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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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8.1.1.	IIpOBepHTb, KaIme H3 CJIe.rr.yIOIIl;HX cPYHKD;Hii jlBJIjllOTCjI OpHrHHa-
	JIaMH, a KaKHe -		HeT.		= t.X(t);
	a) f(t)	= 2e3t • cos 2t . X(t);		6) f(t)	= t.X(t);
					O'	t < 0,
	B) f(t)	= e t2 . X(t);		r) f(t)	= { 1,	°~ t < 2,
					t,	t ~ 2.
a a) YCJIOBHe 1 B orrpe.n;eJIeHHH OpHrHHaJIa, OqeBH.n;HO, BbIIIOJIHeHO. ,il;aJIee,
rrpH t ~	°cPYHKD;HjI f(t) -		HerrpepbIBHa, a CJIe.n;OBaTeJIbHO H a6COJIIOTHO

HHTerpHpyeMa Ha JII060M oTpe3Ke [0, a]. 3HaqHT, YCJIOBHe 2 TaK)J(e BbIIIOJIHjIeTCjI. HaKoHeD;, 12e3t cos 2tl ~ 2e3t , H B KaqeCTBe KOHCTaHT M H S B YCJIOBHH

3 orrpe.n;eJIeHHjI OpHrHHaJIa MO)J(HO BbI6paTb JII060e M > 2 H S = 3. CJIe.n;oBaTeJIbHO, f (t) jlBJIjleTCjI 0pHrHHaJIOM.

6) f(t) He jlBJIjleTCjI OpHrHHaJIOM, rrOCKOJIbKY HHTerpaJI

o/t dt

pacXO.n;HTCjI, a CJIe.n;OBaTeJIbHO, He BbIrrOJIHeHO YCJIOBHe 2 orrpe.n;eJIeHHjI OPHrHHaJIa.

B) f(t) He jlBJIjleTCjI OPHrHHaJIOM, rrOCKOJIbKY HepaBeHCTBO e t2			< Me st
He MO)J(eT BbIIIOJIHjlTbCjI HH rrpH KaKHX S .n;JIjI Bcex t > 0, T. K.
lim _e	t 2	1
		= lim _et(t-s) = 00.
t-too M est		t-too M
OTclO.n;a CJIe.rr.yeT, 'ITO.n;JIjI JII06oro S BbIIIOJIHeHO HepaBeHCTBO e t2			> M est,

HaqHHaji C HeKOToporo 3HaqeHHjI t (HHbIMH CJIOBaMH, cPYHKD;HjI et2 paCTeT 6bICTpee JII060ii cPYHKD;HH M est). °

r) YCJIOBHe 1, OqeBH.n;HO, BbIrrOJIHeHO. IIPH t ~ cPYHKD;HjI HerrpepbIBHa

BCIO.rr.y, KpOMe TOqKH t = 1, B KOTOpoii OHa HMeeT pa3pbIB 1-ro po.n;a. CJIe.n;o-


BaTeJIbHO, f(t)		- HHTerpHPyeMa Ha JII060M OTpe3Ke [0, a].				T. K. If(t)1	~ et ,
TO H YCJIOBHe 3 TO)J(e BbIIIOJIHeHO. CJIe.n;oBaTeJIbHO, f(t) -						OpHrHHaJI.	•
8.1.2.	f(t)	= 3t . X(t).	8.1.3.	f(t)	= t 3 . X(t).
8.1.4.	f(t)	= e it . X(t).	8.1.5.	f(t)	= e-t2 . X(t).
8.1.6.	f(t)	= (t _1 1)2 • X(t).	8.1.7.	f(t)	=In t . X(t).
8.1.8.	f(t)	= tgt· X(t).	8.1.9.	f(t)	= e(2+i)t . X(t - 1).
8.1.10.	f(t)	= sin t· X(t + 1).	8.1.11.	f(t)	= t.X(t - 1).
8.1.12.	f(t)	= e it2 . X(t).	8.1.13.	f(t)	= 2 V't+t4 . X(t).

