北京邮电大学：复变函数（2006年）：1H

BDE8G1 BXGIDQBXBVF8GI
September 20,2006
1/89
FODRCBC6:
1,BXGIARCECVAXBHEFBLB0DY.
2,BXGIARBTG3C4EREXBLBXGIDSERB0DYARBTG3FBFE.
3,BXBVF8GIBLG1BGDLDQGJCHARCECV.
2/89
FODRCBC6:
1,BXGIARCECVAXBHEFBLB0DY.
2,BXGIARBTG3C4EREXBLBXGIDSERB0DYARBTG3FBFE.
3,BXBVF8GIBLG1BGDLDQGJCHARCECV.
2/89
FODRCBC6:
1,BXGIARCECVAXBHEFBLB0DY.
2,BXGIARBTG3C4EREXBLBXGIDSERB0DYARBTG3FBFE.
3,BXBVF8GIBLG1BGDLDQGJCHARCECV.
2/89
FODRCBC6:
1,BXGIARCECVAXBHEFBLB0DY.
2,BXGIARBTG3C4EREXBLBXGIDSERB0DYARBTG3FBFE.
3,BXBVF8GIBLG1BGDLDQGJCHARCECV.
2/89
BDE8AX BXGIBLE3GIB0DY
1,BXGIARCECV
C2GIH0EG,i2 =?1.
BXGI,z = x + iy,G1EL,EIG4,x = Re(z),C2G4:
y = Im(z).
CGC2GI,z = iy.
H0DRBXGIDNAVENDMAREIG4G2C2G4DNAV.
DQEIGIG0FV,E8DOCOBH,H0DRBXGIG0CDB3AMDWA2.
3/89
BDE8AX BXGIBLE3GIB0DY
1,BXGIARCECV
C2GIH0EG,i2 =?1.
BXGI,z = x + iy,G1EL,EIG4,x = Re(z),C2G4:
y = Im(z).
CGC2GI,z = iy.
H0DRBXGIDNAVENDMAREIG4G2C2G4DNAV.
DQEIGIG0FV,E8DOCOBH,H0DRBXGIG0CDB3AMDWA2.
3/89
BDE8AX BXGIBLE3GIB0DY
1,BXGIARCECV
C2GIH0EG,i2 =?1.
BXGI,z = x + iy,G1EL,EIG4,x = Re(z),C2G4:
y = Im(z).
CGC2GI,z = iy.
H0DRBXGIDNAVENDMAREIG4G2C2G4DNAV.
DQEIGIG0FV,E8DOCOBH,H0DRBXGIG0CDB3AMDWA2.
3/89
BDE8AX BXGIBLE3GIB0DY
1,BXGIARCECV
C2GIH0EG,i2 =?1.
BXGI,z = x + iy,G1EL,EIG4,x = Re(z),C2G4:
y = Im(z).
CGC2GI,z = iy.
H0DRBXGIDNAVENDMAREIG4G2C2G4DNAV.
DQEIGIG0FV,E8DOCOBH,H0DRBXGIG0CDB3AMDWA2.
3/89
BDE8AX BXGIBLE3GIB0DY
1,BXGIARCECV
C2GIH0EG,i2 =?1.
BXGI,z = x + iy,G1EL,EIG4,x = Re(z),C2G4:
y = Im(z).
CGC2GI,z = iy.
H0DRBXGIDNAVENDMAREIG4G2C2G4DNAV.
DQEIGIG0FV,E8DOCOBH,H0DRBXGIG0CDB3AMDWA2.
3/89
BDE8AX BXGIBLE3GIB0DY
1,BXGIARCECV
C2GIH0EG,i2 =?1.
BXGI,z = x + iy,G1EL,EIG4,x = Re(z),C2G4:
y = Im(z).
CGC2GI,z = iy.
H0DRBXGIDNAVENDMAREIG4G2C2G4DNAV.
DQEIGIG0FV,E8DOCOBH,H0DRBXGIG0CDB3AMDWA2.
3/89
CQANBXGICDB3AMDWA2,FR?BXGIF0EIGIARGLBE,C3CGC2
BXGICZDWA2,G1DWA2B3CGBTDQEIGIELARDWA2B3CGA2EXE8E7.
AIBXGI 0 G2 i CKEXBRG0:
FR? i negationslash= 0,AN i > 0,C3
i·i > i·0 = 0 =1 > 0.
CVGP.
AN i < 0,C3
i·i > i·0 = 0,(G0AVEQH0G2FVEEEMEXA2D9AYARGI)
EKEX?1 > 0,CVGP.
4/89
CQANBXGICDB3AMDWA2,FR?BXGIF0EIGIARGLBE,C3CGC2
BXGICZDWA2,G1DWA2B3CGBTDQEIGIELARDWA2B3CGA2EXE8E7.
AIBXGI 0 G2 i CKEXBRG0:
FR? i negationslash= 0,AN i > 0,C3
i·i > i·0 = 0 =1 > 0.
CVGP.
AN i < 0,C3
i·i > i·0 = 0,(G0AVEQH0G2FVEEEMEXA2D9AYARGI)
EKEX?1 > 0,CVGP.
4/89
CQANBXGICDB3AMDWA2,FR?BXGIF0EIGIARGLBE,C3CGC2
BXGICZDWA2,G1DWA2B3CGBTDQEIGIELARDWA2B3CGA2EXE8E7.
AIBXGI 0 G2 i CKEXBRG0:
FR? i negationslash= 0,AN i > 0,C3
i·i > i·0 = 0 =1 > 0.
CVGP.
AN i < 0,C3
i·i > i·0 = 0,(G0AVEQH0G2FVEEEMEXA2D9AYARGI)
EKEX?1 > 0,CVGP.
4/89
CQANBXGICDB3AMDWA2,FR?BXGIF0EIGIARGLBE,C3CGC2
BXGICZDWA2,G1DWA2B3CGBTDQEIGIELARDWA2B3CGA2EXE8E7.
AIBXGI 0 G2 i CKEXBRG0:
FR? i negationslash= 0,AN i > 0,C3
i·i > i·0 = 0 =1 > 0.
CVGP.
AN i < 0,C3
i·i > i·0 = 0,(G0AVEQH0G2FVEEEMEXA2D9AYARGI)
EKEX?1 > 0,CVGP.
