А.Н.Шерстнев - Математический анализ
..pdfx137. nEKOTORYE SPECIALXNYE FUNKCII
pRIMENIM POLU^ENNYE REZULXTATY K ANALIZU WAVNYH W PRILOVENIQH SPECIALXNYH FUNKCIJ, ZADANNYH INTEGRALAMI.
1. b\TA-FUNKCIQ |JLERA ZADA•ETSQ INTEGRALOM
B(a; b) = Z 1 xa,1(1 , x)b,1 dx (a; b > 0):
0
w UKAZANNOJ OBLASTI INTEGRAL SHODITSQ. iNTEGRAL QWLQETSQ SOBSTWENNYM W OBLASTI f(a; b)j a 1; b 1g. s POMO]X@ FORMULY nX@TONA-lEJBNICA INTEGRAL MOVET BYTX WY^ISLEN LI[X PRI NEKOTORYH a; b. pO\TOMU FUNK- CI@ B(a; b) PRIHODITSQ IZU^ATX KAK INTEGRAL (WOOB]E, NESOBSTWENNYJ), ZAWISQ]IJ OT PARAMETRA. pOKAVEM SNA^ALA, ^TO B(a; b) NEPRERYWNA.
pREDSTAWIM B(a; b) W WIDE B(a; b) = B0 (a; b) + B1(a; b), GDE
1=2 |
|
1 |
|
B0(a; b) = Z0 |
xa,1 |
(1 , x)b,1 dx; B1 (a; b) = Z1=2xa,1(1 |
, x)b,1 dx: |
kAVDYJ IZ INTEGRALOW B0; B1 IMEET OSOBENNOSTX NE BOLEE ^EM W ODNOJ TO^KE I DOSTATO^NO USTANOWITX NEPRERYWNOSTX KAVDOGO IZ NIH. uTWERV- DENIE 136.1 NEPOSREDSTWENNO NE PRIMENIMO, TAK KAK MNOVESTWO PARAMET- ROW OTKRYTO W R2. pOSKOLXKU NEPRERYWNOSTX FUNKCII W TO^KE ESTX SWOJ- STWO LOKALXNOE, MOVNO USTRANITX \TO ZATRUDNENIE. pUSTX a0; b0 > 0 PRO- IZWOLXNY. pOGRUZIM TO^KU (a0; b0 ) W NEKOTORYJ ZAMKNUTYJ PRQMOUGOLX-
NIK = [a1; a2] [b1; b2] TAK, ^TOBY 0 < a1 < a0 < a2; 0 < b1 < b0 < b2. pOKAVEM, NAPRIMER, ^TO FUNKCIQ B0(a; b) NEPRERYWNA W TO^KE (a0; b0 ).
w SILU 136.1 DOSTATO^NO USTANOWITX, ^TO INTEGRAL |
Z01=2xa,1(1 , x)b,1 dx |
||||||||||||||
SHODITSQ RAWNOMERNO W . pOLAGAQ c = |
|
|
max |
|
|
|
(1 |
, |
x)b,1 , IMEEM |
||||||
|
|
|
|
|
(x;b) |
2 |
[0;1=2] |
|
[b1;b2] |
|
|
||||
|
, x)b,1 |
|
|
|
|
2 |
|
|
|
|
|
|
|
||
xa,1(1 |
cxa1,1 (0 < x < 1=2; (a; b) |
). pO PRIZNAKU wEJER- |
|||||||||||||
[TRASSA OTS@DA SLEDUET RAWNOMERNAQ SHODIMOSTX. |
|
|
> |
|
|
|
|||||||||
|
|
|
|
||||||||||||
wY^ISLIM |
@B(a; b). fORMALXNO DIFFERENCIRUQ POD ZNAKOM INTEGRA- |
||||||||||||||
LA, IMEEM |
@a |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
@B(a; b) |
= Z0 |
1 |
|
|
|
|
|
|
|
|
|
|
|
|
(1) |
|
@a |
xa,1 (1 , x)b,1 lnx dx |
|
(a; b > 0): |
|
|||||||||
pOKAVEM, ^TO DIFFERENCIROWANIE ZAKONNO. dOSTATO^NO UBEDITXSQ (136.2), ^TO INTEGRAL (1) SHODITSQ RAWNOMERNO NA L@BOM OTREZKE [a1; a2]; a1 > 0:
221