B .n;aJIbHeiillIeM BMecTO f(t)· X(t) MbI, KaK rrpaBHJIO, 6y.n;eM rrpOCTO rrHcaTb

f(t). '

490

8.1.14.HaiiTH H306proKeHHe <PYHKlI,HH f(t) = e(3+i)t, HCrrOJIb3YH rrpe06pa-

30BaHHe JIarrJIaca.

a f(t) HBJIHeTCH 0PHrHHaJIOM. TaK KaK le(3+i)tl < Me 3t .II.JlH M > 1, TO

H306proKeHHe F(P) 9TOii <PYHKlI,HH 6y.n;eT orrpe.n;eJIeHO H aHaJIHTHqHO B rrOJIYrrJIOCKOCTH Re p > 3. )J;aJIee, HaxO.n;HM

f(t)

-;-t

F(p) = !00e(3+i)t e -pt dt =

!00e-(p-3-1)t dt =

_(P_3_i)tI K

•

- K~oo P -

(3 + i) e

- p -

(3 + i)"

Hmto

ucnO.l/,'b3Y.R. npeo6pa30eaHU.R. JIan.!taca,

Haftmu

u306paJICeHU.R. c.!teoy'lO-

~UX opUZUHa.!toe:

8.1.15.

f(t)

= 2.

8.1.16.

f(t)

= e2t •

8.1.17.

f(t)

= cos4t.

8.1.18.

f(t)

= t.

f(t)

= {I,

E[0,1],

E [0,1],

8.1.19.

8.1.20.

f(t)

= { 1,

E (1,2]'

~ [0,1].

~ [0,2].

8.1.21.f(t) = et . x(t - 1).

8.1.22.llcrroJIb3YH Ta6JIHIJ,y H306proKeHHii H CBoiicTBO JIHHeiiHOCTH rrpe-

06pa30BaHHH JIarrJIaca, HaiiTH H306proKeHHH CJIe.n;yIOIllHx opHrH-

HaJIOB:		= 2 + t3 + t cos 2t;				= 3t ;
a)	f(t)	= 2 + t3 + t cos 2t;		6)	f(t)	= 3t ;
B)	f(t)	= cos2 t; ,		r)	f(t)	= sin 2t cos3t;
)1;) f(t) = et +5.
a a) 110 Ta6JIHlI,e HaxO.n;HM:
1 -;-t	1		= ~ (III),			p2 -4
	1			tcos2t-;-t (p2+4)2 (X).
	P (I), t 3 -;-t 31			tcos2t-;-t (p2+4)2 (X).
		p	p

CJIe.n;OBaTeJIbHO, rro cBoiicTBY JIHHeiiHOcTH (TeopeMa 1) rrpe06pa30BaHHH JIa-

IIJIaca rrOJIyqHM: 2 + t3 + t cos 2t = 2 . 1 + t3 + t cos 2t					2	6	+	p2	4
					-;-t p-	+ 4"	+	2	- 2.
6) 110CKOJIbKY 3t = et In 3, TO 3t -;-t						P		(P	+ 4)
6) 110CKOJIbKY 3t = et In 3, TO 3t -;-t		1		(II).
		p-ln3
B) llcrroJIb3YH H3BeCTHYIO TpHrOHOMeTpHqeCKYIO <P0PMYJIY rrOHIDKeHHH
CTerreHH, HMeeM:
2t _	1 + cos 2t	_ 1	1 + 1		2t
cos -	2	-2·		2 cos .
T. K. 1 -;-t p.! (I), a cos 2t -;-t	- p2 (VI), TO rro CBoiicTBY JIHHeiiHOCTH rrpe-
,	p +4
06pa30BaHHH JIarrJIaca rrOJIyqaeM: cos2 t -;-t			l	+	P
			2p	2(p2 + 4)

