Добавил:

Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

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

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

Сборник задач по высшей математике 2 том

.pdf

Скачиваний:

Добавлен:

15.03.2015

Размер:

27.38 Mб

Скачать

☆

<<< < Предыдущая 34 35 36 37 38 39 40 41 42 43 44 4546 / 6046 47 48 49 50 51 52 53 54 55 56 57 58 > Следующая >>>

1R.2 .

Llv = 82v + 82v = 0

8x2 8y2

)J,JIa JIlo6oit napbI .D:eitcTBHTeJIbHbIX qHCeJI x H y, T. e. cPYHKU.Ha V(X, y) aBJIaeTca rapMOHHqeCKoit BO Bceit nJIOCKOCTH 3HaqHT, cyru.eCTByeT TaKM aHaJIHTHqeCKM BO Bceit KOMnJIeKcHoit nJIOCKOCTH C cPYHKU.Ha J(z ), qTO

J(z) = J(x + iy) = u(x, y) + iv(x, y).

2. Hait.D:eM .D:eitcTBHTeJIbHYIO qacTb u(x, y). IIepBoe H3 YCJIOBHit KOWHPHMaHa (2.1) .D:aeT paBeHCTBO g~ = g~ = 2x, H3 KOTOPOro B03M02KHO onpe-

.D:eJIHTb cPYHKU.HIO u(x, y) C TOqHOCTblO .D:O CJIaraeMoro <p(y) - cPYHKU.HH, 3aBHcaru.eit TOJIbKO OT Y H He 3aBHcaru.eit OT x, - CJIe.D:YIOru.HM 06pa30M:

u(x, y) = ! g~ dx + <p(y) = ! 2x dx + <p(y) = x 2 + <p(y).

TaK KaK no BTOPOMY H3 YCJIOBHit (2.1) BbIllOJIHaeTCa paBeHCTBO g~ = - g~ =

= -2y,TO

8u(x,y) =2...(

())=-2

T.e.

~(x 2) + ty (<p(y)) = 0 + <p'(y) = -2y.

113 nOCJIe.D:HerO paBeHCTBa HMeeM:

<p(y) = !(-2y) dy = _y2 + C,

r.D:e C = const, C E lit

3HaqHT, u(x, y) = x 2 -

y2 + C, OTKY.D:a OKOHqaTeJIbHO

J(z) = J(x + iy) = u(x, y) + iv(x, y) = (x2 -

y2 + C) + i(2xy - 3) =

= (x 2 - y2 + i

. 2xy) + C -

3i = (x + iy)2 + C -

3i = Z2 + C - 3i.

•

7.2.18.

HaitTH MH02KeCTBO TOqeK, B KOTOPbIX cPYHKU.Ha u(x,y) = eX cosy

Y.D:OBJIeTBOpaeT YCJIOBHIO

Llu

O. Onpe.D:eJIHTb, cyru.ecTByeT

JIH

aHaJIHTHqeCKM B HeKoTopoit 06JIaCTH D cPYHKU.Ha J(z) (z=x+iv),

)J,JIa KOTOPOit Re J = u. ECJIH TaKM cPYHKU.Ha J(z) cyru.eCTByeT, TO

HaitTH ee.

a 1. Hait.D:eM qaCTHble npOH3BO.D:Hb1e:

8u = 2... (eX cos y)

= eX cos y,

82 u

= 2... (eX cos y)

= eX cos y

8x2

CJIe.D:OBaTeJIbHO,

Llu = 82u + 82u = eX cosy -

eX cosy = 0

V(x,y) E IR2.

8y2

450

451

vex, y).

TaKHM 06pa3oM, <PYHKD:HH U(X, y) rapMOHHqeCKM B IIJIOCKOCTH C, H 3HaqHT, cym;ecTByeT TaKM aHaJIHTHqeCKM B C <PYHKD:HH fez), qTO fez) = f(x+iy) =

==u(x,y) +iv(x,y).

