北京邮电大学：复变函数（2006年）：3H

BDD2G1 BXBVF8GIARFKEP
October 18,2006
1/127
FODRCBC6
1,BXBVF8GIFKEPARCECV,BHEFG2C7DYE0DF.
2,BXBVF8GIFKEPARETDRGYEQ ——– DPG4AKCXF9D2,BX
G4BGF0F9D2,DPG4FKEPGYEQ.
3,B5G3F8GIG2C6G2F8GIARB3CG.
2/127
FODRCBC6
1,BXBVF8GIFKEPARCECV,BHEFG2C7DYE0DF.
2,BXBVF8GIFKEPARETDRGYEQ ——– DPG4AKCXF9D2,BX
G4BGF0F9D2,DPG4FKEPGYEQ.
3,B5G3F8GIG2C6G2F8GIARB3CG.
2/127
FODRCBC6
1,BXBVF8GIFKEPARCECV,BHEFG2C7DYE0DF.
2,BXBVF8GIFKEPARETDRGYEQ ——– DPG4AKCXF9D2,BX
G4BGF0F9D2,DPG4FKEPGYEQ.
3,B5G3F8GIG2C6G2F8GIARB3CG.
2/127
FODRCBC6
1,BXBVF8GIFKEPARCECV,BHEFG2C7DYE0DF.
2,BXBVF8GIFKEPARETDRGYEQ ——– DPG4AKCXF9D2,BX
G4BGF0F9D2,DPG4FKEPGYEQ.
3,B5G3F8GIG2C6G2F8GIARB3CG.
2/127
BDE8AX BXBVF8GIFKEPARCECV
1,FKEPARF9FE
CZE5ETDM
AN C?EYEHH2DTF9ARE8CMBDDB (F9EPGBBDDB) ETDM,CG
C2FCF9 C ARH0DRDWCDE0E5ELARE8DRDU?AKE0E5,BSD3
EH C F0CZE5ETDM.
CGC2D6D0 A AK B ARE0E5DU? C ARAKE0E5,BSD3D0 B
AK A ARE0E5FUF0 C ARC3E0E5,C8? C?.
A
B
3/127
BDE8AX BXBVF8GIFKEPARCECV
1,FKEPARF9FE
CZE5ETDM
AN C?EYEHH2DTF9ARE8CMBDDB (F9EPGBBDDB) ETDM,CG
C2FCF9 C ARH0DRDWCDE0E5ELARE8DRDU?AKE0E5,BSD3
EH C F0CZE5ETDM.
CGC2D6D0 A AK B ARE0E5DU? C ARAKE0E5,BSD3D0 B
AK A ARE0E5FUF0 C ARC3E0E5,C8? C?.
A
B
3/127
F9FEANF8GI w = f(z) F9FEBFERDX D CB,C?BFER
DX D CBGDBM? A EQBM? B ARE8CMBDDBARCZE5ETDM,D6ET
DM C BRFBEPEK n DRD0GB,ANEPBM?
A = z0,z1,z2,···,zk?1,zk,···,zn = B,
A
B
ζ1
ζ2
z1
z2
z3
zk?1 z
k
ζk
y
x O
D
ζ3
BFDEDRD0GBhatwiderzk?1zk(k = 1,2,···,n) H2BRFBAIE8BM ζk,FB
DUG2EQ
Sn =
nsummationdisplay
k=1
f(ζk)(zk?zk?1) =
nsummationdisplay
k=1
f(ζk)△zk.
4/127
F9FEANF8GI w = f(z) F9FEBFERDX D CB,C?BFER
DX D CBGDBM? A EQBM? B ARE8CMBDDBARCZE5ETDM,D6ET
DM C BRFBEPEK n DRD0GB,ANEPBM?
A = z0,z1,z2,···,zk?1,zk,···,zn = B,
A
B
ζ1
ζ2
z1
z2
z3
zk?1 z
k
ζk
y
x O
D
ζ3
BFDEDRD0GBhatwiderzk?1zk(k = 1,2,···,n) H2BRFBAIE8BM ζk,FB
DUG2EQ
Sn =
nsummationdisplay
k=1
f(ζk)(zk?zk?1) =
nsummationdisplay
k=1
f(ζk)△zk.
4/127
C8 △sk =hatwiderzk?1zk ARDJG5,δ = max
1lessorequalslantklessorequalslantn
{△sk}.
CGC2 δ EQD9AYEE,CGC2FFG0GH C AREPDFBL ζk ARAIDFCG
G3,Sn CZDXE8ARBGDL,BSD3EHGXBGDLDU?F8GI f(z) GVET
DM C ARFKEP,C8DU
integraldisplay
C
f(z)dz = limn→∞
nsummationdisplay
k=1
f(ζk)△zk.
5/127
C8 △sk =hatwiderzk?1zk ARDJG5,δ = max
1lessorequalslantklessorequalslantn
{△sk}.
CGC2 δ EQD9AYEE,CGC2FFG0GH C AREPDFBL ζk ARAIDFCG
G3,Sn CZDXE8ARBGDL,BSD3EHGXBGDLDU?F8GI f(z) GVET
DM C ARFKEP,C8DU
integraldisplay
C
f(z)dz = limn→∞
nsummationdisplay
k=1
f(ζk)△zk.
5/127
2,FKEPDJBFARCMDRBLG1C7DYE0DF
FODRB4G0,CGC2 f(z) = u(x,y) + iv(x,y) BFBDDBET
DM C H2GJCH,C3
integraldisplay
C
f(z)dz DJBF,FBDPCZ
integraldisplay
C
f(z)dz =
integraldisplay
C
udx?vdy
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
+i
integraldisplay
C
vdx + udy
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
.
6/127
2,FKEPDJBFARCMDRBLG1C7DYE0DF
FODRB4G0,CGC2 f(z) = u(x,y) + iv(x,y) BFBDDBET
DM C H2GJCH,C3
integraldisplay
C
f(z)dz DJBF,FBDPCZ
integraldisplay
C
f(z)dz =
integraldisplay
C
udx?vdy
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
+i
integraldisplay
C
vdx + udy
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
.
6/127
GLAJ,ANBDDBETDM C ARGJGIE0EN
z = z(t) = x(t) + iy(t),α lessorequalslant t lessorequalslant β
AKE0E5?GJGIC7CKARE0E5,GJGI α BL β GHBTD9GDBM A BL
EQBM B,FBDP z′(t) negationslash= 0,α < t < β.
CGC2 f(z) = u(x,y) + iv(x,y) BF D CBBRBRGJCH,
AN ζk = ξk + iηk,CSD9
△zk = zk?zk?1 = xk + iyk?(xk?1 + iyk?1)
= xk?xk?1 + i(yk?yk?1) = △xk + i△yk,
7/127
GLAJ,ANBDDBETDM C ARGJGIE0EN
z = z(t) = x(t) + iy(t),α lessorequalslant t lessorequalslant β
AKE0E5?GJGIC7CKARE0E5,GJGI α BL β GHBTD9GDBM A BL
EQBM B,FBDP z′(t) negationslash= 0,α < t < β.
CGC2 f(z) = u(x,y) + iv(x,y) BF D CBBRBRGJCH,
AN ζk = ξk + iηk,CSD9
△zk = zk?zk?1 = xk + iyk?(xk?1 + iyk?1)
= xk?xk?1 + i(yk?yk?1) = △xk + i△yk,
7/127
EKEX
nsummationdisplay
k=1
f(ζk)△zk =
nsummationdisplay
k=1
[u(ξk,ηk) + iv(ξk,ηk)](△xk + i△yk)
=
nsummationdisplay
k=1
[u(ξk,ηk)△xk?v(ξk,ηk)△yk]
+i
nsummationdisplay
k=1
[v(ξk,ηk)△xk + u(ξk,ηk)△yk].
CSD9 u,v FTF0GJCHF8GI,DUGADMFKEPARDJBFF9D2,H2EQD1
G8ARH0DRG2EQARBGDLFTF0DJBFAR,FRCS
integraldisplay
C
f(z)dz =
integraldisplay
C
udx?vdy + i
integraldisplay
C
vdx + udy.
8/127
EKEX
nsummationdisplay
k=1
f(ζk)△zk =
nsummationdisplay
k=1
[u(ξk,ηk) + iv(ξk,ηk)](△xk + i△yk)
=
nsummationdisplay
k=1
[u(ξk,ηk)△xk?v(ξk,ηk)△yk]
+i
nsummationdisplay
k=1
[v(ξk,ηk)△xk + u(ξk,ηk)△yk].
CSD9 u,v FTF0GJCHF8GI,DUGADMFKEPARDJBFF9D2,H2EQD1
G8ARH0DRG2EQARBGDLFTF0DJBFAR,FRCS
integraldisplay
C
f(z)dz =
integraldisplay
C
udx?vdy + i
integraldisplay
C
vdx + udy.
8/127
EKEX
nsummationdisplay
k=1
f(ζk)△zk =
nsummationdisplay
k=1
[u(ξk,ηk) + iv(ξk,ηk)](△xk + i△yk)
=
nsummationdisplay
k=1
[u(ξk,ηk)△xk?v(ξk,ηk)△yk]
+i
nsummationdisplay
k=1
[v(ξk,ηk)△xk + u(ξk,ηk)△yk].
CSD9 u,v FTF0GJCHF8GI,DUGADMFKEPARDJBFF9D2,H2EQD1
G8ARH0DRG2EQARBGDLFTF0DJBFAR,FRCS
integraldisplay
C
f(z)dz =
integraldisplay
C
udx?vdy + i
integraldisplay
C
vdx + udy.
8/127
EKEX
nsummationdisplay
k=1
f(ζk)△zk =
nsummationdisplay
k=1
[u(ξk,ηk) + iv(ξk,ηk)](△xk + i△yk)
=
nsummationdisplay
k=1
[u(ξk,ηk)△xk?v(ξk,ηk)△yk]
+i
nsummationdisplay
k=1
[v(ξk,ηk)△xk + u(ξk,ηk)△yk].
CSD9 u,v FTF0GJCHF8GI,DUGADMFKEPARDJBFF9D2,H2EQD1
G8ARH0DRG2EQARBGDLFTF0DJBFAR,FRCS
integraldisplay
C
f(z)dz =
integraldisplay
C
udx?vdy + i
integraldisplay
C
vdx + udy.
8/127
H2EQARB8E8C4ERBBEQ
FR?ETDM C ARGJGIE0EN
z(t) = x(t) + iy(t) (α lessorequalslant t lessorequalslant β),
DUGADMFKEPARC7DYE0DF,EKEXintegraldisplay
C
f(z)dz =
integraldisplay
C
u(x,y)dx?v(x,y)dy + i
integraldisplay
C
v(x,y)dx + u(x,y)dy
=
integraldisplay β
α
{u[x(t),y(t)]x′(t)?v[x(t),y(t)]y′(t)}dt
+i
integraldisplay β
α
{v[x(t),y(t)]x′(t) + u[x(t),y(t)]y′(t)}dt
H2EQD1G8DWEXALEK
integraldisplay β
α
{u[x(t),y(t)] + iv[x(t),y(t)]}·{x′(t) + iy′(t)}dt
=
integraldisplay β
α
f[z(t)]z′(t)dt.
EKEX
integraldisplay integraldisplay β9/127
H2EQARB8E8C4ERBBEQ
FR?ETDM C ARGJGIE0EN
z(t) = x(t) + iy(t) (α lessorequalslant t lessorequalslant β),
DUGADMFKEPARC7DYE0DF,EKEXintegraldisplay
C
f(z)dz =
integraldisplay
C
u(x,y)dx?v(x,y)dy + i
integraldisplay
C
v(x,y)dx + u(x,y)dy
=
integraldisplay β
α
{u[x(t),y(t)]x′(t)?v[x(t),y(t)]y′(t)}dt
+i
integraldisplay β
α
{v[x(t),y(t)]x′(t) + u[x(t),y(t)]y′(t)}dt
H2EQD1G8DWEXALEK
integraldisplay β
α
{u[x(t),y(t)] + iv[x(t),y(t)]}·{x′(t) + iy′(t)}dt
=
integraldisplay β
α
f[z(t)]z′(t)dt.
EKEX
integraldisplay integraldisplay β9/127
H2EQARB8E8C4ERBBEQ
FR?ETDM C ARGJGIE0EN
z(t) = x(t) + iy(t) (α lessorequalslant t lessorequalslant β),
DUGADMFKEPARC7DYE0DF,EKEXintegraldisplay
C
f(z)dz =
integraldisplay
C
u(x,y)dx?v(x,y)dy + i
integraldisplay
C
v(x,y)dx + u(x,y)dy
=
integraldisplay β
α
{u[x(t),y(t)]x′(t)?v[x(t),y(t)]y′(t)}dt
+i
integraldisplay β
α
{v[x(t),y(t)]x′(t) + u[x(t),y(t)]y′(t)}dt
H2EQD1G8DWEXALEK
integraldisplay β
α
{u[x(t),y(t)] + iv[x(t),y(t)]}·{x′(t) + iy′(t)}dt
=
integraldisplay β
α
f[z(t)]z′(t)dt.
EKEX
integraldisplay integraldisplay β9/127
H2EQARB8E8C4ERBBEQ
FR?ETDM C ARGJGIE0EN
z(t) = x(t) + iy(t) (α lessorequalslant t lessorequalslant β),
DUGADMFKEPARC7DYE0DF,EKEXintegraldisplay
C
f(z)dz =
integraldisplay
C
u(x,y)dx?v(x,y)dy + i
integraldisplay
C
v(x,y)dx + u(x,y)dy
=
integraldisplay β
α
{u[x(t),y(t)]x′(t)?v[x(t),y(t)]y′(t)}dt
+i
integraldisplay β
α
{v[x(t),y(t)]x′(t) + u[x(t),y(t)]y′(t)}dt
H2EQD1G8DWEXALEK
integraldisplay β
α
{u[x(t),y(t)] + iv[x(t),y(t)]}·{x′(t) + iy′(t)}dt
=
integraldisplay β
α
f[z(t)]z′(t)dt.
EKEX
integraldisplay integraldisplay β9/127
H2EQARB8E8C4ERBBEQ
FR?ETDM C ARGJGIE0EN
z(t) = x(t) + iy(t) (α lessorequalslant t lessorequalslant β),
DUGADMFKEPARC7DYE0DF,EKEXintegraldisplay
C
f(z)dz =
integraldisplay
C
u(x,y)dx?v(x,y)dy + i
integraldisplay
C
v(x,y)dx + u(x,y)dy
=
integraldisplay β
α
{u[x(t),y(t)]x′(t)?v[x(t),y(t)]y′(t)}dt
+i
integraldisplay β
α
{v[x(t),y(t)]x′(t) + u[x(t),y(t)]y′(t)}dt
H2EQD1G8DWEXALEK
integraldisplay β
α
{u[x(t),y(t)] + iv[x(t),y(t)]}·{x′(t) + iy′(t)}dt
=
integraldisplay β
α
f[z(t)]z′(t)dt.
EKEX
integraldisplay integraldisplay β9/127
H2EQARB8E8C4ERBBEQ
FR?ETDM C ARGJGIE0EN
z(t) = x(t) + iy(t) (α lessorequalslant t lessorequalslant β),
DUGADMFKEPARC7DYE0DF,EKEXintegraldisplay
C
f(z)dz =
integraldisplay
C
u(x,y)dx?v(x,y)dy + i
integraldisplay
C
v(x,y)dx + u(x,y)dy
=
integraldisplay β
α
{u[x(t),y(t)]x′(t)?v[x(t),y(t)]y′(t)}dt
+i
integraldisplay β
α
{v[x(t),y(t)]x′(t) + u[x(t),y(t)]y′(t)}dt
H2EQD1G8DWEXALEK
integraldisplay β
α
{u[x(t),y(t)] + iv[x(t),y(t)]}·{x′(t) + iy′(t)}dt
=
integraldisplay β
α
f[z(t)]z′(t)dt.
EKEX
integraldisplay integraldisplay β9/127
A8 C F0EPGBBDDBETDM,C = C1 + C2 +···+ Cn EE,CZ
integraldisplay
C
f(z)dz =
integraldisplay
C1
f(z)dz +
integraldisplay
C2
f(z)dz +···+
integraldisplay
Cn
f(z)dz.
10/127
D3B4GDBH,C7DYGVBDDBETDMARBXBVF8GIFKEPARE0DFCZ:
1,DE?H0DREIAJA3F8GIARDMFKEPBHC7DY.
integraldisplay
C
f(z)dz =
integraldisplay
C
udx?vdy + i
integraldisplay
C
vdx + udy.
2,ALFTC ARGJGIE0EN,z = z(t) = x(t)+iy(t),B8A0EM
integraldisplay
C
f(z)dz =
integraldisplay β
α
f[z(t)]z′(t)dt.
11/127
D3B4GDBH,C7DYGVBDDBETDMARBXBVF8GIFKEPARE0DFCZ:
1,DE?H0DREIAJA3F8GIARDMFKEPBHC7DY.
integraldisplay
C
f(z)dz =
integraldisplay
C
udx?vdy + i
integraldisplay
C
vdx + udy.
2,ALFTC ARGJGIE0EN,z = z(t) = x(t)+iy(t),B8A0EM
integraldisplay
C
f(z)dz =
integraldisplay β
α
f[z(t)]z′(t)dt.
11/127
D3B4GDBH,C7DYGVBDDBETDMARBXBVF8GIFKEPARE0DFCZ:
1,DE?H0DREIAJA3F8GIARDMFKEPBHC7DY.
integraldisplay
C
f(z)dz =
integraldisplay
C
udx?vdy + i
integraldisplay
C
vdx + udy.
2,ALFTC ARGJGIE0EN,z = z(t) = x(t)+iy(t),B8A0EM
integraldisplay
C
f(z)dz =
integraldisplay β
α
f[z(t)]z′(t)dt.
11/127
3,FKEPARBHEF
1)
integraldisplay
C
f(z)dz =?
integraldisplay
C?
f(z)dz;
2)
integraldisplay
C
kf(z)dz = k
integraldisplay
C
f(z)dz;(k?DIGI)
3)
integraldisplay
C
[f(z)±g(z)]dz =
integraldisplay
C
f(z)dz ±
integraldisplay
C
g(z)dz.
12/127
4) ANETDM C ARDJG5? L,F8GI f(z) BF C H2GV
DA |f(z)|lessorequalslant M,BSD3vextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
C
f(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C
|f(z)|ds lessorequalslant ML.
DDEY AN |△zk| = |zk?zk?1|,DP △sk?GXH0BMDND1AR
D0GBARDJG5,DBB8 |△zk|lessorequalslant△sk,EKEXvextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
nsummationdisplay
k=1
f(ζk)△zk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
nsummationdisplay
k=1
|f(ζk)△zk|lessorequalslant
nsummationdisplay
k=1
|f(ζk)|△sk
B9 δ = max
1lessorequalslantklessorequalslantn
{△sk} → 0,AQ
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
f(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C
|f(z)|ds.
FR?BF C H2 |f(z)|lessorequalslant M,AQAKintegraldisplay
C
|f(z)|ds lessorequalslant M
integraldisplay
C
ds = ML.
13/127
4) ANETDM C ARDJG5? L,F8GI f(z) BF C H2GV
DA |f(z)|lessorequalslant M,BSD3vextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
C
f(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C
|f(z)|ds lessorequalslant ML.
DDEY AN |△zk| = |zk?zk?1|,DP △sk?GXH0BMDND1AR
D0GBARDJG5,DBB8 |△zk|lessorequalslant△sk,EKEXvextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
nsummationdisplay
k=1
f(ζk)△zk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
nsummationdisplay
k=1
|f(ζk)△zk|lessorequalslant
nsummationdisplay
k=1
|f(ζk)|△sk
B9 δ = max
1lessorequalslantklessorequalslantn
{△sk} → 0,AQ
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
f(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C
|f(z)|ds.
FR?BF C H2 |f(z)|lessorequalslant M,AQAKintegraldisplay
C
|f(z)|ds lessorequalslant M
integraldisplay
C
ds = ML.
13/127
4) ANETDM C ARDJG5? L,F8GI f(z) BF C H2GV
DA |f(z)|lessorequalslant M,BSD3vextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
C
f(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C
|f(z)|ds lessorequalslant ML.
DDEY AN |△zk| = |zk?zk?1|,DP △sk?GXH0BMDND1AR
D0GBARDJG5,DBB8 |△zk|lessorequalslant△sk,EKEXvextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
nsummationdisplay
k=1
f(ζk)△zk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
nsummationdisplay
k=1
|f(ζk)△zk|lessorequalslant
nsummationdisplay
k=1
|f(ζk)|△sk
B9 δ = max
1lessorequalslantklessorequalslantn
{△sk} → 0,AQ
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
f(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C
|f(z)|ds.
FR?BF C H2 |f(z)|lessorequalslant M,AQAKintegraldisplay
C
|f(z)|ds lessorequalslant M
integraldisplay
C
ds = ML.
13/127
4) ANETDM C ARDJG5? L,F8GI f(z) BF C H2GV
DA |f(z)|lessorequalslant M,BSD3vextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
C
f(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C
|f(z)|ds lessorequalslant ML.
DDEY AN |△zk| = |zk?zk?1|,DP △sk?GXH0BMDND1AR
D0GBARDJG5,DBB8 |△zk|lessorequalslant△sk,EKEXvextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
nsummationdisplay
k=1
f(ζk)△zk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
nsummationdisplay
k=1
|f(ζk)△zk|lessorequalslant
nsummationdisplay
k=1
|f(ζk)|△sk
B9 δ = max
1lessorequalslantklessorequalslantn
{△sk} → 0,AQ
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
f(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C
|f(z)|ds.
FR?BF C H2 |f(z)|lessorequalslant M,AQAKintegraldisplay
C
|f(z)|ds lessorequalslant M
integraldisplay
C
ds = ML.
13/127
4) ANETDM C ARDJG5? L,F8GI f(z) BF C H2GV
DA |f(z)|lessorequalslant M,BSD3vextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
C
f(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C
|f(z)|ds lessorequalslant ML.
DDEY AN |△zk| = |zk?zk?1|,DP △sk?GXH0BMDND1AR
D0GBARDJG5,DBB8 |△zk|lessorequalslant△sk,EKEXvextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
nsummationdisplay
k=1
f(ζk)△zk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
nsummationdisplay
k=1
|f(ζk)△zk|lessorequalslant
nsummationdisplay
k=1
|f(ζk)|△sk
B9 δ = max
1lessorequalslantklessorequalslantn
{△sk} → 0,AQ
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
f(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C
|f(z)|ds.
FR?BF C H2 |f(z)|lessorequalslant M,AQAKintegraldisplay
C
|f(z)|ds lessorequalslant M
integraldisplay
C
ds = ML.
13/127
4) ANETDM C ARDJG5? L,F8GI f(z) BF C H2GV
DA |f(z)|lessorequalslant M,BSD3vextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
C
f(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C
|f(z)|ds lessorequalslant ML.
DDEY AN |△zk| = |zk?zk?1|,DP △sk?GXH0BMDND1AR
D0GBARDJG5,DBB8 |△zk|lessorequalslant△sk,EKEXvextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
nsummationdisplay
k=1
f(ζk)△zk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
nsummationdisplay
k=1
|f(ζk)△zk|lessorequalslant
nsummationdisplay
k=1
|f(ζk)|△sk
B9 δ = max
1lessorequalslantklessorequalslantn
{△sk} → 0,AQ
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
f(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C
|f(z)|ds.
FR?BF C H2 |f(z)|lessorequalslant M,AQAKintegraldisplay
C
|f(z)|ds lessorequalslant M
integraldisplay
C
ds = ML.
13/127
DI 1 C7DYFKEP
integraldisplay
C1
(x2 + iy)dz,
integraldisplay
C2
(x2 + iy)dz,
G1EL C1,C2 EPC7F0GV y = x DQ y = x2 D0A6BMAKBM 1 + i
ARETDM.
y
O x
1+i
y=x
y=x2
14/127
y
O x
1+i
y=x
y=x2
B5F0DM C1 ARGJGIE0EN:
z = t + it,tD00AK1,
D9F0 dz = (1 + i)dt.
EKEXintegraldisplay
C1
(x2 + iy)dz =
integraldisplay 1
0
(t2 + it)(1 + i)dt
= (1 + i)
integraldisplay 1
0
t2dt + i(1 + i)
integraldisplay 1
0
tdt
=?16 + 56i.
15/127
y
O x
1+i
y=x
y=x2
B5F0DM C1 ARGJGIE0EN:
z = t + it,tD00AK1,
D9F0 dz = (1 + i)dt.
EKEXintegraldisplay
C1
(x2 + iy)dz =
integraldisplay 1
0
(t2 + it)(1 + i)dt
= (1 + i)
integraldisplay 1
0
t2dt + i(1 + i)
integraldisplay 1
0
tdt
=?16 + 56i.
15/127
y
O x
1+i
y=x
y=x2
F0DM C2 ARGJGIE0EN:
z = t + it2,tD00AK1,
D9F0 dz = (1 + 2it)dt,EKEX
integraldisplay
C2
(x2 + iy)dz =
integraldisplay 1
0
(t2 + it2)(1 + 2it)dt
= (1 + i)
integraldisplay 1
0
t2dt + 2i(1 + i)
integraldisplay 1
0
t2dt
=?16 + 56i.
16/127
y
O x
1+i
y=x
y=x2
F0DM C2 ARGJGIE0EN:
z = t + it2,tD00AK1,
D9F0 dz = (1 + 2it)dt,EKEX
integraldisplay
C2
(x2 + iy)dz =
integraldisplay 1
0
(t2 + it2)(1 + 2it)dt
= (1 + i)
integraldisplay 1
0
t2dt + 2i(1 + i)
integraldisplay 1
0
t2dt
=?16 + 56i.
16/127
y
O x
1+i
y=x
y=x2
F0DM C2 ARGJGIE0EN:
z = t + it2,tD00AK1,
D9F0 dz = (1 + 2it)dt,EKEX
integraldisplay
C2
(x2 + iy)dz =
integraldisplay 1
0
(t2 + it2)(1 + 2it)dt
= (1 + i)
integraldisplay 1
0
t2dt + 2i(1 + i)
integraldisplay 1
0
t2dt
=?16 + 56i.
16/127
y
O x
1+i
y=x
y=x2
F0DM C2 ARGJGIE0EN:
z = t + it2,tD00AK1,
D9F0 dz = (1 + 2it)dt,EKEX
integraldisplay
C2
(x2 + iy)dz =
integraldisplay 1
0
(t2 + it2)(1 + 2it)dt
= (1 + i)
integraldisplay 1
0
t2dt + 2i(1 + i)
integraldisplay 1
0
t2dt
=?16 + 56i.
16/127
y
O x
1+i
y=x
y=x2
F0DM C2 ARGJGIE0EN:
z = t + it2,tD00AK1,
D9F0 dz = (1 + 2it)dt,EKEX
integraldisplay
C2
(x2 + iy)dz =
integraldisplay 1
0
(t2 + it2)(1 + 2it)dt
= (1 + i)
integraldisplay 1
0
t2dt + 2i(1 + i)
integraldisplay 1
0
t2dt
=?16 + 56i.
16/127
DI 2 C7DY I =
integraldisplay
C
z
zdz,G1EL C F0ABDM{z|1 lessorequalslant|z|lessorequalslant 2}
H2DWG4EPARBQBA,G1E0E5CGGBEKER.
C1
C2
O?2?1 1 2
y
x
C3 C4
17/127
C1
C2
O?2?1 1 2
y
x
C3 C4
B5 AN C1,C2 F0H0DRDWABEX,ENDMARE0ENEPC7?:
C1,z(θ) = 1eiθ,θD0piAK0,
C2,z(θ) = 2eiθ,θD00AKpi,
BSD3integraldisplay
C1
z
zdz =
integraldisplay 0
pi
eiθ
eiθ ·ie
iθdθ =
integraldisplay 0
pi
ie3iθdθ = 23,
integraldisplay
C2
z
zdz =
integraldisplay pi
0
2eiθ
2eiθ ·2ie
iθdθ =
integraldisplay pi
0
2ie3iθdθ =?43.
18/127
C1
C2
O?2?1 1 2
y
x
C3 C4
B5 AN C1,C2 F0H0DRDWABEX,ENDMARE0ENEPC7?:
C1,z(θ) = 1eiθ,θD0piAK0,
C2,z(θ) = 2eiθ,θD00AKpi,
BSD3integraldisplay
C1
z
zdz =
integraldisplay 0
pi
eiθ
eiθ ·ie
iθdθ =
integraldisplay 0
pi
ie3iθdθ = 23,
integraldisplay
C2
z
zdz =
integraldisplay pi
0
2eiθ
2eiθ ·2ie
iθdθ =
integraldisplay pi
0
2ie3iθdθ =?43.
18/127
C1
C2
O?2?1 1 2
y
x
C3 C4
B5 AN C1,C2 F0H0DRDWABEX,ENDMARE0ENEPC7?:
C1,z(θ) = 1eiθ,θD0piAK0,
C2,z(θ) = 2eiθ,θD00AKpi,
BSD3integraldisplay
C1
z
zdz =
integraldisplay 0
pi
eiθ
eiθ ·ie
iθdθ =
integraldisplay 0
pi
ie3iθdθ = 23,
integraldisplay
C2
z
zdz =
integraldisplay pi
0
2eiθ
2eiθ ·2ie
iθdθ =
integraldisplay pi
0
2ie3iθdθ =?43.