491

r) ITpe06pa3yeM OpliTHHa.JI f (t): sin 2t cos 3t = ~(sin 5t - sin t). Tor).l.a,

HCrrOJIb3Yjl <P0PMYJIY V Ta6JIHIJ;bI H CBOftCTBO JIHHeftHocTH rrpeo6pa30BaHHjI

JIarrJIaca, rrOJIyqaeM: sin 2t cos 3t -;+				1	5 2	- ~).
JIarrJIaca, rrOJIyqaeM: sin 2t cos 3t -;+				-2 ( 2	5 2	- ~).
				p	+5	p	+1
)I.) T. K. et+5 = e5 et H et -;+ ~1 (II),					TO et+5 = e5 ~1.			•
			p-				p-
HcnO.!I:b3Y.R. ma6.1/,u'4Y u306pa;)tCe'Hui:i, 'Hai:imu u306pa;)tCe'HU.R. opUZU'Ha.l/,06:
8.1.23.	f(t) = 3e-t		+ et cos 3t.	8.1.24.		f(t)	= 4sh2t - t2 •
				8.1.26.			1
8.1.25.	f(t)	= te2t -	sin 3t.	8.1.26.		f(t)	= t2 + 1.
8.1.25.	f(t)	= te2t -	sin 3t.			f(t)	= t2 + 1.
8.1.27.	f(t)	= tet - I	+ t2 et - 2 •	8.1.28.		f(t)	= sin3 t.
8.1.29.	f(t)	= cost cos 3t.		8.1.30.		f(t)	= sin 4t sin 2t -	t sin t.

8.1.31.f(t) = et cos2 t.

8.1.32.IIcrroJIb3Yjl TeopeMY H306pIDKeHHjI, HaftTH H306pIDKeHHe OpHrHHaJIa f(t) = e3t cht.

									P	OTCIO).l.a
o ITo <popMYJIe XII Ta6JIHIJ;bI H306pIDKeHHft HMeeM: ch t -;+ - 2 -- .										OTCIO).l.a
rro TeopeMe CMew:eHHjI (Po = 3) rrOJIyqaeM:								p	-1
rro TeopeMe CMew:eHHjI (Po = 3) rrOJIyqaeM:										•
		e3t cht -;+			p-3
		e3t cht -;+		(P - 3)		2	-1
Hai:imu u306pa;)tCe'HU.R. opUZU'Ha.l/,06, UCnO.l/,'b3Y.R. meope.M.y C.M.e~e'HU.R.:
8.1.33.	f(t)	= te2t cos 3t.		8.1.34.			f(t)	= e~ sh 2t.
8.1.35.	f(t)	= e4t cos2 t.		8.1.36.			f(t)	= te-t sin 2t.
8.1.37.	HaftTH H306pIDKeHHe <PYHKIJ;HH g(t) = cos(t - 2)x(t -								2).
o PaccMoTpHM <PYHKIJ;HIO f(t)			= cost· X(t). Tor).l.a
		g(t) = f(t -	2) = cos(t -			2) . X(t - 2).
,I1;JIjI OpHrHHa.JIa f(t) HMeeM: f(t) -;+				.....2	P	(VI). Tor).l.a rro TeopeMe 3arra3-
				p	+1
).l.bIBaHHjI OpHrHHa.JIa rrOJIyqHM: g(t) = f(t -						2) -;+ e-2P • - p - .				•
								p2 + 1
Hai:imu	u306pa;)tCe'Hue c.I/,eay'lO~ux fjJY'Hrt:'4ui:i:
8.1.38.	(t -	3)3 . X(t - 3).		8.1.39.			e2t -	4 • X(t - 2).

8.1.40.ch(2t - 1) . X (t - ~).

8.1.41.(t - ~) sin(3t - 71") . X (t - ~).

8.1.42.HaftTH H306pIDKeHHjI <PYHKIJ;Hft, 3a).l.aHHbIX rpa<pHqeCKH:

a)rpa<pHK <PYHKIJ;HH f(t) rrpHBe).l.eH Ha pHC. 108;

6) rpa<PHK <PYHKIJ;HH f(t) rrpHBe).l.eH Ha pHC. 109.

o a) 1I306pIDKeHHe <PYHKIJ;HH f(t) MOlKHO, KOHeqHO, HaftTH HerrOCpe).l.CTBeHHO, rrpHMeHHB rrpe06pa30BaHHe JIarrJIaca. O).l.HaKO rrpow:e rrpe).l.CTaBHTb ee B