2.Hail:.n:eM MHHMYIO qacTb vex, y). IIepBoe H3 YCJIOBHil: KOllH-PHMaHa

g:= g~ = eX cosy, oTKy.n:a B03MO)l{HO olIpe.n:eJIHTb

cPyHKD:HIO vex, y) C TOqHOCTblO .n:o CJIaraeMOro cp(x) -			<PYHKD:HH, 3aBHcHm;eil:
TOJIbKO OT x H He 3aBHCHm;eil: OT y, -		CJIe.n:yIOm;HM 06Pa30M:
vex, y) = ! g~ dy +	cp(x) = ! eX cos y dy + cp(x) =			eX sin y +	cp(x).
TaK KaK 110 BTOPOMY H3 YCJIOBHil: (2.1) HMeeM
ov =	_ou = -(-eXsiny) =exsiny			,
ox	oy			,
TO
tx (eX siny + cp(x)) = eX siny,
tx (eX sin y) +	tx (cp( x)) =	eX sin y + cp' (x) =		eX sin y.
CJIe.n:oBaTeJIbHO, cp'(x) =	0, oTKy.n:a cp(y) = C, r.n:e C		E lR. 3HaqHT, v(x,y) =
=exsiny+C, H
fez) = f(x + iy) = u(x, y) + iv(x, y) = eX cosy + i(eXsiny + C) =
	= eX (cos y + i sin y) +			iC = eZ	+ iC.	•

7.2.19.BbIHCHHTb, 6y.n:eT JIH CKaJIHpHOe IIOJIe v(x,y) = co;x rapMOHHqeCKHM (CM. c. 243). Cym;ecTByeT JIH aHaJIHTHqeCKM B HeKoTopoil: 06JIacTH D <PYHKn:HH fez) (z = x + iv),.D:JIH KOTOPOil: Imf = v?

ECJIH TaKM <PYHKn:HH fez) cym;ecTByeT, TO Hail:TH ee.

Q Hail:.n:eM qacTHbIe IIPOH3Bo.n:HbIe:

ov = ~ (COSX) = _sinx					,
ox ox	y			y	,
	02V			cos x
	ox2		--y-,
CJIe.n:OBaTeJIbHO,
~v =	02v +	02V =		_ cos x +		2 cosx =	cos x (2 _ y2).
	ox2	oy2		Y		y3	y3
OTclO.n:a BH.n:HO, qTO ~v =			0 IIpH Y =		±J2 HJIH IIpH X = ~ +rrk, k E Z, y :f. O.

MHO)l{eCTBO Bcex TOqeK, B KOTOPbIX ~v = 0, He 06pa3yeT HHKaKOil: 06JIa-

CTH B ]R2, H 3HaqHT, He cym;ecTByeT HHKaKoil: aHaJIHTHqeCKoil: <PYHKD:HH fez) = f(x + iy) C MHHMOil: qacTblO •

Ha'iJmu M/H.OOICeCm60 mO"l,e'IC,			6 'lCOmOpb/,X fjjy'lt'IC'lJ,'l.lJI U(X, y) (U.ltU V(X, y») .R6AA-
emCR eap.MO'ltU"I,ec'ICo'iJ. B'btRc'ltum'b,				cyw,eCm6yem .ltU a'lta.ltUmU"I,eC'ICM 6 'lte'ICo-
mopo'iJ	o6.1tacmu D fjjy'lt'IC'I..4U.R		J(Z)	(Z = x + iy),	OAA	'lComopo'iJ Re J = U	(CO-
Om6emCm6e'lt'ltO ImJ = V).			Ec.ltu	ma'ICM fjjy'lt'IC'I..4U.R		J(z) cyw,ecm6yem,	mo
'lta'iJmu	ee.
7.2.20.	u(X,y) = x.			7.2.21.	v(x,y) = y2 - x 2 - 2.
7.2.22.		1		7.2.23.	v(x, y) = 3xy2 - x 3 + 7y.
	v(x,y) = -yo
7.2.24.	u(x,y) = cosychx.			7.2.25.	u(x,y) = eXchy.
7.2.26.	v(x, y) =	2 X 2'		7.2.27.	v(x, y) = - cos 2x sh 2y+3x.
	X	+y
7.2.28.	u(x, y) = In(x2 + y2).			7.2.29.	v(x, y) = arctg ~ + 2y.