pEREHODQ K INTEGRALAM S ODNOJ OSOBENNOSTX@, DOKAVEM, NAPRIMER, ^TO NA
OTREZKE [a1; a2 ] RAWNOMERNO SHODITSQ INTEGRAL Z01=2xa,1(1 , x)b,1 ln x dx. |
||||||||
pREOBRAZUQ PODYNTEGRALXNU@ FUNKCI@ K WIDU xa,1,"(1 |
, x)b,1x" ln x |
|||||||
|
|
|
1 |
|
|
|
|
|
(0 |
< x 2), GDE " > 0 TAKOE, ^TO a1 , " > 0, ZAMETIM, |
^TO FUNKCIQ |
||||||
|
" |
|
1 |
|
|
|
||
jx |
|
ln xj OGRANI^ENA NA (0; 2], TO ESTX |
|
|
||||
|
|
|
|
jxa,1(1 , x)b,1 lnxj Mxa1,1," |
1 |
|
||
|
|
|
|
(x 2 (0; 2 ]); |
|
|||
GDE M | PODHODQ]AQ KONSTANTA. tEPERX MOVNO WOSPOLXZOWATXSQ PRIZNA-
KOM 135.3.
2. gAMMA-FUNKCIQ |JLERA ZADA•ETSQ INTEGRALOM
,(a) = Z +1 xa,1e,x dx; a > 0:
0
iNTEGRAL IMEET OSOBENNOSTI W +1 I (PRI a < 1) W TO^KE 0. pRI WSEH a > 0 INTEGRAL SHODITSQ. pOKAVEM, ^TO NA L@BOM OTREZKE [a1; a2] (0 < a1 < a2 < +1) INTEGRAL SHODITSQ RAWNOMERNO. oTS@DA, W ^ASTNOSTI, SLEDUET NEPRERYWNOSTX FUNKCII ,(a).
pREDSTAWIM ,(a) W WIDE ,(a) = |
Z |
|
1 |
Z |
|
+1 |
xa,1e,x dx. iZ |
0 |
xa,1e,x dx + |
1 |
|
||||
OCENOK |
|
|
|
|
|
xa,1e,x xa1,1 (0 < x 1); xa,1e,x xa2,1e,x (x 1)
I PRIZNAKA wEJER[TRASSA SLEDUET RAWNOMERNAQ SHODIMOSTX INTEGRALA NA
[a1; a2]: >
iZ 136.2 I PRIZNAKA wEJER[TRASSA SLEDUET, ^TO FUNKCIQ ,(a) DIFFE- RENCIRUEMA L@BOE ^ISLO RAZ W OBLASTI a > 0:
|
,(k) (a) = Z0+1xa,1 (lnx)ke,x dx; k = 1; 2; : : : : |
|
3. z A M E ^ A N I E. iMEET MESTO FORMULA |
(2) |
,(a) = (a , 1),(a , 1); a > 1: |
w ^ASTNOSTI, ESLI n 2 N, TO
,(n + 1) = n,(n) = n(n , 1),(n , 1) = : : : = n!,(1) = n!:
222
tAKIM OBRAZOM, GAMMA-FUNKCIQ QWLQETSQ ESTESTWENNYM OBOB]ENIEM FAK- TORIALA NA NECELYE ARGUMENTY. ffORMULA (2) SLEDUET IZ WYKLADKI (DLQ a > 1 INTEGRAL ,(a) IMEET EDINSTWENNU@ OSOBENNOSTX W +1):
,(a) = |
+1 |
|
a |
1 |
e, |
x |
dx = |
lim |
|
Z |
N |
a 1 |
e, |
x |
dx |
|||
|
x |
|
, |
|
|
|
|
x |
, |
|
||||||||
|
0 |
|
|
|
|
|
|
|
|
N!+1 |
0 |
|
N xa,2e,xdx] |
|||||
= |
Z lim |
|
[ |
, |
xa,1e,x N |
+ (a |
|
1) |
|
|||||||||
|
N!+1 |
|
|
|
|
j0 |
|
, |
Z0 |
|
|
|
|
|||||
= |
(a , |
1),(a , |
1):g |
|
|
|
|
|
|
|
|
|
||||||
4. u P R A V N E N I E. pOKAZATX, ^TO INTEGRALY
Z 1xa,1(1 , x)b,1 dx; Z 1xa,1(1 , x)b,1 lnx dx (a; b > 0);
0 0
Z 1xa,1e,x dx; Z +1xa,1e,x dx (a > 0)
0 1
NE SHODQTSQ RAWNOMERNO W UKAZANNYH OBLASTQH.
