1
〈〈 ABAAACAH 〉〉 AFAJ?AGAGAI 2006.11
D6A1CRBNCQ (2 × 6 = 12AW)
1,CFAZCL z = 1i? 3i1? i,DJ z ANBXCK,AYBEA7A4CK
.
AD z ANBXCK
√34
2,AYBEA7A4CK?arctan
5
3.
2,AVAC z3 + 8 = 0ANCODBB1CK,
AD z1 = 1 + √3i,z2 =?2,z3 = 1? √3i.
3,Ln(?3 + 4i) =,A7A4CV,
ADln5 + i
parenleftbigg
pi? arctan 43 + 2kpi
parenrightbigg
,kCVA0CLA1ln 5 + i
parenleftbigg
pi? arctan 43
parenrightbigg
.
4,B9AW
integraldisplay
C
|z|dz =,C2A5B9AWBSBI C CVAHDGAOAL 1 +iAN
A3D1ASA1
ADBSBIC ANABCLAVACCVz = (1+i)t,0 lessorequalslant t lessorequalslant 1,AT|z| = |1+i|·|t| =√
2t,COD7 integraldisplay
C
|z|dz =
integraldisplay 1
0
√2t(1 + i)dt = √2
2 (1 + i).
5,B9AW
contintegraldisplay
C
ez
z(z2? 1)dz =,C2A5B9AWBSBI C,|z| = 3,AV
D3C8A1D3A1
ADDI |z| = 3 BYA9CDCSBBAJA5C7D1 C1,C2,C3,AWA7A3B5 0,1 B7?1,
C4B8A9A3B5C4B8A9D2BDA0DJ
contintegraldisplay
C
ez
z(z2? 1)dz =
contintegraldisplay
C1
ez
z(z2? 1)dz +
contintegraldisplay
C2
ez
z(z2? 1)dz +
contintegraldisplay
C3
ez
z(z2? 1)dz
= 2pii e
z
z2? 1
vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
+ 2pii e
z
z(z + 1)
vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
+ 2pii e
z
z(z? 1)
vextendsinglevextendsingle
vextendsinglevextendsingle
z=?1
= pii
parenleftbigg
2 + e + 1e
parenrightbigg
.
2
6,B9AW
integraldisplay
C
ez
(z? α)3dz =,C2A5 |α| < 1,C CKA1D3AJCWDH
A6A1
AD z = αCKf(z) DIAJCWDHA6BYANC3AOA0COD7
integraldisplay
C
ez
(z? α)3dz =
2pii
2! [(e
z)′′]z=α = piieα.
AU (2AW),A2BWf(z) = xx2 + y2? i yx2 + y2 DI z negationslash= 0AFBFCZA0A8C5AK
B6CLA1
ADD8CV u = xx2 + y2,v =? yx2 + y2,COD7
u
x =
y2? x2
(x2 + y2)2,
u
y =
2xy
(x2 + y2)2,
v
x =
2xy
(x2 + y2)2,
v
y =
y2? x2
(x2 + y2)2,
D7CECMB0C0AKCLDI?C9DGAOCTANC1BVCEBQD5A0COD7 u,v? z = 0 CTBM
CUA0C4BTA8 C-RCSBCA0D8AG f(z) = u+iv? z = 0CTBFCZA1C4AKB6CLCV
f′(z) =?u?x + i?v?x = y
2? x2
(x2 + y2)2 + i
2xy
(x2 + y2)2.
3
CD (3AW),CXv(x,y) = 2xy + 3xCKAXBMD7A9CVBFCZB6CLAND4AAA2CV
CHBUA2CCBZA0A9ADD6B0BFCZB6CL f(z),C4CICPBHB4AO iCGA0B6CLA4CV 0.
ADD8CV
v
x = 2y + 3,
2v
x2 = 0,
v
y = 2x,
2v
y2 = 0,
DICAC1BVCEBQD5A0C4
2v
x2 +
2v
y2 = 0,B3vCKAPB7B6CLA0DJCPBMA9CVBFCZ
B6CL f(z) AND4AAA1CF f(z) = u + iv.