,aAA oa'lt'lto'iJ fjjy'lt'IC'I..4uu J(z) Y'ICa3am'b mo"l,'lCu, 6 'lComop'btx cyw,ecm6yem npoU3600'ltM f'(z), u 'lta'iJmu npou3600'lty'lO 6 amux mO"l,'lCax:

7.2.30.	J(z) = Imz.	7.2.31.	J(z) =	1
7.2.30.	J(z) = Imz.	7.2.31.	J(z) =	-=:.
	J(z) = In(iz).			Z
7.2.32.	J(z) = In(iz).	7.2.33.	J(z) =	1;1'
7.2.34.	J(z) = e3z •	7.2.35.	J(z)=ishz.
7.2.36.	,n:oKa3aTb, qTO He cyru;ecTByeT aHaJUlTllqeCKoil: Ha Bceil: IIJIOCKOCTII
	C <PYHKD:IIII, ~JUI KOTOPOil: <PYHKD:IIH x 2 - y HBJIHJIaCb 6bI MHIIMOil:
	qacTblO.

Ha'iJmu .M'ltOOlCeCm60 mO"l,e'IC, 6 'lComopb/,x fjjy'lt'IC'I..4U.R u(x, y) .R6AAemCR		eap.MO-
'ltu"l,ec'ICo'iJ. Ha'iJmu a'lta.ltumu"l,ec'ICY'lO 'Ita amo.M .M'ltOOlCeCm6e fjjy'lt'IC'I..4u'lO		J(z) =
= J(x + iy),	OAA 'lComopo'iJ fjjy'lt'IC'I..4U.R u(x,y) 6yoem .R6AAm'bCR oe'iJcm6um€.!t'b-
'lto'iJ "I,acm'b'lO.	Y'lCa3am'b coom6emcm6Y'lOw,y'lO .M'ltU.My'lO "I,acm'b v(x,y).

7.2.37.u(x, y) = y + x 2 - y2 + 1. 7.2.38. u(x, y) = e- Y cos x - x.

7.2.39.		(	) _	y	2 )	+ X	2	y	2	.
	U		X, Y -	2(x2 + y	2 )	+ X	-	y		.

y2 _ x2

7.2.40.u(x, y) = (x2 + y2)2

KOHTponbHble BonpOCbl III 60nee CnO)f(Hble 3aWlHIIIH

7.2.41. MO:>KeT JIll <PYHKD:IIH 6bITb ~1I<p<pepeHD:llpyeMoil: B TOqKe Zo II He 6bITb aHaJIIITllqeCKoil: B 9TOil: TOqKe?

7.2.42. MO:>KeT JIll <PYHKD:IIH 6bITb aHaJIIITllqeCKoil: TOJIbKO B O~HOil: TOqKe?

7.2.43. MO:>KeT JIll <PYHKD:IIH, aHaJIIITllqeCKaH B 06JIaCTII, 6bITb:

a) CYMMOil: ~BYX <PYHKD:llil:, He aHaJIIITllqeCKIIX B 9TOil: 06JIaCTlli

452

cyw;ecTByeT.

eCJIH 3TOT npe,n;eJI cyw;ecTByeT H He 3aBHCHT OT BbI60pa npOMelKYTO'lHbIXTO'leKZk

Re J(z)

6) "pOH3Be,ll;eHHeM ,ll;BYX cPYHKII,Hti, He aHaJIHTHqeCKHX B 9TOti o6JIa-

CTH;

B)qacTHbIM ,ll;BYX cPYHKII,Hti, He aHaJIHTHqeCKHX B 9TOti 06JIacTH;

r)CYMMOti aHaJIHTHqeCKoti H He aHaJIHTHqeCKoti B 9TOti 06JIacTH cPYHKII,Hti;

~) "pOH3Be,ll;eHHeM HeHYJIeBoti aHaJIHTHqeCKoti H He aHaJIHTHqeCKOti B 9TOti 06JIacTH cPYHKII,Hti?