223
posledowatelxnosti i rqdy funkcij
x138. rAWNOMERNAQ SHODIMOSTX POSLEDOWATELXNOSTI FUNKCIJ
1. pUSTX | MNOVESTWO. pOSLEDOWATELXNOSTX FUNKCIJ fn : ! C NAZYWAETSQ SHODQ]EJSQ K FUNKCII f : ! C W KAVDOJ TO^KE MNOVESTWA(PI[EM fn ! f), ESLI ^ISLOWAQ POSLEDOWATELXNOSTX fn (!) SHODITSQ K f (!) PRI KAVDOM ! 2 :
(1) 8! 2 8" > 0 9N 8n > N (jfn (!) , f (!)j < ")
(ZDESX NATURALXNOE N , KONE^NO, ZAWISIT OT ! 2 ).
bOLX[EE ZNA^ENIE PRI IZU^ENII FUNKCIONALXNYH POSLEDOWATELXNOS- TEJ IGRAET INOJ, BOLEE SILXNYJ WID SHODIMOSTI.
|
2. pOSLEDOWATELXNOSTX fn : ! C NAZYWAETSQ RAWNOMERNO SHODQ]EJSQ |
K FUNKCII f : ! C (BUDEM PISATX fn =) f ), ESLI |
|
(2) |
8" > 0 9N 8n > N 8! 2 (jfn(!) , f(!)j < "): |
(w \TOM OPREDELENII ^ISLO N UVE NE ZAWISIT OT !!) |
|
3. z A M E ^ A N I E. eSLI fn =) f, TO fn ! f . oBRATNOE, WOOB]E, NEWERNO fDLQ POSLEDOWATELXNOSTI FUNKCIJ fn (t) (0 t 1), ZADANNYH RAWENSTWAMI fn (t) = 0 (t 6= 1=n); fn (1=n) = 1, IMEEM: fn ! 0, NO fn NE SHODITSQ K 0 RAWNOMERNOg.
4. dLQ OGRANI^ENNOJ FUNKCII f : ! C POLOVIM
kfk sup jf(!)j:
!2
wWED•ENNAQ WELI^INA NAZYWAETSQ RAWNOMERNOJ NORMOJ OGRANI^ENNOJ FUNK- CII. oNA OBLADAET WSEMI SWOJSTWAMI NORMY:
kfk = 0 ) f = 0; k fk = j jkfk ( 2 C ), kf + gk kfk + kgk .
w TERMINAH \TOJ NORMY UDOBNO SFORMULIROWATX USLOWIQ RAWNOMERNOJ SHODIMOSTI.