4/89
CQANBXGICDB3AMDWA2,FR?BXGIF0EIGIARGLBE,C3CGC2
BXGICZDWA2,G1DWA2B3CGBTDQEIGIELARDWA2B3CGA2EXE8E7.
AIBXGI 0 G2 i CKEXBRG0:
FR? i negationslash= 0,AN i > 0,C3
i·i > i·0 = 0 =1 > 0.
CVGP.
AN i < 0,C3
i·i > i·0 = 0,(G0AVEQH0G2FVEEEMEXA2D9AYARGI)
EKEX?1 > 0,CVGP.
4/89
CQANBXGICDB3AMDWA2,FR?BXGIF0EIGIARGLBE,C3CGC2
BXGICZDWA2,G1DWA2B3CGBTDQEIGIELARDWA2B3CGA2EXE8E7.
AIBXGI 0 G2 i CKEXBRG0:
FR? i negationslash= 0,AN i > 0,C3
i·i > i·0 = 0 =1 > 0.
CVGP.
AN i < 0,C3
i·i > i·0 = 0,(G0AVEQH0G2FVEEEMEXA2D9AYARGI)
EKEX?1 > 0,CVGP.
4/89
CQANBXGICDB3AMDWA2,FR?BXGIF0EIGIARGLBE,C3CGC2
BXGICZDWA2,G1DWA2B3CGBTDQEIGIELARDWA2B3CGA2EXE8E7.
AIBXGI 0 G2 i CKEXBRG0:
FR? i negationslash= 0,AN i > 0,C3
i·i > i·0 = 0 =1 > 0.
CVGP.
AN i < 0,C3
i·i > i·0 = 0,(G0AVEQH0G2FVEEEMEXA2D9AYARGI)
EKEX?1 > 0,CVGP.
4/89
2,BXGIARE3GIB0DY
CK,DI:
z1 ±z2 = (x1 +iy1)±(x2 +iy2) = (x1 ±x2)+i(y1 ±y2).
EM:
z1?z2 = (x1+iy1)?(x2+iy2) = (x1x2?y1y2)+i(x2y1+x1y2).
GZ:
z = z1z
2
= x1x2 + y1y2x2
2 + y22
+ ix2y1?x1y2x2
2 + y22
5/89
2,BXGIARE3GIB0DY
CK,DI:
z1 ±z2 = (x1 +iy1)±(x2 +iy2) = (x1 ±x2)+i(y1 ±y2).
EM:
z1?z2 = (x1+iy1)?(x2+iy2) = (x1x2?y1y2)+i(x2y1+x1y2).
GZ:
z = z1z
2
= x1x2 + y1y2x2
2 + y22
+ ix2y1?x1y2x2
2 + y22
5/89
2,BXGIARE3GIB0DY
CK,DI:
z1 ±z2 = (x1 +iy1)±(x2 +iy2) = (x1 ±x2)+i(y1 ±y2).
EM:
z1?z2 = (x1+iy1)?(x2+iy2) = (x1x2?y1y2)+i(x2y1+x1y2).
GZ:
z = z1z
2
= x1x2 + y1y2x2
2 + y22
+ ix2y1?x1y2x2
2 + y22
5/89
2,BXGIARE3GIB0DY
CK,DI:
z1 ±z2 = (x1 +iy1)±(x2 +iy2) = (x1 ±x2)+i(y1 ±y2).
EM:
z1?z2 = (x1+iy1)?(x2+iy2) = (x1x2?y1y2)+i(x2y1+x1y2).
GZ:
z = z1z
2
= x1x2 + y1y2x2
2 + y22
+ ix2y1?x1y2x2
2 + y22
5/89
A1G5BXGI,EIG4DNFVADC2G4H1GHDUDNFVFQFUDNDRARH0DR
BXGIEH?A1G5BXGI.
DQBXGI z A1G5ARBXGIC8? z,CGC2 z = x + iy,
C3 z = x?iy.
A1G5BXGIARBHEF:
(1),z1 ±z2 = z1 ±z2,z1z2 = z1 z2,(z1z
2
) = z1z
2
(2),z = z;
(3),zz = [Re(z)]2 + [Im(z)]2;
(4),z + z = 2Re(z),z?z = 2iIm(z).
B3D9G0DFB0DY,DGCMA1G5BXGIARBHEF,CZ:
z1
z2 =
z1z2
z2z2 =
z1z2
|z2|2 =
x1x2 + y1y2
x22 + y22 + i
x2y1?x1y2
x22 + y22,
6/89
A1G5BXGI,EIG4DNFVADC2G4H1GHDUDNFVFQFUDNDRARH0DR
BXGIEH?A1G5BXGI.
DQBXGI z A1G5ARBXGIC8? z,CGC2 z = x + iy,
C3 z = x?iy.
A1G5BXGIARBHEF:
(1),z1 ±z2 = z1 ±z2,z1z2 = z1 z2,(z1z
2
) = z1z
2
(2),z = z;
(3),zz = [Re(z)]2 + [Im(z)]2;
(4),z + z = 2Re(z),z?z = 2iIm(z).
B3D9G0DFB0DY,DGCMA1G5BXGIARBHEF,CZ:
z1
z2 =
z1z2
z2z2 =
z1z2
|z2|2 =
x1x2 + y1y2
x22 + y22 + i
x2y1?x1y2
x22 + y22,
6/89
A1G5BXGI,EIG4DNFVADC2G4H1GHDUDNFVFQFUDNDRARH0DR
BXGIEH?A1G5BXGI.
DQBXGI z A1G5ARBXGIC8? z,CGC2 z = x + iy,
C3 z = x?iy.
A1G5BXGIARBHEF:
(1),z1 ±z2 = z1 ±z2,z1z2 = z1 z2,(z1z
2
) = z1z
2
(2),z = z;
(3),zz = [Re(z)]2 + [Im(z)]2;
(4),z + z = 2Re(z),z?z = 2iIm(z).
B3D9G0DFB0DY,DGCMA1G5BXGIARBHEF,CZ:
z1
z2 =
z1z2
z2z2 =
z1z2
|z2|2 =
x1x2 + y1y2
x22 + y22 + i
x2y1?x1y2
x22 + y22,
6/89
A1G5BXGI,EIG4DNFVADC2G4H1GHDUDNFVFQFUDNDRARH0DR
BXGIEH?A1G5BXGI.
DQBXGI z A1G5ARBXGIC8? z,CGC2 z = x + iy,
C3 z = x?iy.