7.2.44. BepHO JIH, 'ITOcPYHKII,HjI J(z) aHaJIHTHqeCKajl B 06JIacTH D, eCJIH HIm J(z) - cPYHKII,HH, rapMOHHqeCKHe B 9TOti 06JIaCTH?

7.2.45. qTO MO:lKHO CKa3aTb O,ll;BYX aHaJIHTHqeCKHX B O,ll;HOCBjl3HOti o6JIaCTH cPYHKII,HjlX, eCJIH HX ,ll;eticTBHTeJIbHble qacTH:

a) COBIIa,D;aIOTj

6) OTJIHqaIOTCjI Ha IIOCTOjlHHOe CJIaraeMOe;

B)OTJIHqaIOTCjI Ha IIOCTOjlHHblti MHQ)KHTeJIb?

7.2.46.~OKa3aTb, 'ITO,ll;JIjI ,ll;HcPcPepeHII,HPyeMoti cPYHKII,HH J(z):

a) gz (ReJ(z») = ~f'(z);

6) gz (ImJ(z») = i/'(z).

7.2.47.~OKa3aTb, 'ITOypaBHeHHe JIaIIJIaca MO:lKeT 6bITb 3aIIHcaHO B CJIe-

,ll;yIOIII,eti cPopMe:

()2 J

8z8z = o.

§3. lilHTErPlilPOBAHlilE CIlYHKU.lilVi KOMnllEKCHOrO nEPEMEHHOrO

~ I1YCTb Ha KOMnJIeKcHoil: nJIOCKOCTH C 3a,n:aHa opHeHTHpOBaHHaH Kyco'lHO-rJIa,n:- KaH KpHBaH l, Ha KOTOPOil: onpe,ll;eJIeHa <PYHKIJ;HH J(z). Pa306beM 3Ty KPHBYIO Ha n

'{aCTeil: (Zk-l, Zk) TO'lKaMHzo, Zl, ... , zn,		npOHYMepOBaHHbIMH B HanpaBJIeHHH OT
Zo - Ha'laJIbHoil:TO'lKHKPHBOil: "	,n;o zn -	KOHe'lHoil:TO'lKH" H Ha KalK,n;Oil: 'IacTH
BbI6epeM KaKYIO-HH6y,n;b TO'lKYCk	(k = 1,2, ... , n). HttmezpaJlo.M om rjjytt'l(;tJ,uu J(z)

no 'l(;pu(Jo1J. l Ha3bIBaeTCH npe,n;eJI:

(3.1)

Ii Ck.

ECJIH <PYHKIJ;HH J(x) HenpepbIBHa Ha KPHBOil: l, TO HHTerpaJI (3.1)

ECJ1H Z = X + iy H J(z) = u(x, y) + iv(x, y), TO

!J(z) dz = !(u(x, y) + iv(x, y») d(x + iy) = !(u(x, y) + iv(x, y»(dx + i dy) =

I	I	I
	= !(u(x,y)dx-v(x,y)dy) + i!(v(x,y)dX+ u(x,y)dy), (3.2)
	I	I

453

T. e. HHTerpan: (3.1) MOlKeT 6b1Tb 3anHcaH B BH,n;e	CYMMbI ,n;BYX KPHBOJIHHei!:HbIX
HHTerpan:oB 2-ro po,n;a.
ECJIH KpHBaH l 3a,n;aHa ypaBHeHHeM y = y(x) HJIH napoi!: napaMeTpH'leCKHX
ypaBHeHHi!: y = yet), x = x(t), TO B cl>0pMYJIaX (3.2)	MOlKHO 3anHcaTb
dy = dy(x) = y'(x)dx
HJIH, COOTBeTCTBeHHO,

dx = dx(t) = x'(t)dt, dy = dy(t) = y'(t)dt.