224
5. pUSTX fn : ! C (n = 1; 2; : : :) | POSLEDOWATELXNOSTX FUNKCIJ. |
||
sLEDU@]IE USLOWIQ \KWIWALENTNY: |
||
(A) fn SHODITSQ RAWNOMERNO (K f), |
||
(B) |
kfn , fk ! 0 (n ! 1), |
|
(W) |
8" > 0 |
9N 8n; m > N (kfn , fmk < "), |
(G) |
8" > 0 |
9N 8n; m > N 8! 2 (jfn (!) , fm (!)j < "). |
(A) ) (B). dOSTATO^NO ZAMETITX, ^TO IZ (2) SLEDUET, ^TO kfn , fk < " |
||
PRI n > N. uMESTNO OBRATITX WNIMANIE ^ITATELQ, ^TO W USLOWII (A) |
||
NE TREBUETSQ OGRANI^ENNOSTI FUNKCIJ. tEM NE MENEE, PRI DOSTATO^NO
BOLX[IH n RAZNOSTX fn , f UVE OBQZANA BYTX OGRANI^ENNOJ FUNKCIEJ. |
|||||||
(B) ) (W). pUSTX N 2 N TAKOWO, ^TO kfn ,fk < "=2 PRI n > N . tOGDA |
|||||||
|
kfn , fmk kfn , fk + kfm , fk < " (n; m > N): |
|
|
||||
(W) ) |
(G) (!!). |
|
|
|
|
|
|
(G) ) |
(A). uSLOWIE (G) OZNA^AET, ^TO PRI KAVDOM ! 2 ^ISLOWAQ |
||||||
POSLEDOWATELXNOSTX fn (!) FUNDAMENTALXNA, A ZNA^IT, SU]ESTWUET f (!) |
|||||||
n |
|
j |
|
, |
j |
2 |
|
lim fn (!). pUSTX N TAKOWO, ^TO |
|
fn (!) |
|
fm(!) < " (n; m > N; ! |
|
). |
|
pEREHODQ W \TOM NERAWENSTWE K PREDELU PO m, IMEEM jfn (!) , f(!)j
" (n > N; ! 2 ): >
x139. rAWNOMERNAQ SHODIMOSTX I NEPRERYWNOSTX
pRODEMONSTRIRUEM, KAK RABOTAET PONQTIE RAWNOMERNOJ SHODIMOSTI POSLEDOWATELXNOSTI FUNKCIJ, ZADANNYH NA TOPOLOGI^ESKOM PROSTRANST- WE.
1. pUSTX E | TOPOLOGI^ESKOE PROSTRANSTWO, FUNKCII fn : E ! C
(n = 1; 2; : : :) NEPRERYWNY W TO^KE !0 2 E I fn =) f. tOGDA f TAKVE NEPRERYWNA W TO^KE !0.
pUSTX " > 0 PROIZWOLXNO I N 2 N TAKOWO, ^TO jfN (!),f (!)j < "=3 (! 2 E). tAK KAK fN NEPRERYWNA W !0 , NAJDETSQ• OKRESTNOSTX U TO^KI !0 TAKAQ, ^TO jfN (!) , fN (!0)j < "=3(! 2 U). sLEDOWATELXNO, DLQ L@BOJ TO^KI
! 2 U:
jf (!) , f (!0 )j jf(!) , fN (!)j + jfN (!) , fN (!0)j
+ jfN (!0 ) , f (!0)j < ": >
225
2.s L E D S T W I E. pUSTX fn : E ! C | POSLEDOWATELXNOSTX FUNKCIJ, NEPRERYWNYH NA TOPOLOGI^ESKOM PROSTRANSTWE E, I fn =) f. tOGDA f NEPRERYWNA NA E.
3.p R I M E R. rASSMOTRIM POSLEDOWATELXNOSTX ^ISLOWYH FUNKCIJ
fn(t) = (1 |
, nt) [0;1=n] (t) (0 t |
1). |
o^EWIDNO |
fn |
NEPRERYWNY |
, |
PRI^EM |
|
|
|
|
• |
|||||
DLQ L@BOJ TO^KI t 2 [0; 1] SU]ESTWUET PREDEL |
|
|
|
|
||||
|
f (t) = limfn (t) = |
0; |
ESLI 0 < t |
1, |
|
|
||
|
n |
1; |
ESLI t = 0. |
|
|
|||
oDNAKO, \TA PREDELXNAQ FUNKCIQ UVE NE NEPRERYWNA.
x140. rAWNOMERNAQ SHODIMOSTX RQDOW FUNKCIJ
1. pUSTX uj : ! C | POSLEDOWATELXNOSTX ^ISLOWYH FUNKCIJ, ZADAN- NYH NA ABSTRAKTNOM MNOVESTWE , TAK ^TO KAVDOJ TO^KE ! 2 MOVNO SOPOSTAWITX ^ISLOWOJ RQD
1
( ) X uj (!):
j=1
rQD ( ) NAZYWAETSQ RAWNOMERNO SHODQ]IMSQ, ESLI RAWNOMERNO SHODITSQ
n
POSLEDOWATELXNOSTX P uj (!) EGO ^ASTNYH SUMM.
j=1
oTMETIM NEPOSREDSTWENNOE SLEDSTWIE 139.1.
