	PROPN
ma-25	367	98	)	)	PUNCT
ma-25	367	99	.	.	PUNCT
ma-25	368	1	then	then	ADV
ma-25	368	2	λ[p+1,q](f	λ[p+1,q](f	X
ma-25	368	3	)	)	PUNCT
ma-25	368	4	=	=	PUNCT
ma-25	368	5	λ[p+1,q](f	λ[p+1,q](f	X
ma-25	368	6	)	)	PUNCT
ma-25	368	7	=	=	SYM
ma-25	368	8	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	368	9	]	]	PUNCT
ma-25	368	10	(	(	PUNCT
ma-25	368	11	f	f	PROPN
ma-25	368	12	)	)	PUNCT
ma-25	368	13	.	.	PUNCT
ma-25	369	1	lemma	lemma	PROPN
ma-25	369	2	2.13	2.13	NUM
ma-25	369	3	let	let	VERB
ma-25	369	4	h	h	PROPN
ma-25	369	5	⊂	⊂	PROPN
ma-25	369	6	(	(	PUNCT
ma-25	369	7	1,+∞	1,+∞	NUM
ma-25	369	8	)	)	PUNCT
ma-25	369	9	be	be	AUX
ma-25	369	10	a	a	DET
ma-25	369	11	set	set	NOUN
ma-25	369	12	with	with	ADP
ma-25	369	13	a	a	DET
ma-25	369	14	positive	positive	ADJ
ma-25	369	15	upper	upper	ADJ
ma-25	369	16	logarithmic	logarithmic	ADJ
ma-25	369	17	density	density	NOUN
ma-25	369	18	(	(	PUNCT
ma-25	369	19	or	or	CCONJ
ma-25	369	20	infinite	infinite	VERB
ma-25	369	21	logarithmic	logarithmic	ADJ
ma-25	369	22	measure	measure	NOUN
ma-25	369	23	)	)	PUNCT
ma-25	369	24	,	,	PUNCT
ma-25	369	25	and	and	CCONJ
ma-25	369	26	let	let	VERB
ma-25	369	27	aj	aj	PROPN
ma-25	369	28	(	(	PUNCT
ma-25	369	29	z	z	NOUN
ma-25	369	30	)	)	PUNCT
ma-25	369	31	(	(	PUNCT
ma-25	369	32	j	j	NOUN
ma-25	369	33	=	=	SYM
ma-25	369	34	0	0	NUM
ma-25	369	35	,	,	PUNCT
ma-25	369	36	1	1	NUM
ma-25	369	37	,	,	PUNCT
ma-25	369	38	...	...	PUNCT
ma-25	369	39	,	,	PUNCT
ma-25	369	40	k	k	X
ma-25	369	41	)	)	PUNCT
ma-25	369	42	with	with	ADP
ma-25	369	43	ak	ak	PROPN
ma-25	369	44	(	(	PUNCT
ma-25	369	45	z	z	PROPN
ma-25	369	46	)	)	PUNCT
ma-25	369	47	6≡	6≡	NUM
ma-25	369	48	0	0	NUM
ma-25	369	49	and	and	CCONJ
ma-25	369	50	f	f	PROPN
ma-25	369	51	(	(	PUNCT
ma-25	369	52	z	z	NOUN
ma-25	369	53	)	)	PUNCT
ma-25	369	54	6≡	6≡	NUM
ma-25	369	55	0	0	NUM
ma-25	369	56	be	be	AUX
ma-25	369	57	meromorphic	meromorphic	ADJ
ma-25	369	58	functions	function	NOUN
ma-25	369	59	with	with	ADP
ma-25	369	60	finite	finite	NOUN
ma-25	369	61	[	[	X
ma-25	369	62	p	p	X
ma-25	369	63	,	,	PUNCT
ma-25	369	64	q]-order	q]-order	NOUN
ma-25	369	65	.	.	PUNCT
ma-25	370	1	if	if	SCONJ
ma-25	370	2	there	there	PRON
ma-25	370	3	exist	exist	VERB
ma-25	370	4	a	a	DET
ma-25	370	5	positive	positive	ADJ
ma-25	370	6	constant	constant	ADJ
ma-25	370	7	σ	σ	NOUN
ma-25	370	8	>	>	X
ma-25	370	9	0	0	PUNCT
ma-25	370	10	and	and	CCONJ
ma-25	370	11	an	an	DET
ma-25	370	12	integer	integer	NOUN
ma-25	370	13	s	s	NOUN
ma-25	370	14	,	,	PUNCT
ma-25	370	15	0	0	NUM
ma-25	370	16	≤	≤	NUM
ma-25	370	17	s	s	PART
ma-25	370	18	≤	≤	NUM
ma-25	370	19	k	k	NOUN
ma-25	370	20	,	,	PUNCT
ma-25	370	21	such	such	ADJ
ma-25	370	22	that	that	PRON
ma-25	370	23	for	for	ADP
ma-25	370	24	sufficiently	sufficiently	ADV
ma-25	370	25	small	small	ADJ
ma-25	370	26	ε	ε	PROPN
ma-25	370	27	>	>	X
ma-25	370	28	0	0	PROPN
ma-25	370	29	,	,	PUNCT
ma-25	370	30	we	we	PRON
ma-25	370	31	have	have	VERB
ma-25	370	32	|as	|as	NUM
ma-25	370	33	(	(	PUNCT
ma-25	370	34	z	z	NOUN
ma-25	370	35	)	)	PUNCT
ma-25	371	1	|	|	ADV
ma-25	371	2	≥	≥	NOUN
ma-25	371	3	expp+1	expp+1	PROPN
ma-25	371	4	{	{	PUNCT
ma-25	371	5	(	(	PUNCT
ma-25	371	6	σ	σ	PROPN
ma-25	371	7	−	−	PROPN
ma-25	371	8	ε	ε	PROPN
ma-25	371	9	)	)	PUNCT
ma-25	371	10	logq	logq	VERB
ma-25	371	11	r	r	NOUN
ma-25	371	12	}	}	PUNCT
ma-25	371	13	as	as	ADP
ma-25	371	14	|z	|z	PROPN
ma-25	371	15	|	|	ADV
ma-25	371	16	=	=	SYM
ma-25	371	17	r	r	NOUN
ma-25	371	18	∈	∈	PROPN
ma-25	371	19	h	h	NOUN
ma-25	371	20	,	,	PUNCT
ma-25	371	21	r	r	NOUN
ma-25	371	22	→	→	SYM
ma-25	371	23	+	+	NOUN
ma-25	371	24	∞	∞	NUM
ma-25	371	25	and	and	CCONJ
ma-25	371	26	max	max	PROPN
ma-25	371	27	{	{	PUNCT
ma-25	371	28	ρ[p	ρ[p	PROPN
ma-25	371	29	,	,	PUNCT
ma-25	371	30	q	q	X
ma-25	371	31	]	]	X
ma-25	371	32	(	(	PUNCT
ma-25	371	33	aj	aj	PROPN
ma-25	371	34	)	)	PUNCT
ma-25	371	35	(	(	PUNCT
ma-25	371	36	j	j	PROPN
ma-25	371	37	6=	6=	NUM
ma-25	371	38	s	s	PROPN
ma-25	371	39	)	)	PUNCT
ma-25	371	40	,	,	PUNCT
ma-25	371	41	ρ[p	ρ[p	PROPN
ma-25	371	42	,	,	PUNCT
ma-25	371	43	q	q	X
ma-25	371	44	]	]	X
ma-25	371	45	(	(	PUNCT
ma-25	371	46	f	f	PROPN
ma-25	371	47	)	)	PUNCT
ma-25	371	48	}	}	PUNCT
ma-25	371	49	<	<	X
ma-25	371	50	σ	σ	PROPN
ma-25	371	51	,	,	PUNCT
ma-25	371	52	then	then	ADV
ma-25	371	53	every	every	DET
ma-25	371	54	transcendental	transcendental	ADJ
ma-25	371	55	meromorphic	meromorphic	ADJ
ma-25	371	56	solution	solution	NOUN
ma-25	371	57	f	f	PROPN
ma-25	371	58	of	of	ADP
ma-25	371	59	equation	equation	NOUN
ma-25	371	60	(	(	PUNCT
ma-25	371	61	1.4	1.4	NUM
ma-25	371	62	)	)	PUNCT
ma-25	371	63	satisfies	satisfie	NOUN
ma-25	371	64	ρ[p	ρ[p	NOUN
ma-25	371	65	,	,	PUNCT
ma-25	371	66	q](f	q](f	NOUN
ma-25	371	67	)	)	PUNCT
ma-25	371	68	≥	≥	PROPN
ma-25	371	69	σ	σ	PROPN
ma-25	371	70	.	.	PUNCT
ma-25	371	71	proof	proof	NOUN
ma-25	371	72	.	.	PUNCT
ma-25	372	1	assume	assume	VERB
ma-25	372	2	that	that	SCONJ
ma-25	372	3	f	f	PROPN
ma-25	372	4	is	be	AUX
ma-25	372	5	a	a	DET
ma-25	372	6	transcendental	transcendental	ADJ
ma-25	372	7	meromorphic	meromorphic	ADJ
ma-25	372	8	solution	solution	NOUN
ma-25	372	9	of	of	ADP
ma-25	372	10	equation	equation	NOUN
ma-25	372	11	(	(	PUNCT
ma-25	372	12	1.4	1.4	NUM
ma-25	372	13	)	)	PUNCT
ma-25	372	14	with	with	ADP
ma-25	372	15	ρ[p	ρ[p	NOUN
ma-25	372	16	,	,	PUNCT
ma-25	372	17	q](f	q](f	NOUN
ma-25	372	18	)	)	PUNCT
ma-25	372	19	<	<	X
ma-25	372	20	σ.from	σ.from	ADP
ma-25	372	21	(	(	PUNCT
ma-25	372	22	1.4	1.4	NUM
ma-25	372	23	)	)	PUNCT
ma-25	372	24	,	,	PUNCT
ma-25	372	25	we	we	PRON
ma-25	372	26	have	have	VERB
ma-25	372	27	as	as	ADP
ma-25	372	28	=	=	PUNCT
ma-25	372	29	f	f	X
ma-25	372	30	f	f	PROPN
ma-25	372	31	(	(	PUNCT
ma-25	372	32	s	s	NOUN
ma-25	372	33	)	)	PUNCT
ma-25	372	34	−	−	PROPN
ma-25	372	35	k∑	k∑	PROPN
ma-25	372	36	j=0	j=0	PROPN
ma-25	372	37	j	j	PROPN
ma-25	372	38	6	6	NUM
ma-25	372	39	=	=	SYM
ma-25	372	40	s	s	X
ma-25	372	41	aj	aj	PROPN
ma-25	372	42	f	f	PROPN
ma-25	372	43	(	(	PUNCT
ma-25	372	44	j	j	PROPN
ma-25	372	45	)	)	PUNCT
ma-25	372	46	f	f	PROPN
ma-25	372	47	(	(	PUNCT
ma-25	372	48	s	s	NOUN
ma-25	372	49	)	)	PUNCT
ma-25	372	50	.	.	PUNCT
ma-25	373	1	(	(	PUNCT
ma-25	373	2	2.30	2.30	NUM
ma-25	373	3	)	)	PUNCT
ma-25	373	4	since	since	SCONJ
ma-25	373	5	max	max	PROPN
ma-25	373	6	{	{	PUNCT
ma-25	373	7	ρ[p	ρ[p	PROPN
ma-25	373	8	,	,	PUNCT
ma-25	373	9	q	q	X
ma-25	373	10	]	]	X
ma-25	373	11	(	(	PUNCT
ma-25	373	12	aj	aj	PROPN
ma-25	373	13	)	)	PUNCT
ma-25	373	14	(	(	PUNCT
ma-25	373	15	j	j	PROPN
ma-25	373	16	6=	6=	NUM
ma-25	373	17	s	s	PROPN
ma-25	373	18	)	)	PUNCT
ma-25	373	19	,	,	PUNCT
ma-25	373	20	ρ[p	ρ[p	PROPN
ma-25	373	21	,	,	PUNCT
ma-25	373	22	q	q	X
ma-25	373	23	]	]	X
ma-25	373	24	(	(	PUNCT
ma-25	373	25	f	f	PROPN
ma-25	373	26	)	)	PUNCT
ma-25	373	27	}	}	PUNCT
ma-25	373	28	<	<	X
ma-25	373	29	σ	σ	PROPN
ma-25	373	30	and	and	CCONJ
ma-25	373	31	ρ[p	ρ[p	NOUN
ma-25	373	32	,	,	PUNCT
ma-25	373	33	q	q	X
ma-25	373	34	]	]	X
ma-25	373	35	(	(	PUNCT
ma-25	373	36	f	f	X
ma-25	373	37	)	)	PUNCT
ma-25	373	38	<	<	X
ma-25	374	1	σ	σ	PROPN
ma-25	374	2	,	,	PUNCT
ma-25	374	3	then	then	ADV
ma-25	374	4	from	from	ADP
ma-25	374	5	(	(	PUNCT
ma-25	374	6	2.30	2.30	NUM
ma-25	374	7	)	)	PUNCT
ma-25	374	8	we	we	PRON
ma-25	374	9	obtainthat	obtainthat	VERB
ma-25	374	10	ρ1	ρ1	NOUN
ma-25	374	11	=	=	PUNCT
ma-25	374	12	ρ[p	ρ[p	PROPN
ma-25	374	13	,	,	PUNCT
ma-25	374	14	q	q	X
ma-25	374	15	]	]	X
ma-25	374	16	(	(	PUNCT
ma-25	374	17	as	as	ADP
ma-25	374	18	)	)	PUNCT
ma-25	374	19	≤	≤	NUM
ma-25	374	20	max	max	PROPN
ma-25	374	21	{	{	PUNCT
ma-25	374	22	ρ[p	ρ[p	PROPN
ma-25	374	23	,	,	PUNCT
ma-25	374	24	q	q	X
ma-25	374	25	]	]	X
ma-25	374	26	(	(	PUNCT
ma-25	374	27	aj	aj	PROPN
ma-25	374	28	)	)	PUNCT
ma-25	374	29	(	(	PUNCT
ma-25	374	30	j	j	PROPN
ma-25	374	31	6=	6=	NUM
ma-25	374	32	s	s	PROPN
ma-25	374	33	)	)	PUNCT
ma-25	374	34	,	,	PUNCT
ma-25	374	35	ρ[p	ρ[p	PROPN
ma-25	374	36	,	,	PUNCT
ma-25	374	37	q	q	X
ma-25	374	38	]	]	X
ma-25	374	39	(	(	PUNCT
ma-25	374	40	f	f	PROPN
ma-25	374	41	)	)	PUNCT
ma-25	374	42	,	,	PUNCT
ma-25	374	43	ρ[p	ρ[p	PROPN
ma-25	374	44	,	,	PUNCT
ma-25	374	45	q	q	X
ma-25	374	46	]	]	X
ma-25	374	47	(	(	PUNCT
ma-25	374	48	f	f	PROPN
ma-25	374	49	)	)	PUNCT
ma-25	374	50	}	}	PUNCT
ma-25	375	1	<	<	X
ma-25	375	2	σ.by	σ.by	PROPN
ma-25	375	3	lemma	lemma	PROPN
ma-25	375	4	2.3	2.3	NUM
ma-25	375	5	,	,	PUNCT
ma-25	375	6	for	for	ADP
ma-25	375	7	any	any	DET
ma-25	375	8	ε	ε	PROPN
ma-25	375	9	(	(	PUNCT
ma-25	375	10	0	0	PUNCT
ma-25	375	11	<	<	X
ma-25	375	12	2ε	2ε	PROPN
ma-25	375	13	<	<	X
ma-25	375	14	σ	σ	PROPN
ma-25	375	15	−	−	PROPN
ma-25	375	16	ρ1	ρ1	PROPN
ma-25	375	17	)	)	PUNCT
ma-25	375	18	,	,	PUNCT
ma-25	375	19	there	there	PRON
ma-25	375	20	exists	exist	VERB
ma-25	375	21	a	a	DET
ma-25	375	22	set	set	NOUN
ma-25	375	23	e3	e3	NOUN
ma-25	375	24	⊂	⊂	X
ma-25	375	25	(	(	PUNCT
ma-25	375	26	1,+∞	1,+∞	NUM
ma-25	375	27	)	)	PUNCT
ma-25	375	28	with	with	ADP
ma-25	375	29	a	a	DET
ma-25	375	30	finite	finite	ADJ
ma-25	375	31	linearmeasure	linearmeasure	NOUN
ma-25	375	32	such	such	ADJ
ma-25	375	33	that	that	SCONJ
ma-25	375	34	|as	|as	PROPN
ma-25	375	35	(	(	PUNCT
ma-25	375	36	z)|	z)|	ADP
ma-25	375	37	≤	≤	NUM
ma-25	375	38	expp+1	expp+1	PROPN
ma-25	375	39	{	{	PUNCT
ma-25	375	40	(	(	PUNCT
ma-25	375	41	ρ(p	ρ(p	PROPN
ma-25	375	42	,	,	PUNCT
ma-25	375	43	q	q	NOUN
ma-25	375	44	)	)	PUNCT
ma-25	375	45	(	(	PUNCT
ma-25	375	46	as	as	ADP
ma-25	375	47	)	)	PUNCT
ma-25	375	48	+	+	CCONJ
ma-25	375	49	ε	ε	PROPN
ma-25	375	50	)	)	PUNCT
ma-25	375	51	logq	logq	VERB
ma-25	375	52	r	r	NOUN
ma-25	375	53	}	}	PUNCT
ma-25	375	54	=	=	SYM
ma-25	375	55	expp+1	expp+1	NOUN
ma-25	375	56	{	{	PUNCT
ma-25	375	57	(	(	PUNCT
ma-25	375	58	ρ1	ρ1	NOUN
ma-25	375	59	+	+	CCONJ
ma-25	375	60	ε	ε	PROPN
ma-25	375	61	)	)	PUNCT
ma-25	375	62	logq	logq	VERB
ma-25	375	63	r	r	NOUN
ma-25	375	64	}	}	PUNCT
ma-25	375	65	(	(	PUNCT
ma-25	375	66	2.31	2.31	NUM
ma-25	375	67	)	)	PUNCT
ma-25	375	68	holds	hold	VERB
ma-25	375	69	for	for	ADP
ma-25	375	70	all	all	DET
ma-25	375	71	z	z	NOUN
ma-25	375	72	satisfying	satisfy	VERB
ma-25	375	73	|z	|z	PROPN
ma-25	376	1	|	|	ADV
ma-25	376	2	=	=	NOUN
ma-25	376	3	r	r	NOUN
ma-25	376	4	/∈	/∈	PUNCT
ma-25	376	5	e3	e3	NOUN
ma-25	376	6	.	.	PUNCT
ma-25	377	1	from	from	ADP
ma-25	377	2	the	the	DET
ma-25	377	3	hypotheses	hypothesis	NOUN
ma-25	377	4	of	of	ADP
ma-25	377	5	lemma	lemma	PROPN
ma-25	377	6	2.13	2.13	NUM
ma-25	377	7	,	,	PUNCT
ma-25	377	8	there	there	PRON
ma-25	377	9	exists	exist	VERB
ma-25	377	10	a	a	DET
ma-25	377	11	set	set	NOUN
ma-25	377	12	hwith	hwith	NOUN
ma-25	377	13	log	log	NOUN
ma-25	377	14	densh	densh	ADJ
ma-25	377	15	>	>	X
ma-25	377	16	0	0	PUNCT
ma-25	378	1	(	(	PUNCT
ma-25	378	2	or	or	CCONJ
ma-25	378	3	ml	ml	INTJ
ma-25	378	4	(	(	PUNCT
ma-25	378	5	h	h	NOUN
ma-25	378	6	)	)	PUNCT
ma-25	378	7	=	=	PUNCT
ma-25	379	1	+	+	NUM
ma-25	379	2	∞	∞	NOUN
ma-25	379	3	)	)	PUNCT
ma-25	379	4	such	such	ADJ
ma-25	379	5	that	that	PRON
ma-25	379	6	|as	|as	PROPN
ma-25	379	7	(	(	PUNCT
ma-25	379	8	z)|	z)|	PRON
ma-25	379	9	≥	≥	NOUN
ma-25	379	10	expp+1	expp+1	PROPN
ma-25	379	11	{	{	PUNCT
ma-25	379	12	(	(	PUNCT
ma-25	379	13	σ	σ	PROPN
ma-25	379	14	−	−	PROPN
ma-25	379	15	ε	ε	PROPN
ma-25	379	16	)	)	PUNCT
ma-25	379	17	logq	logq	VERB
ma-25	379	18	r	r	NOUN
ma-25	379	19	}	}	PUNCT
ma-25	379	20	(	(	PUNCT
ma-25	379	21	2.32	2.32	NUM
ma-25	379	22	)	)	PUNCT
ma-25	379	23	eur	eur	NOUN
ma-25	379	24	.	.	PUNCT
ma-25	380	1	j.	j.	PROPN
ma-25	380	2	math	math	PROPN
ma-25	380	3	.	.	PUNCT
ma-25	381	1	anal	anal	ADJ
ma-25	381	2	.	.	PUNCT
ma-25	382	1	1	1	NUM
ma-25	382	2	(	(	PUNCT
ma-25	382	3	2021	2021	NUM
ma-25	382	4	)	)	PUNCT
ma-25	382	5	98holds	98hold	NOUN
ma-25	382	6	for	for	ADP
ma-25	382	7	all	all	DET
ma-25	382	8	z	z	NOUN
ma-25	382	9	satisfying	satisfy	VERB
ma-25	382	10	|z	|z	PROPN
ma-25	383	1	|	|	ADV
ma-25	383	2	=	=	SYM
ma-25	383	3	r	r	NOUN
ma-25	383	4	∈	∈	PROPN
ma-25	383	5	h	h	NOUN
ma-25	383	6	,	,	PUNCT
ma-25	383	7	r	r	NOUN
ma-25	383	8	→	→	SYM
ma-25	383	9	+	+	ADJ
ma-25	383	10	∞.	∞.	PROPN
ma-25	383	11	by	by	ADP
ma-25	383	12	(	(	PUNCT
ma-25	383	13	2.31	2.31	NUM
ma-25	383	14	)	)	PUNCT
ma-25	383	15	and	and	CCONJ
ma-25	383	16	(	(	PUNCT
ma-25	383	17	2.32	2.32	NUM
ma-25	383	18	)	)	PUNCT
ma-25	383	19	,	,	PUNCT
ma-25	383	20	we	we	PRON
ma-25	383	21	conclude	conclude	VERB
ma-25	383	22	that	that	SCONJ
ma-25	383	23	for	for	ADP
ma-25	383	24	all	all	DET
ma-25	383	25	zsatisfying	zsatisfye	VERB
ma-25	383	26	|z	|z	PROPN
ma-25	384	1	|	|	ADV
ma-25	384	2	=	=	SYM
ma-25	384	3	r	r	NOUN
ma-25	384	4	∈	∈	NOUN
ma-25	384	5	h	h	NOUN
ma-25	384	6	r	r	NOUN
ma-25	384	7	e3	e3	NOUN
ma-25	384	8	,	,	PUNCT
ma-25	384	9	r	r	NOUN
ma-25	384	10	→	→	SYM
ma-25	384	11	+	+	NOUN
ma-25	384	12	∞	∞	PROPN
ma-25	384	13	,	,	PUNCT
ma-25	384	14	we	we	PRON
ma-25	384	15	have	have	VERB
ma-25	384	16	expp+1	expp+1	NOUN
ma-25	384	17	{	{	PUNCT
ma-25	384	18	(	(	PUNCT
ma-25	384	19	σ	σ	PROPN
ma-25	384	20	−	−	PROPN
ma-25	384	21	ε	ε	PROPN
ma-25	384	22	)	)	PUNCT
ma-25	384	23	logq	logq	VERB
ma-25	384	24	r	r	NOUN
ma-25	384	25	}	}	PUNCT
ma-25	384	26	≤	≤	NUM
ma-25	384	27	expp+1	expp+1	NOUN
ma-25	384	28	{	{	PUNCT
ma-25	384	29	(	(	PUNCT
ma-25	384	30	ρ1	ρ1	NOUN
ma-25	384	31	+	+	CCONJ
ma-25	384	32	ε	ε	PROPN
ma-25	384	33	)	)	PUNCT
ma-25	384	34	logq	logq	VERB
ma-25	384	35	r	r	NOUN
ma-25	384	36	}	}	PUNCT
ma-25	384	37	and	and	CCONJ
ma-25	384	38	by	by	ADP
ma-25	384	39	ε	ε	PROPN
ma-25	384	40	(	(	PUNCT
ma-25	384	41	0	0	PUNCT
ma-25	384	42	<	<	X
ma-25	384	43	2ε	2ε	PROPN
ma-25	384	44	<	<	X
ma-25	384	45	σ	σ	PROPN
ma-25	384	46	−	−	PROPN
ma-25	384	47	ρ1	ρ1	PROPN
ma-25	384	48	)	)	PUNCT
ma-25	384	49	this	this	PRON
ma-25	384	50	is	be	AUX
ma-25	384	51	a	a	DET
ma-25	384	52	contradiction	contradiction	NOUN
ma-25	384	53	as	as	ADP
ma-25	384	54	r	r	NOUN
ma-25	384	55	→	→	SYM
ma-25	384	56	+	+	PROPN
ma-25	384	57	∞.	∞.	PROPN
ma-25	384	58	consequently	consequently	ADV
ma-25	384	59	,	,	PUNCT
ma-25	384	60	any	any	DET
ma-25	384	61	transcendentalmeromorphic	transcendentalmeromorphic	ADJ
ma-25	384	62	solution	solution	NOUN
ma-25	384	63	f	f	PROPN
ma-25	384	64	of	of	ADP
ma-25	384	65	equation	equation	NOUN
ma-25	384	66	(	(	PUNCT
ma-25	384	67	1.4	1.4	NUM
ma-25	384	68	)	)	PUNCT
ma-25	384	69	satisfies	satisfie	NOUN
ma-25	384	70	ρ[p	ρ[p	NOUN
ma-25	384	71	,	,	PUNCT
ma-25	384	72	q	q	X
ma-25	384	73	]	]	X
ma-25	384	74	(	(	PUNCT
ma-25	384	75	f	f	PROPN
ma-25	384	76	)	)	PUNCT
ma-25	384	77	≥	≥	PROPN
ma-25	385	1	σ	σ	PROPN
ma-25	385	2	.	.	PUNCT
ma-25	386	1	lemma	lemma	PROPN
ma-25	386	2	2.14	2.14	NUM
ma-25	386	3	(	(	PUNCT
ma-25	386	4	[	[	X
ma-25	386	5	23	23	NUM
ma-25	386	6	]	]	PUNCT
ma-25	386	7	)	)	PUNCT
ma-25	386	8	let	let	VERB
ma-25	386	9	p	p	PRON
ma-25	386	10	≥	≥	PRON
ma-25	386	11	q	q	NOUN
ma-25	386	12	≥	≥	NUM
ma-25	386	13	1	1	NUM
ma-25	386	14	be	be	AUX
ma-25	386	15	integers	integer	NOUN
ma-25	386	16	.	.	PUNCT
ma-25	387	1	let	let	VERB
ma-25	387	2	f	f	PRON
ma-25	387	3	be	be	AUX
ma-25	387	4	a	a	DET
ma-25	387	5	meromorphic	meromorphic	ADJ
ma-25	387	6	function	function	NOUN
ma-25	387	7	for	for	ADP
ma-25	387	8	which	which	PRON
ma-25	387	9	ρ[p	ρ[p	NOUN
ma-25	387	10	,	,	PUNCT
ma-25	387	11	q	q	X
ma-25	387	12	]	]	X
ma-25	387	13	(	(	PUNCT
ma-25	387	14	f	f	X
ma-25	387	15	)	)	PUNCT
ma-25	387	16	=	=	PUNCT
ma-25	388	1	β	β	X
ma-25	388	2	<	<	X
ma-25	388	3	+	+	PROPN
ma-25	388	4	∞	∞	PROPN
ma-25	388	5	,	,	PUNCT
ma-25	388	6	and	and	CCONJ
ma-25	388	7	let	let	VERB
ma-25	388	8	k	k	PROPN
ma-25	388	9	≥	≥	NUM
ma-25	388	10	1	1	NUM
ma-25	388	11	be	be	AUX
ma-25	388	12	an	an	DET
ma-25	388	13	integer	integer	NOUN
ma-25	388	14	.	.	PUNCT
ma-25	389	1	then	then	ADV
ma-25	389	2	for	for	ADP
ma-25	389	3	any	any	DET
ma-25	389	4	ε	ε	PROPN
ma-25	389	5	>	>	X
ma-25	389	6	0	0	PROPN
ma-25	389	7	,	,	PUNCT
ma-25	389	8	m	m	VERB
ma-25	389	9	(	(	PUNCT
ma-25	389	10	r	r	NOUN
ma-25	389	11	,	,	PUNCT
ma-25	389	12	f	f	PROPN
ma-25	389	13	(	(	PUNCT
ma-25	389	14	k	k	NOUN
ma-25	389	15	)	)	PUNCT
ma-25	389	16	f	f	NOUN
ma-25	389	17	)	)	PUNCT
ma-25	390	1	=	=	PUNCT
ma-25	390	2	o	o	NOUN
ma-25	390	3	(	(	PUNCT
ma-25	390	4	expp−1	expp−1	AUX
ma-25	390	5	{	{	PUNCT
ma-25	390	6	(	(	PUNCT
ma-25	390	7	β	β	X
ma-25	390	8	+	+	CCONJ
ma-25	390	9	ε	ε	PROPN
ma-25	390	10	)	)	PUNCT
ma-25	390	11	logq	logq	VERB
ma-25	390	12	r	r	NOUN
ma-25	390	13	}	}	PUNCT
ma-25	390	14	)	)	PUNCT
ma-25	390	15	,	,	PUNCT
ma-25	390	16	holds	hold	VERB
ma-25	390	17	outside	outside	ADV
ma-25	390	18	of	of	ADP
ma-25	390	19	a	a	DET
ma-25	390	20	possible	possible	ADJ
ma-25	390	21	exceptional	exceptional	ADJ
ma-25	390	22	set	set	NOUN
ma-25	390	23	e8	e8	PROPN
ma-25	390	24	of	of	ADP
ma-25	390	25	finite	finite	PROPN
ma-25	390	26	linear	linear	PROPN
ma-25	390	27	measure	measure	NOUN
ma-25	390	28	.	.	PUNCT
ma-25	391	1	lemma	lemma	PROPN
ma-25	391	2	2.15	2.15	NUM
ma-25	391	3	let	let	VERB
ma-25	391	4	a0	a0	PROPN
ma-25	391	5	,	,	PUNCT
ma-25	391	6	a1	a1	PROPN
ma-25	391	7	,	,	PUNCT
ma-25	391	8	...	...	PUNCT
ma-25	391	9	,	,	PUNCT
ma-25	391	10	ak	ak	PROPN
ma-25	391	11	6≡	6≡	NUM
ma-25	391	12	0	0	NUM
ma-25	391	13	,	,	PUNCT
ma-25	391	14	f	f	PROPN
ma-25	391	15	6≡	6≡	NUM
ma-25	391	16	0	0	NUM
ma-25	391	17	be	be	AUX
ma-25	391	18	finite	finite	NOUN
ma-25	391	19	[	[	X
ma-25	391	20	p	p	X
ma-25	391	21	,	,	PUNCT
ma-25	391	22	q]-order	q]-order	NOUN
ma-25	391	23	meromorphic	meromorphic	ADJ
ma-25	391	24	functions	function	NOUN
ma-25	391	25	.	.	PUNCT
ma-25	392	1	if	if	SCONJ
ma-25	392	2	f	f	PROPN
ma-25	392	3	is	be	AUX
ma-25	392	4	a	a	DET
ma-25	392	5	meromorphic	meromorphic	ADJ
ma-25	392	6	solution	solution	NOUN
ma-25	392	7	with	with	ADP
ma-25	392	8	ρ[p	ρ[p	NOUN
ma-25	392	9	,	,	PUNCT
ma-25	392	10	q	q	X
ma-25	392	11	]	]	X
ma-25	392	12	(	(	PUNCT
ma-25	392	13	f	f	X
ma-25	392	14	)	)	PUNCT
ma-25	393	1	=	=	PUNCT
ma-25	394	1	+	+	PUNCT
ma-25	394	2	∞	∞	NUM
ma-25	394	3	and	and	CCONJ
ma-25	394	4	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	394	5	]	]	PUNCT
ma-25	394	6	(	(	PUNCT
ma-25	394	7	f	f	X