,IJ,OBOJIbHO 'lacTOB Ka'leCTBenapaMeTpa t Bbl6HpaeTcH yrOJI t.p = arg z.

~ct>YHKIJ;HH F(z) Ha3b1BaeTCH nepaoo6pa3'tto1:t cl>YHKIJ;HH fez) B 06JIaCTH D, eCJIH

F(z) ,n;HcI>cI>epeHIJ;HPyeMa B aToi!: 06JIaCTH, H F'(z) = fez)								Vz E D.	$
TeopeMa 7.2. nYCTb <PYHKLlIIIJI	fez) aHaIlIllTIII'leCKaJlB OAHOCBJl3HOin 061laCTill D. a
l - HeKOTOpaJi KplIIBaJi (C Ha'lallbHoinTO'lKOinZl III KOHe'lHoinTO'lKOinZ2). LlelllllKoM
lle>Kal..l..laJi B D. TorAa							AIlJl IIIHTerpalla !fez) dz
1) cYl..l..leCTByeT nepBoo6pa3HaJi F(z) AIlJl fez)				B	D. III		AIlJl IIIHTerpalla !fez) dz
BepHa cjJopMyna HbIOToHa-f1eiA6HHL/a:									,
!fez) dz =F(z2) -				F(Zl),					(3.3)
,
T. e. 3TOT IIIHTerpall He 3aBIIICillT OT BIllAa			KPIllBOin		l.	a	3aBIIICillT OT		Ha'lallbHoin III
KOHe'lHoinTO'leKZl III Z2;
2) eC1l1ll l - 3aMKHYTaJi KpIIIBaJi. TO BepHa			TeopeMa KOWH:
	ff(z) dz = 0								(3.4)
('lepe3f 0603Ha'laeTCJIIIIHTerpall'no3aMKHYToin KplllBOin l);
,							l.
3) eC1l1ll TO'lKa Zo lle>KIIIT BHYTPIll		3aMKHYToin		KPIllBOin			l.	TO BepHa	HHTerpanbHaR
cjJopMyna KOWH:		1		fez)		dz
f(zo) =		1	f	fez)		dz			(3.5)
f(zo) =		-2.	f						(3.5)
j<n)(zo) = 2:i	f,	1TZ	, z -		Zo
j<n)(zo) = 2:i	f,	(z _ ZO)n+l			Z,
		fez)		d			n = 1,2, ...		(3.6)
							n = 1,2, ...		(3.6)
(06XOA KplllBOin l COBepWaeTCJI	npOTIllB 'laCoBoinCTpeIlKIII).

7.3.1. BbI'IHCJIHTbHHTerpaJI J = !Imzdz, r,ll;e l:

a) OTpe30K rrpHMoi!: OT TO'lKHO,ll;O TO'lKH1 + 2i;

6) .n.yra rrapa60JIbI y = 2X2 OT TO'lKH0 ,n;o TO'lKH1 + 2i.

454

o a) Tax Kax l -	OTpe30K rrpHMoil: y = 2x (pHC. 103) H 1m z = y, TO
	1	1
J= !yd(X+iy) = !2xd(x+i.2x) = !2x.(dx+2idx) =
I	0	0
1	1	1	1
= !2x dx + !2x . 2i dx = 2 !x dx + 4i !x dx =
o	0	0	0
	=	2 11	+ 2i) . (12 - 02 ) = 1 + 2i.
	=	(2 + 4i) . ~ 0 = (1	+ 2i) . (12 - 02 ) = 1 + 2i.

