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 36 37 38 39 40 41 42 43 44 45 46 47 4849 / 6049 50 51 52 53 54 55 56 57 58 59 60 > Следующая >>>

o. TaK

yrrpOrn;aH 9TO BbIpaJKeHMe, rrOJIy'lMM

2(3 + 4i)

6 + 8i

3 -

= ..,------..,-----'--

(3 - 4i)(3

+ 4i)

9 + 16

25·

6 + 8i

IITaK, res.f(z) = - 34·

- 25 .

-2t

- Z

Haft,l1.eM BbI'IeTB TO'lKe1, 3arrMcaB <PYHKIIMIO fez)

B BM,l1.e

z+l

fez) =

~((zz))

(z + 2i)2

' r,l1.e

(

)

= 0,

)

...J.

'f'

Z _ 1

1 ;- 0,

TOr,l1.a

z+l

resf(z) =

cp(z)

(z + 2i)21

z + 1

."/( )

z=l

z=l -

(z + 2i)2

z=l

'f'Z

6 + 8i

(1 + 2i)2

-3 + 4i

-25·

Haft,l1.eM BbI'IeTB 6eCKOHe'lHOy,l1.aJIeHHoft TO'lKe:

•

r!sf (z) = - (~~~f(z) + r?sf(z))

6+8i

6+8i)

= o.

= - ( 25 - 25

7.5.2. HaftTM BbI'IeTbI<PYHKIIMM fez) = si~z BO Bcex OC06bIX TO'lKaX, orrpe,l1.eJIMTb MX TMrr, HaiiTM BbI'IeTB 6eCKOHe'lHOY,l1.aJIeHHoft TO'lKe.

a Oco6oft TO'lKOft,l1.aHHoft <PYHKIIMM, O'leBM,l1.HO,.HBJI.HeTC.H TO'lKaz =

KaK cyrn;ecTByeT KOHe'lHbIftrrpe,l1.eJI

lim f (z) =	lim si~z = 1 (1-ft 3aMe'laTeJIbHbIftrrpe,l1.eJI),
z-+zo	z-+zo

TO 0 - YCTpaHMMaH oco6aH TO'lKaM, 3Ha'lMT,

resf(z) = O. o

OTCIO,l1.a B CMJIY TeopeMbI 7.4 rrOJIy'lMM

resf(z) = -resf(z) = o.

		00		0				•
7.5.3.								•
7.5.3.		HaftTM BbI'IeTbI<PYHKIIMM
		fez) =		cosz
			(2z -		11")2
		BO Bcex OC06bIX TO'lKax,orrpe,l1.eJIMTb MX TMrr, HaftTM BbI'IeTB oec-
		KOHe'lHOy,l1.aJIeHHoft TO'lKe.
a Oco6oft TO'lKOft,l1.aHHoft <PYHKIIMM,				O'leBM,l1.HO, .HBJI.HeTC.H TO'lKaz				- ~.
TaKKaK
l-+im_7r		[fez) (z - 7£2)2] = lim	cosz		. (z _ 7£)2 = lim		cosz = 0
Z	2	z-+~ (2z -		11")2	2	z-+~	4	'
	2	z-+~ (2z -		11")2	2	z-+~	4	'

480

TO ~ He HBJUleTCH cyrn;ecTBeHHO oc060it TOqKOit HJIH IIOJIIOCOM IIOpH.D:Ka BbIlIIe

l-ro H, CTaJIO 6bITb, MO)KeT 6bITb JIH60 YCTpaHHMoit oc060it TOqKOit, JIH60 1I0JIIOCOM l-ro IIOpH.D:Ka. Hait.D:eM IIpe.D:eJI

lim [J(z) (z -	zr.)]	= lim	cos z . (z _ zr.)		= lim	cos z
z~1!:	2	z~1!:	(2z _11")2	2	z~1!:	2(2z -	11")
2		2			2
[HeoIIPe.D:eJIeHHOCTb BH.D:a g, BOCIIOJIb3yeMcH IIpaBHJIOM JIOIIHTaJIH]
	= lim			= lim -		•	7r
	= lim		(cosz)'	= lim -	sinz = _ sm"2 = _1.
		Z~~ (2(2z - 11"))'		Z~~	4	4	4
OTCIO.D:a CJIe.n:yeT, 'ITOTOqKa ~ HBJIHeTCH IIOJIIOCOM l-ro IIOpH.D:Ka H
	r~sJ(z) = }~~ [J(z) (z - ~)] = -~.
	2		2
CJIe.D:OBaTeJIbHO,			"2	1			•
			"2	1			•
		resJ(z) = -resJ(z) = -4 •
		00	11"