2.pUSTX | TOPOLOGI^ESKOE PROSTRANSTWO, I FUNKCII uj : ! C (j = 1; 2; : : :) NEPRERYWNY W TO^KE !0. pUSTX RQD ( ) SHODITSQ RAWNOMER- NO K FUNKCII v : ! C . tOGDA v NEPRERYWNA W TO^KE !0. eSLI, KROME TOGO, WSE uj NEPRERYWNY NA , TO I SUMMA RQDA v NEPRERYWNA NA .
3.[kRITERIJ kO[I]. rQD ( ) SHODITSQ RAWNOMERNO TTOGDA
|
8" > 0 9N 8n > N 8p 8! 2 0 |
|
n+p |
uj(!) |
< "1 : |
|
@ |
|
X |
|
A |
|
|
j=n+1 |
|
|
|
n |
4. [pRIZNAK wEJER[TRASSA]. pUSTX j |
> 0; |
juj (!)j j (! 2 ) I |
||
j=1 |
j < +1. tOGDA RQD ( ) SHODITSQ RAWNOMERNO. |
|
|||
P |
|
|
|
|
|
226
p. 3 QWLQETSQ PEREFORMULIROWKOJ DLQ RQDOW KRITERIQ 138.5(G), P. 4 SLEDUET IZ P. 3 (!!). >
5. u P R A V N E N I E. iSSLEDOWATX NA RAWNOMERNU@ SHODIMOSTX RQD
1 2 ,nx
P x e (0 x < +1). f pRIMENITE P. 4.g
n=1
x141. pRIZNAKI SHODIMOSTI dIRIHLE I aBELQ
sLEDU@]IE NIVE PRIZNAKI PRIGODNY DLQ NEABSOL@TNO SHODQ]IHSQ RQDOW WE]ESTWENNYH FUNKCIJ. rASSMOTRIM RQD WIDA
|
|
X |
|
|
|
|
|
|
(1) |
|
1 uj(!)vj (!); |
|
|
|
|
||
|
|
j=1 |
|
|
|
|
|
|
GDE uj ; vj : ! R | WE]ESTWENNYE FUNKCII. |
|
|
|
|
||||
pRIZNAK dIRIHLE |
]. |
pUSTX |
u1(!) u2 |
(!n) |
: : : (! 2 |
), |
PRI^EM |
|
1. [ |
|
|
• |
|||||
uk =) 0, I SU]ESTWUET M > 0 |
TAKOE, ^TO j j=1 vj (!)j M |
(! 2 ; n 2 |
||||||
N). tOGDA RQD (1) SHODITSQ RAWNOMERNO. |
P |
|
|
|
||||
2. [pRIZNAK aBELQ]. pUSTX u1(!) u2(!) : : : (! 2 ), PRI^•EM SU]ESTWUET M > 0 TAKOE, ^TO juj (!)j M (j = 1; 2; : : :). pUSTX, KROME
TOGO, RQD 1 v (!) SHODITSQ RAWNOMERNO. tOGDA RQD (1) TAKVE SHODITSQ
jP=1 j
RAWNOMERNO.