ma-25	394	8	)	)	PUNCT
ma-25	394	9	=	=	SYM
ma-25	395	1	ρ	ρ	X
ma-25	395	2	<	<	X
ma-25	396	1	+	+	NOUN
ma-25	396	2	∞	∞	NUM
ma-25	396	3	of	of	ADP
ma-25	396	4	equation	equation	NOUN
ma-25	396	5	(	(	PUNCT
ma-25	396	6	1.4	1.4	NUM
ma-25	396	7	)	)	PUNCT
ma-25	396	8	,	,	PUNCT
ma-25	396	9	then	then	ADV
ma-25	396	10	λ[p	λ[p	PROPN
ma-25	396	11	,	,	PUNCT
ma-25	396	12	q	q	X
ma-25	396	13	]	]	X
ma-25	396	14	(	(	PUNCT
ma-25	396	15	f	f	X
ma-25	396	16	)	)	PUNCT
ma-25	397	1	=	=	SYM
ma-25	397	2	λ[p	λ[p	ADJ
ma-25	397	3	,	,	PUNCT
ma-25	397	4	q](f	q](f	NOUN
ma-25	397	5	)	)	PUNCT
ma-25	397	6	=	=	SYM
ma-25	397	7	ρ[p	ρ[p	NOUN
ma-25	397	8	,	,	PUNCT
ma-25	397	9	q](f	q](f	NOUN
ma-25	397	10	)	)	PUNCT
ma-25	397	11	=	=	PUNCT
ma-25	398	1	+	+	PUNCT
ma-25	398	2	∞	∞	NUM
ma-25	398	3	and	and	CCONJ
ma-25	398	4	λ[p+1,q	λ[p+1,q	NOUN
ma-25	398	5	]	]	X
ma-25	398	6	(	(	PUNCT
ma-25	398	7	f	f	X
ma-25	398	8	)	)	PUNCT
ma-25	398	9	=	=	SYM
ma-25	398	10	λ[p+1,q](f	λ[p+1,q](f	X
ma-25	398	11	)	)	PUNCT
ma-25	398	12	=	=	SYM
ma-25	398	13	ρ[p+1,q](f	ρ[p+1,q](f	X
ma-25	398	14	)	)	PUNCT
ma-25	398	15	=	=	SYM
ma-25	398	16	ρ	ρ	PROPN
ma-25	398	17	.	.	PUNCT
ma-25	398	18	proof	proof	NOUN
ma-25	398	19	let	let	VERB
ma-25	398	20	f	f	PRON
ma-25	398	21	be	be	AUX
ma-25	398	22	a	a	DET
ma-25	398	23	meromorphic	meromorphic	ADJ
ma-25	398	24	solution	solution	NOUN
ma-25	398	25	of	of	ADP
ma-25	398	26	(	(	PUNCT
ma-25	398	27	1.4	1.4	NUM
ma-25	398	28	)	)	PUNCT
ma-25	398	29	with	with	ADP
ma-25	398	30	infinite	infinite	ADJ
ma-25	398	31	[	[	X
ma-25	398	32	p	p	NOUN
ma-25	398	33	,	,	PUNCT
ma-25	398	34	q]-order	q]-order	NOUN
ma-25	398	35	and	and	CCONJ
ma-25	398	36	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	398	37	]	]	PUNCT
ma-25	398	38	(	(	PUNCT
ma-25	398	39	f	f	X
ma-25	398	40	)	)	PUNCT
ma-25	398	41	=	=	SYM
ma-25	399	1	ρ	ρ	X
ma-25	399	2	<	<	X
ma-25	399	3	+	+	NOUN
ma-25	399	4	∞.	∞.	PROPN
ma-25	399	5	note	note	NOUN
ma-25	399	6	first	first	ADV
ma-25	399	7	that	that	SCONJ
ma-25	399	8	by	by	ADP
ma-25	399	9	definition	definition	NOUN
ma-25	399	10	,	,	PUNCT
ma-25	399	11	we	we	PRON
ma-25	399	12	have	have	VERB
ma-25	399	13	λ[p+1,q	λ[p+1,q	NOUN
ma-25	399	14	]	]	X
ma-25	399	15	(	(	PUNCT
ma-25	399	16	f	f	PROPN
ma-25	399	17	)	)	PUNCT
ma-25	399	18	≤	≤	NUM
ma-25	399	19	λ[p+1,q	λ[p+1,q	PROPN
ma-25	399	20	]	]	X
ma-25	399	21	(	(	PUNCT
ma-25	399	22	f	f	PROPN
ma-25	399	23	)	)	PUNCT
ma-25	399	24	≤	≤	NUM
ma-25	399	25	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	399	26	]	]	PUNCT
ma-25	399	27	(	(	PUNCT
ma-25	399	28	f	f	PROPN
ma-25	399	29	)	)	PUNCT
ma-25	399	30	.	.	PUNCT
ma-25	400	1	then	then	ADV
ma-25	400	2	,	,	PUNCT
ma-25	400	3	itremains	itremain	VERB
ma-25	400	4	to	to	PART
ma-25	400	5	show	show	VERB
ma-25	400	6	that	that	DET
ma-25	400	7	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	400	8	]	]	PUNCT
ma-25	400	9	(	(	PUNCT
ma-25	400	10	f	f	PROPN
ma-25	400	11	)	)	PUNCT
ma-25	400	12	≤	≤	NUM
ma-25	400	13	λ[p+1,q	λ[p+1,q	PROPN
ma-25	400	14	]	]	X
ma-25	400	15	(	(	PUNCT
ma-25	400	16	f	f	PROPN
ma-25	400	17	)	)	PUNCT
ma-25	400	18	≤	≤	NUM
ma-25	400	19	λ[p+1,q	λ[p+1,q	PROPN
ma-25	400	20	]	]	X
ma-25	400	21	(	(	PUNCT
ma-25	400	22	f	f	PROPN
ma-25	400	23	)	)	PUNCT
ma-25	400	24	.	.	PUNCT
ma-25	401	1	we	we	PRON
ma-25	401	2	rewrite	rewrite	VERB
ma-25	401	3	(	(	PUNCT
ma-25	401	4	1.4	1.4	NUM
ma-25	401	5	)	)	PUNCT
ma-25	401	6	as	as	ADP
ma-25	401	7	1	1	NUM
ma-25	401	8	f	f	NOUN
ma-25	401	9	=	=	SYM
ma-25	401	10	1	1	NUM
ma-25	401	11	f	f	PROPN
ma-25	401	12	(	(	PUNCT
ma-25	401	13	ak	ak	PROPN
ma-25	401	14	(	(	PUNCT
ma-25	401	15	z	z	PROPN
ma-25	401	16	)	)	PUNCT
ma-25	401	17	f	f	NOUN
ma-25	401	18	(	(	PUNCT
ma-25	401	19	k	k	NOUN
ma-25	401	20	)	)	PUNCT
ma-25	401	21	f	f	PROPN
ma-25	402	1	+	+	CCONJ
ma-25	402	2	ak−1	ak−1	PROPN
ma-25	402	3	(	(	PUNCT
ma-25	402	4	z	z	NOUN
ma-25	402	5	)	)	PUNCT
ma-25	402	6	f	f	PROPN
ma-25	402	7	(	(	PUNCT
ma-25	402	8	k−1	k−1	PROPN
ma-25	402	9	)	)	PUNCT
ma-25	402	10	f	f	PROPN
ma-25	403	1	+	+	CCONJ
ma-25	403	2	·	·	PUNCT
ma-25	403	3	·	·	PUNCT
ma-25	403	4	·	·	PUNCT
ma-25	403	5	+	+	NUM
ma-25	403	6	a1	a1	NOUN
ma-25	403	7	(	(	PUNCT
ma-25	403	8	z	z	NOUN
ma-25	403	9	)	)	PUNCT
ma-25	403	10	f	f	NOUN
ma-25	404	1	′	′	NUM
ma-25	404	2	f	f	PROPN
ma-25	404	3	+	+	CCONJ
ma-25	404	4	a0	a0	PROPN
ma-25	404	5	(	(	PUNCT
ma-25	404	6	z	z	NOUN
ma-25	404	7	)	)	PUNCT
ma-25	404	8	)	)	PUNCT
ma-25	404	9	.	.	PUNCT
ma-25	405	1	(	(	PUNCT
ma-25	405	2	2.33	2.33	NUM
ma-25	405	3	)	)	PUNCT
ma-25	405	4	by	by	ADP
ma-25	405	5	using	use	VERB
ma-25	405	6	lemma	lemma	PROPN
ma-25	405	7	2.14	2.14	NUM
ma-25	405	8	and	and	CCONJ
ma-25	405	9	(	(	PUNCT
ma-25	405	10	2.33	2.33	NUM
ma-25	405	11	)	)	PUNCT
ma-25	405	12	,	,	PUNCT
ma-25	405	13	for	for	ADP
ma-25	405	14	|z	|z	PROPN
ma-25	406	1	|	|	ADV
ma-25	406	2	=	=	PUNCT
ma-25	406	3	r	r	NOUN
ma-25	406	4	outside	outside	ADP
ma-25	406	5	a	a	DET
ma-25	406	6	set	set	NOUN
ma-25	406	7	e8	e8	PROPN
ma-25	406	8	of	of	ADP
ma-25	406	9	a	a	DET
ma-25	406	10	finite	finite	ADJ
ma-25	406	11	linear	linear	NOUN
ma-25	406	12	measure	measure	NOUN
ma-25	406	13	and	and	CCONJ
ma-25	406	14	anygiven	anygiven	ADJ
ma-25	406	15	ε	ε	PROPN
ma-25	406	16	>	>	X
ma-25	406	17	0	0	PROPN
ma-25	406	18	,	,	PUNCT
ma-25	406	19	we	we	PRON
ma-25	406	20	get	get	VERB
ma-25	406	21	m	m	VERB
ma-25	406	22	(	(	PUNCT
ma-25	406	23	r	r	NOUN
ma-25	406	24	,	,	PUNCT
ma-25	406	25	1	1	NUM
ma-25	406	26	f	f	NOUN
ma-25	406	27	)	)	PUNCT
ma-25	406	28	≤	≤	NUM
ma-25	407	1	m	m	VERB
ma-25	407	2	(	(	PUNCT
ma-25	407	3	r	r	NOUN
ma-25	407	4	,	,	PUNCT
ma-25	407	5	1	1	NUM
ma-25	407	6	f	f	NOUN
ma-25	407	7	)	)	PUNCT
ma-25	408	1	+	+	CCONJ
ma-25	409	1	k∑	k∑	X
ma-25	410	1	j=1	j=1	ADJ
ma-25	410	2	m	m	VERB
ma-25	410	3	(	(	PUNCT
ma-25	410	4	r	r	NOUN
ma-25	410	5	,	,	PUNCT
ma-25	410	6	f	f	PROPN
ma-25	410	7	(	(	PUNCT
ma-25	410	8	j	j	PROPN
ma-25	410	9	)	)	PUNCT
ma-25	410	10	f	f	PROPN
ma-25	410	11	)	)	PUNCT
ma-25	411	1	+	+	CCONJ
ma-25	411	2	k∑	k∑	VERB
ma-25	411	3	j=0	j=0	PROPN
ma-25	411	4	m	m	PROPN
ma-25	411	5	(	(	PUNCT
ma-25	411	6	r	r	PROPN
ma-25	411	7	,	,	PUNCT
ma-25	411	8	aj	aj	PROPN
ma-25	411	9	)	)	PUNCT
ma-25	412	1	+	+	NOUN
ma-25	412	2	o	o	X
ma-25	412	3	(	(	PUNCT
ma-25	412	4	1	1	NUM
ma-25	412	5	)	)	PUNCT
ma-25	412	6	≤	≤	NUM
ma-25	412	7	m	m	VERB
ma-25	412	8	(	(	PUNCT
ma-25	412	9	r	r	NOUN
ma-25	412	10	,	,	PUNCT
ma-25	412	11	1	1	NUM
ma-25	412	12	f	f	NOUN
ma-25	412	13	)	)	PUNCT
ma-25	413	1	+	+	CCONJ
ma-25	413	2	k∑	k∑	VERB
ma-25	413	3	j=0	j=0	PROPN
ma-25	413	4	m	m	PROPN
ma-25	413	5	(	(	PUNCT
ma-25	413	6	r	r	PROPN
ma-25	413	7	,	,	PUNCT
ma-25	413	8	aj	aj	PROPN
ma-25	413	9	)	)	PUNCT
ma-25	414	1	+	+	NOUN
ma-25	414	2	o	o	X
ma-25	414	3	(	(	PUNCT
ma-25	414	4	expp	expp	ADJ
ma-25	414	5	{	{	PUNCT
ma-25	414	6	(	(	PUNCT
ma-25	414	7	ρ+	ρ+	NUM
ma-25	414	8	ε	ε	NOUN
ma-25	414	9	)	)	PUNCT
ma-25	414	10	logq	logq	VERB
ma-25	414	11	r	r	NOUN
ma-25	414	12	}	}	PUNCT
ma-25	414	13	)	)	PUNCT
ma-25	414	14	.	.	PUNCT
ma-25	415	1	(	(	PUNCT
ma-25	415	2	2.34	2.34	NUM
ma-25	415	3	)	)	PUNCT
ma-25	415	4	on	on	ADP
ma-25	415	5	the	the	DET
ma-25	415	6	other	other	ADJ
ma-25	415	7	hand	hand	NOUN
ma-25	415	8	,	,	PUNCT
ma-25	415	9	by	by	ADP
ma-25	415	10	(	(	PUNCT
ma-25	415	11	1.4	1.4	NUM
ma-25	415	12	)	)	PUNCT
ma-25	415	13	,	,	PUNCT
ma-25	415	14	if	if	SCONJ
ma-25	415	15	f	f	PROPN
ma-25	415	16	has	have	VERB
ma-25	415	17	a	a	DET
ma-25	415	18	zero	zero	NUM
ma-25	415	19	at	at	ADP
ma-25	415	20	z0	z0	PROPN
ma-25	415	21	of	of	ADP
ma-25	415	22	order	order	NOUN
ma-25	415	23	α	α	PROPN
ma-25	415	24	(	(	PUNCT
ma-25	415	25	α	α	NOUN
ma-25	415	26	>	>	X
ma-25	415	27	k	k	PROPN
ma-25	415	28	)	)	PUNCT
ma-25	415	29	,	,	PUNCT
ma-25	415	30	and	and	CCONJ
ma-25	415	31	a0	a0	NOUN
ma-25	415	32	,	,	PUNCT
ma-25	415	33	a1	a1	PROPN
ma-25	415	34	,	,	PUNCT
ma-25	415	35	...	...	PUNCT
ma-25	415	36	,	,	PUNCT
ma-25	415	37	ak	ak	PROPN
ma-25	415	38	are	be	AUX
ma-25	415	39	allanalytic	allanalytic	ADJ
ma-25	415	40	at	at	ADP
ma-25	415	41	z0	z0	PROPN
ma-25	415	42	,	,	PUNCT
ma-25	415	43	then	then	ADV
ma-25	415	44	f	f	PROPN
ma-25	415	45	must	must	AUX
ma-25	415	46	have	have	VERB
ma-25	415	47	a	a	DET
ma-25	415	48	zero	zero	NUM
ma-25	415	49	at	at	ADP
ma-25	415	50	z0	z0	PROPN
ma-25	415	51	of	of	ADP
ma-25	415	52	order	order	NOUN
ma-25	415	53	at	at	ADP
ma-25	415	54	least	least	ADJ
ma-25	415	55	α−	α−	ADP
ma-25	415	56	k	k	PROPN
ma-25	415	57	.	.	PUNCT
ma-25	416	1	hence	hence	ADV
ma-25	416	2	,	,	PUNCT
ma-25	416	3	n	n	CCONJ
ma-25	416	4	(	(	PUNCT
ma-25	416	5	r	r	NOUN
ma-25	416	6	,	,	PUNCT
ma-25	416	7	1	1	NUM
ma-25	416	8	f	f	NOUN
ma-25	416	9	)	)	PUNCT
ma-25	416	10	≤	≤	NOUN
ma-25	416	11	kn	kn	NOUN
ma-25	416	12	(	(	PUNCT
ma-25	416	13	r	r	NOUN
ma-25	416	14	,	,	PUNCT
ma-25	416	15	1	1	NUM
ma-25	416	16	f	f	NOUN
ma-25	416	17	)	)	PUNCT
ma-25	417	1	+	+	CCONJ
ma-25	417	2	n	n	CCONJ
ma-25	417	3	(	(	PUNCT
ma-25	417	4	r	r	NOUN
ma-25	417	5	,	,	PUNCT
ma-25	417	6	1	1	NUM
ma-25	417	7	f	f	NOUN
ma-25	417	8	)	)	PUNCT
ma-25	418	1	+	+	CCONJ
ma-25	418	2	k∑	k∑	ADJ
ma-25	418	3	j=0	j=0	PROPN
ma-25	418	4	n	n	CCONJ
ma-25	418	5	(	(	PUNCT
ma-25	418	6	r	r	PROPN
ma-25	418	7	,	,	PUNCT
ma-25	418	8	aj	aj	PROPN
ma-25	418	9	)	)	PUNCT
ma-25	418	10	eur	eur	PROPN
ma-25	418	11	.	.	PUNCT
ma-25	419	1	j.	j.	PROPN
ma-25	419	2	math	math	PROPN
ma-25	419	3	.	.	PUNCT
ma-25	420	1	anal	anal	ADJ
ma-25	420	2	.	.	PUNCT
ma-25	421	1	1	1	NUM
ma-25	421	2	(	(	PUNCT
ma-25	421	3	2021	2021	NUM
ma-25	421	4	)	)	PUNCT
ma-25	422	1	99and	99and	NOUN
ma-25	423	1	n	n	NOUN
ma-25	423	2	(	(	PUNCT
ma-25	423	3	r	r	NOUN
ma-25	423	4	,	,	PUNCT
ma-25	423	5	1	1	NUM
ma-25	423	6	f	f	NOUN
ma-25	423	7	)	)	PUNCT
ma-25	423	8	≤	≤	NOUN
ma-25	424	1	kn	kn	NOUN
ma-25	424	2	(	(	PUNCT
ma-25	424	3	r	r	NOUN
ma-25	424	4	,	,	PUNCT
ma-25	424	5	1	1	NUM
ma-25	424	6	f	f	NOUN
ma-25	424	7	)	)	PUNCT
ma-25	425	1	+	+	CCONJ
ma-25	425	2	n	n	CCONJ
ma-25	425	3	(	(	PUNCT
ma-25	425	4	r	r	NOUN
ma-25	425	5	,	,	PUNCT
ma-25	425	6	1	1	NUM
ma-25	425	7	f	f	NOUN
ma-25	425	8	)	)	PUNCT
ma-25	426	1	+	+	CCONJ
ma-25	426	2	k∑	k∑	ADJ
ma-25	426	3	j=0	j=0	PROPN
ma-25	426	4	n	n	CCONJ
ma-25	426	5	(	(	PUNCT
ma-25	426	6	r	r	PROPN
ma-25	426	7	,	,	PUNCT
ma-25	426	8	aj	aj	PROPN
ma-25	426	9	)	)	PUNCT
ma-25	426	10	.	.	PUNCT
ma-25	427	1	(	(	PUNCT
ma-25	427	2	2.35	2.35	NUM
ma-25	427	3	)	)	PUNCT
ma-25	427	4	therefore	therefore	ADV
ma-25	427	5	,	,	PUNCT
ma-25	427	6	by	by	ADP
ma-25	427	7	(	(	PUNCT
ma-25	427	8	2.34	2.34	NUM
ma-25	427	9	)	)	PUNCT
ma-25	427	10	and	and	CCONJ
ma-25	427	11	(	(	PUNCT
ma-25	427	12	2.35	2.35	NUM
ma-25	427	13	)	)	PUNCT
ma-25	427	14	,	,	PUNCT
ma-25	427	15	for	for	ADP
ma-25	427	16	all	all	DET
ma-25	427	17	sufficiently	sufficiently	ADV
ma-25	427	18	large	large	ADJ
ma-25	427	19	r	r	NOUN
ma-25	427	20	/∈	/∈	PUNCT
ma-25	427	21	e8	e8	PROPN
ma-25	427	22	and	and	CCONJ
ma-25	427	23	any	any	DET
ma-25	427	24	given	give	VERB
ma-25	427	25	ε	ε	PROPN
ma-25	427	26	>	>	X
ma-25	427	27	0	0	PROPN
ma-25	427	28	,	,	PUNCT
ma-25	427	29	we	we	PRON
ma-25	427	30	have	have	VERB
ma-25	427	31	t	t	NOUN
ma-25	427	32	(	(	PUNCT
ma-25	427	33	r	r	NOUN
ma-25	427	34	,	,	PUNCT
ma-25	427	35	f	f	NOUN
ma-25	427	36	)	)	PUNCT
ma-25	428	1	=	=	SYM
ma-25	428	2	t	t	PROPN
ma-25	428	3	(	(	PUNCT
ma-25	428	4	r	r	NOUN
ma-25	428	5	,	,	PUNCT
ma-25	428	6	1	1	NUM
ma-25	428	7	f	f	NOUN
ma-25	428	8	)	)	PUNCT
ma-25	429	1	+	+	NOUN
ma-25	429	2	o	o	X
ma-25	429	3	(	(	PUNCT
ma-25	429	4	1	1	NUM
ma-25	429	5	)	)	PUNCT
ma-25	429	6	≤	≤	NOUN
ma-25	429	7	t	t	NOUN
ma-25	429	8	(	(	PUNCT
ma-25	429	9	r	r	NOUN
ma-25	429	10	,	,	PUNCT
ma-25	429	11	f	f	PROPN
ma-25	429	12	)	)	PUNCT
ma-25	430	1	+	+	CCONJ
ma-25	430	2	k∑	k∑	ADJ
ma-25	430	3	j=0	j=0	PROPN
ma-25	430	4	t	t	PROPN
ma-25	430	5	(	(	PUNCT
ma-25	430	6	r	r	PROPN
ma-25	430	7	,	,	PUNCT
ma-25	430	8	aj	aj	PROPN
ma-25	430	9	)	)	PUNCT
ma-25	431	1	+	+	CCONJ
ma-25	431	2	kn	kn	PROPN
ma-25	431	3	(	(	PUNCT
ma-25	431	4	r	r	NOUN
ma-25	431	5	,	,	PUNCT
ma-25	431	6	1	1	NUM
ma-25	431	7	f	f	NOUN
ma-25	431	8	)	)	PUNCT
ma-25	432	1	+	+	NOUN
ma-25	432	2	o	o	X
ma-25	432	3	(	(	PUNCT
ma-25	432	4	expp	expp	ADJ
ma-25	432	5	{	{	PUNCT
ma-25	432	6	(	(	PUNCT
ma-25	432	7	ρ+	ρ+	NUM
ma-25	432	8	ε	ε	NOUN
ma-25	432	9	)	)	PUNCT
ma-25	432	10	logq	logq	VERB
ma-25	432	11	r	r	NOUN
ma-25	432	12	}	}	PUNCT
ma-25	432	13	)	)	PUNCT
ma-25	432	14	.	.	PUNCT
ma-25	433	1	(	(	PUNCT
ma-25	433	2	2.36	2.36	NUM
ma-25	433	3	)	)	PUNCT
ma-25	433	4	noting	note	VERB
ma-25	433	5	c	c	NOUN
ma-25	433	6	=	=	SYM
ma-25	433	7	max	max	PROPN
ma-25	433	8	{	{	PUNCT
ma-25	433	9	ρ[p	ρ[p	PROPN
ma-25	433	10	,	,	PUNCT
ma-25	433	11	q	q	X
ma-25	433	12	]	]	X
ma-25	433	13	(	(	PUNCT
ma-25	433	14	aj	aj	PROPN
ma-25	433	15	)	)	PUNCT
ma-25	433	16	(	(	PUNCT
ma-25	433	17	j	j	NOUN
ma-25	433	18	=	=	SYM
ma-25	433	19	0	0	NUM
ma-25	433	20	,	,	PUNCT
ma-25	433	21	1	1	NUM
ma-25	433	22	,	,	PUNCT
ma-25	433	23	...	...	PUNCT
ma-25	433	24	,	,	PUNCT
ma-25	433	25	k	k	NOUN
ma-25	433	26	)	)	PUNCT
ma-25	433	27	,	,	PUNCT
ma-25	433	28	ρ[p	ρ[p	PROPN
ma-25	433	29	,	,	PUNCT
ma-25	433	30	q	q	X
ma-25	433	31	]	]	X
ma-25	433	32	(	(	PUNCT
ma-25	433	33	f	f	PROPN
ma-25	433	34	)	)	PUNCT
ma-25	433	35	}	}	PUNCT
ma-25	433	36	.	.	PUNCT
ma-25	434	1	then	then	ADV
ma-25	434	2	,	,	PUNCT
ma-25	434	3	by	by	ADP
ma-25	434	4	using	use	VERB
ma-25	434	5	the	the	DET
ma-25	434	6	definition	definition	NOUN
ma-25	434	7	of	of	ADP
ma-25	434	8	the	the	DET
ma-25	434	9	[	[	X
ma-25	434	10	p	p	X
ma-25	434	11	,	,	PUNCT
ma-25	434	12	q]-order	q]-order	NOUN
ma-25	434	13	,	,	PUNCT
ma-25	434	14	for	for	ADP
ma-25	434	15	the	the	DET
ma-25	434	16	above	above	ADJ
ma-25	434	17	ε	ε	PROPN
ma-25	434	18	and	and	CCONJ
ma-25	434	19	sufficiently	sufficiently	ADV
ma-25	434	20	large	large	ADJ
ma-25	434	21	r	r	NOUN
ma-25	434	22	,	,	PUNCT
ma-25	434	23	we	we	PRON
ma-25	434	24	have	have	VERB
ma-25	434	25	t	t	NOUN
ma-25	434	26	(	(	PUNCT
ma-25	434	27	r	r	NOUN
ma-25	434	28	,	,	PUNCT
ma-25	434	29	f	f	PROPN
ma-25	434	30	)	)	PUNCT
ma-25	434	31	≤	≤	X
ma-25	434	32	expp	expp	ADJ
ma-25	434	33	{	{	PUNCT
ma-25	434	34	(	(	PUNCT
ma-25	434	35	c	c	NOUN
ma-25	434	36	+	+	CCONJ
ma-25	434	37	ε	ε	AUX
ma-25	434	38	)	)	PUNCT
ma-25	434	39	logq	logq	VERB
ma-25	434	40	r	r	NOUN
ma-25	434	41	}	}	PUNCT
ma-25	434	42	,	,	PUNCT
ma-25	434	43	(	(	PUNCT
ma-25	434	44	2.37	2.37	NUM
ma-25	434	45	)	)	PUNCT
ma-25	434	46	t	t	NOUN
ma-25	434	47	(	(	PUNCT
ma-25	434	48	r	r	PROPN
ma-25	434	49	,	,	PUNCT
ma-25	434	50	aj	aj	PROPN
ma-25	434	51	)	)	PUNCT
ma-25	434	52	≤	≤	X
ma-25	434	53	expp	expp	ADJ
ma-25	434	54	{	{	PUNCT
ma-25	434	55	(	(	PUNCT
ma-25	434	56	c	c	NOUN
ma-25	434	57	+	+	CCONJ
ma-25	434	58	ε	ε	AUX
ma-25	434	59	)	)	PUNCT
ma-25	434	60	logq	logq	VERB
ma-25	434	61	r	r	NOUN
ma-25	434	62	}	}	PUNCT
ma-25	434	63	,	,	PUNCT
ma-25	434	64	j	j	PROPN
ma-25	434	65	=	=	SYM
ma-25	434	66	0	0	NUM
ma-25	434	67	,	,	PUNCT
ma-25	434	68	1	1	NUM
ma-25	434	69	,	,	PUNCT
ma-25	434	70	...	...	PUNCT
ma-25	434	71	,	,	PUNCT
ma-25	434	72	k.	k.	PROPN
ma-25	434	73	(	(	PUNCT
ma-25	434	74	2.38	2.38	NUM
ma-25	434	75	)	)	PUNCT
ma-25	435	1	replacing	replace	VERB
ma-25	435	2	(	(	PUNCT
ma-25	435	3	2.37	2.37	NUM
ma-25	435	4	)	)	PUNCT
ma-25	435	5	and	and	CCONJ
ma-25	435	6	(	(	PUNCT
ma-25	435	7	2.38	2.38	NUM
ma-25	435	8	)	)	PUNCT
ma-25	435	9	into	into	ADP
ma-25	435	10	(	(	PUNCT
ma-25	435	11	2.36	2.36	NUM
ma-25	435	12	)	)	PUNCT
ma-25	435	13	,	,	PUNCT
ma-25	435	14	for	for	ADP
ma-25	435	15	r	r	PROPN
ma-25	435	16	/∈	/∈	PUNCT
ma-25	436	1	e8	e8	PROPN
ma-25	436	2	sufficiently	sufficiently	ADV
ma-25	436	3	large	large	ADJ
ma-25	436	4	and	and	CCONJ
ma-25	436	5	any	any	DET
ma-25	436	6	given	give	VERB
ma-25	436	7	ε	ε	PROPN
ma-25	436	8	>	>	X
ma-25	436	9	0	0	PROPN
ma-25	436	10	,	,	PUNCT
ma-25	436	11	weobtain	weobtain	NOUN
ma-25	436	12	t	t	PROPN
ma-25	436	13	(	(	PUNCT
ma-25	436	14	r	r	PROPN
ma-25	436	15	,	,	PUNCT
ma-25	436	16	f	f	PROPN
ma-25	436	17	)	)	PUNCT
ma-25	436	18	≤	≤	NOUN
ma-25	437	1	kn	kn	NOUN
ma-25	437	2	(	(	PUNCT
ma-25	437	3	r	r	NOUN
ma-25	437	4	,	,	PUNCT
ma-25	437	5	1	1	NUM
ma-25	437	6	f	f	NOUN
ma-25	437	7	)	)	PUNCT
ma-25	438	1	+	+	CCONJ
ma-25	438	2	(	(	PUNCT
ma-25	438	3	k	k	X
ma-25	438	4	+	+	PROPN
ma-25	438	5	2	2	X
ma-25	438	6	)	)	PUNCT
ma-25	438	7	expp	expp	ADJ
ma-25	438	8	{	{	PUNCT
ma-25	438	9	(	(	PUNCT
ma-25	438	10	c	c	NOUN
ma-25	438	11	+	+	CCONJ
ma-25	438	12	ε	ε	PROPN
ma-25	438	13	)	)	PUNCT
ma-25	438	14	logq	logq	VERB
ma-25	438	15	r	r	NOUN
ma-25	438	16	}	}	PUNCT
ma-25	438	17	+	+	NOUN
ma-25	438	18	o	o	X
ma-25	438	19	(	(	PUNCT
ma-25	438	20	expp	expp	ADJ
ma-25	438	21	{	{	PUNCT
ma-25	438	22	(	(	PUNCT
ma-25	438	23	ρ+	ρ+	NUM
ma-25	438	24	ε	ε	NOUN
ma-25	438	25	)	)	PUNCT
ma-25	438	26	logq	logq	VERB
ma-25	438	27	r	r	NOUN
ma-25	438	28	}	}	PUNCT
ma-25	438	29	)	)	PUNCT
ma-25	438	30	.	.	PUNCT
ma-25	439	1	(	(	PUNCT
ma-25	439	2	2.39	2.39	NUM
ma-25	439	3	)	)	PUNCT
ma-25	439	4	hence	hence	ADV
ma-25	439	5	,	,	PUNCT
ma-25	439	6	for	for	ADP
ma-25	439	7	any	any	DET
ma-25	439	8	f	f	NOUN
ma-25	439	9	with	with	ADP
ma-25	439	10	ρ[p	ρ[p	NOUN
ma-25	439	11	,	,	PUNCT
ma-25	439	12	q	q	X
ma-25	439	13	]	]	X
ma-25	439	14	(	(	PUNCT
ma-25	439	15	f	f	X
ma-25	439	16	)	)	PUNCT
ma-25	439	17	=	=	PUNCT
ma-25	440	1	+	+	PUNCT
ma-25	440	2	∞	∞	NUM
ma-25	440	3	and	and	CCONJ
ma-25	440	4	ρ[p+1,q](f	ρ[p+1,q](f	NOUN
ma-25	440	5	)	)	PUNCT
ma-25	440	6	=	=	SYM
ma-25	440	7	ρ	ρ	PROPN
ma-25	440	8	,	,	PUNCT
ma-25	440	9	by	by	ADP
ma-25	440	10	(	(	PUNCT
ma-25	440	11	2.39	2.39	NUM
ma-25	440	12	)	)	PUNCT
ma-25	440	13	,	,	PUNCT
ma-25	440	14	we	we	PRON
ma-25	440	15	have	have	VERB
ma-25	440	16	λ[p	λ[p	NOUN
ma-25	440	17	,	,	PUNCT
ma-25	440	18	q	q	X
ma-25	440	19	]	]	X
ma-25	440	20	(	(	PUNCT
ma-25	440	21	f	f	PROPN
ma-25	440	22	)	)	PUNCT
ma-25	440	23	≥	≥	NOUN
ma-25	440	24	ρ[p	ρ[p	NOUN
ma-25	440	25	,	,	PUNCT
ma-25	440	26	q	q	X
ma-25	440	27	]	]	X
ma-25	440	28	(	(	PUNCT
ma-25	440	29	f	f	X
ma-25	440	30	)	)	PUNCT
ma-25	440	31	=	=	PUNCT
ma-25	441	1	+	+	NUM
ma-25	441	2	∞	∞	PROPN