492

f(t)				f(t)
				1
o	1	t						1		2	t
	Puc. 108							Puc. 109
BH.n;e f(t) =	X(t) - X(t - 1).	110 Ta6JIHIJ;e			X(t)		-;+		~	(I). OTcIO.n;a rro Teo-
peMe 3arra3.n;bIBaHIUI OpHrHHana HMeeM X(t					-	1)		-;+	e-P
peMe 3arra3.n;bIBaHIUI OpHrHHana HMeeM X(t					-	1)		-;+	p' CJIe.n;oBaTeJIbHO,
1	e-P
f(t) -;+ p -	p;
6) I1pe.n;cTaBHM <PYHKII,HIO		f(t)	B BH.n;e:
	f(t) = t . X(t)	- (t - 1) . X(t -				1) -		X(t - 2).
110 Ta6JIHIJ;e HaxO.n;HM g(t) = t'X(t)			-;+ --\- (III) H X(t)						-;+	~. ,I1;anee, comaCHO
				p
TeopeMe 3arra3.n;bIBaHHH opHrHHana, rrOJIyqaeM:
g(t - 1) = (t - 1) . X(t - 1)				e-P			X(t -			2)	e- 2p
g(t - 1) = (t - 1) . X(t - 1)				-;+ -p2		H	X(t -			2)	-;+ - p .
		p		p	-2p						•
OKOHqaTeJIbHO HMeeM: f(t) -;+		1		-p	-2p
OKOHqaTeJIbHO HMeeM: f(t) -;+		2" -		~ -	T'

Hai1mu u306paJICeHUJI opUaUHa.l!06, 3aaaHH'btX apatjju"tec'ICu:

8.1.43.rpa<PHK <PYHKIJ;HH f(t) rrpHBe.n;eH Ha pHC. 110.

f(t)

	I			f(t)
	I
	I
	I
1 -----!......--				1
	I	I
	I	I
	I	I
	I	I
o	I	I
o	1	2	t	2	t
	Puc. 110			Puc. 111

8.1.44.rpa<PHK <PYHKIJ;HH f(t) rrpHBe.n;eH Ha pHC. 111.

8.1.45. f(t)	-_ {Sin t,	t E [0,11'], r nrh	rh	f(	t	)	rrpHBe.n;eH Ha
8.1.45. f(t)	0,	PClA±'HK 'i'YHKIJ;HH			t		rrpHBe.n;eH Ha
	0,	t ¢ [0,11'].
pHc.112.

493

f(t)

					t			2			t
	Puc.	112				Puc. 113
8.1.46.	rpa<PHK <PYHKlI,HH f(t) npHBe).!.eH Ha pHC. 113.
									(Xl
8.1.47.	Hail:TH H306pIDKeHlfe <PYHKlI,HH, 3a).l.aHHoil: PH).!.OM:								2: X(t -		n).
								n=O
8.1.48.	Hail:TH H306pIDKeHHe nepHo).!.HqeCKoil: <PYHKlI,HH							f(t)		= {t}	(3).!.ecb
	{t} - ).!.P06HM qacTb qHCJIa t).
a cDYHKlI,HH f(t) -		nepHO).!.HqeCKM C nepHo).!.oM T				= 1.		Ha oTpe3Ke [0,1]
OHa 3a).!.aeTCH paBeHCTBOM f(t)			= t. PacCMOTPHM <PYHKlI,HIO
		()	= {	t,	t E [0,1],
		()
		cp t		0,	t f/. [0,1].
				0,	t f/. [0,1].
Ee MO)J{HO 3anHcaTb TaK)J{e B BH).!.e cp(t) = t· X(t) - (t -							l)X(t -			1) - X(t - 1).
					1		-P			-P
Tor).!.a ee H306pIDKeHHeM 6y).!.eT <PYHKlI,HH cD(p) = 2" -							~ -		e p . cDYHKlI,HH
					P		P
f(t) MO)J{eT 6bITb npe).!.CTaBJIeHa B BH).!.e pH).!.a