7.5.4. HaitTH BbIqeTbI <PYHKlIHH J(z) = z4 ez BO Bcex OC06bIX TOqKax, OIIpe.D:eJIHTb HX THII, HaitTH BbIqeT B 6ecKoHeqHO Y.D:aJIeHHoit TOqKe. a Oc060it TOqKOit .D:aHHoit <PYHKlIHH, OqeBH.D:HO, HBJIHeTCH TOqKa z = O.

OIIpe.D:eJIHM THII 9TOit oco60it TOqKH C IIOMOrn;blO pa3JIO)KeHHH B pH.D: JIopaHa 110 CTeIIeHHM z:

J()	4!	4 ~ 1	4 3 1 2 1 1 1 1 1 1
z =zez=z		·L.",---n=z +z +2'z +3,z+4'+5':Z+6,2"+'"
		n=O n.z	..... z

~aHHblit pH.D: CXO.D:HTCH B KOJIblle 0 < Iz I < 00 H CO.D:ep)KHT 6eCKOHeqHOe 'fHCJIOqJIeHOB C oTpHlIaTeJIbHbIMH CTeIIeHHMH z. CJIe.D:OBaTeJIbHO, TOqKa 0 - cyrn;ecTBeHHo oc06aH TOqKa <PYHKlIHH J(z). K09<P<PHlIHeHT C-l IIpH l paBeH

1	1
5!	= 120' 3HaqHT,

1 r~sJ(z) = 120'

TaK KaK IIOJIyqeHHbIit pH.D: CXO.D:HTCH B OKpeCTHOCTH 6ecKOHeqHO Y.D:aJIeHHoit

1
TOqKH, H C-l = 120' TO
r::J(z)	1	•
r::J(z)	= -r~sJ(z) = -120'	•

HaiJ.mu 6W"I.emW oaHHwx !PYH'lCqUiJ. 60 6cex oc06wx mO"l.'lCax u onpeoe.t&um'b ux mun, HaiJ.mu 6W"I.em 6 6eC'lCOHe"l.HO yOa.tteHHoiJ. mO"l.'lCe.

7.5.5.	J(z)	3	7.5.6.	eZ
		= - 2'		J(z) = --.,
		z-		z -	1I"Z
7.5.7.	J(z)	= chz + 3	7.5.8.	Z2	+ 1
		1- 2z .		J(z) = (z + i)2

16 CooPHHK 38,IUIq no BWeweA Mll1CMaTIIKe. 2 "YJ!C

481

7.5.9.	J(z) =	e2z		7.5.10.	J(z) =	chz
	J(z) =	(2iz -1)2				(Z - 1)3
7.5.11.	J(z) =	z -	1	7.5.12.	J(z) =	z	.
		(z + 1)(z - 2i)				(z + 3i)(z	+ 2)2
7.5.13.	J(z) =	cosz		7.5.14.	J(z) =	1
7.5.13.	J(z) =	z2(3z -	1T) .	7.5.14.	J(z) =	z3 - Z5'
7.5.15.	J(z) =	1		7.5.16.	J(z) =	cos;;..
7.5.15.	J(z) =	z3(z2 -	9)2			Z -"2
7.5.17.	J(z) =	si~z.		7.5.1S.	J(z) =	1 - cosz
7.5.19.	J(z) =	Z		7.5.20.	J(z) =	z4
7.5.19.	J(z) =	COSZ·		7.5.20.	J(z) =	si~z'
		1
		1			J(Z) =	ZCOS}.
7.5.21.	J(z)=e z.			7.5.22.	J(Z) =	ZCOS}.