	Puc. 109			Puc.	104
6) Tax KaK .n:JIH Bcex TO'leKl		HMeeM y = 2X2, TO (pHC. 104)
	1			1
J = !yd(x + iy) = !2x2 d(x + i· 2X2) = !2x2(dx + 2id(X2») =
I	0		0
	1		1		1
= !2X2 (dx + 2i . 2x dx) = 2			!x 2 dx + 8i !x 3 dx =
	o		0		0
	x311	x411	=	2	2	+ 2i.
	= 2. 3 0 + 8i· 4 0		=	3(1- 0)	+ 2i(1- 0) = 3	+ 2i.
9TOT rrpHMep rrOKa3bIBaeT, 'ITOeCJIH l			-	KpHBM B 06JIacTH D C Ha'la.rrb-
HOil: TO'lKOil:Z1	H KOHe'lHoil:TO'lKOil:Z2, a J(z) He aHa.rrHTH'IeCKM<PYHKD:HH B
D, TO HHTerpa.rr		!J(z) dz,
		!J(z) dz,

Boo61I1e rOBopH, 3aBHCHT He TOJIbKO OT TO'leKZ1 H Z2, a TaIOKe H OT BH.n:a

KPHBOil: l.		•
7.3.2.	BbI'IHCJIHTbHHTerpa.rr
	J = !<iz + z2) dz,
	I
	r.n:e l - '1acTbOKP}')KHOCTHIzI = 2, arg z E [~, 'IT] .
Q TaK Kax .n:JIH Bcex TO'leKl BbIIIOJIHHeTCH paBeHcTBo r		= Izl = 2 (pHC. 105),
TO
	z = rei'P = 2ei'P, z = 2e- i'P, Z2 = (2e i'P)2	= 4ei.2'P,

455

	dz = d(2ei'l')	= 2iei'l'd<p,		<p = arg z E [~,7r] .
OTCIO,ll;a IIOJIY'IaeM:
J =	!~ (i . 2e- i'l' + 4e2i'l') 2ie i'l' d<p =		!~ (i2		. 4e-i'l'+i'l' + 8ie2i'l'+i'l') d<p	=
	~		~
	2		2		.
~		~		~	.
= !	(-4 + 8ie3i'l') d<p = -4 ! d<p + 8i !			e3i'l' d<p = -4<P1: + 8i· e;:'I'I: =
~		~	~		2	2
222
= -4 (7r - ~)+i(e3i~_e3i~) = -4·~+i(-I-(-i)) = -27r-i+ii.						•
	y
		x
	Puc. 105				Puc. 106

7.3.3.BbI'IJilCJIJilTb JilHTerpaJI

J = !Sinzdz,

r,ll;e 1 - oTpe30K IIpjlMofi OT TO'IKJiI 0 ,ll;O TO'IKJiI 7r + i7r.

Q TaK KaK BCIO,IJ;y Ha 1HMeeM y = X (PJilc. 106), asinz=sinxchy+i cosxshy

(CM. 3a,IJ;a'IY	7.1.1), TO
J= !(SinxChy+icosxShy)d(x+iy) =
I
~		~
= !(sinxchx + icosxshx) d(x + ix) = !(sinxchx -			cosxshx) dx+
o		0
	~
	+ i !(sinxchx + cosxshx) dx = (J1 -		h) + i(JI + J2 ),
	o
r,ll;e	~	~
	Jl = ! sinxchxdx;	J2 = ! cosxshxdx.
	o	0

IIpJilMeHjIjI ,ll;BroK,ll;bI <POPMYJIY HHTerpHpOBaHHjI IIO 'IaCTjlM JiI Y'IJilTbIBM, 'ITO d(sinx) = cosxdx, d(cosx) = -sinxdx, d(shx) = chxdx, d(chx) = shxdx,

Hafi,ll;eM J1 JiI J2:

