.
1. f(z)= (-1)nz2n= (-z2)n=1/(1+z2); , | z|<1. .
z1,2=i. , f(x)= 1/(1+x2) x=0 -1<x<1 |x|≥1, "x " .
2. , ,
.
- |z|<1 ( - ).
z=1 . , |zk,m|=1;
k-, m=0,1,2....; m=0, zk,0=1; m=2k, =1, m=1,...,2k-1 2k .
n³k =1=> . k¥ zk,m => , (.. ϵ- |z|=1 ) => !!. (.. " z: |z|<1 . 13.1.
-, |z|<1 ( )
. f(z) .z0= -0,9:
f(z)= . .. |z|<1.
z1 |z-z0|<|1-z0|, , |z-z1|<|1-z1|
. , , z=1. f(z) z=1. , - , z=1. , .
3. f(z) g. z0 . z0- f(z).
. Recn≥0 z=1
. z=1
, .
.3. .
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13.2
fi(z)ÎC¥(gi), i=1,2 fi(z)ÎC (gi+G) f1|G= f2|G.
$F(z)= ÎC¥(g=g1+g2+G).
. , "z0ÎG F(z) ( G). "z0ÎG C=C1(Ìg1)+C2(Ìg2); CÌg- - .
F(z)= ÎC¥(g'Ìg). zÏC F(z) C, z0- F(z). z1Îg'1. F(z1)= + + . 0 .. z1Îg'1. => F(z1)=F(z1). , z2Îg'2 F(z2)=F(z2). , F(z0)=F(z0) => z0ÎG F(z). n
F(z) f1(z) g.
14. .
.1. .
[a,b]Ìg z. g $! f(z)ÎC¥(g), f(x) xÎ[a,b]. f(z) $, , . f(x)- , . f(x) x!!.
.
sin x= ; cos x= ; ex= - "x
( ) , , - sin x, cos x, ex z. . : eiz=cos z+ isin z. , , , , , .
.2. .
. , .
. F(w1,w2,,wn), wiÎDi, , F ∂F/∂wi w1,w2,,wn.
. F(z1,z2,,zn), ziÎDi, , i(zi)=F(z10, ,zi-10,zi, zi+10,zn0) zi .
14.1 wi=fi(z)ÎC¥(g) [a,b]Ìg, wiÎDi. F(z)=F[w1,..., wn] wiÎDi. F[f1(x),...,fn(x)]=0, xÎ[a,b] =>F[f1(z),...,fn(z)]º0, zÎg.
. , F(z)ÎC¥(g). n=2. .
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DF=F[f1(z+Dz),f2(z+Dz)]-F[f1(z),f2(z)]=F[f1(z+Dz),f2(z+Dz)]-F[f1(z),f2(z+Dz)]+ +F[f1(z),f2(z+Dz)]-F[f1(z),f2(z)]= .. F = =>$ , Ԓ(z)- =>F(z)ÎC¥(g) n.
14.1 .
.
1. eix=cos x+ isin x => eiz=cos z+ isin z
2. sin2x+cos2x=1 => sin2z+cos2z=1, |cos z| |sin z| .
3. ; elnx=x, x>0. D0 f(z)= ; 1/xÎC¥(x¹0). " , ( !) $ f(z)ÎC¥(D0)-
ln x (x>0). , lnz= , zÎC¥(D0). 14.1 eln z=z, " zÎD0. lnz
14.2 wi=fi(zi)ÎC¥(gi) [ai,bi]Ìgi, wiÎDi. F(z1,,zn)=F[w1,..., wn] wiÎDi. F[f1(x1),...,fn(xn)]=0, xiÎ[ai,bi] (*) =>F[f1(z1),...,fn(zn)]=0, "ziÎgi.
. n=2. . x20Î[a2,b2] F1(z1)=F[f1(z1),f2(x20)]ÎC¥(g1). 14.1 (*) => F1(z1)º0, "z1Îg1. .. x20- , => F[f1(z1),f2(x2)]=0, "z1Îg1, "x2Î[a2,b2] (**). z10Îg1 F2(z2)=F[f1(z10),f2(z2)]ÎC¥(g2). (**)=>F2(z2)º0, "z2Îg2 => F[f1(z10),f2(z2)]=0, "z2Îg2 .. z10- g1 F[f1(z1),f2(z2)]=0, "z1Îg1, "z2Îg2. n
.