(Xl

f(t) = L cp(t - n), n=O

a ee H306pIDKeHHeM (no TeopeMe 3ana3).l.bIBaHHH OpHrHHaJIa) 6y).!.eT <PYHKlI,HH

(Xl
F(p) = 2: e- pn . cD(p). ITOJIyqeHHbIil: pH).!. npe).!.CTaBJIHeT c060il: 6eCKOHeqHO
n=O
y6bIBaIOIIJ,yIO (npH p > 0) reOMeTpHqeCKYIO nporpeCCHIOj H nOTOMY										) =
F(p) = <P(p)(1+e- +e-	2p	+ ... ) =	cD(p~ =	1 _ (l_ e-		P	_	e-	p	) =
P	2p		1-e P 1-e P p2			P			p
			1-e P 1-e P p2		p2			P

			eP			(1 - e- P - pe-P)	eP - 1 - p	•
			eP			(1 - e- P - pe-P)	eP - 1 - p
			eP - 1 .			p2	p2 (eP - 1) .
Hai1mu		u306pa:»ee'H.'IJ.R nepuoou"tec'ICux rPY'H.'IC'4ui1:
8.1.49.		f(t) = Isin tl·
8.1.50.		cDYHKlI,HH f(t)	3a).l.aHa rPa<PHKoM, KOTOPbIil: npHBe).!.eH Ha pHC. 114.
8.1.51.		cDYHKlI,HH f(t)	3a).l.aHa rPa<PHKoM, KOTOPbIil: npHBe).!.eH Ha pHC. 115.
8.1.52.		Hail:TH H306pIDKeHHe <PYHKlI,HH f(t) = t2 sin t.
a	ITo Ta6JIHlI,e H306pIDKeHHil: HMeeM:
			.	t	-tt	1
			sm		-tt	- 2 -- .
						P + 1

494

f(t)	-,	, -,		f(t)~
f(t)
			,
1	,	, ,	,	;~.,
	,	, ,	,	;~.,
	,	, ,	,
	,	, ,	,
0	I', 2',	3: 4'	5:
	, ,	,
-1	----~
	Puc.	114		Puc. 115

OTCIO)l;a no TeopeMe 0 )l;H<p<pepeHIJ;HpOBaHl1H H306pIDKeHHjI nOJIYQHM:

t2 sin t -;-t		(+-)"
		p	+	1
HaxO)l;HM
(		1	) "		(	-~		~ -2
(		+ 1	) "		(	(P2	+ 1)2	)'.....2
	p2	+ 1		=		(P2	+ 1)2	- (p2 + 1)3·

•

Hai1.mu u306pa~e'H'IJ.R tj)y'H'IC'IJ,ui1., ucno.l/,'b3Y.H. meope.My 0 autj)tj)epe'H'IJ,Up06a'HUU u306pa~e'H'IJ.R:

8.1.53.

f(t)

= t 2 cos2t.

8.1.54.

f(t)

= t3 sin t.

8.1.55.

f(t)

= tsh3t.

8.1.56.

f(t)

= t ch 2t.

8.1.57.

f(t)

= tet

sin t.

8.1.58.

HanTH H306pIDKeHHe <PYHKIJ;HH f(t)

1- et

= - t - .

Q ITo Ta6JIHIJ;e H306pIDKeHHn Han)l;eM H306pIDKeHHe <PYHKIJ;HH

cp(t) = 1 -

et -;-t

~ -

p ~ 1.

Tor)l;a no TeopeMe 06 HHTerpHpOBaHHH H306pIDKeHHjI HMeeM:

f(t) =

1 - et -;-t

/00(1 __1_) dp = [lnp -In(p _ 1)]1 00 =

p-1

•

p-1

hm In --

=O-ln-- =In--.

M -+00

P - 1

p -

Hai1.mu

u306pa~e'H'IJ.R c.lteaY70UJ,UX opW'lU'Ha.lt06:

8.1.59.

f(t)=sint.

8.1.60.

f(t) =

1 - cos t.

f(t) = cos3t -

cost.

8.1.61.

f(t)=~.

8.1.62.

