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Санкт-Петербургский государственный университет телекоммуникаций им. проф. М.А. Бонч-Бруевича

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Сборник задач по высшей математике 2 том

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7.4.3.	HaihH p83JIO:>KeHHe <PyHKI.J;HH	2	B PH,n: JIopaHa B oco6oit
		z(3 -z)

TOqKe Zo = O. YKa3aTb rJIaBHYIO H rrpaBHJIbHYIO qacTH pH,n:a H em 06JIaCTb CXO,n:HMOCTH.

a IIpe,n:BapHTeJIbHo rrpe,n:CTaBHM ,n:aHHYIO ,n:p06b B BH,n:e CYMMbI ,n:BYX rrpa-

cTeitwHx ,n:po6eit

2 _A+-.lL

			z(3-z) - z		3-z'
Hait,n:eM qHCJIa A H B:
			2	A(3 -	z) + Bz
			z(3 - z)	z(3 - z)
CJIe,n:OBaTeJIbHO, o· z + 2 = (B - A)z + 3A, oTKy,n:a
		{o= B - A,		T. e.	2
		{o= B - A,		T. e.	A = B = -3 •
		2=3A,
2		2/3	2/3
IITaK, (	)	= - z + -3-'
z3-z			- z
		2/3	Y:>Ke rrpe,n:CTaBJIeHa B BH,n:e CYMMbI (CocToHw;eit H3
TaK KaK ,n:p06b --:z-			Y:>Ke rrpe,n:CTaBJIeHa B BH,n:e CYMMbI (CocToHw;eit H3

o,n:Horo CJIaraeMoro) qJIeHOB BH,n:a cnzn , TO OCTaeTCH HaitTH pa3JIO:>KeHHe ,n:po-

2/3			(4.4)		(B Kpyre Iz I < 1) H
6H 3 _ z· ,I1;JIH ::noro BOCrrOJIb3yeMcH pa3JIO:>KeHHeM			(4.4)		(B Kpyre Iz I < 1) H
rrOJIyqHM:
32~3z ~ 3 ( :~f) ~ /3 ( 1 + j + mz + ... + (jr+.) ~
2	2	2	2		2	n	+ ...
= 32	+ 33 Z + 34 Z			+ ... + 3n +2 z			+ ...

8TOT pH,n: (6eCKOHeqHO y6bIBaIOW;M reOMeTpHqeCKaH rrporpeccHH) CXO,n:HTCH rrpH Iql = I~I < 1, T.e.B OTKPbITOM Kpyre I~I < 1, HJIH Izl < 3.

Terrepb 3arrHweM PH,n: JIopaHa ,n:JIH HCXO,n:HOit ,n:p06H:

	2	2/3 2	2	2	2	~ 2	n
	z(3 -	z) = --:z- + 32	+ 33z + 34z			+ ... = n~l 3n +2z .
06JIacTb CXO,n:HMOCTH 9Toro pH,n:a -				KOJIbI.J;O 0 < Izl < 3. IIepBoe CJIaraeMOe,
2/3							rrpaBHJIb-
--:z-, HBJIHeTCH rJIaBHoit qaCTbIO pH,n:a, OCTaBWMCH qacTb pH,n:a -							rrpaBHJIb-
HOit.							•
7.4.4.	HaitTH pa3JIO:>KeHHe <PyHKI.J;HH				(2	) B PH,n: JIopaHa B oKpecTHa-
				z3-z

CTH 6eCKOHeqHO y,n:aJIeHHoit TOqKH. YKa3aTb rJIaBHYIO H rrpaBHJIbHyIO qacTH pH,n:a Hero 06JIaCTb CXO,n:HMOCTH.

470

a 'MbIY)Ke 3HaeM (3a,l1,aqa 7.4.3), QTO								2	2/3	2/3
a 'MbIY)Ke 3HaeM (3a,l1,aqa 7.4.3), QTO								z(3 _ z) = ----z + 3 _ z' 3arrHmeM
pa3JIO)KeHHe <PYHKIJ,HH					2/3	B 6ecKOHeQHO y6bIBaIOIIIyIO reOMeTpHQeCKYIO
pa3JIO)KeHHe <PYHKIJ,HH				-3-		B 6ecKOHeQHO y6bIBaIOIIIyIO reOMeTpHQeCKYIO
					-z			«BCJIHKO», T. e. l- «MaJIO»:
rrporpeCCHIO B 06JIacTH, B KOTOPOil: z -								«BCJIHKO», T. e. l- «MaJIO»:
2/3		2/3	3)	=	2	(	3	(3)2	(3)n - l)
3-z	-z		3)	=	- 3z		1+ z + z		+ ... + z	+ ... =
	-z	(1- z
							2	2	2·3
							= --------			zn
							3z	Z2	z3	zn
OTOT p.H,l1, CXO,l1,HTC.H rrpH					I~I< 1, T. e. B 06JIacTH Izl > 3.