ma-25	441	3	,	,	PUNCT
ma-25	441	4	λ[p+1,q	λ[p+1,q	PROPN
ma-25	441	5	]	]	X
ma-25	441	6	(	(	PUNCT
ma-25	441	7	f	f	PROPN
ma-25	441	8	)	)	PUNCT
ma-25	441	9	≥	≥	NOUN
ma-25	441	10	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	441	11	]	]	PUNCT
ma-25	441	12	(	(	PUNCT
ma-25	441	13	f	f	PROPN
ma-25	441	14	)	)	PUNCT
ma-25	441	15	,	,	PUNCT
ma-25	441	16	so	so	SCONJ
ma-25	441	17	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	441	18	]	]	PUNCT
ma-25	441	19	(	(	PUNCT
ma-25	441	20	f	f	PROPN
ma-25	441	21	)	)	PUNCT
ma-25	441	22	≤	≤	NUM
ma-25	441	23	λ[p+1,q	λ[p+1,q	PROPN
ma-25	441	24	]	]	X
ma-25	441	25	(	(	PUNCT
ma-25	441	26	f	f	PROPN
ma-25	441	27	)	)	PUNCT
ma-25	441	28	≤	≤	NUM
ma-25	441	29	λ[p+1,q	λ[p+1,q	PROPN
ma-25	441	30	]	]	X
ma-25	441	31	(	(	PUNCT
ma-25	441	32	f	f	PROPN
ma-25	441	33	)	)	PUNCT
ma-25	441	34	.	.	PUNCT
ma-25	442	1	and	and	CCONJ
ma-25	442	2	the	the	DET
ma-25	442	3	fact	fact	NOUN
ma-25	442	4	that	that	SCONJ
ma-25	442	5	λ[p+1,q	λ[p+1,q	NOUN
ma-25	442	6	]	]	X
ma-25	442	7	(	(	PUNCT
ma-25	442	8	f	f	PROPN
ma-25	442	9	)	)	PUNCT
ma-25	442	10	≤	≤	NUM
ma-25	442	11	λ[p+1,q	λ[p+1,q	PROPN
ma-25	442	12	]	]	X
ma-25	442	13	(	(	PUNCT
ma-25	442	14	f	f	PROPN
ma-25	442	15	)	)	PUNCT
ma-25	442	16	≤	≤	NUM
ma-25	442	17	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	442	18	]	]	PUNCT
ma-25	442	19	(	(	PUNCT
ma-25	442	20	f	f	PROPN
ma-25	442	21	)	)	PUNCT
ma-25	442	22	,	,	PUNCT
ma-25	442	23	we	we	PRON
ma-25	442	24	obtain	obtain	VERB
ma-25	442	25	λ[p+1,q	λ[p+1,q	NOUN
ma-25	442	26	]	]	PUNCT
ma-25	442	27	(	(	PUNCT
ma-25	442	28	f	f	X
ma-25	442	29	)	)	PUNCT
ma-25	442	30	=	=	SYM
ma-25	442	31	λ[p+1,q	λ[p+1,q	PROPN
ma-25	442	32	]	]	X
ma-25	442	33	(	(	PUNCT
ma-25	442	34	f	f	X
ma-25	442	35	)	)	PUNCT
ma-25	442	36	=	=	SYM
ma-25	442	37	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	442	38	]	]	PUNCT
ma-25	442	39	(	(	PUNCT
ma-25	442	40	f	f	X
ma-25	442	41	)	)	PUNCT
ma-25	443	1	=	=	PROPN
ma-25	443	2	ρ	ρ	PROPN
ma-25	443	3	.	.	NOUN
ma-25	444	1	3	3	NUM
ma-25	444	2	.	.	X
ma-25	444	3	proof	proof	NOUN
ma-25	444	4	of	of	ADP
ma-25	444	5	theorem	theorem	ADJ
ma-25	444	6	1.1	1.1	NUM
ma-25	444	7	assume	assume	VERB
ma-25	444	8	that	that	SCONJ
ma-25	444	9	f	f	PROPN
ma-25	444	10	6≡	6≡	NUM
ma-25	444	11	0	0	NUM
ma-25	444	12	is	be	AUX
ma-25	444	13	a	a	DET
ma-25	444	14	rational	rational	ADJ
ma-25	444	15	solution	solution	NOUN
ma-25	444	16	of	of	ADP
ma-25	444	17	(	(	PUNCT
ma-25	444	18	1.3	1.3	NUM
ma-25	444	19	)	)	PUNCT
ma-25	444	20	.	.	PUNCT
ma-25	445	1	first	first	ADV
ma-25	445	2	,	,	PUNCT
ma-25	445	3	we	we	PRON
ma-25	445	4	will	will	AUX
ma-25	445	5	prove	prove	VERB
ma-25	445	6	that	that	SCONJ
ma-25	445	7	f	f	PROPN
ma-25	445	8	must	must	AUX
ma-25	445	9	be	be	AUX
ma-25	445	10	a	a	DET
ma-25	445	11	polynomialwith	polynomialwith	NOUN
ma-25	445	12	deg	deg	NOUN
ma-25	445	13	f	f	PROPN
ma-25	445	14	≤	≤	PROPN
ma-25	445	15	s	s	PART
ma-25	445	16	−	−	PROPN
ma-25	445	17	1	1	NUM
ma-25	445	18	.	.	PUNCT
ma-25	446	1	for	for	ADP
ma-25	446	2	this	this	PRON
ma-25	446	3	,	,	PUNCT
ma-25	446	4	if	if	SCONJ
ma-25	446	5	f	f	PROPN
ma-25	446	6	is	be	AUX
ma-25	446	7	a	a	DET
ma-25	446	8	rational	rational	ADJ
ma-25	446	9	function	function	NOUN
ma-25	446	10	,	,	PUNCT
ma-25	446	11	which	which	PRON
ma-25	446	12	has	have	VERB
ma-25	446	13	a	a	DET
ma-25	446	14	pole	pole	NOUN
ma-25	446	15	at	at	ADP
ma-25	446	16	z0	z0	PROPN
ma-25	446	17	of	of	ADP
ma-25	446	18	degree	degree	NOUN
ma-25	446	19	m	m	PROPN
ma-25	446	20	≥	≥	NOUN
ma-25	446	21	1,or	1,or	NUM
ma-25	446	22	f	f	NOUN
ma-25	446	23	is	be	AUX
ma-25	446	24	a	a	DET
ma-25	446	25	polynomial	polynomial	NOUN
ma-25	446	26	with	with	ADP
ma-25	446	27	deg	deg	PROPN
ma-25	446	28	f	f	PROPN
ma-25	446	29	≥	≥	NUM
ma-25	446	30	s	s	PROPN
ma-25	446	31	,	,	PUNCT
ma-25	446	32	then	then	ADV
ma-25	446	33	f	f	X
ma-25	446	34	(	(	PUNCT
ma-25	446	35	s)(z	s)(z	PROPN
ma-25	446	36	)	)	PUNCT
ma-25	446	37	6≡	6≡	NUM
ma-25	446	38	0	0	NUM
ma-25	446	39	.	.	PUNCT
ma-25	447	1	by	by	ADP
ma-25	447	2	(	(	PUNCT
ma-25	447	3	1.3	1.3	NUM
ma-25	447	4	)	)	PUNCT
ma-25	447	5	and	and	CCONJ
ma-25	447	6	lemma	lemma	PROPN
ma-25	447	7	2.4	2.4	NUM
ma-25	447	8	,	,	PUNCT
ma-25	447	9	we	we	PRON
ma-25	447	10	obtain	obtain	VERB
ma-25	447	11	σ	σ	NOUN
ma-25	447	12	≤	≤	NUM
ma-25	447	13	ρ[p	ρ[p	NOUN
ma-25	447	14	,	,	PUNCT
ma-25	447	15	q](as	q](as	NOUN
ma-25	447	16	)	)	PUNCT
ma-25	447	17	=	=	PUNCT
ma-25	447	18	ρ[p	ρ[p	PROPN
ma-25	447	19	,	,	PUNCT
ma-25	447	20	q](as	q](as	PROPN
ma-25	447	21	f	f	X
ma-25	447	22	(	(	PUNCT
ma-25	447	23	s	s	NOUN
ma-25	447	24	)	)	PUNCT
ma-25	447	25	)	)	PUNCT
ma-25	447	26	=	=	PUNCT
ma-25	448	1	ρ[p	ρ[p	PROPN
ma-25	448	2	,	,	PUNCT
ma-25	448	3	q	q	X
ma-25	448	4	]	]	X
ma-25	448	5	−	−	X
ma-25	448	6	k∑	k∑	PUNCT
ma-25	448	7	j=0	j=0	PROPN
ma-25	448	8	,	,	PUNCT
ma-25	448	9	j	j	PROPN
ma-25	448	10	6	6	NUM
ma-25	448	11	=	=	SYM
ma-25	448	12	s	s	X
ma-25	448	13	aj	aj	PROPN
ma-25	448	14	f	f	PROPN
ma-25	448	15	(	(	PUNCT
ma-25	448	16	j	j	PROPN
ma-25	448	17	)	)	PUNCT
ma-25	448	18			PROPN
ma-25	448	19	≤	≤	ADJ
ma-25	448	20	max	max	PROPN
ma-25	449	1	j=0,1,	j=0,1,	VERB
ma-25	449	2	...	...	PROPN
ma-25	449	3	,k	,k	PUNCT
ma-25	449	4	,	,	PUNCT
ma-25	449	5	j	j	PROPN
ma-25	449	6	6	6	NUM
ma-25	449	7	=	=	SYM
ma-25	449	8	s	s	X
ma-25	449	9	{	{	PUNCT
ma-25	449	10	ρ[p	ρ[p	PROPN
ma-25	449	11	,	,	PUNCT
ma-25	449	12	q	q	X
ma-25	449	13	]	]	X
ma-25	449	14	(	(	PUNCT
ma-25	449	15	aj	aj	PROPN
ma-25	449	16	)	)	PUNCT
ma-25	449	17	}	}	PUNCT
ma-25	449	18	which	which	PRON
ma-25	449	19	is	be	AUX
ma-25	449	20	a	a	DET
ma-25	449	21	contradiction	contradiction	NOUN
ma-25	449	22	.	.	PUNCT
ma-25	450	1	therefore	therefore	ADV
ma-25	450	2	,	,	PUNCT
ma-25	450	3	f	f	PROPN
ma-25	450	4	must	must	AUX
ma-25	450	5	be	be	AUX
ma-25	450	6	a	a	DET
ma-25	450	7	polynomial	polynomial	NOUN
ma-25	450	8	with	with	ADP
ma-25	450	9	deg	deg	PROPN
ma-25	450	10	f	f	PROPN
ma-25	450	11	≤	≤	PROPN
ma-25	450	12	s	s	PART
ma-25	451	1	−	−	PROPN
ma-25	451	2	1	1	NUM
ma-25	451	3	.	.	PUNCT
ma-25	451	4	eur	eur	PROPN
ma-25	451	5	.	.	PUNCT
ma-25	452	1	j.	j.	PROPN
ma-25	452	2	math	math	PROPN
ma-25	452	3	.	.	PUNCT
ma-25	453	1	anal	anal	ADJ
ma-25	453	2	.	.	PUNCT
ma-25	454	1	1	1	NUM
ma-25	454	2	(	(	PUNCT
ma-25	454	3	2021	2021	NUM
ma-25	454	4	)	)	PUNCT
ma-25	455	1	100now	100now	NUM
ma-25	455	2	,	,	PUNCT
ma-25	455	3	we	we	PRON
ma-25	455	4	assume	assume	VERB
ma-25	455	5	that	that	SCONJ
ma-25	455	6	f	f	PROPN
ma-25	455	7	is	be	AUX
ma-25	455	8	a	a	DET
ma-25	455	9	transcendental	transcendental	ADJ
ma-25	455	10	meromorphic	meromorphic	ADJ
ma-25	455	11	solution	solution	NOUN
ma-25	455	12	of	of	ADP
ma-25	455	13	(	(	PUNCT
ma-25	455	14	1.3	1.3	NUM
ma-25	455	15	)	)	PUNCT
ma-25	456	1	such	such	ADJ
ma-25	456	2	that	that	SCONJ
ma-25	456	3	λ[p	λ[p	ADJ
ma-25	456	4	,	,	PUNCT
ma-25	456	5	q	q	X
ma-25	456	6	]	]	X
ma-25	456	7	(	(	PUNCT
ma-25	456	8	1f	1f	NUM
ma-25	456	9	)	)	PUNCT
ma-25	456	10	<	<	X
ma-25	456	11	µ[p	µ[p	ADJ
ma-25	456	12	,	,	PUNCT
ma-25	456	13	q](f	q](f	NOUN
ma-25	456	14	)	)	PUNCT
ma-25	456	15	.	.	PUNCT
ma-25	457	1	by	by	ADP
ma-25	457	2	lemma	lemma	PROPN
ma-25	457	3	2.3	2.3	NUM
ma-25	457	4	,	,	PUNCT
ma-25	457	5	for	for	ADP
ma-25	457	6	any	any	DET
ma-25	457	7	given	give	VERB
ma-25	457	8	ε	ε	PROPN
ma-25	457	9	(	(	PUNCT
ma-25	457	10	0	0	PUNCT
ma-25	457	11	<	<	X
ma-25	457	12	2ε	2ε	PROPN
ma-25	457	13	<	<	X
ma-25	457	14	σ	σ	PROPN
ma-25	457	15	−	−	PROPN
ma-25	457	16	ρ	ρ	PROPN
ma-25	457	17	)	)	PUNCT
ma-25	457	18	,	,	PUNCT
ma-25	457	19	there	there	PRON
ma-25	457	20	exists	exist	VERB
ma-25	457	21	a	a	DET
ma-25	457	22	set	set	NOUN
ma-25	457	23	e3	e3	NOUN
ma-25	457	24	⊂	⊂	X
ma-25	457	25	(	(	PUNCT
ma-25	457	26	1,+∞	1,+∞	PROPN
ma-25	457	27	)	)	PUNCT
ma-25	457	28	witha	witha	NOUN
ma-25	457	29	finite	finite	PROPN
ma-25	457	30	linear	linear	PROPN
ma-25	457	31	measure	measure	NOUN
ma-25	457	32	(	(	PUNCT
ma-25	457	33	and	and	CCONJ
ma-25	457	34	so	so	ADV
ma-25	457	35	of	of	ADP
ma-25	457	36	finite	finite	ADJ
ma-25	457	37	logarithmic	logarithmic	ADJ
ma-25	457	38	measure	measure	NOUN
ma-25	457	39	)	)	PUNCT
ma-25	457	40	such	such	ADJ
ma-25	457	41	that	that	SCONJ
ma-25	457	42	|aj	|aj	NUM
ma-25	457	43	(	(	PUNCT
ma-25	457	44	z	z	NOUN
ma-25	457	45	)	)	PUNCT
ma-25	457	46	|	|	ADV
ma-25	457	47	≤	≤	NUM
ma-25	457	48	expp+1	expp+1	NOUN
ma-25	457	49	{	{	PUNCT
ma-25	457	50	(	(	PUNCT
ma-25	457	51	ρ+	ρ+	NUM
ma-25	457	52	ε	ε	NOUN
ma-25	457	53	)	)	PUNCT
ma-25	457	54	logq	logq	VERB
ma-25	457	55	r	r	NOUN
ma-25	457	56	}	}	PUNCT
ma-25	457	57	,	,	PUNCT
ma-25	457	58	j	j	PROPN
ma-25	457	59	=	=	SYM
ma-25	457	60	0	0	NUM
ma-25	457	61	,	,	PUNCT
ma-25	457	62	1	1	NUM
ma-25	457	63	,	,	PUNCT
ma-25	457	64	...	...	PUNCT
ma-25	457	65	,	,	PUNCT
ma-25	457	66	k	k	X
ma-25	457	67	,	,	PUNCT
ma-25	457	68	j	j	PROPN
ma-25	457	69	6=	6=	SYM
ma-25	457	70	s	s	X
ma-25	457	71	(	(	PUNCT
ma-25	457	72	3.1	3.1	NUM
ma-25	457	73	)	)	PUNCT
ma-25	457	74	holds	hold	VERB
ma-25	457	75	for	for	ADP
ma-25	457	76	all	all	DET
ma-25	457	77	z	z	NOUN
ma-25	457	78	satisfying	satisfy	VERB
ma-25	457	79	|z	|z	PROPN
ma-25	458	1	|	|	ADV
ma-25	458	2	=	=	NOUN
ma-25	458	3	r	r	NOUN
ma-25	458	4	/∈	/∈	PUNCT
ma-25	458	5	e3	e3	NOUN
ma-25	458	6	.	.	PUNCT
ma-25	459	1	in	in	ADP
ma-25	459	2	view	view	NOUN
ma-25	459	3	of	of	ADP
ma-25	459	4	lemma	lemma	PROPN
ma-25	459	5	2.8	2.8	NUM
ma-25	459	6	,	,	PUNCT
ma-25	459	7	there	there	PRON
ma-25	459	8	exists	exist	VERB
ma-25	459	9	a	a	DET
ma-25	459	10	set	set	NOUN
ma-25	459	11	e6	e6	PROPN
ma-25	459	12	⊂	⊂	PROPN
ma-25	459	13	(	(	PUNCT
ma-25	459	14	1,+∞	1,+∞	NUM
ma-25	459	15	)	)	PUNCT
ma-25	459	16	offinite	offinite	ADJ
ma-25	459	17	logarithmic	logarithmic	ADJ
ma-25	459	18	measure	measure	NOUN
ma-25	459	19	such	such	ADJ
ma-25	459	20	that	that	PRON
ma-25	459	21	|z	|z	PROPN
ma-25	460	1	|	|	ADV
ma-25	460	2	=	=	NOUN
ma-25	460	3	r	r	NOUN
ma-25	460	4	/∈	/∈	PUNCT
ma-25	461	1	[	[	X
ma-25	461	2	0	0	NUM
ma-25	461	3	,	,	PUNCT
ma-25	461	4	1	1	NUM
ma-25	461	5	]	]	PUNCT
ma-25	461	6	∪	∪	X
ma-25	461	7	e6	e6	PROPN
ma-25	461	8	,	,	PUNCT
ma-25	461	9	|g	|g	NOUN
ma-25	461	10	(	(	PUNCT
ma-25	461	11	z	z	NOUN
ma-25	461	12	)	)	PUNCT
ma-25	462	1	|	|	ADV
ma-25	462	2	=	=	SYM
ma-25	462	3	m	m	PROPN
ma-25	462	4	(	(	PUNCT
ma-25	462	5	r	r	NOUN
ma-25	462	6	,	,	PUNCT
ma-25	462	7	g	g	NOUN
ma-25	462	8	)	)	PUNCT
ma-25	462	9	and	and	CCONJ
ma-25	462	10	for	for	ADP
ma-25	462	11	r	r	NOUN
ma-25	462	12	sufficientlylarge	sufficientlylarge	NOUN
ma-25	462	13	,	,	PUNCT
ma-25	462	14	we	we	PRON
ma-25	462	15	have	have	VERB
ma-25	462	16	∣∣∣∣	∣∣∣∣	PROPN
ma-25	462	17	f	f	X
ma-25	462	18	(	(	PUNCT
ma-25	462	19	z	z	NOUN
ma-25	462	20	)	)	PUNCT
ma-25	462	21	f	f	NOUN
ma-25	462	22	(	(	PUNCT
ma-25	462	23	s	s	NOUN
ma-25	462	24	)	)	PUNCT
ma-25	462	25	(	(	PUNCT
ma-25	462	26	z	z	NOUN
ma-25	462	27	)	)	PUNCT
ma-25	462	28	∣∣∣∣	∣∣∣∣	NOUN
ma-25	462	29	≤	≤	ADJ
ma-25	462	30	r2s	r2s	X
ma-25	462	31	(	(	PUNCT
ma-25	462	32	s	s	X
ma-25	462	33	≥	≥	NOUN
ma-25	462	34	1	1	NUM
ma-25	462	35	is	be	AUX
ma-25	462	36	an	an	DET
ma-25	462	37	integer	integer	NOUN
ma-25	462	38	)	)	PUNCT
ma-25	462	39	.	.	PUNCT
ma-25	463	1	(	(	PUNCT
ma-25	463	2	3.2	3.2	NUM
ma-25	463	3	)	)	PUNCT
ma-25	463	4	according	accord	VERB
ma-25	463	5	to	to	ADP
ma-25	463	6	lemma	lemma	PROPN
ma-25	463	7	2.1	2.1	NUM
ma-25	463	8	,	,	PUNCT
ma-25	463	9	there	there	PRON
ma-25	463	10	exist	exist	VERB
ma-25	463	11	a	a	DET
ma-25	463	12	set	set	NOUN
ma-25	463	13	e1	e1	NOUN
ma-25	463	14	⊂	⊂	PROPN
ma-25	463	15	(	(	PUNCT
ma-25	463	16	1,+∞	1,+∞	NUM
ma-25	463	17	)	)	PUNCT
ma-25	463	18	with	with	ADP
ma-25	463	19	ml(e1	ml(e1	NOUN
ma-25	463	20	)	)	PUNCT
ma-25	464	1	<	<	X
ma-25	464	2	∞	∞	NUM
ma-25	464	3	and	and	CCONJ
ma-25	464	4	a	a	DET
ma-25	464	5	constant	constant	ADJ
ma-25	464	6	b	b	NOUN
ma-25	464	7	>	>	X
ma-25	464	8	0,such	0,such	NOUN
ma-25	465	1	that	that	SCONJ
ma-25	465	2	for	for	ADP
ma-25	465	3	all	all	DET
ma-25	465	4	z	z	NOUN
ma-25	465	5	satisfying	satisfy	VERB
ma-25	465	6	|z	|z	PROPN
ma-25	465	7	|	|	ADV
ma-25	465	8	=	=	NOUN
ma-25	465	9	r	r	NOUN
ma-25	465	10	/∈	/∈	PUNCT
ma-25	466	1	[	[	X
ma-25	466	2	0	0	NUM
ma-25	466	3	,	,	PUNCT
ma-25	466	4	1	1	NUM
ma-25	466	5	]	]	PUNCT
ma-25	466	6	∪	∪	NOUN
ma-25	466	7	e1	e1	NOUN
ma-25	466	8	,	,	PUNCT
ma-25	466	9	we	we	PRON
ma-25	466	10	have∣∣∣∣∣	have∣∣∣∣∣	PROPN
ma-25	466	11	f	f	PROPN
ma-25	466	12	(	(	PUNCT
ma-25	466	13	j	j	PROPN
ma-25	466	14	)	)	PUNCT
ma-25	466	15	(	(	PUNCT
ma-25	466	16	z	z	X
ma-25	466	17	)	)	PUNCT
ma-25	466	18	f	f	NOUN
ma-25	466	19	(	(	PUNCT
ma-25	466	20	z	z	NOUN
ma-25	466	21	)	)	PUNCT
ma-25	466	22	∣∣∣∣∣	∣∣∣∣∣	ADP
ma-25	467	1	≤	≤	PROPN
ma-25	467	2	b	b	X
ma-25	468	1	[	[	X
ma-25	468	2	t	t	X
ma-25	468	3	(	(	PUNCT
ma-25	468	4	2r	2r	NUM
ma-25	468	5	,	,	PUNCT
ma-25	468	6	f	f	PROPN
ma-25	468	7	)	)	PUNCT
ma-25	468	8	]	]	PUNCT
ma-25	468	9	k+1	k+1	X
ma-25	468	10	,	,	PUNCT
ma-25	468	11	j	j	PROPN
ma-25	468	12	=	=	SYM
ma-25	468	13	1	1	NUM
ma-25	468	14	,	,	PUNCT
ma-25	468	15	2	2	NUM
ma-25	468	16	,	,	PUNCT
ma-25	468	17	...	...	PUNCT
ma-25	468	18	,	,	PUNCT
ma-25	468	19	k	k	X
ma-25	468	20	,	,	PUNCT
ma-25	468	21	j	j	PROPN
ma-25	468	22	6=	6=	PROPN
ma-25	468	23	s.	s.	PROPN
ma-25	468	24	(	(	PUNCT
ma-25	468	25	3.3	3.3	NUM
ma-25	468	26	)	)	PUNCT
ma-25	468	27	from	from	ADP
ma-25	468	28	the	the	DET
ma-25	468	29	hypotheses	hypothesis	NOUN
ma-25	468	30	of	of	ADP
ma-25	468	31	theorem	theorem	NOUN
ma-25	468	32	1.1	1.1	NUM
ma-25	468	33	,	,	PUNCT
ma-25	468	34	there	there	PRON
ma-25	468	35	exists	exist	VERB
ma-25	468	36	a	a	DET
ma-25	468	37	set	set	NOUN
ma-25	468	38	h	h	NOUN
ma-25	468	39	⊂	⊂	PROPN
ma-25	468	40	(	(	PUNCT
ma-25	468	41	1,+∞	1,+∞	NUM
ma-25	468	42	)	)	PUNCT
ma-25	468	43	with	with	ADP
ma-25	468	44	ml	ml	PROPN
ma-25	468	45	(	(	PUNCT
ma-25	468	46	h	h	NOUN
ma-25	468	47	)	)	PUNCT
ma-25	468	48	=	=	PUNCT
ma-25	469	1	+	+	NUM
ma-25	469	2	∞	∞	NOUN
ma-25	469	3	,	,	PUNCT
ma-25	469	4	suchthat	suchthat	VERB
ma-25	469	5	for	for	ADP
ma-25	469	6	all	all	DET
ma-25	469	7	z	z	NOUN
ma-25	469	8	satisfying	satisfy	VERB
ma-25	469	9	|z	|z	PROPN
ma-25	470	1	|	|	ADV
ma-25	470	2	=	=	SYM
ma-25	470	3	r	r	NOUN
ma-25	470	4	∈	∈	PROPN
ma-25	470	5	h	h	NOUN
ma-25	470	6	,	,	PUNCT
ma-25	470	7	r	r	NOUN
ma-25	470	8	→	→	SYM
ma-25	470	9	+	+	NOUN
ma-25	470	10	∞	∞	NUM
ma-25	470	11	and	and	CCONJ
ma-25	470	12	sufficiently	sufficiently	ADV
ma-25	470	13	small	small	ADJ
ma-25	470	14	ε	ε	PROPN
ma-25	470	15	>	>	X
ma-25	470	16	0	0	PROPN
ma-25	470	17	,	,	PUNCT
ma-25	470	18	we	we	PRON
ma-25	470	19	have	have	AUX
ma-25	470	20	|as	|as	NUM
ma-25	470	21	(	(	PUNCT
ma-25	470	22	z	z	NOUN
ma-25	470	23	)	)	PUNCT
ma-25	471	1	|	|	ADV
ma-25	471	2	≥	≥	NOUN
ma-25	471	3	expp+1	expp+1	PROPN
ma-25	471	4	{	{	PUNCT
ma-25	471	5	(	(	PUNCT
ma-25	471	6	σ	σ	PROPN
ma-25	471	7	−	−	PROPN
ma-25	471	8	ε	ε	PROPN
ma-25	471	9	)	)	PUNCT
ma-25	471	10	logq	logq	VERB
ma-25	471	11	r	r	NOUN
ma-25	471	12	}	}	PUNCT
ma-25	471	13	.	.	PUNCT
ma-25	472	1	(	(	PUNCT
ma-25	472	2	3.4	3.4	NUM
ma-25	472	3	)	)	PUNCT
ma-25	472	4	now	now	ADV
ma-25	472	5	,	,	PUNCT
ma-25	472	6	by	by	ADP
ma-25	472	7	rewriting	rewrite	VERB
ma-25	472	8	equation	equation	NOUN
ma-25	472	9	(	(	PUNCT
ma-25	472	10	1.3	1.3	NUM
ma-25	472	11	)	)	PUNCT
ma-25	472	12	in	in	ADP
ma-25	472	13	the	the	DET
ma-25	472	14	form	form	NOUN
ma-25	472	15	|as	|as	X
ma-25	472	16	|	|	ADV
ma-25	472	17	≤	≤	NUM
ma-25	472	18	∣∣∣∣	∣∣∣∣	NOUN
ma-25	472	19	ff	ff	NOUN
ma-25	472	20	(	(	PUNCT
ma-25	472	21	s	s	NOUN
ma-25	472	22	)	)	PUNCT
ma-25	472	23	∣∣∣∣	∣∣∣∣	NOUN
ma-25	472	24	|a0|+	|a0|+	VERB
ma-25	472	25	k∑	k∑	NOUN
ma-25	473	1	j=1	j=1	PROPN
ma-25	473	2	j	j	PROPN
ma-25	473	3	6	6	NUM
ma-25	473	4	=	=	SYM
ma-25	473	5	s	s	PART
ma-25	473	6	∣∣aj	∣∣aj	NOUN
ma-25	473	7	∣∣	∣∣	NUM
ma-25	473	8	∣∣∣∣∣	∣∣∣∣∣	PROPN
ma-25	473	9	f	f	PROPN
ma-25	473	10	(	(	PUNCT
ma-25	473	11	j)f	j)f	ADJ
ma-25	473	12	∣∣∣∣∣	∣∣∣∣∣	ADJ
ma-25	473	13			NOUN
ma-25	473	14	(	(	PUNCT
ma-25	473	15	3.5	3.5	NUM
ma-25	473	16	)	)	PUNCT
ma-25	473	17	and	and	CCONJ
ma-25	473	18	substituting	substitute	VERB
ma-25	473	19	(	(	PUNCT
ma-25	473	20	3.1	3.1	NUM
ma-25	473	21	)	)	PUNCT
ma-25	473	22	,	,	PUNCT
ma-25	473	23	(	(	PUNCT
ma-25	473	24	3.2	3.2	NUM
ma-25	473	25	)	)	PUNCT
ma-25	473	26	,	,	PUNCT
ma-25	473	27	(	(	PUNCT
ma-25	473	28	3.3	3.3	NUM
ma-25	473	29	)	)	PUNCT
ma-25	473	30	and	and	CCONJ
ma-25	473	31	(	(	PUNCT
ma-25	473	32	3.4	3.4	NUM
ma-25	473	33	)	)	PUNCT
ma-25	473	34	into	into	ADP
ma-25	473	35	(	(	PUNCT
ma-25	473	36	3.5	3.5	NUM
ma-25	473	37	)	)	PUNCT
ma-25	473	38	,	,	PUNCT
ma-25	473	39	for	for	ADP
ma-25	473	40	all	all	DET
ma-25	473	41	z	z	NOUN
ma-25	473	42	satisfying	satisfy	VERB
ma-25	473	43	|z	|z	PROPN
ma-25	474	1	|	|	ADV
ma-25	474	2	=	=	SYM
ma-25	474	3	r	r	NOUN
ma-25	474	4	∈	∈	PROPN
ma-25	474	5	hr	hr	NOUN
ma-25	474	6	(	(	PUNCT
ma-25	474	7	[	[	X
ma-25	474	8	0	0	NUM
ma-25	474	9	,	,	PUNCT
ma-25	474	10	1]∪	1]∪	PROPN
ma-25	474	11	e1	e1	PROPN
ma-25	474	12	∪	∪	VERB
ma-25	474	13	e3	e3	NOUN
ma-25	474	14	∪	∪	ADJ
ma-25	474	15	e6	e6	NOUN
ma-25	474	16	)	)	PUNCT
ma-25	474	17	,	,	PUNCT
ma-25	474	18	r	r	NOUN
ma-25	474	19	→	→	SYM
ma-25	474	20	+	+	NOUN
ma-25	474	21	∞	∞	PROPN
ma-25	474	22	,	,	PUNCT
ma-25	474	23	we	we	PRON
ma-25	474	24	obtain	obtain	VERB
ma-25	474	25	expp+1	expp+1	NOUN
ma-25	474	26	{	{	PUNCT
ma-25	474	27	(	(	PUNCT
ma-25	474	28	σ	σ	PROPN
ma-25	474	29	−	−	PROPN
ma-25	474	30	ε	ε	PROPN
ma-25	474	31	)	)	PUNCT
ma-25	474	32	logq	logq	VERB
ma-25	474	33	r	r	NOUN
ma-25	474	34	}	}	PUNCT
ma-25	474	35	≤	≤	NOUN
ma-25	474	36	bkr2s	bkr2s	X
ma-25	474	37	expp+1	expp+1	NOUN
ma-25	474	38	{	{	PUNCT
ma-25	474	39	(	(	PUNCT
ma-25	474	40	ρ+	ρ+	NUM
ma-25	474	41	ε	ε	NOUN
ma-25	474	42	)	)	PUNCT
ma-25	474	43	logq	logq	VERB
ma-25	474	44	r	r	NOUN
ma-25	474	45	}	}	PUNCT
ma-25	475	1	[	[	X
ma-25	475	2	t	t	X
ma-25	475	3	(	(	PUNCT
ma-25	475	4	2r	2r	NUM
ma-25	475	5	,	,	PUNCT
ma-25	475	6	f	f	PROPN
ma-25	475	7	)	)	PUNCT
ma-25	475	8	]	]	X
