北京邮电大学：复变函数（2006年）：6H

BDBKG1 A1BBC6AJ
December 18,2006
1/127
BKAR,CMBTG3E0DFBHGNFGB5G3F8GIARBHEF.
FODRCBC6:
1,A1BBC6AJARCECV.
2,E8CPBTFRARA1BBC6AJ—– EPEQDMBHC6AJARCECVG2BH
EF.
3,BTDRFSAVF8GIEKA8EKARA1BBC6AJARBHEF.
2/127
BKAR,CMBTG3E0DFBHGNFGB5G3F8GIARBHEF.
FODRCBC6:
1,A1BBC6AJARCECV.
2,E8CPBTFRARA1BBC6AJ—– EPEQDMBHC6AJARCECVG2BH
EF.
3,BTDRFSAVF8GIEKA8EKARA1BBC6AJARBHEF.
2/127
BKAR,CMBTG3E0DFBHGNFGB5G3F8GIARBHEF.
FODRCBC6:
1,A1BBC6AJARCECV.
2,E8CPBTFRARA1BBC6AJ—– EPEQDMBHC6AJARCECVG2BH
EF.
3,BTDRFSAVF8GIEKA8EKARA1BBC6AJARBHEF.
2/127
BKAR,CMBTG3E0DFBHGNFGB5G3F8GIARBHEF.
FODRCBC6:
1,A1BBC6AJARCECV.
2,E8CPBTFRARA1BBC6AJ—– EPEQDMBHC6AJARCECVG2BH
EF.
3,BTDRFSAVF8GIEKA8EKARA1BBC6AJARBHEF.
2/127
BKAR,CMBTG3E0DFBHGNFGB5G3F8GIARBHEF.
FODRCBC6:
1,A1BBC6AJARCECV.
2,E8CPBTFRARA1BBC6AJ—– EPEQDMBHC6AJARCECVG2BH
EF.
3,BTDRFSAVF8GIEKA8EKARA1BBC6AJARBHEF.
2/127
A1BBC6AJFODRB5GZAREVC3:
1,EUDJC6AJ w = f(z) BLERDX D,EMERDX D BFC6
AJ w = f(z) CTARE4DX G.
2,EUDJERDX D BLERDX G,EMGQERDX D A1BBBBC6AJ?
ERDX G ARB5G3F8GI w = f(z).
3/127
A1BBC6AJFODRB5GZAREVC3:
1,EUDJC6AJ w = f(z) BLERDX D,EMERDX D BFC6
AJ w = f(z) CTARE4DX G.
2,EUDJERDX D BLERDX G,EMGQERDX D A1BBBBC6AJ?
ERDX G ARB5G3F8GI w = f(z).
3/127
A1BBC6AJFODRB5GZAREVC3:
1,EUDJC6AJ w = f(z) BLERDX D,EMERDX D BFC6
AJ w = f(z) CTARE4DX G.
2,EUDJERDX D BLERDX G,EMGQERDX D A1BBBBC6AJ?
ERDX G ARB5G3F8GI w = f(z).
3/127
BDE8AX A1BBC6AJARCECV
GJCHETDMARDNDM
AN z EYEHCBARE8CMCZE5GJCHETDM C DWEXC4ER?
z = z(t),α lessorequalslant t lessorequalslant β
ENARAKE5AI t C7DWEEBM z EJFGARE0E5,z(t)?E8GJCHF8GI.
4/127
B3D9 z′(t),CZCTEHARBHEF:
F9D2 CGC2 z′(t0) negationslash= 0,α < t0 < β,BSD3 z′(t0) C4ERAR
E5H2 (GDBMBF z0) DQ C DNDND9BM z0 = z(t0).
O
y
x
P0
z(t0)
P C
z(t0+?t)
DD FRC4 C H2H0BM P0 DQ P ARDJDMARAKE0E5F0:
z(t0 +△t)?z(t0)
△t,
A8BM P GV C EQE5D9 P0 EE,DJDMARBGDLEGE8FUF0 C
H2 P0 BRARDNDM,BP:
z′(t0) = lim
△t→0
z(t0 +△t)?z(t0)
△t
C4ERARE5H2DQ C DNDND9 z0,DPE0E5DQ C ARAKE5E8E7.
5/127
B3D9 z′(t),CZCTEHARBHEF:
F9D2 CGC2 z′(t0) negationslash= 0,α < t0 < β,BSD3 z′(t0) C4ERAR
E5H2 (GDBMBF z0) DQ C DNDND9BM z0 = z(t0).
O
y
x
P0
z(t0)
P C
z(t0+?t)
DD FRC4 C H2H0BM P0 DQ P ARDJDMARAKE0E5F0:
z(t0 +△t)?z(t0)
△t,
A8BM P GV C EQE5D9 P0 EE,DJDMARBGDLEGE8FUF0 C
H2 P0 BRARDNDM,BP:
z′(t0) = lim
△t→0
z(t0 +△t)?z(t0)
△t
C4ERARE5H2DQ C DNDND9 z0,DPE0E5DQ C ARAKE5E8E7.
5/127
B3D9 z′(t),CZCTEHARBHEF:
F9D2 CGC2 z′(t0) negationslash= 0,α < t0 < β,BSD3 z′(t0) C4ERAR
E5H2 (GDBMBF z0) DQ C DNDND9BM z0 = z(t0).
O
y
x
P0
z(t0)
P C
z(t0+?t)
DD FRC4 C H2H0BM P0 DQ P ARDJDMARAKE0E5F0:
z(t0 +△t)?z(t0)
△t,
A8BM P GV C EQE5D9 P0 EE,DJDMARBGDLEGE8FUF0 C
H2 P0 BRARDNDM,BP:
z′(t0) = lim
△t→0
z(t0 +△t)?z(t0)
△t
C4ERARE5H2DQ C DNDND9 z0,DPE0E5DQ C ARAKE5E8E7.
5/127
B3D9 z′(t),CZCTEHARBHEF:
F9D2 CGC2 z′(t0) negationslash= 0,α < t0 < β,BSD3 z′(t0) C4ERAR
E5H2 (GDBMBF z0) DQ C DNDND9BM z0 = z(t0).
O
y
x
P0
z(t0)
P C
z(t0+?t)
DD FRC4 C H2H0BM P0 DQ P ARDJDMARAKE0E5F0:
z(t0 +△t)?z(t0)
△t,
A8BM P GV C EQE5D9 P0 EE,DJDMARBGDLEGE8FUF0 C
H2 P0 BRARDNDM,BP:
z′(t0) = lim
△t→0
z(t0 +△t)?z(t0)
△t
C4ERARE5H2DQ C DNDND9 z0,DPE0E5DQ C ARAKE5E8E7.
5/127
CGC2BHF9E5H2 z′(t0) ARE0E5DU? C H2BM z0 BRARDNDMAR
AKE5,C3CZ
BHEF 1 Argz′(z0) FUF0BF C H2BM z0 BRARDNDMARAKE5
DQ x F2AKE5DND1ARCH?.
O
y
x
P0
z(t0)
C
θ
z’(t0)
BHEF 2 DNA2D9E8BMARH0CMETDM C1 DQ C2 AKE5DND1AR
CH?,FUF0 C1 DQ C2 BFA2BMBRH0CMDNDMAKE5DND1ARCH?.
y
O x
C2
C1
α
z0
(z)
6/127
CGC2BHF9E5H2 z′(t0) ARE0E5DU? C H2BM z0 BRARDNDMAR
AKE5,C3CZ
BHEF 1 Argz′(z0) FUF0BF C H2BM z0 BRARDNDMARAKE5
DQ x F2AKE5DND1ARCH?.
O
y
x
P0
z(t0)
C
θ
z’(t0)
BHEF 2 DNA2D9E8BMARH0CMETDM C1 DQ C2 AKE5DND1AR
CH?,FUF0 C1 DQ C2 BFA2BMBRH0CMDNDMAKE5DND1ARCH?.
y
O x
C2
C1
α
z0
(z)
6/127
CGC2BHF9E5H2 z′(t0) ARE0E5DU? C H2BM z0 BRARDNDMAR
AKE5,C3CZ
BHEF 1 Argz′(z0) FUF0BF C H2BM z0 BRARDNDMARAKE5
DQ x F2AKE5DND1ARCH?.
O
y
x
P0
z(t0)
C
θ
z’(t0)
BHEF 2 DNA2D9E8BMARH0CMETDM C1 DQ C2 AKE5DND1AR
CH?,FUF0 C1 DQ C2 BFA2BMBRH0CMDNDMAKE5DND1ARCH?.
y
O x
C2
C1
α
z0
(z)
6/127
1,B5G3F8GIARAJGI f′(z) ARBTG3FBFE
1),AJGI f′(z) ARFN? Argf′(z0) ARBTG3FBFE
7/127
1,B5G3F8GIARAJGI f′(z) ARBTG3FBFE
1),AJGI f′(z) ARFN? Argf′(z0) ARBTG3FBFE
7/127
ECBB 1,E8CMETDM.
AN C F0 z EYEHCBFRC4BM z0 ARE8CMCZE5BDDBETDM,G1GJ
GIE0EN:
z = z(t),α lessorequalslant t lessorequalslant β,
AKE5GHBTD9GJGI t C7DWARE0E5,DP z0 = z(t0),z′(t0) negationslash= 0,
α < t0 < β.
C6AJ
w = f(z)
GQETDM C C6AJEK w EYEHE8CMCZE5BDDBETDM Γ,
Γ,w = f[z(t)],α lessorequalslant t lessorequalslant β,
AKE5GHBTD9GJGI t C7DWARE0E5.
z0
y
x O
Arg(z’(t0))
(z)
C
(w) v
u O
Γ
w0
Arg(w’(t0))
8/127
ECBB 1,E8CMETDM.
AN C F0 z EYEHCBFRC4BM z0 ARE8CMCZE5BDDBETDM,G1GJ
GIE0EN:
z = z(t),α lessorequalslant t lessorequalslant β,
AKE5GHBTD9GJGI t C7DWARE0E5,DP z0 = z(t0),z′(t0) negationslash= 0,
α < t0 < β.
C6AJ
w = f(z)
GQETDM C C6AJEK w EYEHE8CMCZE5BDDBETDM Γ,
Γ,w = f[z(t)],α lessorequalslant t lessorequalslant β,
AKE5GHBTD9GJGI t C7DWARE0E5.
z0
y
x O
Arg(z’(t0))
(z)
C
(w) v
u O
Γ
w0
Arg(w’(t0))
8/127
ECBB 1,E8CMETDM.
AN C F0 z EYEHCBFRC4BM z0 ARE8CMCZE5BDDBETDM,G1GJ
GIE0EN:
z = z(t),α lessorequalslant t lessorequalslant β,
AKE5GHBTD9GJGI t C7DWARE0E5,DP z0 = z(t0),z′(t0) negationslash= 0,
α < t0 < β.
C6AJ
w = f(z)
GQETDM C C6AJEK w EYEHE8CMCZE5BDDBETDM Γ,
Γ,w = f[z(t)],α lessorequalslant t lessorequalslant β,
AKE5GHBTD9GJGI t C7DWARE0E5.
z0
y
x O
Arg(z’(t0))
(z)
C
(w) v
u O
Γ
w0
Arg(w’(t0))
8/127
ECBB 1,E8CMETDM.
AN C F0 z EYEHCBFRC4BM z0 ARE8CMCZE5BDDBETDM,G1GJ
GIE0EN:
z = z(t),α lessorequalslant t lessorequalslant β,
AKE5GHBTD9GJGI t C7DWARE0E5,DP z0 = z(t0),z′(t0) negationslash= 0,
α < t0 < β.
C6AJ
w = f(z)
GQETDM C C6AJEK w EYEHE8CMCZE5BDDBETDM Γ,
Γ,w = f[z(t)],α lessorequalslant t lessorequalslant β,
AKE5GHBTD9GJGI t C7DWARE0E5.
z0
y
x O
Arg(z’(t0))
(z)
C
(w) v
u O
Γ
w0
Arg(w’(t0))
8/127
ECBB 1,E8CMETDM.
AN C F0 z EYEHCBFRC4BM z0 ARE8CMCZE5BDDBETDM,G1GJ
GIE0EN:
z = z(t),α lessorequalslant t lessorequalslant β,
AKE5GHBTD9GJGI t C7DWARE0E5,DP z0 = z(t0),z′(t0) negationslash= 0,
α < t0 < β.
C6AJ
w = f(z)
GQETDM C C6AJEK w EYEHE8CMCZE5BDDBETDM Γ,
Γ,w = f[z(t)],α lessorequalslant t lessorequalslant β,
AKE5GHBTD9GJGI t C7DWARE0E5.
z0
y
x O
Arg(z’(t0))
(z)
C
(w) v
u O
Γ
w0
Arg(w’(t0))
8/127
ECBB 1,E8CMETDM.
AN C F0 z EYEHCBFRC4BM z0 ARE8CMCZE5BDDBETDM,G1GJ
GIE0EN:
z = z(t),α lessorequalslant t lessorequalslant β,
AKE5GHBTD9GJGI t C7DWARE0E5,DP z0 = z(t0),z′(t0) negationslash= 0,
α < t0 < β.
C6AJ
w = f(z)
GQETDM C C6AJEK w EYEHE8CMCZE5BDDBETDM Γ,
Γ,w = f[z(t)],α lessorequalslant t lessorequalslant β,
AKE5GHBTD9GJGI t C7DWARE0E5.
z0
y
x O
Arg(z’(t0))
(z)
C
(w) v
u O
Γ
w0
Arg(w’(t0))
8/127
CTEHDJFG Argz′(t0) DQ Argw′(t0) DND1ARB3CG:
DUGABXG4F8GIEMAJDF,CZ
w′(t0) = f′(z0)z′(t0) negationslash= 0,
D9F0
Argw′(t0) = Argf′(z0) + Argz′(t0).
BP
Argw′(t0)?Argz′(t0) = Argf′(z0).
z0
y
x O
Arg(z’(t0))
(z)
C
(w) v
u O
Γ
w0
Arg(w’(t0))
9/127
CTEHDJFG Argz′(t0) DQ Argw′(t0) DND1ARB3CG:
DUGABXG4F8GIEMAJDF,CZ
w′(t0) = f′(z0)z′(t0) negationslash= 0,
D9F0
Argw′(t0) = Argf′(z0) + Argz′(t0).
BP
Argw′(t0)?Argz′(t0) = Argf′(z0).
z0
y
x O
Arg(z’(t0))
(z)
C
(w) v
u O
Γ
w0
Arg(w’(t0))
9/127
B4G0 1:
1) AJGI f′(z0) negationslash= 0 ARFN? Argf′(z0) F0ETDM C EY
C4 w = f(z) C6AJA0BF z0 BRARBPFG?.
2) BPFG?ARDWA2DQE0E5DVETDM C ARBBBZDQE0E5FFB3.
10/127
B4G0 1:
1) AJGI f′(z0) negationslash= 0 ARFN? Argf′(z0) F0ETDM C EY
C4 w = f(z) C6AJA0BF z0 BRARBPFG?.
2) BPFG?ARDWA2DQE0E5DVETDM C ARBBBZDQE0E5FFB3.
10/127
B4G0 1:
1) AJGI f′(z0) negationslash= 0 ARFN? Argf′(z0) F0ETDM C EY
C4 w = f(z) C6AJA0BF z0 BRARBPFG?.
2) BPFG?ARDWA2DQE0E5DVETDM C ARBBBZDQE0E5FFB3.
10/127
ECBB 2,H0CMETDM.
ANETDM C1,C2 DNA2D9BM z0,ENDMARGJGIE0ENEPC7?:
C1,z = z1(t); C2,z = z2(t),α lessorequalslant t lessorequalslant β
FBDP z0 = z1(t0) = z2(t′0),z′1(t0) negationslash= 0,z′2(t′0) negationslash= 0.
y
O x
C2
C1
α
z0
(z) v
O u
α
Γ2
w0 Γ
1
(w)
ANC6AJ w = f(z) GQ C1,C2 EPC7C6AJ?DNA2D9
BM w0 = f(z0) ARH0CMETDM Γ1,Γ2:
Γ1,w = w1(t); Γ2,w = w2(t),α lessorequalslant t lessorequalslant β
11/127
ECBB 2,H0CMETDM.
ANETDM C1,C2 DNA2D9BM z0,ENDMARGJGIE0ENEPC7?:
C1,z = z1(t); C2,z = z2(t),α lessorequalslant t lessorequalslant β
FBDP z0 = z1(t0) = z2(t′0),z′1(t0) negationslash= 0,z′2(t′0) negationslash= 0.
y
O x
C2
C1
α
z0
(z) v
O u
α
Γ2
w0 Γ
1
(w)
ANC6AJ w = f(z) GQ C1,C2 EPC7C6AJ?DNA2D9
BM w0 = f(z0) ARH0CMETDM Γ1,Γ2:
Γ1,w = w1(t); Γ2,w = w2(t),α lessorequalslant t lessorequalslant β
11/127
ECBB 2,H0CMETDM.
ANETDM C1,C2 DNA2D9BM z0,ENDMARGJGIE0ENEPC7?:
C1,z = z1(t); C2,z = z2(t),α lessorequalslant t lessorequalslant β
FBDP z0 = z1(t0) = z2(t′0),z′1(t0) negationslash= 0,z′2(t′0) negationslash= 0.
y
O x
C2
C1
α
z0
(z) v
O u
α
Γ2
w0 Γ
1
(w)
ANC6AJ w = f(z) GQ C1,C2 EPC7C6AJ?DNA2D9
BM w0 = f(z0) ARH0CMETDM Γ1,Γ2:
Γ1,w = w1(t); Γ2,w = w2(t),α lessorequalslant t lessorequalslant β
11/127
CTEHDJFG C1,C2 ARCH?DQ Γ1,Γ2 ARCH?DND1ARB3CG:
FR? w1 = f[z1(t)],w2 = f[z2(t)],DGCMDQECBB 1 DNCP
D5ARGLAJ,CZ
Argw′1(t0)?Argz′1(t0) = Argf′(z0)
Argw′2(t′0)?Argz′2(t′0) = Argf′(z0),
C3
Argw′2(t′0)?Argw′1(t0) = Argz′2(t′0)?Argz′1(t0).
y
O x
C2
C1
α
z0
(z) v
O u
α
Γ2
w0 Γ
1
(w)
12/127
CTEHDJFG C1,C2 ARCH?DQ Γ1,Γ2 ARCH?DND1ARB3CG:
FR? w1 = f[z1(t)],w2 = f[z2(t)],DGCMDQECBB 1 DNCP
D5ARGLAJ,CZ
Argw′1(t0)?Argz′1(t0) = Argf′(z0)
Argw′2(t′0)?Argz′2(t′0) = Argf′(z0),
C3
Argw′2(t′0)?Argw′1(t0) = Argz′2(t′0)?Argz′1(t0).
y
O x
C2
C1
α
z0
(z) v
O u
α
Γ2
w0 Γ
1
(w)
12/127
CTEHDJFG C1,C2 ARCH?DQ Γ1,Γ2 ARCH?DND1ARB3CG:
FR? w1 = f[z1(t)],w2 = f[z2(t)],DGCMDQECBB 1 DNCP
D5ARGLAJ,CZ
Argw′1(t0)?Argz′1(t0) = Argf′(z0)
Argw′2(t′0)?Argz′2(t′0) = Argf′(z0),
C3
Argw′2(t′0)?Argw′1(t0) = Argz′2(t′0)?Argz′1(t0).
y
O x
C2
C1
α
z0
(z) v
O u
α
Γ2
w0 Γ
1
(w)
12/127
B4G0 2:
DNA2D9BM z0 ARBRG3ETDM C1,C2 DND1ARCH?,BFG1DWA2
G2E0E5H2AVFVEYC4 w = f(z) C6AJA0E4ETDM Γ1,Γ2 DND1AR
CH?,(A2?BH)
y
O x
C2
C1
α
z0
(z) v
O u
α
Γ2
w0 Γ
1
(w)
13/127
2),AJGIARAS|f′(z0)|ARBTG3FBFE
AN
z?z0 = reiθ,w?w0 = ρei?,
DPCM△s C4ER C H2ARBM z0 DQ z DND1ARE8GBD0DJ,△σ C4
ER Γ H2ARGHBTBM w0 DQ w DND1ARD0DJ,
O
y
x
P0
z0
P
C
z
(z)?s
r
O
v
u
Q0
w0
Q
w
Γ
σ
ρ
(w)
14/127
2),AJGIARAS|f′(z0)|ARBTG3FBFE
AN
z?z0 = reiθ,w?w0 = ρei?,
DPCM△s C4ER C H2ARBM z0 DQ z DND1ARE8GBD0DJ,△σ C4
ER Γ H2ARGHBTBM w0 DQ w DND1ARD0DJ,
O
y
x
P0
z0
P
C
z
(z)?s
r
O
v
u
Q0
w0
Q
w
Γ
σ
ρ
(w)
14/127
2),AJGIARAS|f′(z0)|ARBTG3FBFE
AN
z?z0 = reiθ,w?w0 = ρei?,
DPCM△s C4ER C H2ARBM z0 DQ z DND1ARE8GBD0DJ,△σ C4
ER Γ H2ARGHBTBM w0 DQ w DND1ARD0DJ,
O
y
x
P0
z0
P
C
z
(z)?s
r
O
v
u
Q0
w0
Q
w
Γ
σ
ρ
(w)
14/127
2),AJGIARAS|f′(z0)|ARBTG3FBFE
AN
z?z0 = reiθ,w?w0 = ρei?,
DPCM△s C4ER C H2ARBM z0 DQ z DND1ARE8GBD0DJ,△σ C4
ER Γ H2ARGHBTBM w0 DQ w DND1ARD0DJ,
O
y
x
P0
z0
P
C
z
(z)?s
r
O
v
u
Q0
w0
Q
w
Γ
σ
ρ
(w)
14/127
DUGA
w?w0
z?z0 =
f(z)?f(z0)
z?z0 =
ρei?
reiθ =
△σ
△s ·
ρ
△σ ·
△s
r e
i(θ)
DWAQ |f′(z0)| = limz→z
0
△σ
△s,
GXDRBGDLDUEH?ETDM C BF z0 ARAREGFJ.
15/127
DUGA
w?w0
z?z0 =
f(z)?f(z0)
z?z0 =
ρei?
reiθ =
△σ
△s ·
ρ
△σ ·
△s
r e
i(θ)
DWAQ |f′(z0)| = limz→z
0
△σ
△s,
GXDRBGDLDUEH?ETDM C BF z0 ARAREGFJ.
15/127
DUGA
w?w0
z?z0 =
f(z)?f(z0)
z?z0 =
ρei?
reiθ =
△σ
△s ·
ρ
△σ ·
△s
r e
i(θ)
DWAQ |f′(z0)| = limz→z
0
△σ
△s,
GXDRBGDLDUEH?ETDM C BF z0 ARAREGFJ.
15/127
B4G0 3:
|f′(z0)|F0EYC4C6AJ w = f(z) A0FRC4BM z0 ARBRG3ET
DM C BF z0 ARAREGFJ,ENDQETDM C ARBBBZBLE0E5FFB3(AR
EGFJARG0BVBH ).
16/127
B4G0 3:
|f′(z0)|F0EYC4C6AJ w = f(z) A0FRC4BM z0 ARBRG3ET
DM C BF z0 ARAREGFJ,ENDQETDM C ARBBBZBLE0E5FFB3(AR
EGFJARG0BVBH ).
16/127
F9D2E8 ANF8GI w = f(z) BFERDX D CBB5G3,z0? D CB
ARE8BM,DP f′(z0) negationslash= 0,BSD3C6AJGCCZH0DRBHEF:
1) A2?BH,BPFRC4 z0 ARH0CMETDMD1ARCH?DVEYC4C6AJ
A0EKAQH0ETDMD1ARCH?BFDWA2G2E0E5H2A2EXG0BV.
2) AREGFJARG0BVBH,BPFRC4 z0 ARBRG3E8CMETDMD1ARAR
EGFJH2?|f
′(z0)|
ADDQG1BBBZG2E0E5FFB3.
17/127
2,A1BBC6AJARCECV
F9FE ANF8GI w = f(z) BF z0 ARAPDXCBF0E8E8AR,BF z0
GCCZA2?BHG2AREGFJARG0BVBH,BSD3EH w = f(z) BF z0
F0A1BBC6AJ(BDE8CPA1BBC6AJ).
CGC2C6AJ w = f(z) BF D CBARDEE8BMFTF0A1BBAR,BSD3
EH w = f(z) F0ERDX D CBARA1BBC6AJ.
18/127
2,A1BBC6AJARCECV
F9FE ANF8GI w = f(z) BF z0 ARAPDXCBF0E8E8AR,BF z0
GCCZA2?BHG2AREGFJARG0BVBH,BSD3EH w = f(z) BF z0
F0A1BBC6AJ(BDE8CPA1BBC6AJ).
CGC2C6AJ w = f(z) BF D CBARDEE8BMFTF0A1BBAR,BSD3
EH w = f(z) F0ERDX D CBARA1BBC6AJ.
18/127
2,A1BBC6AJARCECV
F9FE ANF8GI w = f(z) BF z0 ARAPDXCBF0E8E8AR,BF z0
GCCZA2?BHG2AREGFJARG0BVBH,BSD3EH w = f(z) BF z0
F0A1BBC6AJ(BDE8CPA1BBC6AJ).
CGC2C6AJ w = f(z) BF D CBARDEE8BMFTF0A1BBAR,BSD3
EH w = f(z) F0ERDX D CBARA1BBC6AJ.
18/127
F9D2AJ CGC2F8GI w = f(z) BF z0 B5G3,DP f′(z0) negationslash= 0,
BSD3C6AJ w = f(z) BF z0 F0A1BBAR,DP Argf′(z0) C4ERGX
DRC6AJBF z0 ARBPFG?,|f′(z0)|C4ERAREGFJ.
CGC2B5G3F8GI w = f(z) BF D CBBRBRCZ f′(z) negationslash= 0,BS
D3C6AJ w = f(z) F0 D CBARA1BBC6AJ.
19/127
F9D2AJ CGC2F8GI w = f(z) BF z0 B5G3,DP f′(z0) negationslash= 0,
BSD3C6AJ w = f(z) BF z0 F0A1BBAR,DP Argf′(z0) C4ERGX
DRC6AJBF z0 ARBPFG?,|f′(z0)|C4ERAREGFJ.
CGC2B5G3F8GI w = f(z) BF D CBBRBRCZ f′(z) negationslash= 0,BS
D3C6AJ w = f(z) F0 D CBARA1BBC6AJ.
19/127
DI 1 (1),EM f(z) = z2 BF z = i BRARAREGFJG2BPFG?;
(2),EV f(z) = z2 GQEYC4BM z = i DPEYBDD9EIF2
ARDNDME0E5C6AJEK w EYEHH2ARBSE8DRE0E5?
B5 (1) FR?
f′(z) = 2z,
EKEX:
f′(i) = 2i,
D9F0 f(z) BF z = i F0A1BBAR,DPAREGFJF0:
|f′(i)| = |2i| = 2; BPFG?F0,argf′(i) = arg(2i) = pi2.
20/127
DI 1 (1),EM f(z) = z2 BF z = i BRARAREGFJG2BPFG?;
(2),EV f(z) = z2 GQEYC4BM z = i DPEYBDD9EIF2
ARDNDME0E5C6AJEK w EYEHH2ARBSE8DRE0E5?
B5 (1) FR?
f′(z) = 2z,
EKEX:
f′(i) = 2i,
D9F0 f(z) BF z = i F0A1BBAR,DPAREGFJF0:
|f′(i)| = |2i| = 2; BPFG?F0,argf′(i) = arg(2i) = pi2.
20/127
DI 1 (1),EM f(z) = z2 BF z = i BRARAREGFJG2BPFG?;
(2),EV f(z) = z2 GQEYC4BM z = i DPEYBDD9EIF2
ARDNDME0E5C6AJEK w EYEHH2ARBSE8DRE0E5?
B5 (1) FR?
f′(z) = 2z,
EKEX:
f′(i) = 2i,
D9F0 f(z) BF z = i F0A1BBAR,DPAREGFJF0:
|f′(i)| = |2i| = 2; BPFG?F0,argf′(i) = arg(2i) = pi2.
20/127
DI 1 (1),EM f(z) = z2 BF z = i BRARAREGFJG2BPFG?;
(2),EV f(z) = z2 GQEYC4BM z = i DPEYBDD9EIF2
ARDNDME0E5C6AJEK w EYEHH2ARBSE8DRE0E5?
B5 (1) FR?
f′(z) = 2z,
EKEX:
f′(i) = 2i,
D9F0 f(z) BF z = i F0A1BBAR,DPAREGFJF0:
|f′(i)| = |2i| = 2; BPFG?F0,argf′(i) = arg(2i) = pi2.
20/127
DI 1 (1),EM f(z) = z2 BF z = i BRARAREGFJG2BPFG?;
(2),EV f(z) = z2 GQEYC4BM z = i DPEYBDD9EIF2
ARDNDME0E5C6AJEK w EYEHH2ARBSE8DRE0E5?
B5 (1) FR?
f′(z) = 2z,
EKEX:
f′(i) = 2i,
D9F0 f(z) BF z = i F0A1BBAR,DPAREGFJF0:
|f′(i)| = |2i| = 2; BPFG?F0,argf′(i) = arg(2i) = pi2.
20/127
(2),DUGA (1) ARB4G0,DJ f(z) = z2 GQC4BM z = i DPEY
BDD9EIF2ARDNDME0E5,C6AJEK w EYEHH2ARC4BM?1 DPE0E5
CADQE5H2ARE5H2.
y
O x
i
(z)
v
u O?1
(w)
21/127
DI 2 EVC6AJ f(z) = ln(z?1) GQ z EYEHARBPE8G4EPE8
DWAA,BPE8G4EPEGA2AA.
B5 FR?
f′(z) = 1z?1,
EKEX
|f′(z)| =
vextendsinglevextendsingle
vextendsinglevextendsingle 1
z?1
vextendsinglevextendsingle
vextendsinglevextendsingle= 1radicalbig
(x?1)2 + y2,
D9F0,A8|f′(z)| < 1 EE,BPBFERDX (x?1)2 + y2 > 1
EE,GBBBEGA2AA;
A8|f′(z)| > 1 EE,BPBFERDX (x?1)2 + y2 < 1
EE,GBBBE8DWAA.