1. sin(z1+z2)=sinz1cosz2+cosz1sinz2 "z1,z2 .
2. w=f(z)=ez- ex
. .. ex1+x2= ex1 ex2=> ez1+z2= ez1 ez2, ez=ex+iy=ex eiy=w => =>|w|=ex, arg w=y - w= ez y=y0 z arg w=y0 w.
z=lnw=x+iy=ln|w|+iarg w.
.3.
f1(z)ÎC¥(g1) g1Çg2=g12È g21¹Æ f2(z)ÎC¥(g2), f2(z)ºf1(z), zÎg12, f2(z)≠f1(z) zÎg21,. F(z)= ÎC¥(g). . , F(z). , , , , . g1 g2 g12, f1(z) f2(z) , g21 . , .
.
1. w=f(z)=ez. g0(-p<Imz<p)D0(-p<arg w<p)- , Imz=-p , Imz=p- .
, g1(p<Imz<3p)D1(p<arg w<3p). z g0 g1 Imz=p D0 D1 , arg w=p. - , D0 D1.
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gn((2n-1)p<Imz<(2n+1)p)Dn((2n-1)p<arg w<(2n+1)p). z , Dn, Dn Dn+1 Dn-1 arg w . z=Ln w =ln|w|+iArg w, -¥<Arg w<¥.
w z=Ln w (). n- -
lnn w. w=ez=eLnw. w=ez - 2pi: ez= ez+2pi. ez- ().
.4. .
1. w=f(z)=ez. w=0.
w0¹0 , Dn, , , Dn-1(Dn+1). ,
lnn w .
w=0 lnn w (lnn w0 = ln |w0|+i arg w0, (2n-1)p<arg w0<(2n+1)p) lnn-1 w (lnn-1 w0 = ln |w0|+i arg w0, (2n-3)p<arg w0<(2n-1)p). w=0 - z=Ln w. Ln w w=0- .
w=w¥. :
. z0 e-, z0 " Ì e-, , z0 () .
, .
- w=f1(z)= w=f2(z)= z=0 z=1 ( !).
- f(z)=za, a- " , .
f(z)=eaLnz= ea(ln|z|+iArg z).
a=n: ein(arg z+2pk)= ein arg z => f(z)= zn- ( , n-. 2π/n ).
a=n/m - f(z) m () n-.
a f(z) ().
1i=eiLn1= ei(ln|1|+i2pk)= e-2pk, k=0, 1, 2....
: , , n-, n-, ¥-, ¥-.
. .
15. .
cn(z-z0)n= cn(z-z0)n+ =P(z)+Q(z). P(z) , Q(z)- . P(z)ÎC¥(|z-z0|<R1), Q(z)-? 1/(z-z0)=x; Q(z)`Q(x)= c-nxnÎC¥(|x|<1/R2), 1/R2 (.. |z-z0|>R2). R2<R1 - R2<|z-z0|<R1.
:
1. cn(z-z0)nÎC¥(R2<|z-z0|<R1).
2. , ÎC¥(R2<|z-z0|<R1).
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3. R1 {cn}¥n=0: R1=1/L1, L1= , R2- {c-n}¥n=1 : R2= , R2= ( L2).
4. cn z0 ! z0 !
15.1 f(z) ÎC¥(R2<|z-z0|<R1), f(z)= cn(z-z0)n.
. z :
(R2<|z-z0|<R1) CR'1 : |x-z0|=R1 CR'2 : |x-z0|=R2, z0 R'1 R'2 : R2<R'2<|z-z0|<R'1<R1.
f(z)= + =P(z)+Q(z).
CR'1: |x-z0|=R1 . , 1/(x-z)
1/(x-z)=1/[(x-z0)-(z-z0)]= , x CR (. ),
P(z)= cn(z-z0)n, cn= , n³0.