Terrepb 3arrHmeM pa3JIO)KeHHe B p.H,l1, JIopaHa ,l1,JI.H HCXO,l1,HOil: ,l1,P06H:

2 z(3 - z)

06JIacTb CXO,l1,HMOCTH :;noro p.H,l1,a - KOJIblJ,O Izl	> 3.	•

HaiJ.mu pa3,//,(X)/CeHue tfiYH'II:'qUU 6 pSla JIopaHa 6 mO"l,'II:e Zo no cmene'H.SI.M z - ZOo Y'II:a3am'b Z.lta6HY'IO U npa6U.It'bHY'IO "I,acmu pSlaa U ezo o6.1tacm'b CXOaUMocmu.

7.4.5.	1
7.4.5.	z cosz,		Zo = 00.
7.4.6.	a) Zo = 0;	6)	Zo = 00.
7.4.6.	z sirr z,
	a) Zo = 0;	6)	Zo = 00.
	1
7.4.7.	ez + 1,
	a) Zo = -1;	6)	Zo = 00.
	3z +4
7.4.S.	e z+ 1 ,
	a) Zo = -1;	6)	Zo = 00.

7.4.9.sin(2 + z),

	a) Zo = 0;	6)	Zo = 00.
7.4.10.	3z + 10
7.4.10.	cos z + 3 'Zo = -3.
7.4.11.	2
7.4.11.	z -1'
	z -1'		Zo = 00.
	a) Zo = 1;	6)	Zo = 00.
7.4.12.	3 + 2i
7.4.12.	(z+i)2'
	(z+i)2'
	a) Zo = -i;	6)	Zo = 00.
7.4.13.	~,zo=1.	7.4.14.	Z + 2i	2' Zo = 1.
7.4.13.	~,zo=1.	7.4.14.	(Z - 1)	2' Zo = 1.
	z -		(Z - 1)

471

7.4.15.	1	= 1.
7.4.15.	Z' Zo	= 1.
7.4.16.	2
7.4.16.	z +i'
	z +i'
	a) Zo = 0;				6) Zo = 00.
7.4.17.	z2				7.4.18.	z+2
7.4.17.	--1' Zo =-1.				7.4.18.	--3' Zo = 1.
	z-					z-
7.4.19.	z-2
7.4.19.	(z - 1)(z + 2)'
	(z - 1)(z + 2)'				6) Zo = -2.
	a) Zo = 1;				6) Zo = -2.
7.4.20.		1		= 1.			z-2
7.4.20.	(z - 1)(z + 2)' Zo				7.4.21.	(z-1 )( z+2)' Zo =-1.
	(z - 1)(z + 2)' Zo				7.4.21.	(z-1 )( z+2)' Zo =-1.
7.4.22.	HaiiTH Bce oc06ble TO'iKH<PYHKIIHH					.-l2 H Ollpe,Il;eJIHTb HX THII,
	,Il;JI.H 1I0JIloca HaiiTH ero 1I0P.H,Il;OK.					z-
	,Il;JI.H 1I0JIloca HaiiTH ero 1I0P.H,Il;OK.
a OC06bIMH TO'iKaMH<PYHKIIHH .HBJI.HIOTC.H TO'iKHZ1 = 2 H Z2 = 00.
Cnoco61.				= 0, TO lim.-l2		= 00, 3Ha'iHT,TO'iKaZ1
a) TaK KaK		lim (z -	2)	= 0, TO lim.-l2		= 00, 3Ha'iHT,TO'iKaZ1			2
		z--+2			z--+2 Z -
.HBJI.HeTC.H 1I0JIIOCOM.
Ollpe,Il;eJIHM 1I0P.H,Il;OK 9Toro 1I0JIIOCa, )J;JI.H 'ieroHaii,Il;eM IIpe,Il;eJI
	lim J(z) . (z -			2)1 = lim z - 22 = lim 1 = 1 t= O.
	z--+2				z--+2 Z -	z--+2
3Ha'iHT,1I0P.H,Il;OK 1I0JIIOCa paBeH 1.
6) TaK KaK		lim .-l2		= 0, TO TO'iKaZ2		= 00	.HBJI.HeTC.H YCTpaHHMoii
		z--+oo z -
oc060ii TO'iKOii.
~:~~~!~<PYHKIIHIO J(z) = .-l2 B p.H,Il; JIopaHa 110 CTelleH.HM (z -									2):
					z-
.-l2 =	1 . (z	- 2)-1 -	9TO H		eCTb pa3JIO)KeHHe		(COCTO.HlIIee	H3 O,Il;HOrO
z-						- 2),
CJIaraeMoro) B p.H,Il; JIopaHa 110 CTelleH.HM (z						- 2),	CXO)J;.HlIIeeC.H	B 06JIacTH
o< Iz -	21 < +00.