22/127
DI 2 EVC6AJ f(z) = ln(z?1) GQ z EYEHARBPE8G4EPE8
DWAA,BPE8G4EPEGA2AA.
B5 FR?
f′(z) = 1z?1,
EKEX
|f′(z)| =
vextendsinglevextendsingle
vextendsinglevextendsingle 1
z?1
vextendsinglevextendsingle
vextendsinglevextendsingle= 1radicalbig
(x?1)2 + y2,
D9F0,A8|f′(z)| < 1 EE,BPBFERDX (x?1)2 + y2 > 1
EE,GBBBEGA2AA;
A8|f′(z)| > 1 EE,BPBFERDX (x?1)2 + y2 < 1
EE,GBBBE8DWAA.
22/127
DI 2 EVC6AJ f(z) = ln(z?1) GQ z EYEHARBPE8G4EPE8
DWAA,BPE8G4EPEGA2AA.
B5 FR?
f′(z) = 1z?1,
EKEX
|f′(z)| =
vextendsinglevextendsingle
vextendsinglevextendsingle 1
z?1
vextendsinglevextendsingle
vextendsinglevextendsingle= 1radicalbig
(x?1)2 + y2,
D9F0,A8|f′(z)| < 1 EE,BPBFERDX (x?1)2 + y2 > 1
EE,GBBBEGA2AA;
A8|f′(z)| > 1 EE,BPBFERDX (x?1)2 + y2 < 1
EE,GBBBE8DWAA.
22/127
DI 2 EVC6AJ f(z) = ln(z?1) GQ z EYEHARBPE8G4EPE8
DWAA,BPE8G4EPEGA2AA.
B5 FR?
f′(z) = 1z?1,
EKEX
|f′(z)| =
vextendsinglevextendsingle
vextendsinglevextendsingle 1
z?1
vextendsinglevextendsingle
vextendsinglevextendsingle= 1radicalbig
(x?1)2 + y2,
D9F0,A8|f′(z)| < 1 EE,BPBFERDX (x?1)2 + y2 > 1
EE,GBBBEGA2AA;
A8|f′(z)| > 1 EE,BPBFERDX (x?1)2 + y2 < 1
EE,GBBBE8DWAA.
22/127
DI 2 EVC6AJ f(z) = ln(z?1) GQ z EYEHARBPE8G4EPE8
DWAA,BPE8G4EPEGA2AA.
B5 FR?
f′(z) = 1z?1,
EKEX
|f′(z)| =
vextendsinglevextendsingle
vextendsinglevextendsingle 1
z?1
vextendsinglevextendsingle
vextendsinglevextendsingle= 1radicalbig
(x?1)2 + y2,
D9F0,A8|f′(z)| < 1 EE,BPBFERDX (x?1)2 + y2 > 1
EE,GBBBEGA2AA;
A8|f′(z)| > 1 EE,BPBFERDX (x?1)2 + y2 < 1
EE,GBBBE8DWAA.
22/127
DI 3 ERDX Re(z) > 0 BFC6AJ w = iz + i CTC6AJEKEFD3
ERDX?
B5 C6AJ w = iz + i = epi2z + i F0
z1 = epi2z,w = z1 + i
ARBXG4C6AJ.
23/127
DI 3 ERDX Re(z) > 0 BFC6AJ w = iz + i CTC6AJEKEFD3
ERDX?
B5 C6AJ w = iz + i = epi2z + i F0
z1 = epi2z,w = z1 + i
ARBXG4C6AJ.
23/127
DI 3 ERDX Re(z) > 0 BFC6AJ w = iz + i CTC6AJEKEFD3
ERDX?
B5 C6AJ w = iz + i = epi2z + i F0
z1 = epi2z,w = z1 + i
ARBXG4C6AJ.
23/127
CMBPC6AJ z1 = epi2z D6ERDX Re(z) > 0 BV?ER
DX Im(z1) > 0;
EYEJC6AJ w = z1 + i D6ERDX Im(z1) > 0 BV?ER
DX Im(w) > 1.
O
y
x
Re(z)>0
(z) y
1
x1
(z1)
O
v
u
(w)
i
O
D2H2EKGB,ERDX Re(z) > 0 EYC4C6AJ w = iz + i A0BV?
ERDX Im(w) > 1.
24/127
CMBPC6AJ z1 = epi2z D6ERDX Re(z) > 0 BV?ER
DX Im(z1) > 0;
EYEJC6AJ w = z1 + i D6ERDX Im(z1) > 0 BV?ER
DX Im(w) > 1.
O
y
x
Re(z)>0
(z) y
1
x1
(z1)
O
v
u
(w)
i
O
D2H2EKGB,ERDX Re(z) > 0 EYC4C6AJ w = iz + i A0BV?
ERDX Im(w) > 1.
24/127
DI 4 ERDX 0 < Im(z) < 12 BFC6AJ w = 1z CTC6AJEKEF
D3ERDX?
B5 AN z = x + iy,w = u + iv,CS w = 1z DJ
x + iy = z = 1w = 1u + iv,
EKEXx = uu2 + v2,y =?vu2 + v2
D9F0ERDX0 < Im(z) < 12,BP0 < y < 12,DNA8D9
0 <?vu2 + v2 < 12 v < 0DPu2 + (v + 1)2 > 1
Im(w) < 0DP|w + i|2 > 0
EKEXC6AJGQw = 1zGQERDX0 < Im(z) < 12C6AJEKERDX:
Im(w) < 0 DP|w + i|2 > 0.
25/127
DI 4 ERDX 0 < Im(z) < 12 BFC6AJ w = 1z CTC6AJEKEF
D3ERDX?
B5 AN z = x + iy,w = u + iv,CS w = 1z DJ
x + iy = z = 1w = 1u + iv,
EKEXx = uu2 + v2,y =?vu2 + v2
D9F0ERDX0 < Im(z) < 12,BP0 < y < 12,DNA8D9
0 <?vu2 + v2 < 12 v < 0DPu2 + (v + 1)2 > 1
Im(w) < 0DP|w + i|2 > 0
EKEXC6AJGQw = 1zGQERDX0 < Im(z) < 12C6AJEKERDX:
Im(w) < 0 DP|w + i|2 > 0.
25/127
DI 4 ERDX 0 < Im(z) < 12 BFC6AJ w = 1z CTC6AJEKEF
D3ERDX?
B5 AN z = x + iy,w = u + iv,CS w = 1z DJ
x + iy = z = 1w = 1u + iv,
EKEXx = uu2 + v2,y =?vu2 + v2
D9F0ERDX0 < Im(z) < 12,BP0 < y < 12,DNA8D9
0 <?vu2 + v2 < 12 v < 0DPu2 + (v + 1)2 > 1
Im(w) < 0DP|w + i|2 > 0
EKEXC6AJGQw = 1zGQERDX0 < Im(z) < 12C6AJEKERDX:
Im(w) < 0 DP|w + i|2 > 0.
25/127
DI 4 ERDX 0 < Im(z) < 12 BFC6AJ w = 1z CTC6AJEKEF
D3ERDX?
B5 AN z = x + iy,w = u + iv,CS w = 1z DJ
x + iy = z = 1w = 1u + iv,
EKEXx = uu2 + v2,y =?vu2 + v2
D9F0ERDX0 < Im(z) < 12,BP0 < y < 12,DNA8D9
0 <?vu2 + v2 < 12 v < 0DPu2 + (v + 1)2 > 1
Im(w) < 0DP|w + i|2 > 0
EKEXC6AJGQw = 1zGQERDX0 < Im(z) < 12C6AJEKERDX:
Im(w) < 0 DP|w + i|2 > 0.
25/127
DI 4 ERDX 0 < Im(z) < 12 BFC6AJ w = 1z CTC6AJEKEF
D3ERDX?
B5 AN z = x + iy,w = u + iv,CS w = 1z DJ
x + iy = z = 1w = 1u + iv,
EKEXx = uu2 + v2,y =?vu2 + v2
D9F0ERDX0 < Im(z) < 12,BP0 < y < 12,DNA8D9
0 <?vu2 + v2 < 12 v < 0DPu2 + (v + 1)2 > 1
Im(w) < 0DP|w + i|2 > 0
EKEXC6AJGQw = 1zGQERDX0 < Im(z) < 12C6AJEKERDX:
Im(w) < 0 DP|w + i|2 > 0.
25/127
DI 4 ERDX 0 < Im(z) < 12 BFC6AJ w = 1z CTC6AJEKEF
D3ERDX?
B5 AN z = x + iy,w = u + iv,CS w = 1z DJ
x + iy = z = 1w = 1u + iv,
EKEXx = uu2 + v2,y =?vu2 + v2
D9F0ERDX0 < Im(z) < 12,BP0 < y < 12,DNA8D9
0 <?vu2 + v2 < 12 v < 0DPu2 + (v + 1)2 > 1
Im(w) < 0DP|w + i|2 > 0
EKEXC6AJGQw = 1zGQERDX0 < Im(z) < 12C6AJEKERDX:
Im(w) < 0 DP|w + i|2 > 0.
25/127
DI 4 ERDX 0 < Im(z) < 12 BFC6AJ w = 1z CTC6AJEKEF
D3ERDX?
B5 AN z = x + iy,w = u + iv,CS w = 1z DJ
x + iy = z = 1w = 1u + iv,
EKEXx = uu2 + v2,y =?vu2 + v2
D9F0ERDX0 < Im(z) < 12,BP0 < y < 12,DNA8D9
0 <?vu2 + v2 < 12 v < 0DPu2 + (v + 1)2 > 1
Im(w) < 0DP|w + i|2 > 0
EKEXC6AJGQw = 1zGQERDX0 < Im(z) < 12C6AJEKERDX:
Im(w) < 0 DP|w + i|2 > 0.
25/127
DI 4 ERDX 0 < Im(z) < 12 BFC6AJ w = 1z CTC6AJEKEF
D3ERDX?
B5 AN z = x + iy,w = u + iv,CS w = 1z DJ
x + iy = z = 1w = 1u + iv,
EKEXx = uu2 + v2,y =?vu2 + v2
D9F0ERDX0 < Im(z) < 12,BP0 < y < 12,DNA8D9
0 <?vu2 + v2 < 12 v < 0DPu2 + (v + 1)2 > 1
Im(w) < 0DP|w + i|2 > 0
EKEXC6AJGQw = 1zGQERDX0 < Im(z) < 12C6AJEKERDX:
Im(w) < 0 DP|w + i|2 > 0.
25/127
DI 4 ERDX 0 < Im(z) < 12 BFC6AJ w = 1z CTC6AJEKEF
D3ERDX?
B5 AN z = x + iy,w = u + iv,CS w = 1z DJ
x + iy = z = 1w = 1u + iv,
EKEXx = uu2 + v2,y =?vu2 + v2
D9F0ERDX0 < Im(z) < 12,BP0 < y < 12,DNA8D9
0 <?vu2 + v2 < 12 v < 0DPu2 + (v + 1)2 > 1
Im(w) < 0DP|w + i|2 > 0
EKEXC6AJGQw = 1zGQERDX0 < Im(z) < 12C6AJEKERDX:
Im(w) < 0 DP|w + i|2 > 0.
25/127
DI 4 ERDX 0 < Im(z) < 12 BFC6AJ w = 1z CTC6AJEKEF
D3ERDX?
B5 AN z = x + iy,w = u + iv,CS w = 1z DJ
x + iy = z = 1w = 1u + iv,
EKEXx = uu2 + v2,y =?vu2 + v2
D9F0ERDX0 < Im(z) < 12,BP0 < y < 12,DNA8D9
0 <?vu2 + v2 < 12 v < 0DPu2 + (v + 1)2 > 1
Im(w) < 0DP|w + i|2 > 0
EKEXC6AJGQw = 1zGQERDX0 < Im(z) < 12C6AJEKERDX:
Im(w) < 0 DP|w + i|2 > 0.
25/127
DI 5,DDEY,C6AJ w = z + 1z D6ABEX|z| = c C6AJEKH1
AB
u =
parenleftbigg
c + 1c
parenrightbigg
cosθ,v =
parenleftbigg
c? 1c
parenrightbigg
sinθ.
B5 ABEX|z| = c ARGJGIE0EN,z = ceiθ,0 lessorequalslant θ lessorequalslant 2pi.
ABEXBFC6AJ w = z + 1z DNCTE4ETDMARGJGIE0EN?
w = ceiθ + 1ceiθ =
parenleftbigg
c + 1c
parenrightbigg
cosθ + i
parenleftbigg
c? 1c
parenrightbigg
sinθ (0 lessorequalslant θ lessorequalslant 2pi).
BPE4ETDMF0H1AB
u =
parenleftbigg
c + 1c
parenrightbigg
cosθ,v =
parenleftbigg
c? 1c
parenrightbigg
sinθ.
26/127
DI 5,DDEY,C6AJ w = z + 1z D6ABEX|z| = c C6AJEKH1
AB
u =
parenleftbigg
c + 1c
parenrightbigg
cosθ,v =
parenleftbigg
c? 1c
parenrightbigg
sinθ.
B5 ABEX|z| = c ARGJGIE0EN,z = ceiθ,0 lessorequalslant θ lessorequalslant 2pi.
ABEXBFC6AJ w = z + 1z DNCTE4ETDMARGJGIE0EN?
w = ceiθ + 1ceiθ =
parenleftbigg
c + 1c
parenrightbigg
cosθ + i
parenleftbigg
c? 1c
parenrightbigg
sinθ (0 lessorequalslant θ lessorequalslant 2pi).
BPE4ETDMF0H1AB
u =
parenleftbigg
c + 1c
parenrightbigg
cosθ,v =
parenleftbigg
c? 1c
parenrightbigg
sinθ.
26/127
DI 5,DDEY,C6AJ w = z + 1z D6ABEX|z| = c C6AJEKH1
AB
u =
parenleftbigg
c + 1c
parenrightbigg
cosθ,v =
parenleftbigg
c? 1c
parenrightbigg
sinθ.
B5 ABEX|z| = c ARGJGIE0EN,z = ceiθ,0 lessorequalslant θ lessorequalslant 2pi.
ABEXBFC6AJ w = z + 1z DNCTE4ETDMARGJGIE0EN?
w = ceiθ + 1ceiθ =
parenleftbigg
c + 1c
parenrightbigg
cosθ + i
parenleftbigg
c? 1c
parenrightbigg
sinθ (0 lessorequalslant θ lessorequalslant 2pi).
BPE4ETDMF0H1AB
u =
parenleftbigg
c + 1c
parenrightbigg
cosθ,v =
parenleftbigg
c? 1c
parenrightbigg
sinθ.
26/127
DI 5,DDEY,C6AJ w = z + 1z D6ABEX|z| = c C6AJEKH1
AB
u =
parenleftbigg
c + 1c
parenrightbigg
cosθ,v =
parenleftbigg
c? 1c
parenrightbigg
sinθ.
B5 ABEX|z| = c ARGJGIE0EN,z = ceiθ,0 lessorequalslant θ lessorequalslant 2pi.
ABEXBFC6AJ w = z + 1z DNCTE4ETDMARGJGIE0EN?
w = ceiθ + 1ceiθ =
parenleftbigg
c + 1c
parenrightbigg
cosθ + i
parenleftbigg
c? 1c
parenrightbigg
sinθ (0 lessorequalslant θ lessorequalslant 2pi).
BPE4ETDMF0H1AB
u =
parenleftbigg
c + 1c
parenrightbigg
cosθ,v =
parenleftbigg
c? 1c
parenrightbigg
sinθ.
26/127
DI 6,DDEY,BFC6AJ w = eiz CT,D4DNAKA2ARDQDM
DC Re(z) = c1,Im(z) = c2 EDCVC6AJEKD4DNAKA2ARETDM
DC v = utanc1 G2ABEXDC u2 + v2 = e?c2.
DDCZBRG0DQDMDC Re(z) = c1,Im(z) = c2 ARE4.
AN z = x + iy,w = u + iv.
(i) DQDM Re(z) = c1 ARGJGIE0EN?,z = c1 + iy.
BFC6AJ w = eiz CT,G1E4ETDMARGJGIE0EN?
w = eiz = ei(c1+iy) = e?y ·eic1,
BP
u = e?ycosc1,v = e?ysinc1,
BP v = utanc1.
27/127
DI 6,DDEY,BFC6AJ w = eiz CT,D4DNAKA2ARDQDM
DC Re(z) = c1,Im(z) = c2 EDCVC6AJEKD4DNAKA2ARETDM
DC v = utanc1 G2ABEXDC u2 + v2 = e?c2.
DDCZBRG0DQDMDC Re(z) = c1,Im(z) = c2 ARE4.
AN z = x + iy,w = u + iv.
(i) DQDM Re(z) = c1 ARGJGIE0EN?,z = c1 + iy.
BFC6AJ w = eiz CT,G1E4ETDMARGJGIE0EN?
w = eiz = ei(c1+iy) = e?y ·eic1,
BP
u = e?ycosc1,v = e?ysinc1,
BP v = utanc1.
27/127
DI 6,DDEY,BFC6AJ w = eiz CT,D4DNAKA2ARDQDM
DC Re(z) = c1,Im(z) = c2 EDCVC6AJEKD4DNAKA2ARETDM
DC v = utanc1 G2ABEXDC u2 + v2 = e?c2.
DDCZBRG0DQDMDC Re(z) = c1,Im(z) = c2 ARE4.
AN z = x + iy,w = u + iv.
(i) DQDM Re(z) = c1 ARGJGIE0EN?,z = c1 + iy.
BFC6AJ w = eiz CT,G1E4ETDMARGJGIE0EN?
w = eiz = ei(c1+iy) = e?y ·eic1,
BP
u = e?ycosc1,v = e?ysinc1,
BP v = utanc1.
27/127
DI 6,DDEY,BFC6AJ w = eiz CT,D4DNAKA2ARDQDM
DC Re(z) = c1,Im(z) = c2 EDCVC6AJEKD4DNAKA2ARETDM
DC v = utanc1 G2ABEXDC u2 + v2 = e?c2.
DDCZBRG0DQDMDC Re(z) = c1,Im(z) = c2 ARE4.
AN z = x + iy,w = u + iv.
(i) DQDM Re(z) = c1 ARGJGIE0EN?,z = c1 + iy.
BFC6AJ w = eiz CT,G1E4ETDMARGJGIE0EN?
w = eiz = ei(c1+iy) = e?y ·eic1,
BP
u = e?ycosc1,v = e?ysinc1,
BP v = utanc1.
27/127
DI 6,DDEY,BFC6AJ w = eiz CT,D4DNAKA2ARDQDM
DC Re(z) = c1,Im(z) = c2 EDCVC6AJEKD4DNAKA2ARETDM
DC v = utanc1 G2ABEXDC u2 + v2 = e?c2.
DDCZBRG0DQDMDC Re(z) = c1,Im(z) = c2 ARE4.
AN z = x + iy,w = u + iv.
(i) DQDM Re(z) = c1 ARGJGIE0EN?,z = c1 + iy.
BFC6AJ w = eiz CT,G1E4ETDMARGJGIE0EN?
w = eiz = ei(c1+iy) = e?y ·eic1,
BP
u = e?ycosc1,v = e?ysinc1,
BP v = utanc1.
27/127
DI 6,DDEY,BFC6AJ w = eiz CT,D4DNAKA2ARDQDM
DC Re(z) = c1,Im(z) = c2 EDCVC6AJEKD4DNAKA2ARETDM
DC v = utanc1 G2ABEXDC u2 + v2 = e?c2.
DDCZBRG0DQDMDC Re(z) = c1,Im(z) = c2 ARE4.
AN z = x + iy,w = u + iv.
(i) DQDM Re(z) = c1 ARGJGIE0EN?,z = c1 + iy.
BFC6AJ w = eiz CT,G1E4ETDMARGJGIE0EN?
w = eiz = ei(c1+iy) = e?y ·eic1,
BP
u = e?ycosc1,v = e?ysinc1,
BP v = utanc1.
27/127
DI 6,DDEY,BFC6AJ w = eiz CT,D4DNAKA2ARDQDM
DC Re(z) = c1,Im(z) = c2 EDCVC6AJEKD4DNAKA2ARETDM
DC v = utanc1 G2ABEXDC u2 + v2 = e?c2.
DDCZBRG0DQDMDC Re(z) = c1,Im(z) = c2 ARE4.
AN z = x + iy,w = u + iv.
(i) DQDM Re(z) = c1 ARGJGIE0EN?,z = c1 + iy.
BFC6AJ w = eiz CT,G1E4ETDMARGJGIE0EN?
w = eiz = ei(c1+iy) = e?y ·eic1,
BP
u = e?ycosc1,v = e?ysinc1,
BP v = utanc1.
27/127
(ii) DQDM Im(z) = c2 ARGJGIE0EN,z = x + ic2.
BFC6AJ w = eiz CT,G1E4ETDMARGJGIE0EN?
w = eiz = ei(x+ic2) = e?c2 ·eix,
BP
u = e?c2cosx,v = e?c2sinx,
BP u2 + v2 = e?2c2.
28/127
(ii) DQDM Im(z) = c2 ARGJGIE0EN,z = x + ic2.
BFC6AJ w = eiz CT,G1E4ETDMARGJGIE0EN?
w = eiz = ei(x+ic2) = e?c2 ·eix,
BP
u = e?c2cosx,v = e?c2sinx,
BP u2 + v2 = e?2c2.
28/127
(ii) DQDM Im(z) = c2 ARGJGIE0EN,z = x + ic2.
BFC6AJ w = eiz CT,G1E4ETDMARGJGIE0EN?
w = eiz = ei(x+ic2) = e?c2 ·eix,
BP
u = e?c2cosx,v = e?c2sinx,
BP u2 + v2 = e?2c2.
28/127
(ii) DQDM Im(z) = c2 ARGJGIE0EN,z = x + ic2.
BFC6AJ w = eiz CT,G1E4ETDMARGJGIE0EN?
w = eiz = ei(x+ic2) = e?c2 ·eix,
BP
u = e?c2cosx,v = e?c2sinx,
BP u2 + v2 = e?2c2.
28/127
BEDDETDMDC v = utanc1 G2ABEXDC u2 +v2 = e?c2 AKA2.
FR?
dw
dz = ie
iz negationslash= 0
DP eiz BFBXEYEHH2B5G3,EKEXC6AJ w = eiz BFBXEYEHH2GC
CZA2?BH.
FR?DQDMDC Re(z) = c1 DQ Im(z) = c2 D4DNAKA2,APE4
ETDMDC v = utanc1 G2ABEXDC u2 + v2 = e?c2 D4DNAKA2.
29/127
BEDDETDMDC v = utanc1 G2ABEXDC u2 +v2 = e?c2 AKA2.
FR?
dw
dz = ie
iz negationslash= 0
DP eiz BFBXEYEHH2B5G3,EKEXC6AJ w = eiz BFBXEYEHH2GC
CZA2?BH.
FR?DQDMDC Re(z) = c1 DQ Im(z) = c2 D4DNAKA2,APE4
ETDMDC v = utanc1 G2ABEXDC u2 + v2 = e?c2 D4DNAKA2.
29/127
BEDDETDMDC v = utanc1 G2ABEXDC u2 +v2 = e?c2 AKA2.
FR?
dw
dz = ie
iz negationslash= 0
DP eiz BFBXEYEHH2B5G3,EKEXC6AJ w = eiz BFBXEYEHH2GC
CZA2?BH.
FR?DQDMDC Re(z) = c1 DQ Im(z) = c2 D4DNAKA2,APE4
ETDMDC v = utanc1 G2ABEXDC u2 + v2 = e?c2 D4DNAKA2.
29/127
BEDDETDMDC v = utanc1 G2ABEXDC u2 +v2 = e?c2 AKA2.
FR?
dw
dz = ie
iz negationslash= 0
DP eiz BFBXEYEHH2B5G3,EKEXC6AJ w = eiz BFBXEYEHH2GC
CZA2?BH.
FR?DQDMDC Re(z) = c1 DQ Im(z) = c2 D4DNAKA2,APE4
ETDMDC v = utanc1 G2ABEXDC u2 + v2 = e?c2 D4DNAKA2.
29/127
BEDDETDMDC v = utanc1 G2ABEXDC u2 +v2 = e?c2 AKA2.
FR?
dw
dz = ie
iz negationslash= 0
DP eiz BFBXEYEHH2B5G3,EKEXC6AJ w = eiz BFBXEYEHH2GC
CZA2?BH.
FR?DQDMDC Re(z) = c1 DQ Im(z) = c2 D4DNAKA2,APE4
ETDMDC v = utanc1 G2ABEXDC u2 + v2 = e?c2 D4DNAKA2.
29/127
BEDDETDMDC v = utanc1 G2ABEXDC u2 +v2 = e?c2 AKA2.
FR?
dw
dz = ie
iz negationslash= 0
DP eiz BFBXEYEHH2B5G3,EKEXC6AJ w = eiz BFBXEYEHH2GC
CZA2?BH.
FR?DQDMDC Re(z) = c1 DQ Im(z) = c2 D4DNAKA2,APE4
ETDMDC v = utanc1 G2ABEXDC u2 + v2 = e?c2 D4DNAKA2.
29/127
BDAJCPA1BBC6AJ
CGC2C6AJ w = f(z) GCCZAREGFJARG0BVBH,A2BPA2EXCH
ARH1GHDUG0BVADE0E5DNDR,BSD3EHCCC6AJ?BDAJCPA1BB
C6AJ.
DI w = z F0BDAJCPA1BBC6AJ.
y
O x
z
z
30/127
BDAJCPA1BBC6AJ
CGC2C6AJ w = f(z) GCCZAREGFJARG0BVBH,A2BPA2EXCH
ARH1GHDUG0BVADE0E5DNDR,BSD3EHCCC6AJ?BDAJCPA1BB
C6AJ.
DI w = z F0BDAJCPA1BBC6AJ.
y
O x
z
z
30/127
BDAJCPA1BBC6AJ
CGC2C6AJ w = f(z) GCCZAREGFJARG0BVBH,A2BPA2EXCH
ARH1GHDUG0BVADE0E5DNDR,BSD3EHCCC6AJ?BDAJCPA1BB
C6AJ.
DI w = z F0BDAJCPA1BBC6AJ.
y
O x
z
z
30/127
BDAJAX EPEQDMBHC6AJ
F9FE EPEQDMBHC6AJ (CEDMBHC6AJ):
w = az + bcz + d (ad?bc negationslash= 0)
G1EL,a,b,c,d FTF0DIGI.
BH,FR?
dw
dz =
ad?bc
(cz + d)2
CGC2 ad?bc = 0,C3
dw
dz = 0,ENGQAIDR z EYEHC6AJEK w
EYEHH2ARE8BM.
31/127
BDAJAX EPEQDMBHC6AJ
F9FE EPEQDMBHC6AJ (CEDMBHC6AJ):
w = az + bcz + d (ad?bc negationslash= 0)
G1EL,a,b,c,d FTF0DIGI.
BH,FR?
dw
dz =
ad?bc
(cz + d)2
CGC2 ad?bc = 0,C3
dw
dz = 0,ENGQAIDR z EYEHC6AJEK w
EYEHH2ARE8BM.
31/127
BDAJAX EPEQDMBHC6AJ
F9FE EPEQDMBHC6AJ (CEDMBHC6AJ):
w = az + bcz + d (ad?bc negationslash= 0)
G1EL,a,b,c,d FTF0DIGI.
BH,FR?
dw
dz =
ad?bc
(cz + d)2
CGC2 ad?bc = 0,C3
dw
dz = 0,ENGQAIDR z EYEHC6AJEK w
EYEHH2ARE8BM.
31/127
BDAJAX EPEQDMBHC6AJ
F9FE EPEQDMBHC6AJ (CEDMBHC6AJ):
w = az + bcz + d (ad?bc negationslash= 0)
G1EL,a,b,c,d FTF0DIGI.
BH,FR?
dw
dz =
ad?bc
(cz + d)2
CGC2 ad?bc = 0,C3
dw
dz = 0,ENGQAIDR z EYEHC6AJEK w
EYEHH2ARE8BM.
31/127
CTEHEPG3EPEQDMBHC6AJARBHEF:
FGCZ,D6E8DRE8DOAREPEQDMBHC6AJEPB5E8AADFH0C6AJAR
BXG4:
EPEQDMBHC6AJDWDE?
w = ac + bc?adc2 · 1
z + dc
.
ENDWEPB5?
z1 = z + dc; z2 = 1z
1; w = bc?adc2 z2 + ac.
EKEX,E8DOBBEQAREPEQDMBHC6AJF0CTAFD2ERBTFRC6AJBX
G4ADEK:
1) w = z + b; 2) w = az; 3) w = 1z.
32/127
CTEHEPG3EPEQDMBHC6AJARBHEF:
FGCZ,D6E8DRE8DOAREPEQDMBHC6AJEPB5E8AADFH0C6AJAR
BXG4:
EPEQDMBHC6AJDWDE?
w = ac + bc?adc2 · 1
z + dc
.
ENDWEPB5?
z1 = z + dc; z2 = 1z
1; w = bc?adc2 z2 + ac.
EKEX,E8DOBBEQAREPEQDMBHC6AJF0CTAFD2ERBTFRC6AJBX
G4ADEK:
1) w = z + b; 2) w = az; 3) w = 1z.
32/127
CTEHEPG3EPEQDMBHC6AJARBHEF:
FGCZ,D6E8DRE8DOAREPEQDMBHC6AJEPB5E8AADFH0C6AJAR
BXG4:
EPEQDMBHC6AJDWDE?
w = ac + bc?adc2 · 1
z + dc
.
ENDWEPB5?
z1 = z + dc; z2 = 1z
1; w = bc?adc2 z2 + ac.
EKEX,E8DOBBEQAREPEQDMBHC6AJF0CTAFD2ERBTFRC6AJBX
G4ADEK:
1) w = z + b; 2) w = az; 3) w = 1z.
32/127
CTEHEPG3EPEQDMBHC6AJARBHEF:
FGCZ,D6E8DRE8DOAREPEQDMBHC6AJEPB5E8AADFH0C6AJAR
BXG4:
EPEQDMBHC6AJDWDE?
w = ac + bc?adc2 · 1
z + dc
.
ENDWEPB5?
z1 = z + dc; z2 = 1z
1; w = bc?adc2 z2 + ac.