7.5.23.llCIIOJIb3Yfl OCHOBHYIO TeopeMY 0 BbI'ieTax, BbI'UICJIHTb HHTerpaJI

	f z5 dz- z3' r,lJ;e l - OKPY)KHOCTb		1
	f z5 dz- z3' r,lJ;e l - OKPY)KHOCTb	Izl =	-2 .
a	l
a	Oc06bIMH TO'IKaMH <PYHKII;HH J(z) =	5 1	3 =	3	1	flBJIfl-
		Z -	Z	Z (z -	1)(z	+ 1)

IOTCfl TO'IKH 0, -1, I.ll3 HHX BHYTPH KPHBOii l									(BHYTPH OKPY)KHOCTH Izl = ~),
O'IeBH,lJ;HO, HaXO,lJ;HTCfl TOJIbKO TO'IKa Zl							=	O. 3Ha'IHT,
					fJ(z) dz = 21Ti . r~sJ(z).
					l
TO'IKa Zl	=	0	flBJIfleTCfl IIOJIIOCOM				3-ro IIOpfl,lJ;Ka. Haii,lJ;eM BbI'IeT B 3TOii
TO'IKe:
resJ(z) =	2\ lim (f(z) . (z - 0)3)" =						1		( 3;	. z3)"
resJ(z) =	2\ lim (f(z) . (z - 0)3)" =						-2 lim		( 3;	. z3)"
o	. z-tO				(Z2 - 1 )" -			z-tO	Z (z - 1)
	-		2	z-tO	(Z2 - 1 )" -		2	z-tO	((z2 - 1)2 )'
			1	l'	1		1	l'	-2z
	--1m				--	--1m
	1.			-2(z2 _1)2 -		2(Z2 - 1)· 2z· (-2z)				1
= - hm						(Z2 -	1)4			= - ·( - 2)= - 1 .
	2	z-tO				(Z2 -	1)4			2
	f	5dz		3 =	21Ti'res	51	3 =21Ti·(-I)=-21Ti.				•
	I z		-z		0	z -z					•

7.5.24.llCIIOJIb3Yfl OCHOBHYIO ToopeMY 0 Bbl'IeTax, Bbl'IHCJIHTb HHTerpaJI f r,lJ;e l - OKPY)KHOCTb Izl =. 2 3.dz,z+ 1

l (z + z~)	(z - 1)
a cI>YHKII;Hfl J(z) =	z+1
	(z + i)	2	(z - 1)	aHaJIHTH'IecKM BO Bceii KOMIIJIeKCHOii
IIJIOCKOCTH 3a HCKJIIO'IeHHeM TO'IeK Zl				=	-2i H Z2 = 1, KOTopble JI€)KaT BHY-
TpH OKPY)KHOCTH Izl = 3. 3Ha'IHT,
f	J(z) dz = 21Ti (res J(z) + res J(Z)) .
			Zl		Z2

482

TaK KaK

6 + 8i

(

)

()

resJ(z) = resJ(z)

6+ 8i

res J

= res J z

- 25 '

= -- 25

-2.

(CM. 3a,r:J;a'IY 7.5.1), TO

(

z·;;t

) dz = 21Ti (resJ(z) + resJ(z))

f z

+ ZZ

z -

-2.

= 2 . (6 + 8i _ 6 + 8i) = 0

•

1TZ

BU"I,uc.I/,um'b UUme2pa.l/,'bt no 3aaaUUO.MY r;;oumypy l, Ucno.l/,'b3YSl OCU06UY1O meope.My 0 6U"I,emax:

7.5.25.		dz
7.5.25.	f (z - 1)(z		+ 2)'
	f (z - 1)(z		+ 2)'
	I			6) l: Iz + 21 = 1;
	a) l: Iz - 11		= 1;	6) l: Iz + 21 = 1;
	B) l: Izl = 3.
7.5.26.	f	cosz	dz, l:	Izl = 1T.
	f	(2z - 1T)2

Haf1.mu 6'bt"l,emU aauuux fPyur;;v,uf1. 60 6cex oc06ux mO"l,r;;ax, onpeae.l/,um'b ux mun, uaf1.mu 6'bt"l,em 6 6ecr;;oue"l,uo yaa.l/,euuof1. mO"l,r;;e.