p. 1. DLQ FIKSIROWANNOGO n 2 N POLOVIM
wk = vn+1 + : : : + vn+k (k = 1; 2; : : :):
iMEET MESTO TOVDESTWO
p |
|
p,1 |
|
|
X |
un+kvn+k = |
X |
(un+k , un+k+1 )wk + un+pwp: |
|
k=1 |
k=1 |
|||
|
|
sLEDOWATELXNO,
|
p |
|
(2) |
P |
un+k (!)vn+k (!) |
|
|
|
|
k=1 |
|
p,1
kP=1 jun+k (!) , un+k+1(!)jjwk(!)j
+ jun+p (!)wp (!)j:
227
|
|
|
|
|
|
|
|
|
n+k |
|
|
n |
|
|
|
|
|
|
|
||
pO USLOWI@ |
jwk |
(!)j = j j=1 vj (!), j=1 vj (!)j 2M. |
s U^ETOM MONOTONNOS |
- |
|||||||||||||||||
|
|
|
|
|
|
|
|
• |
|
|
|
||||||||||
|
|
|
|
|
|
|
|
|
P |
|
|
|
P |
|
|
|
|
|
|
|
|
TI POSLEDOWATELXNOSTI uj |
(!) IMEEM |
|
|
|
|
|
|
|
|||||||||||||
|
p |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
j k=1 un+k (!)vn+k(!)j 2M(un+1 (!) , un+2(!) + un+2 (!) , un+3(!) |
|
|
|
||||||||||||||||||
P |
|
|
|
|
|
|
+ : : : , un+p(!) + un+p (!)) = 2Mun+1(!) (! 2 ): |
||||||||||||||
|
|
|
|
|
|
|
|
|
|||||||||||||
tAK KAK uk =) |
0, SOGLASNO 140.3 POLU^AEM TREBUEMOE. |
|
|
|
|
|
|||||||||||||||
|
p. 2. w UKAZANNYH WY[E OBOZNA^ENIQH DLQ L@BOGO " > 0 SU]ESTWUET |
||||||||||||||||||||
N = N (") TAKOE, ^TO PRI n > N DLQ WSEH ! 2 |
|
|
|
|
|
|
|||||||||||||||
|
|
|
|
|
jwk(!)j = jvn+1(!) + : : : + vn+k(!)j < " (k = 1; 2; : : :): |
|
|
|
|||||||||||||
\TO SLEDUET IZ |
140.3, |
PRIMENENNOGO K RQDU |
1 |
(!)). |
sLEDOWATELXNO |
, |
S |
||||||||||||||
( |
|
|
|
|
|
|
|
|
|
|
• |
|
|
j=1 vj |
|
|
|
|
|||
U^ETOM |
|
|
I MONOTONNOSTI |
fuj (!)g: |
P |
|
|
|
|
|
|
||||||||||
• |
|
|
(2) |
|
|
|
|
|
|
|
|
|
|
|
|
||||||
|
|
|
p |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
k=1 |
un+k(!)vn+k (!) "(un+1(!) , un+p (!)) + "jun+p(!)j 3"M: |
|
|
|
|||||||||||||||
|
|
X |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
sNOWA W SILU 140.3 POLU^AEM TREBUEMOE. > |
|
|
|
|
|
|
|
||||||||||||||
|
3. p R I M E R. iSSLEDUEM NA RAWNOMERNU@ SHODIMOSTX RQD |
|
|
|
|||||||||||||||||
(3) |
|
|
|
|
|
|
|
|
|
1 sinnx ( > 0): |
|
|
|
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
n=1 |
n |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
X |
|
|
|
|
|
|
|
|
|
|
|
w SLU^AE > 1 RQD SHODITSQ ABSOL@TNO I RAWNOMERNO (SM. 140.4). w SLU- |
|||||||||||||||||||||
^AE |
|
1 PRIMENIM PRIZNAK dIRIHLE, POLAGAQ vn (x) = sin nx; un(x) = |
|||||||||||||||||||
n, |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
n |
|
|
|
||
|
|
|
oSTAETSQ ISSLEDOWATX NA OGRANI^ENNOSTX SUMMY |
j j=1 vj (x)j = |
|||||||||||||||||
|
(=) 0). |
|
• |
|
|
|
|
|
|
|
|
|
|||||||||
j sin x + sin 2x + : : : + sin nxj. s |
U^•ETOM |
TOVDESTWA |
|
P |
|
= |
|||||||||||||||
2 sin sin |
|
||||||||||||||||||||
cos( , ) , cos( + ) IMEEM |
|
|
|
|
|
|
|
|
|
|
|||||||||||
sinx + sin 2x + : : : + sin nx
=(2 sin x2 ),1 [2 sin x2 sinx + : : : + 2 sin x2 sin nx]
=(2 sin x2 ),1 [cos x2 , cos 32x + cos 32x , : : : , cos(n + 12 )x] x2 ),1 [cos x2 , cos(n + 12)x]; x 6= 2 k (k 2 Z).