EKEX,E8DOBBEQAREPEQDMBHC6AJF0CTAFD2ERBTFRC6AJBX
G4ADEK:
1) w = z + b; 2) w = az; 3) w = 1z.
32/127
CTEHEPG3EPEQDMBHC6AJARBHEF:
FGCZ,D6E8DRE8DOAREPEQDMBHC6AJEPB5E8AADFH0C6AJAR
BXG4:
EPEQDMBHC6AJDWDE?
w = ac + bc?adc2 · 1
z + dc
.
ENDWEPB5?
z1 = z + dc; z2 = 1z
1; w = bc?adc2 z2 + ac.
EKEX,E8DOBBEQAREPEQDMBHC6AJF0CTAFD2ERBTFRC6AJBX
G4ADEK:
1) w = z + b; 2) w = az; 3) w = 1z.
32/127
CTEHEPC7BRG0GXD2ERC6AJARBHEF:
1) w = z + b,EYEJC6AJ.
2) w = az(a negationslash= 0),CMBPDQARDJ (F9EGG9) C6AJ.
O
y
x
b
z
w
(z)=(w)
O
y
x
(z)=(w)
w
z α
3) w = 1z,GXDRC6AJDWEXEPB5?
w1 = 1z; w = w1.
33/127
CTEHEPC7BRG0GXD2ERC6AJARBHEF:
1) w = z + b,EYEJC6AJ.
2) w = az(a negationslash= 0),CMBPDQARDJ (F9EGG9) C6AJ.
O
y
x
b
z
w
(z)=(w)
O
y
x
(z)=(w)
w
z α
3) w = 1z,GXDRC6AJDWEXEPB5?
w1 = 1z; w = w1.
33/127
CTEHEPC7BRG0GXD2ERC6AJARBHEF:
1) w = z + b,EYEJC6AJ.
2) w = az(a negationslash= 0),CMBPDQARDJ (F9EGG9) C6AJ.
O
y
x
b
z
w
(z)=(w)
O
y
x
(z)=(w)
w
z α
3) w = 1z,GXDRC6AJDWEXEPB5?
w1 = 1z; w = w1.
33/127
CTEHEPC7BRG0GXD2ERC6AJARBHEF:
1) w = z + b,EYEJC6AJ.
2) w = az(a negationslash= 0),CMBPDQARDJ (F9EGG9) C6AJ.
O
y
x
b
z
w
(z)=(w)
O
y
x
(z)=(w)
w
z α
3) w = 1z,GXDRC6AJDWEXEPB5?
w1 = 1z; w = w1.
33/127
(i) GHD9 w = w1,CZ
y
O x
w1
w=w1
EKEX,w1 DQ w F0B3D9EIF2ARGHEHBM.
34/127
(ii) GHD9 w1 = 1z:
F9FE,AN C?EXA6BM?ELB1,r?DWF7ARABEX,BFEXABB1
GDBMARE8CMDQDMH2,CGC2CZH0BM P DQ P′ GVDAB3CGEQ
OP ·OP′ = r2,
BSD3EHGXH0BM?B3D9GXABEXARGHEHBM.
GHEHBMARDCDF,AN P BF C A9,D0 P DUABEX C ARDN
DM PT,CS T DU OP ARCADM TP
′,
DQ OP A2D9 P′,BS
D3 P DQ P′ D4?GHEHBM.
C T
O P’ P
35/127
(ii) GHD9 w1 = 1z:
F9FE,AN C?EXA6BM?ELB1,r?DWF7ARABEX,BFEXABB1
GDBMARE8CMDQDMH2,CGC2CZH0BM P DQ P′ GVDAB3CGEQ
OP ·OP′ = r2,
BSD3EHGXH0BM?B3D9GXABEXARGHEHBM.
GHEHBMARDCDF,AN P BF C A9,D0 P DUABEX C ARDN
DM PT,CS T DU OP ARCADM TP
′,
DQ OP A2D9 P′,BS
D3 P DQ P′ D4?GHEHBM.
C T
O P’ P
35/127
(ii) GHD9 w1 = 1z:
F9FE,AN C?EXA6BM?ELB1,r?DWF7ARABEX,BFEXABB1
GDBMARE8CMDQDMH2,CGC2CZH0BM P DQ P′ GVDAB3CGEQ
OP ·OP′ = r2,
BSD3EHGXH0BM?B3D9GXABEXARGHEHBM.
GHEHBMARDCDF,AN P BF C A9,D0 P DUABEX C ARDN
DM PT,CS T DU OP ARCADM TP
′,
DQ OP A2D9 P′,BS
D3 P DQ P′ D4?GHEHBM.
C T
O P’ P
35/127
D9F0,CGC2AN z = reiθ,BSD3
w1 = 1z = 1reiθ,
D0AD|w1||z| = 1,DP w1 DQ z BFFVE8CMAJDMH2.
w1
O
z
CSCSDWDJ,z DQ w1 F0B3D9H0EGABEX|z| = 1 ARGHEHBM.
36/127
D9F0,CGC2AN z = reiθ,BSD3
w1 = 1z = 1reiθ,
D0AD|w1||z| = 1,DP w1 DQ z BFFVE8CMAJDMH2.
w1
O
z
CSCSDWDJ,z DQ w1 F0B3D9H0EGABEX|z| = 1 ARGHEHBM.
36/127
D2H2EKGB,GHD9 w = 1z:
(1) FGCZ,DUFTBM z B3D9H0EGABEXGHEHARBM w1;
w1
w
O
z
(2) B8A0,BEDUFTBM w1 B3D9EIF2GHEHARBM.
37/127
D2H2EKGB,GHD9 w = 1z:
(1) FGCZ,DUFTBM z B3D9H0EGABEXGHEHARBM w1;
w1
w
O
z
(2) B8A0,BEDUFTBM w1 B3D9EIF2GHEHARBM.
37/127
EPEQDMBHC6AJARBHEF
E0DF,FR?EPEQDMBHC6AJDWEXEPB5EKCTEHH0ERC6AJAR
BXG4,
w = 1z,w = az + b,
EKEXDWEXFRC4EPG3GXH0ERC6AJARBHEF,AQAKEPEQDMBHC6AJ
ARBHEF.
38/127
EPEQDMBHC6AJARBHEF
E0DF,FR?EPEQDMBHC6AJDWEXEPB5EKCTEHH0ERC6AJAR
BXG4,
w = 1z,w = az + b,
EKEXDWEXFRC4EPG3GXH0ERC6AJARBHEF,AQAKEPEQDMBHC6AJ
ARBHEF.
38/127
1,A2?BH
B4G0 1,C6AJ w = 1z BFB6FBBXEYEHH2F0BRBRA1BBAR.
DD FGCZDDEY,C6AJ w = 1z BFB6FBBXEYEHCBF0E8E8GHBT
AR.
(F0DD).
39/127
1,A2?BH
B4G0 1,C6AJ w = 1z BFB6FBBXEYEHH2F0BRBRA1BBAR.
DD FGCZDDEY,C6AJ w = 1z BFB6FBBXEYEHCBF0E8E8GHBT
AR.
(F0DD).
39/127
1,A2?BH
B4G0 1,C6AJ w = 1z BFB6FBBXEYEHH2F0BRBRA1BBAR.
DD FGCZDDEY,C6AJ w = 1z BFB6FBBXEYEHCBF0E8E8GHBT
AR.
(F0DD).
39/127
B8A0DDEY,C6AJ w = 1z BFB6FBBXEYEHH2F0BRBRA1BBAR:
(i) A8 z negationslash= 0,z negationslash= ∞EE,FR?
w′ =
parenleftbigg1
z
parenrightbigg′
=? 1z2 negationslash= 0,
EKEXG0AM z = 0,z = ∞,C6AJ w = 1z F0A1BBAR.
40/127
B8A0DDEY,C6AJ w = 1z BFB6FBBXEYEHH2F0BRBRA1BBAR:
(i) A8 z negationslash= 0,z negationslash= ∞EE,FR?
w′ =
parenleftbigg1
z
parenrightbigg′
=? 1z2 negationslash= 0,
EKEXG0AM z = 0,z = ∞,C6AJ w = 1z F0A1BBAR.
40/127
(ii) A8 z = 0,z = ∞EE.
CGC2BHF9,H0CMARE5FFEHAFBMARETDMBFFFEHBM∞BRAR
CH?,AVD9ENDMBFC6AJ w = 1z CTC6EKARFRC4A6BM ξ = 0
ARH0CME4ETDMARCH?,
BSD3C6AJ w = 1z = ξ BF ξ = 0 BRB5G3,DP
w′(ξ)|ξ=0 = 1 negationslash= 0,
EKEXC6AJ w = ξ BF ξ = 0 BR,BPC6AJ w = 1z BF z = ∞
F0A1BBAR.
CPD5AR,BF z = 0 BRC6AJ w = 1z?F0A1BBAR.
41/127
(ii) A8 z = 0,z = ∞EE.
CGC2BHF9,H0CMARE5FFEHAFBMARETDMBFFFEHBM∞BRAR
CH?,AVD9ENDMBFC6AJ w = 1z CTC6EKARFRC4A6BM ξ = 0
ARH0CME4ETDMARCH?,
BSD3C6AJ w = 1z = ξ BF ξ = 0 BRB5G3,DP
w′(ξ)|ξ=0 = 1 negationslash= 0,
EKEXC6AJ w = ξ BF ξ = 0 BR,BPC6AJ w = 1z BF z = ∞
F0A1BBAR.
CPD5AR,BF z = 0 BRC6AJ w = 1z?F0A1BBAR.
41/127
(ii) A8 z = 0,z = ∞EE.
CGC2BHF9,H0CMARE5FFEHAFBMARETDMBFFFEHBM∞BRAR
CH?,AVD9ENDMBFC6AJ w = 1z CTC6EKARFRC4A6BM ξ = 0
ARH0CME4ETDMARCH?,
BSD3C6AJ w = 1z = ξ BF ξ = 0 BRB5G3,DP
w′(ξ)|ξ=0 = 1 negationslash= 0,
EKEXC6AJ w = ξ BF ξ = 0 BR,BPC6AJ w = 1z BF z = ∞
F0A1BBAR.
CPD5AR,BF z = 0 BRC6AJ w = 1z?F0A1BBAR.
41/127
(ii) A8 z = 0,z = ∞EE.
CGC2BHF9,H0CMARE5FFEHAFBMARETDMBFFFEHBM∞BRAR
CH?,AVD9ENDMBFC6AJ w = 1z CTC6EKARFRC4A6BM ξ = 0
ARH0CME4ETDMARCH?,
BSD3C6AJ w = 1z = ξ BF ξ = 0 BRB5G3,DP
w′(ξ)|ξ=0 = 1 negationslash= 0,
EKEXC6AJ w = ξ BF ξ = 0 BR,BPC6AJ w = 1z BF z = ∞
F0A1BBAR.
CPD5AR,BF z = 0 BRC6AJ w = 1z?F0A1BBAR.
41/127
(ii) A8 z = 0,z = ∞EE.
CGC2BHF9,H0CMARE5FFEHAFBMARETDMBFFFEHBM∞BRAR
CH?,AVD9ENDMBFC6AJ w = 1z CTC6EKARFRC4A6BM ξ = 0
ARH0CME4ETDMARCH?,
BSD3C6AJ w = 1z = ξ BF ξ = 0 BRB5G3,DP
w′(ξ)|ξ=0 = 1 negationslash= 0,
EKEXC6AJ w = ξ BF ξ = 0 BR,BPC6AJ w = 1z BF z = ∞
F0A1BBAR.
CPD5AR,BF z = 0 BRC6AJ w = 1z?F0A1BBAR.
41/127
B4G0 2,C6AJ w = az + b (a negationslash= 0) BFB6FBBXEYEHH2F0
BRBRA1BBAR.
DD FGCZDDEY,C6AJ w = az + b BFB6FBBXEYEHCBF0E8E8
GHBTAR.
(F0DD)
42/127
B4G0 2,C6AJ w = az + b (a negationslash= 0) BFB6FBBXEYEHH2F0
BRBRA1BBAR.
DD FGCZDDEY,C6AJ w = az + b BFB6FBBXEYEHCBF0E8E8
GHBTAR.
(F0DD)
42/127
B8A0DDEY,C6AJ w = az + b BFB6FBBXEYEHH2F0BRBRA1
BBAR.
(i) A8 z negationslash= ∞EE,FR?
w′ = (az + b)′ = a negationslash= 0,
EKEXF0A1BBAR.
43/127
B8A0DDEY,C6AJ w = az + b BFB6FBBXEYEHH2F0BRBRA1
BBAR.
(i) A8 z negationslash= ∞EE,FR?
w′ = (az + b)′ = a negationslash= 0,
EKEXF0A1BBAR.
43/127
B8A0DDEY,C6AJ w = az + b BFB6FBBXEYEHH2F0BRBRA1
BBAR.
(i) A8 z negationslash= ∞EE,FR?
w′ = (az + b)′ = a negationslash= 0,
EKEXF0A1BBAR.
43/127
(ii) A8 z = ∞EE,B9
ξ = 1z,η = 1w.
CSEE w = az + b DE?
η = ξa + bξ.
ENBF ξ = 0 B5G3,DP η′(ξ) = a(a + bξ)2
vextendsinglevextendsingle
vextendsinglevextendsingle
ξ=0
= 1a negationslash= 0.
FRADBF ξ = 0 BRF0A1BBAR,BP w = az + b BF z = ∞F0
A1BBAR.
44/127
(ii) A8 z = ∞EE,B9
ξ = 1z,η = 1w.
CSEE w = az + b DE?
η = ξa + bξ.
ENBF ξ = 0 B5G3,DP η′(ξ) = a(a + bξ)2
vextendsinglevextendsingle
vextendsinglevextendsingle
ξ=0
= 1a negationslash= 0.
FRADBF ξ = 0 BRF0A1BBAR,BP w = az + b BF z = ∞F0
A1BBAR.
44/127
(ii) A8 z = ∞EE,B9
ξ = 1z,η = 1w.
CSEE w = az + b DE?
η = ξa + bξ.
ENBF ξ = 0 B5G3,DP η′(ξ) = a(a + bξ)2
vextendsinglevextendsingle
vextendsinglevextendsingle
ξ=0
= 1a negationslash= 0.
FRADBF ξ = 0 BRF0A1BBAR,BP w = az + b BF z = ∞F0
A1BBAR.
44/127
(ii) A8 z = ∞EE,B9
ξ = 1z,η = 1w.
CSEE w = az + b DE?
η = ξa + bξ.
ENBF ξ = 0 B5G3,DP η′(ξ) = a(a + bξ)2
vextendsinglevextendsingle
vextendsinglevextendsingle
ξ=0
= 1a negationslash= 0.
FRADBF ξ = 0 BRF0A1BBAR,BP w = az + b BF z = ∞F0
A1BBAR.
44/127
(ii) A8 z = ∞EE,B9
ξ = 1z,η = 1w.
CSEE w = az + b DE?
η = ξa + bξ.
ENBF ξ = 0 B5G3,DP η′(ξ) = a(a + bξ)2
vextendsinglevextendsingle
vextendsinglevextendsingle
ξ=0
= 1a negationslash= 0.
FRADBF ξ = 0 BRF0A1BBAR,BP w = az + b BF z = ∞F0
A1BBAR.
44/127
D2H2EKGB,CZCTEHARB4G0:
F9D2E8 EPEQDMBHC6AJBFB6FBBXEYEHH2F0E8E8GHBTAR,
DPGCCZA2?BH.
45/127
D2H2EKGB,CZCTEHARB4G0:
F9D2E8 EPEQDMBHC6AJBFB6FBBXEYEHH2F0E8E8GHBTAR,
DPGCCZA2?BH.
45/127
2,A2ABBH
B4G0 1,C6AJ w = az + b (a negationslash= 0) BFB6FBBXEYEHH2GC
CZA2ABBH.
DDEY C6AJ w = az + b (a negationslash= 0) F0GQ z EYEHCBARE8BM
EYC4EYEJ,CMBPG2AREGADAQAKARE4BM w.
FRCS,z EYEHCBARE8DRABEXF9E8CMDQDMEYC4C6
AJ w = az + b EKAQARE6ETDMDBB8BXF0E8DRABEXF9E8CMDQ
DM.
CGC2D6DQDMDBEKF0DWF7FFEHDWARABEX,BSD3C6AJ
w = az + b (a negationslash= 0)
BFB6FBBXEYEHH2D6ABEXC6AJEKABEX,(A2ABBH)
46/127
2,A2ABBH
B4G0 1,C6AJ w = az + b (a negationslash= 0) BFB6FBBXEYEHH2GC
CZA2ABBH.
DDEY C6AJ w = az + b (a negationslash= 0) F0GQ z EYEHCBARE8BM
EYC4EYEJ,CMBPG2AREGADAQAKARE4BM w.
FRCS,z EYEHCBARE8DRABEXF9E8CMDQDMEYC4C6
AJ w = az + b EKAQARE6ETDMDBB8BXF0E8DRABEXF9E8CMDQ
DM.
CGC2D6DQDMDBEKF0DWF7FFEHDWARABEX,BSD3C6AJ
w = az + b (a negationslash= 0)
BFB6FBBXEYEHH2D6ABEXC6AJEKABEX,(A2ABBH)
46/127
2,A2ABBH
B4G0 1,C6AJ w = az + b (a negationslash= 0) BFB6FBBXEYEHH2GC
CZA2ABBH.
DDEY C6AJ w = az + b (a negationslash= 0) F0GQ z EYEHCBARE8BM
EYC4EYEJ,CMBPG2AREGADAQAKARE4BM w.
FRCS,z EYEHCBARE8DRABEXF9E8CMDQDMEYC4C6
AJ w = az + b EKAQARE6ETDMDBB8BXF0E8DRABEXF9E8CMDQ
DM.
CGC2D6DQDMDBEKF0DWF7FFEHDWARABEX,BSD3C6AJ
w = az + b (a negationslash= 0)
BFB6FBBXEYEHH2D6ABEXC6AJEKABEX,(A2ABBH)
46/127
2,A2ABBH
B4G0 1,C6AJ w = az + b (a negationslash= 0) BFB6FBBXEYEHH2GC
CZA2ABBH.
DDEY C6AJ w = az + b (a negationslash= 0) F0GQ z EYEHCBARE8BM
EYC4EYEJ,CMBPG2AREGADAQAKARE4BM w.
FRCS,z EYEHCBARE8DRABEXF9E8CMDQDMEYC4C6
AJ w = az + b EKAQARE6ETDMDBB8BXF0E8DRABEXF9E8CMDQ
DM.
CGC2D6DQDMDBEKF0DWF7FFEHDWARABEX,BSD3C6AJ
w = az + b (a negationslash= 0)
BFB6FBBXEYEHH2D6ABEXC6AJEKABEX,(A2ABBH)
46/127
2,A2ABBH
B4G0 1,C6AJ w = az + b (a negationslash= 0) BFB6FBBXEYEHH2GC
CZA2ABBH.
DDEY C6AJ w = az + b (a negationslash= 0) F0GQ z EYEHCBARE8BM
EYC4EYEJ,CMBPG2AREGADAQAKARE4BM w.
FRCS,z EYEHCBARE8DRABEXF9E8CMDQDMEYC4C6
AJ w = az + b EKAQARE6ETDMDBB8BXF0E8DRABEXF9E8CMDQ
DM.
CGC2D6DQDMDBEKF0DWF7FFEHDWARABEX,BSD3C6AJ
w = az + b (a negationslash= 0)
BFB6FBBXEYEHH2D6ABEXC6AJEKABEX,(A2ABBH)
46/127
B4G0 2 C6AJw = 1zBFB6FBBXEYEHH2GCCZA2ABBH.
DDEY B9 z = x + iy,w = 1z = u + iv,
GQ z = x + iy E3CKw = 1z,AQ
u = xx2 + y2,v =?yx2 + y2;
F9 x = uu2 + v2,y =?vu2 + v2,
FRCS,C6AJw = 1zGQE0EN
a(x2 + y2) + bx + cy + d = 0
BV?E0EN d(u
2 + v2) + bu?cv + a = 0.
DUGACCE0EN,CZ:
A8 a negationslash= 0,d negationslash= 0 EE,C6AJD6ABEXC6AJEKABEX;
A8 a negationslash= 0,d = 0 EE,C6AJD6ABEXC6AJEKDQDM;
A8 a = 0,d negationslash= 0 EE,C6AJD6DQDMC6AJEKABEX;
A8 a = 0,d = 0 EE,C6AJD6DQDMC6AJEKDQDM.
47/127
B4G0 2 C6AJw = 1zBFB6FBBXEYEHH2GCCZA2ABBH.
DDEY B9 z = x + iy,w = 1z = u + iv,
GQ z = x + iy E3CKw = 1z,AQ
u = xx2 + y2,v =?yx2 + y2;
F9 x = uu2 + v2,y =?vu2 + v2,
FRCS,C6AJw = 1zGQE0EN
a(x2 + y2) + bx + cy + d = 0
BV?E0EN d(u
2 + v2) + bu?cv + a = 0.
DUGACCE0EN,CZ:
A8 a negationslash= 0,d negationslash= 0 EE,C6AJD6ABEXC6AJEKABEX;
A8 a negationslash= 0,d = 0 EE,C6AJD6ABEXC6AJEKDQDM;
A8 a = 0,d negationslash= 0 EE,C6AJD6DQDMC6AJEKABEX;
A8 a = 0,d = 0 EE,C6AJD6DQDMC6AJEKDQDM.
47/127
B4G0 2 C6AJw = 1zBFB6FBBXEYEHH2GCCZA2ABBH.
DDEY B9 z = x + iy,w = 1z = u + iv,
GQ z = x + iy E3CKw = 1z,AQ
u = xx2 + y2,v =?yx2 + y2;
F9 x = uu2 + v2,y =?vu2 + v2,
FRCS,C6AJw = 1zGQE0EN
a(x2 + y2) + bx + cy + d = 0
BV?E0EN d(u
2 + v2) + bu?cv + a = 0.
DUGACCE0EN,CZ:
A8 a negationslash= 0,d negationslash= 0 EE,C6AJD6ABEXC6AJEKABEX;
A8 a negationslash= 0,d = 0 EE,C6AJD6ABEXC6AJEKDQDM;
A8 a = 0,d negationslash= 0 EE,C6AJD6DQDMC6AJEKABEX;
A8 a = 0,d = 0 EE,C6AJD6DQDMC6AJEKDQDM.
47/127
B4G0 2 C6AJw = 1zBFB6FBBXEYEHH2GCCZA2ABBH.
DDEY B9 z = x + iy,w = 1z = u + iv,
GQ z = x + iy E3CKw = 1z,AQ
u = xx2 + y2,v =?yx2 + y2;
F9 x = uu2 + v2,y =?vu2 + v2,
FRCS,C6AJw = 1zGQE0EN
a(x2 + y2) + bx + cy + d = 0
BV?E0EN d(u
2 + v2) + bu?cv + a = 0.
DUGACCE0EN,CZ:
A8 a negationslash= 0,d negationslash= 0 EE,C6AJD6ABEXC6AJEKABEX;
A8 a negationslash= 0,d = 0 EE,C6AJD6ABEXC6AJEKDQDM;
A8 a = 0,d negationslash= 0 EE,C6AJD6DQDMC6AJEKABEX;
A8 a = 0,d = 0 EE,C6AJD6DQDMC6AJEKDQDM.
47/127
B4G0 2 C6AJw = 1zBFB6FBBXEYEHH2GCCZA2ABBH.
DDEY B9 z = x + iy,w = 1z = u + iv,
GQ z = x + iy E3CKw = 1z,AQ
u = xx2 + y2,v =?yx2 + y2;
F9 x = uu2 + v2,y =?vu2 + v2,
FRCS,C6AJw = 1zGQE0EN
a(x2 + y2) + bx + cy + d = 0
BV?E0EN d(u
2 + v2) + bu?cv + a = 0.
DUGACCE0EN,CZ:
A8 a negationslash= 0,d negationslash= 0 EE,C6AJD6ABEXC6AJEKABEX;
A8 a negationslash= 0,d = 0 EE,C6AJD6ABEXC6AJEKDQDM;
A8 a = 0,d negationslash= 0 EE,C6AJD6DQDMC6AJEKABEX;
A8 a = 0,d = 0 EE,C6AJD6DQDMC6AJEKDQDM.
47/127
B4G0 2 C6AJw = 1zBFB6FBBXEYEHH2GCCZA2ABBH.
DDEY B9 z = x + iy,w = 1z = u + iv,
GQ z = x + iy E3CKw = 1z,AQ
u = xx2 + y2,v =?yx2 + y2;
F9 x = uu2 + v2,y =?vu2 + v2,
FRCS,C6AJw = 1zGQE0EN
a(x2 + y2) + bx + cy + d = 0
BV?E0EN d(u
2 + v2) + bu?cv + a = 0.
DUGACCE0EN,CZ:
A8 a negationslash= 0,d negationslash= 0 EE,C6AJD6ABEXC6AJEKABEX;
A8 a negationslash= 0,d = 0 EE,C6AJD6ABEXC6AJEKDQDM;
A8 a = 0,d negationslash= 0 EE,C6AJD6DQDMC6AJEKABEX;
A8 a = 0,d = 0 EE,C6AJD6DQDMC6AJEKDQDM.
47/127
B4G0 2 C6AJw = 1zBFB6FBBXEYEHH2GCCZA2ABBH.
DDEY B9 z = x + iy,w = 1z = u + iv,
GQ z = x + iy E3CKw = 1z,AQ
u = xx2 + y2,v =?yx2 + y2;
F9 x = uu2 + v2,y =?vu2 + v2,
FRCS,C6AJw = 1zGQE0EN
a(x2 + y2) + bx + cy + d = 0
BV?E0EN d(u
2 + v2) + bu?cv + a = 0.
DUGACCE0EN,CZ:
A8 a negationslash= 0,d negationslash= 0 EE,C6AJD6ABEXC6AJEKABEX;
A8 a negationslash= 0,d = 0 EE,C6AJD6ABEXC6AJEKDQDM;
A8 a = 0,d negationslash= 0 EE,C6AJD6DQDMC6AJEKABEX;
A8 a = 0,d = 0 EE,C6AJD6DQDMC6AJEKDQDM.
47/127
B4G0 2 C6AJw = 1zBFB6FBBXEYEHH2GCCZA2ABBH.
DDEY B9 z = x + iy,w = 1z = u + iv,
GQ z = x + iy E3CKw = 1z,AQ
u = xx2 + y2,v =?yx2 + y2;
F9 x = uu2 + v2,y =?vu2 + v2,
FRCS,C6AJw = 1zGQE0EN
a(x2 + y2) + bx + cy + d = 0
BV?E0EN d(u
2 + v2) + bu?cv + a = 0.
DUGACCE0EN,CZ:
A8 a negationslash= 0,d negationslash= 0 EE,C6AJD6ABEXC6AJEKABEX;
A8 a negationslash= 0,d = 0 EE,C6AJD6ABEXC6AJEKDQDM;
A8 a = 0,d negationslash= 0 EE,C6AJD6DQDMC6AJEKABEX;
A8 a = 0,d = 0 EE,C6AJD6DQDMC6AJEKDQDM.
47/127
B4G0 2 C6AJw = 1zBFB6FBBXEYEHH2GCCZA2ABBH.
DDEY B9 z = x + iy,w = 1z = u + iv,
GQ z = x + iy E3CKw = 1z,AQ
u = xx2 + y2,v =?yx2 + y2;
F9 x = uu2 + v2,y =?vu2 + v2,
FRCS,C6AJw = 1zGQE0EN
a(x2 + y2) + bx + cy + d = 0
BV?E0EN d(u
2 + v2) + bu?cv + a = 0.
DUGACCE0EN,CZ:
A8 a negationslash= 0,d negationslash= 0 EE,C6AJD6ABEXC6AJEKABEX;
A8 a negationslash= 0,d = 0 EE,C6AJD6ABEXC6AJEKDQDM;
A8 a = 0,d negationslash= 0 EE,C6AJD6DQDMC6AJEKABEX;
A8 a = 0,d = 0 EE,C6AJD6DQDMC6AJEKDQDM.
47/127
B4G0 2 C6AJw = 1zBFB6FBBXEYEHH2GCCZA2ABBH.
DDEY B9 z = x + iy,w = 1z = u + iv,
GQ z = x + iy E3CKw = 1z,AQ
u = xx2 + y2,v =?yx2 + y2;
F9 x = uu2 + v2,y =?vu2 + v2,
FRCS,C6AJw = 1zGQE0EN
a(x2 + y2) + bx + cy + d = 0
BV?E0EN d(u
2 + v2) + bu?cv + a = 0.
DUGACCE0EN,CZ:
A8 a negationslash= 0,d negationslash= 0 EE,C6AJD6ABEXC6AJEKABEX;
A8 a negationslash= 0,d = 0 EE,C6AJD6ABEXC6AJEKDQDM;
A8 a = 0,d negationslash= 0 EE,C6AJD6DQDMC6AJEKABEX;
A8 a = 0,d = 0 EE,C6AJD6DQDMC6AJEKDQDM.
47/127
D2H2EKGB,CZCTEHARB4G0:
F9D2AJ EPEQDMBHC6AJGQB6FB z EYEHH2ARABEXC6AJEKB6
FB w EYEHH2ARABEX,BPGCCZA2ABBH.
48/127
D2H2EKGB,CZCTEHARB4G0:
F9D2AJ EPEQDMBHC6AJGQB6FB z EYEHH2ARABEXC6AJEKB6
FB w EYEHH2ARABEX,BPGCCZA2ABBH.
48/127
ETDRB4G0:
(1) BFEPEQDMBHC6AJCT,CGC2DTF9ARABEXF9DQDMH2DACZ
BMC6AJEKFFEHAFBM,BSD3ENFUC6AJEKDWF7?CZDLARABEX;
(2) CGC2CZE8DRBMC6AJEKFFEHAFBM,BSD3ENFUC6AJEKDQ
DM.
49/127
ETDRB4G0:
(1) BFEPEQDMBHC6AJCT,CGC2DTF9ARABEXF9DQDMH2DACZ
BMC6AJEKFFEHAFBM,BSD3ENFUC6AJEKDWF7?CZDLARABEX;
(2) CGC2CZE8DRBMC6AJEKFFEHAFBM,BSD3ENFUC6AJEKDQ
DM.