7.5.27.	J(z) =		sin2;..			7.5.28.	J(z) =		cos3z
			z -"2z						(2z + 1T)2
7.5.29.	J(z)=		zsin2z			7.5.30.	J(z) =		cos3z .
			(z + 1Ti)3						(z + 2)5
7.5.31.	J(z) =		e- 2z			7.5.32.	J(z) =		1
7.5.31.	J(z) =		Z2(Z -	4i)		7.5.32.	J(z) =		(1- Z2)3
			Z2(Z -	4i)					eZ -1
7.5.33.	J(z) =		(z ~l)n' n		E N.	7.5.34.	J(z) =		eZ -1
7.5.33.	J(z) =		(z ~l)n' n		E N.	7.5.34.	J(z) =		- 3 - ·
									z
7.5.35.	J(z) = ctg2 z.					7.5.36.	J(z) = z3 sin l.
BU"I,uc.I/,um'b UUme2pa.l/,'bt				no	3aaaUUO.MY r;;oumypy			l,	UCnO.l/,'b3YSl OCU06UY1O
meope.My 0 6'bt"l,emax:
7.5.37.		dz
7.5.37.	f z(z + 2)3'
	f z(z + 2)3'
	I	Izl = 1;					l: Izl
	a) l:	Izl = 1;				6)	l: Izl	= 3.
7.5.38.			dz
7.5.38.	f (z + 1)3(z -			1)2'
	f (z + 1)3(z -			1)2'
	I	Iz + 11 = 1;					l: Iz - 11		= 1;
	a) l:	Iz + 11 = 1;				6)	l: Iz - 11		= 1;
	B) l:	Izl	= 3.

483

7.5.39.	f sinz dz				'
	I	z
				Iz - 21		= Ij	6) l: Izl = 1.
	a) l:
7.5.40.	fz	4		1		Izl = 2.
			ez dz, l:
	I
7.5.41.	MO)KeT JIH Y <PYHKlI,HH J(z)						B H30JIHpOBaHHoit oc060it TOqKe:

a)6bITb BbIqeT (pOBHO O,n;HH)j 6) He 6bITb HH O,n;HOro BbIqeTaj

B)6bITb 60JIee O,n;HOro BblqeTa?

7.5.42.IIycTb resJ(z) = O. BepHo JIH, 'ITO

	f	J(z)dz = 0
	Iz-zol=R
	,n;JIjI JII060ro R > O?
7.5.43.	IIycTb res J(z) = o. BepHo JIM, 'ITOTOqKa Zo He jlBJIjleTCjI nOJIIOCOM
	Zo
	l-ro nopjI,n;Ka <PYHKD;HH J(z)?
7.5.44.	,II;oKa3aTb, 'ITOres(f + g)(z)		= resJ(z) + resg(z),		eCJIH Bce TpH
	Zo		Zo	Zo
	BbIqeTa cyrn:eCTBYIOT.
7.5.45.	,II;oKa3aTb, 'ITO eCJIH J(z)	-	HeqeTHM <PYHKD;HjI,		TO res J(z)
					Zo
	= - res J(z ), eCJIH XOTjI 6b1 O,n;HH H3 BbIqeTOB cyrn:ecTByeT.
7.5.46.	-Zo
7.5.46.	BblqHCJIHTb	z
	res	z	2.
	res		2.

°chz-l- Z2

7.5.47.

1 + Z8

J(z)=cosz·chz· z4 (z 4 + 1)

BO Bcex OC06bIX TOqKax H B 6eCKOHeqHO y,n;aJIeHHoit TOqKe 00.

KOHTPOl1bHASI PA60TA

BapMaHT 1

1.RaitTH 3HaqeHHe <PYHKD;HH J(z) = cos3z B TOqKe ~i. YKa3aTb TOqKH, B KOTOPbIX cyrn:ecTByeT npOH3Bo,n;HM f' (z).

2.Onpe,n;eJIHTb, MO)KeT JIH <PYHKD;HjI eY sin x+x 6bITb ,n;eitcTBHTeJIbHoit qacTbIO aHaJIHTHqeCKoit <PyHKD;HH J(z)? ECJIH ,n;a, TO HaitTH J(z).

3. BbIqHCJIHTb fRe(z2) dz, r,n;e l - .n;yra napa60JIbI y = 2X2 OT TOqKH Zl = 0

,n;o TOqKH Z2 = 1 + 2i.

484

4. HaihH pa3JIO:>KeHHe cPYHKIJ;HH --L3 B pll,lJ; JIopaHa B TO'fKeZo = 1. YKa-

3aTb rJIaBHYIO H npaBHJIbHYIO 'faCTHpll,lJ;a H 06JIacTb ero CXO,lJ;HMOCTH.