18/127
C1
C2
O?2?1 1 2
y
x
C3 C4
ADBF C3,C4 H2,CZ z = z = x,C3
integraldisplay
C3
z
zdz =
integraldisplay?1
2
dx = 1,
integraldisplay
C4
z
zdz =
integraldisplay 2
1
dx = 1,
D2H2EKGB,CZ
I =
integraldisplay
C1
+
integraldisplay
C2
+
integraldisplay
C3
+
integraldisplay
C4
= 23? 43 + 1 + 1 = 43.
19/127
C1
C2
O?2?1 1 2
y
x
C3 C4
ADBF C3,C4 H2,CZ z = z = x,C3
integraldisplay
C3
z
zdz =
integraldisplay?1
2
dx = 1,
integraldisplay
C4
z
zdz =
integraldisplay 2
1
dx = 1,
D2H2EKGB,CZ
I =
integraldisplay
C1
+
integraldisplay
C2
+
integraldisplay
C3
+
integraldisplay
C4
= 23? 43 + 1 + 1 = 43.
19/127
DI 3 C7DY
contintegraldisplay
C
dz
(z?z0)n+1,G1EL,C?EX z0?ELB1,r?
DWF7ARAKE5ABEX,n?AIGI.
B5 C ARGJGIE0EN:
z = z0 + reiθ,0 lessorequalslant θ lessorequalslant 2pi,
EKEX
contintegraldisplay
C
dz
(z?z0)n+1dz =
integraldisplay 2pi
0
ireiθ
rn+1ei(n+1)θdθ
=
integraldisplay 2pi
0
i
rneinθdθ =
i
rn
integraldisplay 2pi
0
e?inθdθ.
20/127
DI 3 C7DY
contintegraldisplay
C
dz
(z?z0)n+1,G1EL,C?EX z0?ELB1,r?
DWF7ARAKE5ABEX,n?AIGI.
B5 C ARGJGIE0EN:
z = z0 + reiθ,0 lessorequalslant θ lessorequalslant 2pi,
EKEX
contintegraldisplay
C
dz
(z?z0)n+1dz =
integraldisplay 2pi
0
ireiθ
rn+1ei(n+1)θdθ
=
integraldisplay 2pi
0
i
rneinθdθ =
i
rn
integraldisplay 2pi
0
e?inθdθ.
20/127
DI 3 C7DY
contintegraldisplay
C
dz
(z?z0)n+1,G1EL,C?EX z0?ELB1,r?
DWF7ARAKE5ABEX,n?AIGI.
B5 C ARGJGIE0EN:
z = z0 + reiθ,0 lessorequalslant θ lessorequalslant 2pi,
EKEX
contintegraldisplay
C
dz
(z?z0)n+1dz =
integraldisplay 2pi
0
ireiθ
rn+1ei(n+1)θdθ
=
integraldisplay 2pi
0
i
rneinθdθ =
i
rn
integraldisplay 2pi
0
e?inθdθ.
20/127
DI 3 C7DY
contintegraldisplay
C
dz
(z?z0)n+1,G1EL,C?EX z0?ELB1,r?
DWF7ARAKE5ABEX,n?AIGI.
B5 C ARGJGIE0EN:
z = z0 + reiθ,0 lessorequalslant θ lessorequalslant 2pi,
EKEX
contintegraldisplay
C
dz
(z?z0)n+1dz =
integraldisplay 2pi
0
ireiθ
rn+1ei(n+1)θdθ
=
integraldisplay 2pi
0
i
rneinθdθ =
i
rn
integraldisplay 2pi
0
e?inθdθ.
20/127
A8 n = 0 EE,B4C2?
i
integraldisplay 2pi
0
dθ = 2pii;
A8 n negationslash= 0 EE,B4C2?:
i
rn
integraldisplay 2pi
0
e?inθdθ = irn
integraldisplay 2pi
0
(cosnθ?isinnθ)dθ = 0.
EKEX
contintegraldisplay
C
dz
(z?z0)n+1 =
braceleftbigg 2pii,n = 0,
0,n negationslash= 0.
21/127
A8 n = 0 EE,B4C2?
i
integraldisplay 2pi
0
dθ = 2pii;
A8 n negationslash= 0 EE,B4C2?:
i
rn
integraldisplay 2pi
0
e?inθdθ = irn
integraldisplay 2pi
0
(cosnθ?isinnθ)dθ = 0.
EKEX
contintegraldisplay
C
dz
(z?z0)n+1 =
braceleftbigg 2pii,n = 0,
0,n negationslash= 0.
21/127
A8 n = 0 EE,B4C2?
i
integraldisplay 2pi
0
dθ = 2pii;
A8 n negationslash= 0 EE,B4C2?:
i
rn
integraldisplay 2pi
0
e?inθdθ = irn
integraldisplay 2pi
0
(cosnθ?isinnθ)dθ = 0.
EKEX
contintegraldisplay
C
dz
(z?z0)n+1 =
braceleftbigg 2pii,n = 0,
0,n negationslash= 0.
21/127
DI 4 DDEY:
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
z + 1
z?1dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 8pi,
G1EL C F0ABEX |z?1| = 2.
DDDUGAAFDUF9D2,CZ
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
z + 1
z?1dz
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
integraldisplay
C
vextendsinglevextendsingle
vextendsinglevextendsinglez + 1
z?1
vextendsinglevextendsingle
vextendsinglevextendsingleds =
integraldisplay
C
vextendsinglevextendsingle
vextendsinglevextendsingle(z?1) + 2
2
vextendsinglevextendsingle
vextendsinglevextendsingleds
lessorequalslant
integraldisplay
C
|z?1|+ 2
2 ds =
integraldisplay
C
2 + 2
2 ds
=
integraldisplay
C
2ds = 2·4pi = 8pi.
22/127
DI 4 DDEY:
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
z + 1
z?1dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 8pi,
G1EL C F0ABEX |z?1| = 2.
DDDUGAAFDUF9D2,CZ
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
z + 1
z?1dz
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
integraldisplay
C
vextendsinglevextendsingle
vextendsinglevextendsinglez + 1
z?1
vextendsinglevextendsingle
vextendsinglevextendsingleds =
integraldisplay
C
vextendsinglevextendsingle
vextendsinglevextendsingle(z?1) + 2
2
vextendsinglevextendsingle
vextendsinglevextendsingleds
lessorequalslant
integraldisplay
C
|z?1|+ 2
2 ds =
integraldisplay
C
2 + 2
2 ds
=
integraldisplay
C
2ds = 2·4pi = 8pi.
22/127
DI 4 DDEY:
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
z + 1
z?1dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 8pi,
G1EL C F0ABEX |z?1| = 2.
DDDUGAAFDUF9D2,CZ
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
z + 1
z?1dz
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
integraldisplay
C
vextendsinglevextendsingle
vextendsinglevextendsinglez + 1
z?1
vextendsinglevextendsingle
vextendsinglevextendsingleds =
integraldisplay
C
vextendsinglevextendsingle
vextendsinglevextendsingle(z?1) + 2
2
vextendsinglevextendsingle
vextendsinglevextendsingleds
lessorequalslant
integraldisplay
C
|z?1|+ 2
2 ds =
integraldisplay
C
2 + 2
2 ds
=
integraldisplay
C
2ds = 2·4pi = 8pi.
22/127
DI 4 DDEY:
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
z + 1
z?1dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant 8pi,
G1EL C F0ABEX |z?1| = 2.
DDDUGAAFDUF9D2,CZ
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C
z + 1
z?1dz
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
integraldisplay
C
vextendsinglevextendsingle
vextendsinglevextendsinglez + 1
z?1
vextendsinglevextendsingle
vextendsinglevextendsingleds =
integraldisplay
C
vextendsinglevextendsingle
vextendsinglevextendsingle(z?1) + 2
2
vextendsinglevextendsingle
vextendsinglevextendsingleds
lessorequalslant
integraldisplay
C
|z?1|+ 2
2 ds =
integraldisplay
C
2 + 2
2 ds
=
integraldisplay
C
2ds = 2·4pi = 8pi.
22/127
DI 5 AN I =
integraldisplay
C
eiz
z dz,G1EL,C F0D0 r AK?r ARH2DWAB
EX |z| = r,DDEY,limr→+∞I = 0.
y
x O
r?r
C:|z|=r
23/127
y
x O
r?r
C:|z|=r
DD ETDM C ARGJGIE0EN:
z = reiθ = r(cosθ + isinθ),0 lessorequalslant θ lessorequalslant pi.
EKEX
I =
integraldisplay
C
eiz
z dz =
integraldisplay pi
0
ie?r(sinθ+icosθ)dθ,
24/127
y
x O
r?r
C:|z|=r
DD ETDM C ARGJGIE0EN:
z = reiθ = r(cosθ + isinθ),0 lessorequalslant θ lessorequalslant pi.
EKEX
I =
integraldisplay
C
eiz
z dz =
integraldisplay pi
0
ie?r(sinθ+icosθ)dθ,
24/127
D9F0
|I| lessorequalslant
integraldisplay pi
0
vextendsinglevextendsingle
vextendsingleie?rsinθ+ircosθ
vextendsinglevextendsingle
vextendsingledθ
=
integraldisplay pi
0
e?rsinθdθ = 2
integraldisplay pi2
0
e?rsinθdθ
parenleftBigg
DGCMF9FKEPARBVH2E3DQ,CZ
integraldisplay pi2
0
e?rsinθdθ =
integraldisplay pi
pi
2
e?rsinθdθ
parenrightBigg
lessorequalslant 2
integraldisplay pi2
0
e?2rθpi dθ
parenleftbigg
FR?A80 lessorequalslant θ lessorequalslant pi2EE,CZ
2θ
pi lessorequalslant sinθ lessorequalslant θ
parenrightbigg
= pir(1?e?r)
EKEX limr→+∞I = 0.
25/127
D9F0
|I| lessorequalslant
integraldisplay pi
0
vextendsinglevextendsingle
vextendsingleie?rsinθ+ircosθ
vextendsinglevextendsingle
vextendsingledθ
=
integraldisplay pi
0
e?rsinθdθ = 2
integraldisplay pi2
0
e?rsinθdθ
parenleftBigg
DGCMF9FKEPARBVH2E3DQ,CZ
integraldisplay pi2
0
e?rsinθdθ =
integraldisplay pi
pi
2
e?rsinθdθ
parenrightBigg
lessorequalslant 2
integraldisplay pi2
0
e?2rθpi dθ
parenleftbigg
FR?A80 lessorequalslant θ lessorequalslant pi2EE,CZ
2θ
pi lessorequalslant sinθ lessorequalslant θ
parenrightbigg
= pir(1?e?r)
EKEX limr→+∞I = 0.
25/127
D9F0
|I| lessorequalslant
integraldisplay pi
0
vextendsinglevextendsingle
vextendsingleie?rsinθ+ircosθ
vextendsinglevextendsingle
vextendsingledθ
=
integraldisplay pi
0
e?rsinθdθ = 2
integraldisplay pi2
0
e?rsinθdθ
parenleftBigg
DGCMF9FKEPARBVH2E3DQ,CZ
integraldisplay pi2
0
e?rsinθdθ =
integraldisplay pi
pi
2
e?rsinθdθ
parenrightBigg
lessorequalslant 2
integraldisplay pi2
0
e?2rθpi dθ
parenleftbigg
FR?A80 lessorequalslant θ lessorequalslant pi2EE,CZ
2θ
pi lessorequalslant sinθ lessorequalslant θ
parenrightbigg
= pir(1?e?r)
EKEX limr→+∞I = 0.
25/127
D9F0
|I| lessorequalslant
integraldisplay pi
0
vextendsinglevextendsingle
vextendsingleie?rsinθ+ircosθ
vextendsinglevextendsingle
vextendsingledθ
=
integraldisplay pi
0
e?rsinθdθ = 2
integraldisplay pi2
0
e?rsinθdθ
parenleftBigg
DGCMF9FKEPARBVH2E3DQ,CZ
integraldisplay pi2
0
e?rsinθdθ =
integraldisplay pi
pi
2
e?rsinθdθ
parenrightBigg
lessorequalslant 2
integraldisplay pi2
0
e?2rθpi dθ
parenleftbigg
FR?A80 lessorequalslant θ lessorequalslant pi2EE,CZ
2θ
pi lessorequalslant sinθ lessorequalslant θ
parenrightbigg
= pir(1?e?r)
EKEX limr→+∞I = 0.
25/127
D9F0
|I| lessorequalslant
integraldisplay pi
0
vextendsinglevextendsingle
vextendsingleie?rsinθ+ircosθ
vextendsinglevextendsingle
vextendsingledθ
=
integraldisplay pi
0
e?rsinθdθ = 2
integraldisplay pi2
0
e?rsinθdθ
parenleftBigg
DGCMF9FKEPARBVH2E3DQ,CZ
integraldisplay pi2
0
e?rsinθdθ =
integraldisplay pi
pi
2
e?rsinθdθ
parenrightBigg
lessorequalslant 2
integraldisplay pi2
0
e?2rθpi dθ
parenleftbigg
FR?A80 lessorequalslant θ lessorequalslant pi2EE,CZ
2θ
pi lessorequalslant sinθ lessorequalslant θ
parenrightbigg
= pir(1?e?r)
EKEX limr→+∞I = 0.
25/127
BDAJAX DPG4 - AKCXFGATF9D2
DPG4 - AKCXFGATF9D2CGC2F8GI f(z) BFH0GJFRDX B CB
BRBRB5G3,BSD3F8GI f(z) GV B CBARBRG3E8CMF1BGETDM C
ARFKEP?AY,contintegraldisplay
C
f(z)dz = 0;
B
C
26/127
COEY DUGACRE8AXARB4G0,F3DMCZ
contintegraldisplay
C
f(z)dz =
contintegraldisplay
C
udx?vdy + i
contintegraldisplay
C
vdx + udy.
BFCKD6ARCMDRCT,CGC2 u,v AREGAJGIBF B CBF0GJCHAR,BS
D3DUGADMALGYEQDQDPG4 - CXGXE0ENAQ
contintegraldisplay
C
udx?vdy =
integraldisplayintegraldisplay
D
(?vx?uy)dσ = 0,
contintegraldisplay
C
vdx + udy =
integraldisplayintegraldisplay
D
(ux?vy)dσ = 0.
G1EL,D F0 C EKDWARERDX.
EKEX
contintegraldisplay
C
f(z)dz = 0.
27/127
COEY DUGACRE8AXARB4G0,F3DMCZ
contintegraldisplay
C
f(z)dz =
contintegraldisplay
C
udx?vdy + i
contintegraldisplay
C
vdx + udy.
BFCKD6ARCMDRCT,CGC2 u,v AREGAJGIBF B CBF0GJCHAR,BS
D3DUGADMALGYEQDQDPG4 - CXGXE0ENAQ
contintegraldisplay
C
udx?vdy =
integraldisplayintegraldisplay
D
(?vx?uy)dσ = 0,
contintegraldisplay
C
vdx + udy =
integraldisplayintegraldisplay
D
(ux?vy)dσ = 0.
G1EL,D F0 C EKDWARERDX.
EKEX
contintegraldisplay
C
f(z)dz = 0.
27/127
COEY DUGACRE8AXARB4G0,F3DMCZ
contintegraldisplay
C
f(z)dz =
contintegraldisplay
C
udx?vdy + i
contintegraldisplay
C
vdx + udy.
BFCKD6ARCMDRCT,CGC2 u,v AREGAJGIBF B CBF0GJCHAR,BS
D3DUGADMALGYEQDQDPG4 - CXGXE0ENAQ
contintegraldisplay
C
udx?vdy =
integraldisplayintegraldisplay
D
(?vx?uy)dσ = 0,
contintegraldisplay
C
vdx + udy =
integraldisplayintegraldisplay
D
(ux?vy)dσ = 0.
G1EL,D F0 C EKDWARERDX.
EKEX
contintegraldisplay
C
f(z)dz = 0.
27/127
DPG4 - AKCXFGATF9D2ARGLBE:
CGC2 C F0ERDX B ARBQBA,f(z) BF B CBB5G3,BFBGER
DX B H2GJCH,BSD3F9D2BXB8EKG9,BP
contintegraldisplay
C
f(z)dz = 0;
B
C
28/127
DPG4 - AKCXFGATF9D2ARGLBE:
CGC2 C F0ERDX B ARBQBA,f(z) BF B CBB5G3,BFBGER
DX B H2GJCH,BSD3F9D2BXB8EKG9,BP
contintegraldisplay
C
f(z)dz = 0;
B
C
28/127
DI 1 C7DYCTAFFKEPARDU,G1EL C? |z| = 1 AKE5.
(1)
integraldisplay
C
1
coszdz (2)
integraldisplay
C
zezdz
B5 (1) BHFBAK cosz = 0 ARDUFTBFBGETDM C A9,EK
EX f(z) = 1cosz BF C BL C ARCBG4B5G3,EKEX
integraldisplay
C
1
coszdz = 0.
(2) FR? f(z) = zez BF C BL C ARCBG4B5G3,EKEX
integraldisplay
C
zezdz = 0.
29/127
DI 1 C7DYCTAFFKEPARDU,G1EL C? |z| = 1 AKE5.
(1)
integraldisplay
C
1
coszdz (2)
integraldisplay
C
zezdz
B5 (1) BHFBAK cosz = 0 ARDUFTBFBGETDM C A9,EK
EX f(z) = 1cosz BF C BL C ARCBG4B5G3,EKEX
integraldisplay
C
1
coszdz = 0.
(2) FR? f(z) = zez BF C BL C ARCBG4B5G3,EKEX
integraldisplay
C
zezdz = 0.
29/127
DI 1 C7DYCTAFFKEPARDU,G1EL C? |z| = 1 AKE5.
(1)
integraldisplay
C
1
coszdz (2)
integraldisplay
C
zezdz
B5 (1) BHFBAK cosz = 0 ARDUFTBFBGETDM C A9,EK
EX f(z) = 1cosz BF C BL C ARCBG4B5G3,EKEX
integraldisplay
C
1
coszdz = 0.
(2) FR? f(z) = zez BF C BL C ARCBG4B5G3,EKEX
integraldisplay
C
zezdz = 0.
29/127
DI 1 C7DYCTAFFKEPARDU,G1EL C? |z| = 1 AKE5.
(1)
integraldisplay
C
1
coszdz (2)
integraldisplay
C
zezdz
B5 (1) BHFBAK cosz = 0 ARDUFTBFBGETDM C A9,EK
EX f(z) = 1cosz BF C BL C ARCBG4B5G3,EKEX
integraldisplay
C
1
coszdz = 0.
(2) FR? f(z) = zez BF C BL C ARCBG4B5G3,EKEX
integraldisplay
C
zezdz = 0.
29/127
BDD2AX FGATF9D2ARGLBE – BXG4BGF0F9D2
BKAR,D6DPG4 - AKCXF9D2GLBEAKGTGJFRDXARECF5.
EVC3,ANF8GI f(z) BFGTGJFRDX D CBB5G3,C? D CB
ARBRFBE8CMDFH0BGETDM.
(1) CGC2 C ARCBG4AHASF5D9 D,DICG C1,D0AD f(z)
BF C1 H2BLG1CBG4B5G3,APDJcontintegraldisplay
C1
f(z)dz = 0.
(2) CGC2 C ARCBG4G0AHASF5D9 D,DICG C2,C3H2EQG0BE
EKG9.
C1
C2
D
30/127
BDD2AX FGATF9D2ARGLBE – BXG4BGF0F9D2
BKAR,D6DPG4 - AKCXF9D2GLBEAKGTGJFRDXARECF5.
EVC3,ANF8GI f(z) BFGTGJFRDX D CBB5G3,C? D CB
ARBRFBE8CMDFH0BGETDM.
(1) CGC2 C ARCBG4AHASF5D9 D,DICG C1,D0AD f(z)
BF C1 H2BLG1CBG4B5G3,APDJcontintegraldisplay
C1
f(z)dz = 0.
(2) CGC2 C ARCBG4G0AHASF5D9 D,DICG C2,C3H2EQG0BE
EKG9.
C1
C2
D
30/127
BGF0BVBBA6D2E8DRB5G3F8GIGVBGETDMARFKEP,G0FR?BG
ETDMBFERDXCBDUGJCHBVBBADCDBVENARDU,E0DRBFBVBBC4EN
ELETDMG0EYC4F8GI f(z) G0B5G3ARBM,CGGB,BP
integraldisplay
C
f(z)dz =
integraldisplay
C1
f(z)dz
C
C1
D
31/127
DDEYCGGB,DUBLFRDM
hatwidestAA′
BL
hatwidestBB′,
EMAQH0CMDFH0BGET
DM AEBB
′E′A′A
G2 AA′F′B′BFA ARCBG4ASF5D9 D.
A A’
F’
F
E’
B’ B
E
C
C1
D
D9F0contintegraldisplay
AEBB′E′A′A
f(z)dz = 0,
contintegraldisplay
AA′F′B′BFA
f(z)dz = 0.
GQH2EHH0AVEQDNCK,AQcontintegraldisplay
C
f(z)dz +
contintegraldisplay
C?1
f(z)dz +
integraldisplay
hatwidestAA′
f(z)dz
+
integraldisplay
hatwidestA′A
f(z)dz +
integraldisplay
hatwidestB′B
f(z)dz +
integraldisplay
hatwidestBB′
f(z)dz = 0
32/127
DDEYCGGB,DUBLFRDM
hatwidestAA′
BL
hatwidestBB′,
EMAQH0CMDFH0BGET
DM AEBB
′E′A′A
G2 AA′F′B′BFA ARCBG4ASF5D9 D.
A A’
F’
F
E’
B’ B
E
C
C1
D
D9F0contintegraldisplay
AEBB′E′A′A
f(z)dz = 0,
contintegraldisplay
AA′F′B′BFA
f(z)dz = 0.
GQH2EHH0AVEQDNCK,AQcontintegraldisplay
C
f(z)dz +
contintegraldisplay
C?1
f(z)dz +
integraldisplay
hatwidestAA′
f(z)dz
+
integraldisplay
hatwidestA′A
f(z)dz +
integraldisplay
hatwidestB′B
f(z)dz +
integraldisplay
hatwidestBB′
f(z)dz = 0
32/127
DDEYCGGB,DUBLFRDM
hatwidestAA′
BL
hatwidestBB′,
EMAQH0CMDFH0BGET
DM AEBB
′E′A′A
G2 AA′F′B′BFA ARCBG4ASF5D9 D.
A A’
F’
F
E’
B’ B
E
C
C1
D
D9F0contintegraldisplay
AEBB′E′A′A
f(z)dz = 0,
contintegraldisplay
AA′F′B′BFA
f(z)dz = 0.
GQH2EHH0AVEQDNCK,AQcontintegraldisplay
C
f(z)dz +
contintegraldisplay
C?1
f(z)dz +
integraldisplay
hatwidestAA′
f(z)dz
+
integraldisplay
hatwidestA′A
f(z)dz +
integraldisplay
hatwidestB′B
f(z)dz +
integraldisplay
hatwidestBB′
f(z)dz = 0
32/127
A A’
F’
F
E’
B’ B
E
C
C1
D
FR? integraldisplay
hatwidestAA′
f(z)dz +
integraldisplay
hatwidestA′A
f(z)dz = 0;
integraldisplay
hatwidestB′B
f(z)dz +
integraldisplay
hatwidestBB′
f(z)dz = 0,
C3CZ
contintegraldisplay
C
f(z)dz +
contintegraldisplay
C?1
f(z)dz = 0
BP
contintegraldisplay
C
f(z)dz =
contintegraldisplay
C1
f(z)dz.
33/127
A A’
F’
F
E’
B’ B
E
C
C1
D
FR? integraldisplay
hatwidestAA′
f(z)dz +
integraldisplay
hatwidestA′A
f(z)dz = 0;
integraldisplay
hatwidestB′B
f(z)dz +
integraldisplay
hatwidestBB′
f(z)dz = 0,
C3CZ
contintegraldisplay
C
f(z)dz +
contintegraldisplay
C?1
f(z)dz = 0
BP
contintegraldisplay
C
f(z)dz =
contintegraldisplay
C1
f(z)dz.
33/127
A A’
F’
F
E’
B’ B
E
C
C1
D
FR? integraldisplay
hatwidestAA′
f(z)dz +
integraldisplay
hatwidestA′A
f(z)dz = 0;
integraldisplay
hatwidestB′B
f(z)dz +
integraldisplay
hatwidestBB′
f(z)dz = 0,
C3CZ
contintegraldisplay
C
f(z)dz +
contintegraldisplay
C?1
f(z)dz = 0
BP
contintegraldisplay
C
f(z)dz =
contintegraldisplay
C1
f(z)dz.
33/127
F9D2 (BXG4BGF0F9D2) AN C?GTGJFRDX D CBARE8CMDF
H0BGETDM,C1,C2,···,Cn F0BF C ARCBG4ARDFH0BGETDM,
ENDMD4G0EDF5?D4G0DNA2,FBDPEX C,C1,C2,···,Cn?BQ
BAARERDXASF5D9 D,CGC2 f(z) BF D CBB5G3,BSD3
1)
contintegraldisplay
C
f(z)dz =
nsummationdisplay
k=1
contintegraldisplay
Ck
f(z)dz,
G1EL,C BL Ck H2AIAKE0E5.
D
C2
C1 C3
C
2)
contintegraldisplay
Γ
f(z)dz = 0,
G1EL,Γ?CS C BL Ck(k = 1,2,···,n) EKDGEKARBXG4BG
F0 (G1E0E5F0:C CDCNEEA2BRBD,Ck CDCMEEA2BRBD).
34/127
F9D2 (BXG4BGF0F9D2) AN C?GTGJFRDX D CBARE8CMDF
H0BGETDM,C1,C2,···,Cn F0BF C ARCBG4ARDFH0BGETDM,
ENDMD4G0EDF5?D4G0DNA2,FBDPEX C,C1,C2,···,Cn?BQ
BAARERDXASF5D9 D,CGC2 f(z) BF D CBB5G3,BSD3
1)
contintegraldisplay
C
f(z)dz =
nsummationdisplay
k=1
contintegraldisplay
Ck
f(z)dz,
G1EL,C BL Ck H2AIAKE0E5.
D
C2
C1 C3
C
2)
contintegraldisplay
Γ
f(z)dz = 0,
G1EL,Γ?CS C BL Ck(k = 1,2,···,n) EKDGEKARBXG4BG
F0 (G1E0E5F0:C CDCNEEA2BRBD,Ck CDCMEEA2BRBD).
34/127
F9D2 (BXG4BGF0F9D2) AN C?GTGJFRDX D CBARE8CMDF
H0BGETDM,C1,C2,···,Cn F0BF C ARCBG4ARDFH0BGETDM,
ENDMD4G0EDF5?D4G0DNA2,FBDPEX C,C1,C2,···,Cn?BQ
BAARERDXASF5D9 D,CGC2 f(z) BF D CBB5G3,BSD3
1)
contintegraldisplay
C
f(z)dz =
nsummationdisplay
k=1
contintegraldisplay
Ck
f(z)dz,
G1EL,C BL Ck H2AIAKE0E5.
D
C2
C1 C3
C
2)
contintegraldisplay
Γ
f(z)dz = 0,
G1EL,Γ?CS C BL Ck(k = 1,2,···,n) EKDGEKARBXG4BG
F0 (G1E0E5F0:C CDCNEEA2BRBD,Ck CDCMEEA2BRBD).
34/127
DDEY,DGCMDQBGF0BVBBA6D2FVAOARE0DF.
D
C2
C1 C3
C
35/127
ETDRB4G0:
D0ATG1BDE8AXARDI 2 DJ,A8 C?EX z0?ELB1ARAKE5AB
EXEE,contintegraldisplay
C
dz
z?z0 = 2pii,
EKEX,DUGABGF0BVBBA6D2,GHD9EDF5 z0 ARBRG3E8CMAKE5DF
H0BGETDM Γ FTCZ
contintegraldisplay
Γ
dz
z?z0 = 2pii.
36/127
ETDRB4G0:
D0ATG1BDE8AXARDI 2 DJ,A8 C?EX z0?ELB1ARAKE5AB
EXEE,contintegraldisplay
C
dz
z?z0 = 2pii,
EKEX,DUGABGF0BVBBA6D2,GHD9EDF5 z0 ARBRG3E8CMAKE5DF
H0BGETDM Γ FTCZ
contintegraldisplay
Γ
dz
z?z0 = 2pii.
36/127
ETDRB4G0:
D0ATG1BDE8AXARDI 2 DJ,A8 C?EX z0?ELB1ARAKE5AB
EXEE,contintegraldisplay
C
dz
z?z0 = 2pii,
EKEX,DUGABGF0BVBBA6D2,GHD9EDF5 z0 ARBRG3E8CMAKE5DF
H0BGETDM Γ FTCZ
contintegraldisplay
Γ
dz
z?z0 = 2pii.
36/127
DI 1 C7DYFKEP
contintegraldisplay
C
1
(2z?1)(z + 2)dz,G1EL C
|z| = 1 AKE5.
B5 AQFKF8GI f(z) = 1(2z?1)(z + 2) CZH0DRG3BM
1
2
G2?2,G1EL
1
2 BF C CB,AD?2 BF C A9,
y
x O
2 1/2
C
37/127
DI 1 C7DYFKEP
contintegraldisplay
C
1
(2z?1)(z + 2)dz,G1EL C
|z| = 1 AKE5.