49/127
ETDRB4G0:
(1) BFEPEQDMBHC6AJCT,CGC2DTF9ARABEXF9DQDMH2DACZ
BMC6AJEKFFEHAFBM,BSD3ENFUC6AJEKDWF7?CZDLARABEX;
(2) CGC2CZE8DRBMC6AJEKFFEHAFBM,BSD3ENFUC6AJEKDQ
DM.
49/127
3,A2GHEHBH
F9FE,A2EXGHEHBMG0BVARBHEF,EH?A2GHEHBH.
F9D2D2 ANBM z1,z2 F0B3D9ABEX C ARE8GHGHEHBM,BSD3
BFEPEQDMBHC6AJCT,ENDMARE4BM w1 DQ w2?F0B3D9 C AR
E6ETDM Γ ARE8GHGHEHBM.
50/127
3,A2GHEHBH
F9FE,A2EXGHEHBMG0BVARBHEF,EH?A2GHEHBH.
F9D2D2 ANBM z1,z2 F0B3D9ABEX C ARE8GHGHEHBM,BSD3
BFEPEQDMBHC6AJCT,ENDMARE4BM w1 DQ w2?F0B3D9 C AR
E6ETDM Γ ARE8GHGHEHBM.
50/127
3,A2GHEHBH
F9FE,A2EXGHEHBMG0BVARBHEF,EH?A2GHEHBH.
F9D2D2 ANBM z1,z2 F0B3D9ABEX C ARE8GHGHEHBM,BSD3
BFEPEQDMBHC6AJCT,ENDMARE4BM w1 DQ w2?F0B3D9 C AR
E6ETDM Γ ARE8GHGHEHBM.
50/127
DI 1 ANEPEQDMBHC6AJ w = f(z) = az + bcz + d GQH0EG
AB|z| < 1 C6AJEKH0EGAB|w| < 1,a F0 z EYEHH2H0EG
AB|z| < 1 CBG4ARE8BM,DP f(a) = b.
EM f
parenleftbigg1
a
parenrightbigg
.
B5 FR? w,z = a?→ w = b;
BHFBAK
1
a F0 a BF z EYEHCBB3D9H0EGABEX|z| = 1 ARGHEH
BM.
CGC2
w,z = 1a?→ w = c.
DUGAA2GHEHBH,b G2 c F0 w EYEHCBB3D9H0EGABEX|w| = 1
ARE8GHGHEHBM.
EKEXCZ f
parenleftbigg1
a
parenrightbigg
= 1b.
51/127
DI 1 ANEPEQDMBHC6AJ w = f(z) = az + bcz + d GQH0EG
AB|z| < 1 C6AJEKH0EGAB|w| < 1,a F0 z EYEHH2H0EG
AB|z| < 1 CBG4ARE8BM,DP f(a) = b.
EM f
parenleftbigg1
a
parenrightbigg
.
B5 FR? w,z = a?→ w = b;
BHFBAK
1
a F0 a BF z EYEHCBB3D9H0EGABEX|z| = 1 ARGHEH
BM.
CGC2
w,z = 1a?→ w = c.
DUGAA2GHEHBH,b G2 c F0 w EYEHCBB3D9H0EGABEX|w| = 1
ARE8GHGHEHBM.
EKEXCZ f
parenleftbigg1
a
parenrightbigg
= 1b.
51/127
DI 1 ANEPEQDMBHC6AJ w = f(z) = az + bcz + d GQH0EG
AB|z| < 1 C6AJEKH0EGAB|w| < 1,a F0 z EYEHH2H0EG
AB|z| < 1 CBG4ARE8BM,DP f(a) = b.
EM f
parenleftbigg1
a
parenrightbigg
.
B5 FR? w,z = a?→ w = b;
BHFBAK
1
a F0 a BF z EYEHCBB3D9H0EGABEX|z| = 1 ARGHEH
BM.
CGC2
w,z = 1a?→ w = c.
DUGAA2GHEHBH,b G2 c F0 w EYEHCBB3D9H0EGABEX|w| = 1
ARE8GHGHEHBM.
EKEXCZ f
parenleftbigg1
a
parenrightbigg
= 1b.
51/127
DI 1 ANEPEQDMBHC6AJ w = f(z) = az + bcz + d GQH0EG
AB|z| < 1 C6AJEKH0EGAB|w| < 1,a F0 z EYEHH2H0EG
AB|z| < 1 CBG4ARE8BM,DP f(a) = b.
EM f
parenleftbigg1
a
parenrightbigg
.
B5 FR? w,z = a?→ w = b;
BHFBAK
1
a F0 a BF z EYEHCBB3D9H0EGABEX|z| = 1 ARGHEH
BM.
CGC2
w,z = 1a?→ w = c.
DUGAA2GHEHBH,b G2 c F0 w EYEHCBB3D9H0EGABEX|w| = 1
ARE8GHGHEHBM.
EKEXCZ f
parenleftbigg1
a
parenrightbigg
= 1b.
51/127
DI 1 ANEPEQDMBHC6AJ w = f(z) = az + bcz + d GQH0EG
AB|z| < 1 C6AJEKH0EGAB|w| < 1,a F0 z EYEHH2H0EG
AB|z| < 1 CBG4ARE8BM,DP f(a) = b.
EM f
parenleftbigg1
a
parenrightbigg
.
B5 FR? w,z = a?→ w = b;
BHFBAK
1
a F0 a BF z EYEHCBB3D9H0EGABEX|z| = 1 ARGHEH
BM.
CGC2
w,z = 1a?→ w = c.
DUGAA2GHEHBH,b G2 c F0 w EYEHCBB3D9H0EGABEX|w| = 1
ARE8GHGHEHBM.
EKEXCZ f
parenleftbigg1
a
parenrightbigg
= 1b.
51/127
DI 1 ANEPEQDMBHC6AJ w = f(z) = az + bcz + d GQH0EG
AB|z| < 1 C6AJEKH0EGAB|w| < 1,a F0 z EYEHH2H0EG
AB|z| < 1 CBG4ARE8BM,DP f(a) = b.
EM f
parenleftbigg1
a
parenrightbigg
.
B5 FR? w,z = a?→ w = b;
BHFBAK
1
a F0 a BF z EYEHCBB3D9H0EGABEX|z| = 1 ARGHEH
BM.
CGC2
w,z = 1a?→ w = c.
DUGAA2GHEHBH,b G2 c F0 w EYEHCBB3D9H0EGABEX|w| = 1
ARE8GHGHEHBM.
EKEXCZ f
parenleftbigg1
a
parenrightbigg
= 1b.
51/127
BTC7AR,CGC2
w,z = a?→ w = 0;
BHFBAK 0 G2∞F0 w EYEHCBB3D9H0EGABEX|w| = 1 ARE8
GHGHEHBM,EKEXCZ
w,z = 1a?→ w = ∞.
52/127
BTC7AR,CGC2
w,z = a?→ w = 0;
BHFBAK 0 G2∞F0 w EYEHCBB3D9H0EGABEX|w| = 1 ARE8
GHGHEHBM,EKEXCZ
w,z = 1a?→ w = ∞.
52/127
BTC7AR,CGC2
w,z = a?→ w = 0;
BHFBAK 0 G2∞F0 w EYEHCBB3D9H0EGABEX|w| = 1 ARE8
GHGHEHBM,EKEXCZ
w,z = 1a?→ w = ∞.
52/127
G2D2 z1,z2 F0B3D9ABEX C,|z?z0| = R ARE8GHGHEH
BMEYC4 z1,z2 ARBRG3ABEX Γ DQ C AKA2.
C z0 z
1 z2
R
z’ Γ
DD:,=?,
D0 z0 DU Γ ARDNDM,ANDNBM? z′,DUGAEYEHBTG3ARDJEJ
|z′?z0|2 = |z2?z0|·|z1?z0|
D5FR? z1,z2 F0B3D9ABEX C ARGHEHBM,EKEX
|z2?z0|·|z1?z0| = R2
EKEX|z′?z0| = R,GXC4EY z′ BF C H2,DP Γ ARDNDMFU
F0 C ARDWF7,FRCS Γ DQ C AKA2.
53/127
G2D2 z1,z2 F0B3D9ABEX C,|z?z0| = R ARE8GHGHEH
BMEYC4 z1,z2 ARBRG3ABEX Γ DQ C AKA2.
C z0 z
1 z2
R
z’ Γ
DD:,=?,
D0 z0 DU Γ ARDNDM,ANDNBM? z′,DUGAEYEHBTG3ARDJEJ
|z′?z0|2 = |z2?z0|·|z1?z0|
D5FR? z1,z2 F0B3D9ABEX C ARGHEHBM,EKEX
|z2?z0|·|z1?z0| = R2
EKEX|z′?z0| = R,GXC4EY z′ BF C H2,DP Γ ARDNDMFU
F0 C ARDWF7,FRCS Γ DQ C AKA2.
53/127
G2D2 z1,z2 F0B3D9ABEX C,|z?z0| = R ARE8GHGHEH
BMEYC4 z1,z2 ARBRG3ABEX Γ DQ C AKA2.
C z0 z
1 z2
R
z’ Γ
DD:,=?,
D0 z0 DU Γ ARDNDM,ANDNBM? z′,DUGAEYEHBTG3ARDJEJ
|z′?z0|2 = |z2?z0|·|z1?z0|
D5FR? z1,z2 F0B3D9ABEX C ARGHEHBM,EKEX
|z2?z0|·|z1?z0| = R2
EKEX|z′?z0| = R,GXC4EY z′ BF C H2,DP Γ ARDNDMFU
F0 C ARDWF7,FRCS Γ DQ C AKA2.
53/127
G2D2 z1,z2 F0B3D9ABEX C,|z?z0| = R ARE8GHGHEH
BMEYC4 z1,z2 ARBRG3ABEX Γ DQ C AKA2.
C z0 z
1 z2
R
z’ Γ
DD:,=?,
D0 z0 DU Γ ARDNDM,ANDNBM? z′,DUGAEYEHBTG3ARDJEJ
|z′?z0|2 = |z2?z0|·|z1?z0|
D5FR? z1,z2 F0B3D9ABEX C ARGHEHBM,EKEX
|z2?z0|·|z1?z0| = R2
EKEX|z′?z0| = R,GXC4EY z′ BF C H2,DP Γ ARDNDMFU
F0 C ARDWF7,FRCS Γ DQ C AKA2.
53/127
G2D2 z1,z2 F0B3D9ABEX C,|z?z0| = R ARE8GHGHEH
BMEYC4 z1,z2 ARBRG3ABEX Γ DQ C AKA2.
C z0 z
1 z2
R
z’ Γ
DD:,=?,
D0 z0 DU Γ ARDNDM,ANDNBM? z′,DUGAEYEHBTG3ARDJEJ
|z′?z0|2 = |z2?z0|·|z1?z0|
D5FR? z1,z2 F0B3D9ABEX C ARGHEHBM,EKEX
|z2?z0|·|z1?z0| = R2
EKEX|z′?z0| = R,GXC4EY z′ BF C H2,DP Γ ARDNDMFU
F0 C ARDWF7,FRCS Γ DQ C AKA2.
53/127
G2D2 z1,z2 F0B3D9ABEX C,|z?z0| = R ARE8GHGHEH
BMEYC4 z1,z2 ARBRG3ABEX Γ DQ C AKA2.
C z0 z
1 z2
R
z’ Γ
DD:,=?,
D0 z0 DU Γ ARDNDM,ANDNBM? z′,DUGAEYEHBTG3ARDJEJ
|z′?z0|2 = |z2?z0|·|z1?z0|
D5FR? z1,z2 F0B3D9ABEX C ARGHEHBM,EKEX
|z2?z0|·|z1?z0| = R2
EKEX|z′?z0| = R,GXC4EY z′ BF C H2,DP Γ ARDNDMFU
F0 C ARDWF7,FRCS Γ DQ C AKA2.
53/127
G2D2 z1,z2 F0B3D9ABEX C,|z?z0| = R ARE8GHGHEH
BMEYC4 z1,z2 ARBRG3ABEX Γ DQ C AKA2.
C z0 z
1 z2
R
z’ Γ
DD:,=?,
D0 z0 DU Γ ARDNDM,ANDNBM? z′,DUGAEYEHBTG3ARDJEJ
|z′?z0|2 = |z2?z0|·|z1?z0|
D5FR? z1,z2 F0B3D9ABEX C ARGHEHBM,EKEX
|z2?z0|·|z1?z0| = R2
EKEX|z′?z0| = R,GXC4EY z′ BF C H2,DP Γ ARDNDMFU
F0 C ARDWF7,FRCS Γ DQ C AKA2.
53/127
G2D2 z1,z2 F0B3D9ABEX C,|z?z0| = R ARE8GHGHEH
BMEYC4 z1,z2 ARBRG3ABEX Γ DQ C AKA2.
C z0 z
1 z2
R
z’ Γ
DD:,=?,
D0 z0 DU Γ ARDNDM,ANDNBM? z′,DUGAEYEHBTG3ARDJEJ
|z′?z0|2 = |z2?z0|·|z1?z0|
D5FR? z1,z2 F0B3D9ABEX C ARGHEHBM,EKEX
|z2?z0|·|z1?z0| = R2
EKEX|z′?z0| = R,GXC4EY z′ BF C H2,DP Γ ARDNDMFU
F0 C ARDWF7,FRCS Γ DQ C AKA2.
53/127
G2D2 z1,z2 F0B3D9ABEX C,|z?z0| = R ARE8GHGHEH
BMEYC4 z1,z2 ARBRG3ABEX Γ DQ C AKA2.
C z0 z
1 z2
R
z’ Γ
DD:,=?,
D0 z0 DU Γ ARDNDM,ANDNBM? z′,DUGAEYEHBTG3ARDJEJ
|z′?z0|2 = |z2?z0|·|z1?z0|
D5FR? z1,z2 F0B3D9ABEX C ARGHEHBM,EKEX
|z2?z0|·|z1?z0| = R2
EKEX|z′?z0| = R,GXC4EY z′ BF C H2,DP Γ ARDNDMFU
F0 C ARDWF7,FRCS Γ DQ C AKA2.
53/127
,?=,
FR? Γ DQ C D9A2BM z′ BRAKA2,FRCS,C ARDWF7FUF0 Γ
ARDNDM,EKEXCZ
|z2?z0|·|z1?z0| = |z′?z0|2 = R2,
BP z1 DQ z2 F0B3D9ABEX C ARE8GHGHEHBM.
C z0 z
1 z2
R
z’ Γ
54/127
,?=,
FR? Γ DQ C D9A2BM z′ BRAKA2,FRCS,C ARDWF7FUF0 Γ
ARDNDM,EKEXCZ
|z2?z0|·|z1?z0| = |z′?z0|2 = R2,
BP z1 DQ z2 F0B3D9ABEX C ARE8GHGHEHBM.
C z0 z
1 z2
R
z’ Γ
54/127
,?=,
FR? Γ DQ C D9A2BM z′ BRAKA2,FRCS,C ARDWF7FUF0 Γ
ARDNDM,EKEXCZ
|z2?z0|·|z1?z0| = |z′?z0|2 = R2,
BP z1 DQ z2 F0B3D9ABEX C ARE8GHGHEHBM.
C z0 z
1 z2
R
z’ Γ
54/127
,?=,
FR? Γ DQ C D9A2BM z′ BRAKA2,FRCS,C ARDWF7FUF0 Γ
ARDNDM,EKEXCZ
|z2?z0|·|z1?z0| = |z′?z0|2 = R2,
BP z1 DQ z2 F0B3D9ABEX C ARE8GHGHEHBM.
C z0 z
1 z2
R
z’ Γ
54/127
F9D2D2ARDDEY,ANEYC4 w1 DQ w2 ARBRE8ABEX Γ′ F0EY
C4 z1 DQ z2 ARABEX Γ CSEPEQDMBHC6AJC6AJC4BHAR.
CSD9 Γ DQ C AKA2,ADEPEQDMBHC6AJGCCZA2?BH,EK
EX Γ′ DQ C′(C ARE6)?BJAKA2,FRCS,w1 DQ w2 F0E8GHB3
D9 C
′
ARGHEHBM.
55/127
F9D2D2ARDDEY,ANEYC4 w1 DQ w2 ARBRE8ABEX Γ′ F0EY
C4 z1 DQ z2 ARABEX Γ CSEPEQDMBHC6AJC6AJC4BHAR.
CSD9 Γ DQ C AKA2,ADEPEQDMBHC6AJGCCZA2?BH,EK
EX Γ′ DQ C′(C ARE6)?BJAKA2,FRCS,w1 DQ w2 F0E8GHB3
D9 C
′
ARGHEHBM.
55/127
F9D2D2ARDDEY,ANEYC4 w1 DQ w2 ARBRE8ABEX Γ′ F0EY
C4 z1 DQ z2 ARABEX Γ CSEPEQDMBHC6AJC6AJC4BHAR.
CSD9 Γ DQ C AKA2,ADEPEQDMBHC6AJGCCZA2?BH,EK
EX Γ′ DQ C′(C ARE6)?BJAKA2,FRCS,w1 DQ w2 F0E8GHB3
D9 C
′
ARGHEHBM.
55/127
F9D2D2ARDDEY,ANEYC4 w1 DQ w2 ARBRE8ABEX Γ′ F0EY
C4 z1 DQ z2 ARABEX Γ CSEPEQDMBHC6AJC6AJC4BHAR.
CSD9 Γ DQ C AKA2,ADEPEQDMBHC6AJGCCZA2?BH,EK
EX Γ′ DQ C′(C ARE6)?BJAKA2,FRCS,w1 DQ w2 F0E8GHB3
D9 C
′
ARGHEHBM.
55/127
BDD2AX DXE8GZF9EPEQDMBHC6AJARCMDR
EPEQDMBHC6AJ w = az + bcz + d (ad?bc negationslash= 0) EICBE0CZD2
DRFXG9ARDIGI,FRCS,E0C1DTF9D2DRCMDR,FUCDGZF9E8DREP
EQDMBHC6AJ.
F9D2 BF z EYEHH2BRFBDTF9D2DRDNFLARBM z1,z2,z3,
BF w EYEHH2?DTF9D2DRDNFLARBM w1,w2,w3,BSD3FUDJBF
DXE8AREPEQDMBHC6AJ,GQ zk(k = 1,2,3) EDCVC6AJ
EK wk(k = 1,2,3).
56/127
BDD2AX DXE8GZF9EPEQDMBHC6AJARCMDR
EPEQDMBHC6AJ w = az + bcz + d (ad?bc negationslash= 0) EICBE0CZD2
DRFXG9ARDIGI,FRCS,E0C1DTF9D2DRCMDR,FUCDGZF9E8DREP
EQDMBHC6AJ.
F9D2 BF z EYEHH2BRFBDTF9D2DRDNFLARBM z1,z2,z3,
BF w EYEHH2?DTF9D2DRDNFLARBM w1,w2,w3,BSD3FUDJBF
DXE8AREPEQDMBHC6AJ,GQ zk(k = 1,2,3) EDCVC6AJ
EK wk(k = 1,2,3).
56/127
BDD2AX DXE8GZF9EPEQDMBHC6AJARCMDR
EPEQDMBHC6AJ w = az + bcz + d (ad?bc negationslash= 0) EICBE0CZD2
DRFXG9ARDIGI,FRCS,E0C1DTF9D2DRCMDR,FUCDGZF9E8DREP
EQDMBHC6AJ.
F9D2 BF z EYEHH2BRFBDTF9D2DRDNFLARBM z1,z2,z3,
BF w EYEHH2?DTF9D2DRDNFLARBM w1,w2,w3,BSD3FUDJBF
DXE8AREPEQDMBHC6AJ,GQ zk(k = 1,2,3) EDCVC6AJ
EK wk(k = 1,2,3).
56/127
DD AN w = az + bcz + d (ad?bc negationslash= 0),GQ zk(k = 1,2,3) ED
CVC6AJEK wk(k = 1,2,3),BP
wk = azk + bcz
k + d
(k = 1,2,3)
FRADCZ
w?wk = (z?zk)(ad?bc)(cz + d)(cz
k + d)
,(k = 1,2,3)
w3?wk = (z3?zk)(ad?bc)(cz
3 + d)(czk + d)
,(k = 1,2)
FRCSAQ
w?w1
w?w2 ·
w3?w2
w3?w1 =
z?z1
z?z2 ·
z3?z2
z3?z1
F0DDENF0DXE8AR.
57/127
DD AN w = az + bcz + d (ad?bc negationslash= 0),GQ zk(k = 1,2,3) ED
CVC6AJEK wk(k = 1,2,3),BP
wk = azk + bcz
k + d
(k = 1,2,3)
FRADCZ
w?wk = (z?zk)(ad?bc)(cz + d)(cz
k + d)
,(k = 1,2,3)
w3?wk = (z3?zk)(ad?bc)(cz
3 + d)(czk + d)
,(k = 1,2)
FRCSAQ
w?w1
w?w2 ·
w3?w2
w3?w1 =
z?z1
z?z2 ·
z3?z2
z3?z1
F0DDENF0DXE8AR.
57/127
DD AN w = az + bcz + d (ad?bc negationslash= 0),GQ zk(k = 1,2,3) ED
CVC6AJEK wk(k = 1,2,3),BP
wk = azk + bcz
k + d
(k = 1,2,3)
FRADCZ
w?wk = (z?zk)(ad?bc)(cz + d)(cz
k + d)
,(k = 1,2,3)
w3?wk = (z3?zk)(ad?bc)(cz
3 + d)(czk + d)
,(k = 1,2)
FRCSAQ
w?w1
w?w2 ·
w3?w2
w3?w1 =
z?z1
z?z2 ·
z3?z2
z3?z1
F0DDENF0DXE8AR.
57/127
DD AN w = az + bcz + d (ad?bc negationslash= 0),GQ zk(k = 1,2,3) ED
CVC6AJEK wk(k = 1,2,3),BP
wk = azk + bcz
k + d
(k = 1,2,3)
FRADCZ
w?wk = (z?zk)(ad?bc)(cz + d)(cz
k + d)
,(k = 1,2,3)
w3?wk = (z3?zk)(ad?bc)(cz
3 + d)(czk + d)
,(k = 1,2)
FRCSAQ
w?w1
w?w2 ·
w3?w2
w3?w1 =
z?z1
z?z2 ·
z3?z2
z3?z1
F0DDENF0DXE8AR.
57/127
DD AN w = az + bcz + d (ad?bc negationslash= 0),GQ zk(k = 1,2,3) ED
CVC6AJEK wk(k = 1,2,3),BP
wk = azk + bcz
k + d
(k = 1,2,3)
FRADCZ
w?wk = (z?zk)(ad?bc)(cz + d)(cz
k + d)
,(k = 1,2,3)
w3?wk = (z3?zk)(ad?bc)(cz
3 + d)(czk + d)
,(k = 1,2)
FRCSAQ
w?w1
w?w2 ·
w3?w2
w3?w1 =
z?z1
z?z2 ·
z3?z2
z3?z1
F0DDENF0DXE8AR.
57/127
DD AN w = az + bcz + d (ad?bc negationslash= 0),GQ zk(k = 1,2,3) ED
CVC6AJEK wk(k = 1,2,3),BP
wk = azk + bcz
k + d
(k = 1,2,3)
FRADCZ
w?wk = (z?zk)(ad?bc)(cz + d)(cz
k + d)
,(k = 1,2,3)
w3?wk = (z3?zk)(ad?bc)(cz
3 + d)(czk + d)
,(k = 1,2)
FRCSAQ
w?w1
w?w2 ·
w3?w2
w3?w1 =
z?z1
z?z2 ·
z3?z2
z3?z1
F0DDENF0DXE8AR.
57/127
B4G0,DUGAF9D2,BFH0DREUDJARABEX C DQ C′ H2,EPC7AI
F9D2DRG0FVBMEXA0,BJCDGHAKE8DREPEQDMBHC6AJGQ C C6AJ
EK C′.
C C’
58/127
CTEHAREVC3,C6AJESD6 C ARCBG4C6AJEKEFD3?
B4G0 1 BFGXDREPEQDMBHC6AJCT,C ARCBG4G0F0C6AJ
EK C′ ARCBG4,FUF0C6AJEK C′ ARA9G4.
z2
z1 w2
w1
C C’
Q
59/127
CTEHAREVC3,C6AJESD6 C ARCBG4C6AJEKEFD3?
B4G0 1 BFGXDREPEQDMBHC6AJCT,C ARCBG4G0F0C6AJ
EK C′ ARCBG4,FUF0C6AJEK C′ ARA9G4.
z2
z1 w2
w1
C C’
Q
59/127
DD,DRDDDF.
z2
z1 w2
w1
C C’
Q
AN z1,z2 F0 C CBARBRFBH0BM,CMDQDMD6GXH0BMGJAQGD
BH,ANDMGB z1z2 ARE4F0ABD0hatwidew1w2.
CGC2 w1 BF C′ DNA9,w2 BF C′ DNCB,BSD3ABD0hatwidew1w2 BJ
DQ C′ A2D9E8BM Q,Q BMBF C′ H2,EKEXBJC4F0ABEX C H2
B8BMARE4.
B8E8E0EH,Q F0 z1z2 H2B8BMARE4.
D0ADFUCZH0DRG0FVARBMAQC6AJEKFVE8DRBM,GXDQEPEQDM
BHC6AJARE8E8GHBTCVGP.
60/127
DD,DRDDDF.
z2
z1 w2
w1
C C’
Q
AN z1,z2 F0 C CBARBRFBH0BM,CMDQDMD6GXH0BMGJAQGD
BH,ANDMGB z1z2 ARE4F0ABD0hatwidew1w2.
CGC2 w1 BF C′ DNA9,w2 BF C′ DNCB,BSD3ABD0hatwidew1w2 BJ
DQ C′ A2D9E8BM Q,Q BMBF C′ H2,EKEXBJC4F0ABEX C H2
B8BMARE4.
B8E8E0EH,Q F0 z1z2 H2B8BMARE4.
D0ADFUCZH0DRG0FVARBMAQC6AJEKFVE8DRBM,GXDQEPEQDM
BHC6AJARE8E8GHBTCVGP.
60/127
DD,DRDDDF.
z2
z1 w2
w1
C C’
Q
AN z1,z2 F0 C CBARBRFBH0BM,CMDQDMD6GXH0BMGJAQGD
BH,ANDMGB z1z2 ARE4F0ABD0hatwidew1w2.
CGC2 w1 BF C′ DNA9,w2 BF C′ DNCB,BSD3ABD0hatwidew1w2 BJ
DQ C′ A2D9E8BM Q,Q BMBF C′ H2,EKEXBJC4F0ABEX C H2
B8BMARE4.
B8E8E0EH,Q F0 z1z2 H2B8BMARE4.
D0ADFUCZH0DRG0FVARBMAQC6AJEKFVE8DRBM,GXDQEPEQDM
BHC6AJARE8E8GHBTCVGP.
60/127
DD,DRDDDF.
z2
z1 w2
w1
C C’
Q
AN z1,z2 F0 C CBARBRFBH0BM,CMDQDMD6GXH0BMGJAQGD
BH,ANDMGB z1z2 ARE4F0ABD0hatwidew1w2.
CGC2 w1 BF C′ DNA9,w2 BF C′ DNCB,BSD3ABD0hatwidew1w2 BJ
DQ C′ A2D9E8BM Q,Q BMBF C′ H2,EKEXBJC4F0ABEX C H2
B8BMARE4.
B8E8E0EH,Q F0 z1z2 H2B8BMARE4.
D0ADFUCZH0DRG0FVARBMAQC6AJEKFVE8DRBM,GXDQEPEQDM
BHC6AJARE8E8GHBTCVGP.
60/127
DD,DRDDDF.
z2
z1 w2
w1
C C’
Q
AN z1,z2 F0 C CBARBRFBH0BM,CMDQDMD6GXH0BMGJAQGD
BH,ANDMGB z1z2 ARE4F0ABD0hatwidew1w2.
CGC2 w1 BF C′ DNA9,w2 BF C′ DNCB,BSD3ABD0hatwidew1w2 BJ
DQ C′ A2D9E8BM Q,Q BMBF C′ H2,EKEXBJC4F0ABEX C H2
B8BMARE4.
B8E8E0EH,Q F0 z1z2 H2B8BMARE4.
D0ADFUCZH0DRG0FVARBMAQC6AJEKFVE8DRBM,GXDQEPEQDM
BHC6AJARE8E8GHBTCVGP.
60/127
DD,DRDDDF.
z2
z1 w2
w1
C C’
Q
AN z1,z2 F0 C CBARBRFBH0BM,CMDQDMD6GXH0BMGJAQGD
BH,ANDMGB z1z2 ARE4F0ABD0hatwidew1w2.
CGC2 w1 BF C′ DNA9,w2 BF C′ DNCB,BSD3ABD0hatwidew1w2 BJ
DQ C′ A2D9E8BM Q,Q BMBF C′ H2,EKEXBJC4F0ABEX C H2
B8BMARE4.
B8E8E0EH,Q F0 z1z2 H2B8BMARE4.
D0ADFUCZH0DRG0FVARBMAQC6AJEKFVE8DRBM,GXDQEPEQDM
BHC6AJARE8E8GHBTCVGP.
60/127
DD,DRDDDF.
z2
z1 w2
w1
C C’
Q
AN z1,z2 F0 C CBARBRFBH0BM,CMDQDMD6GXH0BMGJAQGD
BH,ANDMGB z1z2 ARE4F0ABD0hatwidew1w2.
CGC2 w1 BF C′ DNA9,w2 BF C′ DNCB,BSD3ABD0hatwidew1w2 BJ
DQ C′ A2D9E8BM Q,Q BMBF C′ H2,EKEXBJC4F0ABEX C H2
B8BMARE4.
B8E8E0EH,Q F0 z1z2 H2B8BMARE4.
D0ADFUCZH0DRG0FVARBMAQC6AJEKFVE8DRBM,GXDQEPEQDM
BHC6AJARE8E8GHBTCVGP.
60/127
DD,DRDDDF.
z2
z1 w2
w1
C C’
Q
AN z1,z2 F0 C CBARBRFBH0BM,CMDQDMD6GXH0BMGJAQGD
BH,ANDMGB z1z2 ARE4F0ABD0hatwidew1w2.
CGC2 w1 BF C′ DNA9,w2 BF C′ DNCB,BSD3ABD0hatwidew1w2 BJ
DQ C′ A2D9E8BM Q,Q BMBF C′ H2,EKEXBJC4F0ABEX C H2
B8BMARE4.