A1G5BXGIARBHEF:
(1),z1 ±z2 = z1 ±z2,z1z2 = z1 z2,(z1z
2
) = z1z
2
(2),z = z;
(3),zz = [Re(z)]2 + [Im(z)]2;
(4),z + z = 2Re(z),z?z = 2iIm(z).
B3D9G0DFB0DY,DGCMA1G5BXGIARBHEF,CZ:
z1
z2 =
z1z2
z2z2 =
z1z2
|z2|2 =
x1x2 + y1y2
x22 + y22 + i
x2y1?x1y2
x22 + y22,
6/89
A1G5BXGI,EIG4DNFVADC2G4H1GHDUDNFVFQFUDNDRARH0DR
BXGIEH?A1G5BXGI.
DQBXGI z A1G5ARBXGIC8? z,CGC2 z = x + iy,
C3 z = x?iy.
A1G5BXGIARBHEF:
(1),z1 ±z2 = z1 ±z2,z1z2 = z1 z2,(z1z
2
) = z1z
2
(2),z = z;
(3),zz = [Re(z)]2 + [Im(z)]2;
(4),z + z = 2Re(z),z?z = 2iIm(z).
B3D9G0DFB0DY,DGCMA1G5BXGIARBHEF,CZ:
z1
z2 =
z1z2
z2z2 =
z1z2
|z2|2 =
x1x2 + y1y2
x22 + y22 + i
x2y1?x1y2
x22 + y22,
6/89
A1G5BXGI,EIG4DNFVADC2G4H1GHDUDNFVFQFUDNDRARH0DR
BXGIEH?A1G5BXGI.
DQBXGI z A1G5ARBXGIC8? z,CGC2 z = x + iy,
C3 z = x?iy.
A1G5BXGIARBHEF:
(1),z1 ±z2 = z1 ±z2,z1z2 = z1 z2,(z1z
2
) = z1z
2
(2),z = z;
(3),zz = [Re(z)]2 + [Im(z)]2;
(4),z + z = 2Re(z),z?z = 2iIm(z).
B3D9G0DFB0DY,DGCMA1G5BXGIARBHEF,CZ:
z1
z2 =
z1z2
z2z2 =
z1z2
|z2|2 =
x1x2 + y1y2
x22 + y22 + i
x2y1?x1y2
x22 + y22,
6/89
A1G5BXGI,EIG4DNFVADC2G4H1GHDUDNFVFQFUDNDRARH0DR
BXGIEH?A1G5BXGI.
DQBXGI z A1G5ARBXGIC8? z,CGC2 z = x + iy,
C3 z = x?iy.
A1G5BXGIARBHEF:
(1),z1 ±z2 = z1 ±z2,z1z2 = z1 z2,(z1z
2
) = z1z
2
(2),z = z;
(3),zz = [Re(z)]2 + [Im(z)]2;
(4),z + z = 2Re(z),z?z = 2iIm(z).
B3D9G0DFB0DY,DGCMA1G5BXGIARBHEF,CZ:
z1
z2 =
z1z2
z2z2 =
z1z2
|z2|2 =
x1x2 + y1y2
x22 + y22 + i
x2y1?x1y2
x22 + y22,
6/89
A1G5BXGI,EIG4DNFVADC2G4H1GHDUDNFVFQFUDNDRARH0DR
BXGIEH?A1G5BXGI.
DQBXGI z A1G5ARBXGIC8? z,CGC2 z = x + iy,
C3 z = x?iy.
A1G5BXGIARBHEF:
(1),z1 ±z2 = z1 ±z2,z1z2 = z1 z2,(z1z
2
) = z1z
2
(2),z = z;
(3),zz = [Re(z)]2 + [Im(z)]2;
(4),z + z = 2Re(z),z?z = 2iIm(z).
B3D9G0DFB0DY,DGCMA1G5BXGIARBHEF,CZ:
z1
z2 =
z1z2
z2z2 =
z1z2
|z2|2 =
x1x2 + y1y2
x22 + y22 + i
x2y1?x1y2
x22 + y22,
6/89
BDAJAX BXGIARBTG3C4ER
1,BXEYEH
CSD9E8DRBXGI z = x + iy CSE8GHCZCBEIGI (x,y) DXE8
B4F9,EKEXGHD9EYEHH2DTF9ARDQ?DVC1CG,BXGIARASC6DQ
CCEYEHH2ARBMARASC6EKE8E8GHBTARB3CG.
D0ADBXGI z = x + iy DWEXCMCCEYEHH2DVC1? (x,y) AR
BMC4ER.
O
y
x
P(x,y)
y
x
|z|=r
z = x + iy
θ
CSEE,x F2EH?EIF2; y F2EH?C2F2; H0F2EKBFAREYEHEH
BXEYEH.
BXGI z?DWEXCME5H2 OP C4ER.
7/89
BDAJAX BXGIARBTG3C4ER
1,BXEYEH
CSD9E8DRBXGI z = x + iy CSE8GHCZCBEIGI (x,y) DXE8
B4F9,EKEXGHD9EYEHH2DTF9ARDQ?DVC1CG,BXGIARASC6DQ
CCEYEHH2ARBMARASC6EKE8E8GHBTARB3CG.
D0ADBXGI z = x + iy DWEXCMCCEYEHH2DVC1? (x,y) AR
BMC4ER.
O
y
x
P(x,y)
y
x
|z|=r
z = x + iy
θ
CSEE,x F2EH?EIF2; y F2EH?C2F2; H0F2EKBFAREYEHEH
BXEYEH.
BXGI z?DWEXCME5H2 OP C4ER.
7/89
BDAJAX BXGIARBTG3C4ER
1,BXEYEH
CSD9E8DRBXGI z = x + iy CSE8GHCZCBEIGI (x,y) DXE8
B4F9,EKEXGHD9EYEHH2DTF9ARDQ?DVC1CG,BXGIARASC6DQ
CCEYEHH2ARBMARASC6EKE8E8GHBTARB3CG.
D0ADBXGI z = x + iy DWEXCMCCEYEHH2DVC1? (x,y) AR
BMC4ER.
O
y
x
P(x,y)
y
x
|z|=r
z = x + iy
θ
CSEE,x F2EH?EIF2; y F2EH?C2F2; H0F2EKBFAREYEHEH
BXEYEH.