a) TaK KaK 9TO pa3JIO)KeHHe CO,Il;ep)KHT KOHe'iHOe'iHCJIOCJIaraeMblX (a HMeHHO, O,Il;HO CJIaraeMoe) C OTpHIIaTeJIbHbIMH CTelleH.HMH (z - 2), TO TO'iKa

Z1 = 2 .HBJI.HeTC.H 1I0JIIOCOM. TaK KaK HaH60JIblIIM CTelleHb CJIaraeMblX BH,Il;a

B pa3JIO)KeHHH paBHa 1, TO 1I0P.H,Il;OK 1I0JIIOCa paBeH 1.

6) IIoCKOJIbKY pa3JIO)KeHHe <PYHKIIHH B p.H,Il; JIopaHa 110 CTelleH.HM (z - 2),

CXO)J;.HlIIeeC.H B OKpeCTHOCTH TO'iKHZ2		= 00, He CO,Il;ep)KHT 1I0JIO)KHTeJIb~bIX
CTelleHeii	BH,Il;a (z - 2)n, TO TO'iKaZ2	.HBJI.HeTC.H YCTpaHHMoii oc060ii TO'i-
Koii.		•
7.4.23.	HaiiTH Bce oc06ble TO'iKH<PYHKIIHH

z+2

(Z2 - 4)(z - 2)2

H Ollpe,Il;eJIHTb HX THII, )J;JI.H 1I0JIIOCa HaiiTH ero 1I0P.H,Il;OK.

472

o TaK KaK
z+2				z+2			z+2
(Z2 -	4)(z - 2)2		(z + 2)(z -		2)(z -	2)2 -	(z + 2)(z -	2)3'
TO OC06bIMH TOqKaMH <PYHKIJ;HH .HBJUIIOTC.H TOqKH Zl							= -2, Z2 = 2, Z3 = 00.
a) IIoCKOJlbKY cYlllecTByeT KOHeqHbIii rrpe,l1.eJl
·	z + 2		1·		1
1			= 1m
1m	+ 2)(z -	2)3	= 1m
z--+( -2) (z	+ 2)(z -	2)3	z--+( -2) (z - 2)3
TO TOqKa Zl = -2 .HBJl.HeTC.H YCTpaHHMoii oco6oii TOqKOii.
6) TaK KaK lim		(z -	2) = 0, TO
	z--+( -2)
	1·		z + 2		1·	1	= 00.
	1m	(z + 2)(z -		= 1m			= 00.
	z--+2	(z + 2)(z -		2)3	z--+2 (z - 2)3
CJle,l1.0BaTeJlbHO, TOqKa Z2 = 2 .HBJl.HeTC.H rrOJlIOCOM.
Orrpe,l1.eJlHM rrOp.H,l1.0K 3TOro rrOJllOca. OqeBH,l1.HO,
lim J(z) . (z -		2)3 = lim		1	3· (z - 2)3 = lim 1 = 1 ¥- 0.
z--+2			z--+2	(z -	2)		z--+2
TaKHM o6pa30M, TOqKa Z2 = 2 .HBJl.HeTC.H rrOJlIOCOM 3-1'0rrOp.H,l1.Ka.
B) B CHJlY Toro, 'ITO
	lim		z + 2	2)3	= lim	1	= 0,
	z--+oo (z + 2)(z -			2)3	z--+oo	(z - 2)3
TOqKa Z3 = 00 .HBJl.HeTC.H YCTpaHHMoii oco6oii TOqKOii.								•

7.4.24. HaiiTH Bce oco6bIe TOqKH <PYHKIJ;HH sin ~2 H orrpe,l1.eJlHTb HX THrr,
			z-
a	,l1.Jl.H rrOJlIOCa HaiiTH ero rrOp.H,l1.0K.
a	OC06bIMH TOqKaMH <PYHKIJ;HH .HBJl.HIOTC.H TOqKH Zl = 2 H Z2 = 00.
	Cnoco61.	2) = 0, TO lim ~2 = 00, H 3HaqHT,
	a) TaK KaK lim (z -	2) = 0, TO lim ~2 = 00, H 3HaqHT,
	z--+2		z--+2 Z -
		lim (sin~2)
		z--+2	z -
He cYlllecTByeT. OTCIO,l1.a CJle,l1.JeT, 'ITOTOqKa Zl = 2 .HBJl.HeTC.H cYlllecTBeHHo
co6oii TOqKOii.		~2 = 0, TO
	6) IIoCKOJlbKY lim	~2 = 0, TO
	z--+oo Z -
		lim (sin ~2) = 0,
		z--+oo	Z -

CJle,l1.0BaTeJlbHO, TOqKa Z2 = 00 .HBJl.HeTC.H YCTpaHHMoii oco6oii TOqKOii.