5. HanTH Bce oco6ble TO'fKHcPYHKIJ;HH --Ll sin --L , onpe,lJ;eJIHTb HX THn,

Z+ z- 1

,nJIlI nOJIlOca HanTH ero nOpll,lJ;OK. HanTH BbI'feTbIBO Bcex OC06bIX TO'fKaXH B 6eCKOHe'fHOY,lJ;aJIeHHOn TO'fKe.

BapMaHT 2

1. HanTH 3Ha'feHHecPYHKIJ;HH J(z) = ch Z B TO'fKe2 + Tri. YKa3aTb TO'fKH,B KOTOPbIX cyrn:eCTByeT npOH3BO,lJ;HM f' (z).

2.	Onpe,lJ;eJIHTb, MO:>KeT JIH cPYHKIJ;HlI	cos x sh y - 2y 6bITb MHHMOn 'faCTblO
aHaJIHTH'feCKOncPYHKIJ;HH J(z) ? ECJIH ,lJ;a, TO HanTH J(z ).
3.	BbI'fHCJIHTb1(2Z + l)z dz, r,lJ;e l -	,IJ;yra OKPY:>KHOCTH Izl = 1 OT TO'fKH
	I
Zl	= 1 = ei .O ,lJ;O TO'fKHZ2 = -1 = ei .1r •

4. HanTH pa3JIO:>KeHHe cPYHKIJ;HH cos(z - 1) B pll,lJ; JIopaHa B TO'fKeZo = o.

YKa3aTb rJIaBHYIO H npaBHJIbHYIO 'facTHpll,lJ;a H 06JIaCTb ero CXO,lJ;HMOCTH.

5. HanTH Bce oco6ble TO'fKHcPYHKIJ;HH

__Z_e z + 1

Z2 -1 '

onpe,lJ;eJIHTb HX THn, MlI nOJIlOca HanTH ero nOp1l,lJ;OK. HanTH BbI'feTbIBO Bcex OC06bIX TO'fKaXH B 6ecKoHe'fHOY,lJ;aJIeHHOn TO'fKe.

BapMaHT 3

1.	HanTH 3Ha'feHHecPYHKIJ;HH	J(z) =	~ B TO'fKe 1 - i. YKa3aTb TO'fKH, B
KOTOPbIX cyrn:ecTByeT npOH3BO,lJ;HM f' (z).
2.	Onpe,lJ;eJIHTb, MO:>KeT JIH cPYHKIJ;HlI		e-z cos Y + 2x 6bITb ,lJ;enCTBHTeJIbHOn
qacTblO aHaJIHTH'feCKOncPYHKIJ;HH J(z)? ECJIH ,na, TO HanTH J(z).
3. BbI'fHCJIHTbIlzl dz, r,lJ;e l -		OTpe30K npllMOn OT TO'fKHZl = 0 ,lJ;O TO'fKH
	I
Z2	= 3 - 2i.

4. HanTH pa3JIO:>KeHHe cPYHKIJ;HH z! 1 B pll,lJ; JIopaHa B TO'fKeZo = 2. YKa3aTb rJIaBHYIO H npaBHJIbHYIO 'facTHpll,lJ;a H 06JIacTb ero CXO,lJ;HMOCTH.

5	Z - i	1
. HanTH Bce oco6ble TO'fKHcPYHKIJ;HH	z + 1 cos Z' onpe,lJ;eJIHTb HX THn, MlI

nOJIIoca HanTH ero nOpll,lJ;OK. HanTH BbI'feTblBO Bcex OC06bIX TO'fKaXH B 6eCKOHe'fHOY,lJ;aJIeHHOn TO'fKe.

485

BapMaHT 4

1. HaitTH'3Ha'IeHlle<PYHKIJ;HH J(z) = sh iz B TO'IKe~ - i. Y Ka3aTb TO'IKH,B KOTOPbIX CYIIJ;eCTByeT "POH3BO,n;HruI f' (z ).

2. OIIpe,n;eJIHTb, MO)J(eT JIH <PYHKIJ;HjI cos Y ch x - y 6bITb MHHMOit 'IaCTbIOaHaJIHTH'IeCKoit<PYHKIJ;HH J(z)? ECJIH ,n;a, TO HaitTH J(Z).