228
pUSTX " > 0 PROIZWOLXNO MALO. tOGDA NA OTREZKE ["; 2 |
, |
"] MY IMEEM |
|||||||||||||||||||||||
n |
|
|
|
|
|
|
" |
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
j j=1 |
vj (x) |
|
(sin 2), |
|
|
(n = 1; 2; : : :). |
tAKIM OBRAZOM |
, |
RQD |
(3) |
PRI |
|
|
1 |
|||||||||||
|
j |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
P |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
SHODITSQ RAWNOMERNO NA L@BOM OTREZKE WIDA ["; 2 , "]; " > 0: |
|
|
|
||||||||||||||||||||||
u P R A V N E N I Q. iSSLEDOWATX NA RAWNOMERNU@ SHODIMOSTX |
|
|
|
||||||||||||||||||||||
4. |
1 |
|
n |
|
1 |
|
|
|
(0 |
|
x < + ), |
|
|
|
|
|
|
|
|
|
|||||
|
( 1) |
x |
+ n |
|
|
|
|
|
|
|
|
|
|||||||||||||
|
|
n=1 |
, |
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
||
|
|
P |
|
|
1 |
|
|
|
|
x |
n |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
5. |
1 |
|
n |
|
|
|
|
|
|
|
(0 < x < 1). |
|
|
|
|
|
|
|
|
|
|||||
|
( 1) |
n |
1 + xn |
|
|
|
|
|
|
|
|
|
|||||||||||||
|
|
n=1 |
, |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
P |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
6. pOKAZATX, ^TO RQD (3) NA OTREZKE [0; 2 ] SHODITSQ NERAWNOMERNO PRI
0 < 1.
x142. oPERACII NAD RAWNOMERNO SHODQ]IMISQ RQDAMI |
|||||
1. pUSTX ( Rn ) J-IZMERIMO I ZAMKNUTO, fn : ! R NEPRERYWNY I |
|||||
) |
n |
Z |
|
Z |
|
fn = f. tOGDA |
lim |
|
fn(x) dx = |
|
f(x)dx. |
|
|
|
|
|
|
w SILU 139.2 FUNKCIQ f NEPRERYWNA I, W ^ASTNOSTI, INTEGRIRUEMA NA |
|||||
. sOGLASNO 138.5 kfn , fk ! 0 (n ! +1). pO\TOMU |
|||||
|
Z |
fn (x) dx , Z |
f (x) dx Z jfn(x) , f (x)j dx |
||
|
|
|
|||
|
|
|
kfn , fk m( ) ! 0 (n ! +1): |
|
> |
|
|
|
|||
|
|
|
|||
w KA^ESTWE SLEDSTWIQ PRIWEDEM• TEOREMU O PO^LENNOM INTEGRIROWANII RAWNOMERNO SHODQ]EGOSQ RQDA.
2. pUSTX ( Rn) J-IZMERIMO I ZAMKNUTO, un : ! R NEPRERYWNY, I RQD un (x) SHODITSQ RAWNOMERNO NA . tOGDA
|
|
X |
X |
|
|
Z |
( |
1 un(x)) dx = |
1 |
|
un(x) dx: |
|
n=1 |
n=1 Z |
|
||
pOLEZNO WYDELITX SLU^AJ, KOGDA FUNKCII ZADANY NA OTREZKE ^ISLOWOJ PRQMOJ.
229
|
|
|
|
|
|
|
|
1 |
|
3. pUSTX un(t) (a t b) NEPRERYWNY I RQD n=1 un(t) SHODITSQ RAW- |
|||||||
NOMERNO NA [a; b]. tOGDA |
|
|
P |
|||||
|
Z |
x |
|
1 |
|
1 |
|
x |
(1) |
a |
( |
X |
un (t))dt = |
X Z |
a |
un(t)dt (a x b); |
|
|
|
|
n=1 |
|
n=1 |
|
|
|
PRI^•EM RQD W PRAWOJ ^ASTI SHODITSQ RAWNOMERNO.