BXGI z?DWEXCME5H2 OP C4ER.
7/89
BDAJAX BXGIARBTG3C4ER
1,BXEYEH
CSD9E8DRBXGI z = x + iy CSE8GHCZCBEIGI (x,y) DXE8
B4F9,EKEXGHD9EYEHH2DTF9ARDQ?DVC1CG,BXGIARASC6DQ
CCEYEHH2ARBMARASC6EKE8E8GHBTARB3CG.
D0ADBXGI z = x + iy DWEXCMCCEYEHH2DVC1? (x,y) AR
BMC4ER.
O
y
x
P(x,y)
y
x
|z|=r
z = x + iy
θ
CSEE,x F2EH?EIF2; y F2EH?C2F2; H0F2EKBFAREYEHEH
BXEYEH.
BXGI z?DWEXCME5H2 OP C4ER.
7/89
BDAJAX BXGIARBTG3C4ER
1,BXEYEH
CSD9E8DRBXGI z = x + iy CSE8GHCZCBEIGI (x,y) DXE8
B4F9,EKEXGHD9EYEHH2DTF9ARDQ?DVC1CG,BXGIARASC6DQ
CCEYEHH2ARBMARASC6EKE8E8GHBTARB3CG.
D0ADBXGI z = x + iy DWEXCMCCEYEHH2DVC1? (x,y) AR
BMC4ER.
O
y
x
P(x,y)
y
x
|z|=r
z = x + iy
θ
CSEE,x F2EH?EIF2; y F2EH?C2F2; H0F2EKBFAREYEHEH
BXEYEH.
BXGI z?DWEXCME5H2 OP C4ER.
7/89
BXGIARAS
O
y
x
P(x,y)
y
x
|z|=r
z = x + iy
θ
E5H2ARDJG5EH? z ARASF9H1GHDU,C8DU
|z| = r =
radicalbig
x2 + y2.
DBB8CZ,
|x|lessorequalslant|z|,|y|lessorequalslant|z|,
|z|lessorequalslant|x|+ |y|,
z z = |z|2 = |z2|.
8/89
BXGIARAS
O
y
x
P(x,y)
y
x
|z|=r
z = x + iy
θ
E5H2ARDJG5EH? z ARASF9H1GHDU,C8DU
|z| = r =
radicalbig
x2 + y2.
DBB8CZ,
|x|lessorequalslant|z|,|y|lessorequalslant|z|,
|z|lessorequalslant|x|+ |y|,
z z = |z|2 = |z2|.
8/89
BXGIARAS
O
y
x
P(x,y)
y
x
|z|=r
z = x + iy
θ
E5H2ARDJG5EH? z ARASF9H1GHDU,C8DU
|z| = r =
radicalbig
x2 + y2.
DBB8CZ,
|x|lessorequalslant|z|,|y|lessorequalslant|z|,
|z|lessorequalslant|x|+ |y|,
z z = |z|2 = |z2|.
8/89
BXGIARAS
O
y
x
P(x,y)
y
x
|z|=r
z = x + iy
θ
E5H2ARDJG5EH? z ARASF9H1GHDU,C8DU
|z| = r =
radicalbig
x2 + y2.
DBB8CZ,
|x|lessorequalslant|z|,|y|lessorequalslant|z|,
|z|lessorequalslant|x|+ |y|,
z z = |z|2 = |z2|.
8/89
BXGIARAS
O
y
x
P(x,y)
y
x
|z|=r
z = x + iy
θ
E5H2ARDJG5EH? z ARASF9H1GHDU,C8DU
|z| = r =
radicalbig
x2 + y2.
DBB8CZ,
|x|lessorequalslant|z|,|y|lessorequalslant|z|,
|z|lessorequalslant|x|+ |y|,
z z = |z|2 = |z2|.
8/89
BXGIARFN?
O
y
x
P(x,y)
y
x
|z|=r
z = x + iy
θ
BF z negationslash= 0 ARECF5,EXAKEIF2?EPBQ,EXC4ER z ARE5
H2 OP?EQBQAR?ARD0G5GI θ EH? z ARFN?,C8DU
Argz = θ.
CSEECZ,
tg(Argz) = yx.
9/89
BXGIARFN?
O
y
x
P(x,y)
y
x
|z|=r
z = x + iy
θ
BF z negationslash= 0 ARECF5,EXAKEIF2?EPBQ,EXC4ER z ARE5
H2 OP?EQBQAR?ARD0G5GI θ EH? z ARFN?,C8DU
Argz = θ.
CSEECZ,
tg(Argz) = yx.
9/89
BXGIARFN?
O
y
x
P(x,y)
y
x
|z|=r
z = x + iy
θ
BF z negationslash= 0 ARECF5,EXAKEIF2?EPBQ,EXC4ER z ARE5
H2 OP?EQBQAR?ARD0G5GI θ EH? z ARFN?,C8DU
Argz = θ.
CSEECZ,
tg(Argz) = yx.
9/89
BRG3BXGI z negationslash= 0 CZFFEHGTDRFN?.CGC2 θ1 F0G1ELE8DR,
BSD3
Argz = θ1 + 2kpi(k?BRFBAIGI)
FUDTFTAA z ARASG4FN?.
BF z(negationslash= 0) ARFN?EL,D6GVDA?pi < θ0 lessorequalslant pibracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipuprightAR θ0 EH
Argz ARFODU,C8DU θ0 = argz.
A8 z = 0 EE,|z| = 0,ADFN?G0B4F9.
10/89
BRG3BXGI z negationslash= 0 CZFFEHGTDRFN?.CGC2 θ1 F0G1ELE8DR,
BSD3
Argz = θ1 + 2kpi(k?BRFBAIGI)
FUDTFTAA z ARASG4FN?.
BF z(negationslash= 0) ARFN?EL,D6GVDA?pi < θ0 lessorequalslant pibracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipuprightAR θ0 EH
Argz ARFODU,C8DU θ0 = argz.
A8 z = 0 EE,|z| = 0,ADFN?G0B4F9.
10/89
BRG3BXGI z negationslash= 0 CZFFEHGTDRFN?.CGC2 θ1 F0G1ELE8DR,
BSD3
Argz = θ1 + 2kpi(k?BRFBAIGI)
FUDTFTAA z ARASG4FN?.