495

					t
8.1.63.	HaiiTH H306pIDKeHHe <PYHKlI,HH f(t) =				freT dr.
					o
Q MO:lKHO, BbI'IHCJIHBHHTerpaJI, HaiiTH H306pIDKeHHe IIO Ta6JIHlI,e H306pa-
:lKeHHii. O,l.l.HaKO IIpOlll,e B ,l.l.aHHOM CJIY'IaeBOCIIOJIb30BaTbCjI TeopeMoii 06 HH-
TerpHpOBaHHH OpHrHHaJIa. ,I1;eiicTBHTeJIbHO, HMeeM: tet -;+							1 2.	Tor,l.l.a
						(p -1)
IIO TeopeMe 06 HHTerpHpOBaHHH OpHrHHaJIa IIOJIY'IHM:
	t		2: p =					•
	f reT dr -;+	1	2: p =		1	2.
	o	(p -	1)	p(p -		1)
He 6bt"tUC.II.RJl UHmeZpaJl.bt, Hai1mu u306paJICeH'U.R					cJl.eay'/Ow,ux fjjYHr;;u,ui1:
	t				t
8.1.64.	f r sin 2r dr.		8.1.65.		f r cos 3r dr.
	o				o
	t				t
8.1.66.	f r 2e2T dr.		8.1.67.		frcos 2rdr.
	o				o
	t

8.1.68.freTsin2rdr.

8.1.69.	f(t)	= 1. _ 2t.		8.1.70.	f(t)	= et sin 2t cos 4t.
	f(t)	5t	3t		f(t)	= e-t sin2 t.
8.1.71.		= te2tcos3t.		8.1.72.

8.1.73.f(t) = tX(t - 1). 8.1.74. f(t) = sin(2t - 4)x(t - 2).

8.1.75.f(t) = t2cost.

8.1.77.f(t) = t34t .

8.1.79.	f(t) = et	- e2t
		t

8.1.76.f(t) = t2sh 2t.

8.1.78. f(t) = sin 3t - sin t t

8.1.80.f(t) = frsin22rdr.

8.1.81.	f(t) = f r3 eTdr.			8.1.82.	f(t) = f re3T cos4rdr.
		o			o
		I,	tE[O,I],
8.1.83.	f(t)	= { 2 - t,	t	E (1,2],
		0,	t	~ [0,2].
Hai1mu u306paJICeH'U.R nepuoau"tecr;;ux fjjYHr;;u,ui1:
8.1.84.	f(t)	= Icostl·		8.1.85.	f(t) = arcsin(sin t).

496

(Xl

8.1.86.f(t) = L: (-1)n . X(t - n).

n=O

8.1.87.f(t) - rrepMo,n;MqeCKaJI <PYHKU:MjI C rrepMo,n;oM T = 1 M Ha rrpOMe-

:lKYTKe [0,1) 3a,n;aHHaJI paBeHCTBOM f(t) = t2 •

60nee CnO)KHble 3aAa'"lM

8.1.88.	MO:lKeT JIM <PYHKU:MjI F(p) = ~ 6bITb M306pa:lKeHMeM HeKOTOpO-
	I'DOpMrMHaJIa?				sm p
8.1.89. ECJIM f(t)		- OpMrHHaJI, TO 6y,n;eT JIM OPMrHHaJIOM <PYHKU:MjI:
	1) !t f(t) dt;				2) f'(t).
	o	M g(t) -
8.1.90.	ECJIM f(t)	M g(t) -	OpMrHHaJIbI, TO jlBJIjleTCjI JIM OpMrMHaJIOM
	<PYHKU:MjI	f(t)g(t)?
8.1.91.	nOKa3aTb, 'ITOeCJIM f(t) -				rrepMo,n;MqeCKajl <PYHKU:MjI, jlBJIjlIOru;aJI-
	CjI OPMrHHaJIOM, TO ee M306pa:lKeHMe F(p) orrpe,n;eJIeHO B rrOJIyrrJIOC-
	KOCTM Re p > O.
8.1.92.	,I1;oKa3aTb, 'ITOeCJIM f(t)			-tt	F(p), TO
	f(t)x(t - s)		-tt	F(p) - !s f(t)e- pt dt.
					o

§ 2. CSEPTKA cnYHKLJ.lJIl4. OTblCKAHlJIE OPlJIrlJlHAnA no lJI306PA>KEHlJIlO

~ Csepm'ICo1:1 <PYHKIJ;Hit f(t) H g(t) (o603Ha'leHHef(t)*g(t)) Ha3bIBaeTCH <PYHKIJ;HH