456

n n

J1 = jSinxd(shx) = sinxshxl: -		jshx d(sin x) =
o		0
	n		n
= sin 11" sh 11" - sinOshO - j shxcosxdx = 0 - j cosxd(chx) =
	o		0
= -(cosxchxl: - jChXd(COSX») = -(cos1l"ch1l"-cosOchO)+
o
n			n
+ j ch x( - sin x) dx = - (-1 . ch 11" -			1 . 1) - j sin x ch x dx =
o			0
				=	ch1l" + 1- J1,
OTKY,Il;a 2J1 = ch 11" + 1, CJIe,Il;OBaTeJIbHO, J1 =			~(ch11" + 1).
AHaJIOrH'IHO IIOJIY'IaeM
n		n
h = j cosxd(chx) = cosxchxl: -		j chx d(cosx) =
o		0
	n			n	sin x d(sh x) =
= cos 11" ch 11" - cos 0 ch 0 +	j ch x sin x dx =		- ch 11" - 1 +	j	sin x d(sh x) =
	o			0
			n
=-Ch1l"-I+sinxshxl: -			jshxd(sinx) =
			o
		n
= - ch 11" - 1 + (sin 1I"sh 11" -	sinOshO) - j shxcosxdx =			-	ch1l" - 1 -	J2 ,
		o
oTKY,Il;a 2h = -(ch 11" + 1), T. e. h =		-~(ch11" + 1) = -J1 .
llTaK,
J = j sin z dz = (J1 -	J2) + i( J1 + J2) = 2J1 + i . 0 = ch 11" + 1.					•

7.3.4.BbI'IHCJIHTb HHTerpaJI

J = jzkdz,

r,Il;e:

a) 1 - OKPY)I{HOCTb Izl = 1, arg z E [0,211"] (TO'lKa z cOBepIIIaeT IIOJIHblfi 060pOT IIO OKPJ)KHOCTH Izl = 1);

6) 1 - OKPY)I{HOCTb Izl = 1, argz E [0,411"] (TO'lKa z COBepIIIaeT ,Il;Ba IIOJIHbIX o6opoTa IIO OKPY)I{HOCTH Izl = 1).

Q TaK KaK MH Bcex TO'leK 1BbIlIOJIHHeTCH paBeHcTBor = Izl = 1 (pHC. 107),

z = reicp = eicp , zk = eikcp , dz = d(eicp ) = ieicp dcp.

457

							Puc. 107
a) BblqHCJIHM HHTerpa.rr IIpH cp = argz E [0,211"]:
		271"				271"
J =		/	eikcp . ieicp dcp = i			/ ei(k+1)cp dcp =
		o			0
= {i			1:0 dcp = iCPI:7I" = 211"i,				IIpH k + 1 = 0, T. e. k = -1;
		.ei(k+1)cp 1271" __1_ ( i(k+1)·271" _						i(k+1)'O)_		0,	IIpH k + 1 :f. O.
		zi(k + 1) 0		-	k +	1 e		e	-	0,	IIpH k + 1 :f. O.
		/~ = 211"i		(B cJIyqae k = -1), H				/	zk dz = 0	(IIpH k :f. -1),
		I						I
B qacTHOCTH,
	/	zdz = /z 2 dz = /zlOdz = /dz dz = /dz dz = /									dz dz = O.
	/						z2		z3		ZIOO
	I		I		I		I		I	I
6) BblqHCJIHM HHTerpa.rr IIpH cp = arg z E [0,411"]:
		471"				471"
J =		/	eikcp . ieicp dcp = i			/ ei(k+1)cp dcp =
		o	i j:o dcp = iCPI:7I"			0
			i j:o dcp = iCPI:7I"			= 411"i,	IIpH k = -1;
= { ie~(k+1)CP1471"					= _1_ (e i(k+1).471"			_ ei(k+1).O)		= 0	IIpH k:f. "":l.
			i(k + 1)	0	k + 1					'
TaKHM 06P3.30M, ,rr,JUI					TaKoit KPHBOit 1 IIOJIyqHM
			/d: = 411"i	(B cJIyqae k = -1) H				/	zk dz = 0	(IIpH k:f. -1).
			I					I