Cnoco62.

Pa3JlO:>KHM <PYHKIJ;HIO J(z) = sin ~2 B p.H,l1. JIopaHa rro CTerreH.HM (z-2):

sin _1_	= _1__ .1.	1	+.1.	1	+...+ (_l)n .	1	+...
z - 2	z - 2 3!	(z - 2)3	5!	(z - 2)5	(2n + I)! (z -	2)2nH
06JlacTb CXO,l1.HMOCTH 9Toro p.H,l1.a -				KOJlbIJ;O °< Iz - 21 < +00.

473

= 00,

a) IIoJIY'IeHHoepa3JIO)KeHHe, CXOMrn;eeCfl B rrpOKOJIOTOii OKpeCTHOCTH TO'lKHZl = 2, COAep)KHT 6eCKOHe'lHoe'IHCJIOCJIaraeMhlX C OTpHlIaTeJIhHhIMH CTerreHflMH (z - 2), rr09TOMY TO'lKaZl flBJIfleTCfl cyrn;ecTBeHHO oco6oii TO'lKOii.

6) TaK KaK 9TO pa3JIO)KeHHe, CXOMrn;eeCfl B OKpeCTHOCTH TO'lKHZ2

He COAep)KHT CJIaraeMhIX C rrOJIO)KHTeJIhHhIMH CTerreHflMH (Z - 2), TO TO'lKa

Z2 flBJIfleTCfl YCTpaHHMoii oco6oii TO'lKOii.											•
7.4.25. HaiiTH Bce oco6hle TO'lKH<PYHKlIHH						CO~Z		H orrpeAeJIHTh HX THrr,
AJIfl rrOJIIOCa HaiiTH ero rropflAOK.
a OC06hIMH TO'lKaMH<PYHKlIHH flBJIflIOTCfl Bce TO'lKH,B KOTOPhIX cos Z = 0,
T. e. TO'lKHZk = ~ + rrk		(k = 0, ±1, ±2, ...), H TO'lKaZ = 00.
a) TaK KaK	lim	cos Z = 0, TO
	z--+zk=~+11"k
	lim	CO~Z = 00, k = 0, ±1, ±2, .•• ;
	Z--+Zk
3Ha'lHT,K~AM TO'lKaZk = ~ + rrk flBJIfleTCfl rrOJIIOCOM.
OrrpeAeJIHM rropflAOK K~AOro rrOJIIOca. HaiiAeM rrpeAeJI
lim	(I + rrk) r-		[t =		2			= t+		2 + rrk,]	=
[Z -		-		Z -	zr. - rrk ¢} Z				t -+	zr.
z--+2"+11"k	COSZ	-			Z -+ zr.2 + rrk				t -+	°
11"	COSZ						¢}
= lim	t		= lim		t				1	lim _t_ =
t--+o cos	(t + I + rrk)		HO (_l)k+l sin t				(_l)k+l HO sint
				=	1	#0 (k = 0, ±1, ±2,					••. ).
					(_l)k+l
CJIeAOBareJIhHO, K~AM H3 TO'leKZk :;=					~ +rrk (k = 0, ±1, ±2, ...) flBJIfleTCfl

rrOJIIOCOM 1-ro rropflAKa.

6) TO'lKaZ = 00 flBJIfleTCfl rrpeAeJIhHOii AJIfl rrOCJIeAOBaTeJIhHOCTH rrOJIIOCOB - TO'leKZk = ~ +rrk, CJIeAOBaTeJIhHO, Z = 00 He flBJIfleTCfl H30JIHpOBaH-

Hoii oco6oii TO'lKOii. •

HafJ.mu Bce oco6'bte mo"t'ICu !PYH'IC't!UU J(z) U onpeae.aumb UX mun, a.!IJI. nO.lI,'IOCa HafJ.mu e20 nop.Rao'IC.