B8E8E0EH,Q F0 z1z2 H2B8BMARE4.
D0ADFUCZH0DRG0FVARBMAQC6AJEKFVE8DRBM,GXDQEPEQDM
BHC6AJARE8E8GHBTCVGP.
60/127
DD,DRDDDF.
z2
z1 w2
w1
C C’
Q
AN z1,z2 F0 C CBARBRFBH0BM,CMDQDMD6GXH0BMGJAQGD
BH,ANDMGB z1z2 ARE4F0ABD0hatwidew1w2.
CGC2 w1 BF C′ DNA9,w2 BF C′ DNCB,BSD3ABD0hatwidew1w2 BJ
DQ C′ A2D9E8BM Q,Q BMBF C′ H2,EKEXBJC4F0ABEX C H2
B8BMARE4.
B8E8E0EH,Q F0 z1z2 H2B8BMARE4.
D0ADFUCZH0DRG0FVARBMAQC6AJEKFVE8DRBM,GXDQEPEQDM
BHC6AJARE8E8GHBTCVGP.
60/127
B4G0 2 CGC2BF C CBBRAIE8BM z0,AD z0 ARE4BF C′ CB
G4,BSD3 C ARCBG4FUC6AJEK C′ ARCBG4;
CGC2 z0 ARE4BF C′ ARA9G4,BSD3 C ARCBG4FUC6AJEK C′
ARA9G4.
C C’
z0
w0
61/127
B4G0 2 CGC2BF C CBBRAIE8BM z0,AD z0 ARE4BF C′ CB
G4,BSD3 C ARCBG4FUC6AJEK C′ ARCBG4;
CGC2 z0 ARE4BF C′ ARA9G4,BSD3 C ARCBG4FUC6AJEK C′
ARA9G4.
C C’
z0
w0
61/127
B4G0 3 BF C H2AIF9D2DRBM z1,z2,z3,ENDMBF C′ H2AR
E4EPC7? w1,w2,w3.
(i) CGC2 C ED z1 → z2 → z3 ARBJE5DQ C′
ED w1 → w2 → w3 ARBJE5DNFVEE,BSD3 C ARCBG4FUC6AJ
EK C′ ARCBG4;
z1
z2
z3
w1
w2 w3 w
1
w2
w3
(ii) DNDREE,C ARCBG4FUC6AJEK C′ ARA9G4.
62/127
B4G0 3 BF C H2AIF9D2DRBM z1,z2,z3,ENDMBF C′ H2AR
E4EPC7? w1,w2,w3.
(i) CGC2 C ED z1 → z2 → z3 ARBJE5DQ C′
ED w1 → w2 → w3 ARBJE5DNFVEE,BSD3 C ARCBG4FUC6AJ
EK C′ ARCBG4;
z1
z2
z3
w1
w2 w3 w
1
w2
w3
(ii) DNDREE,C ARCBG4FUC6AJEK C′ ARA9G4.
62/127
B4G0 3 BF C H2AIF9D2DRBM z1,z2,z3,ENDMBF C′ H2AR
E4EPC7? w1,w2,w3.
(i) CGC2 C ED z1 → z2 → z3 ARBJE5DQ C′
ED w1 → w2 → w3 ARBJE5DNFVEE,BSD3 C ARCBG4FUC6AJ
EK C′ ARCBG4;
z1
z2
z3
w1
w2 w3 w
1
w2
w3
(ii) DNDREE,C ARCBG4FUC6AJEK C′ ARA9G4.
62/127
B4G0 4 BF C?ABEX,C′?DQDMARECF5CT,H2GBEPEQ
DMBHC6AJGQ C ARCBG4C6AJEK C′ ARB8E8H1ARDWEYEH.
(i) CGC2 C ED z1 → z2 → z3 ARBJE5DQ C′
ED w1 → w2 → w3 ARBJE5DNFVEE,BSD3 C ARCBG4FUC6AJ
EK C′ AR (a) H1;
z1
z2
z3 C
w1
w2
w3 (a)
(b) C’
w1
w2
w3
(a)
(b) C’
(ii) DNDREE,C ARCBG4FUC6AJEK C′ AR (b) H1.
63/127
B4G0 4 BF C?ABEX,C′?DQDMARECF5CT,H2GBEPEQ
DMBHC6AJGQ C ARCBG4C6AJEK C′ ARB8E8H1ARDWEYEH.
(i) CGC2 C ED z1 → z2 → z3 ARBJE5DQ C′
ED w1 → w2 → w3 ARBJE5DNFVEE,BSD3 C ARCBG4FUC6AJ
EK C′ AR (a) H1;
z1
z2
z3 C
w1
w2
w3 (a)
(b) C’
w1
w2
w3
(a)
(b) C’
(ii) DNDREE,C ARCBG4FUC6AJEK C′ AR (b) H1.
63/127
B4G0 4 BF C?ABEX,C′?DQDMARECF5CT,H2GBEPEQ
DMBHC6AJGQ C ARCBG4C6AJEK C′ ARB8E8H1ARDWEYEH.
(i) CGC2 C ED z1 → z2 → z3 ARBJE5DQ C′
ED w1 → w2 → w3 ARBJE5DNFVEE,BSD3 C ARCBG4FUC6AJ
EK C′ AR (a) H1;
z1
z2
z3 C
w1
w2
w3 (a)
(b) C’
w1
w2
w3
(a)
(b) C’
(ii) DNDREE,C ARCBG4FUC6AJEK C′ AR (b) H1.
63/127
B4G0 4 BF C?ABEX,C′?DQDMARECF5CT,H2GBEPEQ
DMBHC6AJGQ C ARCBG4C6AJEK C′ ARB8E8H1ARDWEYEH.
(i) CGC2 C ED z1 → z2 → z3 ARBJE5DQ C′
ED w1 → w2 → w3 ARBJE5DNFVEE,BSD3 C ARCBG4FUC6AJ
EK C′ AR (a) H1;
z1
z2
z3 C
w1
w2
w3 (a)
(b) C’
w1
w2
w3
(a)
(b) C’
(ii) DNDREE,C ARCBG4FUC6AJEK C′ AR (b) H1.
63/127
B3D9EPEQDMBHC6AJARBTDRETDRB4G0:
(1) A8H0DRABEXH2DACZBMC6AJEKFFEHAFBMEE,GXH0DRAB
EXARD0EKDWEKARERDXC6AJEKH0DRABD0EKDWEKARERDX.
D D’
64/127
B3D9EPEQDMBHC6AJARBTDRETDRB4G0:
(1) A8H0DRABEXH2DACZBMC6AJEKFFEHAFBMEE,GXH0DRAB
EXARD0EKDWEKARERDXC6AJEKH0DRABD0EKDWEKARERDX.
D D’
64/127
(2) A8H0DRABEXH2CZE8DRBMC6AJEKFFEHAFBMEE,GXH0DR
ABEXARD0EKDWEKARERDXC6AJEKE8DRABD0DQE8DQDMEKDWEKAR
ERDX.
D D’
∞ z0
65/127
(3) A8H0DRABEXA2BMELARE8DRC6AJEKFFEHAFBMEE,GXH0
DRABEXARD0EKDWEKARERDXC6AJEK?BBERDX.
D D’
z0 ∞
66/127
EPEQDMBHC6AJGNFGARH0CPEVC3:
EVC3 1,EUDJ z EYEHH2ARERDX D G2EPEQDMBHC6
AJ w = f(z),EMBF w = f(z) ARC6AJCT D ARE4DX G.
EVC3 2,EUDJ z EYEHH2ARERDX D G2 w EYEHH2ERDX G,
GHAKGQ D C6AJ? G AREPEQDMBHC6AJ w = f(z).
67/127
EPEQDMBHC6AJGNFGARH0CPEVC3:
EVC3 1,EUDJ z EYEHH2ARERDX D G2EPEQDMBHC6
AJ w = f(z),EMBF w = f(z) ARC6AJCT D ARE4DX G.
EVC3 2,EUDJ z EYEHH2ARERDX D G2 w EYEHH2ERDX G,
GHAKGQ D C6AJ? G AREPEQDMBHC6AJ w = f(z).
67/127
EPEQDMBHC6AJGNFGARH0CPEVC3:
EVC3 1,EUDJ z EYEHH2ARERDX D G2EPEQDMBHC6
AJ w = f(z),EMBF w = f(z) ARC6AJCT D ARE4DX G.
EVC3 2,EUDJ z EYEHH2ARERDX D G2 w EYEHH2ERDX G,
GHAKGQ D C6AJ? G AREPEQDMBHC6AJ w = f(z).
67/127
EVC3 1,EUDJ z EYEHH2ARERDX D G2EPEQDMBHC6
AJ w = f(z),EMBF w = f(z) ARC6AJCT D ARE4DX G.
E0DF,H0DRG2F9:
1,EMFTERDX D ARBQBAARE4;
2,DUGABQBABJE5B4F9E4DX G,F9GUFRC4DJFGE8AABTFRAR
BMARE4B4F9E4DX G.
68/127
EVC3 1,EUDJ z EYEHH2ARERDX D G2EPEQDMBHC6
AJ w = f(z),EMBF w = f(z) ARC6AJCT D ARE4DX G.
E0DF,H0DRG2F9:
1,EMFTERDX D ARBQBAARE4;
2,DUGABQBABJE5B4F9E4DX G,F9GUFRC4DJFGE8AABTFRAR
BMARE4B4F9E4DX G.
68/127
EVC3 1,EUDJ z EYEHH2ARERDX D G2EPEQDMBHC6
AJ w = f(z),EMBF w = f(z) ARC6AJCT D ARE4DX G.
E0DF,H0DRG2F9:
1,EMFTERDX D ARBQBAARE4;
2,DUGABQBABJE5B4F9E4DX G,F9GUFRC4DJFGE8AABTFRAR
BMARE4B4F9E4DX G.
68/127
EVC3 1,EUDJ z EYEHH2ARERDX D G2EPEQDMBHC6
AJ w = f(z),EMBF w = f(z) ARC6AJCT D ARE4DX G.
E0DF,H0DRG2F9:
1,EMFTERDX D ARBQBAARE4;
2,DUGABQBABJE5B4F9E4DX G,F9GUFRC4DJFGE8AABTFRAR
BMARE4B4F9E4DX G.
68/127
DI 1 EPEQDMBHC6AJ w = z?iz + i GQ?BB
DX D,Im(z) > 0∩Re(z) > 0 C6AJ?EFD3ERDX?
y
O x
i
1
(z)
69/127
B5FGCZ,EMFT D ARBQBAARE4.
BFH2DWC2F2H2AID2DRBM,0,i,∞,ENDMARE4:
w,0?→?1;
i?→ 0;
∞?→ 1;
DUGAEPEQDMBHC6AJARA2ABBH,w = z?iz + i D6H2DWC2F2C6AJ
w EYEHH2 u F2H2ARERD1 [?1,1].
y
O x
i
1
(z) v
O
u
1 1
i
(w)
70/127
B5FGCZ,EMFT D ARBQBAARE4.
BFH2DWC2F2H2AID2DRBM,0,i,∞,ENDMARE4:
w,0?→?1;
i?→ 0;
∞?→ 1;
DUGAEPEQDMBHC6AJARA2ABBH,w = z?iz + i D6H2DWC2F2C6AJ
w EYEHH2 u F2H2ARERD1 [?1,1].
y
O x
i
1
(z) v
O
u
1 1
i
(w)
70/127
B5FGCZ,EMFT D ARBQBAARE4.
BFH2DWC2F2H2AID2DRBM,0,i,∞,ENDMARE4:
w,0?→?1;
i?→ 0;
∞?→ 1;
DUGAEPEQDMBHC6AJARA2ABBH,w = z?iz + i D6H2DWC2F2C6AJ
w EYEHH2 u F2H2ARERD1 [?1,1].
y
O x
i
1
(z) v
O
u
1 1
i
(w)
70/127
BFD1DWEIF2H2AID2DRBM,0,1,∞,ENDMARE4:
w,0?→?1;
1?→?i;
∞?→ 1;
DUGAEPEQDMBHC6AJARA2ABBH,w = z?iz + i D6D1DWEIF2C6AJ
w EYEHH2C4?1,?i,1 ARABEX.
y
O x
i
1
(z) v
O
u
1 1
i
(w)
71/127
BFD1DWEIF2H2AID2DRBM,0,1,∞,ENDMARE4:
w,0?→?1;
1?→?i;
∞?→ 1;
DUGAEPEQDMBHC6AJARA2ABBH,w = z?iz + i D6D1DWEIF2C6AJ
w EYEHH2C4?1,?i,1 ARABEX.
y
O x
i
1
(z) v
O
u
1 1
i
(w)
71/127
B8A0,DUGABJE5B4F9E4DX G:
DUGAEPEQDMBHC6AJARA2ABBH,EDBQBABJE5,E4ERDX G
F0 w EYEHH2ARCTDWDRH0EGABCBG4:
|w| < 1,Im(w) < 0.
y
O x
i
1
(z) v
O
u
1 1
i
(w)
72/127
B8A0,DUGABJE5B4F9E4DX G:
DUGAEPEQDMBHC6AJARA2ABBH,EDBQBABJE5,E4ERDX G
F0 w EYEHH2ARCTDWDRH0EGABCBG4:
|w| < 1,Im(w) < 0.
y
O x
i
1
(z) v
O
u
1 1
i
(w)
72/127
DI 2 ELB1EPC7BF z = 1 DQ z =?1,DWF7?
√2
ARAJAB
D0EKDWEKARERDX D,BFC6AJ w = z?iz + i CTC6AJEKEFD3ER
DX?
73/127
B5 FGCZ,EMFTBQBA D ARE4.
BFD1DWABD0H2GHD2DRBM,?i,√2?1,i,ENDMARE4:
w,?i?→ ∞,
√2?1?→ (1?√2) + i(1?√2)
2?√2,
i?→ 0,
GXAAE4BMFTBFBDD2E6DLAR?EYEPDM C′1 H2.
(z) (w) y
x
v
u
C2 C1
C2’
C1’
DUGAEPEQDMBHC6AJARA2ABBH,C1 C6AJEKBDD2E6DLAR?
EYEPDM C′1.
74/127
B5 FGCZ,EMFTBQBA D ARE4.
BFD1DWABD0H2GHD2DRBM,?i,√2?1,i,ENDMARE4:
w,?i?→ ∞,
√2?1?→ (1?√2) + i(1?√2)
2?√2,
i?→ 0,
GXAAE4BMFTBFBDD2E6DLAR?EYEPDM C′1 H2.
(z) (w) y
x
v
u
C2 C1
C2’
C1’
DUGAEPEQDMBHC6AJARA2ABBH,C1 C6AJEKBDD2E6DLAR?
EYEPDM C′1.
74/127
B5 FGCZ,EMFTBQBA D ARE4.
BFD1DWABD0H2GHD2DRBM,?i,√2?1,i,ENDMARE4:
w,?i?→ ∞,
√2?1?→ (1?√2) + i(1?√2)
2?√2,
i?→ 0,
GXAAE4BMFTBFBDD2E6DLAR?EYEPDM C′1 H2.
(z) (w) y
x
v
u
C2 C1
C2’
C1’
DUGAEPEQDMBHC6AJARA2ABBH,C1 C6AJEKBDD2E6DLAR?
EYEPDM C′1.
74/127
B5 FGCZ,EMFTBQBA D ARE4.
BFD1DWABD0H2GHD2DRBM,?i,√2?1,i,ENDMARE4:
w,?i?→ ∞,
√2?1?→ (1?√2) + i(1?√2)
2?√2,
i?→ 0,
GXAAE4BMFTBFBDD2E6DLAR?EYEPDM C′1 H2.
(z) (w) y
x
v
u
C2 C1
C2’
C1’
DUGAEPEQDMBHC6AJARA2ABBH,C1 C6AJEKBDD2E6DLAR?
EYEPDM C′1.
74/127
B5 FGCZ,EMFTBQBA D ARE4.
BFD1DWABD0H2GHD2DRBM,?i,√2?1,i,ENDMARE4:
w,?i?→ ∞,
√2?1?→ (1?√2) + i(1?√2)
2?√2,
i?→ 0,
GXAAE4BMFTBFBDD2E6DLAR?EYEPDM C′1 H2.
(z) (w) y
x
v
u
C2 C1
C2’
C1’
DUGAEPEQDMBHC6AJARA2ABBH,C1 C6AJEKBDD2E6DLAR?
EYEPDM C′1.
74/127
FVD2,BFDQDWABD0H2GHD2DRBM,?i,?(√2?1),i,ENDM
ARE4:
w,?i?→ ∞,
(√2?1)?→ (1?
√2)?i(1?√2)
2?√2,
i?→ 0,
GXAAE4BMFTBFBDAJE6DLAR?EYEPDM C′2 H2.
(z) (w) y
x
v
u
C2 C1
C2’
C1’
DUGAEPEQDMBHC6AJARA2ABBH,C2 C6AJEKBDAJE6DLAR?
EYEPDM C′2.
75/127
FVD2,BFDQDWABD0H2GHD2DRBM,?i,?(√2?1),i,ENDM
ARE4:
w,?i?→ ∞,
(√2?1)?→ (1?
√2)?i(1?√2)
2?√2,
i?→ 0,
GXAAE4BMFTBFBDAJE6DLAR?EYEPDM C′2 H2.
(z) (w) y
x
v
u
C2 C1
C2’
C1’
DUGAEPEQDMBHC6AJARA2ABBH,C2 C6AJEKBDAJE6DLAR?
EYEPDM C′2.
75/127
FVD2,BFDQDWABD0H2GHD2DRBM,?i,?(√2?1),i,ENDM
ARE4:
w,?i?→ ∞,
(√2?1)?→ (1?
√2)?i(1?√2)
2?√2,
i?→ 0,
GXAAE4BMFTBFBDAJE6DLAR?EYEPDM C′2 H2.
(z) (w) y
x
v
u
C2 C1
C2’
C1’
DUGAEPEQDMBHC6AJARA2ABBH,C2 C6AJEKBDAJE6DLAR?
EYEPDM C′2.
75/127
FVD2,BFDQDWABD0H2GHD2DRBM,?i,?(√2?1),i,ENDM
ARE4:
w,?i?→ ∞,
(√2?1)?→ (1?
√2)?i(1?√2)
2?√2,
i?→ 0,
GXAAE4BMFTBFBDAJE6DLAR?EYEPDM C′2 H2.
(z) (w) y
x
v
u
C2 C1
C2’
C1’
DUGAEPEQDMBHC6AJARA2ABBH,C2 C6AJEKBDAJE6DLAR?
EYEPDM C′2.
75/127
DLA0,FRC4DJFGB8AABTFRBMARE4B4F9E4DX G:
BFERDX D CBG4FCC2BM z = 0,ENARE4:
w,0?→?1
E4BMBF x F2ARC3DWF2H2,EKEX G DUE5 x F2ARC3DWF2.
(z) (w) y
x
v
u
C2 C1
C2’
C1’
D2H2EKGB,ERDXBFC6AJCTARE4ERDXF0E8DR?BBERDX,BQ
BAF0BDAJG2BDD2E6DLAR?EYEPDM,DPDUE5 x F2ARC3DWF2.
76/127
DLA0,FRC4DJFGB8AABTFRBMARE4B4F9E4DX G:
BFERDX D CBG4FCC2BM z = 0,ENARE4:
w,0?→?1
E4BMBF x F2ARC3DWF2H2,EKEX G DUE5 x F2ARC3DWF2.
(z) (w) y
x
v
u
C2 C1
C2’
C1’
D2H2EKGB,ERDXBFC6AJCTARE4ERDXF0E8DR?BBERDX,BQ
BAF0BDAJG2BDD2E6DLAR?EYEPDM,DPDUE5 x F2ARC3DWF2.
76/127
DLA0,FRC4DJFGB8AABTFRBMARE4B4F9E4DX G:
BFERDX D CBG4FCC2BM z = 0,ENARE4:
w,0?→?1
E4BMBF x F2ARC3DWF2H2,EKEX G DUE5 x F2ARC3DWF2.
(z) (w) y
x
v
u
C2 C1
C2’
C1’
D2H2EKGB,ERDXBFC6AJCTARE4ERDXF0E8DR?BBERDX,BQ
BAF0BDAJG2BDD2E6DLAR?EYEPDM,DPDUE5 x F2ARC3DWF2.
76/127
DLA0,FRC4DJFGB8AABTFRBMARE4B4F9E4DX G:
BFERDX D CBG4FCC2BM z = 0,ENARE4:
w,0?→?1
E4BMBF x F2ARC3DWF2H2,EKEX G DUE5 x F2ARC3DWF2.
(z) (w) y
x
v
u
C2 C1
C2’
C1’
D2H2EKGB,ERDXBFC6AJCTARE4ERDXF0E8DR?BBERDX,BQ
BAF0BDAJG2BDD2E6DLAR?EYEPDM,DPDUE5 x F2ARC3DWF2.
76/127
DI 3 ERDX D,Re(z) > 0,0 < lm(z) < 1 BFC6
AJ w =
i
z CTC6AJEKEFD3ERDX?
O
i
y
x
(z)
C3
C1
C2
77/127
B5 FGCZ,EMFTBQBA D ARE4:
BF C1 GHD2DRBM,0,i2,i,ENDMARE4:
w = iz,0?→ ∞,
i
2?→ 2,
i?→ 1,
DUGAEPEQDMBHC6AJARA2ABBH,C1 C6AJEK w EYEHH2AR C′1,
BP w EYEH u F2H2ARERD1 [1,+∞).
O
i
y
x
(z)
C3
C1
C2 v
O u 1 1/2
C1’
C2’ C3’
(w)
78/127
B5 FGCZ,EMFTBQBA D ARE4:
BF C1 GHD2DRBM,0,i2,i,ENDMARE4:
w = iz,0?→ ∞,
i
2?→ 2,
i?→ 1,
DUGAEPEQDMBHC6AJARA2ABBH,C1 C6AJEK w EYEHH2AR C′1,
BP w EYEH u F2H2ARERD1 [1,+∞).
O
i
y
x
(z)
C3
C1
C2 v
O u 1 1/2
C1’
C2’ C3’
(w)
78/127
B5 FGCZ,EMFTBQBA D ARE4:
BF C1 GHD2DRBM,0,i2,i,ENDMARE4:
w = iz,0?→ ∞,
i
2?→ 2,
i?→ 1,
DUGAEPEQDMBHC6AJARA2ABBH,C1 C6AJEK w EYEHH2AR C′1,
BP w EYEH u F2H2ARERD1 [1,+∞).
O
i
y
x
(z)
C3
C1
C2 v
O u 1 1/2
C1’
C2’ C3’
(w)
78/127
BF C2 GHD2DRBM,i,1 + i,∞,ENDMARE4:
w = iz,i?→ 1,
1 + i?→ 12 + 12i,
∞?→ 0,
DUGAEPEQDMBHC6AJARA2ABBH,C2 C6AJEK w EYEHH2ARFRC4
BM 1,12 + 12i,0 ARDWABEX C′2.
O
i
y
x
(z)
C3
C1
C2 v
O u 1 1/2
C1’
C2’ C3’
(w)
79/127
BF C2 GHD2DRBM,i,1 + i,∞,ENDMARE4:
w = iz,i?→ 1,
1 + i?→ 12 + 12i,
∞?→ 0,
DUGAEPEQDMBHC6AJARA2ABBH,C2 C6AJEK w EYEHH2ARFRC4
BM 1,12 + 12i,0 ARDWABEX C′2.
O
i
y
x
(z)
C3
C1
C2 v
O u 1 1/2
C1’
C2’ C3’
(w)
79/127
BF C2 GHD2DRBM,i,1 + i,∞,ENDMARE4:
w = iz,i?→ 1,
1 + i?→ 12 + 12i,
∞?→ 0,
DUGAEPEQDMBHC6AJARA2ABBH,C2 C6AJEK w EYEHH2ARFRC4
BM 1,12 + 12i,0 ARDWABEX C′2.
O
i
y
x
(z)
C3
C1
C2 v
O u 1 1/2
C1’
C2’ C3’
(w)
79/127
BF C3 GHD2DRBM,0,1,∞,ENDMARE4:
w = iz,0?→ ∞,
1?→ i,
∞?→ 0,
DUGAEPEQDMBHC6AJARA2ABBH,C3 C6AJEK w EYEHH2ARH2DW
C2F2 C′3.
O
i
y
x
(z)
C3
C1
C2 v
O u 1 1/2
C1’
C2’ C3’
(w)
80/127
BF C3 GHD2DRBM,0,1,∞,ENDMARE4:
w = iz,0?→ ∞,
1?→ i,
∞?→ 0,
DUGAEPEQDMBHC6AJARA2ABBH,C3 C6AJEK w EYEHH2ARH2DW
C2F2 C′3.
O
i
y
x
(z)
C3
C1
C2 v
O u 1 1/2
C1’
C2’ C3’
(w)
80/127
BF C3 GHD2DRBM,0,1,∞,ENDMARE4:
w = iz,0?→ ∞,
1?→ i,
∞?→ 0,
DUGAEPEQDMBHC6AJARA2ABBH,C3 C6AJEK w EYEHH2ARH2DW
C2F2 C′3.
O
i
y
x
(z)
C3
C1
C2 v
O u 1 1/2
C1’
C2’ C3’
(w)
80/127
DLA0,FRC4DJFGBQBAARBJE5B4F9E4DX G:
O
i
y
x
(z)
C3
C1
C2 v
O 1/2 1 u
C1’
C2’ C3’
(w)
81/127
EVC3 2,EUDJ z EYEHH2ARERDX D G2 w EYEHH2ERDX G,
GHAKGQ D C6AJ? G AREPEQDMBHC6AJ w = f(z).
E0DF,DUGA D G2 G ARBBBZDQB3CG,GHFTDRC6H0DRERDX
D1B3CGARF8GI.
CCB3CGE8DOG0CDCME8DRDFH0F8GIC4ER,C1DRDGCME8AABN
BABVDQDUCPDQ,DLA0GHCGGDBH,DGCMBXG4B3CGEMAQF8GI.
82/127
EVC3 2,EUDJ z EYEHH2ARERDX D G2 w EYEHH2ERDX G,
GHAKGQ D C6AJ? G AREPEQDMBHC6AJ w = f(z).
E0DF,DUGA D G2 G ARBBBZDQB3CG,GHFTDRC6H0DRERDX
D1B3CGARF8GI.
CCB3CGE8DOG0CDCME8DRDFH0F8GIC4ER,C1DRDGCME8AABN
BABVDQDUCPDQ,DLA0GHCGGDBH,DGCMBXG4B3CGEMAQF8GI.
82/127
EVC3 2,EUDJ z EYEHH2ARERDX D G2 w EYEHH2ERDX G,
GHAKGQ D C6AJ? G AREPEQDMBHC6AJ w = f(z).
E0DF,DUGA D G2 G ARBBBZDQB3CG,GHFTDRC6H0DRERDX
D1B3CGARF8GI.
CCB3CGE8DOG0CDCME8DRDFH0F8GIC4ER,C1DRDGCME8AABN
BABVDQDUCPDQ,DLA0GHCGGDBH,DGCMBXG4B3CGEMAQF8GI.
82/127
H0ERBNBAERDXDND1ARA1BBC6AJ
CPBA 1 EMGQH2DWEYEH Im(z) > 0 C6AJEKH0EG
AB|w| < 1 AREPEQDMBHC6AJ.
y
x
O
(z) v
u O
(w)
1
i
i
B5 H2DWEYEHDWEXDBDUF0DWF7FFEHDWARABDX,ADEPEQDM
BHC6AJGCCZA2ABBH,FRCSENBJCDGQH2DWEYEH Im(z) > 0 C6
AJEKH0EGAB|w| < 1.
83/127
H0ERBNBAERDXDND1ARA1BBC6AJ
CPBA 1 EMGQH2DWEYEH Im(z) > 0 C6AJEKH0EG
AB|w| < 1 AREPEQDMBHC6AJ.
y
x
O
(z) v
u O
(w)
1
i
i
B5 H2DWEYEHDWEXDBDUF0DWF7FFEHDWARABDX,ADEPEQDM
BHC6AJGCCZA2ABBH,FRCSENBJCDGQH2DWEYEH Im(z) > 0 C6
AJEKH0EGAB|w| < 1.
83/127
H0ERBNBAERDXDND1ARA1BBC6AJ
CPBA 1 EMGQH2DWEYEH Im(z) > 0 C6AJEKH0EG
AB|w| < 1 AREPEQDMBHC6AJ.
y
x
O
(z) v
u O
(w)
1
i
i
B5 H2DWEYEHDWEXDBDUF0DWF7FFEHDWARABDX,ADEPEQDM
BHC6AJGCCZA2ABBH,FRCSENBJCDGQH2DWEYEH Im(z) > 0 C6
AJEKH0EGAB|w| < 1.
83/127
H0ERBNBAERDXDND1ARA1BBC6AJ
CPBA 1 EMGQH2DWEYEH Im(z) > 0 C6AJEKH0EG
AB|w| < 1 AREPEQDMBHC6AJ.
y
x
O
(z) v
u O
(w)
1
i
i
B5 H2DWEYEHDWEXDBDUF0DWF7FFEHDWARABDX,ADEPEQDM
BHC6AJGCCZA2ABBH,FRCSENBJCDGQH2DWEYEH Im(z) > 0 C6
AJEKH0EGAB|w| < 1.
83/127
H0ERBNBAERDXDND1ARA1BBC6AJ
CPBA 1 EMGQH2DWEYEH Im(z) > 0 C6AJEKH0EG
AB|w| < 1 AREPEQDMBHC6AJ.
y
x
O
(z) v
u O
(w)
1
i
i
B5 H2DWEYEHDWEXDBDUF0DWF7FFEHDWARABDX,ADEPEQDM
BHC6AJGCCZA2ABBH,FRCSENBJCDGQH2DWEYEH Im(z) > 0 C6
AJEKH0EGAB|w| < 1.
83/127
CSD9H2DWEYEHD3CZE8BM z = λ C6AJEKH0EGABEX|w| = 1
ARABB1 w = 0,BP
w,z = λ?→ w = 0;
D5EIF2C6AJEKH0EGAB,AD z = λ DQ z = λ F0B3D9EIF2AR
E8GHGHEHBM,D9F0DUGAA2GHEHBH,ENDMARE4 (EPC7F0 w = 0
DQ w = ∞) F0B3D9ABEX|w| = 1 ARE8GHGHEHBM.