B5 AQFKF8GI f(z) = 1(2z?1)(z + 2) CZH0DRG3BM
1
2
G2?2,G1EL
1
2 BF C CB,AD?2 BF C A9,
y
x O
2 1/2
C
37/127
y
x O
2 1/2
C
FR? f(z) = 1(2z?1)(z + 2) =
1
5
z? 12
+
15
z + 2
EKEXcontintegraldisplay
C
1
(2z?1)(z?2)dz =
1
5
contintegraldisplay
C
1
z? 12
dz? 15
contintegraldisplay
C
1
z + 2dz
= 15 ·2pii? 15 ·0 = 25pii.
38/127
y
x O
2 1/2
C
FR? f(z) = 1(2z?1)(z + 2) =
1
5
z? 12
+
15
z + 2
EKEXcontintegraldisplay
C
1
(2z?1)(z?2)dz =
1
5
contintegraldisplay
C
1
z? 12
dz? 15
contintegraldisplay
C
1
z + 2dz
= 15 ·2pii? 15 ·0 = 25pii.
38/127
y
x O
2 1/2
C
FR? f(z) = 1(2z?1)(z + 2) =
1
5
z? 12
+
15
z + 2
EKEXcontintegraldisplay
C
1
(2z?1)(z?2)dz =
1
5
contintegraldisplay
C
1
z? 12
dz? 15
contintegraldisplay
C
1
z + 2dz
= 15 ·2pii? 15 ·0 = 25pii.
38/127
BTCMDPG4 - AKCXF9D2G2BXG4BGF0F9D2C7DYBXFKEPARE8
DOG2F9:
(1) FGCZ,EPG3 f(z) ARB5G3BH,GHFT f(z) AREKCZG3BM;
(2) B8A0,G9E8 f(z) ARG3BMG2FKEPETDM C AREGE8B3CG;
(3) DLA0,FRC4FREQEPB5,F9GUDUFTD4G0DNA2D4G0EDF5AR
DFH0BGETDM,DGCMF9D2EMAQB4C2.
39/127
BTCMDPG4 - AKCXF9D2G2BXG4BGF0F9D2C7DYBXFKEPARE8
DOG2F9:
(1) FGCZ,EPG3 f(z) ARB5G3BH,GHFT f(z) AREKCZG3BM;
(2) B8A0,G9E8 f(z) ARG3BMG2FKEPETDM C AREGE8B3CG;
(3) DLA0,FRC4FREQEPB5,F9GUDUFTD4G0DNA2D4G0EDF5AR
DFH0BGETDM,DGCMF9D2EMAQB4C2.
39/127
BTCMDPG4 - AKCXF9D2G2BXG4BGF0F9D2C7DYBXFKEPARE8
DOG2F9:
(1) FGCZ,EPG3 f(z) ARB5G3BH,GHFT f(z) AREKCZG3BM;
(2) B8A0,G9E8 f(z) ARG3BMG2FKEPETDM C AREGE8B3CG;
(3) DLA0,FRC4FREQEPB5,F9GUDUFTD4G0DNA2D4G0EDF5AR
DFH0BGETDM,DGCMF9D2EMAQB4C2.
39/127
BTCMDPG4 - AKCXF9D2G2BXG4BGF0F9D2C7DYBXFKEPARE8
DOG2F9:
(1) FGCZ,EPG3 f(z) ARB5G3BH,GHFT f(z) AREKCZG3BM;
(2) B8A0,G9E8 f(z) ARG3BMG2FKEPETDM C AREGE8B3CG;
(3) DLA0,FRC4FREQEPB5,F9GUDUFTD4G0DNA2D4G0EDF5AR
DFH0BGETDM,DGCMF9D2EMAQB4C2.
39/127
DI 2 C7DY
contintegraldisplay
Γ
2z?1
z2?zdz ARDU,G1EL Γ?
(1) ABEX |z| = r < 1 ARAKE5.
(2) EDF5ABEX|z| = 1 BFCBARBRG3AKE5DFH0BGETDM.
y
x O 1
Γ:|z|=r<1
40/127
B5 (1) FR? f(z) = 2z?1z2?z CZH0DRG3BM z = 0
G2 z = 1,G1EL z = 0 BF Γ CB,AD z = 1 BF Γ A9.
y
x O 1
Γ:|z|=r<1
D5FR? f(z) = 2z?1z2?z = 1z + 1z?1
EKEX contintegraldisplay
Γ
2z?1
z2?zdz =
contintegraldisplay
Γ
1
zdz +
contintegraldisplay
Γ
1
z?1dz
= 2pii + 0 = 2pii.
41/127
B5 (1) FR? f(z) = 2z?1z2?z CZH0DRG3BM z = 0
G2 z = 1,G1EL z = 0 BF Γ CB,AD z = 1 BF Γ A9.
y
x O 1
Γ:|z|=r<1
D5FR? f(z) = 2z?1z2?z = 1z + 1z?1
EKEX contintegraldisplay
Γ
2z?1
z2?zdz =
contintegraldisplay
Γ
1
zdz +
contintegraldisplay
Γ
1
z?1dz
= 2pii + 0 = 2pii.
41/127
B5 (1) FR? f(z) = 2z?1z2?z CZH0DRG3BM z = 0
G2 z = 1,G1EL z = 0 BF Γ CB,AD z = 1 BF Γ A9.
y
x O 1
Γ:|z|=r<1
D5FR? f(z) = 2z?1z2?z = 1z + 1z?1
EKEX contintegraldisplay
Γ
2z?1
z2?zdz =
contintegraldisplay
Γ
1
zdz +
contintegraldisplay
Γ
1
z?1dz
= 2pii + 0 = 2pii.
41/127
B5 (1) FR? f(z) = 2z?1z2?z CZH0DRG3BM z = 0
G2 z = 1,G1EL z = 0 BF Γ CB,AD z = 1 BF Γ A9.
y
x O 1
Γ:|z|=r<1
D5FR? f(z) = 2z?1z2?z = 1z + 1z?1
EKEX contintegraldisplay
Γ
2z?1
z2?zdz =
contintegraldisplay
Γ
1
zdz +
contintegraldisplay
Γ
1
z?1dz
= 2pii + 0 = 2pii.
41/127
(2) A8 Γ F0EDF5ABEX|z| = 1 BFCBARBRG3AKE5DFH0BGET
DMEE,f(z) ARH0DRG3BMFTEDF5BF Γ CB.
CGGB,BF Γ CBDUH0DRD4G0EDF5?D4G0DNA2ARAKE5AB
EX C1 G2 C2.
Γ
C1 C2
y
x
DGCM f(z) = 2z?1z2?z = 1z + 1z?1
FBDUGABGF0BVBBA6D2,AQcontintegraldisplay
Γ
2z?1
z2?zdz =
contintegraldisplay
Γ
1
zdz +
contintegraldisplay
Γ
1
z?1dz
=
contintegraldisplay
C1
1
zdz +
contintegraldisplay
C2
1
z?1dz = 2pii + 2pii = 4pii.
42/127
(2) A8 Γ F0EDF5ABEX|z| = 1 BFCBARBRG3AKE5DFH0BGET
DMEE,f(z) ARH0DRG3BMFTEDF5BF Γ CB.
CGGB,BF Γ CBDUH0DRD4G0EDF5?D4G0DNA2ARAKE5AB
EX C1 G2 C2.
Γ
C1 C2
y
x
DGCM f(z) = 2z?1z2?z = 1z + 1z?1
FBDUGABGF0BVBBA6D2,AQcontintegraldisplay
Γ
2z?1
z2?zdz =
contintegraldisplay
Γ
1
zdz +
contintegraldisplay
Γ
1
z?1dz
=
contintegraldisplay
C1
1
zdz +
contintegraldisplay
C2
1
z?1dz = 2pii + 2pii = 4pii.
42/127
(2) A8 Γ F0EDF5ABEX|z| = 1 BFCBARBRG3AKE5DFH0BGET
DMEE,f(z) ARH0DRG3BMFTEDF5BF Γ CB.
CGGB,BF Γ CBDUH0DRD4G0EDF5?D4G0DNA2ARAKE5AB
EX C1 G2 C2.
Γ
C1 C2
y
x
DGCM f(z) = 2z?1z2?z = 1z + 1z?1
FBDUGABGF0BVBBA6D2,AQcontintegraldisplay
Γ
2z?1
z2?zdz =
contintegraldisplay
Γ
1
zdz +
contintegraldisplay
Γ
1
z?1dz
=
contintegraldisplay
C1
1
zdz +
contintegraldisplay
C2
1
z?1dz = 2pii + 2pii = 4pii.
42/127
(2) A8 Γ F0EDF5ABEX|z| = 1 BFCBARBRG3AKE5DFH0BGET
DMEE,f(z) ARH0DRG3BMFTEDF5BF Γ CB.
CGGB,BF Γ CBDUH0DRD4G0EDF5?D4G0DNA2ARAKE5AB
EX C1 G2 C2.
Γ
C1 C2
y
x
DGCM f(z) = 2z?1z2?z = 1z + 1z?1
FBDUGABGF0BVBBA6D2,AQcontintegraldisplay
Γ
2z?1
z2?zdz =
contintegraldisplay
Γ
1
zdz +
contintegraldisplay
Γ
1
z?1dz
=
contintegraldisplay
C1
1
zdz +
contintegraldisplay
C2
1
z?1dz = 2pii + 2pii = 4pii.
42/127
DI 3 C7DYI =
contintegraldisplay
C
2z + 3
z(z2 + 1)dz ARDU,G1ELC,
vextendsinglevextendsingle
vextendsinglevextendsinglez? 1
2
vextendsinglevextendsingle
vextendsinglevextendsingle= 2.
B5 FR?f(z) = 2z + 3z(z2 + 1)CZD2DRG3BM z = 0,±i,DPFTED
F5BF C CB.
y
x
i
i
C1
C2
C3
C
D5FR? f(z) =
2z + 3
z(z2 + 1) =
A
z +
B
z + i +
C
z?i
G1EL,A = 3,B =?32 + i,C =?32?i.
43/127
DI 3 C7DYI =
contintegraldisplay
C
2z + 3
z(z2 + 1)dz ARDU,G1ELC,
vextendsinglevextendsingle
vextendsinglevextendsinglez? 1
2
vextendsinglevextendsingle
vextendsinglevextendsingle= 2.
B5 FR?f(z) = 2z + 3z(z2 + 1)CZD2DRG3BM z = 0,±i,DPFTED
F5BF C CB.
y
x
i
i
C1
C2
C3
C
D5FR? f(z) =
2z + 3
z(z2 + 1) =
A
z +
B
z + i +
C
z?i
G1EL,A = 3,B =?32 + i,C =?32?i.
43/127
DI 3 C7DYI =
contintegraldisplay
C
2z + 3
z(z2 + 1)dz ARDU,G1ELC,
vextendsinglevextendsingle
vextendsinglevextendsinglez? 1
2
vextendsinglevextendsingle
vextendsinglevextendsingle= 2.
B5 FR?f(z) = 2z + 3z(z2 + 1)CZD2DRG3BM z = 0,±i,DPFTED
F5BF C CB.
y
x
i
i
C1
C2
C3
C
D5FR? f(z) =
2z + 3
z(z2 + 1) =
A
z +
B
z + i +
C
z?i
G1EL,A = 3,B =?32 + i,C =?32?i.
43/127
y
x
i
i
C1
C2
C3
C
DUBLFRDM C1,C2,C3,C3DUGABGF0BVBBA6D2,CZ
I =
integraldisplay
C
A
z dz +
integraldisplay
C
B
z + idz +
integraldisplay
C
C
z?idz
=
integraldisplay
C2
A
z dz +
integraldisplay
C3
B
z + idz +
integraldisplay
C1
C
z?idz
= A·2pii + B ·2pii + C ·2pii.
44/127
y
x
i
i
C1
C2
C3
C
DUBLFRDM C1,C2,C3,C3DUGABGF0BVBBA6D2,CZ
I =
integraldisplay
C
A
z dz +
integraldisplay
C
B
z + idz +
integraldisplay
C
C
z?idz
=
integraldisplay
C2
A
z dz +
integraldisplay
C3
B
z + idz +
integraldisplay
C1
C
z?idz
= A·2pii + B ·2pii + C ·2pii.
44/127
y
x
i
i
C1
C2
C3
C
DUBLFRDM C1,C2,C3,C3DUGABGF0BVBBA6D2,CZ
I =
integraldisplay
C
A
z dz +
integraldisplay
C
B
z + idz +
integraldisplay
C
C
z?idz
=
integraldisplay
C2
A
z dz +
integraldisplay
C3
B
z + idz +
integraldisplay
C1
C
z?idz
= A·2pii + B ·2pii + C ·2pii.
44/127
DI 4 CGC2F8GI f(z) BF 0 < |z?a| < R CBB5G3,DP
limz→a(z?a)f(z) = A,
C3
integraldisplay
C
f(z)dz = 2piiA,
G1EL C F0BRFBABEX |z?a| = r,0 < r < R,E0E5AIAKE5.
B5 FR?limz→a(z?a)f(z) = A,EKEX
(z?a)f(z) = A + α(z),
G1ELlimz→aα(z) = 0,EKEX
f(z) = Az?a + α(z)z?a.
D9F0
integraldisplay
C
f(z)dz =
integraldisplay
C
A
z?adz +
integraldisplay
C
α(z)
z?adz
= 2piiA +
integraldisplay
C
α(z)
z?adz
45/127
DI 4 CGC2F8GI f(z) BF 0 < |z?a| < R CBB5G3,DP
limz→a(z?a)f(z) = A,
C3
integraldisplay
C
f(z)dz = 2piiA,
G1EL C F0BRFBABEX |z?a| = r,0 < r < R,E0E5AIAKE5.
B5 FR?limz→a(z?a)f(z) = A,EKEX
(z?a)f(z) = A + α(z),
G1ELlimz→aα(z) = 0,EKEX
f(z) = Az?a + α(z)z?a.
D9F0
integraldisplay
C
f(z)dz =
integraldisplay
C
A
z?adz +
integraldisplay
C
α(z)
z?adz
= 2piiA +
integraldisplay
C
α(z)
z?adz
45/127
DI 4 CGC2F8GI f(z) BF 0 < |z?a| < R CBB5G3,DP
limz→a(z?a)f(z) = A,
C3
integraldisplay
C
f(z)dz = 2piiA,
G1EL C F0BRFBABEX |z?a| = r,0 < r < R,E0E5AIAKE5.
B5 FR?limz→a(z?a)f(z) = A,EKEX
(z?a)f(z) = A + α(z),
G1ELlimz→aα(z) = 0,EKEX
f(z) = Az?a + α(z)z?a.
D9F0
integraldisplay
C
f(z)dz =
integraldisplay
C
A
z?adz +
integraldisplay
C
α(z)
z?adz
= 2piiA +
integraldisplay
C
α(z)
z?adz
45/127
DI 4 CGC2F8GI f(z) BF 0 < |z?a| < R CBB5G3,DP
limz→a(z?a)f(z) = A,
C3
integraldisplay
C
f(z)dz = 2piiA,
G1EL C F0BRFBABEX |z?a| = r,0 < r < R,E0E5AIAKE5.
B5 FR?limz→a(z?a)f(z) = A,EKEX
(z?a)f(z) = A + α(z),
G1ELlimz→aα(z) = 0,EKEX
f(z) = Az?a + α(z)z?a.
D9F0
integraldisplay
C
f(z)dz =
integraldisplay
C
A
z?adz +
integraldisplay
C
α(z)
z?adz
= 2piiA +
integraldisplay
C
α(z)
z?adz
45/127
DI 4 CGC2F8GI f(z) BF 0 < |z?a| < R CBB5G3,DP
limz→a(z?a)f(z) = A,
C3
integraldisplay
C
f(z)dz = 2piiA,
G1EL C F0BRFBABEX |z?a| = r,0 < r < R,E0E5AIAKE5.
B5 FR?limz→a(z?a)f(z) = A,EKEX
(z?a)f(z) = A + α(z),
G1ELlimz→aα(z) = 0,EKEX
f(z) = Az?a + α(z)z?a.
D9F0
integraldisplay
C
f(z)dz =
integraldisplay
C
A
z?adz +
integraldisplay
C
α(z)
z?adz
= 2piiA +
integraldisplay
C
α(z)
z?adz
45/127
CTEHDDEY
integraldisplay
C
α(z)
z?adz = 0:
FR?limz→aα(z) = 0,EKEX?ε > 0,?δ > 0,A80 < |z?a| < δ,CZ
|α(z)| < ε.
DUABEX C0,|z?a| = r0,0 < r0 < δ,C3 C0 AHASG9BF
0 < |z?a| < δCB,DPBF C0 H2CZ
|α(z)| < ε.
FBDPDUGABGF0BVBBA6D2G2AFDUG0AVEQ,CZvextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
C
α(z)
z?adz
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C0
α(z)
z?adz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C0
|α(z)|
|z?a|ds lessorequalslant
integraldisplay
C0
ε
r0ds = 2piε.
DUGA ε ARBRFBBH,CZ
integraldisplay
C
α(z)
z?adz = 0,D9F0integraldisplay
C
f(z)dz = 2piiA.
46/127
CTEHDDEY
integraldisplay
C
α(z)
z?adz = 0:
FR?limz→aα(z) = 0,EKEX?ε > 0,?δ > 0,A80 < |z?a| < δ,CZ
|α(z)| < ε.
DUABEX C0,|z?a| = r0,0 < r0 < δ,C3 C0 AHASG9BF
0 < |z?a| < δCB,DPBF C0 H2CZ
|α(z)| < ε.
FBDPDUGABGF0BVBBA6D2G2AFDUG0AVEQ,CZvextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
C
α(z)
z?adz
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C0
α(z)
z?adz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C0
|α(z)|
|z?a|ds lessorequalslant
integraldisplay
C0
ε
r0ds = 2piε.
DUGA ε ARBRFBBH,CZ
integraldisplay
C
α(z)
z?adz = 0,D9F0integraldisplay
C
f(z)dz = 2piiA.
46/127
CTEHDDEY
integraldisplay
C
α(z)
z?adz = 0:
FR?limz→aα(z) = 0,EKEX?ε > 0,?δ > 0,A80 < |z?a| < δ,CZ
|α(z)| < ε.
DUABEX C0,|z?a| = r0,0 < r0 < δ,C3 C0 AHASG9BF
0 < |z?a| < δCB,DPBF C0 H2CZ
|α(z)| < ε.
FBDPDUGABGF0BVBBA6D2G2AFDUG0AVEQ,CZvextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
C
α(z)
z?adz
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C0
α(z)
z?adz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C0
|α(z)|
|z?a|ds lessorequalslant
integraldisplay
C0
ε
r0ds = 2piε.
DUGA ε ARBRFBBH,CZ
integraldisplay
C
α(z)
z?adz = 0,D9F0integraldisplay
C
f(z)dz = 2piiA.
46/127
CTEHDDEY
integraldisplay
C
α(z)
z?adz = 0:
FR?limz→aα(z) = 0,EKEX?ε > 0,?δ > 0,A80 < |z?a| < δ,CZ
|α(z)| < ε.
DUABEX C0,|z?a| = r0,0 < r0 < δ,C3 C0 AHASG9BF
0 < |z?a| < δCB,DPBF C0 H2CZ
|α(z)| < ε.
FBDPDUGABGF0BVBBA6D2G2AFDUG0AVEQ,CZvextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
C
α(z)
z?adz
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C0
α(z)
z?adz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C0
|α(z)|
|z?a|ds lessorequalslant
integraldisplay
C0
ε
r0ds = 2piε.
DUGA ε ARBRFBBH,CZ
integraldisplay
C
α(z)
z?adz = 0,D9F0integraldisplay
C
f(z)dz = 2piiA.
46/127
CTEHDDEY
integraldisplay
C
α(z)
z?adz = 0:
FR?limz→aα(z) = 0,EKEX?ε > 0,?δ > 0,A80 < |z?a| < δ,CZ
|α(z)| < ε.
DUABEX C0,|z?a| = r0,0 < r0 < δ,C3 C0 AHASG9BF
0 < |z?a| < δCB,DPBF C0 H2CZ
|α(z)| < ε.
FBDPDUGABGF0BVBBA6D2G2AFDUG0AVEQ,CZvextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
C
α(z)
z?adz
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C0
α(z)
z?adz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C0
|α(z)|
|z?a|ds lessorequalslant
integraldisplay
C0
ε
r0ds = 2piε.
DUGA ε ARBRFBBH,CZ
integraldisplay
C
α(z)
z?adz = 0,D9F0integraldisplay
C
f(z)dz = 2piiA.
46/127
CTEHDDEY
integraldisplay
C
α(z)
z?adz = 0:
FR?limz→aα(z) = 0,EKEX?ε > 0,?δ > 0,A80 < |z?a| < δ,CZ
|α(z)| < ε.
DUABEX C0,|z?a| = r0,0 < r0 < δ,C3 C0 AHASG9BF
0 < |z?a| < δCB,DPBF C0 H2CZ
|α(z)| < ε.
FBDPDUGABGF0BVBBA6D2G2AFDUG0AVEQ,CZvextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
C
α(z)
z?adz
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
C0
α(z)
z?adz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
C0
|α(z)|
|z?a|ds lessorequalslant
integraldisplay
C0
ε
r0ds = 2piε.
DUGA ε ARBRFBBH,CZ
integraldisplay
C
α(z)
z?adz = 0,D9F0integraldisplay
C
f(z)dz = 2piiA.
46/127
DI 5 CGC2F8GI f(z) BF R < |z| < +∞CBB5G3,DP
limz→∞zf(z) = A,
C3
integraldisplay
C
f(z)dz = 2piiA,
G1EL C F0BRFBABEX |z| = r,r > R,E0E5AIAKE5.
47/127
BDD3AX A6F8GIDQG0F9FKEP
FR?ETDMFKEPGVF1BGETDMARFKEP?AYETDMFKEPDQ
F0F7FFB3,DUGADPG4 - AKCXFGATF9D2,CZ
F9D2E8CGC2F8GI f(z) BFH0GJFRDX D CBBRBRB5G3,BS
D3FKEP
integraldisplay
C
f(z)dz DQGJAQGDBMBLEQBMARF0F7 C FFB3.
D
C1
z1 z2
C2
DUGAF9D2E8,ARF9 z0,BG z BF D CBBVFG,D9F0F8GI
F(z) =
integraldisplay z
z
f(ζ)dζ
B4F9AAE8DRH0DUF8GI
48/127
BDD3AX A6F8GIDQG0F9FKEP
FR?ETDMFKEPGVF1BGETDMARFKEP?AYETDMFKEPDQ
F0F7FFB3,DUGADPG4 - AKCXFGATF9D2,CZ
F9D2E8CGC2F8GI f(z) BFH0GJFRDX D CBBRBRB5G3,BS
D3FKEP
integraldisplay
C
f(z)dz DQGJAQGDBMBLEQBMARF0F7 C FFB3.
D
C1
z1 z2
C2
DUGAF9D2E8,ARF9 z0,BG z BF D CBBVFG,D9F0F8GI
F(z) =
integraldisplay z
z
f(ζ)dζ
B4F9AAE8DRH0DUF8GI
48/127
BDD3AX A6F8GIDQG0F9FKEP
FR?ETDMFKEPGVF1BGETDMARFKEP?AYETDMFKEPDQ
F0F7FFB3,DUGADPG4 - AKCXFGATF9D2,CZ
F9D2E8CGC2F8GI f(z) BFH0GJFRDX D CBBRBRB5G3,BS
D3FKEP
integraldisplay
C
f(z)dz DQGJAQGDBMBLEQBMARF0F7 C FFB3.
D
C1
z1 z2
C2
DUGAF9D2E8,ARF9 z0,BG z BF D CBBVFG,D9F0F8GI
F(z) =
integraldisplay z
z
f(ζ)dζ
B4F9AAE8DRH0DUF8GI
48/127
F9D2AJCGC2F8GI f(z) BFH0GJFRDX B CBBRBRB5G3,BSD3
BVH2DLFKEPB4F9ARF8GI
F(z) =
integraldisplay z
z0
f(ζ)dζ
BF B CBB5G3,FBDP
F′(z) = f(z).
49/127
DDAN z? B ARBRFBE8BM,DJBF z ARAPDX K(z;δ)? B.
AI |△z| FBEPA2EMAQ z +△z ∈ K CB,D9F0
F(z +△z)?F(z) =
integraldisplay z+△z
z0
f(ζ)dζ?
integraldisplay z
z0
f(ζ)dζ.
z0
z
z+δz
K
ζ
CSD9FKEPDQF0F7FFB3,FRCS
integraldisplay z+△z
z0
f(ζ)dζ ARFKEPF0DM
DWAICGGBEKER,FBDPD0 z0 AK z ARFKEPF0DMDVFK
EP
integraldisplay z
z0
f(ζ)dζ DNFV,D9F0CZ
F(z +△z)?F(z) =
integraldisplay z+△z
z
f(ζ)dζ
50/127
DDAN z? B ARBRFBE8BM,DJBF z ARAPDX K(z;δ)? B.
AI |△z| FBEPA2EMAQ z +△z ∈ K CB,D9F0
F(z +△z)?F(z) =
integraldisplay z+△z
z0
f(ζ)dζ?
integraldisplay z
z0
f(ζ)dζ.
z0
z
z+δz
K
ζ
CSD9FKEPDQF0F7FFB3,FRCS
integraldisplay z+△z
z0
f(ζ)dζ ARFKEPF0DM
DWAICGGBEKER,FBDPD0 z0 AK z ARFKEPF0DMDVFK
EP
integraldisplay z
z0
f(ζ)dζ DNFV,D9F0CZ
F(z +△z)?F(z) =
integraldisplay z+△z
z
f(ζ)dζ
50/127
DDAN z? B ARBRFBE8BM,DJBF z ARAPDX K(z;δ)? B.
AI |△z| FBEPA2EMAQ z +△z ∈ K CB,D9F0
F(z +△z)?F(z) =
integraldisplay z+△z
z0
f(ζ)dζ?
integraldisplay z
z0
f(ζ)dζ.
z0
z
z+δz
K
ζ
CSD9FKEPDQF0F7FFB3,FRCS
integraldisplay z+△z
z0
f(ζ)dζ ARFKEPF0DM
DWAICGGBEKER,FBDPD0 z0 AK z ARFKEPF0DMDVFK
EP
integraldisplay z
z0
f(ζ)dζ DNFV,D9F0CZ
F(z +△z)?F(z) =
integraldisplay z+△z
z
f(ζ)dζ
50/127
DDAN z? B ARBRFBE8BM,DJBF z ARAPDX K(z;δ)? B.
AI |△z| FBEPA2EMAQ z +△z ∈ K CB,D9F0
F(z +△z)?F(z) =
integraldisplay z+△z
z0
f(ζ)dζ?
integraldisplay z
z0
f(ζ)dζ.
z0
z
z+δz
K
ζ
CSD9FKEPDQF0F7FFB3,FRCS
integraldisplay z+△z
z0
f(ζ)dζ ARFKEPF0DM
DWAICGGBEKER,FBDPD0 z0 AK z ARFKEPF0DMDVFK
EP
integraldisplay z
z0
f(ζ)dζ DNFV,D9F0CZ
F(z +△z)?F(z) =
integraldisplay z+△z
z
f(ζ)dζ
50/127
D5FR?
integraldisplay z+△z
z
f(z)dζ = f(z)
integraldisplay z+△z
z
dζ = f(z)△z,
EKEX
F(z +△z)?F(z)
△z?f(z) =
1
△z
integraldisplay z+△z
z
[f(ζ)?f(z)]dζ.
51/127
D5FR?
integraldisplay z+△z
z
f(z)dζ = f(z)
integraldisplay z+△z
z
dζ = f(z)△z,
EKEX
F(z +△z)?F(z)
△z?f(z) =
1
△z
integraldisplay z+△z
z
[f(ζ)?f(z)]dζ.
51/127
FR? f(z) BF B CBGJCH,EKEX,?ε > 0,?δ > 0,EMAQ
A8 |ζ?z| < δ EE,CZ
|f(ζ)?f(z)| < ε.
DUGAFKEPARAFDUBHEF
vextendsinglevextendsingle
vextendsinglevextendsingleF(z +△z)?F(z)
△z?f(z)
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
|△z|
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay z+△z
z
[f(ζ)?f(z)]dζ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lessorequalslant 1|△z|
integraldisplay z+△z
z
|f(ζ)?f(z)||dζ|
lessorequalslant 1|△z| ·ε|△z| = ε.
EKEX lim
△z→0
vextendsinglevextendsingle
vextendsinglevextendsingleF(z +△z)?F(z)
△z?f(z)
vextendsinglevextendsingle
vextendsinglevextendsingle= 0.
BP F′(z) = f(z).
52/127
FR? f(z) BF B CBGJCH,EKEX,?ε > 0,?δ > 0,EMAQ
A8 |ζ?z| < δ EE,CZ
|f(ζ)?f(z)| < ε.
DUGAFKEPARAFDUBHEF
vextendsinglevextendsingle
vextendsinglevextendsingleF(z +△z)?F(z)
△z?f(z)
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
|△z|
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay z+△z
z
[f(ζ)?f(z)]dζ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lessorequalslant 1|△z|
integraldisplay z+△z
z
|f(ζ)?f(z)||dζ|
lessorequalslant 1|△z| ·ε|△z| = ε.
EKEX lim
△z→0
vextendsinglevextendsingle
vextendsinglevextendsingleF(z +△z)?F(z)
△z?f(z)
vextendsinglevextendsingle
vextendsinglevextendsingle= 0.