7.4.26.	z2 -	4		7.4.27.	J(z) =	Z2 +4
7.4.26.	J(z) = -- 2'			7.4.27.	J(z) =	-- 2"
	z-			7.4.29.		z -		z
7.4.28.	1			7.4.29.	J(z) = (z _ l)(Z + 2)2'
7.4.28.	J(z) =~.				J(z) = (z _ l)(Z + 2)2'
	z	z
7.4.30.		1				1
7.4.30.	J(z) = sh - 2 - '			7.4.31.	J(z)=e z.
	z	-1		7.4.31.	J(z)=e z.
7.4.32.	J(z) = si~z'			7.4.33.	J(z) =	1 -	cosz			. (Z5 + 3).
7.4.32.	J(z) = si~z'			7.4.33.	J(z) =		z5			. (Z5 + 3).
7.4.34.	z-2		1	7.4.35.	J(z) =		1			1
7.4.34.	J(z) = -- 2 cos z'			7.4.35.	J(z) =	2	-	4)	2 sh -- 1 •
	l+z					(z	-	4)		z-

474

7.4.36.	eZ -1	7.4.37. J(z) = z3 ctg z ! l'
	J(z) = - z - '

7.4.38.J(z) = eZ-~.

Hai1.mu pa3.!tcr.JfCeHUe !PYH'IC'qUU J(z) a p.Ra JIopaHa a mO"l.'lCe Zo no CmeneH.RM z - zoo Y'lCa3am'b 2.!taaHY'IO U npaau.!t'bHy'lO "I.acmu p.Raa U e20 o6.!tacm'b cxoau-

.Mocmu.

7.4.39.	J(z) =	\ eZ ,
		z		6) Zo = 00.
	a) Zo = 0,
		1
7.4.40.	J(z) = ze z -		i , Zo = i.
7.4.41.	J(z) = (z + 2)2 sin z2 + 4z ~ 5, Zo = -2.
		z + 22i,		(z + 2)
7.4.42.	J(z) =
		z-		6) Zo = 00.
	a) Zo = 2,
7.4.43.		z -	2i	.
	J(z) = (z +		2i)3' Zo = -2z.
7.4.44.	(	2 + 3i		.

	Jz)= z+ 1 -z.,zo=z.
7.4.45.			1	+ 2)' Zo = 00.
	J(z) = (z _ l)(z
7.4.46.		z-2		.
	J(z) = (z _ l)(z			+ 2) , Zo = z.

Hai1.mu ace oco6'bte mo"l.'lCu !PYH'IC'qUU J(z), onpeae.!tum'b ux mun, a.!t.R no.!t'lOca HafJ,mu e20 nop.Rao'IC.

7.4.47.	z2 - Z - 6	7.4.48.	2+i
	J(z) =		J(z) = (z _ i)2(Z + 3)5
	(z + 2)2
7.4.49.	z2 + 1	7.4.50.	J(z) = e- z2 •
	J(z) = cos -- 2 .
	z-		1	1
7.4.51.	J(z) = tgz.	7.4.52.
			J(z) = sinz -	Z·

7.4.53. MO:>KeT JIM pa3JIO:>KeHMe HeKOTopoii <PYHKIJ;MM B p.H,l1. JIopaHa CO,l1.ep-

:>KaTb:

a) KOHe'lHOe 'IMCJIO CJ1araeMbIX C OTpMIJ;aTeJIbHbIMM CTeneH.HMM

(z - zo);

6) KOHe'lHOe 'IMCJIO CJIaraeMbIX C nOJIO:>KMTeJIbHbIMM CTeneH.HMM

(z - zo);