EKEXCZ
w,z = λ?→ w = 0;
z = λ?→ w = ∞.
EKEXEPEQDMBHC6AJGCCZCTAFARBBEQ:
w = k
parenleftbiggz?λ
z?λ
parenrightbigg
.
G1EL k F0DIGI.
84/127
CSD9H2DWEYEHD3CZE8BM z = λ C6AJEKH0EGABEX|w| = 1
ARABB1 w = 0,BP
w,z = λ?→ w = 0;
D5EIF2C6AJEKH0EGAB,AD z = λ DQ z = λ F0B3D9EIF2AR
E8GHGHEHBM,D9F0DUGAA2GHEHBH,ENDMARE4 (EPC7F0 w = 0
DQ w = ∞) F0B3D9ABEX|w| = 1 ARE8GHGHEHBM.
EKEXCZ
w,z = λ?→ w = 0;
z = λ?→ w = ∞.
EKEXEPEQDMBHC6AJGCCZCTAFARBBEQ:
w = k
parenleftbiggz?λ
z?λ
parenrightbigg
.
G1EL k F0DIGI.
84/127
CSD9H2DWEYEHD3CZE8BM z = λ C6AJEKH0EGABEX|w| = 1
ARABB1 w = 0,BP
w,z = λ?→ w = 0;
D5EIF2C6AJEKH0EGAB,AD z = λ DQ z = λ F0B3D9EIF2AR
E8GHGHEHBM,D9F0DUGAA2GHEHBH,ENDMARE4 (EPC7F0 w = 0
DQ w = ∞) F0B3D9ABEX|w| = 1 ARE8GHGHEHBM.
EKEXCZ
w,z = λ?→ w = 0;
z = λ?→ w = ∞.
EKEXEPEQDMBHC6AJGCCZCTAFARBBEQ:
w = k
parenleftbiggz?λ
z?λ
parenrightbigg
.
G1EL k F0DIGI.
84/127
CSD9H2DWEYEHD3CZE8BM z = λ C6AJEKH0EGABEX|w| = 1
ARABB1 w = 0,BP
w,z = λ?→ w = 0;
D5EIF2C6AJEKH0EGAB,AD z = λ DQ z = λ F0B3D9EIF2AR
E8GHGHEHBM,D9F0DUGAA2GHEHBH,ENDMARE4 (EPC7F0 w = 0
DQ w = ∞) F0B3D9ABEX|w| = 1 ARE8GHGHEHBM.
EKEXCZ
w,z = λ?→ w = 0;
z = λ?→ w = ∞.
EKEXEPEQDMBHC6AJGCCZCTAFARBBEQ:
w = k
parenleftbiggz?λ
z?λ
parenrightbigg
.
G1EL k F0DIGI.
84/127
CSD9H2DWEYEHD3CZE8BM z = λ C6AJEKH0EGABEX|w| = 1
ARABB1 w = 0,BP
w,z = λ?→ w = 0;
D5EIF2C6AJEKH0EGAB,AD z = λ DQ z = λ F0B3D9EIF2AR
E8GHGHEHBM,D9F0DUGAA2GHEHBH,ENDMARE4 (EPC7F0 w = 0
DQ w = ∞) F0B3D9ABEX|w| = 1 ARE8GHGHEHBM.
EKEXCZ
w,z = λ?→ w = 0;
z = λ?→ w = ∞.
EKEXEPEQDMBHC6AJGCCZCTAFARBBEQ:
w = k
parenleftbiggz?λ
z?λ
parenrightbigg
.
G1EL k F0DIGI.
84/127
CTEHB4F9DIGI k:
FR?|w| = |k|
vextendsinglevextendsingle
vextendsinglevextendsinglez?λ
z?λ
vextendsinglevextendsingle
vextendsinglevextendsingle,
ADEIF2H2ARBM z GHBTCGAB
EX|w| = 1 H2ARBM.
AI z = 0 E3CKH2EQ,AQ
1 = |w| = |k|
vextendsinglevextendsingle
vextendsinglevextendsingle0?λ
0?λ
vextendsinglevextendsingle
vextendsinglevextendsingle= |k|,
EKEX k = eiθ,G1EL θ F0BRFBEIGI.
FRCS,EKEMAREPEQDMBHC6AJARE8DOBBEQ?
w = eiθ
parenleftbiggz?λ
z?λ
parenrightbigg
,(Im(λ) > 0).
DRDN,BBCGH2EQAREPEQDMBHC6AJBJGQH2DWEY
EH Im(z) > 0 C6AJEKH0EGAB|w| < 1.
85/127
CTEHB4F9DIGI k:
FR?|w| = |k|
vextendsinglevextendsingle
vextendsinglevextendsinglez?λ
z?λ
vextendsinglevextendsingle
vextendsinglevextendsingle,
ADEIF2H2ARBM z GHBTCGAB
EX|w| = 1 H2ARBM.
AI z = 0 E3CKH2EQ,AQ
1 = |w| = |k|
vextendsinglevextendsingle
vextendsinglevextendsingle0?λ
0?λ
vextendsinglevextendsingle
vextendsinglevextendsingle= |k|,
EKEX k = eiθ,G1EL θ F0BRFBEIGI.
FRCS,EKEMAREPEQDMBHC6AJARE8DOBBEQ?
w = eiθ
parenleftbiggz?λ
z?λ
parenrightbigg
,(Im(λ) > 0).
DRDN,BBCGH2EQAREPEQDMBHC6AJBJGQH2DWEY
EH Im(z) > 0 C6AJEKH0EGAB|w| < 1.
85/127
CTEHB4F9DIGI k:
FR?|w| = |k|
vextendsinglevextendsingle
vextendsinglevextendsinglez?λ
z?λ
vextendsinglevextendsingle
vextendsinglevextendsingle,
ADEIF2H2ARBM z GHBTCGAB
EX|w| = 1 H2ARBM.
AI z = 0 E3CKH2EQ,AQ
1 = |w| = |k|
vextendsinglevextendsingle
vextendsinglevextendsingle0?λ
0?λ
vextendsinglevextendsingle
vextendsinglevextendsingle= |k|,
EKEX k = eiθ,G1EL θ F0BRFBEIGI.
FRCS,EKEMAREPEQDMBHC6AJARE8DOBBEQ?
w = eiθ
parenleftbiggz?λ
z?λ
parenrightbigg
,(Im(λ) > 0).
DRDN,BBCGH2EQAREPEQDMBHC6AJBJGQH2DWEY
EH Im(z) > 0 C6AJEKH0EGAB|w| < 1.
85/127
CTEHB4F9DIGI k:
FR?|w| = |k|
vextendsinglevextendsingle
vextendsinglevextendsinglez?λ
z?λ
vextendsinglevextendsingle
vextendsinglevextendsingle,
ADEIF2H2ARBM z GHBTCGAB
EX|w| = 1 H2ARBM.
AI z = 0 E3CKH2EQ,AQ
1 = |w| = |k|
vextendsinglevextendsingle
vextendsinglevextendsingle0?λ
0?λ
vextendsinglevextendsingle
vextendsinglevextendsingle= |k|,
EKEX k = eiθ,G1EL θ F0BRFBEIGI.
FRCS,EKEMAREPEQDMBHC6AJARE8DOBBEQ?
w = eiθ
parenleftbiggz?λ
z?λ
parenrightbigg
,(Im(λ) > 0).
DRDN,BBCGH2EQAREPEQDMBHC6AJBJGQH2DWEY
EH Im(z) > 0 C6AJEKH0EGAB|w| < 1.
85/127
CTEHB4F9DIGI k:
FR?|w| = |k|
vextendsinglevextendsingle
vextendsinglevextendsinglez?λ
z?λ
vextendsinglevextendsingle
vextendsinglevextendsingle,
ADEIF2H2ARBM z GHBTCGAB
EX|w| = 1 H2ARBM.
AI z = 0 E3CKH2EQ,AQ
1 = |w| = |k|
vextendsinglevextendsingle
vextendsinglevextendsingle0?λ
0?λ
vextendsinglevextendsingle
vextendsinglevextendsingle= |k|,
EKEX k = eiθ,G1EL θ F0BRFBEIGI.
FRCS,EKEMAREPEQDMBHC6AJARE8DOBBEQ?
w = eiθ
parenleftbiggz?λ
z?λ
parenrightbigg
,(Im(λ) > 0).
DRDN,BBCGH2EQAREPEQDMBHC6AJBJGQH2DWEY
EH Im(z) > 0 C6AJEKH0EGAB|w| < 1.
85/127
CTEHB4F9DIGI k:
FR?|w| = |k|
vextendsinglevextendsingle
vextendsinglevextendsinglez?λ
z?λ
vextendsinglevextendsingle
vextendsinglevextendsingle,
ADEIF2H2ARBM z GHBTCGAB
EX|w| = 1 H2ARBM.
AI z = 0 E3CKH2EQ,AQ
1 = |w| = |k|
vextendsinglevextendsingle
vextendsinglevextendsingle0?λ
0?λ
vextendsinglevextendsingle
vextendsinglevextendsingle= |k|,
EKEX k = eiθ,G1EL θ F0BRFBEIGI.
FRCS,EKEMAREPEQDMBHC6AJARE8DOBBEQ?
w = eiθ
parenleftbiggz?λ
z?λ
parenrightbigg
,(Im(λ) > 0).
DRDN,BBCGH2EQAREPEQDMBHC6AJBJGQH2DWEY
EH Im(z) > 0 C6AJEKH0EGAB|w| < 1.
85/127
CVDJC3:
1,GQCTDWEYEH Im(z) < 0 C6AJEKH0EGAB|w| < 1 AREP
EQDMBHC6AJ.
2,GQH2DWEYEH Im(z) > 0 C6AJEKH0EGABEX|w| = 1 AR
ARA9G4AREPEQDMBHC6AJ.
3,GQCTDWEYEH Im(z) < 0 C6AJEKH0EGABEX|w| = 1 AR
ARA9G4AREPEQDMBHC6AJ.
86/127
CVDJC3:
1,GQCTDWEYEH Im(z) < 0 C6AJEKH0EGAB|w| < 1 AREP
EQDMBHC6AJ.
2,GQH2DWEYEH Im(z) > 0 C6AJEKH0EGABEX|w| = 1 AR
ARA9G4AREPEQDMBHC6AJ.
3,GQCTDWEYEH Im(z) < 0 C6AJEKH0EGABEX|w| = 1 AR
ARA9G4AREPEQDMBHC6AJ.
86/127
CVDJC3:
1,GQCTDWEYEH Im(z) < 0 C6AJEKH0EGAB|w| < 1 AREP
EQDMBHC6AJ.
2,GQH2DWEYEH Im(z) > 0 C6AJEKH0EGABEX|w| = 1 AR
ARA9G4AREPEQDMBHC6AJ.
3,GQCTDWEYEH Im(z) < 0 C6AJEKH0EGABEX|w| = 1 AR
ARA9G4AREPEQDMBHC6AJ.
86/127
CVDJC3:
1,GQCTDWEYEH Im(z) < 0 C6AJEKH0EGAB|w| < 1 AREP
EQDMBHC6AJ.
2,GQH2DWEYEH Im(z) > 0 C6AJEKH0EGABEX|w| = 1 AR
ARA9G4AREPEQDMBHC6AJ.
3,GQCTDWEYEH Im(z) < 0 C6AJEKH0EGABEX|w| = 1 AR
ARA9G4AREPEQDMBHC6AJ.
86/127
DI 1 EMGQH2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1
DPGVDACMDR w(2i) = 0,argw′(2i) = 0 AREPEQDMBHC6AJ.
B5 EKEMAREPEQDMBHC6AJARE8DOBBEQ?
w = eiθ
parenleftbiggz?λ
z?λ
parenrightbigg
,(Im(λ) > 0).
FGCZB4F9 λ:
CSCMDR w(2i) = 0 DJ,EKEMARC6AJDRGQBM z = 2i C6AJ
EKH0EGABEXARABB1 w = 0,EKEX
w = eiθ
parenleftbiggz?2i
z + 2i
parenrightbigg
.
87/127
DI 1 EMGQH2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1
DPGVDACMDR w(2i) = 0,argw′(2i) = 0 AREPEQDMBHC6AJ.
B5 EKEMAREPEQDMBHC6AJARE8DOBBEQ?
w = eiθ
parenleftbiggz?λ
z?λ
parenrightbigg
,(Im(λ) > 0).
FGCZB4F9 λ:
CSCMDR w(2i) = 0 DJ,EKEMARC6AJDRGQBM z = 2i C6AJ
EKH0EGABEXARABB1 w = 0,EKEX
w = eiθ
parenleftbiggz?2i
z + 2i
parenrightbigg
.
87/127
DI 1 EMGQH2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1
DPGVDACMDR w(2i) = 0,argw′(2i) = 0 AREPEQDMBHC6AJ.
B5 EKEMAREPEQDMBHC6AJARE8DOBBEQ?
w = eiθ
parenleftbiggz?λ
z?λ
parenrightbigg
,(Im(λ) > 0).
FGCZB4F9 λ:
CSCMDR w(2i) = 0 DJ,EKEMARC6AJDRGQBM z = 2i C6AJ
EKH0EGABEXARABB1 w = 0,EKEX
w = eiθ
parenleftbiggz?2i
z + 2i
parenrightbigg
.
87/127
DI 1 EMGQH2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1
DPGVDACMDR w(2i) = 0,argw′(2i) = 0 AREPEQDMBHC6AJ.
B5 EKEMAREPEQDMBHC6AJARE8DOBBEQ?
w = eiθ
parenleftbiggz?λ
z?λ
parenrightbigg
,(Im(λ) > 0).
FGCZB4F9 λ:
CSCMDR w(2i) = 0 DJ,EKEMARC6AJDRGQBM z = 2i C6AJ
EKH0EGABEXARABB1 w = 0,EKEX
w = eiθ
parenleftbiggz?2i
z + 2i
parenrightbigg
.
87/127
DI 1 EMGQH2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1
DPGVDACMDR w(2i) = 0,argw′(2i) = 0 AREPEQDMBHC6AJ.
B5 EKEMAREPEQDMBHC6AJARE8DOBBEQ?
w = eiθ
parenleftbiggz?λ
z?λ
parenrightbigg
,(Im(λ) > 0).
FGCZB4F9 λ:
CSCMDR w(2i) = 0 DJ,EKEMARC6AJDRGQBM z = 2i C6AJ
EKH0EGABEXARABB1 w = 0,EKEX
w = eiθ
parenleftbiggz?2i
z + 2i
parenrightbigg
.
87/127
B8A0B4F9 θ:
FR? w′(z) = eiθ 4i(z + 2i)2,
APCZ w′(2i) = eiθ(?i4).
EKEX
argw′(2i) = argeiθ + arg(?i4) = θ + (?pi2) = 0,θ = pi2.
D9F0EKEMC6AJ?
w = i
parenleftbiggz?2i
z + 2i
parenrightbigg
.
88/127
B8A0B4F9 θ:
FR? w′(z) = eiθ 4i(z + 2i)2,
APCZ w′(2i) = eiθ(?i4).
EKEX
argw′(2i) = argeiθ + arg(?i4) = θ + (?pi2) = 0,θ = pi2.
D9F0EKEMC6AJ?
w = i
parenleftbiggz?2i
z + 2i
parenrightbigg
.
88/127
B8A0B4F9 θ:
FR? w′(z) = eiθ 4i(z + 2i)2,
APCZ w′(2i) = eiθ(?i4).
EKEX
argw′(2i) = argeiθ + arg(?i4) = θ + (?pi2) = 0,θ = pi2.
D9F0EKEMC6AJ?
w = i
parenleftbiggz?2i
z + 2i
parenrightbigg
.
88/127
B8A0B4F9 θ:
FR? w′(z) = eiθ 4i(z + 2i)2,
APCZ w′(2i) = eiθ(?i4).
EKEX
argw′(2i) = argeiθ + arg(?i4) = θ + (?pi2) = 0,θ = pi2.
D9F0EKEMC6AJ?
w = i
parenleftbiggz?2i
z + 2i
parenrightbigg
.
88/127
B8A0B4F9 θ:
FR? w′(z) = eiθ 4i(z + 2i)2,
APCZ w′(2i) = eiθ(?i4).
EKEX
argw′(2i) = argeiθ + arg(?i4) = θ + (?pi2) = 0,θ = pi2.
D9F0EKEMC6AJ?
w = i
parenleftbiggz?2i
z + 2i
parenrightbigg
.
88/127
B8A0B4F9 θ:
FR? w′(z) = eiθ 4i(z + 2i)2,
APCZ w′(2i) = eiθ(?i4).
EKEX
argw′(2i) = argeiθ + arg(?i4) = θ + (?pi2) = 0,θ = pi2.
D9F0EKEMC6AJ?
w = i
parenleftbiggz?2i
z + 2i
parenrightbigg
.
88/127
DI 2,EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDR f(i) = 0,f(?1) = 1 AREPEQDMBHC6AJ.
B5 D6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1 AREP
EQDMBHC6AJARE8DOC4EREQ?
w = f(z) = ei?
parenleftbiggz?λ
z?λ
parenrightbigg
(Im(λ) > 0)
FGCZB4F9 λ:
DUGA f(i) = 0,C3 λ = i,C3 w = ei?
parenleftbiggz?i
z?i
parenrightbigg
.
B8A0B4F9?:
DUGA f(?1) = 1,C3
ei?
parenleftbigg?1?i
1?i
parenrightbigg
= 1
EKEX ei? = 1.
D2H2EKGB,APEKEMC6AJ?w =?i
parenleftbiggz?i
z + i
parenrightbigg
.
89/127
DI 2,EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDR f(i) = 0,f(?1) = 1 AREPEQDMBHC6AJ.
B5 D6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1 AREP
EQDMBHC6AJARE8DOC4EREQ?
w = f(z) = ei?
parenleftbiggz?λ
z?λ
parenrightbigg
(Im(λ) > 0)
FGCZB4F9 λ:
DUGA f(i) = 0,C3 λ = i,C3 w = ei?
parenleftbiggz?i
z?i
parenrightbigg
.
B8A0B4F9?:
DUGA f(?1) = 1,C3
ei?
parenleftbigg?1?i
1?i
parenrightbigg
= 1
EKEX ei? = 1.
D2H2EKGB,APEKEMC6AJ?w =?i
parenleftbiggz?i
z + i
parenrightbigg
.
89/127
DI 2,EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDR f(i) = 0,f(?1) = 1 AREPEQDMBHC6AJ.
B5 D6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1 AREP
EQDMBHC6AJARE8DOC4EREQ?
w = f(z) = ei?
parenleftbiggz?λ
z?λ
parenrightbigg
(Im(λ) > 0)
FGCZB4F9 λ:
DUGA f(i) = 0,C3 λ = i,C3 w = ei?
parenleftbiggz?i
z?i
parenrightbigg
.
B8A0B4F9?:
DUGA f(?1) = 1,C3
ei?
parenleftbigg?1?i
1?i
parenrightbigg
= 1
EKEX ei? = 1.
D2H2EKGB,APEKEMC6AJ?w =?i
parenleftbiggz?i
z + i
parenrightbigg
.
89/127
DI 2,EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDR f(i) = 0,f(?1) = 1 AREPEQDMBHC6AJ.
B5 D6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1 AREP
EQDMBHC6AJARE8DOC4EREQ?
w = f(z) = ei?
parenleftbiggz?λ
z?λ
parenrightbigg
(Im(λ) > 0)
FGCZB4F9 λ:
DUGA f(i) = 0,C3 λ = i,C3 w = ei?
parenleftbiggz?i
z?i
parenrightbigg
.
B8A0B4F9?:
DUGA f(?1) = 1,C3
ei?
parenleftbigg?1?i
1?i
parenrightbigg
= 1
EKEX ei? = 1.
D2H2EKGB,APEKEMC6AJ?w =?i
parenleftbiggz?i
z + i
parenrightbigg
.
89/127
DI 2,EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDR f(i) = 0,f(?1) = 1 AREPEQDMBHC6AJ.
B5 D6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1 AREP
EQDMBHC6AJARE8DOC4EREQ?
w = f(z) = ei?
parenleftbiggz?λ
z?λ
parenrightbigg
(Im(λ) > 0)
FGCZB4F9 λ:
DUGA f(i) = 0,C3 λ = i,C3 w = ei?
parenleftbiggz?i
z?i
parenrightbigg
.
B8A0B4F9?:
DUGA f(?1) = 1,C3
ei?
parenleftbigg?1?i
1?i
parenrightbigg
= 1
EKEX ei? = 1.
D2H2EKGB,APEKEMC6AJ?w =?i
parenleftbiggz?i
z + i
parenrightbigg
.
89/127
DI 2,EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDR f(i) = 0,f(?1) = 1 AREPEQDMBHC6AJ.
B5 D6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1 AREP
EQDMBHC6AJARE8DOC4EREQ?
w = f(z) = ei?
parenleftbiggz?λ
z?λ
parenrightbigg
(Im(λ) > 0)
FGCZB4F9 λ:
DUGA f(i) = 0,C3 λ = i,C3 w = ei?
parenleftbiggz?i
z?i
parenrightbigg
.
B8A0B4F9?:
DUGA f(?1) = 1,C3
ei?
parenleftbigg?1?i
1?i
parenrightbigg
= 1
EKEX ei? = 1.
D2H2EKGB,APEKEMC6AJ?w =?i
parenleftbiggz?i
z + i
parenrightbigg
.
89/127
DI 2,EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDR f(i) = 0,f(?1) = 1 AREPEQDMBHC6AJ.
B5 D6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1 AREP
EQDMBHC6AJARE8DOC4EREQ?
w = f(z) = ei?
parenleftbiggz?λ
z?λ
parenrightbigg
(Im(λ) > 0)
FGCZB4F9 λ:
DUGA f(i) = 0,C3 λ = i,C3 w = ei?
parenleftbiggz?i
z?i
parenrightbigg
.
B8A0B4F9?:
DUGA f(?1) = 1,C3
ei?
parenleftbigg?1?i
1?i
parenrightbigg
= 1
EKEX ei? = 1.
D2H2EKGB,APEKEMC6AJ?w =?i
parenleftbiggz?i
z + i
parenrightbigg
.
89/127
DI 2,EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDR f(i) = 0,f(?1) = 1 AREPEQDMBHC6AJ.
B5 D6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1 AREP
EQDMBHC6AJARE8DOC4EREQ?
w = f(z) = ei?
parenleftbiggz?λ
z?λ
parenrightbigg
(Im(λ) > 0)
FGCZB4F9 λ:
DUGA f(i) = 0,C3 λ = i,C3 w = ei?
parenleftbiggz?i
z?i
parenrightbigg
.
B8A0B4F9?:
DUGA f(?1) = 1,C3
ei?
parenleftbigg?1?i
1?i
parenrightbigg
= 1
EKEX ei? = 1.
D2H2EKGB,APEKEMC6AJ?w =?i
parenleftbiggz?i
z + i
parenrightbigg
.
89/127
DI 2,EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDR f(i) = 0,f(?1) = 1 AREPEQDMBHC6AJ.
B5 D6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1 AREP
EQDMBHC6AJARE8DOC4EREQ?
w = f(z) = ei?
parenleftbiggz?λ
z?λ
parenrightbigg
(Im(λ) > 0)
FGCZB4F9 λ:
DUGA f(i) = 0,C3 λ = i,C3 w = ei?
parenleftbiggz?i
z?i
parenrightbigg
.
B8A0B4F9?:
DUGA f(?1) = 1,C3
ei?
parenleftbigg?1?i
1?i
parenrightbigg
= 1
EKEX ei? = 1.
D2H2EKGB,APEKEMC6AJ?w =?i
parenleftbiggz?i
z + i
parenrightbigg
.
89/127
DI 3 EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDRf(1) = 1,f(i) = 15AREPEQDMBHC6AJ.
B5 CSD9 f(i) = 1√5,
BHFBAK,z = i B3D9H2DWEYEHARBQBA Im(z) = 0 ARGHEH
BMF0,i =?i.
ADw = 1√5B3D9H0EGABEX|w| = 1 ARGHEHBMF0,√5.
DUGAEPEQDMBHC6AJARA2GHEHBHAQ
f(?i) = √5.
EKEX,EPEQDMBHC6AJ w = f(z) GQD2BM 1,i,?i EDCVC6AJ
EK1,√5,1√5.
DUGADXE8B4F9EPEQDMBHC6AJARCMDRDWAQparenleftBigg
w?1/√5
w?√5
parenrightBiggslashbiggparenleftBigg
1?1/√5
1?√5
parenrightBigg
=
parenleftbiggz?i
z + i
parenrightbigg
/
parenleftbigg1?i
1 + i
parenrightbigg
.
90/127
DI 3 EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDRf(1) = 1,f(i) = 15AREPEQDMBHC6AJ.
B5 CSD9 f(i) = 1√5,
BHFBAK,z = i B3D9H2DWEYEHARBQBA Im(z) = 0 ARGHEH
BMF0,i =?i.
ADw = 1√5B3D9H0EGABEX|w| = 1 ARGHEHBMF0,√5.
DUGAEPEQDMBHC6AJARA2GHEHBHAQ
f(?i) = √5.
EKEX,EPEQDMBHC6AJ w = f(z) GQD2BM 1,i,?i EDCVC6AJ
EK1,√5,1√5.
DUGADXE8B4F9EPEQDMBHC6AJARCMDRDWAQparenleftBigg
w?1/√5
w?√5
parenrightBiggslashbiggparenleftBigg
1?1/√5
1?√5
parenrightBigg
=
parenleftbiggz?i
z + i
parenrightbigg
/
parenleftbigg1?i
1 + i
parenrightbigg
.
90/127
DI 3 EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDRf(1) = 1,f(i) = 15AREPEQDMBHC6AJ.
B5 CSD9 f(i) = 1√5,
BHFBAK,z = i B3D9H2DWEYEHARBQBA Im(z) = 0 ARGHEH
BMF0,i =?i.
ADw = 1√5B3D9H0EGABEX|w| = 1 ARGHEHBMF0,√5.
DUGAEPEQDMBHC6AJARA2GHEHBHAQ
f(?i) = √5.
EKEX,EPEQDMBHC6AJ w = f(z) GQD2BM 1,i,?i EDCVC6AJ
EK1,√5,1√5.
DUGADXE8B4F9EPEQDMBHC6AJARCMDRDWAQparenleftBigg
w?1/√5
w?√5
parenrightBiggslashbiggparenleftBigg
1?1/√5
1?√5
parenrightBigg
=
parenleftbiggz?i
z + i
parenrightbigg
/
parenleftbigg1?i
1 + i
parenrightbigg
.
90/127
DI 3 EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDRf(1) = 1,f(i) = 15AREPEQDMBHC6AJ.
B5 CSD9 f(i) = 1√5,
BHFBAK,z = i B3D9H2DWEYEHARBQBA Im(z) = 0 ARGHEH
BMF0,i =?i.
ADw = 1√5B3D9H0EGABEX|w| = 1 ARGHEHBMF0,√5.
DUGAEPEQDMBHC6AJARA2GHEHBHAQ
f(?i) = √5.
EKEX,EPEQDMBHC6AJ w = f(z) GQD2BM 1,i,?i EDCVC6AJ
EK1,√5,1√5.
DUGADXE8B4F9EPEQDMBHC6AJARCMDRDWAQparenleftBigg
w?1/√5
w?√5
parenrightBiggslashbiggparenleftBigg
1?1/√5
1?√5
parenrightBigg
=
parenleftbiggz?i
z + i
parenrightbigg
/
parenleftbigg1?i
1 + i
parenrightbigg
.
90/127
DI 3 EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDRf(1) = 1,f(i) = 15AREPEQDMBHC6AJ.
B5 CSD9 f(i) = 1√5,
BHFBAK,z = i B3D9H2DWEYEHARBQBA Im(z) = 0 ARGHEH
BMF0,i =?i.
ADw = 1√5B3D9H0EGABEX|w| = 1 ARGHEHBMF0,√5.
DUGAEPEQDMBHC6AJARA2GHEHBHAQ
f(?i) = √5.
EKEX,EPEQDMBHC6AJ w = f(z) GQD2BM 1,i,?i EDCVC6AJ
EK1,√5,1√5.
DUGADXE8B4F9EPEQDMBHC6AJARCMDRDWAQparenleftBigg
w?1/√5
w?√5
parenrightBiggslashbiggparenleftBigg
1?1/√5
1?√5
parenrightBigg
=
parenleftbiggz?i
z + i
parenrightbigg
/
parenleftbigg1?i
1 + i
parenrightbigg
.
90/127
DI 3 EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDRf(1) = 1,f(i) = 15AREPEQDMBHC6AJ.
B5 CSD9 f(i) = 1√5,
BHFBAK,z = i B3D9H2DWEYEHARBQBA Im(z) = 0 ARGHEH
BMF0,i =?i.
ADw = 1√5B3D9H0EGABEX|w| = 1 ARGHEHBMF0,√5.
DUGAEPEQDMBHC6AJARA2GHEHBHAQ
f(?i) = √5.
EKEX,EPEQDMBHC6AJ w = f(z) GQD2BM 1,i,?i EDCVC6AJ
EK1,√5,1√5.
DUGADXE8B4F9EPEQDMBHC6AJARCMDRDWAQparenleftBigg
w?1/√5
w?√5
parenrightBiggslashbiggparenleftBigg
1?1/√5
1?√5
parenrightBigg
=
parenleftbiggz?i
z + i
parenrightbigg
/
parenleftbigg1?i
1 + i
parenrightbigg
.
90/127
DI 3 EMD6H2DWEYEH Im(z) > 0 C6AJEKH0EGAB|w| < 1,
FBGVDACMDRf(1) = 1,f(i) = 15AREPEQDMBHC6AJ.
B5 CSD9 f(i) = 1√5,
BHFBAK,z = i B3D9H2DWEYEHARBQBA Im(z) = 0 ARGHEH
BMF0,i =?i.
ADw = 1√5B3D9H0EGABEX|w| = 1 ARGHEHBMF0,√5.
DUGAEPEQDMBHC6AJARA2GHEHBHAQ
f(?i) = √5.
EKEX,EPEQDMBHC6AJ w = f(z) GQD2BM 1,i,?i EDCVC6AJ
EK1,√5,1√5.