BF z(negationslash= 0) ARFN?EL,D6GVDA?pi < θ0 lessorequalslant pibracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipuprightAR θ0 EH
Argz ARFODU,C8DU θ0 = argz.
A8 z = 0 EE,|z| = 0,ADFN?G0B4F9.
10/89
BRG3BXGI z negationslash= 0 CZFFEHGTDRFN?.CGC2 θ1 F0G1ELE8DR,
BSD3
Argz = θ1 + 2kpi(k?BRFBAIGI)
FUDTFTAA z ARASG4FN?.
BF z(negationslash= 0) ARFN?EL,D6GVDA?pi < θ0 lessorequalslant pibracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipuprightAR θ0 EH
Argz ARFODU,C8DU θ0 = argz.
A8 z = 0 EE,|z| = 0,ADFN?G0B4F9.
10/89
FN?FODU argz(z negationslash= 0) DWCDCTAFB3CGB4F9,(BH
FB?pi < argz lessorequalslant pi)
argz =






arc tgyx,z BFBDE8E6DL;
arc tgyx + pi,z BFBDAJE6DL;
arc tgyx?pi,z BFBDD2E6DL;
arc tgyx + 2pi,z BFBDD3E6DL.
G1EL,?pi2 < arc tgyx < pi2.
11/89
DUGABXGIARB0DYDFC3,H0DRBXGI z1 G2 z2 ARCK,DIDFB0
DYG2DNBTE5H2CK,DIB0DYE8E7.
y
O
z2
z1
z1+z2
x
y
z2
z1
z1? z2
z2
O
x
12/89
BXGIARD2?C4EREQ
DGCMDQ?DVC1DQBGDVC1ARB3CG:
x = rcosθ,y = rsinθ,
DWAQBXGI z = x + iy ARD2?C4EREQ
z = r(cosθ + isinθ).
13/89
BXGIARD2?C4EREQ
DGCMDQ?DVC1DQBGDVC1ARB3CG:
x = rcosθ,y = rsinθ,
DWAQBXGI z = x + iy ARD2?C4EREQ
z = r(cosθ + isinθ).
13/89
BXGIARD2?C4EREQ
DGCMDQ?DVC1DQBGDVC1ARB3CG:
x = rcosθ,y = rsinθ,
DWAQBXGI z = x + iy ARD2?C4EREQ
z = r(cosθ + isinθ).
13/89
BXGIARDXGIC4EREQ
DGCMGMBAGYEQ:
eiθ = cosθ + isinθ,
DWAQBXGI z ARDXGIC4EREQ
z = reiθ.
14/89
BXGIARDXGIC4EREQ
DGCMGMBAGYEQ:
eiθ = cosθ + isinθ,
DWAQBXGI z ARDXGIC4EREQ
z = reiθ.
14/89
BXGIARDXGIC4EREQ
DGCMGMBAGYEQ:
eiθ = cosθ + isinθ,
DWAQBXGI z ARDXGIC4EREQ
z = reiθ.
14/89
DI 1 AN 0 lessorequalslant? lessorequalslant pi2,FDGQBXGI z = 1 + sin? + icos? DE
D2?C4EREQDQDXGIC4EREQ.
B5FR? 0 lessorequalslant? lessorequalslant pi2,DGCMD2?GOAVEQ,CZC7DYBXGIARAS
|z| =
radicalbig
(1 + sin?)2 + cos2?
=
radicalbig
2(1 + sin?)
= √2
radicalbigg
sin2?2 + cos2?2 + 2sin?2cos?2
= √2
radicalbigg
(sin?2 + cos?2)2
= √2(sin?2 + cos?2)
= 2cos(pi42) = 2cospi? 2?4,
15/89
DI 1 AN 0 lessorequalslant? lessorequalslant pi2,FDGQBXGI z = 1 + sin? + icos? DE
D2?C4EREQDQDXGIC4EREQ.
B5FR? 0 lessorequalslant? lessorequalslant pi2,DGCMD2?GOAVEQ,CZC7DYBXGIARAS
|z| =
radicalbig
(1 + sin?)2 + cos2?
=
radicalbig
2(1 + sin?)
= √2
radicalbigg
sin2?2 + cos2?2 + 2sin?2cos?2
= √2
radicalbigg
(sin?2 + cos?2)2
= √2(sin?2 + cos?2)
= 2cos(pi42) = 2cospi? 2?4,
15/89
DI 1 AN 0 lessorequalslant? lessorequalslant pi2,FDGQBXGI z = 1 + sin? + icos? DE
D2?C4EREQDQDXGIC4EREQ.
B5FR? 0 lessorequalslant? lessorequalslant pi2,DGCMD2?GOAVEQ,CZC7DYBXGIARAS
|z| =
radicalbig
(1 + sin?)2 + cos2?
=
radicalbig
2(1 + sin?)
= √2
radicalbigg
sin2?2 + cos2?2 + 2sin?2cos?2
= √2
radicalbigg
(sin?2 + cos?2)2
= √2(sin?2 + cos?2)
= 2cos(pi42) = 2cospi? 2?4,
15/89
DI 1 AN 0 lessorequalslant? lessorequalslant pi2,FDGQBXGI z = 1 + sin? + icos? DE
D2?C4EREQDQDXGIC4EREQ.
B5FR? 0 lessorequalslant? lessorequalslant pi2,DGCMD2?GOAVEQ,CZC7DYBXGIARAS
|z| =
radicalbig
(1 + sin?)2 + cos2?
=
radicalbig
2(1 + sin?)
= √2
radicalbigg
sin2?2 + cos2?2 + 2sin?2cos?2
= √2
radicalbigg
(sin?2 + cos?2)2
= √2(sin?2 + cos?2)
= 2cos(pi42) = 2cospi? 2?4,
15/89
DI 1 AN 0 lessorequalslant? lessorequalslant pi2,FDGQBXGI z = 1 + sin? + icos? DE
D2?C4EREQDQDXGIC4EREQ.
B5FR? 0 lessorequalslant? lessorequalslant pi2,DGCMD2?GOAVEQ,CZC7DYBXGIARAS
|z| =
radicalbig
(1 + sin?)2 + cos2?
=
radicalbig
2(1 + sin?)