3. BbI'IHCJIHTb

fZ;z,

r,n;e l - ,n;yra OKPY)J(HOCTH Izl = 1 OT TO'IKHZl = 1 = ei .O ,n;o TO'IKHZ2 = -i =

= ei(-~).

4. HaitTH pa3JIo)J(eHHe <PYHKIJ;HH sin(2 - z) B PjI,n; JIopaHa B TO'IKeZo = o.

YKa3aTb rJIaBHYIO H IIpaBHJIbHYIO 'IaCTHPMa H 06JIaCTb ero CXO,n;HMOCTH.

5.	z	+ 1	1
	HaitTH Bce oc06bIe TO'IKH<PYHKIJ;HH z	_ 1 sh Z' OIIpe,n;eJIHTb HX THII, ,n;JIjI

IIOJIIOCa HaitTH ero IIOPjI,n;OK. HaitTH BbI'IeTbIBO Bcex OC06bIX TO'IKaX H B 6ecKoHe'IHOy,n;aJIeHHoit TO'IKe.

060-

f(t)· X(t):

rllasa 8. onEPAu.l'IOHHOEl'IC'-Il'ICllEHl'IE

§1. OPVlnlHAll Vl306PA>KEHVlSI. nPE06PA30BAHVlE llAnllACA. HAXO>KAEHVlE Vl306PA>KEHVI~

Opl>'1rIl1Han111 npeo6pa30aaHII1e llannaca

~KOMnJIeKCH03Ha'lHaH<PYHKD;HH f(t) .n;eii:cTBHTeJIbHOrO nepeMeHHoro Ha3bIBaeT-

0pU2U'H.a.n.OM,
CH		eCJIH OHa y.n;OBJIeTBOpHeT CJIe.n;yIOrn;HM TpeM YCJIOBHHM:
1.	f(t)	= °)J.JIH Bcex t < OJ
2.	f(t)	- a6COJIIOTHO HHTerpHPyeMa6 Ha JII060M oTpe3Ke [0, a] nOJIOlKHTeJIbHoit
nOJIYOCHj
3.	cyrn;eCTBYIOT .n;eitcTBHTeJIbHble '1HCJIaM > 0, to ~ °H S TaKHe, 'ITOIf(t)1		<
< M est npH Bcex t > to ~ 0.			~

IIpocTeitIIIHM OpHrHHaJIOM HBJIHeTCH fPY'H.J(;v,W/, Xe6ucaiiaa, onpe.n;eJIHeMaH CJIe-

.n;yIOrn;HM 06pa30M:
x(t) = {O,	npH t < 0,
1,	npH t ~ 0.

ECJIH <PYHKD;HH cp(t) y.n;oBJIeTBOpHeT YCJIOBHHM 2 H 3, TO <PYHKD;HH f(t) = cp(t) . x(t) y.n;OBJIeTBOpHeT H YCJIOBHIO 1, T. e. HBJIHeTCH 0pHrHHaJIOM. Pa.n;H ynporn;eHHH 3anHCH B .n;aJIbHeitIIIeM, 3a He60JIbIIIHM HCKJIIO'IeHHeM,6y.n;eM nHcaTb f(t) BMeCTO

~ IIycTb f(t) - OpHrHHaJI, a p = Q +ifJ - KOMnJIeKCHOe '1HCJIO.H306pa-:>tCe'H.UeM

OpHrHHaJIa f(t) Ha3bIBaeTCH <PYHKD;HH F(P), onpe.n;eJIHeMaH paBeHcTBoM:

F(P) = jf(t)e- Pt dt. o

<I>yHKD;HH F(P) Ha3bIBaeTCH TaKlKe npeo6pa306a'H.UeM JIan.n.aca OT <PyHKD;HH f(t). ~

MOlKHO nOKa3aTb, 'ITOHec06cTBeHHbIit HHTerpaJI B onpe.n;eJIeHHH H306palKeHHH CXO.n;HTCH )J.JIH 3Ha'leHHitp, y.n;OBJIeTBOpHIOrn;HX YCJIOBHIO Rep> s, a onpe.n;eJIHeMaH HM <PyHKD;HH F(p) HBJIHeTCH aHaJIHTH'IeCKoit<pYHKD;Heit B nOJIynJIOCKOCTH Rep> s.