DA C-RAVAC
u
x =
v
y,AM
u
x = 2x,u =
integraldisplay
2xdx = x2 +?(y),
COD7
u
y =?
′(y),
DA
u
y =?
v
x,AM
′(y) =?(2y + 3).
D8AG?(y) =?y2? 3y + C,B3
u = x2? y2? 3y + C,
DDCKf(z) = x2?y2?3y +C +i(2xy +3x),AICBf(i) = 0,AMC = 4,B3
f(z) = x2? y2? 3y + 4 + i(2xy + 3x) = z2 + 4 + 3iz.
4
CM (2 AW),BACNB9AW
1
2pii
integraldisplay
C
ez
z(1? z)3dz,C2A5 C CVA9BHB4AO 0 DE 1
ANA1D3BBAJA5C7D1A1
AD (1)CCAO 0,1BKA9DI C BYAACGA0DJA4B9B6CL f(z) =
ez
z(1? z)3 DI
D7C CVA6BGANDBBGA5C6DF D CEBFCZA0DABLD0B9AWAQBO
1
2pii
integraldisplay
C
ez
z(1? z)3dz = 0.
(2)CCAO 0 DIC BYAAA0AO 1A9DI C BYAACGA0DABLD0B9AWAQBO
1
2pii
integraldisplay
C
ez
z(1? z)3dz =
1
2pii
integraldisplay
C
ez
(1? z)3
z dz =
ez
(1? z)3
vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 1,
(3)CCAO 1 DIC BYAAA0AO 0A9DI C BYAACGA0DABLD0B9AWAQBO
1
2pii
integraldisplay
C
ez
z(1? z)3dz =
1
2pii
integraldisplay
C
ez
z
(z? 1)3dz =
1
2!
parenleftbigg
e
z
z
parenrightbigg′′ vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
=?e2,
(4)CCAO 0,1ARDI C BYAAA0DI C BYA9B8A9D2BDA0B8A9A3B5ANBRB0DH
A6 C0,C1,AWA7A3B5 0,1,DABLD0B9AWAQBO
1
2pii
integraldisplay
C
ez
z(1? z)3dz =
1
2pii
integraldisplay
C0
ez
z(1? z)3dz +
1
2pii
integraldisplay
C1
ez
z(1? z)3dz
= 1? e2.
5
CY (1 AW),CF f(z) DI|z| < R(R > 1)BFCZA0C4 f(0) = 1,BACNB9AW
1
2pii
contintegraldisplay
C
bracketleftbigg
2 +
parenleftbigg
z + 1z
parenrightbiggbracketrightbigg
f(z)dzz,
C2A5 C CKA1D3AJCWDHA6A1A8BPD9B9AWA2BW
2
pi
integraldisplay 2pi
0
f(eiθ)cos2
parenleftbiggθ
2
parenrightbigg
dθ = 2 + f′(0).
ADB1BJBLD0B9AWB2CJA0DB
1
2pii
contintegraldisplay
C
bracketleftbigg
2 +
parenleftbigg
z + 1z
parenrightbiggbracketrightbigg
f(z)dzz = 12pii
bracketleftbiggcontintegraldisplay
C
2f(z)dzz +
contintegraldisplay
C
(z2 + 1)f(z)dzz2
bracketrightbigg
= 12pii
bracketleftbigg
2 · 2pii[f(z)]
vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
+ [(z2 + 1)f(z)]′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
bracketrightbigg
= 2f(0) + f′(0) = 2 + f′(0).
DCDAAZB9AWBMAM
1
2pii
contintegraldisplay
C
bracketleftbigg
2 +
parenleftbigg
z + 1z
parenrightbiggbracketrightbigg
f(z)dzz = 12pii
integraldisplay 2pi
0
bracketleftbig2 + (eiθ + e?iθ)f(eiθ)dθbracketrightbig
= 12pii
integraldisplay 2pi
0
(1 + cosθ)f(eiθ)dθ
= 1pi
integraldisplay 2pi
0
2cos2
parenleftbiggθ
2
parenrightbigg
f(eiθ)dθ,
COD7
2
pi
integraldisplay 2pi
0
f(eiθ)cos2
parenleftbiggθ
2
parenrightbigg
dθ = 2 + f′(0).