ma-25	475	9	k+1	k+1	X
ma-25	475	10	.	.	PUNCT
ma-25	476	1	since	since	SCONJ
ma-25	476	2	0	0	NUM
ma-25	476	3	<	<	X
ma-25	476	4	2ε	2ε	PROPN
ma-25	476	5	<	<	X
ma-25	476	6	σ	σ	PROPN
ma-25	476	7	−	−	PROPN
ma-25	476	8	ρ	ρ	PROPN
ma-25	476	9	,	,	PUNCT
ma-25	476	10	then	then	ADV
ma-25	476	11	we	we	PRON
ma-25	476	12	have	have	VERB
ma-25	476	13	exp	exp	NOUN
ma-25	476	14	{	{	PUNCT
ma-25	476	15	(	(	PUNCT
ma-25	476	16	1−	1−	NUM
ma-25	476	17	o	o	NOUN
ma-25	476	18	(	(	PUNCT
ma-25	476	19	1	1	NUM
ma-25	476	20	)	)	PUNCT
ma-25	476	21	)	)	PUNCT
ma-25	476	22	expp	expp	ADJ
ma-25	476	23	{	{	PUNCT
ma-25	476	24	(	(	PUNCT
ma-25	476	25	σ	σ	PROPN
ma-25	476	26	−	−	PROPN
ma-25	476	27	ε	ε	PROPN
ma-25	476	28	)	)	PUNCT
ma-25	476	29	logq	logq	VERB
ma-25	476	30	r	r	NOUN
ma-25	476	31	}	}	PUNCT
ma-25	476	32	}	}	PUNCT
ma-25	476	33	≤	≤	NUM
ma-25	476	34	bkr2s	bkr2s	PUNCT
ma-25	477	1	[	[	X
ma-25	477	2	t	t	X
ma-25	477	3	(	(	PUNCT
ma-25	477	4	2r	2r	NUM
ma-25	477	5	,	,	PUNCT
ma-25	477	6	f	f	PROPN
ma-25	477	7	)	)	PUNCT
ma-25	477	8	]	]	X
ma-25	477	9	k+1	k+1	X
ma-25	477	10	.	.	PUNCT
ma-25	478	1	(	(	PUNCT
ma-25	478	2	3.6	3.6	NUM
ma-25	478	3	)	)	PUNCT
ma-25	478	4	from	from	ADP
ma-25	478	5	(	(	PUNCT
ma-25	478	6	3.6	3.6	NUM
ma-25	478	7	)	)	PUNCT
ma-25	478	8	and	and	CCONJ
ma-25	478	9	lemma	lemma	PROPN
ma-25	478	10	2.9	2.9	NUM
ma-25	478	11	,	,	PUNCT
ma-25	478	12	for	for	ADP
ma-25	478	13	any	any	DET
ma-25	478	14	given	give	VERB
ma-25	478	15	γ	γ	NOUN
ma-25	478	16	>	>	SYM
ma-25	478	17	1	1	NUM
ma-25	478	18	and	and	CCONJ
ma-25	478	19	sufficiently	sufficiently	ADV
ma-25	478	20	large	large	ADJ
ma-25	478	21	r	r	NOUN
ma-25	478	22	>	>	X
ma-25	478	23	r	r	NOUN
ma-25	478	24	,	,	PUNCT
ma-25	478	25	we	we	PRON
ma-25	478	26	get	get	VERB
ma-25	478	27	exp	exp	NOUN
ma-25	478	28	{	{	PUNCT
ma-25	478	29	(	(	PUNCT
ma-25	478	30	1−	1−	NUM
ma-25	478	31	o	o	NOUN
ma-25	478	32	(	(	PUNCT
ma-25	478	33	1	1	NUM
ma-25	478	34	)	)	PUNCT
ma-25	478	35	)	)	PUNCT
ma-25	479	1	expp	expp	ADJ
ma-25	479	2	{	{	PUNCT
ma-25	479	3	(	(	PUNCT
ma-25	479	4	σ	σ	PROPN
ma-25	479	5	−	−	PROPN
ma-25	479	6	ε	ε	PROPN
ma-25	479	7	)	)	PUNCT
ma-25	479	8	logq	logq	VERB
ma-25	479	9	r	r	NOUN
ma-25	479	10	}	}	PUNCT
ma-25	479	11	}	}	PUNCT
ma-25	479	12	≤	≤	NOUN
ma-25	480	1	bk	bk	ADP
ma-25	480	2	(	(	PUNCT
ma-25	480	3	γr)2s	γr)2s	X
ma-25	480	4	[	[	X
ma-25	480	5	t	t	X
ma-25	480	6	(	(	PUNCT
ma-25	480	7	2γr	2γr	NOUN
ma-25	480	8	,	,	PUNCT
ma-25	480	9	f	f	PROPN
ma-25	480	10	)	)	PUNCT
ma-25	480	11	]	]	PUNCT
ma-25	480	12	k+1	k+1	X
ma-25	480	13	which	which	PRON
ma-25	480	14	gives	give	VERB
ma-25	480	15	ρ[p	ρ[p	NOUN
ma-25	480	16	,	,	PUNCT
ma-25	480	17	q](f	q](f	NOUN
ma-25	480	18	)	)	PUNCT
ma-25	480	19	=	=	SYM
ma-25	480	20	µ[p	µ[p	ADJ
ma-25	480	21	,	,	PUNCT
ma-25	480	22	q	q	X
ma-25	480	23	]	]	X
ma-25	480	24	(	(	PUNCT
ma-25	480	25	f	f	X
ma-25	480	26	)	)	PUNCT
ma-25	480	27	=	=	PUNCT
ma-25	481	1	+	+	NUM
ma-25	481	2	∞	∞	PROPN
ma-25	481	3	,	,	PUNCT
ma-25	481	4	σ	σ	PROPN
ma-25	481	5	≤	≤	NUM
ma-25	481	6	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	481	7	]	]	PUNCT
ma-25	481	8	(	(	PUNCT
ma-25	481	9	f	f	PROPN
ma-25	481	10	)	)	PUNCT
ma-25	481	11	.	.	PUNCT
ma-25	482	1	(	(	PUNCT
ma-25	482	2	3.7	3.7	NUM
ma-25	482	3	)	)	PUNCT
ma-25	482	4	by	by	ADP
ma-25	482	5	using	use	VERB
ma-25	482	6	lemma	lemma	PROPN
ma-25	482	7	2.4	2.4	NUM
ma-25	482	8	,	,	PUNCT
ma-25	482	9	we	we	PRON
ma-25	482	10	have	have	VERB
ma-25	482	11	max	max	PROPN
ma-25	482	12	{	{	PUNCT
ma-25	482	13	ρ[p	ρ[p	PROPN
ma-25	482	14	,	,	PUNCT
ma-25	482	15	q	q	X
ma-25	482	16	]	]	X
ma-25	482	17	(	(	PUNCT
ma-25	482	18	aj	aj	PROPN
ma-25	482	19	)	)	PUNCT
ma-25	482	20	:	:	PUNCT
ma-25	483	1	j	j	X
ma-25	483	2	=	=	SYM
ma-25	483	3	0	0	PROPN
ma-25	483	4	,	,	PUNCT
ma-25	483	5	1	1	NUM
ma-25	483	6	,	,	PUNCT
ma-25	483	7	...	...	PUNCT
ma-25	483	8	,	,	PUNCT
ma-25	483	9	k	k	NOUN
ma-25	483	10	}	}	PUNCT
ma-25	483	11	=	=	PUNCT
ma-25	483	12	ρ[p	ρ[p	NOUN
ma-25	483	13	,	,	PUNCT
ma-25	483	14	q	q	X
ma-25	483	15	]	]	X
ma-25	483	16	(	(	PUNCT
ma-25	483	17	as	as	ADP
ma-25	483	18	)	)	PUNCT
ma-25	483	19	=	=	PUNCT
ma-25	483	20	β	β	X
ma-25	483	21	<	<	X
ma-25	484	1	+	+	PROPN
ma-25	484	2	∞.	∞.	PROPN
ma-25	484	3	eur	eur	NOUN
ma-25	484	4	.	.	PUNCT
ma-25	485	1	j.	j.	PROPN
ma-25	485	2	math	math	PROPN
ma-25	485	3	.	.	PUNCT
ma-25	486	1	anal	anal	ADJ
ma-25	486	2	.	.	PUNCT
ma-25	487	1	1	1	NUM
ma-25	487	2	(	(	PUNCT
ma-25	487	3	2021	2021	NUM
ma-25	487	4	)	)	PUNCT
ma-25	488	1	101since	101since	PROPN
ma-25	488	2	f	f	PROPN
ma-25	488	3	is	be	AUX
ma-25	488	4	of	of	ADP
ma-25	488	5	infinite	infinite	ADJ
ma-25	488	6	[	[	X
ma-25	488	7	p	p	NOUN
ma-25	488	8	,	,	PUNCT
ma-25	488	9	q]-order	q]-order	NOUN
ma-25	488	10	meromorphic	meromorphic	ADJ
ma-25	488	11	solution	solution	NOUN
ma-25	488	12	of	of	ADP
ma-25	488	13	equation	equation	NOUN
ma-25	488	14	(	(	PUNCT
ma-25	488	15	1.3	1.3	NUM
ma-25	488	16	)	)	PUNCT
ma-25	488	17	satisfying	satisfy	VERB
ma-25	488	18	λ[p	λ[p	NOUN
ma-25	488	19	,	,	PUNCT
ma-25	488	20	q	q	X
ma-25	488	21	]	]	X
ma-25	488	22	(	(	PUNCT
ma-25	488	23	1f	1f	NUM
ma-25	488	24	)	)	PUNCT
ma-25	488	25	<	<	X
ma-25	488	26	µ[p	µ[p	ADJ
ma-25	488	27	,	,	PUNCT
ma-25	488	28	q	q	X
ma-25	488	29	]	]	X
ma-25	488	30	(	(	PUNCT
ma-25	488	31	f	f	PROPN
ma-25	488	32	)	)	PUNCT
ma-25	488	33	,	,	PUNCT
ma-25	488	34	then	then	ADV
ma-25	488	35	by	by	ADP
ma-25	488	36	lemma	lemma	PROPN
ma-25	488	37	2.11	2.11	NUM
ma-25	488	38	,	,	PUNCT
ma-25	488	39	we	we	PRON
ma-25	488	40	obtain	obtain	VERB
ma-25	488	41	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	488	42	]	]	PUNCT
ma-25	488	43	(	(	PUNCT
ma-25	488	44	f	f	PROPN
ma-25	488	45	)	)	PUNCT
ma-25	488	46	≤	≤	PROPN
ma-25	488	47	max	max	PROPN
ma-25	488	48	{	{	PUNCT
ma-25	488	49	ρ[p	ρ[p	PROPN
ma-25	488	50	,	,	PUNCT
ma-25	488	51	q	q	X
ma-25	488	52	]	]	X
ma-25	488	53	(	(	PUNCT
ma-25	488	54	aj	aj	PROPN
ma-25	488	55	)	)	PUNCT
ma-25	488	56	:	:	PUNCT
ma-25	489	1	j	j	X
ma-25	489	2	=	=	SYM
ma-25	489	3	0	0	PROPN
ma-25	489	4	,	,	PUNCT
ma-25	489	5	1	1	NUM
ma-25	489	6	,	,	PUNCT
ma-25	489	7	...	...	PUNCT
ma-25	489	8	,	,	PUNCT
ma-25	489	9	k	k	NOUN
ma-25	489	10	}	}	PUNCT
ma-25	489	11	=	=	PUNCT
ma-25	489	12	ρ[p	ρ[p	NOUN
ma-25	489	13	,	,	PUNCT
ma-25	489	14	q	q	X
ma-25	489	15	]	]	X
ma-25	489	16	(	(	PUNCT
ma-25	489	17	as	as	ADP
ma-25	489	18	)	)	PUNCT
ma-25	489	19	.	.	PUNCT
ma-25	490	1	(	(	PUNCT
ma-25	490	2	3.8	3.8	NUM
ma-25	490	3	)	)	PUNCT
ma-25	490	4	by	by	ADP
ma-25	490	5	(	(	PUNCT
ma-25	490	6	3.7	3.7	NUM
ma-25	490	7	)	)	PUNCT
ma-25	490	8	and	and	CCONJ
ma-25	490	9	(	(	PUNCT
ma-25	490	10	3.8	3.8	NUM
ma-25	490	11	)	)	PUNCT
ma-25	490	12	,	,	PUNCT
ma-25	490	13	we	we	PRON
ma-25	490	14	conclude	conclude	VERB
ma-25	490	15	that	that	PRON
ma-25	490	16	µ[p	µ[p	VERB
ma-25	490	17	,	,	PUNCT
ma-25	490	18	q	q	X
ma-25	490	19	]	]	X
ma-25	490	20	(	(	PUNCT
ma-25	490	21	f	f	X
ma-25	490	22	)	)	PUNCT
ma-25	490	23	=	=	PUNCT
ma-25	491	1	ρ[p	ρ[p	PROPN
ma-25	491	2	,	,	PUNCT
ma-25	491	3	q	q	X
ma-25	491	4	]	]	X
ma-25	491	5	(	(	PUNCT
ma-25	491	6	f	f	X
ma-25	491	7	)	)	PUNCT
ma-25	491	8	=	=	PUNCT
ma-25	492	1	+	+	PUNCT
ma-25	492	2	∞	∞	NUM
ma-25	492	3	and	and	CCONJ
ma-25	492	4	σ	σ	PROPN
ma-25	492	5	≤	≤	PROPN
ma-25	492	6	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	492	7	]	]	PUNCT
ma-25	492	8	(	(	PUNCT
ma-25	492	9	f	f	PROPN
ma-25	492	10	)	)	PUNCT
ma-25	492	11	≤	≤	PROPN
ma-25	492	12	ρ[p	ρ[p	NOUN
ma-25	492	13	,	,	PUNCT
ma-25	492	14	q	q	X
ma-25	492	15	]	]	X
ma-25	492	16	(	(	PUNCT
ma-25	492	17	as	as	ADP
ma-25	492	18	)	)	PUNCT
ma-25	492	19	.	.	PUNCT
ma-25	493	1	4	4	X
ma-25	493	2	.	.	X
ma-25	493	3	proof	proof	NOUN
ma-25	493	4	of	of	ADP
ma-25	493	5	corollary	corollary	ADJ
ma-25	493	6	1.1	1.1	NUM
ma-25	493	7	assume	assume	VERB
ma-25	493	8	that	that	SCONJ
ma-25	493	9	ϕ	ϕ	NOUN
ma-25	493	10	is	be	AUX
ma-25	493	11	a	a	DET
ma-25	493	12	transcendental	transcendental	ADJ
ma-25	493	13	meromorphic	meromorphic	ADJ
ma-25	493	14	function	function	NOUN
ma-25	493	15	such	such	ADJ
ma-25	493	16	that	that	DET
ma-25	493	17	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	493	18	]	]	PUNCT
ma-25	493	19	(	(	PUNCT
ma-25	493	20	ϕ	ϕ	NOUN
ma-25	493	21	)	)	PUNCT
ma-25	493	22	<	<	X
ma-25	493	23	σ	σ	PROPN
ma-25	493	24	.	.	PUNCT
ma-25	493	25	noting	note	VERB
ma-25	493	26	g	g	PROPN
ma-25	493	27	=	=	SYM
ma-25	493	28	f	f	PROPN
ma-25	494	1	−	−	PROPN
ma-25	494	2	ϕ	ϕ	PROPN
ma-25	494	3	,	,	PUNCT
ma-25	494	4	then	then	ADV
ma-25	494	5	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	494	6	]	]	PUNCT
ma-25	494	7	(	(	PUNCT
ma-25	494	8	g	g	NOUN
ma-25	494	9	)	)	PUNCT
ma-25	494	10	=	=	SYM
ma-25	494	11	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	494	12	]	]	PUNCT
ma-25	494	13	(	(	PUNCT
ma-25	494	14	f	f	PROPN
ma-25	494	15	)	)	PUNCT
ma-25	494	16	,	,	PUNCT
ma-25	494	17	so	so	ADV
ma-25	494	18	by	by	ADP
ma-25	494	19	theorem	theorem	NOUN
ma-25	494	20	1.1	1.1	NUM
ma-25	494	21	,	,	PUNCT
ma-25	494	22	σ	σ	NOUN
ma-25	494	23	≤	≤	NUM
ma-25	494	24	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	494	25	]	]	PUNCT
ma-25	494	26	(	(	PUNCT
ma-25	494	27	g	g	NOUN
ma-25	494	28	)	)	PUNCT
ma-25	494	29	≤	≤	NOUN
ma-25	494	30	ρ[p	ρ[p	NOUN
ma-25	494	31	,	,	PUNCT
ma-25	494	32	q	q	X
ma-25	494	33	]	]	X
ma-25	494	34	(	(	PUNCT
ma-25	494	35	as	as	ADP
ma-25	494	36	)	)	PUNCT
ma-25	494	37	.	.	PUNCT
ma-25	495	1	bysubstituting	bysubstitute	VERB
ma-25	495	2	f	f	PROPN
ma-25	495	3	=	=	SYM
ma-25	495	4	g	g	PROPN
ma-25	495	5	+	+	CCONJ
ma-25	495	6	ϕ	ϕ	NOUN
ma-25	495	7	into	into	ADP
ma-25	495	8	(	(	PUNCT
ma-25	495	9	1.3	1.3	NUM
ma-25	495	10	)	)	PUNCT
ma-25	495	11	,	,	PUNCT
ma-25	495	12	we	we	PRON
ma-25	495	13	obtain	obtain	VERB
ma-25	495	14	ak	ak	PROPN
ma-25	495	15	(	(	PUNCT
ma-25	495	16	z	z	NOUN
ma-25	495	17	)	)	PUNCT
ma-25	495	18	g(k	g(k	NOUN
ma-25	495	19	)	)	PUNCT
ma-25	495	20	+	+	CCONJ
ma-25	495	21	ak−1	ak−1	ADV
ma-25	495	22	(	(	PUNCT
ma-25	495	23	z	z	NOUN
ma-25	495	24	)	)	PUNCT
ma-25	495	25	g(k−1	g(k−1	PUNCT
ma-25	495	26	)	)	PUNCT
ma-25	495	27	+	+	CCONJ
ma-25	495	28	·	·	PUNCT
ma-25	495	29	·	·	PUNCT
ma-25	495	30	·	·	PUNCT
ma-25	496	1	+	+	NUM
ma-25	496	2	a1	a1	NOUN
ma-25	496	3	(	(	PUNCT
ma-25	496	4	z	z	NOUN
ma-25	496	5	)	)	PUNCT
ma-25	496	6	g′	g′	NOUN
ma-25	496	7	+	+	CCONJ
ma-25	496	8	a0	a0	PROPN
ma-25	496	9	(	(	PUNCT
ma-25	496	10	z	z	NOUN
ma-25	496	11	)	)	PUNCT
ma-25	496	12	g	g	NOUN
ma-25	496	13	=	=	SYM
ma-25	496	14	−	−	PROPN
ma-25	496	15	(	(	PUNCT
ma-25	496	16	ak	ak	PROPN
ma-25	496	17	(	(	PUNCT
ma-25	496	18	z)ϕ(k	z)ϕ(k	NOUN
ma-25	496	19	)	)	PUNCT
ma-25	497	1	+	+	CCONJ
ma-25	497	2	ak−1	ak−1	ADV
ma-25	497	3	(	(	PUNCT
ma-25	497	4	z)ϕ(k−1	z)ϕ(k−1	PROPN
ma-25	497	5	)	)	PUNCT
ma-25	497	6	+	+	CCONJ
ma-25	497	7	·	·	PUNCT
ma-25	497	8	·	·	PUNCT
ma-25	497	9	·	·	PUNCT
ma-25	497	10	+	+	NUM
ma-25	497	11	a1	a1	NOUN
ma-25	497	12	(	(	PUNCT
ma-25	497	13	z)ϕ′	z)ϕ′	PROPN
ma-25	497	14	+	+	NUM
ma-25	497	15	a0	a0	PROPN
ma-25	497	16	(	(	PUNCT
ma-25	497	17	z)ϕ	z)ϕ	NOUN
ma-25	497	18	)	)	PUNCT
ma-25	498	1	=	=	SYM
ma-25	498	2	g	g	PROPN
ma-25	498	3	(	(	PUNCT
ma-25	498	4	z	z	NOUN
ma-25	498	5	)	)	PUNCT
ma-25	498	6	.	.	PUNCT
ma-25	499	1	(	(	PUNCT
ma-25	499	2	4.1)it	4.1)it	NOUN
ma-25	499	3	is	be	AUX
ma-25	499	4	clear	clear	ADJ
ma-25	499	5	that	that	SCONJ
ma-25	499	6	the	the	DET
ma-25	499	7	right	right	ADJ
ma-25	499	8	side	side	NOUN
ma-25	499	9	g	g	NOUN
ma-25	499	10	of	of	ADP
ma-25	499	11	equation	equation	NOUN
ma-25	499	12	(	(	PUNCT
ma-25	499	13	4.1	4.1	NUM
ma-25	499	14	)	)	PUNCT
ma-25	499	15	is	be	AUX
ma-25	499	16	non	non	ADJ
ma-25	499	17	-	-	ADJ
ma-25	499	18	zero	zero	NUM
ma-25	499	19	,	,	PUNCT
ma-25	499	20	because	because	SCONJ
ma-25	499	21	by	by	ADP
ma-25	499	22	theorem	theorem	NOUN
ma-25	499	23	1.1	1.1	NUM
ma-25	499	24	,	,	PUNCT
ma-25	499	25	ϕ	ϕ	NOUN
ma-25	499	26	is	be	AUX
ma-25	499	27	not	not	PART
ma-25	499	28	asolution	asolution	NOUN
ma-25	499	29	of	of	ADP
ma-25	499	30	equation	equation	NOUN
ma-25	499	31	(	(	PUNCT
ma-25	499	32	1.3	1.3	NUM
ma-25	499	33	)	)	PUNCT
ma-25	499	34	.	.	PUNCT
ma-25	500	1	moreover	moreover	ADV
ma-25	500	2	,	,	PUNCT
ma-25	500	3	the	the	PRON
ma-25	500	4	[	[	X
ma-25	500	5	p	p	X
ma-25	500	6	+	+	NOUN
ma-25	500	7	1	1	NUM
ma-25	500	8	,	,	PUNCT
ma-25	500	9	q]-order	q]-order	NOUN
ma-25	500	10	of	of	ADP
ma-25	500	11	g	g	PROPN
ma-25	500	12	satisfies	satisfy	VERB
ma-25	500	13	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	500	14	]	]	PUNCT
ma-25	500	15	(	(	PUNCT
ma-25	500	16	g	g	NOUN
ma-25	500	17	)	)	PUNCT
ma-25	500	18	≤	≤	NUM
ma-25	501	1	max	max	PROPN
ma-25	501	2	{	{	PUNCT
ma-25	501	3	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	501	4	]	]	PUNCT
ma-25	501	5	(	(	PUNCT
ma-25	501	6	ϕ	ϕ	NOUN
ma-25	501	7	)	)	PUNCT
ma-25	501	8	,	,	PUNCT
ma-25	501	9	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	501	10	]	]	PUNCT
ma-25	501	11	(	(	PUNCT
ma-25	501	12	aj	aj	PROPN
ma-25	501	13	)	)	PUNCT
ma-25	501	14	(	(	PUNCT
ma-25	501	15	j	j	NOUN
ma-25	501	16	=	=	SYM
ma-25	501	17	0	0	NUM
ma-25	501	18	,	,	PUNCT
ma-25	501	19	1	1	NUM
ma-25	501	20	,	,	PUNCT
ma-25	501	21	...	...	PUNCT
ma-25	501	22	,	,	PUNCT
ma-25	501	23	k	k	NOUN
ma-25	501	24	)	)	PUNCT
ma-25	501	25	}	}	PUNCT
ma-25	501	26	<	<	X
ma-25	501	27	σ	σ	PROPN
ma-25	501	28	,	,	PUNCT
ma-25	501	29	which	which	PRON
ma-25	501	30	implies	imply	VERB
ma-25	501	31	max	max	PROPN
ma-25	501	32	{	{	PUNCT
ma-25	501	33	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	501	34	]	]	PUNCT
ma-25	501	35	(	(	PUNCT
ma-25	501	36	g	g	NOUN
ma-25	501	37	)	)	PUNCT
ma-25	501	38	,	,	PUNCT
ma-25	501	39	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	501	40	]	]	PUNCT
ma-25	501	41	(	(	PUNCT
ma-25	501	42	aj	aj	PROPN
ma-25	501	43	)	)	PUNCT
ma-25	501	44	(	(	PUNCT
ma-25	501	45	j	j	NOUN
ma-25	501	46	=	=	SYM
ma-25	501	47	0	0	NUM
ma-25	501	48	,	,	PUNCT
ma-25	501	49	1	1	NUM
ma-25	501	50	,	,	PUNCT
ma-25	501	51	...	...	PUNCT
ma-25	501	52	,	,	PUNCT
ma-25	501	53	k	k	NOUN
ma-25	501	54	)	)	PUNCT
ma-25	501	55	}	}	PUNCT
ma-25	501	56	<	<	X
ma-25	501	57	σ	σ	PROPN
ma-25	501	58	≤	≤	NUM
ma-25	501	59	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	501	60	]	]	PUNCT
ma-25	501	61	(	(	PUNCT
ma-25	501	62	g	g	NOUN
ma-25	501	63	)	)	PUNCT
ma-25	501	64	.	.	PUNCT
ma-25	502	1	then	then	ADV
ma-25	502	2	by	by	ADP
ma-25	502	3	lemma	lemma	PROPN
ma-25	502	4	2.12	2.12	NUM
ma-25	502	5	,	,	PUNCT
ma-25	502	6	we	we	PRON
ma-25	502	7	obtain	obtain	VERB
ma-25	502	8	σ	σ	NOUN
ma-25	502	9	≤	≤	NUM
ma-25	502	10	λ[p+1,q	λ[p+1,q	NOUN
ma-25	502	11	]	]	X
ma-25	502	12	(	(	PUNCT
ma-25	502	13	g	g	NOUN
ma-25	502	14	)	)	PUNCT
ma-25	502	15	=	=	SYM
ma-25	502	16	λ[p+1,q	λ[p+1,q	PROPN
ma-25	502	17	]	]	X
ma-25	502	18	(	(	PUNCT
ma-25	502	19	g	g	NOUN
ma-25	502	20	)	)	PUNCT
ma-25	502	21	=	=	SYM
ma-25	502	22	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	502	23	]	]	PUNCT
ma-25	502	24	(	(	PUNCT
ma-25	502	25	g	g	NOUN
ma-25	502	26	)	)	PUNCT
ma-25	502	27	=	=	SYM
ma-25	502	28	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	502	29	]	]	PUNCT
ma-25	502	30	(	(	PUNCT
ma-25	502	31	f	f	PROPN
ma-25	502	32	)	)	PUNCT
ma-25	502	33	≤	≤	PROPN
ma-25	502	34	ρ[p	ρ[p	NOUN
ma-25	502	35	,	,	PUNCT
ma-25	502	36	q	q	X
ma-25	502	37	]	]	X
ma-25	502	38	(	(	PUNCT
ma-25	502	39	as	as	ADP
ma-25	502	40	)	)	PUNCT
ma-25	502	41	,	,	PUNCT
ma-25	502	42	that	that	ADV
ma-25	502	43	is	is	ADV
ma-25	502	44	σ	σ	PROPN
ma-25	502	45	≤	≤	NUM
ma-25	502	46	λ[p+1,q	λ[p+1,q	NOUN
ma-25	502	47	]	]	X
ma-25	502	48	(	(	PUNCT
ma-25	502	49	f	f	PROPN
ma-25	502	50	−	−	PROPN
ma-25	502	51	ϕ	ϕ	PROPN
ma-25	502	52	)	)	PUNCT
ma-25	502	53	=	=	SYM
ma-25	502	54	λ[p+1,q	λ[p+1,q	PROPN
ma-25	502	55	]	]	X
ma-25	502	56	(	(	PUNCT
ma-25	502	57	f	f	PROPN
ma-25	502	58	−	−	PROPN
ma-25	502	59	ϕ	ϕ	PROPN
ma-25	502	60	)	)	PUNCT
ma-25	502	61	=	=	SYM
ma-25	502	62	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	502	63	]	]	PUNCT
ma-25	502	64	(	(	PUNCT
ma-25	502	65	f	f	X
ma-25	502	66	−	−	PROPN
ma-25	502	67	ϕ	ϕ	PROPN
ma-25	502	68	)	)	PUNCT
ma-25	502	69	=	=	SYM
ma-25	502	70	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	502	71	]	]	PUNCT
ma-25	502	72	(	(	PUNCT
ma-25	502	73	f	f	PROPN
ma-25	502	74	)	)	PUNCT
ma-25	502	75	≤	≤	PROPN
ma-25	502	76	ρ[p	ρ[p	NOUN
ma-25	502	77	,	,	PUNCT
ma-25	502	78	q	q	X
ma-25	502	79	]	]	X
ma-25	502	80	(	(	PUNCT
ma-25	502	81	as	as	ADP
ma-25	502	82	)	)	PUNCT
ma-25	502	83	.	.	PUNCT
ma-25	503	1	5	5	X
ma-25	503	2	.	.	X
ma-25	503	3	proof	proof	NOUN
ma-25	503	4	of	of	ADP
ma-25	503	5	theorem	theorem	ADJ
ma-25	503	6	1.2	1.2	NUM
ma-25	503	7	assume	assume	VERB
ma-25	503	8	that	that	SCONJ
ma-25	503	9	f	f	PROPN
ma-25	503	10	is	be	AUX
ma-25	503	11	a	a	DET
ma-25	503	12	rational	rational	ADJ
ma-25	503	13	solution	solution	NOUN
ma-25	503	14	of	of	ADP
ma-25	503	15	(	(	PUNCT
ma-25	503	16	1.4	1.4	NUM
ma-25	503	17	)	)	PUNCT
ma-25	503	18	.	.	PUNCT
ma-25	504	1	first	first	ADV
ma-25	504	2	,	,	PUNCT
ma-25	504	3	we	we	PRON
ma-25	504	4	will	will	AUX
ma-25	504	5	prove	prove	VERB
ma-25	504	6	that	that	SCONJ
ma-25	504	7	f	f	PROPN
ma-25	504	8	must	must	AUX
ma-25	504	9	be	be	AUX
ma-25	504	10	a	a	DET
ma-25	504	11	polynomial	polynomial	NOUN
ma-25	504	12	with	with	ADP
ma-25	504	13	deg	deg	PROPN
ma-25	504	14	f	f	PROPN
ma-25	504	15	≤	≤	PROPN
ma-25	504	16	s	s	PART
ma-25	504	17	−	−	PROPN
ma-25	504	18	1	1	NUM
ma-25	504	19	.	.	PUNCT
ma-25	505	1	for	for	ADP
ma-25	505	2	this	this	PRON
ma-25	505	3	,	,	PUNCT
ma-25	505	4	if	if	SCONJ
ma-25	505	5	f	f	PROPN
ma-25	505	6	is	be	AUX
ma-25	505	7	a	a	DET
ma-25	505	8	rational	rational	ADJ
ma-25	505	9	function	function	NOUN
ma-25	505	10	,	,	PUNCT
ma-25	505	11	which	which	PRON
ma-25	505	12	has	have	VERB
ma-25	505	13	a	a	DET
ma-25	505	14	pole	pole	NOUN
ma-25	505	15	at	at	ADP
ma-25	505	16	z0	z0	PROPN
ma-25	505	17	of	of	ADP
ma-25	505	18	degree	degree	NOUN
ma-25	505	19	m	m	PROPN
ma-25	505	20	≥	≥	NOUN
ma-25	505	21	1	1	NUM
ma-25	505	22	,	,	PUNCT
ma-25	505	23	or	or	CCONJ
ma-25	505	24	f	f	PROPN