BP F′(z) = f(z).
52/127
FR? f(z) BF B CBGJCH,EKEX,?ε > 0,?δ > 0,EMAQ
A8 |ζ?z| < δ EE,CZ
|f(ζ)?f(z)| < ε.
DUGAFKEPARAFDUBHEF
vextendsinglevextendsingle
vextendsinglevextendsingleF(z +△z)?F(z)
△z?f(z)
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
|△z|
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay z+△z
z
[f(ζ)?f(z)]dζ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lessorequalslant 1|△z|
integraldisplay z+△z
z
|f(ζ)?f(z)||dζ|
lessorequalslant 1|△z| ·ε|△z| = ε.
EKEX lim
△z→0
vextendsinglevextendsingle
vextendsinglevextendsingleF(z +△z)?F(z)
△z?f(z)
vextendsinglevextendsingle
vextendsinglevextendsingle= 0.
BP F′(z) = f(z).
52/127
FR? f(z) BF B CBGJCH,EKEX,?ε > 0,?δ > 0,EMAQ
A8 |ζ?z| < δ EE,CZ
|f(ζ)?f(z)| < ε.
DUGAFKEPARAFDUBHEF
vextendsinglevextendsingle
vextendsinglevextendsingleF(z +△z)?F(z)
△z?f(z)
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
|△z|
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay z+△z
z
[f(ζ)?f(z)]dζ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lessorequalslant 1|△z|
integraldisplay z+△z
z
|f(ζ)?f(z)||dζ|
lessorequalslant 1|△z| ·ε|△z| = ε.
EKEX lim
△z→0
vextendsinglevextendsingle
vextendsinglevextendsingleF(z +△z)?F(z)
△z?f(z)
vextendsinglevextendsingle
vextendsinglevextendsingle= 0.
BP F′(z) = f(z).
52/127
FR? f(z) BF B CBGJCH,EKEX,?ε > 0,?δ > 0,EMAQ
A8 |ζ?z| < δ EE,CZ
|f(ζ)?f(z)| < ε.
DUGAFKEPARAFDUBHEF
vextendsinglevextendsingle
vextendsinglevextendsingleF(z +△z)?F(z)
△z?f(z)
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
|△z|
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay z+△z
z
[f(ζ)?f(z)]dζ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lessorequalslant 1|△z|
integraldisplay z+△z
z
|f(ζ)?f(z)||dζ|
lessorequalslant 1|△z| ·ε|△z| = ε.
EKEX lim
△z→0
vextendsinglevextendsingle
vextendsinglevextendsingleF(z +△z)?F(z)
△z?f(z)
vextendsinglevextendsingle
vextendsinglevextendsingle= 0.
BP F′(z) = f(z).
52/127
FR? f(z) BF B CBGJCH,EKEX,?ε > 0,?δ > 0,EMAQ
A8 |ζ?z| < δ EE,CZ
|f(ζ)?f(z)| < ε.
DUGAFKEPARAFDUBHEF
vextendsinglevextendsingle
vextendsinglevextendsingleF(z +△z)?F(z)
△z?f(z)
vextendsinglevextendsingle
vextendsinglevextendsingle = 1
|△z|
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay z+△z
z
[f(ζ)?f(z)]dζ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lessorequalslant 1|△z|
integraldisplay z+△z
z
|f(ζ)?f(z)||dζ|
lessorequalslant 1|△z| ·ε|△z| = ε.
EKEX lim
△z→0
vextendsinglevextendsingle
vextendsinglevextendsingleF(z +△z)?F(z)
△z?f(z)
vextendsinglevextendsingle
vextendsinglevextendsingle= 0.
BP F′(z) = f(z).
52/127
F9FE CGC2DJBF B CBARB5G3F8GI F(z),EM
AQ F′(z) = f(z),BSD3EH F(z)? f(z) BFERDX B CBARA6
F8GI.
BHEF 1 f(z) ARBRG3H0DRA6F8GIDNABE8DRDIGI.
BHEF 2 CGC2 f(z) BFERDX B CBCZE8DRA6F8GI F(z),
C3 f(z) AREKCZA6F8GIGCCZE8DOC4DSEQ F(z) + C,C?BR
FBDIGI.
F9FEEH f(z) ARA6F8GIARE8DOC4DSEQ F(z) + c(G1EL c
BRFBDIGI)? f(z) ARG0F9FKEP,C8DU
integraldisplay
f(z)dz = F(z) + c.
53/127
F9FE CGC2DJBF B CBARB5G3F8GI F(z),EM
AQ F′(z) = f(z),BSD3EH F(z)? f(z) BFERDX B CBARA6
F8GI.
BHEF 1 f(z) ARBRG3H0DRA6F8GIDNABE8DRDIGI.
BHEF 2 CGC2 f(z) BFERDX B CBCZE8DRA6F8GI F(z),
C3 f(z) AREKCZA6F8GIGCCZE8DOC4DSEQ F(z) + C,C?BR
FBDIGI.
F9FEEH f(z) ARA6F8GIARE8DOC4DSEQ F(z) + c(G1EL c
BRFBDIGI)? f(z) ARG0F9FKEP,C8DU
integraldisplay
f(z)dz = F(z) + c.
53/127
F9FE CGC2DJBF B CBARB5G3F8GI F(z),EM
AQ F′(z) = f(z),BSD3EH F(z)? f(z) BFERDX B CBARA6
F8GI.
BHEF 1 f(z) ARBRG3H0DRA6F8GIDNABE8DRDIGI.
BHEF 2 CGC2 f(z) BFERDX B CBCZE8DRA6F8GI F(z),
C3 f(z) AREKCZA6F8GIGCCZE8DOC4DSEQ F(z) + C,C?BR
FBDIGI.
F9FEEH f(z) ARA6F8GIARE8DOC4DSEQ F(z) + c(G1EL c
BRFBDIGI)? f(z) ARG0F9FKEP,C8DU
integraldisplay
f(z)dz = F(z) + c.
53/127
F9FE CGC2DJBF B CBARB5G3F8GI F(z),EM
AQ F′(z) = f(z),BSD3EH F(z)? f(z) BFERDX B CBARA6
F8GI.
BHEF 1 f(z) ARBRG3H0DRA6F8GIDNABE8DRDIGI.
BHEF 2 CGC2 f(z) BFERDX B CBCZE8DRA6F8GI F(z),
C3 f(z) AREKCZA6F8GIGCCZE8DOC4DSEQ F(z) + C,C?BR
FBDIGI.
F9FEEH f(z) ARA6F8GIARE8DOC4DSEQ F(z) + c(G1EL c
BRFBDIGI)? f(z) ARG0F9FKEP,C8DU
integraldisplay
f(z)dz = F(z) + c.
53/127
BHFBEVE1:
CT B F0GTGJFRDX,F8GI f(z) BF B CBB5G3,f(z) GV B
CBARBGETDM C ARFKEPDWCDG0?AY,FRAD
integraldisplay z
z0
f(ζ)dζ
E8DOCOBHDQ z0 AK z BF B CBARF0F7CZB3,f(z) BF B CBFU
G0DWCDCZA6F8GI.
DICG,f(z) = 1z BF B = C?{0}CBB5G3,A2BF B CBDA
CZA6F8GI.
54/127
BHFBEVE1:
CT B F0GTGJFRDX,F8GI f(z) BF B CBB5G3,f(z) GV B
CBARBGETDM C ARFKEPDWCDG0?AY,FRAD
integraldisplay z
z0
f(ζ)dζ
E8DOCOBHDQ z0 AK z BF B CBARF0F7CZB3,f(z) BF B CBFU
G0DWCDCZA6F8GI.
DICG,f(z) = 1z BF B = C?{0}CBB5G3,A2BF B CBDA
CZA6F8GI.
54/127
BHFBEVE1:
CT B F0GTGJFRDX,F8GI f(z) BF B CBB5G3,f(z) GV B
CBARBGETDM C ARFKEPDWCDG0?AY,FRAD
integraldisplay z
z0
f(ζ)dζ
E8DOCOBHDQ z0 AK z BF B CBARF0F7CZB3,f(z) BF B CBFU
G0DWCDCZA6F8GI.
DICG,f(z) = 1z BF B = C?{0}CBB5G3,A2BF B CBDA
CZA6F8GI.
54/127
F9D2D2 (G2GM - BGG1CICJGYEQ) CGC2 f(z) BFH0GJFR
DX B CBBRBRB5G3,G(z)? f(z) ARE8DRA6F8GI,BSD3
integraldisplay z1
z0
f(z)dz = G(z1)?G(z0),
GXD4 z0,z1?DX B CBARH0BM.
DD FR?
integraldisplay z
z0
f(z)dz?F0 f(z) ARA6F8GI,EKEX
integraldisplay z
z0
f(z)dz = G(z) + c.
A8 z = z0 EE,DUGADPG4 - AKCXF9D2,DWAQ
integraldisplay z0
z0
f(z)dz = 0.
E3CKH2EQAQ c =?G(z0),FRCS
integraldisplay z1
z0
f(z)dz = G(z1)?G(z0),
55/127
F9D2D2 (G2GM - BGG1CICJGYEQ) CGC2 f(z) BFH0GJFR
DX B CBBRBRB5G3,G(z)? f(z) ARE8DRA6F8GI,BSD3
integraldisplay z1
z0
f(z)dz = G(z1)?G(z0),
GXD4 z0,z1?DX B CBARH0BM.
DD FR?
integraldisplay z
z0
f(z)dz?F0 f(z) ARA6F8GI,EKEX
integraldisplay z
z0
f(z)dz = G(z) + c.
A8 z = z0 EE,DUGADPG4 - AKCXF9D2,DWAQ
integraldisplay z0
z0
f(z)dz = 0.
E3CKH2EQAQ c =?G(z0),FRCS
integraldisplay z1
z0
f(z)dz = G(z1)?G(z0),
55/127
F9D2D2 (G2GM - BGG1CICJGYEQ) CGC2 f(z) BFH0GJFR
DX B CBBRBRB5G3,G(z)? f(z) ARE8DRA6F8GI,BSD3
integraldisplay z1
z0
f(z)dz = G(z1)?G(z0),
GXD4 z0,z1?DX B CBARH0BM.
DD FR?
integraldisplay z
z0
f(z)dz?F0 f(z) ARA6F8GI,EKEX
integraldisplay z
z0
f(z)dz = G(z) + c.
A8 z = z0 EE,DUGADPG4 - AKCXF9D2,DWAQ
integraldisplay z0
z0
f(z)dz = 0.
E3CKH2EQAQ c =?G(z0),FRCS
integraldisplay z1
z0
f(z)dz = G(z1)?G(z0),
55/127
F9D2D2 (G2GM - BGG1CICJGYEQ) CGC2 f(z) BFH0GJFR
DX B CBBRBRB5G3,G(z)? f(z) ARE8DRA6F8GI,BSD3
integraldisplay z1
z0
f(z)dz = G(z1)?G(z0),
GXD4 z0,z1?DX B CBARH0BM.
DD FR?
integraldisplay z
z0
f(z)dz?F0 f(z) ARA6F8GI,EKEX
integraldisplay z
z0
f(z)dz = G(z) + c.
A8 z = z0 EE,DUGADPG4 - AKCXF9D2,DWAQ
integraldisplay z0
z0
f(z)dz = 0.
E3CKH2EQAQ c =?G(z0),FRCS
integraldisplay z1
z0
f(z)dz = G(z1)?G(z0),
55/127
F9D2D2 (G2GM - BGG1CICJGYEQ) CGC2 f(z) BFH0GJFR
DX B CBBRBRB5G3,G(z)? f(z) ARE8DRA6F8GI,BSD3
integraldisplay z1
z0
f(z)dz = G(z1)?G(z0),
GXD4 z0,z1?DX B CBARH0BM.
DD FR?
integraldisplay z
z0
f(z)dz?F0 f(z) ARA6F8GI,EKEX
integraldisplay z
z0
f(z)dz = G(z) + c.
A8 z = z0 EE,DUGADPG4 - AKCXF9D2,DWAQ
integraldisplay z0
z0
f(z)dz = 0.
E3CKH2EQAQ c =?G(z0),FRCS
integraldisplay z1
z0
f(z)dz = G(z1)?G(z0),
55/127
F9D2D2 (G2GM - BGG1CICJGYEQ) CGC2 f(z) BFH0GJFR
DX B CBBRBRB5G3,G(z)? f(z) ARE8DRA6F8GI,BSD3
integraldisplay z1
z0
f(z)dz = G(z1)?G(z0),
GXD4 z0,z1?DX B CBARH0BM.
DD FR?
integraldisplay z
z0
f(z)dz?F0 f(z) ARA6F8GI,EKEX
integraldisplay z
z0
f(z)dz = G(z) + c.
A8 z = z0 EE,DUGADPG4 - AKCXF9D2,DWAQ
integraldisplay z0
z0
f(z)dz = 0.
E3CKH2EQAQ c =?G(z0),FRCS
integraldisplay z1
z0
f(z)dz = G(z1)?G(z0),
55/127
F9 integraldisplay z
1
z0
f(z)dz = G(z1)?G(z0),
DDBA.
56/127
DI 1,C7DYCTAFDSC3.
1)
integraldisplay i
0
zez2dz; 2)
integraldisplay pii
pii
sin2zdz.
B5 (1) FR? f(z) = zez2 BFBXEYEHH2BRBRB5G3,FBDPCZ
bracketleftbigg1
2e
z2
bracketrightbigg′
= zez2
EKEX
integraldisplay i
0
zez2dz =
bracketleftbigg1
2e
z2
bracketrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
i
0
= 12ei2? 12 = 12(e?1?1).
57/127
DI 1,C7DYCTAFDSC3.
1)
integraldisplay i
0
zez2dz; 2)
integraldisplay pii
pii
sin2zdz.
B5 (1) FR? f(z) = zez2 BFBXEYEHH2BRBRB5G3,FBDPCZ
bracketleftbigg1
2e
z2
bracketrightbigg′
= zez2
EKEX
integraldisplay i
0
zez2dz =
bracketleftbigg1
2e
z2
bracketrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
i
0
= 12ei2? 12 = 12(e?1?1).
57/127
DI 1,C7DYCTAFDSC3.
1)
integraldisplay i
0
zez2dz; 2)
integraldisplay pii
pii
sin2zdz.
B5 (1) FR? f(z) = zez2 BFBXEYEHH2BRBRB5G3,FBDPCZ
bracketleftbigg1
2e
z2
bracketrightbigg′
= zez2
EKEX
integraldisplay i
0
zez2dz =
bracketleftbigg1
2e
z2
bracketrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
i
0
= 12ei2? 12 = 12(e?1?1).
57/127
(2) FR? f(z) = sin2z = 12(1?cos2z) BFBXEYEHH2BRBR
B5G3,FBDPCZ
1
2
parenleftbigg
z? 12sin2z
parenrightbigg
F0 f(z) ARA6F8GI.
EKEX
integraldisplay pii
pii
sin2zdz =
bracketleftbigg1
2
parenleftbigg
z? 12sin2z
parenrightbiggbracketrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
pii
pii
= 12
parenleftbigg
pii? 12sin(2pii)
parenrightbigg
12
parenleftbigg
pii? 12sin(?2pii)
parenrightbigg
= pii? 12sin(2pii).
58/127
(2) FR? f(z) = sin2z = 12(1?cos2z) BFBXEYEHH2BRBR
B5G3,FBDPCZ
1
2
parenleftbigg
z? 12sin2z
parenrightbigg
F0 f(z) ARA6F8GI.
EKEX
integraldisplay pii
pii
sin2zdz =
bracketleftbigg1
2
parenleftbigg
z? 12sin2z
parenrightbiggbracketrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
pii
pii
= 12
parenleftbigg
pii? 12sin(2pii)
parenrightbigg
12
parenleftbigg
pii? 12sin(?2pii)
parenrightbigg
= pii? 12sin(2pii).
58/127
(2) FR? f(z) = sin2z = 12(1?cos2z) BFBXEYEHH2BRBR
B5G3,FBDPCZ
1
2
parenleftbigg
z? 12sin2z
parenrightbigg
F0 f(z) ARA6F8GI.
EKEX
integraldisplay pii
pii
sin2zdz =
bracketleftbigg1
2
parenleftbigg
z? 12sin2z
parenrightbiggbracketrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
pii
pii
= 12
parenleftbigg
pii? 12sin(2pii)
parenrightbigg
12
parenleftbigg
pii? 12sin(?2pii)
parenrightbigg
= pii? 12sin(2pii).
58/127
(2) FR? f(z) = sin2z = 12(1?cos2z) BFBXEYEHH2BRBR
B5G3,FBDPCZ
1
2
parenleftbigg
z? 12sin2z
parenrightbigg
F0 f(z) ARA6F8GI.
EKEX
integraldisplay pii
pii
sin2zdz =
bracketleftbigg1
2
parenleftbigg
z? 12sin2z
parenrightbiggbracketrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
pii
pii
= 12
parenleftbigg
pii? 12sin(2pii)
parenrightbigg
12
parenleftbigg
pii? 12sin(?2pii)
parenrightbigg
= pii? 12sin(2pii).
58/127
BHFBEVE1:
1,BXFKEPARFKEPH2,CTDLARH0BMBTCDAAEDF5BFE8DRH0GJ
FRDXCB,DPAQFKF8GI f(z) BFCCH0GJFRDXCBB5G3.
2,DWCMCPD5D4AVGIFIELDNB3DRFKEPDJEJEMFT f(z) ARA6
F8GI F(z).
59/127
BHFBEVE1:
1,BXFKEPARFKEPH2,CTDLARH0BMBTCDAAEDF5BFE8DRH0GJ
FRDXCB,DPAQFKF8GI f(z) BFCCH0GJFRDXCBB5G3.
2,DWCMCPD5D4AVGIFIELDNB3DRFKEPDJEJEMFT f(z) ARA6
F8GI F(z).
59/127
BHFBEVE1:
1,BXFKEPARFKEPH2,CTDLARH0BMBTCDAAEDF5BFE8DRH0GJ
FRDXCB,DPAQFKF8GI f(z) BFCCH0GJFRDXCBB5G3.
2,DWCMCPD5D4AVGIFIELDNB3DRFKEPDJEJEMFT f(z) ARA6
F8GI F(z).
59/127
DI 2 C7DYFKEP
integraldisplay i
1
1 + tanz
cosz2 dz.
B5 AQFKF8GI f(z) = 1 + tanzcosz2 BFABEX |z| < pi2 CBBRBR
B5G3,DPGJAQ 1 DQ i ARDMGBBFABEX|z| < pi2 CB.
D5FR?
1 + tanz
cosz2 dz = (1 + tanz)d(tanz) = tanz +
1
2tan
2z
EKEX
integraldisplay i
1
1 + tanz
cosz2 dz =
parenleftbigg
tanz + 12tan2z
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
i
1
= tani + 12tan2i?tan1? 12tan21.
60/127
DI 2 C7DYFKEP
integraldisplay i
1
1 + tanz
cosz2 dz.
B5 AQFKF8GI f(z) = 1 + tanzcosz2 BFABEX |z| < pi2 CBBRBR
B5G3,DPGJAQ 1 DQ i ARDMGBBFABEX|z| < pi2 CB.
D5FR?
1 + tanz
cosz2 dz = (1 + tanz)d(tanz) = tanz +
1
2tan
2z
EKEX
integraldisplay i
1
1 + tanz
cosz2 dz =
parenleftbigg
tanz + 12tan2z
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
i
1
= tani + 12tan2i?tan1? 12tan21.
60/127
DI 2 C7DYFKEP
integraldisplay i
1
1 + tanz
cosz2 dz.
B5 AQFKF8GI f(z) = 1 + tanzcosz2 BFABEX |z| < pi2 CBBRBR
B5G3,DPGJAQ 1 DQ i ARDMGBBFABEX|z| < pi2 CB.
D5FR?
1 + tanz
cosz2 dz = (1 + tanz)d(tanz) = tanz +
1
2tan
2z
EKEX
integraldisplay i
1
1 + tanz
cosz2 dz =
parenleftbigg
tanz + 12tan2z
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
i
1
= tani + 12tan2i?tan1? 12tan21.
60/127
DI 2 C7DYFKEP
integraldisplay i
1
1 + tanz
cosz2 dz.
B5 AQFKF8GI f(z) = 1 + tanzcosz2 BFABEX |z| < pi2 CBBRBR
B5G3,DPGJAQ 1 DQ i ARDMGBBFABEX|z| < pi2 CB.
D5FR?
1 + tanz
cosz2 dz = (1 + tanz)d(tanz) = tanz +
1
2tan
2z
EKEX
integraldisplay i
1
1 + tanz
cosz2 dz =
parenleftbigg
tanz + 12tan2z
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
i
1
= tani + 12tan2i?tan1? 12tan21.
60/127
BXBVF8GIARFKEPEL?CZDQEIE8A3F8GIF9FKEPCPD5AREP
G4FKEPGYEQ.
F9D2 (EPG4FKEPGYEQ) ANF8GI f(z),g(z) BFH0GJFRDX D
CBB5G3,DP z0,z F0 D CBARH0BM,C3
integraldisplay z
z0
f(ζ)g′(ζ)dζ = f(ζ)g(ζ)
vextendsinglevextendsingle
vextendsinglevextendsingle
z
z0
integraldisplay z
z0
f′(ζ)g(ζ)dζ.
61/127
BXBVF8GIARFKEPEL?CZDQEIE8A3F8GIF9FKEPCPD5AREP
G4FKEPGYEQ.
F9D2 (EPG4FKEPGYEQ) ANF8GI f(z),g(z) BFH0GJFRDX D
CBB5G3,DP z0,z F0 D CBARH0BM,C3
integraldisplay z
z0
f(ζ)g′(ζ)dζ = f(ζ)g(ζ)
vextendsinglevextendsingle
vextendsinglevextendsingle
z
z0
integraldisplay z
z0
f′(ζ)g(ζ)dζ.
61/127
DI 3 C7DYFKEP
integraldisplay 1
0
zsinzdz;
B5 FR? f(z) = zsinz BFBXEYEHH2BRBRB5G3,DGCMDQEI
FKEPCPD5AREPG4FKEPGYEQ,CZ
integraldisplay 1
0
zsinzdz =?
integraldisplay 1
0
zdcosz
=?[zcosz]
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
+
integraldisplay 1
0
coszdz
=?cos1 + sinz
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
= sin1?cos1.
62/127
DI 3 C7DYFKEP
integraldisplay 1
0
zsinzdz;
B5 FR? f(z) = zsinz BFBXEYEHH2BRBRB5G3,DGCMDQEI
FKEPCPD5AREPG4FKEPGYEQ,CZ
integraldisplay 1
0
zsinzdz =?
integraldisplay 1
0
zdcosz
=?[zcosz]
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
+
integraldisplay 1
0
coszdz
=?cos1 + sinz
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
= sin1?cos1.
62/127
DI 3 C7DYFKEP
integraldisplay 1
0
zsinzdz;
B5 FR? f(z) = zsinz BFBXEYEHH2BRBRB5G3,DGCMDQEI
FKEPCPD5AREPG4FKEPGYEQ,CZ
integraldisplay 1
0
zsinzdz =?
integraldisplay 1
0
zdcosz
=?[zcosz]
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
+
integraldisplay 1
0
coszdz
=?cos1 + sinz
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
= sin1?cos1.
62/127
DI 3 C7DYFKEP
integraldisplay 1
0
zsinzdz;
B5 FR? f(z) = zsinz BFBXEYEHH2BRBRB5G3,DGCMDQEI
FKEPCPD5AREPG4FKEPGYEQ,CZ
integraldisplay 1
0
zsinzdz =?
integraldisplay 1
0
zdcosz
=?[zcosz]
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
+
integraldisplay 1
0
coszdz
=?cos1 + sinz
vextendsinglevextendsingle
vextendsinglevextendsingle
1
0
= sin1?cos1.
62/127
BDFMAX DPG4FKEPGYEQ
F9D2 (DPG4FKEPGYEQ) CGC2 f(z) BFERDX D CBBRBRB5
G3,C? D CBARBRG3E8CMAKE5DFH0BGETDM,ENARCBG4AHAS
F5D9 D,z0? C CBARBRE8BM,BSD3
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz.
z0
C
D
63/127
DDAN z0?DX D CBBRFBE8BM,EX z0?ELB1,r?DWF7
BF D CBDUE8A2ABEX cr,G1CBG4G9D9 D,AN D1 F0CS C
G2 Cr EKDWEKARDX.
z0 C
D
Cr D1
64/127
FR?F8GI
f(z)
z?z0 BF D1 CBB5G3,DUGABGF0BVBBA6D2,CZ
contintegraldisplay
C
f(z)
z?z0dz =
contintegraldisplay
Cr
f(z)
z?z0dz
CSD9 limz→z
0
(z?z0) f(z)z?z
0
= f(z0),EXBL 3.3 AXDI 4,CZ
contintegraldisplay
Cr
f(z)
z?z0dz = 2piif(z0).
EKEX
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz.
65/127
FR?F8GI
f(z)
z?z0 BF D1 CBB5G3,DUGABGF0BVBBA6D2,CZ
contintegraldisplay
C
f(z)
z?z0dz =
contintegraldisplay
Cr
f(z)
z?z0dz
CSD9 limz→z
0
(z?z0) f(z)z?z
0
= f(z0),EXBL 3.3 AXDI 4,CZ
contintegraldisplay
Cr
f(z)
z?z0dz = 2piif(z0).
EKEX
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz.
65/127
FR?F8GI
f(z)
z?z0 BF D1 CBB5G3,DUGABGF0BVBBA6D2,CZ
contintegraldisplay
C
f(z)
z?z0dz =
contintegraldisplay
Cr
f(z)
z?z0dz
CSD9 limz→z
0
(z?z0) f(z)z?z
0
= f(z0),EXBL 3.3 AXDI 4,CZ
contintegraldisplay
Cr
f(z)
z?z0dz = 2piif(z0).
EKEX
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz.
65/127
DPG4FKEPGYEQARGLBE:
CGC2 D F0AKE5DFH0BGETDM C ARCBG4,CGC2 f(z) BFER
DX D CBB5G3,BF D H2GJCH,z0? C CBARBRE8BM,BSD3
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz.
C
D
DPG4FKEPGYEQC4ER,f(z) BFBGETDM C H2ARDUGZF9AA
ENBF C CBG4BRFBE8BMARDU.
66/127
DPG4FKEPGYEQARGLBE:
CGC2 D F0AKE5DFH0BGETDM C ARCBG4,CGC2 f(z) BFER
DX D CBB5G3,BF D H2GJCH,z0? C CBARBRE8BM,BSD3
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz.
C
D
DPG4FKEPGYEQC4ER,f(z) BFBGETDM C H2ARDUGZF9AA
ENBF C CBG4BRFBE8BMARDU.
66/127
DPG4FKEPGYEQARGLBE:
CGC2 D F0AKE5DFH0BGETDM C ARCBG4,CGC2 f(z) BFER
DX D CBB5G3,BF D H2GJCH,z0? C CBARBRE8BM,BSD3
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz.
C
D
DPG4FKEPGYEQC4ER,f(z) BFBGETDM C H2ARDUGZF9AA
ENBF C CBG4BRFBE8BMARDU.
66/127
DI 1 ANF8GI f(z) BFERDX D B5G3,BF D H2GJCH,BQ
BA C CSFHGBBDDBETDMDGEK,CT f(z) BF C H2GO?DIGI,DD
EY,f(z) BF D H2?GO?DIGI.
C
D z0
z
B5 AN f(z) = M,z ∈ C.
GHD9BRFB z0 ∈ D,DUGADPG4FKEPGYEQ,CZ
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz =
1
2pii
contintegraldisplay
C
M
z?z0dz
= M · 12pii
contintegraldisplay
C
1
z?z0dz = M.
EKEX f(z) BF D H2?GO?DIGI.
67/127
DI 1 ANF8GI f(z) BFERDX D B5G3,BF D H2GJCH,BQ
BA C CSFHGBBDDBETDMDGEK,CT f(z) BF C H2GO?DIGI,DD
EY,f(z) BF D H2?GO?DIGI.
C
D z0
z
B5 AN f(z) = M,z ∈ C.
GHD9BRFB z0 ∈ D,DUGADPG4FKEPGYEQ,CZ
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz =
1
2pii
contintegraldisplay
C
M
z?z0dz
= M · 12pii
contintegraldisplay
C
1
z?z0dz = M.
EKEX f(z) BF D H2?GO?DIGI.
67/127
DI 1 ANF8GI f(z) BFERDX D B5G3,BF D H2GJCH,BQ
BA C CSFHGBBDDBETDMDGEK,CT f(z) BF C H2GO?DIGI,DD
EY,f(z) BF D H2?GO?DIGI.
C
D z0
z
B5 AN f(z) = M,z ∈ C.
GHD9BRFB z0 ∈ D,DUGADPG4FKEPGYEQ,CZ
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz =
1
2pii
contintegraldisplay
C
M
z?z0dz
= M · 12pii
contintegraldisplay
C
1
z?z0dz = M.
EKEX f(z) BF D H2?GO?DIGI.
67/127
DI 1 ANF8GI f(z) BFERDX D B5G3,BF D H2GJCH,BQ
BA C CSFHGBBDDBETDMDGEK,CT f(z) BF C H2GO?DIGI,DD
EY,f(z) BF D H2?GO?DIGI.
C
D z0
z
B5 AN f(z) = M,z ∈ C.