475

	B) 6eCKOHeqHOe qHCJIO CJIaraeMbIX C OTpHIJ;aTeJIbHbIMH CTerreH5IMH
	(Z -	Zo);
	r) 6eCKOHeqHOe qHCJIO CJIaraeMbIX C rrOJIO)KHTeJIbHbIMH CTerreH5IMH
	(Z -	zo)?		=f. 00 -
7.4.54.	IIycTb Zo			=f. 00 -	H30JIHpOBaHHruJ oco6ruJ TOqKa cPYHKIJ;HH J(z).
	Orrpe,l1.eJIHTb THrr 9TOft oco6oft TOqKH, eCJIH pa3JIO)KeHHe cPYHKIJ;HH
	J(z)	B p5l,l1. JIopaHa B OKpeCTHOCTH Zo CO,l1.ep)KHT:
	a) KOHeqHOe qHCJIO CJIaraeMbIX C rrOJIO)KHTeJIbHbIMH CTerreH5IMH
	(z -	zo) H KOHeqHOe (=f.				0)	qHCJIO CJIaraeMbIX C OTpHIJ;aTeJIbHbIMH
	CTerreH5IMH (z - zo);
	6) 6eCKOHeqHOe qHCJIO CJIaraeMbIX C rrOJIO)KHTeJIbHbIMH CTerreH5I-
	MH (z -		zo) H 6eCKOHeqHoe qHCJIO CJIaraeMblX C OTpHIJ;aTeJIbHbIMH
	CTerreH5IMH (z - zo);
	B) TOJIbKO 6eCKOHeqHOe qHCJIO CJIaraeMbIX C rrOJIO)KHTeJIbHbIMH
	CTerreH5IMH (z - zo).
7.4.55.	IIycTb Cn			(n = 0, ±l, ±2, ...) -				K09cPcPHIJ;HeHTbI pa3JIO)KeHH5I B
	p5l,l1. JIopaHa cPYHKIJ;HH
						+00
				J(z) = L			cn(z -	zo)n.
					n=-oo
	HaftTH K09cPcPHIJ;HeHTbI c~ pa3JIO)KeHH5I B p5l,l1. JIopaHa cPYHKIJ;HH:
	a) (z -		zo)J(z);				6) (z -		zo)3 J(z);
	B) _l-J(z);
	z -		Zo
	r) (	z -	1	)m J(z)	(m -	HaTYPaJIbHOe qHCJIO).
7.4.56.		z -	Zo
7.4.56.	HaftTH MHO)KeCTBO TOqeK, B KOTOPbIX CXO,l1.HTC5I p5l,l1. JIopaHa:
		+00		zn				+00	zn
	a) n=~oo			3n + 1;			6) n=~oo n 2 + 1 ;
	B)	+00		2n z n .
	B)	L		2n z n .
	n=-oo

7.4.57. "YKa3aTb THrr oco6oft TOqKH Zo ,l1.JI5I cPYHKIJ;HH J(z) + g(z), eCJIH TOqKa Zo 51BJI5IeTC5I:

a) YCTpaHHMoft oco6oft TOqKOft ,l1.JI5I J(z) H YCTpaHHMofi oco6ofi TOqKOfi ,l1.JI5I g(z);

6) YCTpaHHMoft oco6ofi TOqKOfi ,l1.JI5I J(z) H rrOJIIOCOM ,lI..JI5I g(z);

B) YCTpaHHMoft oco6ofi TOqKOfi ,l1.JI5I J(z) H cymecTBeHHO oco6oft TOqKOfi ,l1.JI5I g(z);

r) rrOJIIOCOM ,l1.JI5I J(z) H cymecTBeHHO oco6ofi TOqKOfi ,lI..JI5I g(z);

)I.) rrOJIIOCOM n-ro rrOp5l,l1.Ka ,lI..JI5I J(z) H rrOJIIOCOM m-ro rrOp5l,l1.Ka ,lI..JI5I g(z).

7.4.58. MO)KeT JIH TOqKa Zo 6bITb oco6ofi TOqKOft YKa3aHHbIX THrrOB ,l1.JI5I

,l1.aHHbIX cPYHKIJ;Hfi:

a) rrOJIIOCOM ,l1.JI5I J(z) H rrOJIIOCOM ,lI..JI5I (z - zo)J(z);

476

J(z).

6) nOJIIOCOM )l,JI5I J(z) H YCTpaHHMoit oc060it TOqKOit

)l,JI5I (z - zo)J(z)j

B) nOJIIOCOM )l,JI5I J(Z) H Cyrn;ecTBeHHO oc060it TOqKOit

)l,JI5I (Z - ZO)J(Z)j

	r)	Cyrn;ecTBeHHO oc060it TOqKOit )l,JI5I J(Z) H Cyrn;ecTBeHHO oc060it
	TOqKOit )l,JI5I		(Z -	ZO)J(Z)j
	~) YCTpaHHMoit oc060it TOqKOit )l,JI5I J(Z) H YCTpaHHMoit oc060it
	TOqKOit )l,JI5I		_1_ J(z) j
			Z -	Zo
	e) YCTpaHHMoit oc060it TOqKOit )l,JI5I J(z) H nOJIIOCOM
	)l,JI5I _1_ J(z)j
		Z - Zo
	>K) YCTpaHHMoit oc060it TOqKOit )l,JI5I J(z) H cyrn;ecTBeHHo oc060it
	TOqKOit )l,JI5I _1-J(z)?
			Z -	Zo
7.4.59.	IIycTb TOqKa Zo			51BJI5IeTC5I nOJIIOCOM k-ro nOp5l)l,Ka )l,JI5I <PyHKIJ;HH
	J(z). YKa3aTb THn oc060it TOqKH Zo )l,JI5I <PYHKIJ;HH:
	a)	(z - zo)J(z)j				6) (z - zO)3 J(Z)j
	B)	Z! Zo J(Z)j				r)	(z -\0)5 J(z).
7.4.60.	Onpe)l,eJIHTb THn oc060it TOqKH Zo = 0 )l,JI5I <PyHKIJ;HH ~1-_._1_.
								e -	SlllZ
7.4.61.	Onpe)l,eJIHTb THn oc060it TOqKH Zo = ~ )l,JI5I <PyHKIJ;HH
				1	(	7r)-2		.
				COSZ -		Z - "2		.
7.4.62.	IIycTb Zo -		H30JIHpOBaHHa51 oc06a51 TOqKa <PyHKIJ;HH J(z). ,Il;oKa-
	3aTb, 'ITOeCJIH J(z) OrpaHHqeHa B OKpeCTHOCTH TOqKH Zo, TO Zo -
	YCTpaHHMM oc06a51 TOqKa )l,JI5I J(z).
7.4.63.	IIycTb Zo -		H30JIHpOBaHHa51 OC06M TOqKa <PyHKIJ;HH J(z). ,Il;oKa-
	3aTb, 'ITOeCJIH
				IJ(z)1 <		1
				IJ(z)1 <		(z _ zo)m