ma-25	505	25	isa	isa	NOUN
ma-25	505	26	polynomial	polynomial	ADJ
ma-25	505	27	with	with	ADP
ma-25	505	28	deg	deg	PROPN
ma-25	505	29	f	f	PROPN
ma-25	505	30	≥	≥	NUM
ma-25	505	31	s	s	PROPN
ma-25	505	32	,	,	PUNCT
ma-25	505	33	then	then	ADV
ma-25	505	34	f	f	X
ma-25	505	35	(	(	PUNCT
ma-25	505	36	s)(z	s)(z	PROPN
ma-25	505	37	)	)	PUNCT
ma-25	505	38	6≡	6≡	NUM
ma-25	505	39	0	0	NUM
ma-25	505	40	.	.	PUNCT
ma-25	505	41	by	by	ADP
ma-25	505	42	(	(	PUNCT
ma-25	505	43	1.4	1.4	NUM
ma-25	505	44	)	)	PUNCT
ma-25	505	45	and	and	CCONJ
ma-25	505	46	lemma	lemma	PROPN
ma-25	505	47	2.4	2.4	NUM
ma-25	505	48	,	,	PUNCT
ma-25	505	49	we	we	PRON
ma-25	505	50	obtain	obtain	VERB
ma-25	505	51	σ	σ	NOUN
ma-25	505	52	≤	≤	NUM
ma-25	505	53	ρ[p	ρ[p	NOUN
ma-25	505	54	,	,	PUNCT
ma-25	505	55	q](as	q](as	NOUN
ma-25	505	56	)	)	PUNCT
ma-25	505	57	=	=	PUNCT
ma-25	505	58	ρ[p	ρ[p	PROPN
ma-25	505	59	,	,	PUNCT
ma-25	505	60	q](as	q](as	PROPN
ma-25	505	61	f	f	X
ma-25	505	62	(	(	PUNCT
ma-25	505	63	s	s	NOUN
ma-25	505	64	)	)	PUNCT
ma-25	505	65	)	)	PUNCT
ma-25	505	66	=	=	PUNCT
ma-25	506	1	ρ[p	ρ[p	PROPN
ma-25	506	2	,	,	PUNCT
ma-25	506	3	q	q	X
ma-25	506	4	]	]	X
ma-25	506	5	f	f	PROPN
ma-25	506	6	−	−	PROPN
ma-25	506	7	k∑	k∑	PROPN
ma-25	506	8	j=0	j=0	PROPN
ma-25	506	9	j	j	PROPN
ma-25	506	10	6	6	NUM
ma-25	506	11	=	=	SYM
ma-25	506	12	s	s	X
ma-25	506	13	aj	aj	PROPN
ma-25	506	14	(	(	PUNCT
ma-25	506	15	z	z	PROPN
ma-25	506	16	)	)	PUNCT
ma-25	506	17	f	f	PROPN
ma-25	506	18	(	(	PUNCT
ma-25	506	19	j	j	NOUN
ma-25	506	20	)	)	PUNCT
ma-25	506	21			NOUN
ma-25	506	22	≤	≤	NUM
ma-25	506	23	max	max	PROPN
ma-25	507	1	j=0,1,	j=0,1,	VERB
ma-25	507	2	...	...	PROPN
ma-25	507	3	,k	,k	PUNCT
ma-25	507	4	,	,	PUNCT
ma-25	507	5	j	j	PROPN
ma-25	507	6	6	6	NUM
ma-25	507	7	=	=	SYM
ma-25	507	8	s	s	X
ma-25	507	9	{	{	PUNCT
ma-25	507	10	ρ[p	ρ[p	PROPN
ma-25	507	11	,	,	PUNCT
ma-25	507	12	q	q	X
ma-25	507	13	]	]	X
ma-25	507	14	(	(	PUNCT
ma-25	507	15	aj	aj	PROPN
ma-25	507	16	)	)	PUNCT
ma-25	507	17	,	,	PUNCT
ma-25	507	18	ρ[p	ρ[p	PROPN
ma-25	507	19	,	,	PUNCT
ma-25	507	20	q	q	X
ma-25	507	21	]	]	X
ma-25	507	22	(	(	PUNCT
ma-25	507	23	f	f	PROPN
ma-25	507	24	)	)	PUNCT
ma-25	507	25	}	}	PUNCT
ma-25	507	26	,	,	PUNCT
ma-25	507	27	eur	eur	PROPN
ma-25	507	28	.	.	PUNCT
ma-25	508	1	j.	j.	PROPN
ma-25	508	2	math	math	PROPN
ma-25	508	3	.	.	PUNCT
ma-25	509	1	anal	anal	ADJ
ma-25	509	2	.	.	PUNCT
ma-25	510	1	1	1	NUM
ma-25	510	2	(	(	PUNCT
ma-25	510	3	2021	2021	NUM
ma-25	510	4	)	)	PUNCT
ma-25	511	1	102which	102which	PROPN
ma-25	511	2	is	be	AUX
ma-25	511	3	a	a	DET
ma-25	511	4	contradiction	contradiction	NOUN
ma-25	511	5	.	.	PUNCT
ma-25	512	1	therefore	therefore	ADV
ma-25	512	2	,	,	PUNCT
ma-25	512	3	f	f	PROPN
ma-25	512	4	must	must	AUX
ma-25	512	5	be	be	AUX
ma-25	512	6	a	a	DET
ma-25	512	7	polynomial	polynomial	NOUN
ma-25	512	8	with	with	ADP
ma-25	512	9	deg	deg	PROPN
ma-25	512	10	f	f	PROPN
ma-25	512	11	≤	≤	PROPN
ma-25	512	12	s	s	PART
ma-25	512	13	−	−	PROPN
ma-25	512	14	1	1	NUM
ma-25	512	15	.	.	PUNCT
ma-25	513	1	now	now	ADV
ma-25	513	2	,	,	PUNCT
ma-25	513	3	we	we	PRON
ma-25	513	4	assume	assume	VERB
ma-25	513	5	that	that	SCONJ
ma-25	513	6	f	f	PROPN
ma-25	513	7	is	be	AUX
ma-25	513	8	a	a	DET
ma-25	513	9	transcendental	transcendental	ADJ
ma-25	513	10	meromorphic	meromorphic	ADJ
ma-25	513	11	solution	solution	NOUN
ma-25	513	12	of	of	ADP
ma-25	513	13	(	(	PUNCT
ma-25	513	14	1.4	1.4	NUM
ma-25	513	15	)	)	PUNCT
ma-25	514	1	such	such	ADJ
ma-25	514	2	that	that	SCONJ
ma-25	514	3	λ[p	λ[p	ADJ
ma-25	514	4	,	,	PUNCT
ma-25	514	5	q	q	X
ma-25	514	6	]	]	X
ma-25	514	7	(	(	PUNCT
ma-25	514	8	1f	1f	NUM
ma-25	514	9	)	)	PUNCT
ma-25	514	10	<	<	X
ma-25	514	11	µ[p	µ[p	ADJ
ma-25	514	12	,	,	PUNCT
ma-25	514	13	q](f	q](f	NOUN
ma-25	514	14	)	)	PUNCT
ma-25	514	15	.	.	PUNCT
ma-25	515	1	from	from	ADP
ma-25	515	2	lemma	lemma	PROPN
ma-25	515	3	2.13	2.13	NUM
ma-25	515	4	,	,	PUNCT
ma-25	515	5	we	we	PRON
ma-25	515	6	know	know	VERB
ma-25	515	7	that	that	SCONJ
ma-25	515	8	f	f	PROPN
ma-25	515	9	satisfies	satisfy	VERB
ma-25	515	10	ρ[p	ρ[p	NOUN
ma-25	515	11	,	,	PUNCT
ma-25	515	12	q	q	X
ma-25	515	13	]	]	X
ma-25	515	14	(	(	PUNCT
ma-25	515	15	f	f	PROPN
ma-25	515	16	)	)	PUNCT
ma-25	515	17	≥	≥	PROPN
ma-25	515	18	σ	σ	PROPN
ma-25	515	19	.	.	PUNCT
ma-25	516	1	by	by	ADP
ma-25	516	2	the	the	DET
ma-25	516	3	hypothesis	hypothesis	NOUN
ma-25	516	4	λ[p	λ[p	NOUN
ma-25	516	5	,	,	PUNCT
ma-25	516	6	q	q	X
ma-25	516	7	]	]	X
ma-25	516	8	(	(	PUNCT
ma-25	516	9	1f	1f	NUM
ma-25	516	10	)	)	PUNCT
ma-25	516	11	<	<	X
ma-25	516	12	min{µ[p	min{µ[p	NOUN
ma-25	516	13	,	,	PUNCT
ma-25	516	14	q](f	q](f	NOUN
ma-25	516	15	)	)	PUNCT
ma-25	516	16	,	,	PUNCT
ma-25	516	17	σ	σ	PROPN
ma-25	516	18	}	}	PUNCT
ma-25	516	19	and	and	CCONJ
ma-25	516	20	hadamard	hadamard	ADJ
ma-25	516	21	factorization	factorization	NOUN
ma-25	516	22	theorem	theorem	NOUN
ma-25	516	23	,	,	PUNCT
ma-25	516	24	we	we	PRON
ma-25	516	25	can	can	AUX
ma-25	516	26	write	write	VERB
ma-25	516	27	f	f	PROPN
ma-25	516	28	as	as	ADP
ma-25	516	29	f	f	PROPN
ma-25	516	30	(	(	PUNCT
ma-25	516	31	z	z	NOUN
ma-25	516	32	)	)	PUNCT
ma-25	516	33	=	=	SYM
ma-25	516	34	g(z	g(z	ADJ
ma-25	516	35	)	)	PUNCT
ma-25	516	36	d(z	d(z	PROPN
ma-25	516	37	)	)	PUNCT
ma-25	516	38	,	,	PUNCT
ma-25	516	39	where	where	SCONJ
ma-25	516	40	g	g	PROPN
ma-25	516	41	(	(	PUNCT
ma-25	516	42	z)and	z)and	PROPN
ma-25	516	43	d	d	PROPN
ma-25	516	44	(	(	PUNCT
ma-25	516	45	z	z	NOUN
ma-25	516	46	)	)	PUNCT
ma-25	516	47	are	be	AUX
ma-25	516	48	entire	entire	ADJ
ma-25	516	49	functions	function	NOUN
ma-25	516	50	satisfying	satisfy	VERB
ma-25	516	51	µ[p	µ[p	ADJ
ma-25	516	52	,	,	PUNCT
ma-25	516	53	q](g	q](g	NOUN
ma-25	516	54	)	)	PUNCT
ma-25	516	55	=	=	PUNCT
ma-25	517	1	µ[p	µ[p	ADJ
ma-25	517	2	,	,	PUNCT
ma-25	517	3	q](f	q](f	NOUN
ma-25	517	4	)	)	PUNCT
ma-25	518	1	=	=	SYM
ma-25	518	2	µ	µ	PRON
ma-25	518	3	≤	≤	NUM
ma-25	518	4	ρ[p	ρ[p	NUM
ma-25	518	5	,	,	PUNCT
ma-25	518	6	q](g	q](g	NOUN
ma-25	518	7	)	)	PUNCT
ma-25	518	8	=	=	PUNCT
ma-25	519	1	ρ[p	ρ[p	NOUN
ma-25	519	2	,	,	PUNCT
ma-25	519	3	q](f	q](f	NOUN
ma-25	519	4	)	)	PUNCT
ma-25	519	5	,	,	PUNCT
ma-25	519	6	ρ[p	ρ[p	NOUN
ma-25	519	7	,	,	PUNCT
ma-25	519	8	q](d	q](d	NOUN
ma-25	519	9	)	)	PUNCT
ma-25	520	1	=	=	SYM
ma-25	520	2	λ[p	λ[p	NUM
ma-25	520	3	,	,	PUNCT
ma-25	520	4	q	q	X
ma-25	520	5	]	]	X
ma-25	520	6	(	(	PUNCT
ma-25	520	7	1	1	NUM
ma-25	520	8	f	f	NOUN
ma-25	520	9	)	)	PUNCT
ma-25	520	10	=	=	PUNCT
ma-25	521	1	β	β	X
ma-25	521	2	<	<	X
ma-25	521	3	min{µ[p	min{µ[p	NOUN
ma-25	521	4	,	,	PUNCT
ma-25	521	5	q](f	q](f	NOUN
ma-25	521	6	)	)	PUNCT
ma-25	521	7	,	,	PUNCT
ma-25	521	8	σ	σ	PROPN
ma-25	521	9	}	}	PUNCT
ma-25	521	10	.	.	PUNCT
ma-25	522	1	the	the	DET
ma-25	522	2	definition	definition	NOUN
ma-25	522	3	of	of	ADP
ma-25	522	4	the	the	DET
ma-25	522	5	lower	low	ADJ
ma-25	522	6	[	[	X
ma-25	522	7	p	p	X
ma-25	522	8	,	,	PUNCT
ma-25	522	9	q]-order	q]-order	NOUN
ma-25	522	10	assures	assure	VERB
ma-25	522	11	us	we	PRON
ma-25	522	12	that	that	SCONJ
ma-25	522	13	|g	|g	VERB
ma-25	522	14	(	(	PUNCT
ma-25	522	15	z)|	z)|	NOUN
ma-25	522	16	=	=	SYM
ma-25	522	17	m(r	m(r	PROPN
ma-25	522	18	,	,	PUNCT
ma-25	522	19	g	g	NOUN
ma-25	522	20	)	)	PUNCT
ma-25	522	21	≥	≥	NOUN
ma-25	522	22	expp+1	expp+1	NOUN
ma-25	522	23	{	{	PUNCT
ma-25	522	24	(	(	PUNCT
ma-25	522	25	µ[p	µ[p	ADJ
ma-25	522	26	,	,	PUNCT
ma-25	522	27	q	q	X
ma-25	522	28	]	]	X
ma-25	522	29	(	(	PUNCT
ma-25	522	30	g)−	g)−	PROPN
ma-25	522	31	ε	ε	PROPN
ma-25	522	32	)	)	PUNCT
ma-25	522	33	logq	logq	VERB
ma-25	522	34	r	r	NOUN
ma-25	522	35	}	}	PUNCT
ma-25	522	36	.	.	PUNCT
ma-25	523	1	(	(	PUNCT
ma-25	523	2	5.1	5.1	NUM
ma-25	523	3	)	)	PUNCT
ma-25	523	4	putting	put	VERB
ma-25	523	5	ρ1	ρ1	NOUN
ma-25	523	6	=	=	SYM
ma-25	523	7	max	max	X
ma-25	523	8	{	{	PUNCT
ma-25	523	9	ρ[p	ρ[p	PROPN
ma-25	523	10	,	,	PUNCT
ma-25	523	11	q	q	X
ma-25	523	12	]	]	X
ma-25	523	13	(	(	PUNCT
ma-25	523	14	aj	aj	PROPN
ma-25	523	15	)	)	PUNCT
ma-25	523	16	(	(	PUNCT
ma-25	523	17	j	j	PROPN
ma-25	523	18	6=	6=	NUM
ma-25	523	19	s	s	PART
ma-25	523	20	)	)	PUNCT
ma-25	523	21	,	,	PUNCT
ma-25	523	22	ρ[p	ρ[p	PROPN
ma-25	523	23	,	,	PUNCT
ma-25	523	24	q	q	X
ma-25	523	25	]	]	X
ma-25	523	26	(	(	PUNCT
ma-25	523	27	f	f	PROPN
ma-25	523	28	)	)	PUNCT
ma-25	523	29	}	}	PUNCT
ma-25	523	30	<	<	X
ma-25	523	31	σ.then	σ.then	ADV
ma-25	523	32	,	,	PUNCT
ma-25	523	33	by	by	ADP
ma-25	523	34	lemma	lemma	PROPN
ma-25	523	35	2.3	2.3	NUM
ma-25	523	36	and	and	CCONJ
ma-25	523	37	(	(	PUNCT
ma-25	523	38	5.1	5.1	NUM
ma-25	523	39	)	)	PUNCT
ma-25	523	40	,	,	PUNCT
ma-25	523	41	for	for	ADP
ma-25	523	42	any	any	DET
ma-25	523	43	given	give	VERB
ma-25	523	44	ε	ε	PROPN
ma-25	523	45	satisfying	satisfy	VERB
ma-25	523	46	0	0	PUNCT
ma-25	523	47	<	<	X
ma-25	523	48	2ε	2ε	PROPN
ma-25	523	49	<	<	X
ma-25	523	50	min{σ	min{σ	NOUN
ma-25	523	51	−	−	PROPN
ma-25	523	52	ρ1	ρ1	NOUN
ma-25	523	53	,	,	PUNCT
ma-25	523	54	µ[p	µ[p	ADJ
ma-25	523	55	,	,	PUNCT
ma-25	523	56	q	q	X
ma-25	523	57	]	]	X
ma-25	523	58	(	(	PUNCT
ma-25	523	59	g)−	g)−	PROPN
ma-25	523	60	ρ[p	ρ[p	PROPN
ma-25	523	61	,	,	PUNCT
ma-25	523	62	q	q	X
ma-25	523	63	]	]	X
ma-25	523	64	(	(	PUNCT
ma-25	523	65	d	d	NOUN
ma-25	523	66	)	)	PUNCT
ma-25	523	67	}	}	PUNCT
ma-25	523	68	,	,	PUNCT
ma-25	523	69	there	there	PRON
ma-25	523	70	exists	exist	VERB
ma-25	523	71	a	a	DET
ma-25	523	72	set	set	NOUN
ma-25	523	73	e3	e3	NOUN
ma-25	523	74	⊂	⊂	X
ma-25	523	75	(	(	PUNCT
ma-25	523	76	1,+∞	1,+∞	NUM
ma-25	523	77	)	)	PUNCT
ma-25	523	78	with	with	ADP
ma-25	523	79	a	a	DET
ma-25	523	80	finite	finite	ADJ
ma-25	523	81	logarithmic	logarithmic	ADJ
ma-25	523	82	measure	measure	NOUN
ma-25	523	83	such	such	ADJ
ma-25	523	84	that	that	PRON
ma-25	523	85	for	for	ADP
ma-25	523	86	all	all	DET
ma-25	523	87	z	z	NOUN
ma-25	523	88	satisfying	satisfy	VERB
ma-25	523	89	|z	|z	PROPN
ma-25	524	1	|	|	ADV
ma-25	524	2	=	=	NOUN
ma-25	524	3	r	r	NOUN
ma-25	524	4	/∈	/∈	PUNCT
ma-25	524	5	e3	e3	NOUN
ma-25	524	6	at	at	ADP
ma-25	524	7	which	which	PRON
ma-25	524	8	|g	|g	VERB
ma-25	524	9	(	(	PUNCT
ma-25	524	10	z	z	NOUN
ma-25	524	11	)	)	PUNCT
ma-25	524	12	|	|	ADV
ma-25	524	13	=	=	SYM
ma-25	524	14	m(r	m(r	PROPN
ma-25	524	15	,	,	PUNCT
ma-25	524	16	g	g	NOUN
ma-25	524	17	)	)	PUNCT
ma-25	524	18	,	,	PUNCT
ma-25	524	19	we	we	PRON
ma-25	524	20	obtain∣∣∣∣f	obtain∣∣∣∣f	VERB
ma-25	524	21	(	(	PUNCT
ma-25	524	22	z	z	NOUN
ma-25	524	23	)	)	PUNCT
ma-25	524	24	f	f	NOUN
ma-25	524	25	(	(	PUNCT
ma-25	524	26	z	z	NOUN
ma-25	524	27	)	)	PUNCT
ma-25	524	28	∣∣∣∣	∣∣∣∣	PROPN
ma-25	524	29	=	=	SYM
ma-25	524	30	|f	|f	PROPN
ma-25	524	31	(	(	PUNCT
ma-25	524	32	z)|	z)|	INTJ
ma-25	524	33	|g(z)|	|g(z)|	PROPN
ma-25	524	34	|d	|d	NOUN
ma-25	524	35	(	(	PUNCT
ma-25	524	36	z)|	z)|	ADP
ma-25	524	37	≤	≤	NUM
ma-25	524	38	expp+1	expp+1	NOUN
ma-25	524	39	{	{	PUNCT
ma-25	524	40	(	(	PUNCT
ma-25	524	41	ρ[p	ρ[p	NOUN
ma-25	524	42	,	,	PUNCT
ma-25	524	43	q	q	X
ma-25	524	44	]	]	X
ma-25	524	45	(	(	PUNCT
ma-25	524	46	d	d	NOUN
ma-25	524	47	)	)	PUNCT
ma-25	525	1	+	+	CCONJ
ma-25	525	2	ε	ε	PROPN
ma-25	525	3	)	)	PUNCT
ma-25	525	4	logq	logq	VERB
ma-25	525	5	r	r	NOUN
ma-25	525	6	}	}	PUNCT
ma-25	525	7	expp+1	expp+1	NOUN
ma-25	525	8	{	{	PUNCT
ma-25	525	9	(	(	PUNCT
ma-25	525	10	ρ1	ρ1	NOUN
ma-25	525	11	+	+	CCONJ
ma-25	525	12	ε	ε	PROPN
ma-25	525	13	)	)	PUNCT
ma-25	525	14	logq	logq	VERB
ma-25	525	15	r	r	NOUN
ma-25	525	16	}	}	PUNCT
ma-25	525	17	expp+1	expp+1	NOUN
ma-25	525	18	{	{	PUNCT
ma-25	525	19	(	(	PUNCT
ma-25	525	20	µ[p	µ[p	ADJ
ma-25	525	21	,	,	PUNCT
ma-25	525	22	q	q	X
ma-25	525	23	]	]	X
ma-25	525	24	(	(	PUNCT
ma-25	525	25	g)−	g)−	PROPN
ma-25	525	26	ε	ε	PROPN
ma-25	525	27	)	)	PUNCT
ma-25	525	28	logq	logq	VERB
ma-25	525	29	r	r	NOUN
ma-25	525	30	}	}	PUNCT
ma-25	525	31	≤	≤	NUM
ma-25	525	32	expp+1	expp+1	NOUN
ma-25	525	33	{	{	PUNCT
ma-25	525	34	(	(	PUNCT
ma-25	525	35	ρ1	ρ1	NOUN
ma-25	525	36	+	+	CCONJ
ma-25	525	37	ε	ε	PROPN
ma-25	525	38	)	)	PUNCT
ma-25	525	39	logq	logq	VERB
ma-25	525	40	r	r	NOUN
ma-25	525	41	}	}	PUNCT
ma-25	525	42	.	.	PUNCT
ma-25	526	1	(	(	PUNCT
ma-25	526	2	5.2)by	5.2)by	X
ma-25	526	3	using	use	VERB
ma-25	526	4	the	the	DET
ma-25	526	5	same	same	ADJ
ma-25	526	6	arguments	argument	NOUN
ma-25	526	7	as	as	ADP
ma-25	526	8	in	in	ADP
ma-25	526	9	the	the	DET
ma-25	526	10	proof	proof	NOUN
ma-25	526	11	of	of	ADP
ma-25	526	12	theorem	theorem	ADJ
ma-25	526	13	1.1	1.1	NUM
ma-25	526	14	,	,	PUNCT
ma-25	526	15	for	for	ADP
ma-25	526	16	any	any	DET
ma-25	526	17	given	give	VERB
ma-25	526	18	ε	ε	PROPN
ma-25	526	19	(	(	PUNCT
ma-25	526	20	0	0	PUNCT
ma-25	526	21	<	<	X
ma-25	526	22	2ε	2ε	PROPN
ma-25	526	23	<	<	X
ma-25	526	24	min{σ	min{σ	NOUN
ma-25	526	25	−	−	PROPN
ma-25	526	26	ρ1	ρ1	NOUN
ma-25	526	27	,	,	PUNCT
ma-25	526	28	µ[p	µ[p	ADJ
ma-25	526	29	,	,	PUNCT
ma-25	526	30	q	q	X
ma-25	526	31	]	]	X
ma-25	526	32	(	(	PUNCT
ma-25	526	33	g)−	g)−	PROPN
ma-25	526	34	ρ[p	ρ[p	PROPN
ma-25	526	35	,	,	PUNCT
ma-25	526	36	q	q	X
ma-25	526	37	]	]	X
ma-25	526	38	(	(	PUNCT
ma-25	526	39	d	d	NOUN
ma-25	526	40	)	)	PUNCT
ma-25	526	41	}	}	PUNCT
ma-25	526	42	)	)	PUNCT
ma-25	526	43	and	and	CCONJ
ma-25	526	44	all	all	DET
ma-25	526	45	z	z	NOUN
ma-25	526	46	satisfying	satisfy	VERB
ma-25	526	47	|z	|z	PROPN
ma-25	526	48	|	|	ADV
ma-25	527	1	=	=	SYM
ma-25	528	1	r	r	NOUN
ma-25	528	2	∈	∈	PROPN
ma-25	528	3	hr(e1	hr(e1	NOUN
ma-25	528	4	∪	∪	VERB
ma-25	528	5	e3	e3	PROPN
ma-25	528	6	∪	∪	X
ma-25	528	7	e6	e6	NOUN
ma-25	528	8	)	)	PUNCT
ma-25	528	9	,	,	PUNCT
ma-25	528	10	r	r	NOUN
ma-25	528	11	→	→	SYM
ma-25	528	12	+	+	NUM
ma-25	528	13	∞	∞	PROPN
ma-25	528	14	at	at	ADP
ma-25	528	15	which	which	PRON
ma-25	528	16	|g	|g	VERB
ma-25	528	17	(	(	PUNCT
ma-25	528	18	z	z	NOUN
ma-25	528	19	)	)	PUNCT
ma-25	528	20	|	|	ADV
ma-25	528	21	=	=	SYM
ma-25	528	22	m(r	m(r	PROPN
ma-25	528	23	,	,	PUNCT
ma-25	528	24	g	g	NOUN
ma-25	528	25	)	)	PUNCT
ma-25	528	26	,	,	PUNCT
ma-25	528	27	we	we	PRON
ma-25	528	28	have	have	AUX
ma-25	528	29	(	(	PUNCT
ma-25	528	30	3.2	3.2	NUM
ma-25	528	31	)	)	PUNCT
ma-25	528	32	,	,	PUNCT
ma-25	528	33	(	(	PUNCT
ma-25	528	34	3.3	3.3	NUM
ma-25	528	35	)	)	PUNCT
ma-25	528	36	,	,	PUNCT
ma-25	528	37	(	(	PUNCT
ma-25	528	38	3.4	3.4	NUM
ma-25	528	39	)	)	PUNCT
ma-25	528	40	hold	hold	VERB
ma-25	528	41	and	and	CCONJ
ma-25	528	42	|aj	|aj	NUM
ma-25	528	43	(	(	PUNCT
ma-25	528	44	z	z	NOUN
ma-25	528	45	)	)	PUNCT
ma-25	528	46	|	|	ADV
ma-25	528	47	≤	≤	NUM
ma-25	528	48	expp+1	expp+1	NOUN
ma-25	528	49	{	{	PUNCT
ma-25	528	50	(	(	PUNCT
ma-25	528	51	ρ1	ρ1	NOUN
ma-25	528	52	+	+	CCONJ
ma-25	528	53	ε	ε	PROPN
ma-25	528	54	)	)	PUNCT
ma-25	528	55	logq	logq	VERB
ma-25	528	56	r	r	NOUN
ma-25	528	57	}	}	PUNCT
ma-25	528	58	,	,	PUNCT
ma-25	528	59	j	j	PROPN
ma-25	528	60	=	=	SYM
ma-25	528	61	0	0	NUM
ma-25	528	62	,	,	PUNCT
ma-25	528	63	1	1	NUM
ma-25	528	64	,	,	PUNCT
ma-25	528	65	...	...	PUNCT
ma-25	528	66	,	,	PUNCT
ma-25	528	67	k	k	X
ma-25	528	68	,	,	PUNCT
ma-25	528	69	j	j	PROPN
ma-25	528	70	6=	6=	PROPN
ma-25	528	71	s.	s.	PROPN
ma-25	528	72	(	(	PUNCT
ma-25	528	73	5.3	5.3	NUM
ma-25	528	74	)	)	PUNCT
ma-25	528	75	by	by	ADP
ma-25	528	76	(	(	PUNCT
ma-25	528	77	1.4	1.4	NUM
ma-25	528	78	)	)	PUNCT
ma-25	528	79	,	,	PUNCT
ma-25	528	80	we	we	PRON
ma-25	528	81	have	have	VERB
ma-25	528	82	|as	|as	NUM
ma-25	528	83	|	|	ADV
ma-25	528	84	≤	≤	NUM
ma-25	528	85	∣∣∣∣	∣∣∣∣	NOUN
ma-25	528	86	ff	ff	NOUN
ma-25	528	87	(	(	PUNCT
ma-25	528	88	s	s	NOUN
ma-25	528	89	)	)	PUNCT
ma-25	528	90	∣∣∣∣	∣∣∣∣	NOUN
ma-25	528	91	|a0|+	|a0|+	VERB
ma-25	528	92	k∑	k∑	NOUN
ma-25	529	1	j=1	j=1	PROPN
ma-25	529	2	j	j	PROPN
ma-25	529	3	6	6	NUM
ma-25	529	4	=	=	SYM
ma-25	529	5	s	s	PART
ma-25	529	6	∣∣aj	∣∣aj	NOUN
ma-25	529	7	∣∣	∣∣	NUM
ma-25	529	8	∣∣∣∣∣	∣∣∣∣∣	PROPN
ma-25	529	9	f	f	X
ma-25	529	10	(	(	PUNCT
ma-25	529	11	j)f	j)f	PROPN
ma-25	529	12	∣∣∣∣∣+	∣∣∣∣∣+	PROPN
ma-25	529	13	∣∣∣∣ff	∣∣∣∣ff	VERB
ma-25	529	14	∣∣∣∣	∣∣∣∣	NOUN
ma-25	529	15			NOUN
ma-25	529	16	.	.	PUNCT
ma-25	530	1	(	(	PUNCT
ma-25	530	2	5.4	5.4	NUM
ma-25	530	3	)	)	PUNCT
ma-25	530	4	hence	hence	ADV
ma-25	530	5	,	,	PUNCT
ma-25	530	6	by	by	ADP
ma-25	530	7	substituting	substitute	VERB
ma-25	530	8	(	(	PUNCT
ma-25	530	9	3.2	3.2	NUM
ma-25	530	10	)	)	PUNCT
ma-25	530	11	,	,	PUNCT
ma-25	530	12	(	(	PUNCT
ma-25	530	13	3.3	3.3	NUM
ma-25	530	14	)	)	PUNCT
ma-25	530	15	,	,	PUNCT
ma-25	530	16	(	(	PUNCT
ma-25	530	17	3.4	3.4	NUM
ma-25	530	18	)	)	PUNCT
ma-25	530	19	,	,	PUNCT
ma-25	530	20	(	(	PUNCT
ma-25	530	21	5.2	5.2	NUM
ma-25	530	22	)	)	PUNCT
ma-25	530	23	and	and	CCONJ
ma-25	530	24	(	(	PUNCT
ma-25	530	25	5.3	5.3	NUM
ma-25	530	26	)	)	PUNCT
ma-25	530	27	into	into	ADP
ma-25	530	28	(	(	PUNCT
ma-25	530	29	5.4	5.4	NUM
ma-25	530	30	)	)	PUNCT
ma-25	530	31	,	,	PUNCT
ma-25	530	32	for	for	ADP
ma-25	530	33	all	all	DET
ma-25	530	34	z	z	NOUN
ma-25	530	35	satisfying	satisfy	VERB
ma-25	530	36	|z	|z	PROPN
ma-25	531	1	|	|	ADV
ma-25	531	2	=	=	SYM
ma-25	531	3	r	r	NOUN
ma-25	531	4	∈	∈	NOUN
ma-25	531	5	h	h	NOUN
ma-25	531	6	r	r	NOUN
ma-25	531	7	(	(	PUNCT
ma-25	531	8	e1	e1	NOUN
ma-25	531	9	∪	∪	VERB
ma-25	531	10	e3	e3	NOUN
ma-25	531	11	∪	∪	ADJ
ma-25	531	12	e6	e6	NOUN
ma-25	531	13	)	)	PUNCT
ma-25	531	14	,	,	PUNCT
ma-25	531	15	r	r	NOUN
ma-25	531	16	→	→	SYM
ma-25	531	17	+	+	NOUN
ma-25	531	18	∞	∞	PROPN
ma-25	531	19	,	,	PUNCT
ma-25	531	20	at	at	ADP
ma-25	531	21	which	which	PRON
ma-25	531	22	|g	|g	VERB
ma-25	531	23	(	(	PUNCT
ma-25	531	24	z	z	NOUN
ma-25	531	25	)	)	PUNCT
ma-25	532	1	|	|	ADV
ma-25	532	2	=	=	SYM
ma-25	532	3	m	m	PROPN
ma-25	532	4	(	(	PUNCT
ma-25	532	5	r	r	NOUN
ma-25	532	6	,	,	PUNCT
ma-25	532	7	g	g	NOUN
ma-25	532	8	)	)	PUNCT
ma-25	532	9	and	and	CCONJ
ma-25	532	10	any	any	DET
ma-25	532	11	given	give	VERB
ma-25	532	12	ε	ε	PROPN
ma-25	532	13	(	(	PUNCT
ma-25	532	14	0	0	PUNCT
ma-25	532	15	<	<	X
ma-25	532	16	2ε	2ε	PROPN
ma-25	532	17	<	<	X
ma-25	532	18	min{σ	min{σ	NOUN
ma-25	532	19	−	−	PROPN
ma-25	532	20	ρ1	ρ1	NOUN
ma-25	532	21	,	,	PUNCT
ma-25	532	22	µ[p	µ[p	ADJ
ma-25	532	23	,	,	PUNCT
ma-25	532	24	q	q	X