GHD9BRFB z0 ∈ D,DUGADPG4FKEPGYEQ,CZ
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz =
1
2pii
contintegraldisplay
C
M
z?z0dz
= M · 12pii
contintegraldisplay
C
1
z?z0dz = M.
EKEX f(z) BF D H2?GO?DIGI.
67/127
DI 1 ANF8GI f(z) BFERDX D B5G3,BF D H2GJCH,BQ
BA C CSFHGBBDDBETDMDGEK,CT f(z) BF C H2GO?DIGI,DD
EY,f(z) BF D H2?GO?DIGI.
C
D z0
z
B5 AN f(z) = M,z ∈ C.
GHD9BRFB z0 ∈ D,DUGADPG4FKEPGYEQ,CZ
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz =
1
2pii
contintegraldisplay
C
M
z?z0dz
= M · 12pii
contintegraldisplay
C
1
z?z0dz = M.
EKEX f(z) BF D H2?GO?DIGI.
67/127
DI 2 AN f(z) DQ g(z) BFERDX D CBBRBRB5G3,C? D
CBARBRG3E8CMDFH0BGETDM,ENARCBG4ASF5D9 D.
CGC2 f(z) = g(z) BF C H2EKCZARBMBREKG9,FDDDBF C
CBAREKCZBMBR f(z) = g(z)?EKG9.
z0 C
D
z
DD DUGAC3BKCMDR,CZ
f(z)?g(z) = 0,z ∈ C,
D9F0,GH C CBBRE8BM z0,CSDPG4FKEPGYEQAQ
f(z0)?g(z0) = 12pii
contintegraldisplay
C
f(z)?g(z)
z?z0 dz =
1
2pii
contintegraldisplay
C
0
z?z0dz =
EKEXBF C CBBRFBBMBRCZ f(z0) = g(z0).
68/127
DI 2 AN f(z) DQ g(z) BFERDX D CBBRBRB5G3,C? D
CBARBRG3E8CMDFH0BGETDM,ENARCBG4ASF5D9 D.
CGC2 f(z) = g(z) BF C H2EKCZARBMBREKG9,FDDDBF C
CBAREKCZBMBR f(z) = g(z)?EKG9.
z0 C
D
z
DD DUGAC3BKCMDR,CZ
f(z)?g(z) = 0,z ∈ C,
D9F0,GH C CBBRE8BM z0,CSDPG4FKEPGYEQAQ
f(z0)?g(z0) = 12pii
contintegraldisplay
C
f(z)?g(z)
z?z0 dz =
1
2pii
contintegraldisplay
C
0
z?z0dz =
EKEXBF C CBBRFBBMBRCZ f(z0) = g(z0).
68/127
DI 2 AN f(z) DQ g(z) BFERDX D CBBRBRB5G3,C? D
CBARBRG3E8CMDFH0BGETDM,ENARCBG4ASF5D9 D.
CGC2 f(z) = g(z) BF C H2EKCZARBMBREKG9,FDDDBF C
CBAREKCZBMBR f(z) = g(z)?EKG9.
z0 C
D
z
DD DUGAC3BKCMDR,CZ
f(z)?g(z) = 0,z ∈ C,
D9F0,GH C CBBRE8BM z0,CSDPG4FKEPGYEQAQ
f(z0)?g(z0) = 12pii
contintegraldisplay
C
f(z)?g(z)
z?z0 dz =
1
2pii
contintegraldisplay
C
0
z?z0dz =
EKEXBF C CBBRFBBMBRCZ f(z0) = g(z0).
68/127
DI 2 AN f(z) DQ g(z) BFERDX D CBBRBRB5G3,C? D
CBARBRG3E8CMDFH0BGETDM,ENARCBG4ASF5D9 D.
CGC2 f(z) = g(z) BF C H2EKCZARBMBREKG9,FDDDBF C
CBAREKCZBMBR f(z) = g(z)?EKG9.
z0 C
D
z
DD DUGAC3BKCMDR,CZ
f(z)?g(z) = 0,z ∈ C,
D9F0,GH C CBBRE8BM z0,CSDPG4FKEPGYEQAQ
f(z0)?g(z0) = 12pii
contintegraldisplay
C
f(z)?g(z)
z?z0 dz =
1
2pii
contintegraldisplay
C
0
z?z0dz =
EKEXBF C CBBRFBBMBRCZ f(z0) = g(z0).
68/127
BTCMDPG4 - AKCXF9D2,BXG4BGF0F9D2G2DPG4FKEPGYEQ
C7DYBXFKEPARE8DOG2F9:
(1) FGCZ,EPG3 f(z) ARB5G3BH,GHFT f(z) AREKCZG3BM;
(2) B8A0,G9E8 f(z) ARG3BMG2FKEPETDM C AREGE8B3CG;
(3) DLA0,FRC4FREQEPB5,F9GUDUFTD4G0DNA2D4G0EDF5AR
DFH0BGETDM,DGCMF9D2EMAQB4C2.
69/127
BTCMDPG4 - AKCXF9D2,BXG4BGF0F9D2G2DPG4FKEPGYEQ
C7DYBXFKEPARE8DOG2F9:
(1) FGCZ,EPG3 f(z) ARB5G3BH,GHFT f(z) AREKCZG3BM;
(2) B8A0,G9E8 f(z) ARG3BMG2FKEPETDM C AREGE8B3CG;
(3) DLA0,FRC4FREQEPB5,F9GUDUFTD4G0DNA2D4G0EDF5AR
DFH0BGETDM,DGCMF9D2EMAQB4C2.
69/127
BTCMDPG4 - AKCXF9D2,BXG4BGF0F9D2G2DPG4FKEPGYEQ
C7DYBXFKEPARE8DOG2F9:
(1) FGCZ,EPG3 f(z) ARB5G3BH,GHFT f(z) AREKCZG3BM;
(2) B8A0,G9E8 f(z) ARG3BMG2FKEPETDM C AREGE8B3CG;
(3) DLA0,FRC4FREQEPB5,F9GUDUFTD4G0DNA2D4G0EDF5AR
DFH0BGETDM,DGCMF9D2EMAQB4C2.
69/127
BTCMDPG4 - AKCXF9D2,BXG4BGF0F9D2G2DPG4FKEPGYEQ
C7DYBXFKEPARE8DOG2F9:
(1) FGCZ,EPG3 f(z) ARB5G3BH,GHFT f(z) AREKCZG3BM;
(2) B8A0,G9E8 f(z) ARG3BMG2FKEPETDM C AREGE8B3CG;
(3) DLA0,FRC4FREQEPB5,F9GUDUFTD4G0DNA2D4G0EDF5AR
DFH0BGETDM,DGCMF9D2EMAQB4C2.
69/127
DI 1 C7DYFKEP
contintegraldisplay
C
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
,G1EL,
1) C F0AKE5ABEX|z| = 14.
2) C F0AKE5ABEX|z| = 1.
3) C F0AKE5ABEX|z| = 3.
B5AQFKF8GI f(z) = 1parenleftbigg
z? i2
parenrightbigg
(z + 2)
CZH0DRG3
BM z2 = i2,z1 =?2.
70/127
DI 1 C7DYFKEP
contintegraldisplay
C
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
,G1EL,
1) C F0AKE5ABEX|z| = 14.
2) C F0AKE5ABEX|z| = 1.
3) C F0AKE5ABEX|z| = 3.
B5AQFKF8GI f(z) = 1parenleftbigg
z? i2
parenrightbigg
(z + 2)
CZH0DRG3
BM z2 = i2,z1 =?2.
70/127
1) H0DRG3BMFTBFABEX |z| = 14 A9,AQFKF8GIBFAB
EX |z| = 14 CBB5G3,DUGADPG4 - AKCXF9D2,DJ
contintegraldisplay
C
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
= 0.
71/127
2) G3BM z1 = i2 BFABEX |z| = 1 CB,G3BM z2 =?2 BFAB
EX |z| = 1 A9.
2 1/2
C
y
x
CSDPG4FKEPGYEQ
contintegraldisplay
C
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
=
contintegraldisplay
C
1
z + 2
z? i2
dz
= 2pii· 1z + 2
vextendsinglevextendsingle
vextendsinglevextendsingle
z= i2
= 4pii4 + i
72/127
2) G3BM z1 = i2 BFABEX |z| = 1 CB,G3BM z2 =?2 BFAB
EX |z| = 1 A9.
2 1/2
C
y
x
CSDPG4FKEPGYEQ
contintegraldisplay
C
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
=
contintegraldisplay
C
1
z + 2
z? i2
dz
= 2pii· 1z + 2
vextendsinglevextendsingle
vextendsinglevextendsingle
z= i2
= 4pii4 + i
72/127
2) G3BM z1 = i2 BFABEX |z| = 1 CB,G3BM z2 =?2 BFAB
EX |z| = 1 A9.
2 1/2
C
y
x
CSDPG4FKEPGYEQ
contintegraldisplay
C
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
=
contintegraldisplay
C
1
z + 2
z? i2
dz
= 2pii· 1z + 2
vextendsinglevextendsingle
vextendsinglevextendsingle
z= i2
= 4pii4 + i
72/127
3) H0DRG3BMFTBFBFABEX|z| = 3 CB,CGGB,BF C CBEPC7
EX
i
2 G2?2?ABB1DUH0DRAKE5ABEX C1,C2.
DUGABXG4BGF0F9D2,
contintegraldisplay
C
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
=
contintegraldisplay
C1
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
+
contintegraldisplay
C2
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
y
x
1/2
C1 C
2
2
C
73/127
3) H0DRG3BMFTBFBFABEX|z| = 3 CB,CGGB,BF C CBEPC7
EX
i
2 G2?2?ABB1DUH0DRAKE5ABEX C1,C2.
DUGABXG4BGF0F9D2,
contintegraldisplay
C
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
=
contintegraldisplay
C1
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
+
contintegraldisplay
C2
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
y
x
1/2
C1 C
2
2
C
73/127
y
x
1/2
C1 C
2
2
C
CSDPG4FKEPGYEQ
contintegraldisplay
C1
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
=
contintegraldisplay
C1
1
z + 2
z? i2
dz
= 2pii· 1z + 2
vextendsinglevextendsingle
vextendsinglevextendsingle
z= i2
= 4pii4 + i
74/127
y
x
1/2
C1 C
2
2
C
FVAOCZ
contintegraldisplay
C2
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
=
contintegraldisplay
C2
1
z? i2
z + 2 dz
= 2pii· 1z? i
2
vextendsinglevextendsingle
vextendsinglevextendsingle
z=?2
=? 4pii4 + i
EKEX
contintegraldisplay
C
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
= 4pii4 + i? 4pii4 + i = 0.
75/127
y
x
1/2
C1 C
2
2
C
FVAOCZ
contintegraldisplay
C2
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
=
contintegraldisplay
C2
1
z? i2
z + 2 dz
= 2pii· 1z? i
2
vextendsinglevextendsingle
vextendsinglevextendsingle
z=?2
=? 4pii4 + i
EKEX
contintegraldisplay
C
dzparenleftbigg
z? i2
parenrightbigg
(z + 2)
= 4pii4 + i? 4pii4 + i = 0.
75/127
DI 2 C7DYFKEP
contintegraldisplay
C
e2z
z2 + zdz,G1EL,C,|z| =
1
4,AKE5.
B5 AQFKF8GI f(z) = e
2z
z2 + z CZH0DRG3BM z1 = 0,z
2 =?1,G1EL,z1 = 0 BF C CB,z2 =?1 BF C A9.
y
x O
2
DUGADPG4FKEPGYEQ,CZ
contintegraldisplay
C
e2z
z2 + zdz =
contintegraldisplay
C
e2z
z + 1
z dz = 2pii·
e2z
z + 1
vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 2pii
76/127
DI 2 C7DYFKEP
contintegraldisplay
C
e2z
z2 + zdz,G1EL,C,|z| =
1
4,AKE5.
B5 AQFKF8GI f(z) = e
2z
z2 + z CZH0DRG3BM z1 = 0,z
2 =?1,G1EL,z1 = 0 BF C CB,z2 =?1 BF C A9.
y
x O
2
DUGADPG4FKEPGYEQ,CZ
contintegraldisplay
C
e2z
z2 + zdz =
contintegraldisplay
C
e2z
z + 1
z dz = 2pii·
e2z
z + 1
vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 2pii
76/127
DI 2 C7DYFKEP
contintegraldisplay
C
e2z
z2 + zdz,G1EL,C,|z| =
1
4,AKE5.
B5 AQFKF8GI f(z) = e
2z
z2 + z CZH0DRG3BM z1 = 0,z
2 =?1,G1EL,z1 = 0 BF C CB,z2 =?1 BF C A9.
y
x O
2
DUGADPG4FKEPGYEQ,CZ
contintegraldisplay
C
e2z
z2 + zdz =
contintegraldisplay
C
e2z
z + 1
z dz = 2pii·
e2z
z + 1
vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 2pii
76/127
DI 3 C7DYFKEP
contintegraldisplay
C
sinpi4z
z2?1dz,G1EL,C,|z| = 2,AKE5.
B5 AQFKF8GI f(z) =
sinpi4z
z2?1 CZH0DRG3BM z1 = 1,z
2 =?1,DPFTBF C CB.
DUBLFRETDM C1,C2(CGGB),DUGABXG4BGF0F9D2,CZ
contintegraldisplay
C
sinpi4z
z2?1dz =
contintegraldisplay
C1
sinpi4z
z2?1dz +
contintegraldisplay
C2
sinpi4z
z2?1dz
y
x
C
C1 C2
O
1 1
77/127
DI 3 C7DYFKEP
contintegraldisplay
C
sinpi4z
z2?1dz,G1EL,C,|z| = 2,AKE5.
B5 AQFKF8GI f(z) =
sinpi4z
z2?1 CZH0DRG3BM z1 = 1,z
2 =?1,DPFTBF C CB.
DUBLFRETDM C1,C2(CGGB),DUGABXG4BGF0F9D2,CZ
contintegraldisplay
C
sinpi4z
z2?1dz =
contintegraldisplay
C1
sinpi4z
z2?1dz +
contintegraldisplay
C2
sinpi4z
z2?1dz
y
x
C
C1 C2
O
1 1
77/127
DI 3 C7DYFKEP
contintegraldisplay
C
sinpi4z
z2?1dz,G1EL,C,|z| = 2,AKE5.
B5 AQFKF8GI f(z) =
sinpi4z
z2?1 CZH0DRG3BM z1 = 1,z
2 =?1,DPFTBF C CB.
DUBLFRETDM C1,C2(CGGB),DUGABXG4BGF0F9D2,CZ
contintegraldisplay
C
sinpi4z
z2?1dz =
contintegraldisplay
C1
sinpi4z
z2?1dz +
contintegraldisplay
C2
sinpi4z
z2?1dz
y
x
C
C1 C2
O
1 1
77/127
y
x
C
C1 C2
O
1 1
G1EL,DUGADPG4FKEPGYEQ,CZ
contintegraldisplay
C1
sinpi4z
z2?1dz =
contintegraldisplay
C1
sinpi4z
z?1 ·
dz
z + 1 = 2pii
sinpi4z
z?1
vextendsinglevextendsingle
vextendsinglevextendsingle
z=?1
=
√2
2 pii;
contintegraldisplay
C2
sinpi4z
z2?1dz =
contintegraldisplay
C2
sinpi4z
z + 1 ·
dz
z?1 = 2pii
sinpi4z
z + 1
vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
=
√2
2 pii.
EKEX
contintegraldisplay
C
sinpi4z
z2?1dz =
√2
2 pii +
√2
2 pii =
√2pii.
78/127
y
x
C
C1 C2
O
1 1
G1EL,DUGADPG4FKEPGYEQ,CZ
contintegraldisplay
C1
sinpi4z
z2?1dz =
contintegraldisplay
C1
sinpi4z
z?1 ·
dz
z + 1 = 2pii
sinpi4z
z?1
vextendsinglevextendsingle
vextendsinglevextendsingle
z=?1
=
√2
2 pii;
contintegraldisplay
C2
sinpi4z
z2?1dz =
contintegraldisplay
C2
sinpi4z
z + 1 ·
dz
z?1 = 2pii
sinpi4z
z + 1
vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
=
√2
2 pii.
EKEX
contintegraldisplay
C
sinpi4z
z2?1dz =
√2
2 pii +
√2
2 pii =
√2pii.
78/127
y
x
C
C1 C2
O
1 1
G1EL,DUGADPG4FKEPGYEQ,CZ
contintegraldisplay
C1
sinpi4z
z2?1dz =
contintegraldisplay
C1
sinpi4z
z?1 ·
dz
z + 1 = 2pii
sinpi4z
z?1
vextendsinglevextendsingle
vextendsinglevextendsingle
z=?1
=
√2
2 pii;
contintegraldisplay
C2
sinpi4z
z2?1dz =
contintegraldisplay
C2
sinpi4z
z + 1 ·
dz
z?1 = 2pii
sinpi4z
z + 1
vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
=
√2
2 pii.
EKEX
contintegraldisplay
C
sinpi4z
z2?1dz =
√2
2 pii +
√2
2 pii =
√2pii.
78/127
y
x
C
C1 C2
O
1 1
G1EL,DUGADPG4FKEPGYEQ,CZ
contintegraldisplay
C1
sinpi4z
z2?1dz =
contintegraldisplay
C1
sinpi4z
z?1 ·
dz
z + 1 = 2pii
sinpi4z
z?1
vextendsinglevextendsingle
vextendsinglevextendsingle
z=?1
=
√2
2 pii;
contintegraldisplay
C2
sinpi4z
z2?1dz =
contintegraldisplay
C2
sinpi4z
z + 1 ·
dz
z?1 = 2pii
sinpi4z
z + 1
vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
=
√2
2 pii.
EKEX
contintegraldisplay
C
sinpi4z
z2?1dz =
√2
2 pii +
√2
2 pii =
√2pii.
78/127
DI 4(DPG4GYEQARGLBE) ANERDX D F0ABDMDX,f(z) BF D
CBB5G3,EXABDMARELB1?ELB1DUAKE5ABEX K1 DQ K2,K2 ED
F5 K1,z0? K1,K2 DND1BRE8BM,FDDD
f(z0) = 12pii
contintegraldisplay
K?1 +K2
f(z)
z?z0dz
K2
K1
B
A
z0
79/127
K2
K1
B
A
z0
DD DUBLFRDM AB.
D9F0BFF0DM A → K?1 → A → B → K2 → B → A(C8
L) EKDWEKARH0GJFRERDXCBBL L H2 f(z) H2B5G3,DP z0
BF L CBG4,CSDPG4FKEPGYEQAQ
f(z0) = 12pii
contintegraldisplay
L
f(z)
z?z0dz
BP
f(z0) = 12pii
contintegraldisplay
K?1 +K2
f(z)
z?z0dz +
contintegraldisplay
AB
f(z)
z?z0dz +
contintegraldisplay
BA
f(z)
z?z0dz
80/127
K2
K1
B
A
z0
DD DUBLFRDM AB.
D9F0BFF0DM A → K?1 → A → B → K2 → B → A(C8
L) EKDWEKARH0GJFRERDXCBBL L H2 f(z) H2B5G3,DP z0
BF L CBG4,CSDPG4FKEPGYEQAQ
f(z0) = 12pii
contintegraldisplay
L
f(z)
z?z0dz
BP
f(z0) = 12pii
contintegraldisplay
K?1 +K2
f(z)
z?z0dz +
contintegraldisplay
AB
f(z)
z?z0dz +
contintegraldisplay
BA
f(z)
z?z0dz
80/127
K2
K1
B
A
z0
DD DUBLFRDM AB.
D9F0BFF0DM A → K?1 → A → B → K2 → B → A(C8
L) EKDWEKARH0GJFRERDXCBBL L H2 f(z) H2B5G3,DP z0
BF L CBG4,CSDPG4FKEPGYEQAQ
f(z0) = 12pii
contintegraldisplay
L
f(z)
z?z0dz
BP
f(z0) = 12pii
contintegraldisplay
K?1 +K2
f(z)
z?z0dz +
contintegraldisplay
AB
f(z)
z?z0dz +
contintegraldisplay
BA
f(z)
z?z0dz
80/127
K2
K1
B
A
z0
DD DUBLFRDM AB.
D9F0BFF0DM A → K?1 → A → B → K2 → B → A(C8
L) EKDWEKARH0GJFRERDXCBBL L H2 f(z) H2B5G3,DP z0
BF L CBG4,CSDPG4FKEPGYEQAQ
f(z0) = 12pii
contintegraldisplay
L
f(z)
z?z0dz
BP
f(z0) = 12pii
contintegraldisplay
K?1 +K2
f(z)
z?z0dz +
contintegraldisplay
AB
f(z)
z?z0dz +
contintegraldisplay
BA
f(z)
z?z0dz
80/127
K2
K1
B
A
z0
FR?
contintegraldisplay
AB
f(z)
z?z0dz +
contintegraldisplay
BA
f(z)
z?z0dz = 0,
EKEX
f(z0) = 12pii
contintegraldisplay
K?1 +K2
f(z)
z?z0dz.
81/127
K2
K1
B
A
z0
FR?
contintegraldisplay
AB
f(z)
z?z0dz +
contintegraldisplay
BA
f(z)
z?z0dz = 0,
EKEX
f(z0) = 12pii
contintegraldisplay
K?1 +K2
f(z)
z?z0dz.
81/127
DI 5(DPG4GYEQARGLBE) AN C?GTGJFRDX D CBARE8CM
DFH0BGETDM,C1,C2,···,Cn F0BF C ARCBG4ARDFH0BGET
DM,ENDMD4G0EDF5?D4G0DNA2,FBDPEX C,C1,C2,···,Cn
BQBAARERDX G ASF5D9 D.
C1
C2
C3
z0
C D
CGC2 f(z) BF D CBB5G3,C3GHBRFB z0 ∈ G,CZ
f(z0) = 12pii
contintegraldisplay
Γ
f(z)
z?z0dz
G1EL,Γ?CS C BL Ck(k = 1,2,···,n) EKDGEKARBXG4BG
F0,C AICNEEA2,Ck AICMEEA2E0E5.
82/127
DI 5(DPG4GYEQARGLBE) AN C?GTGJFRDX D CBARE8CM
DFH0BGETDM,C1,C2,···,Cn F0BF C ARCBG4ARDFH0BGET
DM,ENDMD4G0EDF5?D4G0DNA2,FBDPEX C,C1,C2,···,Cn
BQBAARERDX G ASF5D9 D.
C1
C2
C3
z0
C D
CGC2 f(z) BF D CBB5G3,C3GHBRFB z0 ∈ G,CZ
f(z0) = 12pii
contintegraldisplay
Γ
f(z)
z?z0dz
G1EL,Γ?CS C BL Ck(k = 1,2,···,n) EKDGEKARBXG4BG
F0,C AICNEEA2,Ck AICMEEA2E0E5.
82/127
DPG4FKEPGYEQARE8ERBTFRC4ERBBEQ
F9D2 (EYH2DUGYEQ) CTF8GI f(z) BFAB |z?a| < R CBB5
G3,C3
f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ,0 < r < R
DDAN C,z = a + reiθ(0 lessorequalslant θ lessorequalslant 2pi),CSDPG4GYEQ
f(a) = 12pii
integraldisplay
C
f(z)
z?adz
= 12pii
integraldisplay 2pi
0
f(a + reiθ)
reiθ rie
iθdθ
BP,f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ.
EYH2DUGYEQCOEY,E8DRB5G3F8GIBFABB1BRARDUAVD9ENBF
ABEXH2AREYH2DU.
83/127
DPG4FKEPGYEQARE8ERBTFRC4ERBBEQ
F9D2 (EYH2DUGYEQ) CTF8GI f(z) BFAB |z?a| < R CBB5
G3,C3
f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ,0 < r < R
DDAN C,z = a + reiθ(0 lessorequalslant θ lessorequalslant 2pi),CSDPG4GYEQ
f(a) = 12pii
integraldisplay
C
f(z)
z?adz
= 12pii
integraldisplay 2pi
0
f(a + reiθ)
reiθ rie
iθdθ
BP,f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ.
EYH2DUGYEQCOEY,E8DRB5G3F8GIBFABB1BRARDUAVD9ENBF
ABEXH2AREYH2DU.
83/127
DPG4FKEPGYEQARE8ERBTFRC4ERBBEQ
F9D2 (EYH2DUGYEQ) CTF8GI f(z) BFAB |z?a| < R CBB5
G3,C3
f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ,0 < r < R
DDAN C,z = a + reiθ(0 lessorequalslant θ lessorequalslant 2pi),CSDPG4GYEQ
f(a) = 12pii
integraldisplay
C
f(z)
z?adz
= 12pii
integraldisplay 2pi
0
f(a + reiθ)
reiθ rie
iθdθ
BP,f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ.
EYH2DUGYEQCOEY,E8DRB5G3F8GIBFABB1BRARDUAVD9ENBF
ABEXH2AREYH2DU.
83/127
DPG4FKEPGYEQARE8ERBTFRC4ERBBEQ
F9D2 (EYH2DUGYEQ) CTF8GI f(z) BFAB |z?a| < R CBB5
G3,C3
f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ,0 < r < R
DDAN C,z = a + reiθ(0 lessorequalslant θ lessorequalslant 2pi),CSDPG4GYEQ
f(a) = 12pii
integraldisplay
C
f(z)
z?adz
= 12pii
integraldisplay 2pi
0
f(a + reiθ)
reiθ rie
iθdθ
BP,f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ.
EYH2DUGYEQCOEY,E8DRB5G3F8GIBFABB1BRARDUAVD9ENBF
ABEXH2AREYH2DU.
83/127
DPG4FKEPGYEQARE8ERBTFRC4ERBBEQ
F9D2 (EYH2DUGYEQ) CTF8GI f(z) BFAB |z?a| < R CBB5
G3,C3
f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ,0 < r < R
DDAN C,z = a + reiθ(0 lessorequalslant θ lessorequalslant 2pi),CSDPG4GYEQ
f(a) = 12pii
integraldisplay
C
f(z)
z?adz
= 12pii
integraldisplay 2pi
0
f(a + reiθ)
reiθ rie
iθdθ
BP,f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ.
EYH2DUGYEQCOEY,E8DRB5G3F8GIBFABB1BRARDUAVD9ENBF
ABEXH2AREYH2DU.
83/127
DPG4FKEPGYEQARE8ERBTFRC4ERBBEQ
F9D2 (EYH2DUGYEQ) CTF8GI f(z) BFAB |z?a| < R CBB5
G3,C3
f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ,0 < r < R
DDAN C,z = a + reiθ(0 lessorequalslant θ lessorequalslant 2pi),CSDPG4GYEQ
f(a) = 12pii
integraldisplay
C
f(z)
z?adz
= 12pii
integraldisplay 2pi
0
f(a + reiθ)
reiθ rie
iθdθ
BP,f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ.
EYH2DUGYEQCOEY,E8DRB5G3F8GIBFABB1BRARDUAVD9ENBF
ABEXH2AREYH2DU.
83/127
DPG4FKEPGYEQARE8ERBTFRC4ERBBEQ
F9D2 (EYH2DUGYEQ) CTF8GI f(z) BFAB |z?a| < R CBB5
G3,C3
f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ,0 < r < R
DDAN C,z = a + reiθ(0 lessorequalslant θ lessorequalslant 2pi),CSDPG4GYEQ
f(a) = 12pii
integraldisplay
C
f(z)
z?adz
= 12pii
integraldisplay 2pi
0
f(a + reiθ)
reiθ rie
iθdθ
BP,f(a) = 12pi
integraldisplay 2pi
0
f(a + reiθ)dθ.
EYH2DUGYEQCOEY,E8DRB5G3F8GIBFABB1BRARDUAVD9ENBF
ABEXH2AREYH2DU.
83/127
BDBKAX B5G3F8GIARD4AUAJGI
F9D2 1 B5G3F8GI f(z) ARAJGIBX?B5G3F8GI,ENAR n AU
AJGI?:
f(n)(z0) = n!2pii
contintegraldisplay
C
f(z)
(z?z0)n+1dz (n = 1,2,···)
G1EL,C?BFF8GI f(z) ARB5G3ERDX D CBDWBJ z0 ARBRG3
E8CMAKE5DFH0BGETDM,ADDPENARCBG4ASF5D9 D.
DD CZDD n = 1 ARECBB,BP
f′(z0) = 12pii
contintegraldisplay
C
f(z)
(z?z0)2dz.
DUGAF9FECZ
f′(z0) = lim
△x→0
f(z0 +△z)?f(z0)
△z,
84/127
BDBKAX B5G3F8GIARD4AUAJGI
F9D2 1 B5G3F8GI f(z) ARAJGIBX?B5G3F8GI,ENAR n AU
AJGI?:
f(n)(z0) = n!2pii
contintegraldisplay
C
f(z)
(z?z0)n+1dz (n = 1,2,···)
G1EL,C?BFF8GI f(z) ARB5G3ERDX D CBDWBJ z0 ARBRG3
E8CMAKE5DFH0BGETDM,ADDPENARCBG4ASF5D9 D.
DD CZDD n = 1 ARECBB,BP
f′(z0) = 12pii
contintegraldisplay
C
f(z)
(z?z0)2dz.