TOT <paKT, 'ITO<PyHKD;HH F(P) HBJIHeTCH H306palKeHHeM OpHrHHaJIa f(t),

3Ha'laIOTTaK: f(t) -+t F(P) HJIH F(P) = L{f(t)}.

Pa3HbIM OpHrHHaJIaM COOTBeTCTBYIOT pa3Hble H306palKeHHH, TO'lHeeHMeeT MeCTO CJIe.n;yIOrn;aH

TeopeMa 8.1 (€!AII1HCTBeHHOCTII1 II1s06pa)l(eHII1H). ECI1I11 oplllrlllHal1bl f(t) III g(t)

HenpepblBHbl III IIIMelOT OAIliHaKOBoe 111306pa)l(eHille F(P), TO 3TIII cPYHKlIlII1II COBna-

AaIOT.

6To'lHee:HHTerpa.rr ,n;OJIlKeH CYIIIeCTBOBaTb XOTH 6bI B Hec06cTBeHHOM CMbICJIe.

487

CBoiiicTBa npe06pa30BaHMR flannaca

B ,ll;aJIbHethneM BCIO,!I;y, eCJIH He oroBopeHo npoTHBHoe, f(t) 0603Ha'iaeTHeKHti: OPHPHHaJI. Ih06paJKeHHe OpHPHHaJIa 0603Ha'iaeTCHTOti: lKe 6YKBOti:, TOJIbKO 3arJIaBHOti:, HarrpHMep: f(t) ~ F(p), g2(t) ~ G2(P) H T.,II;. BaJKHeti:mHe CBoti:cTBa rrpe06pa30BaHHH JIarrJIaca OTpaJKeHbI B CJIe,!l;yIOm;HX BOCbMH TeopeMax.

TeopeMa 8.2 (CBOMCTBO nIllHeMHocTIII). Ami npOlll3BOl1bHbiX KOMnl1eKCHblX noCTOHHHblX a III f3 cnpaBeAl1l11BO COOTHoweHllle: a· f(t) +f3. g(t) ~ aF(p) + f3G(p).

TeopeMa 8.3 (nOA06I11R). Al1H 11Io6oro Ae"CTBIllTel1bHOrO r > 0 cnpaBeAl1l11BO coOTHoweHllle: f(r· t) = ~. F (~).

TeopeMa 8.4 (cMe~HIIIR). Al1H 11Io6oro KOMnl1eKCHoro '1I11Cl1a po IIIMeeT MeCTO COOTHoweHllle: epot • f(t) ~ F(P - po).

TeopeMY CMem;eHHH (H306paJKeHHH) Ha3b1BaIOT HHOP,II;a TeopeMoti: C,!I,BHPa (H30-

6paJKeHHH).

TeopeMa 8.5 (3anaSAbIBaHIIIR). Al1H 11Io6oro Ae"CTBIIITeJ1bHOrO n0110>KIIITeJ1bHoro '1I11Cl1a r IIIMeeT MeCTO COOTHoweHllle: f(t - r) ~ e-p-r· F(P).

TeopeMY 3arra3,11;blBaHHH (OpHPHHaJIa) pelKe Ha3b1BaIOT TeopeMoti: CMem;eHHH

HJIH C,II;BHPa OpHPHHaJIa.

CJIe,!l;yeT OTMeTHTb, 'ITOrrpH rrpHMeHeHHH TeopeMbI 3arra3,11;bIBaHHH HYlKHO rroMHHTb, 'ITOrro HameMY COPJIameHHIO rro,ll; <PYHKD;Heti:-opHPHHaJIOM f(t) rrOHHMaeTCH

<PYHKD;HH f(t)· X(t); rr03TOMY rro,ll; <PYHKD;Heti: f(t - r) CJIe,!l;yeT rrOHHMaTb <PYHKD;HIO f(t - r) . X(t - r), a He f(t - r) . x(t). IIpH HCrrOJIb30BaHHH 3TOti: TeopeMbI YMeCTHO He HCrrOJIb30BaTb COKpam;eHHYIO 3aIIHCb ,!I;JIH OpHPHHaJIa H rrpHrrHCblBaTb <PYHKD;HIO XeBHcati:,II;a B Ka'leCTBeCOMHOlKHTeJIH. TeopeMY 3arra3,11;blBaHHH '1acTOHCrrOJIb3yIOT

,!I;JIH HaxolK,II;eHHH H306paJKeHHti: rrepHO,ll;H'IeCKHX<PYHKD;Hti:.