ma-25	532	25	]	]	X
ma-25	532	26	(	(	PUNCT
ma-25	532	27	g)−	g)−	PROPN
ma-25	532	28	ρ[p	ρ[p	PROPN
ma-25	532	29	,	,	PUNCT
ma-25	532	30	q	q	X
ma-25	532	31	]	]	X
ma-25	532	32	(	(	PUNCT
ma-25	532	33	d	d	NOUN
ma-25	532	34	)	)	PUNCT
ma-25	532	35	}	}	PUNCT
ma-25	532	36	)	)	PUNCT
ma-25	532	37	,	,	PUNCT
ma-25	532	38	we	we	PRON
ma-25	532	39	obtain	obtain	VERB
ma-25	532	40	expp+1	expp+1	NOUN
ma-25	532	41	{	{	PUNCT
ma-25	532	42	(	(	PUNCT
ma-25	532	43	σ	σ	PROPN
ma-25	532	44	−	−	PROPN
ma-25	532	45	ε	ε	PROPN
ma-25	532	46	)	)	PUNCT
ma-25	532	47	logq	logq	VERB
ma-25	532	48	r	r	NOUN
ma-25	532	49	}	}	PUNCT
ma-25	532	50	≤	≤	ADJ
ma-25	532	51	r2s	r2s	X
ma-25	532	52	(	(	PUNCT
ma-25	532	53	expp+1	expp+1	NOUN
ma-25	532	54	{	{	PUNCT
ma-25	532	55	(	(	PUNCT
ma-25	532	56	ρ1	ρ1	NOUN
ma-25	532	57	+	+	CCONJ
ma-25	532	58	ε	ε	PROPN
ma-25	532	59	)	)	PUNCT
ma-25	532	60	logq	logq	VERB
ma-25	532	61	r	r	NOUN
ma-25	532	62	}	}	PUNCT
ma-25	532	63	eur	eur	PROPN
ma-25	532	64	.	.	PUNCT
ma-25	533	1	j.	j.	PROPN
ma-25	533	2	math	math	PROPN
ma-25	533	3	.	.	PUNCT
ma-25	534	1	anal	anal	ADJ
ma-25	534	2	.	.	PUNCT
ma-25	535	1	1	1	NUM
ma-25	535	2	(	(	PUNCT
ma-25	535	3	2021	2021	NUM
ma-25	535	4	)	)	PUNCT
ma-25	535	5	103	103	NUM
ma-25	536	1	+	+	CCONJ
ma-25	536	2	k∑	k∑	VERB
ma-25	536	3	j=1,j	j=1,j	NOUN
ma-25	536	4	6	6	NUM
ma-25	536	5	=	=	SYM
ma-25	536	6	s	s	NOUN
ma-25	536	7	expp+1	expp+1	NOUN
ma-25	536	8	{	{	PUNCT
ma-25	536	9	(	(	PUNCT
ma-25	536	10	ρ1	ρ1	NOUN
ma-25	536	11	+	+	CCONJ
ma-25	536	12	ε	ε	PROPN
ma-25	536	13	)	)	PUNCT
ma-25	536	14	logq	logq	VERB
ma-25	536	15	r	r	NOUN
ma-25	536	16	}	}	PUNCT
ma-25	536	17	b	b	PROPN
ma-25	537	1	[	[	X
ma-25	537	2	t	t	X
ma-25	537	3	(	(	PUNCT
ma-25	537	4	2r	2r	NUM
ma-25	537	5	,	,	PUNCT
ma-25	537	6	f	f	PROPN
ma-25	537	7	)	)	PUNCT
ma-25	537	8	]	]	PUNCT
ma-25	537	9	k+1	k+1	X
ma-25	537	10	+	+	X
ma-25	537	11	expp+1	expp+1	NOUN
ma-25	537	12	{	{	PUNCT
ma-25	537	13	(	(	PUNCT
ma-25	537	14	ρ1	ρ1	NOUN
ma-25	537	15	+	+	CCONJ
ma-25	537	16	ε	ε	PROPN
ma-25	537	17	)	)	PUNCT
ma-25	537	18	logq	logq	VERB
ma-25	537	19	r	r	NOUN
ma-25	537	20	}	}	PUNCT
ma-25	537	21	)	)	PUNCT
ma-25	537	22	≤	≤	NUM
ma-25	537	23	b	b	X
ma-25	537	24	(	(	PUNCT
ma-25	537	25	k	k	NOUN
ma-25	537	26	+	+	PROPN
ma-25	537	27	1	1	X
ma-25	537	28	)	)	PUNCT
ma-25	537	29	r2s	r2s	NOUN
ma-25	538	1	[	[	X
ma-25	538	2	t	t	X
ma-25	538	3	(	(	PUNCT
ma-25	538	4	2r	2r	NUM
ma-25	538	5	,	,	PUNCT
ma-25	538	6	f	f	PROPN
ma-25	538	7	)	)	PUNCT
ma-25	538	8	]	]	PUNCT
ma-25	538	9	k+1	k+1	X
ma-25	538	10	expp+1	expp+1	NOUN
ma-25	538	11	{	{	PUNCT
ma-25	538	12	(	(	PUNCT
ma-25	538	13	ρ1	ρ1	NOUN
ma-25	538	14	+	+	CCONJ
ma-25	538	15	ε	ε	PROPN
ma-25	538	16	)	)	PUNCT
ma-25	538	17	logq	logq	VERB
ma-25	538	18	r	r	NOUN
ma-25	538	19	}	}	PUNCT
ma-25	538	20	.	.	PUNCT
ma-25	539	1	(	(	PUNCT
ma-25	539	2	5.5	5.5	NUM
ma-25	539	3	)	)	PUNCT
ma-25	539	4	since	since	SCONJ
ma-25	539	5	0	0	NUM
ma-25	539	6	<	<	X
ma-25	539	7	2ε	2ε	PROPN
ma-25	539	8	<	<	X
ma-25	539	9	σ	σ	PROPN
ma-25	539	10	−	−	PROPN
ma-25	539	11	ρ1	ρ1	NOUN
ma-25	539	12	,	,	PUNCT
ma-25	539	13	then	then	ADV
ma-25	539	14	we	we	PRON
ma-25	539	15	can	can	AUX
ma-25	539	16	use	use	VERB
ma-25	539	17	lemma	lemma	PROPN
ma-25	539	18	2.9	2.9	NUM
ma-25	539	19	with	with	ADP
ma-25	539	20	(	(	PUNCT
ma-25	539	21	5.5	5.5	NUM
ma-25	539	22	)	)	PUNCT
ma-25	539	23	such	such	ADJ
ma-25	539	24	that	that	PRON
ma-25	539	25	for	for	ADP
ma-25	539	26	any	any	DET
ma-25	539	27	given	give	VERB
ma-25	539	28	γ	γ	X
ma-25	539	29	>	>	X
ma-25	539	30	1	1	NUM
ma-25	539	31	andsufficiently	andsufficiently	ADV
ma-25	539	32	large	large	ADJ
ma-25	539	33	r	r	NOUN
ma-25	539	34	>	>	X
ma-25	539	35	r	r	NOUN
ma-25	539	36	,	,	PUNCT
ma-25	539	37	we	we	PRON
ma-25	539	38	obtain	obtain	VERB
ma-25	539	39	exp	exp	NOUN
ma-25	539	40	{	{	PUNCT
ma-25	539	41	(	(	PUNCT
ma-25	539	42	1−	1−	NUM
ma-25	539	43	o	o	NOUN
ma-25	539	44	(	(	PUNCT
ma-25	539	45	1	1	NUM
ma-25	539	46	)	)	PUNCT
ma-25	539	47	)	)	PUNCT
ma-25	540	1	expp	expp	ADJ
ma-25	540	2	{	{	PUNCT
ma-25	540	3	(	(	PUNCT
ma-25	540	4	σ	σ	PROPN
ma-25	540	5	−	−	PROPN
ma-25	540	6	ε	ε	PROPN
ma-25	540	7	)	)	PUNCT
ma-25	540	8	logq	logq	VERB
ma-25	540	9	r	r	NOUN
ma-25	540	10	}	}	PUNCT
ma-25	540	11	}	}	PUNCT
ma-25	540	12	≤	≤	NUM
ma-25	540	13	b	b	X
ma-25	540	14	(	(	PUNCT
ma-25	540	15	k	k	NOUN
ma-25	540	16	+	+	PROPN
ma-25	540	17	1	1	NUM
ma-25	540	18	)	)	PUNCT
ma-25	540	19	(	(	PUNCT
ma-25	540	20	γr)2s	γr)2s	X
ma-25	541	1	[	[	X
ma-25	541	2	t	t	X
ma-25	541	3	(	(	PUNCT
ma-25	541	4	2γr	2γr	NOUN
ma-25	541	5	,	,	PUNCT
ma-25	541	6	f	f	PROPN
ma-25	541	7	)	)	PUNCT
ma-25	541	8	]	]	PUNCT
ma-25	541	9	k+1	k+1	X
ma-25	541	10	which	which	PRON
ma-25	541	11	gives	give	VERB
ma-25	541	12	ρ[p	ρ[p	NOUN
ma-25	541	13	,	,	PUNCT
ma-25	541	14	q](f	q](f	NOUN
ma-25	541	15	)	)	PUNCT
ma-25	541	16	=	=	SYM
ma-25	541	17	µ[p	µ[p	ADJ
ma-25	541	18	,	,	PUNCT
ma-25	541	19	q	q	X
ma-25	541	20	]	]	X
ma-25	541	21	(	(	PUNCT
ma-25	541	22	f	f	X
ma-25	541	23	)	)	PUNCT
ma-25	541	24	=	=	PUNCT
ma-25	542	1	+	+	NUM
ma-25	542	2	∞	∞	PROPN
ma-25	542	3	,	,	PUNCT
ma-25	542	4	ρ[p+1,q](f	ρ[p+1,q](f	X
ma-25	542	5	)	)	PUNCT
ma-25	542	6	≥	≥	PROPN
ma-25	542	7	σ	σ	X
ma-25	542	8	.	.	PUNCT
ma-25	543	1	(	(	PUNCT
ma-25	543	2	5.6	5.6	NUM
ma-25	543	3	)	)	PUNCT
ma-25	543	4	making	make	VERB
ma-25	543	5	use	use	NOUN
ma-25	543	6	of	of	ADP
ma-25	543	7	lemma	lemma	PROPN
ma-25	543	8	2.4	2.4	NUM
ma-25	543	9	,	,	PUNCT
ma-25	543	10	we	we	PRON
ma-25	543	11	have	have	VERB
ma-25	543	12	max	max	PROPN
ma-25	543	13	{	{	PUNCT
ma-25	543	14	ρ[p	ρ[p	PROPN
ma-25	543	15	,	,	PUNCT
ma-25	543	16	q	q	X
ma-25	543	17	]	]	X
ma-25	543	18	(	(	PUNCT
ma-25	543	19	aj	aj	PROPN
ma-25	543	20	)	)	PUNCT
ma-25	543	21	(	(	PUNCT
ma-25	543	22	j	j	NOUN
ma-25	543	23	=	=	SYM
ma-25	543	24	0	0	NUM
ma-25	543	25	,	,	PUNCT
ma-25	543	26	1	1	NUM
ma-25	543	27	,	,	PUNCT
ma-25	543	28	...	...	PUNCT
ma-25	543	29	,	,	PUNCT
ma-25	543	30	k	k	NOUN
ma-25	543	31	)	)	PUNCT
ma-25	543	32	,	,	PUNCT
ma-25	543	33	ρ[p	ρ[p	PROPN
ma-25	543	34	,	,	PUNCT
ma-25	543	35	q	q	X
ma-25	543	36	]	]	X
ma-25	543	37	(	(	PUNCT
ma-25	543	38	f	f	NOUN
ma-25	543	39	)	)	PUNCT
ma-25	543	40	}	}	PUNCT
ma-25	543	41	=	=	PUNCT
ma-25	543	42	ρ[p	ρ[p	NOUN
ma-25	543	43	,	,	PUNCT
ma-25	543	44	q	q	X
ma-25	543	45	]	]	X
ma-25	543	46	(	(	PUNCT
ma-25	543	47	as	as	ADP
ma-25	543	48	)	)	PUNCT
ma-25	543	49	=	=	PUNCT
ma-25	543	50	β	β	X
ma-25	543	51	<	<	X
ma-25	544	1	+	+	X
ma-25	544	2	∞.	∞.	PROPN
ma-25	544	3	by	by	ADP
ma-25	544	4	lemma	lemma	PROPN
ma-25	544	5	2.11	2.11	NUM
ma-25	544	6	and	and	CCONJ
ma-25	544	7	since	since	SCONJ
ma-25	544	8	f	f	PROPN
ma-25	544	9	is	be	AUX
ma-25	544	10	of	of	ADP
ma-25	544	11	infinite	infinite	ADJ
ma-25	544	12	[	[	X
ma-25	544	13	p	p	NOUN
ma-25	544	14	,	,	PUNCT
ma-25	544	15	q]-order	q]-order	NOUN
ma-25	544	16	meromorphic	meromorphic	ADJ
ma-25	544	17	solution	solution	NOUN
ma-25	544	18	of	of	ADP
ma-25	544	19	equation	equation	NOUN
ma-25	544	20	(	(	PUNCT
ma-25	544	21	1.4)satisfying	1.4)satisfying	NUM
ma-25	544	22	λ[p	λ[p	NUM
ma-25	544	23	,	,	PUNCT
ma-25	544	24	q	q	X
ma-25	544	25	]	]	X
ma-25	544	26	(	(	PUNCT
ma-25	544	27	1f	1f	NUM
ma-25	544	28	)	)	PUNCT
ma-25	544	29	<	<	X
ma-25	544	30	µ[p	µ[p	ADJ
ma-25	544	31	,	,	PUNCT
ma-25	544	32	q	q	X
ma-25	544	33	]	]	X
ma-25	544	34	(	(	PUNCT
ma-25	544	35	f	f	PROPN
ma-25	544	36	)	)	PUNCT
ma-25	544	37	,	,	PUNCT
ma-25	544	38	we	we	PRON
ma-25	544	39	get	get	VERB
ma-25	544	40	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	544	41	]	]	PUNCT
ma-25	544	42	(	(	PUNCT
ma-25	544	43	f	f	PROPN
ma-25	544	44	)	)	PUNCT
ma-25	544	45	≤	≤	PROPN
ma-25	544	46	max	max	PROPN
ma-25	544	47	{	{	PUNCT
ma-25	544	48	ρ[p	ρ[p	PROPN
ma-25	544	49	,	,	PUNCT
ma-25	544	50	q	q	X
ma-25	544	51	]	]	X
ma-25	544	52	(	(	PUNCT
ma-25	544	53	aj	aj	PROPN
ma-25	544	54	)	)	PUNCT
ma-25	544	55	(	(	PUNCT
ma-25	544	56	j	j	NOUN
ma-25	544	57	=	=	SYM
ma-25	544	58	0	0	NUM
ma-25	544	59	,	,	PUNCT
ma-25	544	60	1	1	NUM
ma-25	544	61	,	,	PUNCT
ma-25	544	62	...	...	PUNCT
ma-25	544	63	,	,	PUNCT
ma-25	544	64	k	k	NOUN
ma-25	544	65	)	)	PUNCT
ma-25	544	66	,	,	PUNCT
ma-25	544	67	ρ[p	ρ[p	PROPN
ma-25	544	68	,	,	PUNCT
ma-25	544	69	q	q	X
ma-25	544	70	]	]	X
ma-25	544	71	(	(	PUNCT
ma-25	544	72	f	f	NOUN
ma-25	544	73	)	)	PUNCT
ma-25	544	74	}	}	PUNCT
ma-25	544	75	=	=	PUNCT
ma-25	544	76	ρ[p	ρ[p	NOUN
ma-25	544	77	,	,	PUNCT
ma-25	544	78	q	q	X
ma-25	544	79	]	]	X
ma-25	544	80	(	(	PUNCT
ma-25	544	81	as	as	ADP
ma-25	544	82	)	)	PUNCT
ma-25	544	83	.	.	PUNCT
ma-25	545	1	(	(	PUNCT
ma-25	545	2	5.7	5.7	NUM
ma-25	545	3	)	)	PUNCT
ma-25	545	4	since	since	SCONJ
ma-25	545	5	f	f	PROPN
ma-25	545	6	6≡	6≡	NUM
ma-25	545	7	0	0	NUM
ma-25	545	8	,	,	PUNCT
ma-25	545	9	then	then	ADV
ma-25	545	10	by	by	ADP
ma-25	545	11	lemma	lemma	PROPN
ma-25	545	12	2.15	2.15	NUM
ma-25	545	13	,	,	PUNCT
ma-25	545	14	we	we	PRON
ma-25	545	15	have	have	VERB
ma-25	545	16	λ[p	λ[p	NOUN
ma-25	545	17	,	,	PUNCT
ma-25	545	18	q	q	X
ma-25	545	19	]	]	X
ma-25	545	20	(	(	PUNCT
ma-25	545	21	f	f	X
ma-25	545	22	)	)	PUNCT
ma-25	546	1	=	=	SYM
ma-25	546	2	λ[p	λ[p	ADJ
ma-25	546	3	,	,	PUNCT
ma-25	546	4	q](f	q](f	NOUN
ma-25	546	5	)	)	PUNCT
ma-25	546	6	=	=	SYM
ma-25	546	7	µ[p	µ[p	ADJ
ma-25	546	8	,	,	PUNCT
ma-25	546	9	q	q	X
ma-25	546	10	]	]	X
ma-25	546	11	(	(	PUNCT
ma-25	546	12	f	f	X
ma-25	546	13	)	)	PUNCT
ma-25	546	14	=	=	PUNCT
ma-25	546	15	ρ[p	ρ[p	NOUN
ma-25	546	16	,	,	PUNCT
ma-25	546	17	q](f	q](f	NOUN
ma-25	546	18	)	)	PUNCT
ma-25	546	19	=	=	PUNCT
ma-25	547	1	+	+	NUM
ma-25	547	2	∞	∞	PROPN
ma-25	547	3	(	(	PUNCT
ma-25	547	4	5.8	5.8	NUM
ma-25	547	5	)	)	PUNCT
ma-25	547	6	and	and	CCONJ
ma-25	547	7	σ	σ	PROPN
ma-25	547	8	≤	≤	PROPN
ma-25	547	9	λ[p+1,q	λ[p+1,q	PROPN
ma-25	547	10	]	]	X
ma-25	547	11	(	(	PUNCT
ma-25	547	12	f	f	X
ma-25	547	13	)	)	PUNCT
ma-25	547	14	=	=	SYM
ma-25	547	15	λ[p+1,q](f	λ[p+1,q](f	X
ma-25	547	16	)	)	PUNCT
ma-25	547	17	=	=	SYM
ma-25	547	18	ρ[p+1,q](f	ρ[p+1,q](f	PROPN
ma-25	547	19	)	)	PUNCT
ma-25	547	20	.	.	PUNCT
ma-25	548	1	(	(	PUNCT
ma-25	548	2	5.9	5.9	NUM
ma-25	548	3	)	)	PUNCT
ma-25	548	4	by	by	ADP
ma-25	548	5	(	(	PUNCT
ma-25	548	6	5.7	5.7	NUM
ma-25	548	7	)	)	PUNCT
ma-25	548	8	,	,	PUNCT
ma-25	548	9	(	(	PUNCT
ma-25	548	10	5.8	5.8	NUM
ma-25	548	11	)	)	PUNCT
ma-25	548	12	and	and	CCONJ
ma-25	548	13	(	(	PUNCT
ma-25	548	14	5.9	5.9	NUM
ma-25	548	15	)	)	PUNCT
ma-25	548	16	,	,	PUNCT
ma-25	548	17	we	we	PRON
ma-25	548	18	conclude	conclude	VERB
ma-25	548	19	that	that	PRON
ma-25	548	20	λ[p	λ[p	ADJ
ma-25	548	21	,	,	PUNCT
ma-25	548	22	q	q	X
ma-25	548	23	]	]	X
ma-25	548	24	(	(	PUNCT
ma-25	548	25	f	f	X
ma-25	548	26	)	)	PUNCT
ma-25	549	1	=	=	SYM
ma-25	549	2	λ[p	λ[p	ADJ
ma-25	549	3	,	,	PUNCT
ma-25	549	4	q](f	q](f	NOUN
ma-25	549	5	)	)	PUNCT
ma-25	549	6	=	=	SYM
ma-25	549	7	µ[p	µ[p	ADJ
ma-25	549	8	,	,	PUNCT
ma-25	549	9	q	q	X
ma-25	549	10	]	]	X
ma-25	549	11	(	(	PUNCT
ma-25	549	12	f	f	X
ma-25	549	13	)	)	PUNCT
ma-25	549	14	=	=	PUNCT
ma-25	549	15	ρ[p	ρ[p	NOUN
ma-25	549	16	,	,	PUNCT
ma-25	549	17	q](f	q](f	NOUN
ma-25	549	18	)	)	PUNCT
ma-25	549	19	=	=	PUNCT
ma-25	550	1	+	+	PUNCT
ma-25	550	2	∞	∞	NUM
ma-25	550	3	and	and	CCONJ
ma-25	550	4	σ	σ	PROPN
ma-25	550	5	≤	≤	PROPN
ma-25	550	6	λ[p+1,q	λ[p+1,q	PROPN
ma-25	550	7	]	]	X
ma-25	550	8	(	(	PUNCT
ma-25	550	9	f	f	X
ma-25	550	10	)	)	PUNCT
ma-25	550	11	=	=	SYM
ma-25	550	12	λ[p+1,q](f	λ[p+1,q](f	X
ma-25	550	13	)	)	PUNCT
ma-25	550	14	=	=	SYM
ma-25	550	15	ρ[p+1,q](f	ρ[p+1,q](f	X
ma-25	550	16	)	)	PUNCT
ma-25	550	17	≤	≤	NOUN
ma-25	550	18	ρ[p	ρ[p	NOUN
ma-25	550	19	,	,	PUNCT
ma-25	550	20	q	q	X
ma-25	550	21	]	]	X
ma-25	550	22	(	(	PUNCT
ma-25	550	23	as	as	ADP
ma-25	550	24	)	)	PUNCT
ma-25	550	25	.	.	PUNCT
ma-25	551	1	6	6	X
ma-25	551	2	.	.	X
ma-25	551	3	proof	proof	NOUN
ma-25	551	4	of	of	ADP
ma-25	551	5	corollary	corollary	ADJ
ma-25	551	6	1.2	1.2	NUM
ma-25	551	7	assume	assume	VERB
ma-25	551	8	that	that	SCONJ
ma-25	551	9	ϕ	ϕ	NOUN
ma-25	551	10	is	be	AUX
ma-25	551	11	a	a	DET
ma-25	551	12	transcendental	transcendental	ADJ
ma-25	551	13	meromorphic	meromorphic	ADJ
ma-25	551	14	function	function	NOUN
ma-25	551	15	such	such	ADJ
ma-25	551	16	that	that	DET
ma-25	551	17	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	551	18	]	]	PUNCT
ma-25	551	19	(	(	PUNCT
ma-25	551	20	ϕ	ϕ	NOUN
ma-25	551	21	)	)	PUNCT
ma-25	551	22	<	<	X
ma-25	551	23	σ	σ	PROPN
ma-25	551	24	.	.	PUNCT
ma-25	551	25	noting	note	VERB
ma-25	551	26	h	h	PROPN
ma-25	551	27	=	=	SYM
ma-25	551	28	f	f	PROPN
ma-25	552	1	−	−	PROPN
ma-25	552	2	ϕ	ϕ	PROPN
ma-25	552	3	,	,	PUNCT
ma-25	552	4	then	then	ADV
ma-25	552	5	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	552	6	]	]	PUNCT
ma-25	552	7	(	(	PUNCT
ma-25	552	8	h	h	NOUN
ma-25	552	9	)	)	PUNCT
ma-25	552	10	=	=	SYM
ma-25	552	11	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	552	12	]	]	PUNCT
ma-25	552	13	(	(	PUNCT
ma-25	552	14	f	f	PROPN
ma-25	552	15	)	)	PUNCT
ma-25	552	16	,	,	PUNCT
ma-25	552	17	so	so	ADV
ma-25	552	18	by	by	ADP
ma-25	552	19	theorem	theorem	NOUN
ma-25	552	20	1.2	1.2	NUM
ma-25	552	21	,	,	PUNCT
ma-25	552	22	σ	σ	NOUN
ma-25	552	23	≤	≤	NUM
ma-25	552	24	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	552	25	]	]	PUNCT
ma-25	552	26	(	(	PUNCT
ma-25	552	27	h	h	NOUN
ma-25	552	28	)	)	PUNCT
ma-25	552	29	≤	≤	NOUN
ma-25	552	30	ρ[p	ρ[p	NOUN
ma-25	552	31	,	,	PUNCT
ma-25	552	32	q	q	X
ma-25	552	33	]	]	X
ma-25	552	34	(	(	PUNCT
ma-25	552	35	as	as	ADP
ma-25	552	36	)	)	PUNCT
ma-25	552	37	.	.	PUNCT
ma-25	553	1	bysubstituting	bysubstitute	VERB
ma-25	553	2	f	f	PROPN
ma-25	553	3	=	=	PUNCT
ma-25	553	4	h	h	PROPN
ma-25	553	5	+	+	CCONJ
ma-25	553	6	ϕ	ϕ	NOUN
ma-25	553	7	into	into	ADP
ma-25	553	8	(	(	PUNCT
ma-25	553	9	1.4	1.4	NUM
ma-25	553	10	)	)	PUNCT
ma-25	553	11	,	,	PUNCT
ma-25	553	12	we	we	PRON
ma-25	553	13	obtain	obtain	VERB
ma-25	553	14	ak	ak	PROPN
ma-25	553	15	(	(	PUNCT
ma-25	553	16	z	z	NOUN
ma-25	553	17	)	)	PUNCT
ma-25	553	18	h(k	h(k	PROPN
ma-25	553	19	)	)	PUNCT
ma-25	554	1	+	+	CCONJ
ma-25	554	2	ak−1	ak−1	ADV
ma-25	554	3	(	(	PUNCT
ma-25	554	4	z	z	NOUN
ma-25	554	5	)	)	PUNCT
ma-25	554	6	h(k−1	h(k−1	NOUN
ma-25	554	7	)	)	PUNCT
ma-25	555	1	+	+	CCONJ
ma-25	555	2	·	·	PUNCT
ma-25	555	3	·	·	PUNCT
ma-25	555	4	·	·	PUNCT
ma-25	555	5	+	+	NUM
ma-25	555	6	a1	a1	NOUN
ma-25	555	7	(	(	PUNCT
ma-25	555	8	z	z	NOUN
ma-25	555	9	)	)	PUNCT
ma-25	555	10	h′	h′	PROPN
ma-25	555	11	+	+	CCONJ
ma-25	555	12	a0	a0	PROPN
ma-25	555	13	(	(	PUNCT
ma-25	555	14	z	z	NOUN
ma-25	555	15	)	)	PUNCT
ma-25	555	16	h	h	NOUN
ma-25	555	17	=	=	SYM
ma-25	555	18	f	f	PROPN
ma-25	555	19	(	(	PUNCT
ma-25	555	20	z)−	z)−	PROPN
ma-25	555	21	(	(	PUNCT
ma-25	555	22	ak	ak	PROPN
ma-25	555	23	(	(	PUNCT
ma-25	555	24	z)ϕ(k	z)ϕ(k	NOUN
ma-25	555	25	)	)	PUNCT
ma-25	556	1	+	+	CCONJ
ma-25	556	2	ak−1	ak−1	ADV
ma-25	556	3	(	(	PUNCT
ma-25	556	4	z)ϕ(k−1	z)ϕ(k−1	PROPN
ma-25	556	5	)	)	PUNCT
ma-25	556	6	+	+	CCONJ
ma-25	556	7	·	·	PUNCT
ma-25	556	8	·	·	PUNCT
ma-25	556	9	·	·	PUNCT
ma-25	556	10	+	+	NUM
ma-25	556	11	a1	a1	NOUN
ma-25	556	12	(	(	PUNCT
ma-25	556	13	z)ϕ′	z)ϕ′	PROPN
ma-25	556	14	+	+	NUM
ma-25	556	15	a0	a0	PROPN
ma-25	556	16	(	(	PUNCT
ma-25	556	17	z)ϕ	z)ϕ	NOUN
ma-25	556	18	)	)	PUNCT
ma-25	557	1	=	=	SYM
ma-25	557	2	ψ	ψ	X
ma-25	557	3	(	(	PUNCT
ma-25	557	4	z	z	NOUN
ma-25	557	5	)	)	PUNCT
ma-25	557	6	.	.	PUNCT
ma-25	558	1	(	(	PUNCT
ma-25	558	2	6.1	6.1	NUM
ma-25	558	3	)	)	PUNCT
ma-25	558	4	eur	eur	PROPN
ma-25	558	5	.	.	PUNCT
ma-25	559	1	j.	j.	PROPN
ma-25	559	2	math	math	PROPN
ma-25	559	3	.	.	PUNCT
ma-25	560	1	anal	anal	ADJ
ma-25	560	2	.	.	PUNCT
ma-25	561	1	1	1	NUM
ma-25	561	2	(	(	PUNCT
ma-25	561	3	2021	2021	NUM
ma-25	561	4	)	)	PUNCT
ma-25	561	5	104it	104it	PROPN
ma-25	561	6	is	be	AUX
ma-25	561	7	clear	clear	ADJ
ma-25	561	8	that	that	SCONJ
ma-25	561	9	the	the	DET
ma-25	561	10	right	right	ADJ
ma-25	561	11	side	side	NOUN
ma-25	561	12	ψ	ψ	NOUN
ma-25	561	13	of	of	ADP
ma-25	561	14	the	the	DET
ma-25	561	15	equation	equation	NOUN
ma-25	561	16	(	(	PUNCT
ma-25	561	17	6.1	6.1	NUM
ma-25	561	18	)	)	PUNCT
ma-25	561	19	is	be	AUX
ma-25	561	20	non	non	ADJ
ma-25	561	21	-	-	ADJ
ma-25	561	22	zero	zero	NUM
ma-25	561	23	,	,	PUNCT
ma-25	561	24	because	because	SCONJ
ma-25	561	25	by	by	ADP
ma-25	561	26	theorem	theorem	NOUN
ma-25	561	27	1.2	1.2	NUM
ma-25	561	28	,	,	PUNCT
ma-25	561	29	ϕ	ϕ	PROPN
ma-25	561	30	isnot	isnot	ADV
ma-25	561	31	a	a	DET
ma-25	561	32	solution	solution	NOUN
ma-25	561	33	of	of	ADP
ma-25	561	34	equation	equation	NOUN
ma-25	561	35	(	(	PUNCT
ma-25	561	36	1.4	1.4	NUM
ma-25	561	37	)	)	PUNCT
ma-25	561	38	.	.	PUNCT
ma-25	562	1	moreover	moreover	ADV
ma-25	562	2	,	,	PUNCT
ma-25	562	3	the	the	DET
ma-25	562	4	[	[	X
ma-25	562	5	p	p	X
ma-25	562	6	+	+	NOUN
ma-25	562	7	1	1	NUM
ma-25	562	8	,	,	PUNCT
ma-25	562	9	q]-order	q]-order	NOUN
ma-25	562	10	of	of	ADP
ma-25	562	11	ψ	ψ	PROPN
ma-25	562	12	verifies	verifie	NOUN
ma-25	562	13	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	562	14	]	]	PUNCT
ma-25	562	15	(	(	PUNCT
ma-25	562	16	ψ	ψ	NOUN
ma-25	562	17	)	)	PUNCT
ma-25	562	18	≤	≤	NUM
ma-25	562	19	max	max	PROPN
ma-25	562	20	{	{	PUNCT
ma-25	562	21	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	562	22	]	]	PUNCT
ma-25	562	23	(	(	PUNCT
ma-25	562	24	ϕ	ϕ	NOUN
ma-25	562	25	)	)	PUNCT
ma-25	562	26	,	,	PUNCT
ma-25	562	27	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	562	28	]	]	PUNCT
ma-25	562	29	(	(	PUNCT
ma-25	562	30	aj	aj	PROPN
ma-25	562	31	)	)	PUNCT
ma-25	562	32	(	(	PUNCT
ma-25	562	33	j	j	NOUN
ma-25	562	34	=	=	SYM
ma-25	562	35	0	0	NUM
ma-25	562	36	,	,	PUNCT
ma-25	562	37	1	1	NUM
ma-25	562	38	,	,	PUNCT
ma-25	562	39	...	...	PUNCT
ma-25	562	40	,	,	PUNCT
ma-25	562	41	k	k	NOUN
ma-25	562	42	)	)	PUNCT
ma-25	562	43	}	}	PUNCT
ma-25	562	44	<	<	X
ma-25	562	45	σ	σ	PROPN
ma-25	562	46	,	,	PUNCT
ma-25	562	47	which	which	PRON
ma-25	562	48	leads	lead	VERB
ma-25	562	49	to	to	ADP
ma-25	562	50	max	max	PROPN
ma-25	562	51	{	{	PUNCT
ma-25	562	52	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	562	53	]	]	PUNCT