DUGAF9FECZ
f′(z0) = lim
△x→0
f(z0 +△z)?f(z0)
△z,
84/127
BDBKAX B5G3F8GIARD4AUAJGI
F9D2 1 B5G3F8GI f(z) ARAJGIBX?B5G3F8GI,ENAR n AU
AJGI?:
f(n)(z0) = n!2pii
contintegraldisplay
C
f(z)
(z?z0)n+1dz (n = 1,2,···)
G1EL,C?BFF8GI f(z) ARB5G3ERDX D CBDWBJ z0 ARBRG3
E8CMAKE5DFH0BGETDM,ADDPENARCBG4ASF5D9 D.
DD CZDD n = 1 ARECBB,BP
f′(z0) = 12pii
contintegraldisplay
C
f(z)
(z?z0)2dz.
DUGAF9FECZ
f′(z0) = lim
△x→0
f(z0 +△z)?f(z0)
△z,
84/127
DUGADPG4FKEPGYEQAQ
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz;
f(z0 +△z) = 12pii
contintegraldisplay
C
f(z)
z?z0?△zdz.
D9F0
f(z0 +△z)?f(z0)
△z =
1
2pii△z
bracketleftbiggcontintegraldisplay
C
f(z)
z?z0?△zdz?
contintegraldisplay
C
f(z)
z?z0
bracketrightbigg
dz
= 12pii
contintegraldisplay
C
f(z)
(z?z0)(z?z0?△z)dz
= 12pii
contintegraldisplay
C
f(z)
(z?z0)2dz +
△z
2pii
contintegraldisplay
C
f(z)
(z?z0)2(z?z0?△z)dzbracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
85/127
DUGADPG4FKEPGYEQAQ
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz;
f(z0 +△z) = 12pii
contintegraldisplay
C
f(z)
z?z0?△zdz.
D9F0
f(z0 +△z)?f(z0)
△z =
1
2pii△z
bracketleftbiggcontintegraldisplay
C
f(z)
z?z0?△zdz?
contintegraldisplay
C
f(z)
z?z0
bracketrightbigg
dz
= 12pii
contintegraldisplay
C
f(z)
(z?z0)(z?z0?△z)dz
= 12pii
contintegraldisplay
C
f(z)
(z?z0)2dz +
△z
2pii
contintegraldisplay
C
f(z)
(z?z0)2(z?z0?△z)dzbracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
85/127
DUGADPG4FKEPGYEQAQ
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz;
f(z0 +△z) = 12pii
contintegraldisplay
C
f(z)
z?z0?△zdz.
D9F0
f(z0 +△z)?f(z0)
△z =
1
2pii△z
bracketleftbiggcontintegraldisplay
C
f(z)
z?z0?△zdz?
contintegraldisplay
C
f(z)
z?z0
bracketrightbigg
dz
= 12pii
contintegraldisplay
C
f(z)
(z?z0)(z?z0?△z)dz
= 12pii
contintegraldisplay
C
f(z)
(z?z0)2dz +
△z
2pii
contintegraldisplay
C
f(z)
(z?z0)2(z?z0?△z)dzbracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
85/127
DUGADPG4FKEPGYEQAQ
f(z0) = 12pii
contintegraldisplay
C
f(z)
z?z0dz;
f(z0 +△z) = 12pii
contintegraldisplay
C
f(z)
z?z0?△zdz.
D9F0
f(z0 +△z)?f(z0)
△z =
1
2pii△z
bracketleftbiggcontintegraldisplay
C
f(z)
z?z0?△zdz?
contintegraldisplay
C
f(z)
z?z0
bracketrightbigg
dz
= 12pii
contintegraldisplay
C
f(z)
(z?z0)(z?z0?△z)dz
= 12pii
contintegraldisplay
C
f(z)
(z?z0)2dz +
△z
2pii
contintegraldisplay
C
f(z)
(z?z0)2(z?z0?△z)dzbracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
85/127
CTEHDDEYFKEP
contintegraldisplay
C
f(z)
(z?z0)2(z?z0?△z)dzCZBA:
ANGXDRFKEP? I,BSD3
|I| =
vextendsinglevextendsingle
vextendsinglevextendsingle
contintegraldisplay
C
f(z)
(z?z0)2(z?z0?△z)dz
vextendsinglevextendsingle
vextendsinglevextendsingle
lessorequalslant 12pi
contintegraldisplay
C
|f(z)|
|z?z0|2|z?z0?△z|ds
FR? f(z) BF C H2F0B5G3AR,EKEXBF C H2GJCH,D5FR
C F0BGETDM,C3 f(z) BF C H2CZBA,C3?M > 0,EMAQ
|f(z)|lessorequalslant M,?z ∈ C.
D
C z
0 d
86/127
CTEHDDEYFKEP
contintegraldisplay
C
f(z)
(z?z0)2(z?z0?△z)dzCZBA:
ANGXDRFKEP? I,BSD3
|I| =
vextendsinglevextendsingle
vextendsinglevextendsingle
contintegraldisplay
C
f(z)
(z?z0)2(z?z0?△z)dz
vextendsinglevextendsingle
vextendsinglevextendsingle
lessorequalslant 12pi
contintegraldisplay
C
|f(z)|
|z?z0|2|z?z0?△z|ds
FR? f(z) BF C H2F0B5G3AR,EKEXBF C H2GJCH,D5FR
C F0BGETDM,C3 f(z) BF C H2CZBA,C3?M > 0,EMAQ
|f(z)|lessorequalslant M,?z ∈ C.
D
C z
0 d
86/127
CTEHDDEYFKEP
contintegraldisplay
C
f(z)
(z?z0)2(z?z0?△z)dzCZBA:
ANGXDRFKEP? I,BSD3
|I| =
vextendsinglevextendsingle
vextendsinglevextendsingle
contintegraldisplay
C
f(z)
(z?z0)2(z?z0?△z)dz
vextendsinglevextendsingle
vextendsinglevextendsingle
lessorequalslant 12pi
contintegraldisplay
C
|f(z)|
|z?z0|2|z?z0?△z|ds
FR? f(z) BF C H2F0B5G3AR,EKEXBF C H2GJCH,D5FR
C F0BGETDM,C3 f(z) BF C H2CZBA,C3?M > 0,EMAQ
|f(z)|lessorequalslant M,?z ∈ C.
D
C z
0 d
86/127
CTEHDDEYFKEP
contintegraldisplay
C
f(z)
(z?z0)2(z?z0?△z)dzCZBA:
ANGXDRFKEP? I,BSD3
|I| =
vextendsinglevextendsingle
vextendsinglevextendsingle
contintegraldisplay
C
f(z)
(z?z0)2(z?z0?△z)dz
vextendsinglevextendsingle
vextendsinglevextendsingle
lessorequalslant 12pi
contintegraldisplay
C
|f(z)|
|z?z0|2|z?z0?△z|ds
FR? f(z) BF C H2F0B5G3AR,EKEXBF C H2GJCH,D5FR
C F0BGETDM,C3 f(z) BF C H2CZBA,C3?M > 0,EMAQ
|f(z)|lessorequalslant M,?z ∈ C.
D
C z
0 d
86/127
AN d?D0 z0 AKETDM C H2DSBMARDLG9GDD0,FB
AI |△z| < 12d,D9F0CZ
|z?z0|greaterorequalslant d =? 1|z?z
0|
lessorequalslant 1d;
|z?z0?△z|greaterorequalslant|z?z0|?|△z| > d2 =? 1|z?z
0?△z|
lessorequalslant
EKEX |I|lessorequalslant
ML
pid2,
GXD4 L? C ARDJG5.
CGC2 △z → 0,BSD3 I → 0,D0AD
f′(z0) = lim
△x→0
f(z0 +△z)?f(z0)
△z =
1
2pii
contintegraldisplay
C
f(z)
(z?z0)2dz.
87/127
AN d?D0 z0 AKETDM C H2DSBMARDLG9GDD0,FB
AI |△z| < 12d,D9F0CZ
|z?z0|greaterorequalslant d =? 1|z?z
0|
lessorequalslant 1d;
|z?z0?△z|greaterorequalslant|z?z0|?|△z| > d2 =? 1|z?z
0?△z|
lessorequalslant
EKEX |I|lessorequalslant
ML
pid2,
GXD4 L? C ARDJG5.
CGC2 △z → 0,BSD3 I → 0,D0AD
f′(z0) = lim
△x→0
f(z0 +△z)?f(z0)
△z =
1
2pii
contintegraldisplay
C
f(z)
(z?z0)2dz.
87/127
AN d?D0 z0 AKETDM C H2DSBMARDLG9GDD0,FB
AI |△z| < 12d,D9F0CZ
|z?z0|greaterorequalslant d =? 1|z?z
0|
lessorequalslant 1d;
|z?z0?△z|greaterorequalslant|z?z0|?|△z| > d2 =? 1|z?z
0?△z|
lessorequalslant
EKEX |I|lessorequalslant
ML
pid2,
GXD4 L? C ARDJG5.
CGC2 △z → 0,BSD3 I → 0,D0AD
f′(z0) = lim
△x→0
f(z0 +△z)?f(z0)
△z =
1
2pii
contintegraldisplay
C
f(z)
(z?z0)2dz.
87/127
AN d?D0 z0 AKETDM C H2DSBMARDLG9GDD0,FB
AI |△z| < 12d,D9F0CZ
|z?z0|greaterorequalslant d =? 1|z?z
0|
lessorequalslant 1d;
|z?z0?△z|greaterorequalslant|z?z0|?|△z| > d2 =? 1|z?z
0?△z|
lessorequalslant
EKEX |I|lessorequalslant
ML
pid2,
GXD4 L? C ARDJG5.
CGC2 △z → 0,BSD3 I → 0,D0AD
f′(z0) = lim
△x→0
f(z0 +△z)?f(z0)
△z =
1
2pii
contintegraldisplay
C
f(z)
(z?z0)2dz.
87/127
AN d?D0 z0 AKETDM C H2DSBMARDLG9GDD0,FB
AI |△z| < 12d,D9F0CZ
|z?z0|greaterorequalslant d =? 1|z?z
0|
lessorequalslant 1d;
|z?z0?△z|greaterorequalslant|z?z0|?|△z| > d2 =? 1|z?z
0?△z|
lessorequalslant
EKEX |I|lessorequalslant
ML
pid2,
GXD4 L? C ARDJG5.
CGC2 △z → 0,BSD3 I → 0,D0AD
f′(z0) = lim
△x→0
f(z0 +△z)?f(z0)
△z =
1
2pii
contintegraldisplay
C
f(z)
(z?z0)2dz.
87/127
AN d?D0 z0 AKETDM C H2DSBMARDLG9GDD0,FB
AI |△z| < 12d,D9F0CZ
|z?z0|greaterorequalslant d =? 1|z?z
0|
lessorequalslant 1d;
|z?z0?△z|greaterorequalslant|z?z0|?|△z| > d2 =? 1|z?z
0?△z|
lessorequalslant
EKEX |I|lessorequalslant
ML
pid2,
GXD4 L? C ARDJG5.
CGC2 △z → 0,BSD3 I → 0,D0AD
f′(z0) = lim
△x→0
f(z0 +△z)?f(z0)
△z =
1
2pii
contintegraldisplay
C
f(z)
(z?z0)2dz.
87/127
EDCVCPGL,CMGIFIBKBUDFDWEXDDEY,
f(n)(z0) = n!2pii
contintegraldisplay f(z)
(z?z0)n+1dz (n = 1,2,···)
DDBA.
88/127
DI 1 C7DYFKEP
contintegraldisplay
C
ez
(z?a)3dz,G1EL,a? |a| negationslash= 1 ARBR
G3BXGI,C,|z| = 1?AKE5.
B5 AQFKF8GI f(z) = e
z
(z?a)3 CZE8DRG3BM z = a.
(1) A8 |a| > 1 EE,AQFKF8GIARG3BM a BF C A9,DUGADP
G4 - AKCXF9D2AQCZ
contintegraldisplay
C
ez
(z?a)3dz = 0.
y
a
x O
89/127
DI 1 C7DYFKEP
contintegraldisplay
C
ez
(z?a)3dz,G1EL,a? |a| negationslash= 1 ARBR
G3BXGI,C,|z| = 1?AKE5.
B5 AQFKF8GI f(z) = e
z
(z?a)3 CZE8DRG3BM z = a.
(1) A8 |a| > 1 EE,AQFKF8GIARG3BM a BF C A9,DUGADP
G4 - AKCXF9D2AQCZ
contintegraldisplay
C
ez
(z?a)3dz = 0.
y
a
x O
89/127
DI 1 C7DYFKEP
contintegraldisplay
C
ez
(z?a)3dz,G1EL,a? |a| negationslash= 1 ARBR
G3BXGI,C,|z| = 1?AKE5.
B5 AQFKF8GI f(z) = e
z
(z?a)3 CZE8DRG3BM z = a.
(1) A8 |a| > 1 EE,AQFKF8GIARG3BM a BF C A9,DUGADP
G4 - AKCXF9D2AQCZ
contintegraldisplay
C
ez
(z?a)3dz = 0.
y
a
x O
89/127
(2) A8 |a| < 1 EE,AQFKF8GIARG3BM a BF C CB,CSD4AU
AJGIARDPG4FKEPGYEQAQ
contintegraldisplay
C
ez
(z?a)3dz = 2pii·
1
2!(e
z)′′|z=a = piiea.
y
a
x O
90/127
DI 2
contintegraldisplay
C
cospiz
z3(z?1)2dz,G1EL C F0:
(1) |z| = 12; (2) |z| = 2 FT?AKE5.
B5 AQFKF8GI f(z) = cospizz3(z?1)2 CZH0DRG3
BM z = 0,z = 1.
91/127
DI 2
contintegraldisplay
C
cospiz
z3(z?1)2dz,G1EL C F0:
(1) |z| = 12; (2) |z| = 2 FT?AKE5.
B5 AQFKF8GI f(z) = cospizz3(z?1)2 CZH0DRG3
BM z = 0,z = 1.
91/127
(1) A8 C,|z| = 12 EE,G3BM z = 0 BF C CB,AD z = 1
BF C A9.
O
y
x
1
DUGAD4AUAJGIGYEQ,CZ
contintegraldisplay
C
cospiz
z3(z?1)2dz =
contintegraldisplay
C
cospiz
(z?1)2
z3 dz
= 2pii2!
bracketleftbigg cospiz
(z?1)2
bracketrightbigg′′vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= (6?pi)pii.
92/127
(1) A8 C,|z| = 12 EE,G3BM z = 0 BF C CB,AD z = 1
BF C A9.
O
y
x
1
DUGAD4AUAJGIGYEQ,CZ
contintegraldisplay
C
cospiz
z3(z?1)2dz =
contintegraldisplay
C
cospiz
(z?1)2
z3 dz
= 2pii2!
bracketleftbigg cospiz
(z?1)2
bracketrightbigg′′vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= (6?pi)pii.
92/127
(1) A8 C,|z| = 12 EE,G3BM z = 0 BF C CB,AD z = 1
BF C A9.
O
y
x
1
DUGAD4AUAJGIGYEQ,CZ
contintegraldisplay
C
cospiz
z3(z?1)2dz =
contintegraldisplay
C
cospiz
(z?1)2
z3 dz
= 2pii2!
bracketleftbigg cospiz
(z?1)2
bracketrightbigg′′vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= (6?pi)pii.
92/127
(2) A8 C,|z| = 2 EE,G3BM z = 0,z = 1 FTBF C CB.
y
x
C:|z|=2
1
O
C1 C2
DU C1,C2(CGGB) EPC7EDDWG3BM z = 0 G2 z = 1,DUGABXG4
BGF0F9D2G2D4AUAJGIGYEQ,CZcontintegraldisplay
C
cospiz
z3(z?1)2dz =
contintegraldisplay
C1
cospiz
z3(z?1)2dz +
contintegraldisplay
C2
cospiz
z3(z?1)2dz
=
contintegraldisplay
C1
cospiz
(z?1)2
z3 dz +
contintegraldisplay
C2
cospiz
z3
(z?1)2dz
= 2pii2!
bracketleftbigg cospiz
(z?1)2
bracketrightbigg′′vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
+ 2pii
bracketleftBigcospiz
z3
bracketrightBig′vextendsinglevextendsinglevextendsingle
vextendsinglez=1
= (6?pi)pii + 6pii.
93/127
(2) A8 C,|z| = 2 EE,G3BM z = 0,z = 1 FTBF C CB.
y
x
C:|z|=2
1
O
C1 C2
DU C1,C2(CGGB) EPC7EDDWG3BM z = 0 G2 z = 1,DUGABXG4
BGF0F9D2G2D4AUAJGIGYEQ,CZcontintegraldisplay
C
cospiz
z3(z?1)2dz =
contintegraldisplay
C1
cospiz
z3(z?1)2dz +
contintegraldisplay
C2
cospiz
z3(z?1)2dz
=
contintegraldisplay
C1
cospiz
(z?1)2
z3 dz +
contintegraldisplay
C2
cospiz
z3
(z?1)2dz
= 2pii2!
bracketleftbigg cospiz
(z?1)2
bracketrightbigg′′vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
+ 2pii
bracketleftBigcospiz
z3
bracketrightBig′vextendsinglevextendsinglevextendsingle
vextendsinglez=1
= (6?pi)pii + 6pii.
93/127
(2) A8 C,|z| = 2 EE,G3BM z = 0,z = 1 FTBF C CB.
y
x
C:|z|=2
1
O
C1 C2
DU C1,C2(CGGB) EPC7EDDWG3BM z = 0 G2 z = 1,DUGABXG4
BGF0F9D2G2D4AUAJGIGYEQ,CZcontintegraldisplay
C
cospiz
z3(z?1)2dz =
contintegraldisplay
C1
cospiz
z3(z?1)2dz +
contintegraldisplay
C2
cospiz
z3(z?1)2dz
=
contintegraldisplay
C1
cospiz
(z?1)2
z3 dz +
contintegraldisplay
C2
cospiz
z3
(z?1)2dz
= 2pii2!
bracketleftbigg cospiz
(z?1)2
bracketrightbigg′′vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
+ 2pii
bracketleftBigcospiz
z3
bracketrightBig′vextendsinglevextendsinglevextendsingle
vextendsinglez=1
= (6?pi)pii + 6pii.
93/127
(2) A8 C,|z| = 2 EE,G3BM z = 0,z = 1 FTBF C CB.
y
x
C:|z|=2
1
O
C1 C2
DU C1,C2(CGGB) EPC7EDDWG3BM z = 0 G2 z = 1,DUGABXG4
BGF0F9D2G2D4AUAJGIGYEQ,CZcontintegraldisplay
C
cospiz
z3(z?1)2dz =
contintegraldisplay
C1
cospiz
z3(z?1)2dz +
contintegraldisplay
C2
cospiz
z3(z?1)2dz
=
contintegraldisplay
C1
cospiz
(z?1)2
z3 dz +
contintegraldisplay
C2
cospiz
z3
(z?1)2dz
= 2pii2!
bracketleftbigg cospiz
(z?1)2
bracketrightbigg′′vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
+ 2pii
bracketleftBigcospiz
z3
bracketrightBig′vextendsinglevextendsinglevextendsingle
vextendsinglez=1
= (6?pi)pii + 6pii.
93/127
(2) A8 C,|z| = 2 EE,G3BM z = 0,z = 1 FTBF C CB.
y
x
C:|z|=2
1
O
C1 C2
DU C1,C2(CGGB) EPC7EDDWG3BM z = 0 G2 z = 1,DUGABXG4
BGF0F9D2G2D4AUAJGIGYEQ,CZcontintegraldisplay
C
cospiz
z3(z?1)2dz =
contintegraldisplay
C1
cospiz
z3(z?1)2dz +
contintegraldisplay
C2
cospiz
z3(z?1)2dz
=
contintegraldisplay
C1
cospiz
(z?1)2
z3 dz +
contintegraldisplay
C2
cospiz
z3
(z?1)2dz
= 2pii2!
bracketleftbigg cospiz
(z?1)2
bracketrightbigg′′vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
+ 2pii
bracketleftBigcospiz
z3
bracketrightBig′vextendsinglevextendsinglevextendsingle
vextendsinglez=1
= (6?pi)pii + 6pii.
93/127
(2) A8 C,|z| = 2 EE,G3BM z = 0,z = 1 FTBF C CB.
y
x
C:|z|=2
1
O
C1 C2
DU C1,C2(CGGB) EPC7EDDWG3BM z = 0 G2 z = 1,DUGABXG4
BGF0F9D2G2D4AUAJGIGYEQ,CZcontintegraldisplay
C
cospiz
z3(z?1)2dz =
contintegraldisplay
C1
cospiz
z3(z?1)2dz +
contintegraldisplay
C2
cospiz
z3(z?1)2dz
=
contintegraldisplay
C1
cospiz
(z?1)2
z3 dz +
contintegraldisplay
C2
cospiz
z3
(z?1)2dz
= 2pii2!
bracketleftbigg cospiz
(z?1)2
bracketrightbigg′′vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
+ 2pii
bracketleftBigcospiz
z3
bracketrightBig′vextendsinglevextendsinglevextendsingle
vextendsinglez=1
= (6?pi)pii + 6pii.
93/127
DI 3 AN f(z) BF |z| < R(R > 1) CBB5G3,DP f(0) = 1,
f′(0) = 2,FDC7DYFKEP
contintegraldisplay
|z|=1
(z + 1)2f(z)z2 dz
FBCSCSC7DY
integraldisplay 2pi
0
cos2θ2f(eiθ)dθ.
B5 DUGAD4AUAJGIGYEQ,CZ
contintegraldisplay
|z|=1
(z + 1)2f(z)z2 dz = 2pii·[(z + 1)2f(z)]′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
= 2pii(2f(0) + f′(0)) = 8pii.
94/127
DI 3 AN f(z) BF |z| < R(R > 1) CBB5G3,DP f(0) = 1,
f′(0) = 2,FDC7DYFKEP
contintegraldisplay
|z|=1
(z + 1)2f(z)z2 dz
FBCSCSC7DY
integraldisplay 2pi
0
cos2θ2f(eiθ)dθ.
B5 DUGAD4AUAJGIGYEQ,CZ
contintegraldisplay
|z|=1
(z + 1)2f(z)z2 dz = 2pii·[(z + 1)2f(z)]′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
= 2pii(2f(0) + f′(0)) = 8pii.
94/127
D5DUGABXFKEPC7DYGYEQ,CZ
contintegraldisplay
|z|=1
(z + 1)2f(z)z2 dz =
integraldisplay 2pi
0
(eiθ + 1)2f(e
iθ)
e2iθ ie
iθdθ
= i
integraldisplay 2pi
0
(2 + 2cosθ)f(eiθ)dθ
= 4i
integraldisplay 2pi
0
cos2θ2f(eiθ)dθ
BP
integraldisplay 2pi
0
cos2θ2f(eiθ)dθ = 2pi.
95/127
D5DUGABXFKEPC7DYGYEQ,CZ
contintegraldisplay
|z|=1
(z + 1)2f(z)z2 dz =
integraldisplay 2pi
0
(eiθ + 1)2f(e
iθ)
e2iθ ie
iθdθ
= i
integraldisplay 2pi
0
(2 + 2cosθ)f(eiθ)dθ
= 4i
integraldisplay 2pi
0
cos2θ2f(eiθ)dθ
BP
integraldisplay 2pi
0
cos2θ2f(eiθ)dθ = 2pi.
95/127
D5DUGABXFKEPC7DYGYEQ,CZ
contintegraldisplay
|z|=1
(z + 1)2f(z)z2 dz =
integraldisplay 2pi
0
(eiθ + 1)2f(e
iθ)
e2iθ ie
iθdθ
= i
integraldisplay 2pi
0
(2 + 2cosθ)f(eiθ)dθ
= 4i
integraldisplay 2pi
0
cos2θ2f(eiθ)dθ
BP
integraldisplay 2pi
0
cos2θ2f(eiθ)dθ = 2pi.
95/127
D5DUGABXFKEPC7DYGYEQ,CZ
contintegraldisplay
|z|=1
(z + 1)2f(z)z2 dz =
integraldisplay 2pi
0
(eiθ + 1)2f(e
iθ)
e2iθ ie
iθdθ
= i
integraldisplay 2pi
0
(2 + 2cosθ)f(eiθ)dθ
= 4i
integraldisplay 2pi
0
cos2θ2f(eiθ)dθ
BP
integraldisplay 2pi
0
cos2θ2f(eiθ)dθ = 2pi.
95/127
DI 4 EMFKEP
contintegraldisplay
|z|=1
ez
z dz,D0ADDD
EY
integraldisplay pi
0
ecosθcos(sinθ)dθ = pi.
B5 DUGADPG4FKEPGYEQ,CZ
contintegraldisplay
|z|=1
ez
z dz = 2piie
z
vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 2pii.
96/127
DI 4 EMFKEP
contintegraldisplay
|z|=1
ez
z dz,D0ADDD
EY
integraldisplay pi
0
ecosθcos(sinθ)dθ = pi.
B5 DUGADPG4FKEPGYEQ,CZ
contintegraldisplay
|z|=1
ez
z dz = 2piie
z
vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 2pii.
96/127
D5DUGABXFKEPC7DYGYEQ,CZ
contintegraldisplay
|z|=1
ez
z dz =
integraldisplay 2pi
0
e(cosθ+isinθ)
eiθ ie
iθdθ
= i
parenleftbiggintegraldisplay pi
pi
ecosθcos(sinθ)dθ + i
integraldisplay pi
pi
ecosθsin(sinθ)dθ
parenrightbigg
= i
parenleftbiggintegraldisplay pi
0
ecosθcos(sinθ)dθ + 0
parenrightbigg
(DUGAG3GRF8GIARFKEPBHEF)
BP
integraldisplay pi
0
ecosθcos(sinθ)dθ = pi.
97/127
D5DUGABXFKEPC7DYGYEQ,CZ
contintegraldisplay
|z|=1
ez
z dz =
integraldisplay 2pi
0
e(cosθ+isinθ)
eiθ ie
iθdθ
= i
parenleftbiggintegraldisplay pi
pi
ecosθcos(sinθ)dθ + i
integraldisplay pi
pi
ecosθsin(sinθ)dθ
parenrightbigg
= i
parenleftbiggintegraldisplay pi
0
ecosθcos(sinθ)dθ + 0
parenrightbigg
(DUGAG3GRF8GIARFKEPBHEF)
BP
integraldisplay pi
0
ecosθcos(sinθ)dθ = pi.
97/127
D5DUGABXFKEPC7DYGYEQ,CZ
contintegraldisplay
|z|=1
ez
z dz =
integraldisplay 2pi
0
e(cosθ+isinθ)
eiθ ie
iθdθ
= i
parenleftbiggintegraldisplay pi
pi
ecosθcos(sinθ)dθ + i
integraldisplay pi
pi
ecosθsin(sinθ)dθ
parenrightbigg
= i
parenleftbiggintegraldisplay pi
0
ecosθcos(sinθ)dθ + 0
parenrightbigg
(DUGAG3GRF8GIARFKEPBHEF)
BP
integraldisplay pi
0
ecosθcos(sinθ)dθ = pi.
97/127
D5DUGABXFKEPC7DYGYEQ,CZ
contintegraldisplay
|z|=1
ez
z dz =
integraldisplay 2pi
0
e(cosθ+isinθ)
eiθ ie
iθdθ
= i
parenleftbiggintegraldisplay pi
pi
ecosθcos(sinθ)dθ + i
integraldisplay pi
pi
ecosθsin(sinθ)dθ
parenrightbigg
= i
parenleftbiggintegraldisplay pi
0
ecosθcos(sinθ)dθ + 0
parenrightbigg
(DUGAG3GRF8GIARFKEPBHEF)
BP
integraldisplay pi
0
ecosθcos(sinθ)dθ = pi.
97/127
DUGAD4AUAJGIGYEQ,F0AQCTEHARB4G0:
F9D2 2 CTF8GI f(z) BFDX D CBB5G3,C3 f(z) BF D CB
CZDSAUAJGI.
GXDRF9D2C4EY,ERDX D CBARB5G3F8GI,G1AJF8GIBXB8F0
B5G3F8GI.
98/127
DUGAD4AUAJGIGYEQ,F0AQCTEHARB4G0:
F9D2 2 CTF8GI f(z) BFDX D CBB5G3,C3 f(z) BF D CB
CZDSAUAJGI.
GXDRF9D2C4EY,ERDX D CBARB5G3F8GI,G1AJF8GIBXB8F0
B5G3F8GI.
98/127
DUGAD4AUAJGIGYEQ,F0AQCTEHARB4G0:
F9D2 2 CTF8GI f(z) BFDX D CBB5G3,C3 f(z) BF D CB
CZDSAUAJGI.
GXDRF9D2C4EY,ERDX D CBARB5G3F8GI,G1AJF8GIBXB8F0
B5G3F8GI.
98/127
DI 1 AN f(z) BFH0GJFRDX B CBBRBRB5G3,DPG0?AY.C
B CBBRG3E8CMDFH0BGETDM,EVFKEP
contintegraldisplay
C
f′(z)
f(z)dz F0FG?
AYEFD3?
B5FR? f(z) BF B CBB5G3,C3 f′(z) BF B CBB5G3; D5FR
f(z) negationslash= 0,C3
1
f(z) BF B CBB5G3.
FRCS,f
′(z)
f(z) BF B CBB5G3,DUGADPG4 - AKCXF9D2DJFKEP
DU?AY.
99/127
DI 1 AN f(z) BFH0GJFRDX B CBBRBRB5G3,DPG0?AY.C
B CBBRG3E8CMDFH0BGETDM,EVFKEP
contintegraldisplay
C
f′(z)
f(z)dz F0FG?
