` xhqnq t''yz - zeakexn zeivwpeta mkqn libxz
(zecewp 30) 1 dl`y
xnelk .
n
dlrnn
p(z)
mepilet ici lr dneqgd ,dnly divwpet
f
idz
.1
|f (z)| ≤ |p(z)|
n
.
gxkda
f
m`d
.
D
megza
n
dlrnn
-l deey e` dphw dlrnn mepilet gxkda `id
mepilet
ici-lr
dneqgd
D
megza
zihilp`
f
-y e`xd .
divwpet
z∈C
f :D→C
lkl
idz
.2
!ewnp ?mepilet
oexzt
f`
|z| ≥ 1
m` ik al miyp .
p(z) = an z n + ... + a1 z + a0
onqp
.1
|p(z)| ≤ |an | · |z|n + ... + |a1 | · |z| + |a0 | ≤ C · |z|n
xy`k
C = |an | + ... + |a1 | + |a0 | ≥ 0 .
f
:
|f
(n + 1)
ly zi-
(n+1)
-d zxfbpl iyew zgqepn lawp
(n + 1)!
2πi
Z
≤
(n + 1)!
·
2π
Z
Z
≤
(n + 1)!
·
2π
Z
≤
(n + 1)!
·
2π
(z)| =
≤
Cr,z
Cr,z
Cr,z
Cr,z
r>0
“
r
dneqge
f
ik xekfp
f (n+1) ≡ 0
ik wiqp
xear okle ,dnly
|f (w)|
dw
|w − z|n+2
|p(w)|
dw
|w − z|n+2
C · |wn |
dw
|w − z|n+2
n
(1 + |z|
(n + 1)!
(r + |z|)n
r )
· 2πr · C ·
=
C
·
(n
+
1)!
·
−−−→ 0
r→∞
2π
rn+2
r
n≥
i
z∈C
f (w)dw
(w − z)n+2
.
mepiletd
lkle
zihilp`
ez
divwpetd
,ziy`xd
aiaq
dlrnn mepilet
1
qeicxn
xeck
f
hxtae
D = B1 (0)
m`
!`ly
D
.mepilet dppi` `id j` .
-a
e
i`ceea
.2
reawd
(zecewp 30) 2 dl`y
miiw
w∈C
lkle
ε>0
C
lkl xnelk ,
-a dtetv
f
ly dpenzd ik e`xd .dreaw dpi`y dnly divwpet
f
z∈C
idz
.1
zeniiwnd zenlyd zeivwpetd lk z` `evnl zpn lr '` sirqa eynzyd
.2
y jk
|f (z) − w| < ε
|f (z)|2 > 2Im(f (z))
1
:oexzt
xicbp .
z
lkl
|f (z) − w0 | > ε0
-y jk
w0 ∈ C
miiwe
ε0 > 0
miiw xnelk .
C
-a dtetv dpi`
f
ly dpenzdy gipp
.1
(z`f ewca) dnlyd divwpetd z`
g(z) =
.dgpdl dxizqa ,dreaw
f
1
f (z) − w0
1
ε0
|g(z)| <
mby o`kn .dreaw `id liaeil htyn itl okle
ik dneqg
g
-y xexa
-l lewy oezpd i`pzdy al miyp
.2
(Im(f (z)) − 1)2 + Re(f (z))2 > 1
|z − i| ≤ 1
.dtetv dpi` gxkda `id okle
zeniiwnd
z
zecewpd lk z` dlikn `l
f
ly dpenzd xnelk .
z
lkl
.i`pzd z` miiwl zeleki zereaw zeivwpet wx ,okl
(zecewp 30) 3 dl`y
lkl
f (az) = f (z)
zniiwne
0
zaiaqa zihilp` ,dreaw `l
f (z)
divwpet zniiw mxear
a
ly mikxrd lk z` e`vn
.ef daiaqa
z
oexzt
zewfg xeh dl yi ef daiaqa f`
f (z) =
∞
X
0
ly daiaqa zihilp`
f
m`
an z n
n=0
okle
f (az) =
∞
X
∞
X
an an z n =
n=0
an z n
n=0
y gxkda xxeby dn
an = 1
miiwzn gxkda ,xnelk .
an 6= 0
exear irah
n
lkl
a = eπiq
divwpetd ,efd dxevdn xtqn lkly xexa .ilpeivx xtqn
q
xy`k
f (z) = z m
.i`pzd z` miiwz mi`zn
m
xear
(zecewp 30) 4 dl`y
z
f (z) = e − e
.
idz .
.lawzn `ed oda zecewpd z`e
S
±1 ± πi
-a
|f (z)|
S \ ∂S
?(lebxza mzi`xy menipind oexwra xfridl ulnen) okid ,ok m` ?
eicewcwy xebqd oalnd
ly meniqwnd z` e`vn
-a menipin zlawn
|f (z)|
m`d
S
idi
.1
.2
oexzt
Re(z) ≤ 1
oalnd
zty
lry
al
miyp
.oalnd
zty
lr
lawzn
|f (z)|
ly
meniqwnd
,meniqwnd
oexwr
itl
okle
|f (z)| = |ez − e| ≤ |ez | + e = eRe(z) + e ≤ 2e
.meniqwn zecewp od dl` okle
2
|f (z)| = 2e
miiwzn
z = 1 ± pi
zecewpa
.1
oxear zecewp od zqt`zn
f (z)
oda zecewpd
ez−1 = 1
xnelk
zk = 1 + 2πik
oexwr itly o`kn
.megza zqt`zn dpi` divwpetd okle
S \ ∂S
megzl zkiiy `l l
“
pd zecewpdn zg` s`
lr meniqwnd oexwr z` elirtd ,dgkedd z` mzi`x `l m`) megza menipin zlawn dpi`
|f (z)|
.(
!dglvda
3
1
f
,menipind
divwpetd
.2