ma-25	562	54	(	(	PUNCT
ma-25	562	55	ψ	ψ	NOUN
ma-25	562	56	)	)	PUNCT
ma-25	562	57	,	,	PUNCT
ma-25	562	58	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	562	59	]	]	PUNCT
ma-25	562	60	(	(	PUNCT
ma-25	562	61	aj	aj	PROPN
ma-25	562	62	)	)	PUNCT
ma-25	562	63	(	(	PUNCT
ma-25	562	64	j	j	NOUN
ma-25	562	65	=	=	SYM
ma-25	562	66	0	0	NUM
ma-25	562	67	,	,	PUNCT
ma-25	562	68	1	1	NUM
ma-25	562	69	,	,	PUNCT
ma-25	562	70	...	...	PUNCT
ma-25	562	71	,	,	PUNCT
ma-25	562	72	k	k	NOUN
ma-25	562	73	)	)	PUNCT
ma-25	562	74	}	}	PUNCT
ma-25	562	75	<	<	X
ma-25	562	76	σ	σ	PROPN
ma-25	562	77	≤	≤	NUM
ma-25	562	78	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	562	79	]	]	PUNCT
ma-25	562	80	(	(	PUNCT
ma-25	562	81	h	h	NOUN
ma-25	562	82	)	)	PUNCT
ma-25	562	83	.	.	PUNCT
ma-25	563	1	therefore	therefore	ADV
ma-25	563	2	,	,	PUNCT
ma-25	563	3	by	by	ADP
ma-25	563	4	lemma	lemma	PROPN
ma-25	563	5	2.12	2.12	NUM
ma-25	563	6	,	,	PUNCT
ma-25	563	7	we	we	PRON
ma-25	563	8	obtain	obtain	VERB
ma-25	563	9	σ	σ	NOUN
ma-25	563	10	≤	≤	NUM
ma-25	563	11	λ[p+1,q	λ[p+1,q	NOUN
ma-25	563	12	]	]	X
ma-25	563	13	(	(	PUNCT
ma-25	563	14	h	h	NOUN
ma-25	563	15	)	)	PUNCT
ma-25	563	16	=	=	SYM
ma-25	563	17	λ[p+1,q	λ[p+1,q	PROPN
ma-25	563	18	]	]	X
ma-25	563	19	(	(	PUNCT
ma-25	563	20	h	h	NOUN
ma-25	563	21	)	)	PUNCT
ma-25	563	22	=	=	SYM
ma-25	563	23	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	563	24	]	]	PUNCT
ma-25	563	25	(	(	PUNCT
ma-25	563	26	h	h	NOUN
ma-25	563	27	)	)	PUNCT
ma-25	563	28	=	=	SYM
ma-25	563	29	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	563	30	]	]	PUNCT
ma-25	563	31	(	(	PUNCT
ma-25	563	32	f	f	PROPN
ma-25	563	33	)	)	PUNCT
ma-25	563	34	≤	≤	PROPN
ma-25	563	35	ρ[p	ρ[p	NOUN
ma-25	563	36	,	,	PUNCT
ma-25	563	37	q	q	X
ma-25	563	38	]	]	X
ma-25	563	39	(	(	PUNCT
ma-25	563	40	as	as	ADP
ma-25	563	41	)	)	PUNCT
ma-25	563	42	,	,	PUNCT
ma-25	563	43	that	that	ADV
ma-25	563	44	is	is	ADV
ma-25	563	45	σ	σ	PROPN
ma-25	563	46	≤	≤	NUM
ma-25	563	47	λ[p+1,q	λ[p+1,q	NOUN
ma-25	563	48	]	]	X
ma-25	563	49	(	(	PUNCT
ma-25	563	50	f	f	PROPN
ma-25	563	51	−	−	PROPN
ma-25	563	52	ϕ	ϕ	PROPN
ma-25	563	53	)	)	PUNCT
ma-25	563	54	=	=	SYM
ma-25	563	55	λ[p+1,q	λ[p+1,q	PROPN
ma-25	563	56	]	]	X
ma-25	563	57	(	(	PUNCT
ma-25	563	58	f	f	PROPN
ma-25	563	59	−	−	PROPN
ma-25	563	60	ϕ	ϕ	PROPN
ma-25	563	61	)	)	PUNCT
ma-25	563	62	=	=	SYM
ma-25	563	63	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	563	64	]	]	PUNCT
ma-25	563	65	(	(	PUNCT
ma-25	563	66	f	f	X
ma-25	563	67	−	−	PROPN
ma-25	563	68	ϕ	ϕ	PROPN
ma-25	563	69	)	)	PUNCT
ma-25	563	70	=	=	SYM
ma-25	563	71	ρ[p+1,q	ρ[p+1,q	NOUN
ma-25	563	72	]	]	PUNCT
ma-25	563	73	(	(	PUNCT
ma-25	563	74	f	f	PROPN
ma-25	563	75	)	)	PUNCT
ma-25	563	76	≤	≤	PROPN
ma-25	563	77	ρ[p	ρ[p	NOUN
ma-25	563	78	,	,	PUNCT
ma-25	563	79	q	q	X
ma-25	563	80	]	]	X
ma-25	563	81	(	(	PUNCT
ma-25	563	82	as	as	ADP
ma-25	563	83	)	)	PUNCT
ma-25	563	84	.	.	PUNCT
ma-25	564	1	acknowledgements	acknowledgement	NOUN
ma-25	564	2	.	.	PUNCT
ma-25	565	1	this	this	DET
ma-25	565	2	paper	paper	NOUN
ma-25	565	3	was	be	AUX
ma-25	565	4	supported	support	VERB
ma-25	565	5	by	by	ADP
ma-25	565	6	the	the	DET
ma-25	565	7	directorate	directorate	NOUN
ma-25	565	8	-	-	PUNCT
ma-25	565	9	general	general	ADJ
ma-25	565	10	for	for	ADP
ma-25	565	11	scientific	scientific	ADJ
ma-25	565	12	researchand	researchand	NOUN
ma-25	565	13	technological	technological	ADJ
ma-25	565	14	development	development	NOUN
ma-25	565	15	(	(	PUNCT
ma-25	565	16	dgrsdt	dgrsdt	NOUN
ma-25	565	17	)	)	PUNCT
ma-25	565	18	.	.	PUNCT
ma-25	566	1	references	reference	NOUN
ma-25	566	2	[	[	X
ma-25	566	3	1	1	NUM
ma-25	566	4	]	]	PUNCT
ma-25	566	5	b.	b.	PROPN
ma-25	566	6	belaïdi	belaïdi	PROPN
ma-25	566	7	,	,	PUNCT
ma-25	566	8	growth	growth	NOUN
ma-25	566	9	and	and	CCONJ
ma-25	566	10	oscillation	oscillation	NOUN
ma-25	566	11	theory	theory	NOUN
ma-25	566	12	of	of	ADP
ma-25	566	13	[	[	X
ma-25	566	14	p	p	X
ma-25	566	15	,	,	PUNCT
ma-25	566	16	q]-order	q]-order	NOUN
ma-25	566	17	analytic	analytic	ADJ
ma-25	566	18	solutions	solution	NOUN
ma-25	566	19	of	of	ADP
ma-25	566	20	linear	linear	PROPN
ma-25	566	21	differential	differential	ADJ
ma-25	566	22	equations	equation	NOUN
ma-25	566	23	in	in	ADP
ma-25	566	24	theunit	theunit	VERB
ma-25	566	25	disc	disc	PROPN
ma-25	566	26	.	.	PUNCT
ma-25	567	1	j.	j.	PROPN
ma-25	567	2	math	math	PROPN
ma-25	567	3	.	.	PUNCT
ma-25	568	1	anal	anal	PROPN
ma-25	568	2	.	.	PUNCT
ma-25	569	1	3	3	NUM
ma-25	569	2	(	(	PUNCT
ma-25	569	3	2012	2012	NUM
ma-25	569	4	)	)	PUNCT
ma-25	569	5	,	,	PUNCT
ma-25	569	6	1–11	1–11	PROPN
ma-25	569	7	.	.	PUNCT
ma-25	570	1	http://www.ilirias.com/jma/repository/docs/jma3-1-1.pdf.[2	http://www.ilirias.com/jma/repository/docs/jma3-1-1.pdf.[2	PROPN
ma-25	570	2	]	]	X
ma-25	570	3	b.	b.	PROPN
ma-25	570	4	belaïdi	belaïdi	PROPN
ma-25	570	5	,	,	PUNCT
ma-25	570	6	iterated	iterate	VERB
ma-25	570	7	order	order	NOUN
ma-25	570	8	of	of	ADP
ma-25	570	9	meromorphic	meromorphic	ADJ
ma-25	570	10	solutions	solution	NOUN
ma-25	570	11	of	of	ADP
ma-25	570	12	homogeneous	homogeneous	ADJ
ma-25	570	13	and	and	CCONJ
ma-25	570	14	non	non	ADJ
ma-25	570	15	-	-	ADJ
ma-25	570	16	homogeneous	homogeneous	ADJ
ma-25	570	17	linear	linear	NOUN
ma-25	570	18	differentialequations	differentialequation	NOUN
ma-25	570	19	.	.	PUNCT
ma-25	571	1	romai	romai	PROPN
ma-25	571	2	j.	j.	PROPN
ma-25	571	3	11	11	NUM
ma-25	571	4	(	(	PUNCT
ma-25	571	5	2015	2015	NUM
ma-25	571	6	)	)	PUNCT
ma-25	571	7	,	,	PUNCT
ma-25	571	8	33–46	33–46	NUM
ma-25	571	9	.	.	PUNCT
ma-25	572	1	http://rj.romai.ro/arhiva/2015/1/rjv11n1.pdf.[3	http://rj.romai.ro/arhiva/2015/1/rjv11n1.pdf.[3	NOUN
ma-25	572	2	]	]	X
ma-25	572	3	b.	b.	PROPN
ma-25	572	4	belaïdi	belaïdi	PROPN
ma-25	572	5	,	,	PUNCT
ma-25	572	6	differential	differential	ADJ
ma-25	572	7	polynomials	polynomial	NOUN
ma-25	572	8	generated	generate	VERB
ma-25	572	9	by	by	ADP
ma-25	572	10	meromorphic	meromorphic	ADJ
ma-25	572	11	solutions	solution	NOUN
ma-25	572	12	of	of	ADP
ma-25	572	13	[	[	X
ma-25	572	14	p	p	X
ma-25	572	15	,	,	PUNCT
ma-25	572	16	q]-order	q]-order	NOUN
ma-25	572	17	to	to	ADP
ma-25	572	18	complex	complex	ADJ
ma-25	572	19	linear	linear	ADJ
ma-25	572	20	differentialequations	differentialequation	NOUN
ma-25	572	21	.	.	PUNCT
ma-25	573	1	rom	rom	PROPN
ma-25	573	2	.	.	PUNCT
ma-25	574	1	j.	j.	PROPN
ma-25	574	2	math	math	PROPN
ma-25	574	3	.	.	PUNCT
ma-25	575	1	comput	comput	NOUN
ma-25	575	2	.	.	PUNCT
ma-25	576	1	sci	sci	PROPN
ma-25	576	2	.	.	PROPN
ma-25	576	3	5	5	NUM
ma-25	576	4	(	(	PUNCT
ma-25	576	5	2015	2015	NUM
ma-25	576	6	)	)	PUNCT
ma-25	576	7	,	,	PUNCT
ma-25	576	8	46–62	46–62	NUM
ma-25	576	9	.	.	PUNCT
ma-25	576	10	http://rjm-cs.ro/belaidi-2015.pdf.[4	http://rjm-cs.ro/belaidi-2015.pdf.[4	PROPN
ma-25	576	11	]	]	X
ma-25	576	12	a.	a.	NOUN
ma-25	576	13	ferraoun	ferraoun	PROPN
ma-25	576	14	and	and	CCONJ
ma-25	576	15	b.	b.	PROPN
ma-25	576	16	belaïdi	belaïdi	PROPN
ma-25	576	17	,	,	PUNCT
ma-25	576	18	on	on	ADP
ma-25	576	19	the	the	DET
ma-25	576	20	(	(	PUNCT
ma-25	576	21	p	p	NOUN
ma-25	576	22	,	,	PUNCT
ma-25	576	23	q)−order	q)−order	PROPN
ma-25	576	24	of	of	ADP
ma-25	576	25	solutions	solution	NOUN
ma-25	576	26	of	of	ADP
ma-25	576	27	some	some	DET
ma-25	576	28	complex	complex	ADJ
ma-25	576	29	linear	linear	ADJ
ma-25	576	30	differential	differential	NOUN
ma-25	576	31	equations	equation	NOUN
ma-25	576	32	.	.	PUNCT
ma-25	577	1	commun.optim	commun.optim	NOUN
ma-25	577	2	.	.	PUNCT
ma-25	578	1	theory	theory	NOUN
ma-25	578	2	,	,	PUNCT
ma-25	578	3	2017	2017	NUM
ma-25	578	4	(	(	PUNCT
ma-25	578	5	2017	2017	NUM
ma-25	578	6	)	)	PUNCT
ma-25	578	7	,	,	PUNCT
ma-25	578	8	article	article	NOUN
ma-25	578	9	i	i	PROPN
ma-25	578	10	d	d	PROPN
ma-25	578	11	17	17	NUM
ma-25	578	12	,	,	PUNCT
ma-25	578	13	1–23	1–23	NOUN
ma-25	578	14	.	.	PUNCT
ma-25	579	1	https://doi.org/10.23952/cot.2017.17.[5	https://doi.org/10.23952/cot.2017.17.[5	PROPN
ma-25	579	2	]	]	PUNCT
ma-25	579	3	g.	g.	PROPN
ma-25	579	4	g.	g.	PROPN
ma-25	579	5	gundersen	gundersen	PROPN
ma-25	579	6	,	,	PUNCT
ma-25	579	7	estimates	estimate	VERB
ma-25	579	8	for	for	ADP
ma-25	579	9	the	the	DET
ma-25	579	10	logarithmic	logarithmic	ADJ
ma-25	579	11	derivative	derivative	NOUN
ma-25	579	12	of	of	ADP
ma-25	579	13	a	a	DET
ma-25	579	14	meromorphic	meromorphic	ADJ
ma-25	579	15	function	function	NOUN
ma-25	579	16	,	,	PUNCT
ma-25	579	17	plus	plus	CCONJ
ma-25	579	18	similar	similar	ADJ
ma-25	579	19	estimates	estimate	NOUN
ma-25	579	20	.	.	PUNCT
ma-25	580	1	j.london	j.london	ADJ
ma-25	580	2	math	math	NOUN
ma-25	580	3	.	.	PUNCT
ma-25	581	1	soc	soc	PROPN
ma-25	581	2	.	.	PUNCT
ma-25	582	1	(	(	PUNCT
ma-25	582	2	2	2	X
ma-25	582	3	)	)	PUNCT
ma-25	582	4	37	37	NUM
ma-25	582	5	(	(	PUNCT
ma-25	582	6	1988	1988	NUM
ma-25	582	7	)	)	PUNCT
ma-25	582	8	,	,	PUNCT
ma-25	582	9	88–104	88–104	PROPN
ma-25	582	10	.	.	PUNCT
ma-25	583	1	https://doi.org/10.1112/jlms/s2-37.121.88.[6	https://doi.org/10.1112/jlms/s2-37.121.88.[6	PROPN
ma-25	583	2	]	]	X
ma-25	583	3	g.	g.	PROPN
ma-25	583	4	g.	g.	PROPN
ma-25	583	5	gundersen	gundersen	PROPN
ma-25	583	6	,	,	PUNCT
ma-25	583	7	finite	finite	VERB
ma-25	583	8	order	order	NOUN
ma-25	583	9	solutions	solution	NOUN
ma-25	583	10	of	of	ADP
ma-25	583	11	second	second	ADJ
ma-25	583	12	order	order	NOUN
ma-25	583	13	linear	linear	NOUN
ma-25	583	14	differential	differential	NOUN
ma-25	583	15	equations	equation	NOUN
ma-25	583	16	.	.	PUNCT
ma-25	584	1	trans	trans	PROPN
ma-25	584	2	.	.	PUNCT
ma-25	585	1	amer	amer	PROPN
ma-25	585	2	.	.	PUNCT
ma-25	585	3	math	math	PROPN
ma-25	585	4	.	.	PUNCT
ma-25	586	1	soc	soc	PROPN
ma-25	586	2	.	.	PUNCT
ma-25	587	1	305(1988	305(1988	NUM
ma-25	587	2	)	)	PUNCT
ma-25	587	3	,	,	PUNCT
ma-25	587	4	415–429	415–429	NUM
ma-25	587	5	.	.	PUNCT
ma-25	588	1	https://doi.org/10.2307/2001061.[7	https://doi.org/10.2307/2001061.[7	X
ma-25	588	2	]	]	X
ma-25	588	3	a.	a.	NOUN
ma-25	588	4	a.	a.	PROPN
ma-25	588	5	goldberg	goldberg	PROPN
ma-25	588	6	and	and	CCONJ
ma-25	588	7	i.	i.	PROPN
ma-25	588	8	v.	v.	PROPN
ma-25	588	9	ostrovskii	ostrovskii	PROPN
ma-25	588	10	,	,	PUNCT
ma-25	588	11	the	the	DET
ma-25	588	12	distribution	distribution	NOUN
ma-25	588	13	of	of	ADP
ma-25	588	14	values	value	NOUN
ma-25	588	15	of	of	ADP
ma-25	588	16	meromorphic	meromorphic	ADJ
ma-25	588	17	functions	function	NOUN
ma-25	588	18	.	.	PUNCT
ma-25	589	1	irdat	irdat	PROPN
ma-25	589	2	nauk	nauk	PROPN
ma-25	589	3	,	,	PUNCT
ma-25	589	4	moscow	moscow	PROPN
ma-25	589	5	,	,	PUNCT
ma-25	589	6	1970(in	1970(in	NUM
ma-25	589	7	russian	russian	NOUN
ma-25	589	8	)	)	PUNCT
ma-25	589	9	,	,	PUNCT
ma-25	589	10	transl	transl	PROPN
ma-25	589	11	.	.	PUNCT
ma-25	589	12	math	math	PROPN
ma-25	589	13	.	.	PUNCT
ma-25	590	1	monogr	monogr	PROPN
ma-25	590	2	.	.	PUNCT
ma-25	591	1	,	,	PUNCT
ma-25	591	2	vol	vol	NOUN
ma-25	591	3	.	.	PROPN
ma-25	591	4	236	236	NUM
ma-25	591	5	,	,	PUNCT
ma-25	591	6	amer	amer	PROPN
ma-25	591	7	.	.	PROPN
ma-25	591	8	math	math	PROPN
ma-25	591	9	.	.	PUNCT
ma-25	592	1	soc	soc	PROPN
ma-25	592	2	.	.	PUNCT
ma-25	593	1	providence	providence	PROPN
ma-25	593	2	ri	ri	PROPN
ma-25	593	3	,	,	PUNCT
ma-25	593	4	2008.[8	2008.[8	NUM
ma-25	593	5	]	]	PUNCT
ma-25	593	6	k.	k.	PROPN
ma-25	593	7	hamani	hamani	PROPN
ma-25	593	8	and	and	CCONJ
ma-25	593	9	b.	b.	PROPN
ma-25	593	10	belaïdi	belaïdi	PROPN
ma-25	593	11	,	,	PUNCT
ma-25	593	12	growth	growth	NOUN
ma-25	593	13	of	of	ADP
ma-25	593	14	solutions	solution	NOUN
ma-25	593	15	of	of	ADP
ma-25	593	16	complex	complex	ADJ
ma-25	593	17	linear	linear	ADJ
ma-25	593	18	differential	differential	NOUN
ma-25	593	19	equations	equation	NOUN
ma-25	593	20	with	with	ADP
ma-25	593	21	entire	entire	ADJ
ma-25	593	22	coefficientsof	coefficientsof	NOUN
ma-25	593	23	finite	finite	NOUN
ma-25	593	24	iterated	iterate	VERB
ma-25	593	25	order	order	NOUN
ma-25	593	26	.	.	PUNCT
ma-25	594	1	acta	acta	PROPN
ma-25	594	2	univ	univ	PROPN
ma-25	594	3	.	.	PUNCT
ma-25	595	1	apulensis	apulensis	NOUN
ma-25	595	2	math	math	NOUN
ma-25	595	3	.	.	PUNCT
ma-25	596	1	inform	inform	NOUN
ma-25	596	2	.	.	PUNCT
ma-25	597	1	27	27	NUM
ma-25	597	2	(	(	PUNCT
ma-25	597	3	2011	2011	NUM
ma-25	597	4	)	)	PUNCT
ma-25	597	5	,	,	PUNCT
ma-25	597	6	203–216	203–216	NUM
ma-25	597	7	.	.	PUNCT
ma-25	598	1	http://emis.impa.br/emis/	http://emis.impa.br/emis/	PROPN
ma-25	598	2	journals	journals	PROPN
ma-25	598	3	/	/	SYM
ma-25	598	4	aua	aua	PROPN
ma-25	598	5	/	/	SYM
ma-25	598	6	acta27	acta27	PROPN
ma-25	598	7	/	/	PUNCT
ma-25	598	8	paper21	paper21	ADJ
ma-25	598	9	-	-	PUNCT
ma-25	598	10	acta27	acta27	NOUN
ma-25	598	11	-	-	PUNCT
ma-25	598	12	2011.pdf.[9	2011.pdf.[9	PROPN
ma-25	598	13	]	]	PUNCT
ma-25	598	14	w.	w.	PROPN
ma-25	598	15	k.	k.	PROPN
ma-25	598	16	hayman	hayman	PROPN
ma-25	598	17	,	,	PUNCT
ma-25	598	18	meromorphic	meromorphic	ADJ
ma-25	598	19	functions	function	NOUN
ma-25	598	20	.	.	PUNCT
ma-25	599	1	oxford	oxford	PROPN
ma-25	599	2	mathematical	mathematical	PROPN
ma-25	599	3	monographs	monograph	NOUN
ma-25	599	4	,	,	PUNCT
ma-25	599	5	clarendon	clarendon	PROPN
ma-25	599	6	press	press	NOUN
ma-25	599	7	,	,	PUNCT
ma-25	599	8	oxford	oxford	PROPN
ma-25	599	9	1964.[10	1964.[10	NUM
ma-25	599	10	]	]	X
ma-25	599	11	w.	w.	PROPN
ma-25	599	12	k.	k.	PROPN
ma-25	599	13	hayman	hayman	PROPN
ma-25	599	14	,	,	PUNCT
ma-25	599	15	the	the	DET
ma-25	599	16	local	local	ADJ
ma-25	599	17	growth	growth	NOUN
ma-25	599	18	of	of	ADP
ma-25	599	19	power	power	NOUN
ma-25	599	20	series	series	NOUN
ma-25	599	21	:	:	PUNCT
ma-25	599	22	a	a	DET
ma-25	599	23	survey	survey	NOUN
ma-25	599	24	of	of	ADP
ma-25	599	25	the	the	DET
ma-25	599	26	wiman	wiman	PROPN
ma-25	599	27	-	-	PUNCT
ma-25	599	28	valiron	valiron	PROPN
ma-25	599	29	method	method	NOUN
ma-25	599	30	.	.	PUNCT
ma-25	600	1	canad	canad	PROPN
ma-25	600	2	.	.	PUNCT
ma-25	601	1	math	math	NOUN
ma-25	601	2	.	.	PUNCT
ma-25	602	1	bull	bull	NOUN
ma-25	602	2	.	.	PUNCT
ma-25	603	1	17(1974	17(1974	NUM
ma-25	603	2	)	)	PUNCT
ma-25	603	3	,	,	PUNCT
ma-25	604	1	317–358	317–358	NUM
ma-25	604	2	.	.	PUNCT
ma-25	605	1	https://doi.org/10.4153/cmb-1974-064-0.[11	https://doi.org/10.4153/cmb-1974-064-0.[11	ADJ
ma-25	605	2	]	]	PUNCT
ma-25	605	3	h.	h.	PROPN
ma-25	605	4	hu	hu	PROPN
ma-25	605	5	and	and	CCONJ
ma-25	605	6	x.	x.	PROPN
ma-25	605	7	m.	m.	PROPN
ma-25	605	8	zheng	zheng	PROPN
ma-25	605	9	,	,	PUNCT
ma-25	605	10	growth	growth	NOUN
ma-25	605	11	of	of	ADP
ma-25	605	12	solutions	solution	NOUN
ma-25	605	13	to	to	PART
ma-25	605	14	linear	linear	VERB
ma-25	605	15	differential	differential	ADJ
ma-25	605	16	equations	equation	NOUN
ma-25	605	17	with	with	ADP
ma-25	605	18	entire	entire	ADJ
ma-25	605	19	coefficients	coefficient	NOUN
ma-25	605	20	.	.	PUNCT
ma-25	606	1	electron	electron	NOUN
ma-25	606	2	.	.	PUNCT
ma-25	607	1	j.differ	j.differ	NOUN
ma-25	607	2	.	.	PUNCT
ma-25	608	1	equations	equation	NOUN
ma-25	608	2	2012	2012	NUM
ma-25	608	3	(	(	PUNCT
ma-25	608	4	2012	2012	NUM
ma-25	608	5	)	)	PUNCT
ma-25	608	6	,	,	PUNCT
ma-25	608	7	226	226	NUM
ma-25	608	8	,	,	PUNCT
ma-25	608	9	15	15	NUM
ma-25	608	10	pp	pp	NOUN
ma-25	608	11	.	.	PUNCT
ma-25	609	1	https://ejde.math.txstate.edu/volumes/2012/226/hu.pdf	https://ejde.math.txstate.edu/volumes/2012/226/hu.pdf	PROPN
ma-25	609	2	.	.	PUNCT
ma-25	609	3	http://www.ilirias.com/jma/repository/docs/jma3-1-1.pdf	http://www.ilirias.com/jma/repository/docs/jma3-1-1.pdf	PROPN
ma-25	609	4	http://rj.romai.ro/arhiva/2015/1/rjv11n1.pdf	http://rj.romai.ro/arhiva/2015/1/rjv11n1.pdf	PROPN
ma-25	609	5	http://rjm-cs.ro/belaidi-2015.pdf	http://rjm-cs.ro/belaidi-2015.pdf	PROPN
ma-25	609	6	https://doi.org/10.23952/cot.2017.17	https://doi.org/10.23952/cot.2017.17	ADJ
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ma-25	609	8	https://doi.org/10.2307/2001061	https://doi.org/10.2307/2001061	PROPN
ma-25	609	9	http://emis.impa.br/emis/journals/aua/acta27/paper21-acta27-2011.pdf	http://emis.impa.br/emis/journals/aua/acta27/paper21-acta27-2011.pdf	NOUN
ma-25	609	10	http://emis.impa.br/emis/journals/aua/acta27/paper21-acta27-2011.pdf	http://emis.impa.br/emis/journals/aua/acta27/paper21-acta27-2011.pdf	NOUN
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ma-25	609	12	https://ejde.math.txstate.edu/volumes/2012/226/hu.pdf	https://ejde.math.txstate.edu/volumes/2012/226/hu.pdf	PROPN
ma-25	609	13	eur	eur	PROPN
ma-25	609	14	.	.	PUNCT
ma-25	610	1	j.	j.	PROPN
ma-25	610	2	math	math	PROPN
ma-25	610	3	.	.	PUNCT
ma-25	611	1	anal	anal	ADJ
ma-25	611	2	.	.	PUNCT
ma-25	612	1	1	1	NUM
ma-25	612	2	(	(	PUNCT
ma-25	612	3	2021	2021	NUM
ma-25	612	4	)	)	PUNCT
ma-25	612	5	105	105	NUM
ma-25	613	1	[	[	X
ma-25	613	2	12	12	NUM
ma-25	613	3	]	]	PUNCT
ma-25	613	4	o.	o.	PROPN
ma-25	613	5	p.	p.	PROPN
ma-25	613	6	juneja	juneja	PROPN
ma-25	613	7	,	,	PUNCT
ma-25	613	8	g.	g.	PROPN
ma-25	613	9	p.	p.	PROPN
ma-25	613	10	kapoor	kapoor	PROPN
ma-25	613	11	and	and	CCONJ
ma-25	613	12	s.	s.	PROPN
ma-25	613	13	k.	k.	PROPN
ma-25	613	14	bajpai	bajpai	PROPN
ma-25	613	15	,	,	PUNCT
ma-25	613	16	on	on	ADP
ma-25	613	17	the	the	DET
ma-25	613	18	(	(	PUNCT
ma-25	613	19	p	p	NOUN
ma-25	613	20	,	,	PUNCT
ma-25	613	21	q)-order	q)-order	NOUN
ma-25	613	22	and	and	CCONJ
ma-25	613	23	lower	low	ADJ
ma-25	613	24	(	(	PUNCT
ma-25	613	25	p	p	NOUN
ma-25	613	26	,	,	PUNCT
ma-25	613	27	q)-order	q)-order	NOUN
ma-25	613	28	of	of	ADP
ma-25	613	29	an	an	DET
ma-25	613	30	entire	entire	ADJ
ma-25	613	31	function	function	NOUN
ma-25	613	32	.	.	PUNCT
ma-25	614	1	j.reine	j.reine	PROPN
ma-25	614	2	angew	angew	PROPN
ma-25	614	3	.	.	PUNCT
ma-25	615	1	math	math	NOUN
ma-25	615	2	.	.	PUNCT
ma-25	616	1	282	282	NUM
ma-25	616	2	(	(	PUNCT
ma-25	616	3	1976	1976	NUM
ma-25	616	4	)	)	PUNCT
ma-25	616	5	,	,	PUNCT
ma-25	616	6	53–67	53–67	NUM
ma-25	616	7	.	.	PUNCT