!f(r)g(t-r)dr. o

OrrepaIJ;HH CBepTbIBaHHH <PYHKIJ;Hit 06JIa.n;aeT CBOitCTBOM KOMMYTaTHBHOCTH:

f(t) * g(t) = g(t) * f(t), T. e.		!t	f(r)g(t -	r) dr = !t g(r)f(t - r) dr.
		o		0
TeopeMa 8.10 (06 YMHO)KeHlo1Io1 111306pa)KeHIII~. IIInlll TeopeMa 0 cBepTKe).
nycTb f(t)	III g(t) - oplllrlllHallbJ, a	F(P) III G(p), COOTBeTCTBeHHO, IIIX 111306pa)l(e-
HIII~. TorAa	f(t) *g(t) -tt F(P)· G(p).

TaKHM 06Pa30M, u306pa':JICe'H,'ue csepm'ICu iJsyx opUZU'HaJlOS ecm'b npou3seiJe'HUe ux u306pa':JICe'Hu1:1.

497

OTblCKaHMe opMrMHal10B no M306pa)l(eHMHM

AnH HaXOJK,n;eHHH OpHrHHaJIOB rro 3a,n;aHHbIM H306paJKeHHHM MOJKHO HCrrOJIb30-

BaTb HeCKOJIbKO rrpHeMOB.

Q(P)

JIepa'bttt COCTOHT B TOM, 'ITOH306paJKeHHe R(P) rrpe,n;CTaBJIHeTCH B BH,n;e CYM-

MbI ,meMeHTapHbIX ,n;p06efi, HBJIHIOm;HXCH H306paJKeHHHMH rrpocTbIx OPHrHHaJIOB. nOCJIe '1ero,HCrrOJIb3YH Ta6JIH~ OpHrHHaJIOB H CBoficTBO JIHHefiHOCTH rrpe06pa3Q-

BaHHH JIarrJIaca, Haxo,n;HT OpHrHHaJI, COOTBeTcTByIOm;Hfi HCXO,n;HOfi ,n;P06H.

Bmopott crroc06 COCTOHT B TOM, 'ITO,n;p06b rrpe,n;CTaBJIHeTCH B BH,n;e rrpoH3Be-

,n;eHHH ,n;p06efi, HBJIHIOm;HXCH H306paJKeHHHMH HeKOTopbIX <PYHKIJ;Hfi, rrOCJIe '1ero rrpHMeHHeTCH TeopeMa 0 cBepTKe.

Tpemutt crroc06 OCHOBaH Ha CJIe.n;yIOm;efi TeopeMe (rrpHBe,n;eM ee B HeCKOJIbKO

oCJIa6JIeHHOM BapHaHTe):

TeopeMa 8.11 (0 pa3110)l(eHI>11>17).		Q(P)
TeopeMa 8.11 (0 pa3110)l(eHI>11>17).	nYCTb q,YHKl.4l11l1	F(P) = R(p) npeACTaBlllleT
c060iii npaBIIIIlbHYIO pal.4l11OHallbHYIO	AP06b, IIIMelOllIYIO nOlllOCbl B TO'iKaXPko rAe
k = 1,2, ... , n. TorAa oplllrlllHallOM	Allll Hee CIlY)I(IIIT	q,YHKl.4l11l1

rAe CYMMa 6epeTClI no BceM nOlllOcaM.