DUGADXE8B4F9EPEQDMBHC6AJARCMDRDWAQparenleftBigg
w?1/√5
w?√5
parenrightBiggslashbiggparenleftBigg
1?1/√5
1?√5
parenrightBigg
=
parenleftbiggz?i
z + i
parenrightbigg
/
parenleftbigg1?i
1 + i
parenrightbigg
.
90/127
BP
w = 3z + (
√5?2i)
(√5?2i)z + 3.
91/127
CPBA 2 EMGQH0EGAB|z| < 1 C6AJEKH0EGAB|w| < 1 AR
EPEQDMBHC6AJ.
92/127
B5AN z EYEHH2H0EGAB|z| < 1 CBG4ARE8BM a C6AJEK w
EYEHH2ARH0EGAB|w| < 1 ARELB1 w = 0,BP
w,z = a?→ w = 0,
FR?BM a G2BM
1
a F0B3D9H0EGABEX|z| = 1 ARE8GHGHEH
BM,DUGAEPEQDMBHC6AJARA2GHEHBHDJ,C3CZ
w,z = a?→ w = 0;
z = 1a?→ w = ∞.
D9F0EPEQDMBHC6AJBTGCCZCGCTARBBEQ
w = k
z?a
z? 1a
= kaparenleftbigg z?a
az?1
parenrightbigg
= k′
parenleftbigg z?a
1?az
parenrightbigg
,
(B9k′ =?ka)93/127
B5AN z EYEHH2H0EGAB|z| < 1 CBG4ARE8BM a C6AJEK w
EYEHH2ARH0EGAB|w| < 1 ARELB1 w = 0,BP
w,z = a?→ w = 0,
FR?BM a G2BM
1
a F0B3D9H0EGABEX|z| = 1 ARE8GHGHEH
BM,DUGAEPEQDMBHC6AJARA2GHEHBHDJ,C3CZ
w,z = a?→ w = 0;
z = 1a?→ w = ∞.
D9F0EPEQDMBHC6AJBTGCCZCGCTARBBEQ
w = k
z?a
z? 1a
= kaparenleftbigg z?a
az?1
parenrightbigg
= k′
parenleftbigg z?a
1?az
parenrightbigg
,
(B9k′ =?ka)93/127
B5AN z EYEHH2H0EGAB|z| < 1 CBG4ARE8BM a C6AJEK w
EYEHH2ARH0EGAB|w| < 1 ARELB1 w = 0,BP
w,z = a?→ w = 0,
FR?BM a G2BM
1
a F0B3D9H0EGABEX|z| = 1 ARE8GHGHEH
BM,DUGAEPEQDMBHC6AJARA2GHEHBHDJ,C3CZ
w,z = a?→ w = 0;
z = 1a?→ w = ∞.
D9F0EPEQDMBHC6AJBTGCCZCGCTARBBEQ
w = k
z?a
z? 1a
= kaparenleftbigg z?a
az?1
parenrightbigg
= k′
parenleftbigg z?a
1?az
parenrightbigg
,
(B9k′ =?ka)93/127
B5AN z EYEHH2H0EGAB|z| < 1 CBG4ARE8BM a C6AJEK w
EYEHH2ARH0EGAB|w| < 1 ARELB1 w = 0,BP
w,z = a?→ w = 0,
FR?BM a G2BM
1
a F0B3D9H0EGABEX|z| = 1 ARE8GHGHEH
BM,DUGAEPEQDMBHC6AJARA2GHEHBHDJ,C3CZ
w,z = a?→ w = 0;
z = 1a?→ w = ∞.
D9F0EPEQDMBHC6AJBTGCCZCGCTARBBEQ
w = k
z?a
z? 1a
= kaparenleftbigg z?a
az?1
parenrightbigg
= k′
parenleftbigg z?a
1?az
parenrightbigg
,
(B9k′ =?ka)93/127
B5AN z EYEHH2H0EGAB|z| < 1 CBG4ARE8BM a C6AJEK w
EYEHH2ARH0EGAB|w| < 1 ARELB1 w = 0,BP
w,z = a?→ w = 0,
FR?BM a G2BM
1
a F0B3D9H0EGABEX|z| = 1 ARE8GHGHEH
BM,DUGAEPEQDMBHC6AJARA2GHEHBHDJ,C3CZ
w,z = a?→ w = 0;
z = 1a?→ w = ∞.
D9F0EPEQDMBHC6AJBTGCCZCGCTARBBEQ
w = k
z?a
z? 1a
= kaparenleftbigg z?a
az?1
parenrightbigg
= k′
parenleftbigg z?a
1?az
parenrightbigg
,
(B9k′ =?ka)93/127
B5AN z EYEHH2H0EGAB|z| < 1 CBG4ARE8BM a C6AJEK w
EYEHH2ARH0EGAB|w| < 1 ARELB1 w = 0,BP
w,z = a?→ w = 0,
FR?BM a G2BM
1
a F0B3D9H0EGABEX|z| = 1 ARE8GHGHEH
BM,DUGAEPEQDMBHC6AJARA2GHEHBHDJ,C3CZ
w,z = a?→ w = 0;
z = 1a?→ w = ∞.
D9F0EPEQDMBHC6AJBTGCCZCGCTARBBEQ
w = k
z?a
z? 1a
= kaparenleftbigg z?a
az?1
parenrightbigg
= k′
parenleftbigg z?a
1?az
parenrightbigg
,
(B9k′ =?ka)93/127
B5AN z EYEHH2H0EGAB|z| < 1 CBG4ARE8BM a C6AJEK w
EYEHH2ARH0EGAB|w| < 1 ARELB1 w = 0,BP
w,z = a?→ w = 0,
FR?BM a G2BM
1
a F0B3D9H0EGABEX|z| = 1 ARE8GHGHEH
BM,DUGAEPEQDMBHC6AJARA2GHEHBHDJ,C3CZ
w,z = a?→ w = 0;
z = 1a?→ w = ∞.
D9F0EPEQDMBHC6AJBTGCCZCGCTARBBEQ
w = k
z?a
z? 1a
= kaparenleftbigg z?a
az?1
parenrightbigg
= k′
parenleftbigg z?a
1?az
parenrightbigg
,
(B9k′ =?ka)93/127
B5AN z EYEHH2H0EGAB|z| < 1 CBG4ARE8BM a C6AJEK w
EYEHH2ARH0EGAB|w| < 1 ARELB1 w = 0,BP
w,z = a?→ w = 0,
FR?BM a G2BM
1
a F0B3D9H0EGABEX|z| = 1 ARE8GHGHEH
BM,DUGAEPEQDMBHC6AJARA2GHEHBHDJ,C3CZ
w,z = a?→ w = 0;
z = 1a?→ w = ∞.
D9F0EPEQDMBHC6AJBTGCCZCGCTARBBEQ
w = k
z?a
z? 1a
= kaparenleftbigg z?a
az?1
parenrightbigg
= k′
parenleftbigg z?a
1?az
parenrightbigg
,
(B9k′ =?ka)93/127
B5AN z EYEHH2H0EGAB|z| < 1 CBG4ARE8BM a C6AJEK w
EYEHH2ARH0EGAB|w| < 1 ARELB1 w = 0,BP
w,z = a?→ w = 0,
FR?BM a G2BM
1
a F0B3D9H0EGABEX|z| = 1 ARE8GHGHEH
BM,DUGAEPEQDMBHC6AJARA2GHEHBHDJ,C3CZ
w,z = a?→ w = 0;
z = 1a?→ w = ∞.
D9F0EPEQDMBHC6AJBTGCCZCGCTARBBEQ
w = k
z?a
z? 1a
= kaparenleftbigg z?a
az?1
parenrightbigg
= k′
parenleftbigg z?a
1?az
parenrightbigg
,
(B9k′ =?ka)93/127
CTEHB4F9DIGI k′:
CSD9 z EYEHH2H0EGABEXH2ARBMDRC6AJEK w EYEHH2H0EG
ABEXH2ARBM,EKEXABEX|z| = 1 H2ARBM z = 1 ARE4BTGV
DA|w| = 1,BP
|k′|
vextendsinglevextendsingle
vextendsinglevextendsingle1?a
1?a
vextendsinglevextendsingle
vextendsinglevextendsingle= |w| = 1.
D5FR? |1?a| = |1?a|,
EKEX |k′| = 1,BP k′ = eiθ.
EKEX,EKEMEPEQDMBHC6AJARE8DOC4EREQF0
w = eiθ
parenleftbigg z?a
1?az
parenrightbigg
,(|a| < 1)
DRDN,H2EQAREPEQDMBHC6AJBJGQH0EGABEX|z| < 1 C6AJ
EKH0EGABEX|w| < 1.
94/127
CTEHB4F9DIGI k′:
CSD9 z EYEHH2H0EGABEXH2ARBMDRC6AJEK w EYEHH2H0EG
ABEXH2ARBM,EKEXABEX|z| = 1 H2ARBM z = 1 ARE4BTGV
DA|w| = 1,BP
|k′|
vextendsinglevextendsingle
vextendsinglevextendsingle1?a
1?a
vextendsinglevextendsingle
vextendsinglevextendsingle= |w| = 1.
D5FR? |1?a| = |1?a|,
EKEX |k′| = 1,BP k′ = eiθ.
EKEX,EKEMEPEQDMBHC6AJARE8DOC4EREQF0
w = eiθ
parenleftbigg z?a
1?az
parenrightbigg
,(|a| < 1)
DRDN,H2EQAREPEQDMBHC6AJBJGQH0EGABEX|z| < 1 C6AJ
EKH0EGABEX|w| < 1.
94/127
CTEHB4F9DIGI k′:
CSD9 z EYEHH2H0EGABEXH2ARBMDRC6AJEK w EYEHH2H0EG
ABEXH2ARBM,EKEXABEX|z| = 1 H2ARBM z = 1 ARE4BTGV
DA|w| = 1,BP
|k′|
vextendsinglevextendsingle
vextendsinglevextendsingle1?a
1?a
vextendsinglevextendsingle
vextendsinglevextendsingle= |w| = 1.
D5FR? |1?a| = |1?a|,
EKEX |k′| = 1,BP k′ = eiθ.
EKEX,EKEMEPEQDMBHC6AJARE8DOC4EREQF0
w = eiθ
parenleftbigg z?a
1?az
parenrightbigg
,(|a| < 1)
DRDN,H2EQAREPEQDMBHC6AJBJGQH0EGABEX|z| < 1 C6AJ
EKH0EGABEX|w| < 1.
94/127
CTEHB4F9DIGI k′:
CSD9 z EYEHH2H0EGABEXH2ARBMDRC6AJEK w EYEHH2H0EG
ABEXH2ARBM,EKEXABEX|z| = 1 H2ARBM z = 1 ARE4BTGV
DA|w| = 1,BP
|k′|
vextendsinglevextendsingle
vextendsinglevextendsingle1?a
1?a
vextendsinglevextendsingle
vextendsinglevextendsingle= |w| = 1.
D5FR? |1?a| = |1?a|,
EKEX |k′| = 1,BP k′ = eiθ.
EKEX,EKEMEPEQDMBHC6AJARE8DOC4EREQF0
w = eiθ
parenleftbigg z?a
1?az
parenrightbigg
,(|a| < 1)
DRDN,H2EQAREPEQDMBHC6AJBJGQH0EGABEX|z| < 1 C6AJ
EKH0EGABEX|w| < 1.
94/127
CTEHB4F9DIGI k′:
CSD9 z EYEHH2H0EGABEXH2ARBMDRC6AJEK w EYEHH2H0EG
ABEXH2ARBM,EKEXABEX|z| = 1 H2ARBM z = 1 ARE4BTGV
DA|w| = 1,BP
|k′|
vextendsinglevextendsingle
vextendsinglevextendsingle1?a
1?a
vextendsinglevextendsingle
vextendsinglevextendsingle= |w| = 1.
D5FR? |1?a| = |1?a|,
EKEX |k′| = 1,BP k′ = eiθ.
EKEX,EKEMEPEQDMBHC6AJARE8DOC4EREQF0
w = eiθ
parenleftbigg z?a
1?az
parenrightbigg
,(|a| < 1)
DRDN,H2EQAREPEQDMBHC6AJBJGQH0EGABEX|z| < 1 C6AJ
EKH0EGABEX|w| < 1.
94/127
CTEHB4F9DIGI k′:
CSD9 z EYEHH2H0EGABEXH2ARBMDRC6AJEK w EYEHH2H0EG
ABEXH2ARBM,EKEXABEX|z| = 1 H2ARBM z = 1 ARE4BTGV
DA|w| = 1,BP
|k′|
vextendsinglevextendsingle
vextendsinglevextendsingle1?a
1?a
vextendsinglevextendsingle
vextendsinglevextendsingle= |w| = 1.
D5FR? |1?a| = |1?a|,
EKEX |k′| = 1,BP k′ = eiθ.
EKEX,EKEMEPEQDMBHC6AJARE8DOC4EREQF0
w = eiθ
parenleftbigg z?a
1?az
parenrightbigg
,(|a| < 1)
DRDN,H2EQAREPEQDMBHC6AJBJGQH0EGABEX|z| < 1 C6AJ
EKH0EGABEX|w| < 1.
94/127
CTEHB4F9DIGI k′:
CSD9 z EYEHH2H0EGABEXH2ARBMDRC6AJEK w EYEHH2H0EG
ABEXH2ARBM,EKEXABEX|z| = 1 H2ARBM z = 1 ARE4BTGV
DA|w| = 1,BP
|k′|
vextendsinglevextendsingle
vextendsinglevextendsingle1?a
1?a
vextendsinglevextendsingle
vextendsinglevextendsingle= |w| = 1.
D5FR? |1?a| = |1?a|,
EKEX |k′| = 1,BP k′ = eiθ.
EKEX,EKEMEPEQDMBHC6AJARE8DOC4EREQF0
w = eiθ
parenleftbigg z?a
1?az
parenrightbigg
,(|a| < 1)
DRDN,H2EQAREPEQDMBHC6AJBJGQH0EGABEX|z| < 1 C6AJ
EKH0EGABEX|w| < 1.
94/127
CTEHB4F9DIGI k′:
CSD9 z EYEHH2H0EGABEXH2ARBMDRC6AJEK w EYEHH2H0EG
ABEXH2ARBM,EKEXABEX|z| = 1 H2ARBM z = 1 ARE4BTGV
DA|w| = 1,BP
|k′|
vextendsinglevextendsingle
vextendsinglevextendsingle1?a
1?a
vextendsinglevextendsingle
vextendsinglevextendsingle= |w| = 1.
D5FR? |1?a| = |1?a|,
EKEX |k′| = 1,BP k′ = eiθ.
EKEX,EKEMEPEQDMBHC6AJARE8DOC4EREQF0
w = eiθ
parenleftbigg z?a
1?az
parenrightbigg
,(|a| < 1)
DRDN,H2EQAREPEQDMBHC6AJBJGQH0EGABEX|z| < 1 C6AJ
EKH0EGABEX|w| < 1.
94/127
DI 1 EMGQH0EGABC6AJEKH0EGABDPGVDACMDR w(12) = 0,
w′(12) > 0 AREPEQDMBHC6AJ.
95/127
B5 FR?GQH0EGABEXC6AJEKH0EGABEXAREPEQDMBHC6AJAR
E8DOC4EREQF0
w = ei?
parenleftbigg z?a
1?az
parenrightbigg
,(|a| < 1)
FGCZB4F9 a:
CSCMDRw(12) = 0DJ,EKEMARC6AJGQ|z| < 1 CBARBMz = 12C6
AJEK|w| < 1 ARELB1,EKEX
w = ei?
z?
1
2
1? 12z
,
96/127
B5 FR?GQH0EGABEXC6AJEKH0EGABEXAREPEQDMBHC6AJAR
E8DOC4EREQF0
w = ei?
parenleftbigg z?a
1?az
parenrightbigg
,(|a| < 1)
FGCZB4F9 a:
CSCMDRw(12) = 0DJ,EKEMARC6AJGQ|z| < 1 CBARBMz = 12C6
AJEK|w| < 1 ARELB1,EKEX
w = ei?
z?
1
2
1? 12z
,
96/127
B5 FR?GQH0EGABEXC6AJEKH0EGABEXAREPEQDMBHC6AJAR
E8DOC4EREQF0
w = ei?
parenleftbigg z?a
1?az
parenrightbigg
,(|a| < 1)
FGCZB4F9 a:
CSCMDRw(12) = 0DJ,EKEMARC6AJGQ|z| < 1 CBARBMz = 12C6
AJEK|w| < 1 ARELB1,EKEX
w = ei?
z?
1
2
1? 12z
,
96/127
B5 FR?GQH0EGABEXC6AJEKH0EGABEXAREPEQDMBHC6AJAR
E8DOC4EREQF0
w = ei?
parenleftbigg z?a
1?az
parenrightbigg
,(|a| < 1)
FGCZB4F9 a:
CSCMDRw(12) = 0DJ,EKEMARC6AJGQ|z| < 1 CBARBMz = 12C6
AJEK|w| < 1 ARELB1,EKEX
w = ei?
z?
1
2
1? 12z
,
96/127
B8A0B4F9?:
CSCSAQ
w′(12) = ei?
parenleftbigg
1? 12z
parenrightbigg
+
parenleftbigg
z? 12
parenrightbigg1
2parenleftbigg
1? 12z
parenrightbigg2
vextendsinglevextendsingle
vextendsinglevextendsingle
z=12
= ei?43,
APargw′(12) =?.
CSD9w′(12) > 0,FRCSw′(12)?AKEIGI,D0ADargw′(12) = 0,
BP? = 0.
97/127
B8A0B4F9?:
CSCSAQ
w′(12) = ei?
parenleftbigg
1? 12z
parenrightbigg
+
parenleftbigg
z? 12
parenrightbigg1
2parenleftbigg
1? 12z
parenrightbigg2
vextendsinglevextendsingle
vextendsinglevextendsingle
z=12
= ei?43,
APargw′(12) =?.
CSD9w′(12) > 0,FRCSw′(12)?AKEIGI,D0ADargw′(12) = 0,
BP? = 0.
97/127
B8A0B4F9?:
CSCSAQ
w′(12) = ei?
parenleftbigg
1? 12z
parenrightbigg
+
parenleftbigg
z? 12
parenrightbigg1
2parenleftbigg
1? 12z
parenrightbigg2
vextendsinglevextendsingle
vextendsinglevextendsingle
z=12
= ei?43,
APargw′(12) =?.
CSD9w′(12) > 0,FRCSw′(12)?AKEIGI,D0ADargw′(12) = 0,
BP? = 0.
97/127
B8A0B4F9?:
CSCSAQ
w′(12) = ei?
parenleftbigg
1? 12z
parenrightbigg
+
parenleftbigg
z? 12
parenrightbigg1
2parenleftbigg
1? 12z
parenrightbigg2
vextendsinglevextendsingle
vextendsinglevextendsingle
z=12
= ei?43,
APargw′(12) =?.
CSD9w′(12) > 0,FRCSw′(12)?AKEIGI,D0ADargw′(12) = 0,
BP? = 0.
97/127
B8A0B4F9?:
CSCSAQ
w′(12) = ei?
parenleftbigg
1? 12z
parenrightbigg
+
parenleftbigg
z? 12
parenrightbigg1
2parenleftbigg
1? 12z
parenrightbigg2
vextendsinglevextendsingle
vextendsinglevextendsingle
z=12
= ei?43,
APargw′(12) =?.
CSD9w′(12) > 0,FRCSw′(12)?AKEIGI,D0ADargw′(12) = 0,
BP? = 0.
97/127
B8A0B4F9?:
CSCSAQ
w′(12) = ei?
parenleftbigg
1? 12z
parenrightbigg
+
parenleftbigg
z? 12
parenrightbigg1
2parenleftbigg
1? 12z
parenrightbigg2
vextendsinglevextendsingle
vextendsinglevextendsingle
z=12
= ei?43,
APargw′(12) =?.
CSD9w′(12) > 0,FRCSw′(12)?AKEIGI,D0ADargw′(12) = 0,
BP? = 0.
97/127
B8A0B4F9?:
CSCSAQ
w′(12) = ei?
parenleftbigg
1? 12z
parenrightbigg
+
parenleftbigg
z? 12
parenrightbigg1
2parenleftbigg
1? 12z
parenrightbigg2
vextendsinglevextendsingle
vextendsinglevextendsingle
z=12
= ei?43,
APargw′(12) =?.
CSD9w′(12) > 0,FRCSw′(12)?AKEIGI,D0ADargw′(12) = 0,
BP? = 0.
97/127
B8A0B4F9?:
CSCSAQ
w′(12) = ei?
parenleftbigg
1? 12z
parenrightbigg
+
parenleftbigg
z? 12
parenrightbigg1
2parenleftbigg
1? 12z
parenrightbigg2
vextendsinglevextendsingle
vextendsinglevextendsingle
z=12
= ei?43,
APargw′(12) =?.
CSD9w′(12) > 0,FRCSw′(12)?AKEIGI,D0ADargw′(12) = 0,
BP? = 0.
97/127
B8A0B4F9?:
CSCSAQ
w′(12) = ei?
parenleftbigg
1? 12z
parenrightbigg
+
parenleftbigg
z? 12
parenrightbigg1
2parenleftbigg
1? 12z
parenrightbigg2
vextendsinglevextendsingle
vextendsinglevextendsingle
z=12
= ei?43,
APargw′(12) =?.
CSD9w′(12) > 0,FRCSw′(12)?AKEIGI,D0ADargw′(12) = 0,
BP? = 0.
97/127
EKEX,EKEMC6AJ?
w =
z? 12
1? 12z
= 2z?12?z,
98/127
E8AAC1DRDGCMBXG4F8GIARDIC3
ERDX D G2E4ERDX G ARB3CG,GHGTECF5CTG0CDCME8DRDF
H0F8GIC4ER,C1DRDGCME8AABNBABVDQDUCPDQ,DGCMBXG4B3
CGEMAQF8GI.
99/127
E8AAC1DRDGCMBXG4F8GIARDIC3
ERDX D G2E4ERDX G ARB3CG,GHGTECF5CTG0CDCME8DRDF
H0F8GIC4ER,C1DRDGCME8AABNBABVDQDUCPDQ,DGCMBXG4B3
CGEMAQF8GI.
99/127
DI 1,FDEMGQABDX|z| < R C6AJEKABDX|w| < 1 AREPEQ
DMBHC6AJ.
B5FGCZ,ABDX|z| < R EYC4AREGC6AJ
z1 = 1Rz
A0BV?H0EGABDX:|z1| < 1.
B8A0,ABDX|z1| < 1 C6AJEKABDX|w| < 1 AREPEQDMBHC6
AJARE8DOC4EREQ?
w = ei? z1?α1?αz
1
(EIGI,|α| < 1)
D2H2EKGB,GQABDX|z| < R C6AJEKABDX|w| < 1 AREPEQ
DMBHC6AJF0
w = ei?
1
Rz?α
1?α 1Rz
= ei?z?αRR?αz(|α| < 1)
100/127
DI 1,FDEMGQABDX|z| < R C6AJEKABDX|w| < 1 AREPEQ
DMBHC6AJ.
B5FGCZ,ABDX|z| < R EYC4AREGC6AJ
z1 = 1Rz
A0BV?H0EGABDX:|z1| < 1.
B8A0,ABDX|z1| < 1 C6AJEKABDX|w| < 1 AREPEQDMBHC6
AJARE8DOC4EREQ?
w = ei? z1?α1?αz
1
(EIGI,|α| < 1)
D2H2EKGB,GQABDX|z| < R C6AJEKABDX|w| < 1 AREPEQ
DMBHC6AJF0
w = ei?
1
Rz?α
1?α 1Rz
= ei?z?αRR?αz(|α| < 1)
100/127
DI 1,FDEMGQABDX|z| < R C6AJEKABDX|w| < 1 AREPEQ
DMBHC6AJ.
B5FGCZ,ABDX|z| < R EYC4AREGC6AJ
z1 = 1Rz
A0BV?H0EGABDX:|z1| < 1.
B8A0,ABDX|z1| < 1 C6AJEKABDX|w| < 1 AREPEQDMBHC6
AJARE8DOC4EREQ?
w = ei? z1?α1?αz
1
(EIGI,|α| < 1)
D2H2EKGB,GQABDX|z| < R C6AJEKABDX|w| < 1 AREPEQ
DMBHC6AJF0
w = ei?
1
Rz?α
1?α 1Rz
= ei?z?αRR?αz(|α| < 1)
100/127
DI 1,FDEMGQABDX|z| < R C6AJEKABDX|w| < 1 AREPEQ
DMBHC6AJ.
B5FGCZ,ABDX|z| < R EYC4AREGC6AJ
z1 = 1Rz
A0BV?H0EGABDX:|z1| < 1.
B8A0,ABDX|z1| < 1 C6AJEKABDX|w| < 1 AREPEQDMBHC6
AJARE8DOC4EREQ?
w = ei? z1?α1?αz
1
(EIGI,|α| < 1)
D2H2EKGB,GQABDX|z| < R C6AJEKABDX|w| < 1 AREPEQ
DMBHC6AJF0
w = ei?
1
Rz?α
1?α 1Rz
= ei?z?αRR?αz(|α| < 1)
100/127
DI 2,EMFTE8DRD6D1DWEYEH Re(z) > 0 C6AJEKH0EG
AB|w| < 1 ARC6AJ.
B5FGCZ,D1DWEYEH Re(z) > 0 C6AJEKH2DWEY
EH Im(w1) > 0 ARC6AJF0CMBPC6AJ
w1 = epi2z = iz
B8A0,H2DWEYEH Im(w1) > 0 C6AJEKH0EGAB|z| < 1 EPEQ
DMBHC6AJARE8DOC4EREQF0
w = ei?
parenleftbiggw
1?λ
w1?λ
parenrightbigg
(Im(λ) > 0).
D2H2EKGB,EKEMC6AJBP
w = ei?
parenleftbiggiz?λ
iz?λ
parenrightbigg
(Im(λ) > 0).
101/127
DI 2,EMFTE8DRD6D1DWEYEH Re(z) > 0 C6AJEKH0EG
AB|w| < 1 ARC6AJ.
B5FGCZ,D1DWEYEH Re(z) > 0 C6AJEKH2DWEY
EH Im(w1) > 0 ARC6AJF0CMBPC6AJ
w1 = epi2z = iz
B8A0,H2DWEYEH Im(w1) > 0 C6AJEKH0EGAB|z| < 1 EPEQ
DMBHC6AJARE8DOC4EREQF0
w = ei?
parenleftbiggw
1?λ
w1?λ
parenrightbigg
(Im(λ) > 0).
D2H2EKGB,EKEMC6AJBP
w = ei?
parenleftbiggiz?λ
iz?λ
parenrightbigg
(Im(λ) > 0).
101/127
DI 2,EMFTE8DRD6D1DWEYEH Re(z) > 0 C6AJEKH0EG
AB|w| < 1 ARC6AJ.
B5FGCZ,D1DWEYEH Re(z) > 0 C6AJEKH2DWEY
EH Im(w1) > 0 ARC6AJF0CMBPC6AJ
w1 = epi2z = iz
B8A0,H2DWEYEH Im(w1) > 0 C6AJEKH0EGAB|z| < 1 EPEQ
DMBHC6AJARE8DOC4EREQF0
w = ei?
parenleftbiggw
1?λ
w1?λ
parenrightbigg
(Im(λ) > 0).
D2H2EKGB,EKEMC6AJBP
w = ei?
parenleftbiggiz?λ
iz?λ
parenrightbigg
(Im(λ) > 0).
101/127
DI 2,EMFTE8DRD6D1DWEYEH Re(z) > 0 C6AJEKH0EG
AB|w| < 1 ARC6AJ.
B5FGCZ,D1DWEYEH Re(z) > 0 C6AJEKH2DWEY
EH Im(w1) > 0 ARC6AJF0CMBPC6AJ
w1 = epi2z = iz
B8A0,H2DWEYEH Im(w1) > 0 C6AJEKH0EGAB|z| < 1 EPEQ
DMBHC6AJARE8DOC4EREQF0
w = ei?
parenleftbiggw
1?λ
w1?λ
parenrightbigg
(Im(λ) > 0).
D2H2EKGB,EKEMC6AJBP
w = ei?
parenleftbiggiz?λ
iz?λ
parenrightbigg
(Im(λ) > 0).
101/127
DI 3 EMGQ Im(z) > 0 C6AJEK|w?2i| < 2 DPGVDACM
DR w(2i) = 2i,argw
′(2i) =?pi
2 AREPEQDMBHC6AJ.
B5 FGCZ,GQ Im(z) > 0 C6AJEKH0EGAB|ξ| < 1,DPGV
DA ξ(2i) = 0 ARC6AJF0
ξ = eiθ
parenleftbiggz?2i
z + 2i
parenrightbigg
,
B8A0,GQH0EGABEX|ξ| < 1 C6AJEKABEX|w?2i| < 2 AR
C6AJF0
w = 2ξ + 2i.
D2H2EKGB,GQ Im(z) > 0 C6AJEK|w?2i| < 2 DPGVDACM
DR w(2i) = 2i AREPEQDMBHC6AJF0
w = 2eiθ
parenleftbiggz?2i
z + 2i
parenrightbigg
+ 2i,
102/127
DI 3 EMGQ Im(z) > 0 C6AJEK|w?2i| < 2 DPGVDACM
DR w(2i) = 2i,argw
′(2i) =?pi
2 AREPEQDMBHC6AJ.
B5 FGCZ,GQ Im(z) > 0 C6AJEKH0EGAB|ξ| < 1,DPGV
DA ξ(2i) = 0 ARC6AJF0
ξ = eiθ
parenleftbiggz?2i
z + 2i
parenrightbigg
,
B8A0,GQH0EGABEX|ξ| < 1 C6AJEKABEX|w?2i| < 2 AR
C6AJF0
w = 2ξ + 2i.