(m - HaTypaJIbHOe qHCJIO) B HeKoTopoit OKpeCTHOCTH TOqKH Zo,

TO Zo - JIH60 YCTpaHHMa51 oc06a51 TOqKa, JIH60 nOJIIOC )l,JI5I <PYHKIJ;HH

7.4.64. ,Il;oKa3aTb, 'ITO<PYHKIJ;H5I aHaJIHTHqeCKM BO Bceit KOHeqHoit KOMnJIeKcHoit nJIOCKOCTH H HMeIOrn;a51 B TOqKe Zo = 00 nOJIIOC nOp5l)l,Ka n, 51BJI5IeTC5I MHOrOqJIeHOM CTeneHH n.

§ 5. BblYETbl

=? IIycTb <PYHKIJ;HH J(z) aHaJIHTIPIHa B HeKoTopolt rrpOKOJIOTolt OKpeCTHOCTH U

KOHeqHolt TO'lKHzoo B'b/,"I,emOM (/jy'ltICv,uu J(z)			6 mO"l,ICe Zo Ha3bIBaeTCH '1HCJIO
	1		1	j	J(z) dz,
res J(z) = - 2.		jJ(z) dz = - 2.
zo	7rt		7rt	Iz-zol=p
		'Y

477

TeopeMa 7.4.

pa3)IO)l{eHa B pH,D; JIopaHa B OKpeCTHOCTH TO'IKHzo:

I',D;e'Y - HeKoTopbIit 3aMKHYTbdl: KOHTyp, U:eJIHKOM JIe)l{am:Hit B U H CO,D;ep)l{am:Hit BHyTpH TO'IKYzo, a Iz - zol = p - OKPY)l{HOCTb C u:eHTPOM B TO'IKeZo ,D;OCTaTO'I-

HO MaJIOI'Opa,D;Hyca p, U:eJIHKOM JIe)l{am:aH B U. 06xo,D; KOHTypa 'Y H OKPY)l{HOCTH

npOH3BO,D;HTCH npomUB "acoBoit cmpe.ll7CU. ~

3Ha'IeHHH060HX npHBe,D;eHHbIX HHTeI'paJIOBnpH YKa3aHHbIX YCJIOBHHX COBna- ,D;aIOT.

ECJIH <PYHKU:HH J(z)

C-k		k + ... +	C-2	C-l	()	(	Z-ZO	)n	... ,
f ( z ) = ... +	ZO)	k + ... +	(Z - ZO)	2 + --- +CO+Cl Z-ZO .. .+Cn			Z-ZO		... ,
(Z -	ZO)		(Z - ZO)	Z - Zo
TO resJ(z) = C-l.								(5.1)
TO resJ(z) = C-l.
:0

~ IIycTb <pyHKU:HH J(Z) aHaJIHTH'IHaB HeKoTopoit npOKOJIOTOit OKpeCTHOCTH U 6ecKOHe'IHOy,D;aJIeHHoit TO'IKH00. B'b/,,,emo.M ¢YH.7C'IJ,UU J(Z) B mO"7Ce 00 Ha3bIBaeTCH

'IHCJIO	J(z) =		!J(z) dz =		! J(z) dz,
00		1		1
00		7rZ		7rZ
res		- 2.		- 2.
			"I		1:I=p
I',D;e'Y - HeKoTopbIit	3aMKHYTbIit		KOHTyp, a	Izl = p - OKPY)l{HOCTb ,D;OCTaTO'IHO

60JIbllOI'Opa,D;Hyca, U:eJIHKOM JIe)l{am:aH BU. 06xo,D; KOHTypa H OKPY)l{HOCTH npo-

H3BO,D;HTCH no "acoBoit cmpe.ll7Ce. ~

3Ha'IeHHH060HX npHBe,D;eHHbIX HHTeI'paJIOBnpH YKa3aHHbIx YCJIOBHHX COBna- ,D;aIOT.