8.2.1.	HaitTH OpHrHHa.JIbI CJIe.rr.yIOIIIHX H306pIDKeHHii:
				7					4	3
	a) F(p) = p3;						6) F(p) = (p+ 1)4			(p-1)2;
				4					3p-1
	B) F(p) = p2 _ 6p + 13;						r) F (p) = p2 + 4p + 29 ;
	)	F(	) -	e-P
	.n.	p	-	(p _ 2)3
a a) ITo Ta6JIHIle H306pIDKeHHii HMeeM: t						2	-7t	2'	2
a a) ITo Ta6JIHIle H306pIDKeHHii HMeeM: t							-7t	-t =	3"'IT09TOMY, IIpe06pa-
								p	p
3YH F(p), IIOJIY'IHM ~ = ~. 23 , OTCIO,lla IIO CBoiicTBY JIHHeiiHOcTH IIpe06pa-
			p	p
30BaHHH JIaIIJIaca HaXO,llHM OpHrHHa.JI ,llJIH F (p): ~ = ~. 23 +;-										~t2 •
								P	P
6) ITpe06pa3yeM F(p) TaKHM 06pa30M, 'IT06bIMO:JKHO 6blJI0 BOCIIOJIb30-
BaTbCH Ta6JIHIIeii H306pIDKeHHii:
		4		3	4		3!	_ 3.	1
				(p_ 1)2	3i'(p+ 1)4				(p_1)2

73a'lacTYIO9TY TeopeMY Ha3blBaIOT TaKJKe 6mopoit meope.Moit pa3J1OO1CeHUJI (noCKOJIbKY eCTb ew;e H nepBaH TeopeMa pa3JIOJKeHHH).

498

<l>YHKIJ;HH		3! 4 HBJIHeTCH H306pIDKeHHeM OpHrHHa.rra e-t ·t3 , a (p 1 )2
		(p+1)		-1
H306pIDKeHHeM opHrHHa.rra et				. t. TaKHM 06pa30M, OKOHqaTeJIbHO HMeeM:
			F(p) +;- ~e-t. t 3 - 3et . t.
B) IIpe06pa3yeM ,n;P06b, BbI,n;eJIHB nOJIHblft KBa.n;paT B 3HaMeHaTeJIe:
F(	) -	4	(p -	4
p	-	p2 _ 6p+ 13	(p -	3)2 + 4

IIOCJIe,n;HHH ,n;P06b HBJIHeTCH H306pIDKeHHeM <PYHKIJ;HH e3t sin 2t. Tor,n;a no CBOftCTBY JIHHeftHocTH npe06pa30BaHHH JIanJIaca nOJIyqaeM

4 +;- 2e3t sin 2t.

(p - 3)2 + 4

r) ,LLeftcTByeM aHa.rrOrHqHO nyHKTy B):

3p -1	3p -1
F(p) = p2 +4p+29 -	(p+2)2 +25·

IIOKIDKeM, qTO nOCJIe,n;HHH ,n;p06b eCTb JIHHeftHM KOM6HHaIJ;HH H306pIDKeHHft <PYHKIJ;Hft e- 2t sin 5t H e- 2t cos 5t. ,LLeftcTBHTeJIbHO,

3p -	1	3(p + 2) - 7	3(p + 2)		7
--~~----=		(p + 2)2 + 25	= --~~~--		(p + 2)2 + 25
(p + 2)2	+ 25	(p + 2)2 + 25	(p + 2)2	+ 25	(p + 2)2 + 25
			=3.	p+2	7	5
			(p + 2)2 + 25 - 5"			. (p + 2)2 + 25·

CJIe,n;oBaTeJIbHO, COrJIaCHO Ta6JIHIJ;e H306pIDKeHHft H CBOftCTBY JIHHeftHocTH

npe06pa30BaHHH JIanJIaca, HaxO,n;HM opHrHHa.rr:

F(P) - 3·	P + 2	_ 1 .	5	+ 25	+;- 3e-2t cos 5t - l e- 2t sin 5t.
-	(p + 2)2 + 25	5	(p + 2)2	+ 25	5

)I.) IIo Ta6JIHIJ;e H306pIDKeHHft Haxo,n;HM CHaqa.rra OpHrHHa.rr f(t) ,n;JIH

<PYHKIJ;HH (p-1	2)	3 . A HMeHHO:
		1
IIpHMeHHB TeopeMY 3ana3,n;bIBaHHH opHrHHa.rra, HMeeM:
	e-P +;- f(t -		1) = le2 (t-I)(t - 1)2X(t - 1).
(p- 2)3			2

(HanoMHHaeM, qTO no,n; <PYHKIJ;Heft ~e2tt2 MbI nOHHMaeM <PYHKIJ;HIO

~e2tt2x(t». •

499

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