D2H2EKGB,GQ Im(z) > 0 C6AJEK|w?2i| < 2 DPGVDACM
DR w(2i) = 2i AREPEQDMBHC6AJF0
w = 2eiθ
parenleftbiggz?2i
z + 2i
parenrightbigg
+ 2i,
102/127
DI 3 EMGQ Im(z) > 0 C6AJEK|w?2i| < 2 DPGVDACM
DR w(2i) = 2i,argw
′(2i) =?pi
2 AREPEQDMBHC6AJ.
B5 FGCZ,GQ Im(z) > 0 C6AJEKH0EGAB|ξ| < 1,DPGV
DA ξ(2i) = 0 ARC6AJF0
ξ = eiθ
parenleftbiggz?2i
z + 2i
parenrightbigg
,
B8A0,GQH0EGABEX|ξ| < 1 C6AJEKABEX|w?2i| < 2 AR
C6AJF0
w = 2ξ + 2i.
D2H2EKGB,GQ Im(z) > 0 C6AJEK|w?2i| < 2 DPGVDACM
DR w(2i) = 2i AREPEQDMBHC6AJF0
w = 2eiθ
parenleftbiggz?2i
z + 2i
parenrightbigg
+ 2i,
102/127
DI 3 EMGQ Im(z) > 0 C6AJEK|w?2i| < 2 DPGVDACM
DR w(2i) = 2i,argw
′(2i) =?pi
2 AREPEQDMBHC6AJ.
B5 FGCZ,GQ Im(z) > 0 C6AJEKH0EGAB|ξ| < 1,DPGV
DA ξ(2i) = 0 ARC6AJF0
ξ = eiθ
parenleftbiggz?2i
z + 2i
parenrightbigg
,
B8A0,GQH0EGABEX|ξ| < 1 C6AJEKABEX|w?2i| < 2 AR
C6AJF0
w = 2ξ + 2i.
D2H2EKGB,GQ Im(z) > 0 C6AJEK|w?2i| < 2 DPGVDACM
DR w(2i) = 2i AREPEQDMBHC6AJF0
w = 2eiθ
parenleftbiggz?2i
z + 2i
parenrightbigg
+ 2i,
102/127
CTEHB4F9 θ:
FR?
w′(2i) = 2eiθ 14i,
argw′(2i) = arg(2eiθ) + arg( 14i) = θ? pi2.
DUGAEUDJCMDR argw′(2i) =?pi2,EKEX θ = 0.
D9F0EKEMARC6AJF0
w = 2
parenleftbiggz?2i
z + 2i
parenrightbigg
+ 2i.
103/127
CTEHB4F9 θ:
FR?
w′(2i) = 2eiθ 14i,
argw′(2i) = arg(2eiθ) + arg( 14i) = θ? pi2.
DUGAEUDJCMDR argw′(2i) =?pi2,EKEX θ = 0.
D9F0EKEMARC6AJF0
w = 2
parenleftbiggz?2i
z + 2i
parenrightbigg
+ 2i.
103/127
CTEHB4F9 θ:
FR?
w′(2i) = 2eiθ 14i,
argw′(2i) = arg(2eiθ) + arg( 14i) = θ? pi2.
DUGAEUDJCMDR argw′(2i) =?pi2,EKEX θ = 0.
D9F0EKEMARC6AJF0
w = 2
parenleftbiggz?2i
z + 2i
parenrightbigg
+ 2i.
103/127
CTEHB4F9 θ:
FR?
w′(2i) = 2eiθ 14i,
argw′(2i) = arg(2eiθ) + arg( 14i) = θ? pi2.
DUGAEUDJCMDR argw′(2i) =?pi2,EKEX θ = 0.
D9F0EKEMARC6AJF0
w = 2
parenleftbiggz?2i
z + 2i
parenrightbigg
+ 2i.
103/127
CTEHB4F9 θ:
FR?
w′(2i) = 2eiθ 14i,
argw′(2i) = arg(2eiθ) + arg( 14i) = θ? pi2.
DUGAEUDJCMDR argw′(2i) =?pi2,EKEX θ = 0.
D9F0EKEMARC6AJF0
w = 2
parenleftbiggz?2i
z + 2i
parenrightbigg
+ 2i.
103/127
BDD3AX BTDRFSAVF8GIEKA8EKARC6AJ
1,E8F8GI w = zn(n greaterorequalslant 2?CVB8GI)
E8F8GIARA1BBC6AJBHEF
E8F8GIBF z EYEHCBBRBRDWAJ,ENARAJGIF0
dw
dz = nz
n?1,
FRADA8 z negationslash= 0 EE,
dw
dz negationslash= 0,
EKEX,BF z EYEHCBG0AMA6BMA9,CS w = zn EKA8EKARC6AJF0
BRBRA1BBAR.
104/127
BDD3AX BTDRFSAVF8GIEKA8EKARC6AJ
1,E8F8GI w = zn(n greaterorequalslant 2?CVB8GI)
E8F8GIARA1BBC6AJBHEF
E8F8GIBF z EYEHCBBRBRDWAJ,ENARAJGIF0
dw
dz = nz
n?1,
FRADA8 z negationslash= 0 EE,
dw
dz negationslash= 0,
EKEX,BF z EYEHCBG0AMA6BMA9,CS w = zn EKA8EKARC6AJF0
BRBRA1BBAR.
104/127
BDD3AX BTDRFSAVF8GIEKA8EKARC6AJ
1,E8F8GI w = zn(n greaterorequalslant 2?CVB8GI)
E8F8GIARA1BBC6AJBHEF
E8F8GIBF z EYEHCBBRBRDWAJ,ENARAJGIF0
dw
dz = nz
n?1,
FRADA8 z negationslash= 0 EE,
dw
dz negationslash= 0,
EKEX,BF z EYEHCBG0AMA6BMA9,CS w = zn EKA8EKARC6AJF0
BRBRA1BBAR.
104/127
BDD3AX BTDRFSAVF8GIEKA8EKARC6AJ
1,E8F8GI w = zn(n greaterorequalslant 2?CVB8GI)
E8F8GIARA1BBC6AJBHEF
E8F8GIBF z EYEHCBBRBRDWAJ,ENARAJGIF0
dw
dz = nz
n?1,
FRADA8 z negationslash= 0 EE,
dw
dz negationslash= 0,
EKEX,BF z EYEHCBG0AMA6BMA9,CS w = zn EKA8EKARC6AJF0
BRBRA1BBAR.
104/127
BDD3AX BTDRFSAVF8GIEKA8EKARC6AJ
1,E8F8GI w = zn(n greaterorequalslant 2?CVB8GI)
E8F8GIARA1BBC6AJBHEF
E8F8GIBF z EYEHCBBRBRDWAJ,ENARAJGIF0
dw
dz = nz
n?1,
FRADA8 z negationslash= 0 EE,
dw
dz negationslash= 0,
EKEX,BF z EYEHCBG0AMA6BMA9,CS w = zn EKA8EKARC6AJF0
BRBRA1BBAR.
104/127
E8F8GIARC6AJBTBM
CGC2B9 z = reiθ,w = ρei?,
BSD3
ρ = rn,? = nθ.
EKEX,BF w = zn C6AJCT,?BB
DX 0 < θ < θ0
parenleftbigg
θ0 < 2pin
parenrightbigg
C6AJEK?BBDX 0 <? < nθ0.
O
nθ0
(w)
O
θ0
(z)
w=zn
y
x
v
u
DUGAE8F8GIARGXE8BTBM,CGC2DRD6?BBDXC6AJEK?BBDX,
EYDIDGCME8F8GI.
105/127
E8F8GIARC6AJBTBM
CGC2B9 z = reiθ,w = ρei?,
BSD3
ρ = rn,? = nθ.
EKEX,BF w = zn C6AJCT,?BB
DX 0 < θ < θ0
parenleftbigg
θ0 < 2pin
parenrightbigg
C6AJEK?BBDX 0 <? < nθ0.
O
nθ0
(w)
O
θ0
(z)
w=zn
y
x
v
u
DUGAE8F8GIARGXE8BTBM,CGC2DRD6?BBDXC6AJEK?BBDX,
EYDIDGCME8F8GI.
105/127
E8F8GIARC6AJBTBM
CGC2B9 z = reiθ,w = ρei?,
BSD3
ρ = rn,? = nθ.
EKEX,BF w = zn C6AJCT,?BB
DX 0 < θ < θ0
parenleftbigg
θ0 < 2pin
parenrightbigg
C6AJEK?BBDX 0 <? < nθ0.
O
nθ0
(w)
O
θ0
(z)
w=zn
y
x
v
u
DUGAE8F8GIARGXE8BTBM,CGC2DRD6?BBDXC6AJEK?BBDX,
EYDIDGCME8F8GI.
105/127
E8F8GIARC6AJBTBM
CGC2B9 z = reiθ,w = ρei?,
BSD3
ρ = rn,? = nθ.
EKEX,BF w = zn C6AJCT,?BB
DX 0 < θ < θ0
parenleftbigg
θ0 < 2pin
parenrightbigg
C6AJEK?BBDX 0 <? < nθ0.
O
nθ0
(w)
O
θ0
(z)
w=zn
y
x
v
u
DUGAE8F8GIARGXE8BTBM,CGC2DRD6?BBDXC6AJEK?BBDX,
EYDIDGCME8F8GI.
105/127
E8F8GIARC6AJBTBM
CGC2B9 z = reiθ,w = ρei?,
BSD3
ρ = rn,? = nθ.
EKEX,BF w = zn C6AJCT,?BB
DX 0 < θ < θ0
parenleftbigg
θ0 < 2pin
parenrightbigg
C6AJEK?BBDX 0 <? < nθ0.
O
nθ0
(w)
O
θ0
(z)
w=zn
y
x
v
u
DUGAE8F8GIARGXE8BTBM,CGC2DRD6?BBDXC6AJEK?BBDX,
EYDIDGCME8F8GI.
105/127
BTC7AR,
BBDX 0 < θ < 2pin C6AJEKGVAKEIF2DHD1AR w EY
EH 0 <? < 2pi.
ENARE8BQ θ = 0 C6AJEK w EYEHAKEIF2ARH2BQ? = 0;
B8E8BQ θ = 2pin C6AJEK w EYEHAKEIF2ARCTBQ? = 2pi.
(w)
O
v
u O
y
x
2pi
n
106/127
BTC7AR,
BBDX 0 < θ < 2pin C6AJEKGVAKEIF2DHD1AR w EY
EH 0 <? < 2pi.
ENARE8BQ θ = 0 C6AJEK w EYEHAKEIF2ARH2BQ? = 0;
B8E8BQ θ = 2pin C6AJEK w EYEHAKEIF2ARCTBQ? = 2pi.
(w)
O
v
u O
y
x
2pi
n
106/127
BTC7AR,
BBDX 0 < θ < 2pin C6AJEKGVAKEIF2DHD1AR w EY
EH 0 <? < 2pi.
ENARE8BQ θ = 0 C6AJEK w EYEHAKEIF2ARH2BQ? = 0;
B8E8BQ θ = 2pin C6AJEK w EYEHAKEIF2ARCTBQ? = 2pi.
(w)
O
v
u O
y
x
2pi
n
106/127
DI 1 EMD6?BBDX 0 < argz < pi4 C6AJEKH0EGAB|w| < 1
ARE8DRC6AJ.
B5 FGCZ,C6AJ ξ = z4 GQEKDT?BBDX 0 < argz < pi4 C6
AJEKH2DWEYEH Im(ξ) > 0.
O
pi/4
w=
z4? i
z4 + i
(z) (w)
(a)
(b)
(c)
ξ=z4 (ξ)
w=
ξ? i
ξ + i
O
O
107/127
DI 1 EMD6?BBDX 0 < argz < pi4 C6AJEKH0EGAB|w| < 1
ARE8DRC6AJ.
B5 FGCZ,C6AJ ξ = z4 GQEKDT?BBDX 0 < argz < pi4 C6
AJEKH2DWEYEH Im(ξ) > 0.
O
pi/4
w=
z4? i
z4 + i
(z) (w)
(a)
(b)
(c)
ξ=z4 (ξ)
w=
ξ? i
ξ + i
O
O
107/127
B8A0,EPEQDMBHC6AJ w = ξ?iξ + i GQH2DWEYEHC6AJEKH0EG
AB|w| < 1.
O
pi/4
w=
z4? i
z4 + i
(z) (w)
(a)
(b)
(c)
ξ=z4 (ξ)
w=
ξ? i
ξ + i
O
O
DLA0,EKEMARC6AJ?
w = z
4?i
z4 + i.
108/127
B8A0,EPEQDMBHC6AJ w = ξ?iξ + i GQH2DWEYEHC6AJEKH0EG
AB|w| < 1.
O
pi/4
w=
z4? i
z4 + i
(z) (w)
(a)
(b)
(c)
ξ=z4 (ξ)
w=
ξ? i
ξ + i
O
O
DLA0,EKEMARC6AJ?
w = z
4?i
z4 + i.
108/127
DI 2 EMD6CTGBELCSABD0 C1 DQ C2 EKDWEKARA2 α
ARARG7DXC6AJEK?BBDX φ0 < argw < φ0 + α ARE8DRC6AJ.
B5 BDE8G2,GHE8DRD6ARG7DXC6AJEK?BB
DX 0 < argξ < α ARC6AJ.
O
(z) i
i
1
C1 C2
(a)
ξ=i( z?i z+i )
(ξ)
O
α
(b)
α
O
(w)
w=ei φ0 ξ
w=iei φ0 ( z?i z+i )
α
109/127
DI 2 EMD6CTGBELCSABD0 C1 DQ C2 EKDWEKARA2 α
ARARG7DXC6AJEK?BBDX φ0 < argw < φ0 + α ARE8DRC6AJ.
B5 BDE8G2,GHE8DRD6ARG7DXC6AJEK?BB
DX 0 < argξ < α ARC6AJ.
O
(z) i
i
1
C1 C2
(a)
ξ=i( z?i z+i )
(ξ)
O
α
(b)
α
O
(w)
w=ei φ0 ξ
w=iei φ0 ( z?i z+i )
α
109/127
FGCZ,DUGAEPEQDMBHC6AJARBHEF,GVDA
ξ = f(z),z = i?→ ξ = 0,
z =?i?→ ξ = ∞,
FBEMARG7DXC6AJEK?BBDXARC6AJ,DWFCCMEXCTBBEQAREPEQ
DMBHC6AJ:
ξ = k
parenleftbiggz?i
z + i
parenrightbigg
.
G1EL,k?E6F9ARBXDIGI.
110/127
FBDPDUGAEPEQDMBHC6AJARA2?BH,EKAQ?BBDXAR?G5
F0 α.
O
(z) i
i
1
C1 C2
(a)
ξ=i( z?i z+i )
(ξ)
O
α
(b)
α
O
(w)
w=ei φ0 ξ
w=iei φ0 ( z?i z+i )
α
111/127
B8A0,B4F9?BBDXAREGE8.
CGC2DRD6 C1 C6AJEK ξ EYEHH2ARAKEIF2,C3BF C1 H2GH
E8BM,EMAQCCBMARE4BF ξ EYEHH2ARAKEIF2H2,BTC7AR,
ξ = f(z),z = 1?→ ξ = 1,
E3CKAKEPEQDMBHC6AJEL,AQ k = i.
O
(z) i
i
1
C1 C2
(a)
ξ=i( z?i z+i )
(ξ)
O
α
(b)
α
O
(w)
w=ei φ0 ξ
w=iei φ0 ( z?i z+i )
α
112/127
B8A0,B4F9?BBDXAREGE8.
CGC2DRD6 C1 C6AJEK ξ EYEHH2ARAKEIF2,C3BF C1 H2GH
E8BM,EMAQCCBMARE4BF ξ EYEHH2ARAKEIF2H2,BTC7AR,
ξ = f(z),z = 1?→ ξ = 1,
E3CKAKEPEQDMBHC6AJEL,AQ k = i.
O
(z) i
i
1
C1 C2
(a)
ξ=i( z?i z+i )
(ξ)
O
α
(b)
α
O
(w)
w=ei φ0 ξ
w=iei φ0 ( z?i z+i )
α
112/127
B8A0,B4F9?BBDXAREGE8.
CGC2DRD6 C1 C6AJEK ξ EYEHH2ARAKEIF2,C3BF C1 H2GH
E8BM,EMAQCCBMARE4BF ξ EYEHH2ARAKEIF2H2,BTC7AR,
ξ = f(z),z = 1?→ ξ = 1,
E3CKAKEPEQDMBHC6AJEL,AQ k = i.
O
(z) i
i
1
C1 C2
(a)
ξ=i( z?i z+i )
(ξ)
O
α
(b)
α
O
(w)
w=ei φ0 ξ
w=iei φ0 ( z?i z+i )
α
112/127
B8A0,B4F9?BBDXAREGE8.
CGC2DRD6 C1 C6AJEK ξ EYEHH2ARAKEIF2,C3BF C1 H2GH
E8BM,EMAQCCBMARE4BF ξ EYEHH2ARAKEIF2H2,BTC7AR,
ξ = f(z),z = 1?→ ξ = 1,
E3CKAKEPEQDMBHC6AJEL,AQ k = i.
O
(z) i
i
1
C1 C2
(a)
ξ=i( z?i z+i )
(ξ)
O
α
(b)
α
O
(w)
w=ei φ0 ξ
w=iei φ0 ( z?i z+i )
α
112/127
D9F0,AQAKD6ARG7DXC6AJEK?BBDX 0 < argξ < α ARC6
AJ:
ξ = i
parenleftbiggz?i
z + i
parenrightbigg
O
(z) i
i
1
C1 C2
(a)
ξ=i( z?i z+i )
(ξ)
O
α
(b)
α
O
(w)
w=ei φ0 ξ
w=iei φ0 ( z?i z+i )
α
113/127
BDAJG2,?BBDX 0 < argξ < α FRC4C6AJ
w = eiφ0ξ
BPC4E8DR?G5 φ0,AQAK?BBDX,φ0 < argw < φ0 + α.
O
(z) i
i
1
C1 C2
(a)
ξ=i( z?i z+i )
(ξ)
O
α
(b)
α
O
(w)
w=ei φ0 ξ
w=iei φ0 ( z?i z+i )
α
114/127
BDAJG2,?BBDX 0 < argξ < α FRC4C6AJ
w = eiφ0ξ
BPC4E8DR?G5 φ0,AQAK?BBDX,φ0 < argw < φ0 + α.
O
(z) i
i
1
C1 C2
(a)
ξ=i( z?i z+i )
(ξ)
O
α
(b)
α
O
(w)
w=ei φ0 ξ
w=iei φ0 ( z?i z+i )
α
114/127
D2H2EKGB,EKEMC6AJ?
ξ = ieiφ0
parenleftbiggz?i
z + 1
parenrightbigg
.
O
(z) i
i
1
C1 C2
(a)
ξ=i( z?i z+i )
(ξ)
O
α
(b)
α
O
(w)
w=ei φ0 ξ
w=iei φ0 ( z?i z+i )
α
115/127
DI 3 EMD6GCCZDJGG Re(z) = a,0 lessorequalslant Im(z) lessorequalslant h ARH2DW
EYEHC6AJEKH2DWEYEHARE8DRC6AJ.
B5 FGCZ,D6H2DWEYEHE5DQDUE8DRGDD0? a AREY
EJ:z1 = z?a,AQ z1 EYEHH2ARGBBB.
BDAJ,CMC6AJ z2 = z21,AQAKE8DRGCCZDJ
GG?h2 lessorequalslant Re(z2) < +∞,Im(z2) = 0 AR z2 EYEH.
BDD2,D6 z2 EYEHE5D1DUE8GDD0? h2 AREY
EJ:z3 = z2 + h2,BUAQAKAMC3AAAKEIF2AR z3 EYEH.
116/127
DI 3 EMD6GCCZDJGG Re(z) = a,0 lessorequalslant Im(z) lessorequalslant h ARH2DW
EYEHC6AJEKH2DWEYEHARE8DRC6AJ.
B5 FGCZ,D6H2DWEYEHE5DQDUE8DRGDD0? a AREY
EJ:z1 = z?a,AQ z1 EYEHH2ARGBBB.
BDAJ,CMC6AJ z2 = z21,AQAKE8DRGCCZDJ
GG?h2 lessorequalslant Re(z2) < +∞,Im(z2) = 0 AR z2 EYEH.
BDD2,D6 z2 EYEHE5D1DUE8GDD0? h2 AREY
EJ:z3 = z2 + h2,BUAQAKAMC3AAAKEIF2AR z3 EYEH.
116/127
DI 3 EMD6GCCZDJGG Re(z) = a,0 lessorequalslant Im(z) lessorequalslant h ARH2DW
EYEHC6AJEKH2DWEYEHARE8DRC6AJ.
B5 FGCZ,D6H2DWEYEHE5DQDUE8DRGDD0? a AREY
EJ:z1 = z?a,AQ z1 EYEHH2ARGBBB.
BDAJ,CMC6AJ z2 = z21,AQAKE8DRGCCZDJ
GG?h2 lessorequalslant Re(z2) < +∞,Im(z2) = 0 AR z2 EYEH.
BDD2,D6 z2 EYEHE5D1DUE8GDD0? h2 AREY
EJ:z3 = z2 + h2,BUAQAKAMC3AAAKEIF2AR z3 EYEH.
116/127
DI 3 EMD6GCCZDJGG Re(z) = a,0 lessorequalslant Im(z) lessorequalslant h ARH2DW
EYEHC6AJEKH2DWEYEHARE8DRC6AJ.
B5 FGCZ,D6H2DWEYEHE5DQDUE8DRGDD0? a AREY
EJ:z1 = z?a,AQ z1 EYEHH2ARGBBB.
BDAJ,CMC6AJ z2 = z21,AQAKE8DRGCCZDJ
GG?h2 lessorequalslant Re(z2) < +∞,Im(z2) = 0 AR z2 EYEH.
BDD2,D6 z2 EYEHE5D1DUE8GDD0? h2 AREY
EJ:z3 = z2 + h2,BUAQAKAMC3AAAKEIF2AR z3 EYEH.
116/127
O
B
C(a+ih)
D
(z)
a
y
x
z1=z?a
O
(z1)
B
C
D
(z2) (z3)
O
D
B
h2
C
z2=z12 z3=z2+h2
D
B h2
C
O
O
y
x
B C D
h h
v
u O
B C D
a a?h a+h
(z4) (w) z4=z31/2 w=z
4+a
117/127
BDD3,FRC4C6AJ z4 = √z3,BUAQAKH2DWEYEH z4 EYEH.
DLA0,D6 z4 EYEHE5D1DUE8GDD0? a AREYEJ w = z4 + a,
BUAQAK w EYEHELARH2DWEYEH.
D6EKCZARC6AJBXG4GDBHFUAQAKEKEMARC6AJ
w =
radicalbig
(a?a)2 + h2 + a.
118/127
BDD3,FRC4C6AJ z4 = √z3,BUAQAKH2DWEYEH z4 EYEH.
DLA0,D6 z4 EYEHE5D1DUE8GDD0? a AREYEJ w = z4 + a,
BUAQAK w EYEHELARH2DWEYEH.
D6EKCZARC6AJBXG4GDBHFUAQAKEKEMARC6AJ
w =
radicalbig
(a?a)2 + h2 + a.
118/127
BDD3,FRC4C6AJ z4 = √z3,BUAQAKH2DWEYEH z4 EYEH.
DLA0,D6 z4 EYEHE5D1DUE8GDD0? a AREYEJ w = z4 + a,
BUAQAK w EYEHELARH2DWEYEH.
D6EKCZARC6AJBXG4GDBHFUAQAKEKEMARC6AJ
w =
radicalbig
(a?a)2 + h2 + a.
118/127
2,DXGIF8GI w = ez
DXGIF8GI w = ez A1BBC6AJBHEF
CSD9BF z EYEHCB
w′ = (ez)′ = ez negationslash= 0
EKEX,CS w = ez EKA8EKARC6AJF0E8DRASEYEHH2ARA1BBC6
AJ.
119/127
2,DXGIF8GI w = ez
DXGIF8GI w = ez A1BBC6AJBHEF
CSD9BF z EYEHCB
w′ = (ez)′ = ez negationslash= 0
EKEX,CS w = ez EKA8EKARC6AJF0E8DRASEYEHH2ARA1BBC6
AJ.
119/127
2,DXGIF8GI w = ez
DXGIF8GI w = ez A1BBC6AJBHEF
CSD9BF z EYEHCB
w′ = (ez)′ = ez negationslash= 0
EKEX,CS w = ez EKA8EKARC6AJF0E8DRASEYEHH2ARA1BBC6
AJ.
119/127
2,DXGIF8GI w = ez
DXGIF8GI w = ez A1BBC6AJBHEF
CSD9BF z EYEHCB
w′ = (ez)′ = ez negationslash= 0
EKEX,CS w = ez EKA8EKARC6AJF0E8DRASEYEHH2ARA1BBC6
AJ.
119/127
DXGIF8GIARC6AJBTBM
AN z = x + iy,w = ρei?,BSD3
ρ = ex,? = y,
CSCSDWDJ,z EYEHH2ARDQDM y = C(DIGI),AQC6AJEK w EY
EHH2ARAJDM? = C(DIGI).
EKEX,C6AJD6E1BBDX 0 < Im(z) < a(0 < a lessorequalslant 2pi) C6AJ
EK?BBDX 0 < argw < a.
y
x O
ai
argw=a
O
v
u
w=ez (z) (w)
Ci φ=C
120/127
DXGIF8GIARC6AJBTBM
AN z = x + iy,w = ρei?,BSD3
ρ = ex,? = y,
CSCSDWDJ,z EYEHH2ARDQDM y = C(DIGI),AQC6AJEK w EY
EHH2ARAJDM? = C(DIGI).
EKEX,C6AJD6E1BBDX 0 < Im(z) < a(0 < a lessorequalslant 2pi) C6AJ
EK?BBDX 0 < argw < a.
y
x O
ai
argw=a
O
v
u
w=ez (z) (w)
Ci φ=C
120/127
DXGIF8GIARC6AJBTBM
AN z = x + iy,w = ρei?,BSD3
ρ = ex,? = y,
CSCSDWDJ,z EYEHH2ARDQDM y = C(DIGI),AQC6AJEK w EY
EHH2ARAJDM? = C(DIGI).
EKEX,C6AJD6E1BBDX 0 < Im(z) < a(0 < a lessorequalslant 2pi) C6AJ
EK?BBDX 0 < argw < a.
y
x O
ai
argw=a
O
v
u
w=ez (z) (w)
Ci φ=C
120/127
DXGIF8GIARC6AJBTBM
AN z = x + iy,w = ρei?,BSD3
ρ = ex,? = y,
CSCSDWDJ,z EYEHH2ARDQDM y = C(DIGI),AQC6AJEK w EY
EHH2ARAJDM? = C(DIGI).
EKEX,C6AJD6E1BBDX 0 < Im(z) < a(0 < a lessorequalslant 2pi) C6AJ
EK?BBDX 0 < argw < a.
y
x O
ai
argw=a
O
v
u
w=ez (z) (w)
Ci φ=C
120/127
BTC7AR,E1BBDX 0 < Im(z) < 2pi C6AJEK?BB
DX 0 < argw < 2pi.
y
x O
2pii
O
v
u
w=ez (z) (w)
FRCS,CGC2DRD6E1BBDXC6AJEK?BBDX,DIDIDGCMDXGIF8
GI.
121/127
BTC7AR,E1BBDX 0 < Im(z) < 2pi C6AJEK?BB
DX 0 < argw < 2pi.
y
x O
2pii
O
v
u
w=ez (z) (w)
FRCS,CGC2DRD6E1BBDXC6AJEK?BBDX,DIDIDGCMDXGIF8
GI.
121/127
DI 1 EMD6E1BBDX 0 < Im(z) < pi C6AJEKH0EG
AB|w| < 1 ARE8DRC6AJ.
B5 FGCZ,C6AJ ξ = ez GQEKDTARE1BBDXC6AJEK ξ EYEHAR
H2DWEYEH Im(ξ) > 0.
y
x O
pii
(z)
O
(ξ)
(w)
O
122/127
DI 1 EMD6E1BBDX 0 < Im(z) < pi C6AJEKH0EG
AB|w| < 1 ARE8DRC6AJ.
B5 FGCZ,C6AJ ξ = ez GQEKDTARE1BBDXC6AJEK ξ EYEHAR
H2DWEYEH Im(ξ) > 0.
y
x O
pii
(z)
O
(ξ)
(w)
O
122/127
B8A0,EPEQDMBHC6AJ w = ξ?iξ + i GQH2DWEYEH Im(ξ) > 0
C6AJEKH0EGAB|w| < 1.
y
x O
pii
(z)
O
(ξ)
(w)
O
123/127
D2H2EKGB,EKEMARC6AJ?
w = e
z?i
ez + i
y
x O
pii
(z)
O
(ξ)
(w)
O
124/127
DI 2 EMD6E1BBDX a < Re(z) < b C6AJEKH2DWEY
EH Im(w) > 0 ARE8DRC6AJ.
B5 FGCZ,E1BBDX a < Re(z) < b EYC4EYEJ,E8DWBLCMBP
ARC6AJ
ξ = piib?a(z?a)
A0DWC6AJEKE1BBDX 0 < Im(ξ) < pi.
O a b
(z) (W)
O
O
(ξ)
pii
(a)
(b)
(c)
ξ=(pii)/(b?a)(z?a)
w=e(pii)/(b?a)(z?a)
w=eξ
125/127
DI 2 EMD6E1BBDX a < Re(z) < b C6AJEKH2DWEY
EH Im(w) > 0 ARE8DRC6AJ.
B5 FGCZ,E1BBDX a < Re(z) < b EYC4EYEJ,E8DWBLCMBP
ARC6AJ
ξ = piib?a(z?a)
A0DWC6AJEKE1BBDX 0 < Im(ξ) < pi.
O a b
(z) (W)
O
O
(ξ)
pii
(a)
(b)
(c)
ξ=(pii)/(b?a)(z?a)
w=e(pii)/(b?a)(z?a)
w=eξ
125/127
B8A0,C6AJ w = eξ DWEXD6E1BBDX 0 < Im(ξ) < pi C6AJ
EKH2DWEYEH Im(w) > 0.
O a b
(z) (W)
O
O
(ξ)
pii
(a)
(b)
(c)
ξ=(pii)/(b?a)(z?a)
w=e(pii)/(b?a)(z?a)
w=eξ
126/127
D2H2EKGB,EKEMARC6AJ?
ξ = e piib?a(z?a)
O a b
(z) (W)
O
O
(ξ)
pii
(a)
(b)
(c)
ξ=(pii)/(b?a)(z?a)
w=e(pii)/(b?a)(z?a)
w=eξ
127/127