ma-25	617	1	https://doi.org/10.1515/crll.1976.282.53.[13	https://doi.org/10.1515/crll.1976.282.53.[13	PROPN
ma-25	617	2	]	]	X
ma-25	617	3	l.	l.	PROPN
ma-25	617	4	kinnunen	kinnunen	PROPN
ma-25	617	5	,	,	PUNCT
ma-25	617	6	linear	linear	ADJ
ma-25	617	7	differential	differential	ADJ
ma-25	617	8	equations	equation	NOUN
ma-25	617	9	with	with	ADP
ma-25	617	10	solutions	solution	NOUN
ma-25	617	11	of	of	ADP
ma-25	617	12	finite	finite	ADJ
ma-25	617	13	iterated	iterate	VERB
ma-25	617	14	order	order	NOUN
ma-25	617	15	.	.	PUNCT
ma-25	618	1	southeast	southeast	ADJ
ma-25	618	2	asian	asian	ADJ
ma-25	618	3	bull	bull	PROPN
ma-25	618	4	.	.	PUNCT
ma-25	619	1	math	math	NOUN
ma-25	619	2	.	.	PUNCT
ma-25	620	1	22(1998	22(1998	NUM
ma-25	620	2	)	)	PUNCT
ma-25	620	3	,	,	PUNCT
ma-25	620	4	385–405.[14	385–405.[14	PROPN
ma-25	620	5	]	]	PUNCT
ma-25	620	6	i.	i.	PROPN
ma-25	620	7	laine	laine	PROPN
ma-25	620	8	,	,	PUNCT
ma-25	620	9	nevanlinna	nevanlinna	NOUN
ma-25	620	10	theory	theory	NOUN
ma-25	620	11	and	and	CCONJ
ma-25	620	12	complex	complex	ADJ
ma-25	620	13	differential	differential	ADJ
ma-25	620	14	equations	equation	NOUN
ma-25	620	15	.	.	PUNCT
ma-25	621	1	de	de	PROPN
ma-25	621	2	gruyter	gruyter	NOUN
ma-25	621	3	studies	study	NOUN
ma-25	621	4	in	in	ADP
ma-25	621	5	mathematics	mathematic	NOUN
ma-25	621	6	,	,	PUNCT
ma-25	621	7	15	15	NUM
ma-25	621	8	.	.	PUNCT
ma-25	621	9	walter	walter	PROPN
ma-25	621	10	degruyter	degruyter	PROPN
ma-25	621	11	&	&	CCONJ
ma-25	621	12	co.	co.	PROPN
ma-25	621	13	,	,	PUNCT
ma-25	621	14	berlin	berlin	PROPN
ma-25	621	15	,	,	PUNCT
ma-25	621	16	1993	1993	NUM
ma-25	621	17	.	.	PUNCT
ma-25	622	1	https://doi.org/10.1515/9783110863147.[15	https://doi.org/10.1515/9783110863147.[15	PROPN
ma-25	622	2	]	]	PUNCT
ma-25	622	3	l.	l.	PROPN
ma-25	622	4	m.	m.	PROPN
ma-25	622	5	li	li	PROPN
ma-25	622	6	and	and	CCONJ
ma-25	622	7	t.	t.	PROPN
ma-25	622	8	b.	b.	PROPN
ma-25	622	9	cao	cao	PROPN
ma-25	622	10	,	,	PUNCT
ma-25	622	11	solutions	solution	NOUN
ma-25	622	12	for	for	ADP
ma-25	622	13	linear	linear	PROPN
ma-25	622	14	differential	differential	ADJ
ma-25	622	15	equations	equation	NOUN
ma-25	622	16	with	with	ADP
ma-25	622	17	meromorphic	meromorphic	ADJ
ma-25	622	18	coefficients	coefficient	NOUN
ma-25	622	19	of	of	ADP
ma-25	622	20	[	[	X
ma-25	622	21	p	p	X
ma-25	622	22	,	,	PUNCT
ma-25	622	23	q]-order	q]-order	NOUN
ma-25	622	24	inthe	inthe	DET
ma-25	622	25	plane	plane	NOUN
ma-25	622	26	.	.	PUNCT
ma-25	623	1	electron	electron	PROPN
ma-25	623	2	.	.	PUNCT
ma-25	624	1	j.	j.	PROPN
ma-25	624	2	differ	differ	VERB
ma-25	624	3	.	.	PUNCT
ma-25	625	1	equations	equation	NOUN
ma-25	625	2	2012	2012	NUM
ma-25	625	3	(	(	PUNCT
ma-25	625	4	2012	2012	NUM
ma-25	625	5	)	)	PUNCT
ma-25	625	6	,	,	PUNCT
ma-25	625	7	195	195	NUM
ma-25	625	8	,	,	PUNCT
ma-25	625	9	15	15	NUM
ma-25	625	10	pp	pp	NOUN
ma-25	625	11	.	.	PUNCT
ma-25	626	1	https://ejde.math.txstate.edu/volumes/	https://ejde.math.txstate.edu/volumes/	PROPN
ma-25	626	2	2012/195	2012/195	NUM
ma-25	626	3	/	/	SYM
ma-25	626	4	li.pdf.[16	li.pdf.[16	PROPN
ma-25	626	5	]	]	X
ma-25	626	6	j.	j.	PROPN
ma-25	626	7	liu	liu	PROPN
ma-25	626	8	,	,	PUNCT
ma-25	626	9	j.	j.	PROPN
ma-25	626	10	tu	tu	PROPN
ma-25	626	11	and	and	CCONJ
ma-25	626	12	l.	l.	PROPN
ma-25	626	13	z.	z.	PROPN
ma-25	626	14	shi	shi	PROPN
ma-25	626	15	,	,	PUNCT
ma-25	626	16	linear	linear	PROPN
ma-25	626	17	differential	differential	ADJ
ma-25	626	18	equations	equation	NOUN
ma-25	626	19	with	with	ADP
ma-25	626	20	entire	entire	ADJ
ma-25	626	21	coefficients	coefficient	NOUN
ma-25	626	22	of	of	ADP
ma-25	626	23	[	[	X
ma-25	626	24	p	p	X
ma-25	626	25	,	,	PUNCT
ma-25	626	26	q]-order	q]-order	NOUN
ma-25	626	27	in	in	ADP
ma-25	626	28	the	the	DET
ma-25	626	29	complex	complex	ADJ
ma-25	626	30	plane.j	plane.j	PROPN
ma-25	626	31	.	.	PUNCT
ma-25	626	32	math	math	NOUN
ma-25	626	33	.	.	PUNCT
ma-25	627	1	anal	anal	PROPN
ma-25	627	2	.	.	PUNCT
ma-25	627	3	appl	appl	PROPN
ma-25	627	4	.	.	PUNCT
ma-25	628	1	372	372	NUM
ma-25	628	2	(	(	PUNCT
ma-25	628	3	2010	2010	NUM
ma-25	628	4	)	)	PUNCT
ma-25	628	5	,	,	PUNCT
ma-25	628	6	55–67	55–67	NUM
ma-25	628	7	.	.	PUNCT
ma-25	629	1	https://doi.org/10.1016/j.jmaa.2010.05.014.[17	https://doi.org/10.1016/j.jmaa.2010.05.014.[17	PROPN
ma-25	629	2	]	]	X
ma-25	629	3	m.	m.	NOUN
ma-25	629	4	saidani	saidani	PROPN
ma-25	629	5	and	and	CCONJ
ma-25	629	6	b.	b.	PROPN
ma-25	629	7	belaïdi	belaïdi	PROPN
ma-25	629	8	,	,	PUNCT
ma-25	629	9	oscillation	oscillation	NOUN
ma-25	629	10	of	of	ADP
ma-25	629	11	solutions	solution	NOUN
ma-25	629	12	and	and	CCONJ
ma-25	629	13	their	their	PRON
ma-25	629	14	arbitrary	arbitrary	ADJ
ma-25	629	15	order	order	NOUN
ma-25	629	16	derivatives	derivative	NOUN
ma-25	629	17	of	of	ADP
ma-25	629	18	higher	high	ADJ
ma-25	629	19	or	or	CCONJ
ma-25	629	20	-	-	PUNCT
ma-25	629	21	der	der	ADJ
ma-25	629	22	non	non	ADJ
ma-25	629	23	-	-	ADJ
ma-25	629	24	homogeneous	homogeneous	ADJ
ma-25	629	25	lde	lde	PROPN
ma-25	629	26	.	.	PUNCT
ma-25	629	27	scientific	scientific	ADJ
ma-25	629	28	publications	publication	NOUN
ma-25	629	29	of	of	ADP
ma-25	629	30	the	the	DET
ma-25	629	31	state	state	PROPN
ma-25	629	32	university	university	PROPN
ma-25	629	33	of	of	ADP
ma-25	629	34	novi	novi	PROPN
ma-25	629	35	pazar	pazar	PROPN
ma-25	629	36	,	,	PUNCT
ma-25	629	37	ser	ser	NOUN
ma-25	629	38	.	.	PUNCT
ma-25	630	1	a	a	DET
ma-25	630	2	:	:	PUNCT
ma-25	630	3	appl.math	appl.math	NOUN
ma-25	630	4	.	.	PUNCT
ma-25	630	5	inform	inform	NOUN
ma-25	630	6	.	.	PUNCT
ma-25	631	1	and	and	CCONJ
ma-25	631	2	mech	mech	NOUN
ma-25	631	3	.	.	PUNCT
ma-25	632	1	9	9	NUM
ma-25	632	2	(	(	PUNCT
ma-25	632	3	2017	2017	NUM
ma-25	632	4	)	)	PUNCT
ma-25	632	5	,	,	PUNCT
ma-25	633	1	103–126	103–126	NUM
ma-25	633	2	.	.	PUNCT
ma-25	634	1	https://scindeks-clanci.ceon.rs/data/pdf/2217-5539/2017/	https://scindeks-clanci.ceon.rs/data/pdf/2217-5539/2017/	NOUN
ma-25	634	2	2217	2217	NUM
ma-25	634	3	-	-	SYM
ma-25	634	4	55391702103s.pdf.[18	55391702103s.pdf.[18	NUM
ma-25	634	5	]	]	PUNCT
ma-25	634	6	m.	m.	NOUN
ma-25	634	7	saidani	saidani	PROPN
ma-25	634	8	and	and	CCONJ
ma-25	634	9	b.	b.	PROPN
ma-25	634	10	belaïdi	belaïdi	PROPN
ma-25	634	11	,	,	PUNCT
ma-25	634	12	on	on	ADP
ma-25	634	13	the	the	DET
ma-25	634	14	growth	growth	NOUN
ma-25	634	15	of	of	ADP
ma-25	634	16	solutions	solution	NOUN
ma-25	634	17	of	of	ADP
ma-25	634	18	homogeneous	homogeneous	ADJ
ma-25	634	19	and	and	CCONJ
ma-25	634	20	non	non	ADJ
ma-25	634	21	-	-	ADJ
ma-25	634	22	homogeneous	homogeneous	ADJ
ma-25	634	23	linear	linear	PROPN
ma-25	634	24	differentialequations	differentialequation	NOUN
ma-25	634	25	with	with	ADP
ma-25	634	26	meromorphic	meromorphic	ADJ
ma-25	634	27	coefficients	coefficient	NOUN
ma-25	634	28	.	.	PUNCT
ma-25	635	1	sci	sci	PROPN
ma-25	635	2	.	.	PROPN
ma-25	635	3	stud	stud	PROPN
ma-25	635	4	.	.	PUNCT
ma-25	636	1	res	re	NOUN
ma-25	636	2	.	.	PUNCT
ma-25	636	3	ser	ser	PROPN
ma-25	636	4	.	.	PROPN
ma-25	636	5	math	math	PROPN
ma-25	636	6	.	.	PUNCT
ma-25	637	1	inform	inform	NOUN
ma-25	637	2	.	.	PUNCT
ma-25	638	1	28	28	NUM
ma-25	638	2	(	(	PUNCT
ma-25	638	3	2018	2018	NUM
ma-25	638	4	)	)	PUNCT
ma-25	638	5	,	,	PUNCT
ma-25	638	6	131–146	131–146	NUM
ma-25	638	7	.	.	PUNCT
ma-25	639	1	https://pubs	https://pub	NOUN
ma-25	639	2	.	.	PUNCT
ma-25	640	1	ub.ro/?pg=revues&rev=ssrsmi&num=201801&vol=28&aid=4817.[19	ub.ro/?pg=revues&rev=ssrsmi&num=201801&vol=28&aid=4817.[19	PROPN
ma-25	640	2	]	]	PUNCT
ma-25	640	3	m.	m.	NOUN
ma-25	640	4	saidani	saidani	PROPN
ma-25	640	5	and	and	CCONJ
ma-25	640	6	b.	b.	PROPN
ma-25	640	7	belaïdi	belaïdi	PROPN
ma-25	640	8	,	,	PUNCT
ma-25	640	9	meromorphic	meromorphic	ADJ
ma-25	640	10	solutions	solution	NOUN
ma-25	640	11	to	to	PART
ma-25	640	12	linear	linear	VERB
ma-25	640	13	differential	differential	ADJ
ma-25	640	14	equations	equation	NOUN
ma-25	640	15	with	with	ADP
ma-25	640	16	entire	entire	ADJ
ma-25	640	17	coefficients	coefficient	NOUN
ma-25	640	18	of	of	ADP
ma-25	640	19	[	[	X
ma-25	640	20	p	p	X
ma-25	640	21	,	,	PUNCT
ma-25	640	22	q]-order	q]-order	NOUN
ma-25	640	23	.	.	PUNCT
ma-25	641	1	j.	j.	PROPN
ma-25	641	2	dyn	dyn	PROPN
ma-25	641	3	.	.	PUNCT
ma-25	642	1	syst	syst	PROPN
ma-25	642	2	.	.	PUNCT
ma-25	643	1	geom	geom	PROPN
ma-25	643	2	.	.	PUNCT
ma-25	644	1	theor	theor	PROPN
ma-25	644	2	.	.	PUNCT
ma-25	645	1	16	16	NUM
ma-25	645	2	(	(	PUNCT
ma-25	645	3	2018	2018	NUM
ma-25	645	4	)	)	PUNCT
ma-25	645	5	,	,	PUNCT
ma-25	645	6	33–53	33–53	NUM
ma-25	645	7	.	.	PUNCT
ma-25	646	1	https://doi.org/10.1080/1726037x.2017.1413065.[20	https://doi.org/10.1080/1726037x.2017.1413065.[20	PROPN
ma-25	646	2	]	]	PUNCT
ma-25	647	1	j.	j.	PROPN
ma-25	647	2	tu	tu	PROPN
ma-25	647	3	and	and	CCONJ
ma-25	647	4	z.	z.	PROPN
ma-25	647	5	x.	x.	PROPN
ma-25	647	6	chen	chen	PROPN
ma-25	647	7	,	,	PUNCT
ma-25	647	8	growth	growth	NOUN
ma-25	647	9	of	of	ADP
ma-25	647	10	solutions	solution	NOUN
ma-25	647	11	of	of	ADP
ma-25	647	12	complex	complex	ADJ
ma-25	647	13	differential	differential	ADJ
ma-25	647	14	equations	equation	NOUN
ma-25	647	15	with	with	ADP
ma-25	647	16	meromorphic	meromorphic	ADJ
ma-25	647	17	coefficients	coefficient	NOUN
ma-25	647	18	of	of	ADP
ma-25	647	19	finiteiterated	finiteiterate	VERB
ma-25	647	20	order	order	NOUN
ma-25	647	21	.	.	PUNCT
ma-25	648	1	southeast	southeast	ADJ
ma-25	648	2	asian	asian	ADJ
ma-25	648	3	bull	bull	PROPN
ma-25	648	4	.	.	PUNCT
ma-25	649	1	math	math	NOUN
ma-25	649	2	.	.	PUNCT
ma-25	650	1	33	33	NUM
ma-25	650	2	(	(	PUNCT
ma-25	650	3	2009	2009	NUM
ma-25	650	4	)	)	PUNCT
ma-25	650	5	,	,	PUNCT
ma-25	650	6	153–164.[21	153–164.[21	PROPN
ma-25	650	7	]	]	PUNCT
ma-25	650	8	j.	j.	PROPN
ma-25	650	9	tu	tu	PROPN
ma-25	650	10	and	and	CCONJ
ma-25	650	11	t.	t.	PROPN
ma-25	650	12	long	long	ADJ
ma-25	650	13	,	,	PUNCT
ma-25	650	14	oscillation	oscillation	NOUN
ma-25	650	15	of	of	ADP
ma-25	650	16	complex	complex	ADJ
ma-25	650	17	high	high	ADJ
ma-25	650	18	order	order	NOUN
ma-25	650	19	linear	linear	NOUN
ma-25	650	20	differential	differential	NOUN
ma-25	650	21	equations	equation	NOUN
ma-25	650	22	with	with	ADP
ma-25	650	23	coefficients	coefficient	NOUN
ma-25	650	24	of	of	ADP
ma-25	650	25	finite	finite	PROPN
ma-25	650	26	iteratedorder	iteratedorder	NOUN
ma-25	650	27	.	.	PUNCT
ma-25	651	1	electron	electron	PROPN
ma-25	651	2	.	.	PUNCT
ma-25	652	1	j.	j.	PROPN
ma-25	652	2	qual	qual	PROPN
ma-25	652	3	.	.	PROPN
ma-25	652	4	theory	theory	NOUN
ma-25	652	5	differ	differ	VERB
ma-25	652	6	.	.	PUNCT
ma-25	653	1	equ	equ	PROPN
ma-25	653	2	.	.	PROPN
ma-25	653	3	2009	2009	NUM
ma-25	653	4	(	(	PUNCT
ma-25	653	5	2009	2009	NUM
ma-25	653	6	)	)	PUNCT
ma-25	653	7	,	,	PUNCT
ma-25	653	8	66	66	NUM
ma-25	653	9	,	,	PUNCT
ma-25	653	10	1–13	1–13	NOUN
ma-25	653	11	.	.	PUNCT
ma-25	654	1	https://www.math.u-szeged.hu/ejqtde/	https://www.math.u-szeged.hu/ejqtde/	PROPN
ma-25	654	2	p453.pdf.[22	p453.pdf.[22	PROPN
ma-25	654	3	]	]	X
ma-25	654	4	g.	g.	PROPN
ma-25	654	5	valiron	valiron	PROPN
ma-25	654	6	,	,	PUNCT
ma-25	654	7	lectures	lecture	VERB
ma-25	654	8	on	on	ADP
ma-25	654	9	the	the	DET
ma-25	654	10	general	general	ADJ
ma-25	654	11	theory	theory	NOUN
ma-25	654	12	of	of	ADP
ma-25	654	13	integral	integral	ADJ
ma-25	654	14	functions	function	NOUN
ma-25	654	15	.	.	PUNCT
ma-25	655	1	translated	translate	VERB
ma-25	655	2	by	by	ADP
ma-25	655	3	e.	e.	PROPN
ma-25	655	4	f.	f.	PROPN
ma-25	655	5	collingwood	collingwood	PROPN
ma-25	655	6	,	,	PUNCT
ma-25	655	7	chelsea	chelsea	PROPN
ma-25	655	8	,	,	PUNCT
ma-25	655	9	newyork	newyork	NOUN
ma-25	655	10	,	,	PUNCT
ma-25	655	11	1949.[23	1949.[23	NUM
ma-25	655	12	]	]	X
ma-25	655	13	h.	h.	PROPN
ma-25	655	14	y.	y.	PROPN
ma-25	655	15	xu	xu	PROPN
ma-25	655	16	,	,	PUNCT
ma-25	655	17	j.	j.	PROPN
ma-25	655	18	tu	tu	PROPN
ma-25	655	19	and	and	CCONJ
ma-25	655	20	z.	z.	PROPN
ma-25	655	21	x.	x.	PROPN
ma-25	655	22	xuan	xuan	PROPN
ma-25	655	23	,	,	PUNCT
ma-25	655	24	the	the	DET
ma-25	655	25	oscillation	oscillation	NOUN
ma-25	655	26	on	on	ADP
ma-25	655	27	solutions	solution	NOUN
ma-25	655	28	of	of	ADP
ma-25	655	29	some	some	DET
ma-25	655	30	classes	class	NOUN
ma-25	655	31	of	of	ADP
ma-25	655	32	linear	linear	PROPN
ma-25	655	33	differential	differential	ADJ
ma-25	655	34	equations	equation	NOUN
ma-25	655	35	withmeromorphic	withmeromorphic	ADJ
ma-25	655	36	coefficients	coefficient	NOUN
ma-25	655	37	of	of	ADP
ma-25	655	38	finite	finite	NOUN
ma-25	655	39	[	[	X
ma-25	655	40	p	p	X
ma-25	655	41	,	,	PUNCT
ma-25	655	42	q]-order	q]-order	NOUN
ma-25	655	43	.	.	PUNCT
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ma-25	656	2	.	.	PROPN
ma-25	656	3	world	world	PROPN
ma-25	656	4	j.	j.	PROPN
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ma-25	656	6	(	(	PUNCT
ma-25	656	7	2013	2013	NUM
ma-25	656	8	)	)	PUNCT
ma-25	656	9	,	,	PUNCT
ma-25	656	10	article	article	NOUN
ma-25	656	11	i	i	PROPN
ma-25	656	12	d	d	PROPN
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ma-25	656	14	,	,	PUNCT
ma-25	656	15	8	8	NUM
ma-25	656	16	pages	page	NOUN
ma-25	656	17	.	.	PUNCT
ma-25	657	1	https	https	NOUN
ma-25	657	2	:	:	PUNCT
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ma-25	657	4	]	]	X
ma-25	657	5	c.	c.	PROPN
ma-25	657	6	c.	c.	PROPN
ma-25	657	7	yang	yang	PROPN
ma-25	657	8	and	and	CCONJ
ma-25	657	9	h.	h.	PROPN
ma-25	657	10	x.	x.	PROPN
ma-25	657	11	yi	yi	PROPN
ma-25	657	12	,	,	PUNCT
ma-25	657	13	uniqueness	uniqueness	NOUN
ma-25	657	14	theory	theory	NOUN
ma-25	657	15	of	of	ADP
ma-25	657	16	meromorphic	meromorphic	ADJ
ma-25	657	17	functions	function	NOUN
ma-25	657	18	.	.	PUNCT
ma-25	658	1	mathematics	mathematic	NOUN
ma-25	658	2	and	and	CCONJ
ma-25	658	3	its	its	PRON
ma-25	658	4	applications	application	NOUN
ma-25	658	5	,	,	PUNCT
ma-25	658	6	557.kluwer	557.kluwer	NUM
ma-25	658	7	academic	academic	ADJ
ma-25	658	8	publishers	publisher	NOUN
ma-25	658	9	group	group	NOUN
ma-25	658	10	,	,	PUNCT
ma-25	658	11	dordrecht	dordrecht	PROPN
ma-25	658	12	,	,	PUNCT
ma-25	658	13	2003.[25	2003.[25	PROPN
ma-25	658	14	]	]	X
ma-25	658	15	m.	m.	NOUN
ma-25	658	16	l.	l.	PROPN
ma-25	658	17	zhan	zhan	PROPN
ma-25	658	18	and	and	CCONJ
ma-25	658	19	x.	x.	PROPN
ma-25	658	20	m.	m.	PROPN
ma-25	658	21	zheng	zheng	PROPN
ma-25	658	22	,	,	PUNCT
ma-25	658	23	solutions	solution	NOUN
ma-25	658	24	to	to	PART
ma-25	658	25	linear	linear	VERB
ma-25	658	26	differential	differential	ADJ
ma-25	658	27	equations	equation	NOUN
ma-25	658	28	with	with	ADP
ma-25	658	29	some	some	DET
ma-25	658	30	coefficient	coefficient	NOUN
ma-25	658	31	being	be	AUX
ma-25	658	32	lacunary	lacunary	ADJ
ma-25	658	33	seriesof	seriesof	NOUN
ma-25	658	34	[	[	X
ma-25	658	35	p	p	X
ma-25	658	36	,	,	PUNCT
ma-25	658	37	q]-order	q]-order	NOUN
ma-25	658	38	in	in	ADP
ma-25	658	39	the	the	DET
ma-25	658	40	complex	complex	ADJ
ma-25	658	41	plane	plane	NOUN
ma-25	658	42	.	.	PUNCT
ma-25	659	1	ann	ann	PROPN
ma-25	659	2	.	.	PROPN
ma-25	660	1	differ	differ	VERB
ma-25	660	2	.	.	PUNCT
ma-25	661	1	equations	equation	NOUN
ma-25	661	2	30	30	NUM
ma-25	661	3	(	(	PUNCT
ma-25	661	4	2014	2014	NUM
ma-25	661	5	)	)	PUNCT
ma-25	661	6	,	,	PUNCT
ma-25	661	7	364–372	364–372	NUM
ma-25	661	8	.	.	PUNCT
ma-25	662	1	https://doi.org/10.1515/crll.1976.282.53	https://doi.org/10.1515/crll.1976.282.53	NOUN
ma-25	662	2	https://doi.org/10.1515/9783110863147	https://doi.org/10.1515/9783110863147	NUM
ma-25	663	1	https://ejde.math.txstate.edu/volumes/2012/195/li.pdf	https://ejde.math.txstate.edu/volumes/2012/195/li.pdf	PROPN
ma-25	663	2	https://ejde.math.txstate.edu/volumes/2012/195/li.pdf	https://ejde.math.txstate.edu/volumes/2012/195/li.pdf	NOUN
ma-25	663	3	https://doi.org/10.1016/j.jmaa.2010.05.014	https://doi.org/10.1016/j.jmaa.2010.05.014	NOUN
ma-25	663	4	https://scindeks-clanci.ceon.rs/data/pdf/2217-5539/2017/2217-55391702103s.pdf	https://scindeks-clanci.ceon.rs/data/pdf/2217-5539/2017/2217-55391702103s.pdf	ADJ
ma-25	663	5	https://scindeks-clanci.ceon.rs/data/pdf/2217-5539/2017/2217-55391702103s.pdf	https://scindeks-clanci.ceon.rs/data/pdf/2217-5539/2017/2217-55391702103s.pdf	ADJ
ma-25	663	6	https://pubs.ub.ro/?pg=revues&rev=ssrsmi&num=201801&vol=28&aid=4817	https://pubs.ub.ro/?pg=revues&rev=ssrsmi&num=201801&vol=28&aid=4817	PROPN
ma-25	663	7	https://pubs.ub.ro/?pg=revues&rev=ssrsmi&num=201801&vol=28&aid=4817	https://pubs.ub.ro/?pg=revues&rev=ssrsmi&num=201801&vol=28&aid=4817	PROPN
ma-25	664	1	https://doi.org/10.1080/1726037x.2017.1413065	https://doi.org/10.1080/1726037x.2017.1413065	PROPN
ma-25	664	2	https://www.math.u-szeged.hu/ejqtde/p453.pdf	https://www.math.u-szeged.hu/ejqtde/p453.pdf	PROPN
ma-25	664	3	https://www.math.u-szeged.hu/ejqtde/p453.pdf	https://www.math.u-szeged.hu/ejqtde/p453.pdf	PROPN
ma-25	664	4	https://doi.org/10.1155/2013/243873	https://doi.org/10.1155/2013/243873	VERB
ma-25	664	5	https://doi.org/10.1155/2013/243873	https://doi.org/10.1155/2013/243873	VERB
ma-25	664	6	1	1	NUM
ma-25	664	7	.	.	PUNCT
ma-25	664	8	introduction	introduction	NOUN
ma-25	664	9	and	and	CCONJ
ma-25	664	10	main	main	ADJ
ma-25	664	11	results	result	NOUN
ma-25	664	12	2	2	NUM
ma-25	664	13	.	.	X
ma-25	665	1	some	some	DET
ma-25	665	2	auxiliary	auxiliary	NOUN
ma-25	665	3	lemmas	lemma	VERB
ma-25	665	4	3	3	NUM
ma-25	665	5	.	.	PUNCT
ma-25	666	1	proof	proof	NOUN
ma-25	666	2	of	of	ADP
ma-25	666	3	theorem	theorem	ADJ
ma-25	666	4	1.1	1.1	NUM
ma-25	666	5	4	4	NUM
ma-25	666	6	.	.	PUNCT
ma-25	667	1	proof	proof	NOUN
ma-25	667	2	of	of	ADP
ma-25	667	3	corollary	corollary	ADJ
ma-25	667	4	1.1	1.1	NUM
ma-25	667	5	5	5	NUM
ma-25	667	6	.	.	PUNCT
ma-25	668	1	proof	proof	NOUN
ma-25	668	2	of	of	ADP
ma-25	668	3	theorem	theorem	ADJ
ma-25	668	4	1.2	1.2	NUM
ma-25	668	5	6	6	NUM
ma-25	668	6	.	.	PUNCT
ma-25	669	1	proof	proof	NOUN
ma-25	669	2	of	of	ADP
ma-25	669	3	corollary	corollary	ADJ
ma-25	669	4	1.2	1.2	NUM
ma-25	669	5	references	reference	NOUN