= √2
radicalbigg
sin2?2 + cos2?2 + 2sin?2cos?2
= √2
radicalbigg
(sin?2 + cos?2)2
= √2(sin?2 + cos?2)
= 2cos(pi42) = 2cospi? 2?4,
15/89
DI 1 AN 0 lessorequalslant? lessorequalslant pi2,FDGQBXGI z = 1 + sin? + icos? DE
D2?C4EREQDQDXGIC4EREQ.
B5FR? 0 lessorequalslant? lessorequalslant pi2,DGCMD2?GOAVEQ,CZC7DYBXGIARAS
|z| =
radicalbig
(1 + sin?)2 + cos2?
=
radicalbig
2(1 + sin?)
= √2
radicalbigg
sin2?2 + cos2?2 + 2sin?2cos?2
= √2
radicalbigg
(sin?2 + cos?2)2
= √2(sin?2 + cos?2)
= 2cos(pi42) = 2cospi? 2?4,
15/89
DI 1 AN 0 lessorequalslant? lessorequalslant pi2,FDGQBXGI z = 1 + sin? + icos? DE
D2?C4EREQDQDXGIC4EREQ.
B5FR? 0 lessorequalslant? lessorequalslant pi2,DGCMD2?GOAVEQ,CZC7DYBXGIARAS
|z| =
radicalbig
(1 + sin?)2 + cos2?
=
radicalbig
2(1 + sin?)
= √2
radicalbigg
sin2?2 + cos2?2 + 2sin?2cos?2
= √2
radicalbigg
(sin?2 + cos?2)2
= √2(sin?2 + cos?2)
= 2cos(pi42) = 2cospi? 2?4,
15/89
BEC7DYBXGIARFN?
argz = arctan cos?1 + sin? = arctan
cos2?2?sin2?2
parenleftBig
cos?2 + sin?2
parenrightBig2
= arctan
sin(pi42)
cos(pi42)
= pi42 = pi?2?4,
AP z ARD2?C4EREQ
1 + sin? + icos? = 2cospi? 2?4
parenleftbigg
cospi?2?4 + isinpi?2?4
parenrightbigg
z ARDXGIC4EREQ
1 + sin? + icos? = 2cospi?2?4 eipi?2?4
16/89
BEC7DYBXGIARFN?
argz = arctan cos?1 + sin? = arctan
cos2?2?sin2?2
parenleftBig
cos?2 + sin?2
parenrightBig2
= arctan
sin(pi42)
cos(pi42)
= pi42 = pi?2?4,
AP z ARD2?C4EREQ
1 + sin? + icos? = 2cospi? 2?4
parenleftbigg
cospi?2?4 + isinpi?2?4
parenrightbigg
z ARDXGIC4EREQ
1 + sin? + icos? = 2cospi?2?4 eipi?2?4
16/89
BEC7DYBXGIARFN?
argz = arctan cos?1 + sin? = arctan
cos2?2?sin2?2
parenleftBig
cos?2 + sin?2
parenrightBig2
= arctan
sin(pi42)
cos(pi42)
= pi42 = pi?2?4,
AP z ARD2?C4EREQ
1 + sin? + icos? = 2cospi? 2?4
parenleftbigg
cospi?2?4 + isinpi?2?4
parenrightbigg
z ARDXGIC4EREQ
1 + sin? + icos? = 2cospi?2?4 eipi?2?4
16/89
BEC7DYBXGIARFN?
argz = arctan cos?1 + sin? = arctan
cos2?2?sin2?2
parenleftBig
cos?2 + sin?2
parenrightBig2
= arctan
sin(pi42)
cos(pi42)
= pi42 = pi?2?4,
AP z ARD2?C4EREQ
1 + sin? + icos? = 2cospi? 2?4
parenleftbigg
cospi?2?4 + isinpi?2?4
parenrightbigg
z ARDXGIC4EREQ
1 + sin? + icos? = 2cospi?2?4 eipi?2?4
16/89
BEC7DYBXGIARFN?
argz = arctan cos?1 + sin? = arctan
cos2?2?sin2?2
parenleftBig
cos?2 + sin?2
parenrightBig2
= arctan
sin(pi42)
cos(pi42)
= pi42 = pi?2?4,
AP z ARD2?C4EREQ
1 + sin? + icos? = 2cospi? 2?4
parenleftbigg
cospi?2?4 + isinpi?2?4
parenrightbigg
z ARDXGIC4EREQ
1 + sin? + icos? = 2cospi?2?4 eipi?2?4
16/89
BEC7DYBXGIARFN?
argz = arctan cos?1 + sin? = arctan
cos2?2?sin2?2
parenleftBig
cos?2 + sin?2
parenrightBig2
= arctan
sin(pi42)
cos(pi42)
= pi42 = pi?2?4,
AP z ARD2?C4EREQ
1 + sin? + icos? = 2cospi? 2?4
parenleftbigg
cospi?2?4 + isinpi?2?4
parenrightbigg
z ARDXGIC4EREQ
1 + sin? + icos? = 2cospi?2?4 eipi?2?4
16/89
E0DFAJDGCMD2?GYEQ,DWDJ
1 + sin? + icos? = 1 + cos(pi2) + isin(pi2)
= 2cos2(pi42) + 2isin(pi42)cos(pi42)
= 2cospi? 2?4
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
parenleftbigg
cospi? 2?4 + isinpi? 2?4
parenrightbigg
= 2cospi?2?4 eipi?2?4
17/89
E0DFAJDGCMD2?GYEQ,DWDJ
1 + sin? + icos? = 1 + cos(pi2) + isin(pi2)
= 2cos2(pi42) + 2isin(pi42)cos(pi42)
= 2cospi? 2?4
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
parenleftbigg
cospi? 2?4 + isinpi? 2?4
parenrightbigg
= 2cospi?2?4 eipi?2?4
17/89
E0DFAJDGCMD2?GYEQ,DWDJ
1 + sin? + icos? = 1 + cos(pi2) + isin(pi2)
= 2cos2(pi42) + 2isin(pi42)cos(pi42)
= 2cospi? 2?4
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
parenleftbigg
cospi? 2?4 + isinpi? 2?4
parenrightbigg
= 2cospi?2?4 eipi?2?4
17/89
E0DFAJDGCMD2?GYEQ,DWDJ
1 + sin? + icos? = 1 + cos(pi2) + isin(pi2)
= 2cos2(pi42) + 2isin(pi42)cos(pi42)
= 2cospi? 2?4
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
parenleftbigg
cospi? 2?4 + isinpi? 2?4
parenrightbigg
= 2cospi?2?4 eipi?2?4
17/89
E0DFAJDGCMD2?GYEQ,DWDJ
1 + sin? + icos? = 1 + cos(pi2) + isin(pi2)
= 2cos2(pi42) + 2isin(pi42)cos(pi42)
= 2cospi? 2?4
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
parenleftbigg
cospi? 2?4 + isinpi? 2?4
parenrightbigg
= 2cospi?2?4 eipi?2?4
17/89
EYEHBTG3EVC3ARBXGIE0DF
CSD9BXGIDQEYEHH2ARBM (E5H2) DND1ARB3CG,D9F0
GHGTEYEHGBBBCDCMBXGIBBEQARE0EN (F9G0AVEQ) C4ER;
DWEXCSDTF9ARBXGIBBEQARE0EN (F9G0AVEQ) B4F9EN
DMEKC4ERAREYEHGBBB.