ECJIH pa3JIO)l{eHHe (5.1) CXO,D;HTCH B HeKoTopoit OKpeCTHOCTH TO'IKH00, TO

resJ(z) = -C-l.

EClllII <PYHK~lIIfl J(z) aHalllllTIII'lHaHa BCeiii KOMnlleKCHoiii nllOCKOCTIil

C. sa IIICl(JUO'leHllleMIIISOlllllpOBaHHblX OC06blX TO'leKZl, Z2, . .. Zn. TO

	n
resJ(z) = - L res J(z).
00	k=l :k

ECJIH <PYHKU:HH J (Z) aHaJIHTH'IHaB TO'IKeZo HJIH eCJIH Zo - YCTpaHHMaH oco6ax

TO'IKa,D;JIHJ(Z), TO

resJ(z) =0.

TeopeMa 7.5. EClllII	TO'lKa Zo -	nOlllOC	k-ro nopflAKa (k		> 1) Allfl <PYHK~IIIIII
J(z). TO			d'<-l [J(z)(z - Zo)k]
	1	.	d'<-l [J(z)(z - Zo)k]
res J(z) = (		)' hm	dz	k-l	.
:0	k -	1 . :-+:0	dz

478

ECJIH Zo - IIOJIIOC l-ro IIOpH)l;Ka)l;JIH <PYHKU:HH J(Z), TO

resJ(z) = lim [J(z)(z-zo)],

zo z---+zo

rp(z)

a eCJIH em;e H3BeCTHO, 'ITO<PYHKU:HH J(z) IIpe)l;CTaBHMa B BH)l;e J(z) = t/J(z)' r)l;e

<PYHKU:HH rp(z) H t/J(z) - aHaJIHTHqeCKHe B TOqKe zo, t/J(zo) = 0, t/J'(zo) =I 0, TO

	rp(zo)
res J(z) = ---;---().
zo	t/J	Zo

qacTO IIpH BblqHCJIeHHH HHTerpaJIOB OT <PYHKU:Hit KOMIIJIeKCHOro IIepeMeHHoro IIpHMeHHIOT CJIe)l;JIOIIJ;yIO TeopeMY.

TeopeMa 7.6 (OcHoBHaR TeopeMa 0 BbllieTax). nYCTb <PYHK~~"I J(z) -		aHalllll-
TIII'leCKal'lB	OAHOCBI'ISHOA 0611aCTIil D sa IIICKlllO'leHllleM HeKOTopblX IIISOlllllpOBaH-
HblX OC06blX	TO'leK; l - npOCTal'l saMKHYTal'l KpIIIBal'l, ~ellIllKOM lle>Ka~al'l	B D III

He npoxOAI'I~al'l 'lepesoco6ble TO'lKIil <PYHK~IIIIII J(z). TorAa

f	J(Z)dz = 27ri· tresJ(z),
f	Zk
I	k=l

7.5.1.HaiiTIi Bhl'leThI <PYHKIJ;IIII

(

z + 1

f z) =

(z + 2i)2(Z _ 1)

BO Bcex OC06hIX TO'iKax II onpe,n:e.ITIiTh IIX Tlln, HaiiTIi BhlqeT B 6ec-

KOHe'iHO y,n:a.neHHoii TO'iKe.

Q OC06hIMH

TO'iKaMIi

<PYHKIJ;IIII f (z),

O'ieBII,n:HO,

.HB.IT.HIOTC.H C.ITe.n:yIOID;lIe

TO'iKII:

-2i -

nO.ITIOC 2-ro nOp.H,n:Ka,

1 -

nO.ITIOC 1-ro nopH,n:Ka.

Haii,n:eM BhI'ieT B TO'iKe -2i:

res.f(z) = lim. [f(z)(z - (-2i))2]

[(Z+1)(Z+2i)2]'

lim.

(

.)2(

z -

)

-2t

z-+-2t

Z + 2z

(z+l)'

. l·(z-l)-l·(z+l)

-2

(z -

1)2

---=~ =

z-+-2i

Z - 1

z-+-2i

(z - 1)2

-2

(-2i -

1)2

(1 + 2i)2

1 + 4i -

4 = 3 - 4i;

479

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