458

7.3.5. I1CIIOJIb3YH aHaJIHTH'IHOCTbIIO,n;bIHTerpaJIbHOfi <PYHKII,HH, BbI'IHC-

JIHTb HHTerpaJI

				!sinzdz,
				I
a		r,n;e l - OTpe30K IIPHMOfi OT TO'IKH0 ,n;o TO'IKHrr + irr.
	c'[)YHKIJ;HHsin z -		aHaJIHTH'IeCKMHa Bcefi KOMIIJIeKCHOfi IIJIOCKOCTH C, a
<pyHKIJ;HH ( - cos z),			O'IeBH,n;Ho,HBJIHeTCH IIepBo06pa3Hofi ,n;JIH sin z B C, CJIe-
,n;OBaTeJIbHO,
			1I"+i1l"	= -(cos(rr + irr) -	cosO) =
	!sinzdz = (-cosz) 10
	I	[T. K. cos(x		+ iy) = cosxch Y -	i sinx sh y]

		= - (cos rr ch rr - i sin rr sh rr - 1) = - (- ch rr - 1) = ch rr + 1
(cpaBHHTe C 3a,n;a'Iefi7.3.3).					•
7.3.6.		BbI'IHCJIHTbHHTerpaJI

f z+zz2 dz.

110 3aMKHYTofi KPHBOfi l, HCIIOJIb3YH <POPMYJIbI (3.4), (3.5) HJIH (3.6)

(06xo,n; KPHBOfi ocym;eCTBJIHeTCH IIpOTHB 'IacoBofiCTpeJIKH). a) l: Izl = ~; 6) l: Iz + il = 1.

a a) TaK KaK <PYHKIJ;HH

z+i

aHaJIHTH'IHaHa Bcefi KOMIIJIeKCHOfi IIJIOCKOCTH C, KpOMe TO'IKH-i, KOTOPM He JIe:>KHT BHYTPH OKPY:>KHOCTH Izl = ~ H Ha 9Tofi OKPY:>KHOCTH, TO 110 TeopeMe

KOllm (3.4) IIOJIY'IHM:		f z+zz2dz.	= o.
		Izl=~
6) TaK KaK <PYHKIJ;HH j(z) = Z2 aHaJIHTH'IHaHa Bcefi KOMIIJIeKCHOfi IIJIOC-
KOCTH C, H TO'IKaZo =		- i JIe:>KHT BHYTPH OKPY:>KHOCTH Iz + il = 1, TO 110
IiHTerpaJIbHOfi <popMYJIe KOIIIH (3.5) IIOJIY'IHM:
		1	!	Z2
	j(zo = -z). = -2.		!	-- . dz,
		rrz		z + z
		Iz+il=l
OTKy,n;a
	f ·Ldzz+z = 2rri· j(zo = -i) = 2rri· (_i)2 = 2rri· (-1) = -2rri. •
IZ+il=l
7.3.7.	BbI'IHCJIHTbHHTerpaJI

459

<<< < Предыдущая 34 35 36 37 38 39 40 41 42 43 44 4546 / 6046 47 48 49 50 51 52 53 54 55 56 57 58 > Следующая >>>

Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]

#
01.05.2025163.16 Кб0САПР_ЛЕК_C_0413.docx
#
01.07.202510.46 Mб0САПЭАиТ - М 1-6.rtf
#
04.09.2019147.46 Кб12сацо-30.doc
#
01.05.202575.66 Кб7Саша реферат.docx
#
01.04.20252.56 Mб0Сб.з.ТТСС1.doc
#
15.03.201527.38 Mб68Сборник задач по высшей математике 2 том.pdf
#
15.03.20151.11 Mб97сборник иб .doc
#
15.03.201513.6 Кб50сводная таблица налоги.docx
#
15.03.2015140.33 Кб24Связи с общественностью.docx
#
04.09.201995.23 Кб15СГП.doc
#
23.11.2019207.16 Кб4СДЭ.docx