AYEFD3?
B5FR? f(z) BF B CBB5G3,C3 f′(z) BF B CBB5G3; D5FR
f(z) negationslash= 0,C3
1
f(z) BF B CBB5G3.
FRCS,f
′(z)
f(z) BF B CBB5G3,DUGADPG4 - AKCXF9D2DJFKEP
DU?AY.
99/127
DI 1 AN f(z) BFH0GJFRDX B CBBRBRB5G3,DPG0?AY.C
B CBBRG3E8CMDFH0BGETDM,EVFKEP
contintegraldisplay
C
f′(z)
f(z)dz F0FG?
AYEFD3?
B5FR? f(z) BF B CBB5G3,C3 f′(z) BF B CBB5G3; D5FR
f(z) negationslash= 0,C3
1
f(z) BF B CBB5G3.
FRCS,f
′(z)
f(z) BF B CBB5G3,DUGADPG4 - AKCXF9D2DJFKEP
DU?AY.
99/127
F9D2 3 (Morera) CTF8GI f(z) BFDX D CBGJCH,DP
GV D CBBRFBE8CMDFH0BGETDM C CZ
contintegraldisplay
C
f(z)dz = 0,
C3 f(z) BF D CBB5G3.
DDAN z0 F0 D CBARF9E8BM,DUGACQAN,BVH2DLFKEP
F(z) =
integraldisplay z
z0
f(ζ)dζ
B4F9E8H0DUF8GI F(z),ETBXBDD3AXF9D2AJARDDEY,DW
AQ F(z) BF D B5G3.
DUGAF9D2 2,F(z) ARAJF8GI,BP f(z),?BF D CBB5G3.
100/127
F9D2 3 (Morera) CTF8GI f(z) BFDX D CBGJCH,DP
GV D CBBRFBE8CMDFH0BGETDM C CZ
contintegraldisplay
C
f(z)dz = 0,
C3 f(z) BF D CBB5G3.
DDAN z0 F0 D CBARF9E8BM,DUGACQAN,BVH2DLFKEP
F(z) =
integraldisplay z
z0
f(ζ)dζ
B4F9E8H0DUF8GI F(z),ETBXBDD3AXF9D2AJARDDEY,DW
AQ F(z) BF D B5G3.
DUGAF9D2 2,F(z) ARAJF8GI,BP f(z),?BF D CBB5G3.
100/127
F9D2 3 (Morera) CTF8GI f(z) BFDX D CBGJCH,DP
GV D CBBRFBE8CMDFH0BGETDM C CZ
contintegraldisplay
C
f(z)dz = 0,
C3 f(z) BF D CBB5G3.
DDAN z0 F0 D CBARF9E8BM,DUGACQAN,BVH2DLFKEP
F(z) =
integraldisplay z
z0
f(ζ)dζ
B4F9E8H0DUF8GI F(z),ETBXBDD3AXF9D2AJARDDEY,DW
AQ F(z) BF D B5G3.
DUGAF9D2 2,F(z) ARAJF8GI,BP f(z),?BF D CBB5G3.
100/127
F9D2 3 (Morera) CTF8GI f(z) BFDX D CBGJCH,DP
GV D CBBRFBE8CMDFH0BGETDM C CZ
contintegraldisplay
C
f(z)dz = 0,
C3 f(z) BF D CBB5G3.
DDAN z0 F0 D CBARF9E8BM,DUGACQAN,BVH2DLFKEP
F(z) =
integraldisplay z
z0
f(ζ)dζ
B4F9E8H0DUF8GI F(z),ETBXBDD3AXF9D2AJARDDEY,DW
AQ F(z) BF D B5G3.
DUGAF9D2 2,F(z) ARAJF8GI,BP f(z),?BF D CBB5G3.
100/127
F9D2 4(DPG4G0AVEQ) CTF8GI f(z) BFAB |z?a| < R CB
B5G3,FBDP |f(z)|lessorequalslant M,C3
|f(n)(a) lessorequalslant n!MRn,(n = 1,2,···,)
B5 ANC,|z?a| = r(0 < r < R),CSD4AUAJGIGYEQ,CZ
f(n)(a) = n!2pii
contintegraldisplay
C
f(ζ)
(ζ?a)n+1dζ (n = 1,2,···,)
EKEX
vextendsinglevextendsingle
vextendsinglef(n)(a)
vextendsinglevextendsingle
vextendsingle lessorequalslant n!2pi
contintegraldisplay
C
|f(ζ)|
|ζ?a|n+1|dζ|
lessorequalslant n!2pi Mrn+1 ·2pir = n!Mrn
B9 r → R,AQAK
|f(n)(a) lessorequalslant n!MRn,(n = 1,2,···,)
101/127
F9D2 4(DPG4G0AVEQ) CTF8GI f(z) BFAB |z?a| < R CB
B5G3,FBDP |f(z)|lessorequalslant M,C3
|f(n)(a) lessorequalslant n!MRn,(n = 1,2,···,)
B5 ANC,|z?a| = r(0 < r < R),CSD4AUAJGIGYEQ,CZ
f(n)(a) = n!2pii
contintegraldisplay
C
f(ζ)
(ζ?a)n+1dζ (n = 1,2,···,)
EKEX
vextendsinglevextendsingle
vextendsinglef(n)(a)
vextendsinglevextendsingle
vextendsingle lessorequalslant n!2pi
contintegraldisplay
C
|f(ζ)|
|ζ?a|n+1|dζ|
lessorequalslant n!2pi Mrn+1 ·2pir = n!Mrn
B9 r → R,AQAK
|f(n)(a) lessorequalslant n!MRn,(n = 1,2,···,)
101/127
F9D2 4(DPG4G0AVEQ) CTF8GI f(z) BFAB |z?a| < R CB
B5G3,FBDP |f(z)|lessorequalslant M,C3
|f(n)(a) lessorequalslant n!MRn,(n = 1,2,···,)
B5 ANC,|z?a| = r(0 < r < R),CSD4AUAJGIGYEQ,CZ
f(n)(a) = n!2pii
contintegraldisplay
C
f(ζ)
(ζ?a)n+1dζ (n = 1,2,···,)
EKEX
vextendsinglevextendsingle
vextendsinglef(n)(a)
vextendsinglevextendsingle
vextendsingle lessorequalslant n!2pi
contintegraldisplay
C
|f(ζ)|
|ζ?a|n+1|dζ|
lessorequalslant n!2pi Mrn+1 ·2pir = n!Mrn
B9 r → R,AQAK
|f(n)(a) lessorequalslant n!MRn,(n = 1,2,···,)
101/127
F9D2 4(DPG4G0AVEQ) CTF8GI f(z) BFAB |z?a| < R CB
B5G3,FBDP |f(z)|lessorequalslant M,C3
|f(n)(a) lessorequalslant n!MRn,(n = 1,2,···,)
B5 ANC,|z?a| = r(0 < r < R),CSD4AUAJGIGYEQ,CZ
f(n)(a) = n!2pii
contintegraldisplay
C
f(ζ)
(ζ?a)n+1dζ (n = 1,2,···,)
EKEX
vextendsinglevextendsingle
vextendsinglef(n)(a)
vextendsinglevextendsingle
vextendsingle lessorequalslant n!2pi
contintegraldisplay
C
|f(ζ)|
|ζ?a|n+1|dζ|
lessorequalslant n!2pi Mrn+1 ·2pir = n!Mrn
B9 r → R,AQAK
|f(n)(a) lessorequalslant n!MRn,(n = 1,2,···,)
101/127
F9D2 5 (Liouville) CTF8GI f(z) BFCZEHEYEHCH2B5G3,
DPCZBA,C3 f(z) BJ?DIGI.
B5 AN z0 F0BRFBBXGI,DJFGABEX,C,|z?z0| = R.
DUGACMDR,?M,EMAQGHE8DN z,CZ |f(z)|lessorequalslant M.
DUGADPG4G0AVEQ,CZ
|f′(z0)|lessorequalslant MR,
B9 R → ∞,AQAK f′(z0) = 0.
CSD9 z0 F0BRFBAR,EKEXGHD9E8DN z,f′(z) = 0,AP f(z)
F0E8DIGI.
102/127
F9D2 5 (Liouville) CTF8GI f(z) BFCZEHEYEHCH2B5G3,
DPCZBA,C3 f(z) BJ?DIGI.
B5 AN z0 F0BRFBBXGI,DJFGABEX,C,|z?z0| = R.
DUGACMDR,?M,EMAQGHE8DN z,CZ |f(z)|lessorequalslant M.
DUGADPG4G0AVEQ,CZ
|f′(z0)|lessorequalslant MR,
B9 R → ∞,AQAK f′(z0) = 0.
CSD9 z0 F0BRFBAR,EKEXGHD9E8DN z,f′(z) = 0,AP f(z)
F0E8DIGI.
102/127
F9D2 5 (Liouville) CTF8GI f(z) BFCZEHEYEHCH2B5G3,
DPCZBA,C3 f(z) BJ?DIGI.
B5 AN z0 F0BRFBBXGI,DJFGABEX,C,|z?z0| = R.
DUGACMDR,?M,EMAQGHE8DN z,CZ |f(z)|lessorequalslant M.
DUGADPG4G0AVEQ,CZ
|f′(z0)|lessorequalslant MR,
B9 R → ∞,AQAK f′(z0) = 0.
CSD9 z0 F0BRFBAR,EKEXGHD9E8DN z,f′(z) = 0,AP f(z)
F0E8DIGI.
102/127
F9D2 5 (Liouville) CTF8GI f(z) BFCZEHEYEHCH2B5G3,
DPCZBA,C3 f(z) BJ?DIGI.
B5 AN z0 F0BRFBBXGI,DJFGABEX,C,|z?z0| = R.
DUGACMDR,?M,EMAQGHE8DN z,CZ |f(z)|lessorequalslant M.
DUGADPG4G0AVEQ,CZ
|f′(z0)|lessorequalslant MR,
B9 R → ∞,AQAK f′(z0) = 0.
CSD9 z0 F0BRFBAR,EKEXGHD9E8DN z,f′(z) = 0,AP f(z)
F0E8DIGI.
102/127
F9D2 5 (Liouville) CTF8GI f(z) BFCZEHEYEHCH2B5G3,
DPCZBA,C3 f(z) BJ?DIGI.
B5 AN z0 F0BRFBBXGI,DJFGABEX,C,|z?z0| = R.
DUGACMDR,?M,EMAQGHE8DN z,CZ |f(z)|lessorequalslant M.
DUGADPG4G0AVEQ,CZ
|f′(z0)|lessorequalslant MR,
B9 R → ∞,AQAK f′(z0) = 0.
CSD9 z0 F0BRFBAR,EKEXGHD9E8DN z,f′(z) = 0,AP f(z)
F0E8DIGI.
102/127
DI 1 AN f(z) BFBXEYEHH2BRBRB5G3DPCZBA,GHD9BRFBDT
F9ARH0DRBXGI a,b,FDEMBGDL
limR→+∞
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
FBCSCSGLDD f(a) = f(b)(CSCSAQAK Liouville F9D2ARB8E8
DDEY).
B5 FR? |f(z)|lessorequalslant M DPA8 z ∈ C,|z| = R EE,CZ
|z?a|greaterorequalslant||z|?|a|| = |R?|a||,
EKEXvextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle lessorequalslant
contintegraldisplay
|z|=R
|f(z)|
|(z?a)(z?b)||dz|
lessorequalslant
contintegraldisplay
|z|=R
M
|R?|a||·|R?|b|||dz|
= 2piRM|R?|a||·|R?|b||?→ 0(R?→ +∞)
103/127
DI 1 AN f(z) BFBXEYEHH2BRBRB5G3DPCZBA,GHD9BRFBDT
F9ARH0DRBXGI a,b,FDEMBGDL
limR→+∞
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
FBCSCSGLDD f(a) = f(b)(CSCSAQAK Liouville F9D2ARB8E8
DDEY).
B5 FR? |f(z)|lessorequalslant M DPA8 z ∈ C,|z| = R EE,CZ
|z?a|greaterorequalslant||z|?|a|| = |R?|a||,
EKEXvextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle lessorequalslant
contintegraldisplay
|z|=R
|f(z)|
|(z?a)(z?b)||dz|
lessorequalslant
contintegraldisplay
|z|=R
M
|R?|a||·|R?|b|||dz|
= 2piRM|R?|a||·|R?|b||?→ 0(R?→ +∞)
103/127
DI 1 AN f(z) BFBXEYEHH2BRBRB5G3DPCZBA,GHD9BRFBDT
F9ARH0DRBXGI a,b,FDEMBGDL
limR→+∞
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
FBCSCSGLDD f(a) = f(b)(CSCSAQAK Liouville F9D2ARB8E8
DDEY).
B5 FR? |f(z)|lessorequalslant M DPA8 z ∈ C,|z| = R EE,CZ
|z?a|greaterorequalslant||z|?|a|| = |R?|a||,
EKEXvextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle lessorequalslant
contintegraldisplay
|z|=R
|f(z)|
|(z?a)(z?b)||dz|
lessorequalslant
contintegraldisplay
|z|=R
M
|R?|a||·|R?|b|||dz|
= 2piRM|R?|a||·|R?|b||?→ 0(R?→ +∞)
103/127
DI 1 AN f(z) BFBXEYEHH2BRBRB5G3DPCZBA,GHD9BRFBDT
F9ARH0DRBXGI a,b,FDEMBGDL
limR→+∞
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
FBCSCSGLDD f(a) = f(b)(CSCSAQAK Liouville F9D2ARB8E8
DDEY).
B5 FR? |f(z)|lessorequalslant M DPA8 z ∈ C,|z| = R EE,CZ
|z?a|greaterorequalslant||z|?|a|| = |R?|a||,
EKEXvextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle lessorequalslant
contintegraldisplay
|z|=R
|f(z)|
|(z?a)(z?b)||dz|
lessorequalslant
contintegraldisplay
|z|=R
M
|R?|a||·|R?|b|||dz|
= 2piRM|R?|a||·|R?|b||?→ 0(R?→ +∞)
103/127
DI 1 AN f(z) BFBXEYEHH2BRBRB5G3DPCZBA,GHD9BRFBDT
F9ARH0DRBXGI a,b,FDEMBGDL
limR→+∞
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
FBCSCSGLDD f(a) = f(b)(CSCSAQAK Liouville F9D2ARB8E8
DDEY).
B5 FR? |f(z)|lessorequalslant M DPA8 z ∈ C,|z| = R EE,CZ
|z?a|greaterorequalslant||z|?|a|| = |R?|a||,
EKEXvextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle lessorequalslant
contintegraldisplay
|z|=R
|f(z)|
|(z?a)(z?b)||dz|
lessorequalslant
contintegraldisplay
|z|=R
M
|R?|a||·|R?|b|||dz|
= 2piRM|R?|a||·|R?|b||?→ 0(R?→ +∞)
103/127
EKEX lim
R→+∞
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz = 0.
104/127
BEDD f(a) = f(b):
CGC2 a = b,C3DBB8CZ,f(a) = f(b).
FGC3,A8 R > max{|a|,|b|} EE,DUGADPG4FKEPGYEQ
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
= 1a?b
braceleftBiggcontintegraldisplay
|z|=R
f(z)
z?adz?
contintegraldisplay
|z|=R
f(z)
z?bdz
bracerightBigg
= 1a?b{2piif(a)?2piif(b)}.
B9 R?→ +∞,BPAQ f(a) = f(b).
105/127
BEDD f(a) = f(b):
CGC2 a = b,C3DBB8CZ,f(a) = f(b).
FGC3,A8 R > max{|a|,|b|} EE,DUGADPG4FKEPGYEQ
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
= 1a?b
braceleftBiggcontintegraldisplay
|z|=R
f(z)
z?adz?
contintegraldisplay
|z|=R
f(z)
z?bdz
bracerightBigg
= 1a?b{2piif(a)?2piif(b)}.
B9 R?→ +∞,BPAQ f(a) = f(b).
105/127
BEDD f(a) = f(b):
CGC2 a = b,C3DBB8CZ,f(a) = f(b).
FGC3,A8 R > max{|a|,|b|} EE,DUGADPG4FKEPGYEQ
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
= 1a?b
braceleftBiggcontintegraldisplay
|z|=R
f(z)
z?adz?
contintegraldisplay
|z|=R
f(z)
z?bdz
bracerightBigg
= 1a?b{2piif(a)?2piif(b)}.
B9 R?→ +∞,BPAQ f(a) = f(b).
105/127
BEDD f(a) = f(b):
CGC2 a = b,C3DBB8CZ,f(a) = f(b).
FGC3,A8 R > max{|a|,|b|} EE,DUGADPG4FKEPGYEQ
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
= 1a?b
braceleftBiggcontintegraldisplay
|z|=R
f(z)
z?adz?
contintegraldisplay
|z|=R
f(z)
z?bdz
bracerightBigg
= 1a?b{2piif(a)?2piif(b)}.
B9 R?→ +∞,BPAQ f(a) = f(b).
105/127
BEDD f(a) = f(b):
CGC2 a = b,C3DBB8CZ,f(a) = f(b).
FGC3,A8 R > max{|a|,|b|} EE,DUGADPG4FKEPGYEQ
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
= 1a?b
braceleftBiggcontintegraldisplay
|z|=R
f(z)
z?adz?
contintegraldisplay
|z|=R
f(z)
z?bdz
bracerightBigg
= 1a?b{2piif(a)?2piif(b)}.
B9 R?→ +∞,BPAQ f(a) = f(b).
105/127
BEDD f(a) = f(b):
CGC2 a = b,C3DBB8CZ,f(a) = f(b).
FGC3,A8 R > max{|a|,|b|} EE,DUGADPG4FKEPGYEQ
contintegraldisplay
|z|=R
f(z)
(z?a)(z?b)dz
= 1a?b
braceleftBiggcontintegraldisplay
|z|=R
f(z)
z?adz?
contintegraldisplay
|z|=R
f(z)
z?bdz
bracerightBigg
= 1a?b{2piif(a)?2piif(b)}.
B9 R?→ +∞,BPAQ f(a) = f(b).
105/127
BDBKAX B5G3F8GIDQC6G2F8GIARB3CG
C6G2F8GI CGC2AJA3EIBVF8GI?(x,y) BFERDX D CBGCCZ
AJAUGJCHEGAJGIFBDPGVDABAFMBACSE0EN
2?
x2 +
2?
y2 = 0,
BSD3EH?(x,y)?ERDX D CBARC6G2F8GI.
106/127
DI 1 DDEY,CT u?C6G2F8GIDPG0AVD9DIGI,C3 u2 G0F0
C6G2F8GI.
B5 FR? u F0C6G2F8GI
2u
x2 +
2u
y2 = 0,
D5FR?
(u2)
x = 2u
u
x,
2(u2)
x2 = 2
parenleftbigg?u
x
parenrightbigg2
+ 2u?
2u
x2,
FVD2CZ
2(u2)
y2 = 2
parenleftbigg?u
y
parenrightbigg2
+ 2u?
2u
y2,
107/127
DI 1 DDEY,CT u?C6G2F8GIDPG0AVD9DIGI,C3 u2 G0F0
C6G2F8GI.
B5 FR? u F0C6G2F8GI
2u
x2 +
2u
y2 = 0,
D5FR?
(u2)
x = 2u
u
x,
2(u2)
x2 = 2
parenleftbigg?u
x
parenrightbigg2
+ 2u?
2u
x2,
FVD2CZ
2(u2)
y2 = 2
parenleftbigg?u
y
parenrightbigg2
+ 2u?
2u
y2,
107/127
DI 1 DDEY,CT u?C6G2F8GIDPG0AVD9DIGI,C3 u2 G0F0
C6G2F8GI.
B5 FR? u F0C6G2F8GI
2u
x2 +
2u
y2 = 0,
D5FR?
(u2)
x = 2u
u
x,
2(u2)
x2 = 2
parenleftbigg?u
x
parenrightbigg2
+ 2u?
2u
x2,
FVD2CZ
2(u2)
y2 = 2
parenleftbigg?u
y
parenrightbigg2
+ 2u?
2u
y2,
107/127
DI 1 DDEY,CT u?C6G2F8GIDPG0AVD9DIGI,C3 u2 G0F0
C6G2F8GI.
B5 FR? u F0C6G2F8GI
2u
x2 +
2u
y2 = 0,
D5FR?
(u2)
x = 2u
u
x,
2(u2)
x2 = 2
parenleftbigg?u
x
parenrightbigg2
+ 2u?
2u
x2,
FVD2CZ
2(u2)
y2 = 2
parenleftbigg?u
y
parenrightbigg2
+ 2u?
2u
y2,
107/127
EKEX
2(u2)
x2 +
2(u2)
y2 = 2
bracketleftBiggparenleftbigg
u
x
parenrightbigg2
+
parenleftbigg?u
y
parenrightbigg2
+ u
parenleftbigg?2u
x2 +
2u
y2
parenrightbiggbracketrightBigg
= 2
bracketleftBiggparenleftbigg
u
x
parenrightbigg2
+
parenleftbigg?u
y
parenrightbigg2bracketrightBigg
negationslash= 0.
FR? u G0?DIGI,EKEX
u
x,
u
y G0FVEE?AY,EKEXH2EQG0
AY,BP u2 G0F0C6G2F8GI.
108/127
EKEX
2(u2)
x2 +
2(u2)
y2 = 2
bracketleftBiggparenleftbigg
u
x
parenrightbigg2
+
parenleftbigg?u
y
parenrightbigg2
+ u
parenleftbigg?2u
x2 +
2u
y2
parenrightbiggbracketrightBigg
= 2
bracketleftBiggparenleftbigg
u
x
parenrightbigg2
+
parenleftbigg?u
y
parenrightbigg2bracketrightBigg
negationslash= 0.
FR? u G0?DIGI,EKEX
u
x,
u
y G0FVEE?AY,EKEXH2EQG0
AY,BP u2 G0F0C6G2F8GI.
108/127
DI 2 B4F9BBCG u = f
parenleftBigy
x
parenrightBig
AREKCZC6G2F8GI.
B5 B9 t = yx,C3 u = f
parenleftBigy
x
parenrightBig
= f(t),D9F0
u
x = f
′(t)?y
x,
2u
x2 = f
′′(t)y2
x4 + f
′(t)2y
x3,
u
y = f
′(t)1
x,
2u
y2 = f
′′(t) 1
x2,
E3CKAKBAFMBACSE0ENDJ
f′′(t)
parenleftbiggy2
x4 +
1
x2
parenrightbigg
+ f′(t)2yx3 = 0,
BP f′′(t)(1 + t2) + 2tf′(t) = 0,
B5DREPE0EN,AQ f(t) = C1arctant + C2.
109/127
DI 2 B4F9BBCG u = f
parenleftBigy
x
parenrightBig
AREKCZC6G2F8GI.
B5 B9 t = yx,C3 u = f
parenleftBigy
x
parenrightBig
= f(t),D9F0
u
x = f
′(t)?y
x,
2u
x2 = f
′′(t)y2
x4 + f
′(t)2y
x3,
u
y = f
′(t)1
x,
2u
y2 = f
′′(t) 1
x2,
E3CKAKBAFMBACSE0ENDJ
f′′(t)
parenleftbiggy2
x4 +
1
x2
parenrightbigg
+ f′(t)2yx3 = 0,
BP f′′(t)(1 + t2) + 2tf′(t) = 0,
B5DREPE0EN,AQ f(t) = C1arctant + C2.
109/127
DI 2 B4F9BBCG u = f
parenleftBigy
x
parenrightBig
AREKCZC6G2F8GI.
B5 B9 t = yx,C3 u = f
parenleftBigy
x
parenrightBig
= f(t),D9F0
u
x = f
′(t)?y
x,
2u
x2 = f
′′(t)y2
x4 + f
′(t)2y
x3,
u
y = f
′(t)1
x,
2u
y2 = f
′′(t) 1
x2,
E3CKAKBAFMBACSE0ENDJ
f′′(t)
parenleftbiggy2
x4 +
1
x2
parenrightbigg
+ f′(t)2yx3 = 0,
BP f′′(t)(1 + t2) + 2tf′(t) = 0,
B5DREPE0EN,AQ f(t) = C1arctant + C2.
109/127
DI 2 B4F9BBCG u = f
parenleftBigy
x
parenrightBig
AREKCZC6G2F8GI.
B5 B9 t = yx,C3 u = f
parenleftBigy
x
parenrightBig
= f(t),D9F0
u
x = f
′(t)?y
x,
2u
x2 = f
′′(t)y2
x4 + f
′(t)2y
x3,
u
y = f
′(t)1
x,
2u
y2 = f
′′(t) 1
x2,
E3CKAKBAFMBACSE0ENDJ
f′′(t)
parenleftbiggy2
x4 +
1
x2
parenrightbigg
+ f′(t)2yx3 = 0,
BP f′′(t)(1 + t2) + 2tf′(t) = 0,
B5DREPE0EN,AQ f(t) = C1arctant + C2.
109/127
DI 2 B4F9BBCG u = f
parenleftBigy
x
parenrightBig
AREKCZC6G2F8GI.
B5 B9 t = yx,C3 u = f
parenleftBigy
x
parenrightBig
= f(t),D9F0
u
x = f
′(t)?y
x,
2u
x2 = f
′′(t)y2
x4 + f
′(t)2y
x3,
u
y = f
′(t)1
x,
2u
y2 = f
′′(t) 1
x2,
E3CKAKBAFMBACSE0ENDJ
f′′(t)
parenleftbiggy2
x4 +
1
x2
parenrightbigg
+ f′(t)2yx3 = 0,
BP f′′(t)(1 + t2) + 2tf′(t) = 0,
B5DREPE0EN,AQ f(t) = C1arctant + C2.
109/127
DI 2 B4F9BBCG u = f
parenleftBigy
x
parenrightBig
AREKCZC6G2F8GI.
B5 B9 t = yx,C3 u = f
parenleftBigy
x
parenrightBig
= f(t),D9F0
u
x = f
′(t)?y
x,
2u
x2 = f
′′(t)y2
x4 + f
′(t)2y
x3,
u
y = f
′(t)1
x,
2u
y2 = f
′′(t) 1
x2,
E3CKAKBAFMBACSE0ENDJ
f′′(t)
parenleftbiggy2
x4 +
1
x2
parenrightbigg
+ f′(t)2yx3 = 0,
BP f′′(t)(1 + t2) + 2tf′(t) = 0,
B5DREPE0EN,AQ f(t) = C1arctant + C2.
109/127
D9F0AQBBCG u = f
parenleftBigy
x
parenrightBig
AREKCZC6G2F8GI
u = C1arctanyx + C2.
110/127
CTEHARF9D2COEYAAC6G2F8GIDQB5G3F8GIARB3CG.
F9D2 BRG3BFERDX D CBB5G3ARF8GI,ENAREIG4G2C2G4
FTF0 D CBARC6G2F8GI.
DD AN w = f(z) = u + iv? D CBARE8DRB5G3F8GI,BS
D3?u
x =
v
y,
u
y =?
v
x,
D0AD
2u
x2 =
2v
y?x,
2u
y2 =?
2v
x?y,
111/127
CTEHARF9D2COEYAAC6G2F8GIDQB5G3F8GIARB3CG.
F9D2 BRG3BFERDX D CBB5G3ARF8GI,ENAREIG4G2C2G4
FTF0 D CBARC6G2F8GI.
DD AN w = f(z) = u + iv? D CBARE8DRB5G3F8GI,BS
D3?u
x =
v
y,
u
y =?
v
x,
D0AD
2u
x2 =
2v
y?x,
2u
y2 =?
2v
x?y,
111/127
CTEHARF9D2COEYAAC6G2F8GIDQB5G3F8GIARB3CG.
F9D2 BRG3BFERDX D CBB5G3ARF8GI,ENAREIG4G2C2G4
FTF0 D CBARC6G2F8GI.
DD AN w = f(z) = u + iv? D CBARE8DRB5G3F8GI,BS
D3?u
x =
v
y,
u
y =?
v
x,
D0AD
2u
x2 =
2v
y?x,
2u
y2 =?
2v
x?y,
111/127
CTEHARF9D2COEYAAC6G2F8GIDQB5G3F8GIARB3CG.
F9D2 BRG3BFERDX D CBB5G3ARF8GI,ENAREIG4G2C2G4
FTF0 D CBARC6G2F8GI.
DD AN w = f(z) = u + iv? D CBARE8DRB5G3F8GI,BS
D3?u
x =
v
y,
u
y =?
v
x,
D0AD
2u
x2 =
2v
y?x,
2u
y2 =?
2v
x?y,
111/127
DUGAB5G3F8GID4AUAJGIF9D2,u G2 v GCCZBRFBAUGJCHEG
AJGI,EKEX
2v
y?x =
2v
x?y.
EKEX
2u
x2 +
2u
y2 = 0,
FVD2
2v
x2 +
2v
y2 = 0.
112/127
DUGAB5G3F8GID4AUAJGIF9D2,u G2 v GCCZBRFBAUGJCHEG
AJGI,EKEX
2v
y?x =
2v
x?y.
EKEX
2u
x2 +
2u
y2 = 0,
FVD2
2v
x2 +
2v
y2 = 0.
112/127
DUGAB5G3F8GID4AUAJGIF9D2,u G2 v GCCZBRFBAUGJCHEG
AJGI,EKEX
2v
y?x =
2v
x?y.
EKEX
2u
x2 +
2u
y2 = 0,
FVD2
2v
x2 +
2v
y2 = 0.