w SILU P. 2 W DOKAZATELXSTWE NUVDAETSQ LI[X RAWNOMERNAQ SHODIMOSTX RQDA W PRAWOJ ^ASTI (1). tREBUEMOE SLEDUET IZ OCENKI
|
n+p |
x |
|
|
|
|
|
|
|
x n+p |
|
|
|
|
|
|
|
|
n+p |
|
|
|
|
|
|
|
|
|
||||||
|
X |
|
|
|
|
|
|
|
|
|
|
X |
|
|
|
|
|
|
|
|
X |
|
|
|
|
|
|
|
|
|
||||
|
jk=n+1 Za |
uk(t) dtj Za |
jk=n+1 uk (t)jdt (b , a)k k=n+1 ukk[a;b]: |
|
> |
|||||||||||||||||||||||||||||
|
pUSTX |
uj : [a; b] ! |
R |
|
|
GLADKIE FUNKCII |
|
PRI^EM |
|
|
|
|
|
|
|
|
|
|||||||||||||||||
4. |
|
|
|
| |
|
|
|
|
|
, |
|
|
|
|
• |
|
|
|
|
|
|
|
|
|
|
|||||||||
A |
PRI NEKOTOROM |
c (a c |
b) |
SHODITSQ ^ISLOWOJ RQD |
|
|
1 |
|
|
|
|
|
|
|||||||||||||||||||||
( ) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
j=1 uj (c), |
|||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
P |
|
|
|
|
|
|
|
(B) |
RQD 1 u0 (x) SHODITSQ RAWNOMERNO NA |
[a; b]. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||
|
P |
|
j |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
j=1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
tOGDA RQD 1 |
uj(x) |
SHODITSQ RAWNOMERNO NA |
[a; b] |
I |
( |
1 |
|
|
|
|
0 |
|
1 |
u0 (x). |
||||||||||||||||||||
|
P |
|
|
|
|
|
|
|
|
|
|
|
P |
uj (x)) = |
|
P |
||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
j |
|||||||
|
j=1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
j=1 |
|
|
|
|
|
j=1 |
|||||||
pOLOVIM |
|
X |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
, |
|
|
|
|
|
2 |
|
|
|
|
|
|
||||||
|
'(x) = 1 |
|
u0 |
(x) (a |
x |
b); vj(x) = uj (x) |
uj(c) (j |
N): |
|
|||||||||||||||||||||||||
|
|
|
|
|
|
j |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
j=1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
sOGLASNO P. 3 |
Z |
|
x |
'(t)dt = |
|
1 |
( |
|
xu0 (t) dt) = |
1 vj(x), I RQD |
|
1 |
vj (x) SHO- |
|||||||||||||||||||||
|
|
|
c |
|
|
|
|
|
|
P |
Z |
c |
|
j |
P |
|
|
|
|
|
|
|
|
|
|
P |
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
j=1 |
|
|
|
j=1 |
|
|
|
|
|
|
|
|
|
j=1 |
|
|
|
|
|
|
||||
DITSQ RAWNOMERNO. |
pO\TOMU |
|
RAWNOMERNO |
SHODITSQ |
|
RQD |
|
1 |
|
|
|
|
|
|
||||||||||||||||||||
|
|
|
j=1 uj (x) = |
|||||||||||||||||||||||||||||||
1 [vj (x) + uj (c)]. pRI \TOM |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
P |
|
|
|
|
|
|
|||||||||||
P |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
j=1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
X |
|
|
|
|
|
X |
|
|
|
|
|
|
Z |
x |
|
|
|
|
|
|
|
|
X |
|
|
|
|
|
|
|
|
|
|
|
[ 1 uj |
(x)]0 |
= [ |
1 vj (x)]0 = [ '(t)dt]0 |
= '(x) = |
|
1 u0 |
(x): |
|
> |
|
|||||||||||||||||||||||
|
|
|
|
|||||||||||||||||||||||||||||||
|
|
|
|
|||||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
c |
|
|
|
|
|
|
|
|
|
j |
|
|
|
|
|
|
|
|
|
j=1 |
|
|
|
|
|
j=1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
j=1 |
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
230