CR'2: |x-z0|=R2 . , 1/(x-z)
1/(x-z)=1/[(x-z0)-(z-z0)]=
:
Q(z)= , n>0, c-n= . , : c-n= , n>0. cn c-n R2<|z-z0|<R1. . cn= , n=0,1, 2,,
C- , R2<|z-z0|<R1 z0 . f(z) :
f(z)= cn(z-z0)n+ = cn(z-z0)n, cn= . .. z- R2<|z-z0|<R1 => cn(z-z0)n f(z) ,
R2<R'2£|z-z0|£R'1<R1 f(z) .
. , f(z)= c'n(z-z0)n, c'n¹cn. R2<|z-z0|<R1 : c'n(z-z0)n= c'n(z-z0)n. CR, R, R2<R<R1, z0. c'n(z-z0)n c'n(z-z0)n CR . (z-z0)-m-1, m- . . z-z0=Reij., =
= .=> , m c'm=cm, n
R2<|z-z0|<R1, f(z), ( ) . (. 13.1).
16. .
.1. .
. z0 f(z), f(z) ÎC¥(0<|z-z0|<r(z0)), z0 f(z).
, z0 f(z), $ z0, .
.
z0 ( ) .
z0 f(z) . f(z) z0 ,
0<|z-z0|<r(z0). f(z) z0 Q(z)= .
.
.2. .
:
a) "n>0 c-n=0; Q(z)=0; f(z)c0 zz0 ( 0)- . z0 c0, , f(z0)=c0 . 0<|z-z0|<r(z0): |f(z)|<M f(z)=(z-z0)mj(z), m³0- , j(z0)¹0; f(z)=0, z0- m- .
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"n>0 c-n=0 Q(z)=0
16.1 f(z)ÎC¥(0<|z-z0|<r(z0)) (.. z0- f(z)) |f(z)|<M 0<|z- z0|<r(z0), z0- .
. f(z) . c-n= , n>0. z0 r: |x-z0|=r. , x-z0=reij, dx=ireijdj , |einj|=1, :
|c-n|<rMrn-10 r0. .. c-n r, c-n=0. n
b) f(z) ;
Q(z)= ; c-m¹0. f(z)¥ zz0- m,
f(z)= ; y(z)- y(z0)¹0. y(z) :
Q(z)= ;.. .
m- .
16.2 f(z)ÎC¥(0<|z-z0|<r(z0)), z0- f(z) |f(z)|=>¥ zz0 ( z z0), z0- f(z).
. |f(z)|=>¥ zz0 => "A>0 $e: 0<|z-z0|<e, |f(z)|>A; g(z)=1/f(z); g(z)ÎC¥(0<|z-z0|<e); |g(z)|<1/A=M => z0- g(z) ( 16.1) g(z)→0 z→z0 => g(z)=(z-z0)mj(z), m³0 =>
=>f(z)= ; y(z0)¹0 n.
. z0, m f(z), m g(z)=1/f(z)
c) z0 f(z), f(z) z0 (z-z0). ( c-n¹0). f(z) . .
- " B "e>0, "h- z0 0<|z-z0|<h $ z1: |f(z1)-B|<e.
. ( ) z0 f(z). $ e0 h0: "z 0<|z-z0|<h0; |f(z)-B|>e0 B. g(z)=1/[f(z)-B]=> |g(z)|=1/|f(z)-B|<1/e0=M. => z0- g(z) ( 16.1) => g(z)=(z-z0)mj(z), m³0 => f(z)=B+ ; y(z0)¹0 => z0- f(z) m¹0, m=0. .. z0 . . n
1. {hn}0 =>{z(n)1}z0. {f(z(n)1)}B=> {z(n)1}z0 , {f(z(n)1)} " .
2 z 0, f(z)≠0 g(z)=1/f(z) z 0
. f(z)=e1/z z=0 - .
. :
.
z0- f(z)ÎC¥(0<|z-z0|<r(z0)).
a) - z0 - f(z).
b) - z0 - f(z).
c) - z0 - f(z).