112/127
A1G5C6G2F8GI AN u(x,y)?ERDX D CBARC6G2F8GI,F3
DMD6EM
f(z) = u + iv
BF D CBA8EKB5G3F8GIARC6G2F8GI v(x,y) EH? u(x,y)
ARA1G5C6G2F8GI.
113/127
DI 1 DDEY,F8GI u = x2?y2,v = xx2 + y2 FTF0C6G2F8
GI,A2 f(z) = u + iv G0F0B5G3F8GI.
DD FR?
u
x = 2x,
u
y =?2y,
2u
x2 = 2,
2u
y2 =?2,
v
x =
x2?y2
(x2 + y2)2,
v
y =
2xy
(x2 + y2)2,
2v
x2 =
2y3
(x2 + y2)2,
2v
y2 =
2y3
(x2 + y2)2,
EKEX
2u
x2 +
2u
y2 = 0,
2v
x2 +
2v
y2 = 0.
AP u F0ASEYEHH2ARC6G2F8GI,v BFG0A6BMEXA9BFASEYEHH2
C6G2.
114/127
DI 1 DDEY,F8GI u = x2?y2,v = xx2 + y2 FTF0C6G2F8
GI,A2 f(z) = u + iv G0F0B5G3F8GI.
DD FR?
u
x = 2x,
u
y =?2y,
2u
x2 = 2,
2u
y2 =?2,
v
x =
x2?y2
(x2 + y2)2,
v
y =
2xy
(x2 + y2)2,
2v
x2 =
2y3
(x2 + y2)2,
2v
y2 =
2y3
(x2 + y2)2,
EKEX
2u
x2 +
2u
y2 = 0,
2v
x2 +
2v
y2 = 0.
AP u F0ASEYEHH2ARC6G2F8GI,v BFG0A6BMEXA9BFASEYEHH2
C6G2.
114/127
DI 1 DDEY,F8GI u = x2?y2,v = xx2 + y2 FTF0C6G2F8
GI,A2 f(z) = u + iv G0F0B5G3F8GI.
DD FR?
u
x = 2x,
u
y =?2y,
2u
x2 = 2,
2u
y2 =?2,
v
x =
x2?y2
(x2 + y2)2,
v
y =
2xy
(x2 + y2)2,
2v
x2 =
2y3
(x2 + y2)2,
2v
y2 =
2y3
(x2 + y2)2,
EKEX
2u
x2 +
2u
y2 = 0,
2v
x2 +
2v
y2 = 0.
AP u F0ASEYEHH2ARC6G2F8GI,v BFG0A6BMEXA9BFASEYEHH2
C6G2.
114/127
DI 1 DDEY,F8GI u = x2?y2,v = xx2 + y2 FTF0C6G2F8
GI,A2 f(z) = u + iv G0F0B5G3F8GI.
DD FR?
u
x = 2x,
u
y =?2y,
2u
x2 = 2,
2u
y2 =?2,
v
x =
x2?y2
(x2 + y2)2,
v
y =
2xy
(x2 + y2)2,
2v
x2 =
2y3
(x2 + y2)2,
2v
y2 =
2y3
(x2 + y2)2,
EKEX
2u
x2 +
2u
y2 = 0,
2v
x2 +
2v
y2 = 0.
AP u F0ASEYEHH2ARC6G2F8GI,v BFG0A6BMEXA9BFASEYEHH2
C6G2.
114/127
DI 1 DDEY,F8GI u = x2?y2,v = xx2 + y2 FTF0C6G2F8
GI,A2 f(z) = u + iv G0F0B5G3F8GI.
DD FR?
u
x = 2x,
u
y =?2y,
2u
x2 = 2,
2u
y2 =?2,
v
x =
x2?y2
(x2 + y2)2,
v
y =
2xy
(x2 + y2)2,
2v
x2 =
2y3
(x2 + y2)2,
2v
y2 =
2y3
(x2 + y2)2,
EKEX
2u
x2 +
2u
y2 = 0,
2v
x2 +
2v
y2 = 0.
AP u F0ASEYEHH2ARC6G2F8GI,v BFG0A6BMEXA9BFASEYEHH2
C6G2.
114/127
DI 1 DDEY,F8GI u = x2?y2,v = xx2 + y2 FTF0C6G2F8
GI,A2 f(z) = u + iv G0F0B5G3F8GI.
DD FR?
u
x = 2x,
u
y =?2y,
2u
x2 = 2,
2u
y2 =?2,
v
x =
x2?y2
(x2 + y2)2,
v
y =
2xy
(x2 + y2)2,
2v
x2 =
2y3
(x2 + y2)2,
2v
y2 =
2y3
(x2 + y2)2,
EKEX
2u
x2 +
2u
y2 = 0,
2v
x2 +
2v
y2 = 0.
AP u F0ASEYEHH2ARC6G2F8GI,v BFG0A6BMEXA9BFASEYEHH2
C6G2.
114/127
A2F0,FR?
u
x negationslash=
v
y,
G0GVDA C-R E0EN,EKEX f(z) G0F0B5G3F8GI.
115/127
BHFBEVE1,A8 v F0 u ARA1G5C6G2F8GIEE,u G0E8F9
F0 v ARA1G5C6G2F8GI.
DT CT v F0 u ARA1G5C6G2F8GIEE,f(z) = u + iv F0B5
G3F8GI,DPCZ
u
x =
v
y,
u
y =?
v
x.
CT u F0 v ARA1G5C6G2F8GIEE,f(z) = v + iu F0B5G3F8GI,
DPCZ
v
x =
u
y,
v
y =?
u
x.
EXH2H0DGAVEQE8DOG0CDFVEEEKG9.
FRCS,A8 v F0 u ARA1G5C6G2F8GIEE,u G0E8F9F0 v ARA1
G5C6G2F8GI.
116/127
BHFBEVE1,A8 v F0 u ARA1G5C6G2F8GIEE,u G0E8F9
F0 v ARA1G5C6G2F8GI.
DT CT v F0 u ARA1G5C6G2F8GIEE,f(z) = u + iv F0B5
G3F8GI,DPCZ
u
x =
v
y,
u
y =?
v
x.
CT u F0 v ARA1G5C6G2F8GIEE,f(z) = v + iu F0B5G3F8GI,
DPCZ
v
x =
u
y,
v
y =?
u
x.
EXH2H0DGAVEQE8DOG0CDFVEEEKG9.
FRCS,A8 v F0 u ARA1G5C6G2F8GIEE,u G0E8F9F0 v ARA1
G5C6G2F8GI.
116/127
BHFBEVE1,A8 v F0 u ARA1G5C6G2F8GIEE,u G0E8F9
F0 v ARA1G5C6G2F8GI.
DT CT v F0 u ARA1G5C6G2F8GIEE,f(z) = u + iv F0B5
G3F8GI,DPCZ
u
x =
v
y,
u
y =?
v
x.
CT u F0 v ARA1G5C6G2F8GIEE,f(z) = v + iu F0B5G3F8GI,
DPCZ
v
x =
u
y,
v
y =?
u
x.
EXH2H0DGAVEQE8DOG0CDFVEEEKG9.
FRCS,A8 v F0 u ARA1G5C6G2F8GIEE,u G0E8F9F0 v ARA1
G5C6G2F8GI.
116/127
BHFBEVE1,A8 v F0 u ARA1G5C6G2F8GIEE,u G0E8F9
F0 v ARA1G5C6G2F8GI.
DT CT v F0 u ARA1G5C6G2F8GIEE,f(z) = u + iv F0B5
G3F8GI,DPCZ
u
x =
v
y,
u
y =?
v
x.
CT u F0 v ARA1G5C6G2F8GIEE,f(z) = v + iu F0B5G3F8GI,
DPCZ
v
x =
u
y,
v
y =?
u
x.
EXH2H0DGAVEQE8DOG0CDFVEEEKG9.
FRCS,A8 v F0 u ARA1G5C6G2F8GIEE,u G0E8F9F0 v ARA1
G5C6G2F8GI.
116/127
BHFBEVE1,A8 v F0 u ARA1G5C6G2F8GIEE,u G0E8F9
F0 v ARA1G5C6G2F8GI.
DT CT v F0 u ARA1G5C6G2F8GIEE,f(z) = u + iv F0B5
G3F8GI,DPCZ
u
x =
v
y,
u
y =?
v
x.
CT u F0 v ARA1G5C6G2F8GIEE,f(z) = v + iu F0B5G3F8GI,
DPCZ
v
x =
u
y,
v
y =?
u
x.
EXH2H0DGAVEQE8DOG0CDFVEEEKG9.
FRCS,A8 v F0 u ARA1G5C6G2F8GIEE,u G0E8F9F0 v ARA1
G5C6G2F8GI.
116/127
EVC3,C6AOEMA1G5C6G2F8GI?
E0DFE8,DGCMDPG4 - CXGXE0ENEM u ARA1G5C6G2F8GI v.
EVC3,EUDJ u(x,y),EM v(x,y).
(1) FR?
v
y =
u
x,EKEX,
v =
integraldisplay?u
xdy + g(x) = φ(x,y) + g(x)
G1EL,g(x) F0E9DJF8GI,AD φ(x,y) F0EUDJF8GI.
117/127
EVC3,C6AOEMA1G5C6G2F8GI?
E0DFE8,DGCMDPG4 - CXGXE0ENEM u ARA1G5C6G2F8GI v.
EVC3,EUDJ u(x,y),EM v(x,y).
(1) FR?
v
y =
u
x,EKEX,
v =
integraldisplay?u
xdy + g(x) = φ(x,y) + g(x)
G1EL,g(x) F0E9DJF8GI,AD φ(x,y) F0EUDJF8GI.
117/127
EVC3,C6AOEMA1G5C6G2F8GI?
E0DFE8,DGCMDPG4 - CXGXE0ENEM u ARA1G5C6G2F8GI v.
EVC3,EUDJ u(x,y),EM v(x,y).
(1) FRv?y =?u?x,EKEX,
v =
integraldisplay?u
xdy + g(x) = φ(x,y) + g(x)
G1EL,g(x) F0E9DJF8GI,AD φ(x,y) F0EUDJF8GI.
117/127
(2) D9F0CZ
v
x =
x[φ(x,y) + g(x)] =
φ(x,y)
x + g
′(x)
E3CK
v
x =?
u
y,C3CZ
φ(x,y)
x + g
′(x) =u(x,y)
y
CSCSEMFT g′(x),B8A0FKEPAQ g(x),D9F0
v = φ(x,y) + g(x).
118/127
(2) D9F0CZ
v
x =
x[φ(x,y) + g(x)] =
φ(x,y)
x + g
′(x)
E3CK
v
x =?
u
y,C3CZ
φ(x,y)
x + g
′(x) =u(x,y)
y
CSCSEMFT g′(x),B8A0FKEPAQ g(x),D9F0
v = φ(x,y) + g(x).
118/127
(2) D9F0CZ
v
x =
x[φ(x,y) + g(x)] =
φ(x,y)
x + g
′(x)
E3CK
v
x =?
u
y,C3CZ
φ(x,y)
x + g
′(x) =u(x,y)
y
CSCSEMFT g′(x),B8A0FKEPAQ g(x),D9F0
v = φ(x,y) + g(x).
118/127
DI 1 DDEY u(x,y) = y3?3x2y F0C6G2F8GI,FBEMA1G5
C6G2F8GI v(x,y) G2CSENDMA8EKARB5G3F8GI.
B5 1) F0DD u(x,y)?C6G2F8GI.
119/127
DI 1 DDEY u(x,y) = y3?3x2y F0C6G2F8GI,FBEMA1G5
C6G2F8GI v(x,y) G2CSENDMA8EKARB5G3F8GI.
B5 1) F0DD u(x,y)?C6G2F8GI.
119/127
2) CS
v
y =
u
x =?6xy,AQ
v =
integraldisplay
6xydy =?3xy2 + g(x),
D9F0
v
x =?3y
2 + g′(x).
E3CKAK
v
x =?
u
y,AQ
3y2 + g′(x) =?3y2 + 3x2,
AP g(x) =
integraldisplay
x2dx = x3 + c,
FRCS v(x,y) = x3?3xy2 + c.
D0ADAQAKE8DRB5G3F8GI
w = y3?3x2y + i(x3?3xy2 + c).
120/127
2) CS
v
y =
u
x =?6xy,AQ
v =
integraldisplay
6xydy =?3xy2 + g(x),
D9F0
v
x =?3y
2 + g′(x).
E3CKAK
v
x =?
u
y,AQ
3y2 + g′(x) =?3y2 + 3x2,
AP g(x) =
integraldisplay
x2dx = x3 + c,
FRCS v(x,y) = x3?3xy2 + c.
D0ADAQAKE8DRB5G3F8GI
w = y3?3x2y + i(x3?3xy2 + c).
120/127
2) CS
v
y =
u
x =?6xy,AQ
v =
integraldisplay
6xydy =?3xy2 + g(x),
D9F0
v
x =?3y
2 + g′(x).
E3CKAK
v
x =?
u
y,AQ
3y2 + g′(x) =?3y2 + 3x2,
AP g(x) =
integraldisplay
x2dx = x3 + c,
FRCS v(x,y) = x3?3xy2 + c.
D0ADAQAKE8DRB5G3F8GI
w = y3?3x2y + i(x3?3xy2 + c).
120/127
2) CS
v
y =
u
x =?6xy,AQ
v =
integraldisplay
6xydy =?3xy2 + g(x),
D9F0
v
x =?3y
2 + g′(x).
E3CKAK
v
x =?
u
y,AQ
3y2 + g′(x) =?3y2 + 3x2,
AP g(x) =
integraldisplay
x2dx = x3 + c,
FRCS v(x,y) = x3?3xy2 + c.
D0ADAQAKE8DRB5G3F8GI
w = y3?3x2y + i(x3?3xy2 + c).
120/127
2) CS
v
y =
u
x =?6xy,AQ
v =
integraldisplay
6xydy =?3xy2 + g(x),
D9F0
v
x =?3y
2 + g′(x).
E3CKAK
v
x =?
u
y,AQ
3y2 + g′(x) =?3y2 + 3x2,
AP g(x) =
integraldisplay
x2dx = x3 + c,
FRCS v(x,y) = x3?3xy2 + c.
D0ADAQAKE8DRB5G3F8GI
w = y3?3x2y + i(x3?3xy2 + c).
120/127
2) CS
v
y =
u
x =?6xy,AQ
v =
integraldisplay
6xydy =?3xy2 + g(x),
D9F0
v
x =?3y
2 + g′(x).
E3CKAK
v
x =?
u
y,AQ
3y2 + g′(x) =?3y2 + 3x2,
AP g(x) =
integraldisplay
x2dx = x3 + c,
FRCS v(x,y) = x3?3xy2 + c.
D0ADAQAKE8DRB5G3F8GI
w = y3?3x2y + i(x3?3xy2 + c).
120/127
E0DFAJDMFKEPDF.
AN u F0ERDX D CBB5G3F8GI f(z) AREIG4,CSD9ENF0C6G2
F8GI,EKEX
uxx + uyy = 0.
EKEX (?uy)y = (ux)x,C3?uydx+uxdy F0B8DRAJA3EIBV
F8GI v ARASDREP
dv =?uydx + uxdy
D9F0 v =
integraldisplay (x,y)
(x0,y0)
uydx + uxdy + c
DBB8H2EQELAR v GVDA,
vy = ux,vx =?uy
BP u,v GVDA C-R E0EN,EKEX v F0 u ARA1G5C6G2F8GI.
121/127
E0DFAJDMFKEPDF.
AN u F0ERDX D CBB5G3F8GI f(z) AREIG4,CSD9ENF0C6G2
F8GI,EKEX
uxx + uyy = 0.
EKEX (?uy)y = (ux)x,C3?uydx+uxdy F0B8DRAJA3EIBV
F8GI v ARASDREP
dv =?uydx + uxdy
D9F0 v =
integraldisplay (x,y)
(x0,y0)
uydx + uxdy + c
DBB8H2EQELAR v GVDA,
vy = ux,vx =?uy
BP u,v GVDA C-R E0EN,EKEX v F0 u ARA1G5C6G2F8GI.
121/127
E0DFAJDMFKEPDF.
AN u F0ERDX D CBB5G3F8GI f(z) AREIG4,CSD9ENF0C6G2
F8GI,EKEX
uxx + uyy = 0.
EKEX (?uy)y = (ux)x,C3?uydx+uxdy F0B8DRAJA3EIBV
F8GI v ARASDREP
dv =?uydx + uxdy
D9F0 v =
integraldisplay (x,y)
(x0,y0)
uydx + uxdy + c
DBB8H2EQELAR v GVDA,
vy = ux,vx =?uy
BP u,v GVDA C-R E0EN,EKEX v F0 u ARA1G5C6G2F8GI.
121/127
E0DFAJDMFKEPDF.
AN u F0ERDX D CBB5G3F8GI f(z) AREIG4,CSD9ENF0C6G2
F8GI,EKEX
uxx + uyy = 0.
EKEX (?uy)y = (ux)x,C3?uydx+uxdy F0B8DRAJA3EIBV
F8GI v ARASDREP
dv =?uydx + uxdy
D9F0 v =
integraldisplay (x,y)
(x0,y0)
uydx + uxdy + c
DBB8H2EQELAR v GVDA,
vy = ux,vx =?uy
BP u,v GVDA C-R E0EN,EKEX v F0 u ARA1G5C6G2F8GI.
121/127
E0DFAJDMFKEPDF.
AN u F0ERDX D CBB5G3F8GI f(z) AREIG4,CSD9ENF0C6G2
F8GI,EKEX
uxx + uyy = 0.
EKEX (?uy)y = (ux)x,C3?uydx+uxdy F0B8DRAJA3EIBV
F8GI v ARASDREP
dv =?uydx + uxdy
D9F0 v =
integraldisplay (x,y)
(x0,y0)
uydx + uxdy + c
DBB8H2EQELAR v GVDA,
vy = ux,vx =?uy
BP u,v GVDA C-R E0EN,EKEX v F0 u ARA1G5C6G2F8GI.
121/127
DI 2 AN u = yx2 + y2,EMG1A1G5C6G2F8GI,FBALFT f(z)
ARBBEQ.
B5 F0A8DD u = yx2 + y2 F0C6G2F8GI.
FR?
u
x =?
2xy
(x2 + y2)2,
u
y =
x2?y2
(x2 + y2)2,
AI (x0,y0) = (1,0),EKEX
v =
integraldisplay (x,y)
(1,0)
u?ydx +?u?xdy + C
=
integraldisplay x
1
x2
x4dx?x
integraldisplay y
0
2y
(x2 + y2)2dy + C
= xx2 + y2?1 + C.
122/127
DI 2 AN u = yx2 + y2,EMG1A1G5C6G2F8GI,FBALFT f(z)
ARBBEQ.
B5 F0A8DD u = yx2 + y2 F0C6G2F8GI.
FR?
u
x =?
2xy
(x2 + y2)2,
u
y =
x2?y2
(x2 + y2)2,
AI (x0,y0) = (1,0),EKEX
v =
integraldisplay (x,y)
(1,0)
u?ydx +?u?xdy + C
=
integraldisplay x
1
x2
x4dx?x
integraldisplay y
0
2y
(x2 + y2)2dy + C
= xx2 + y2?1 + C.
122/127
DI 2 AN u = yx2 + y2,EMG1A1G5C6G2F8GI,FBALFT f(z)
ARBBEQ.
B5 F0A8DD u = yx2 + y2 F0C6G2F8GI.
FR?
u
x =?
2xy
(x2 + y2)2,
u
y =
x2?y2
(x2 + y2)2,
AI (x0,y0) = (1,0),EKEX
v =
integraldisplay (x,y)
(1,0)
u?ydx +?u?xdy + C
=
integraldisplay x
1
x2
x4dx?x
integraldisplay y
0
2y
(x2 + y2)2dy + C
= xx2 + y2?1 + C.
122/127
DI 2 AN u = yx2 + y2,EMG1A1G5C6G2F8GI,FBALFT f(z)
ARBBEQ.
B5 F0A8DD u = yx2 + y2 F0C6G2F8GI.
FR?
u
x =?
2xy
(x2 + y2)2,
u
y =
x2?y2
(x2 + y2)2,
AI (x0,y0) = (1,0),EKEX
v =
integraldisplay (x,y)
(1,0)
u?ydx +?u?xdy + C
=
integraldisplay x
1
x2
x4dx?x
integraldisplay y
0
2y
(x2 + y2)2dy + C
= xx2 + y2?1 + C.
122/127
DI 2 AN u = yx2 + y2,EMG1A1G5C6G2F8GI,FBALFT f(z)
ARBBEQ.
B5 F0A8DD u = yx2 + y2 F0C6G2F8GI.
FR?
u
x =?
2xy
(x2 + y2)2,
u
y =
x2?y2
(x2 + y2)2,
AI (x0,y0) = (1,0),EKEX
v =
integraldisplay (x,y)
(1,0)
u?ydx +?u?xdy + C
=
integraldisplay x
1
x2
x4dx?x
integraldisplay y
0
2y
(x2 + y2)2dy + C
= xx2 + y2?1 + C.
122/127
D9F0
f(z) = yx2 + y2 + i
parenleftbigg x
x2 + y2?1 + C
parenrightbigg
.
123/127
DI 3 ANF8GI u(x,y) = epxcosy?B8E8B5G3F8GI f(z)
AREIG4,FDB4F9 p ARDUDQB5G3F8
GI f(z) = u(x,y) + iv(x,y).
DD FGCZ,DUGAC6G2F8GIARF9FEEM p ARDU.
u
x = pe
pxcosy,?u
y =?e
pxsiny,
2u
x2 = p
2epxcosy,?
2u
y2 =?e
pxcosy,
D9F0CZ
2u
x2 +
2u
y2 = (p
2?1)epxcosy = 0,
EKEX p = 1 F9 p =?1.
124/127
DI 3 ANF8GI u(x,y) = epxcosy?B8E8B5G3F8GI f(z)
AREIG4,FDB4F9 p ARDUDQB5G3F8
GI f(z) = u(x,y) + iv(x,y).
DD FGCZ,DUGAC6G2F8GIARF9FEEM p ARDU.
u
x = pe
pxcosy,?u
y =?e
pxsiny,
2u
x2 = p
2epxcosy,?
2u
y2 =?e
pxcosy,
D9F0CZ
2u
x2 +
2u
y2 = (p
2?1)epxcosy = 0,
EKEX p = 1 F9 p =?1.
124/127
DI 3 ANF8GI u(x,y) = epxcosy?B8E8B5G3F8GI f(z)
AREIG4,FDB4F9 p ARDUDQB5G3F8
GI f(z) = u(x,y) + iv(x,y).
DD FGCZ,DUGAC6G2F8GIARF9FEEM p ARDU.
u
x = pe
pxcosy,?u
y =?e
pxsiny,
2u
x2 = p
2epxcosy,?
2u
y2 =?e
pxcosy,
D9F0CZ
2u
x2 +
2u
y2 = (p
2?1)epxcosy = 0,
EKEX p = 1 F9 p =?1.
124/127
B8A0,EMB5G3F8GI f(z) = u + iv,(CMH0ERE0DF)
E0DFE8 DGCMDPG4 - CXGXE0EN
DUGADPG4 - CXGXE0EN,CZ
v
x =?
u
y = e
pxsiny,(1)
v
y =
u
x = pe
pxcosy,(2)
DUGA (1) CZ v =
integraldisplay
epxsinydx = 1pepxsiny + g(y),
E3CKAK (2),AQ
1
pe
pxcosy + g′(y) = pepxcosy,
DWAQ,g′(y) = 0,DWDJ g(y) = c,EKEXCZ
v = 1pepxsiny + c
125/127
B8A0,EMB5G3F8GI f(z) = u + iv,(CMH0ERE0DF)
E0DFE8 DGCMDPG4 - CXGXE0EN
DUGADPG4 - CXGXE0EN,CZ
v
x =?
u
y = e
pxsiny,(1)
v
y =
u
x = pe
pxcosy,(2)
DUGA (1) CZ v =
integraldisplay
epxsinydx = 1pepxsiny + g(y),
E3CKAK (2),AQ
1
pe
pxcosy + g′(y) = pepxcosy,
DWAQ,g′(y) = 0,DWDJ g(y) = c,EKEXCZ
v = 1pepxsiny + c
125/127
B8A0,EMB5G3F8GI f(z) = u + iv,(CMH0ERE0DF)
E0DFE8 DGCMDPG4 - CXGXE0EN
DUGADPG4 - CXGXE0EN,CZ
v
x =?
u
y = e
pxsiny,(1)
v
y =
u
x = pe
pxcosy,(2)
DUGA (1) CZ v =
integraldisplay
epxsinydx = 1pepxsiny + g(y),
E3CKAK (2),AQ
1
pe
pxcosy + g′(y) = pepxcosy,
DWAQ,g′(y) = 0,DWDJ g(y) = c,EKEXCZ
v = 1pepxsiny + c
125/127
B8A0,EMB5G3F8GI f(z) = u + iv,(CMH0ERE0DF)
E0DFE8 DGCMDPG4 - CXGXE0EN
DUGADPG4 - CXGXE0EN,CZ
v
x =?
u
y = e
pxsiny,(1)
v
y =
u
x = pe
pxcosy,(2)
DUGA (1) CZ v =
integraldisplay
epxsinydx = 1pepxsiny + g(y),
E3CKAK (2),AQ
1
pe
pxcosy + g′(y) = pepxcosy,
DWAQ,g′(y) = 0,DWDJ g(y) = c,EKEXCZ
v = 1pepxsiny + c
125/127
B8A0,EMB5G3F8GI f(z) = u + iv,(CMH0ERE0DF)
E0DFE8 DGCMDPG4 - CXGXE0EN
DUGADPG4 - CXGXE0EN,CZ
v
x =?
u
y = e
pxsiny,(1)
v
y =
u
x = pe
pxcosy,(2)
DUGA (1) CZ v =
integraldisplay
epxsinydx = 1pepxsiny + g(y),
E3CKAK (2),AQ
1
pe
pxcosy + g′(y) = pepxcosy,
DWAQ,g′(y) = 0,DWDJ g(y) = c,EKEXCZ
v = 1pepxsiny + c
125/127
EKEX
f(z) = epxcosy + i
parenleftbigg1
pe
pxsiny + c
parenrightbigg
.
126/127
E0DFAJ DGCMDMFKEPDF
DUGA
v =
integraldisplay (x,y)
(x0,y0)
uy(x,y)dx + ux(x,y)dy + C
AI (x0,y0) = (0,0),FBAIGVDVC1F2EYBDARF0F7AQ
v =
integraldisplay x
0
uy(x,0)dx +
integraldisplay y
0
ux(x,y)dy + C
=
integraldisplay y
0
pepxcosydy + C
= pepxsiny + C
EKEX f(z) = epxcosy + i(pepxsiny + C).
127/127
E0DFAJ DGCMDMFKEPDF
DUGA
v =
integraldisplay (x,y)
(x0,y0)
uy(x,y)dx + ux(x,y)dy + C
AI (x0,y0) = (0,0),FBAIGVDVC1F2EYBDARF0F7AQ
v =
integraldisplay x
0
uy(x,0)dx +
integraldisplay y
0
ux(x,y)dy + C
=
integraldisplay y
0
pepxcosydy + C
= pepxsiny + C
EKEX f(z) = epxcosy + i(pepxsiny + C).
127/127
E0DFAJ DGCMDMFKEPDF
DUGA
v =
integraldisplay (x,y)
(x0,y0)
uy(x,y)dx + ux(x,y)dy + C
AI (x0,y0) = (0,0),FBAIGVDVC1F2EYBDARF0F7AQ
v =
integraldisplay x
0
uy(x,0)dx +
integraldisplay y
0
ux(x,y)dy + C
=
integraldisplay y
0
pepxcosydy + C
= pepxsiny + C
EKEX f(z) = epxcosy + i(pepxsiny + C).
127/127
E0DFAJ DGCMDMFKEPDF
DUGA
v =
integraldisplay (x,y)
(x0,y0)
uy(x,y)dx + ux(x,y)dy + C
AI (x0,y0) = (0,0),FBAIGVDVC1F2EYBDARF0F7AQ
v =
integraldisplay x
0
uy(x,0)dx +
integraldisplay y
0
ux(x,y)dy + C
=
integraldisplay y
0
pepxcosydy + C
= pepxsiny + C
EKEX f(z) = epxcosy + i(pepxsiny + C).
127/127
E0DFAJ DGCMDMFKEPDF
DUGA
v =
integraldisplay (x,y)
(x0,y0)
uy(x,y)dx + ux(x,y)dy + C
AI (x0,y0) = (0,0),FBAIGVDVC1F2EYBDARF0F7AQ
v =
integraldisplay x
0
uy(x,0)dx +
integraldisplay y
0
ux(x,y)dy + C
=
integraldisplay y
0
pepxcosydy + C
= pepxsiny + C
EKEX f(z) = epxcosy + i(pepxsiny + C).
127/127
E0DFAJ DGCMDMFKEPDF
DUGA
v =
integraldisplay (x,y)
(x0,y0)
uy(x,y)dx + ux(x,y)dy + C
AI (x0,y0) = (0,0),FBAIGVDVC1F2EYBDARF0F7AQ
v =
integraldisplay x
0
uy(x,0)dx +
integraldisplay y
0
ux(x,y)dy + C
=
integraldisplay y
0
pepxcosydy + C
= pepxsiny + C
EKEX f(z) = epxcosy + i(pepxsiny + C).
127/127