TeopeMa 8.6 (0 AlllclJclJepeH4IIIPOBaHIIIIII		oplllrIllHana). ECl1l11 ct>YHKLlIIIH f(t) III ee
npOlll3BOAHbie HBl1HIOTCH oplllrlllHal1aMIll III		f(t)	~ F(P), TO
!,(t) ~ pF(P) -			f(O),
!"(t) ~ p2 F(P) - pf(O) - !'(0),
f(n) (t) ~ pn F(P) _ pn-1 f(O) -	pn-2!' (0) -			... - pf(n-2) (0) -	/n-1) (0);
nplll 3TOM nOA f(k)(O) nOHIllMaeTCH	lim	f(k)(t),		k = 1,2, ... , n -	1.
	t~+O

488

9TH <P0PMYJIbI 3aMeTHO yrrpOm;aIOTCR, eCJIH f(O) = 1'(0) = ... = t<n-l)(o) = O.

B aTOM CJIY'Iae:

!'(t) -+t pF(P) , !"(t) -+t p2 F(P), ... ,

TeopeMa 8.7 (0 AlllclJclJepeH~poBaHIIIIII 111306pa)l(eHIIIR).

ECl1l11 f(t)

-+t

F(P) TO

-tf(t) -+t

F' (P). B 6011ee 06U4eM Cl1YLfae: (-lttn f(t) -+t

F(n) (p).

1'13aToit TeopeMbI, B '1acTHOCTH,rrOJIY'IaeM:t -+t

2 .

t7t.

-+t

pn+l'

p2 '

p3' ... ,

TeopeMa

8.8 (06 IIIHTerplllpOBaHlII1II oplllrIllHana).

ECl1l11

f(t)

-+t F(P), TO

F~).

/f(T)dT-+t

TeopeMa 8.9 (06 IIIHTerplllpOBaHlII1II 111306pa)l(eHIIIR).			ECl1l11 f(t) -+t F(P) III IIIH-
00		fW	/00
Terpal1 /	F(P) dp RBl1ReTCR CXOARU4I11MCR, TO	-t- -+t	F(P) dp.
p			p

TeopeMbI 0 ,n;H<p<pepeHIJ;HpOBaHHH H HHTerpHpOBaHHH OpHrHHaJIa ,n;eMOHCTpHPYlOT TOT <paKT, 'ITOOrrepaIJ;HH ,n;H<p<pepeHIJ;HpOBaHHR H HHTerpHpoBaHHR OpHrHHaJIOB CBO,n;RTCR COOTBeTCTBeHHO K OrrepaIJ;HRM YMHO)KeHHR H ,n;eJIeHHR Ha p HX H306p~e

HHit.

B 3aKJIIO'IeHHerrpHBe,n;eM Ta6JIHU;Y H306p~eHHit HeKOTopbIX OCHOBHbIX <PYHKIJ;Hit (HH)Ke rro,n;pa3YMeBaeTCR, 'ITOa H (3 - KOMrrJIeKCHbIe '1HCJIa,n - HaTypaJIbHoe):

Nt	f(t)	F(P)	Nt	f(t)	F(P)
I	1	1	VII	eat sin (3t	(3
		p			(p_a)2+(32


II	eat	1	VIII	eat cos (3t	(p- a)
		p-a			(p - a? + (32

III	tn	n!	IX	t . sin (3t	2p(3
		--			(p2 + (32)2
		pn+l
IV	tneat	n!	X	t . cos (3t	p2 _ (32
					(p2 + (32)2
		(p - at+1
V	sin(3t	(3	XI	sh(3t	(3
		p2 + (32			---
					p2 _ (32
VI	cos (3t	P	XII	ch(3t	P
		p2 + (32			---
					p2 _ (32

489

<<< < Предыдущая 36 37 38 39 40 41 42 43 44 45 46 47 4849 / 6049 50 51 52 53 54 55 56 57 58 59 60 > 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