DXBRE8G2AR,F3DMDWEXCMBXGIE0DFBHEPG3AXBRG0G2EMDD
EYEHBTG3EVC3.
DICMARE0DF,GQEYEHH2ARBMCMBXGIC4ER,GQEYEHETDMDE
BXGIBBEQ; F9GUGQBMCME5H2C4ER,DGCME5H2B0DYEMB5.
18/89
EYEHBTG3EVC3ARBXGIE0DF
CSD9BXGIDQEYEHH2ARBM (E5H2) DND1ARB3CG,D9F0
GHGTEYEHGBBBCDCMBXGIBBEQARE0EN (F9G0AVEQ) C4ER;
DWEXCSDTF9ARBXGIBBEQARE0EN (F9G0AVEQ) B4F9EN
DMEKC4ERAREYEHGBBB.
DXBRE8G2AR,F3DMDWEXCMBXGIE0DFBHEPG3AXBRG0G2EMDD
EYEHBTG3EVC3.
DICMARE0DF,GQEYEHH2ARBMCMBXGIC4ER,GQEYEHETDMDE
BXGIBBEQ; F9GUGQBMCME5H2C4ER,DGCME5H2B0DYEMB5.
18/89
EYEHBTG3EVC3ARBXGIE0DF
CSD9BXGIDQEYEHH2ARBM (E5H2) DND1ARB3CG,D9F0
GHGTEYEHGBBBCDCMBXGIBBEQARE0EN (F9G0AVEQ) C4ER;
DWEXCSDTF9ARBXGIBBEQARE0EN (F9G0AVEQ) B4F9EN
DMEKC4ERAREYEHGBBB.
DXBRE8G2AR,F3DMDWEXCMBXGIE0DFBHEPG3AXBRG0G2EMDD
EYEHBTG3EVC3.
DICMARE0DF,GQEYEHH2ARBMCMBXGIC4ER,GQEYEHETDMDE
BXGIBBEQ; F9GUGQBMCME5H2C4ER,DGCME5H2B0DYEMB5.
18/89
EYEHBTG3EVC3ARBXGIE0DF
CSD9BXGIDQEYEHH2ARBM (E5H2) DND1ARB3CG,D9F0
GHGTEYEHGBBBCDCMBXGIBBEQARE0EN (F9G0AVEQ) C4ER;
DWEXCSDTF9ARBXGIBBEQARE0EN (F9G0AVEQ) B4F9EN
DMEKC4ERAREYEHGBBB.
DXBRE8G2AR,F3DMDWEXCMBXGIE0DFBHEPG3AXBRG0G2EMDD
EYEHBTG3EVC3.
DICMARE0DF,GQEYEHH2ARBMCMBXGIC4ER,GQEYEHETDMDE
BXGIBBEQ; F9GUGQBMCME5H2C4ER,DGCME5H2B0DYEMB5.
18/89
EYEHBTG3EVC3ARBXGIE0DF
CSD9BXGIDQEYEHH2ARBM (E5H2) DND1ARB3CG,D9F0
GHGTEYEHGBBBCDCMBXGIBBEQARE0EN (F9G0AVEQ) C4ER;
DWEXCSDTF9ARBXGIBBEQARE0EN (F9G0AVEQ) B4F9EN
DMEKC4ERAREYEHGBBB.
DXBRE8G2AR,F3DMDWEXCMBXGIE0DFBHEPG3AXBRG0G2EMDD
EYEHBTG3EVC3.
DICMARE0DF,GQEYEHH2ARBMCMBXGIC4ER,GQEYEHETDMDE
BXGIBBEQ; F9GUGQBMCME5H2C4ER,DGCME5H2B0DYEMB5.
18/89
EYEHBTG3EVC3ARBXGIE0DF
CSD9BXGIDQEYEHH2ARBM (E5H2) DND1ARB3CG,D9F0
GHGTEYEHGBBBCDCMBXGIBBEQARE0EN (F9G0AVEQ) C4ER;
DWEXCSDTF9ARBXGIBBEQARE0EN (F9G0AVEQ) B4F9EN
DMEKC4ERAREYEHGBBB.
DXBRE8G2AR,F3DMDWEXCMBXGIE0DFBHEPG3AXBRG0G2EMDD
EYEHBTG3EVC3.
DICMARE0DF,GQEYEHH2ARBMCMBXGIC4ER,GQEYEHETDMDE
BXGIBBEQ; F9GUGQBMCME5H2C4ER,DGCME5H2B0DYEMB5.
18/89
DI 3 GQFRC4H0BM z1 = x1 + iy1 DQ z2 = x2 + iy2 ARDQ
DMCMBXGIBBEQARE0ENBHC4ER.
B5FRC4H0BM (x1,y1) DQ (x2,y2) ARDQDMDWEXCMGJGIE0EN
C4ER?
braceleftbigg x = x
1 + t(x2?x1),y = y
1 + t(y2?y1),(?∞ < t < +∞).
FRCS,ENARBXGIBBEQARGJGIE0EN?
z = x + iy
= x1 + t(x2?x1)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright+i(y1 + t(y2?y1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright))
= x1 + iy1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright+t[(x2 + iy2bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright)? (x1 + iy1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright)]
= z1 + t(z2?z1),(?∞ < t < +∞).
19/89
DI 3 GQFRC4H0BM z1 = x1 + iy1 DQ z2 = x2 + iy2 ARDQ
DMCMBXGIBBEQARE0ENBHC4ER.
B5FRC4H0BM (x1,y1) DQ (x2,y2) ARDQDMDWEXCMGJGIE0EN
C4ER?
braceleftbigg x = x
1 + t(x2?x1),y = y
1 + t(y2?y1),(?∞ < t < +∞).
FRCS,ENARBXGIBBEQARGJGIE0EN?
z = x + iy
= x1 + t(x2?x1)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright+i(y1 + t(y2?y1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright))
= x1 + iy1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright+t[(x2 + iy2bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright)? (x1 + iy1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright)]
= z1 + t(z2?z1),(?∞ < t < +∞).
19/89
DI 3 GQFRC4H0BM z1 = x1 + iy1 DQ z2 = x2 + iy2 ARDQ
DMCMBXGIBBEQARE0ENBHC4ER.
B5FRC4H0BM (x1,y1) DQ (x2,y2) ARDQDMDWEXCMGJGIE0EN
C4ER?
braceleftbigg x = x
1 + t(x2?x1),y = y
1 + t(y2?y1),(?∞ < t < +∞).
FRCS,ENARBXGIBBEQARGJGIE0EN?
z = x + iy
= x1 + t(x2?x1)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright+i(y1 + t(y2?y1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright))
= x1 + iy1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright+t[(x2 + iy2bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright)? (x1 + iy1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright)]
= z1 + t(z2?z1),(?∞ < t < +∞).
19/89
DI 3 GQFRC4H0BM z1 = x1 + iy1 DQ z2 = x2 + iy2 ARDQ
DMCMBXGIBBEQARE0ENBHC4ER.
B5FRC4H0BM (x1,y1) DQ (x2,y2) ARDQDMDWEXCMGJGIE0EN
C4ER?
braceleftbigg x = x
1 + t(x2?x1),y = y
1 + t(y2?y1),(?∞ < t < +∞).
FRCS,ENARBXGIBBEQARGJGIE0EN?
z = x + iy
= x1 + t(x2?x1)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright+i(y1 + t(y2?y1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright))
= x1 + iy1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright+t[(x2 + iy2bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright)? (x1 + iy1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright)]
= z1 + t(z2?z1),(?∞ < t < +∞).
19/89
DI 3 GQFRC4H0BM z1 = x1 + iy1 DQ z2 = x2 + iy2 ARDQ
DMCMBXGIBBEQARE0ENBHC4ER.
B5FRC4H0BM (x1,y1) DQ (x2,y2) ARDQDMDWEXCMGJGIE0EN
C4ER?
braceleftbigg x = x
1 + t(x2?x1),y = y
1 + t(y2?y1),(?∞ < t < +∞).
FRCS,ENARBXGIBBEQARGJGIE0EN?
z = x + iy
= x1 + t(x2?x1)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright+i(y1 + t(y2?y1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright))
= x1 + iy1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright+t[(x2 + iy2bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright)? (x1 + iy1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright)]
= z1 + t(z2?z1),(?∞ < t < +∞).
19/89
DI 3 GQFRC4H0BM z1 = x1 + iy1 DQ z2 = x2 + iy2 ARDQ
DMCMBXGIBBEQARE0ENBHC4ER.
B5FRC4H0BM (x1,y1) DQ (x2,y2) ARDQDMDWEXCMGJGIE0EN
C4ER?
braceleftbigg x = x
1 + t(x2?x1),y = y
1 + t(y2?y1),(?∞ < t < +∞).
FRCS,ENARBXGIBBEQARGJGIE0EN?
z = x + iy
= x1 + t(x2?x1)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright+i(y1 + t(y2?y1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright))
= x1 + iy1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright+t[(x2 + iy2bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright)? (x1 + iy1bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright)]
= z1 + t(z2?z1),(?∞ < t < +∞).
19/89
DI 4 EMCTAFE0ENEKC4ERARETDM:
1) |z + i| = 2; 2) |z?2i| = |z + 2|;
B5 1) D0BTG3H2DB,E0EN |z + i| = 2 C4EREKCZDQBM?i
ARGDD0? 2 ARBMARBNB8,BPELB1i,DWF7? 2 ARAB.
i 2
O
y
x
D0E3GIH2DB,AN z = x + iy,E0ENBV?
|x + (y + 1)i| = 2.
FUF0
radicalbigx2 + (y + 1)2 = 2.
F9 x2 + (y + 1)2 = 4.
20/89
DI 4 EMCTAFE0ENEKC4ERARETDM:
1) |z + i| = 2; 2) |z?2i| = |z + 2|;
B5 1) D0BTG3H2DB,E0EN |z + i| = 2 C4EREKCZDQBM?i
ARGDD0? 2 ARBMARBNB8,BPELB1i,DWF7? 2 ARAB.
i 2
O
y
x
D0E3GIH2DB,AN z = x + iy,E0ENBV?
|x + (y + 1)i| = 2.
FUF0
radicalbigx2 + (y + 1)2 = 2.
F9 x2 + (y + 1)2 = 4.
20/89
DI 4 EMCTAFE0ENEKC4ERARETDM:
1) |z + i| = 2; 2) |z?2i| = |z + 2|;
B5 1) D0BTG3H2DB,E0EN |z + i| = 2 C4EREKCZDQBM?i
ARGDD0? 2 ARBMARBNB8,BPELB1i,DWF7? 2 ARAB.
i 2
O
y
x
D0E3GIH2DB,AN z = x + iy,E0ENBV?
|x + (y + 1)i| = 2.
FUF0
radicalbigx2 + (y + 1)2 = 2.
F9 x2 + (y + 1)2 = 4.
20/89
DI 4 EMCTAFE0ENEKC4ERARETDM:
1) |z + i| = 2; 2) |z?2i| = |z + 2|;
B5 1) D0BTG3H2DB,E0EN |z + i| = 2 C4EREKCZDQBM?i
ARGDD0? 2 ARBMARBNB8,BPELB1i,DWF7? 2 ARAB.
i 2
O
y
x
D0E3GIH2DB,AN z = x + iy,E0ENBV?
|x + (y + 1)i| = 2.
FUF0
radicalbigx2 + (y + 1)2 = 2.
F9 x2 + (y + 1)2 = 4.
20/89
DI 4 EMCTAFE0ENEKC4ERARETDM:
1) |z + i| = 2; 2) |z?2i| = |z + 2|;
B5 1) D0BTG3H2DB,E0EN |z + i| = 2 C4EREKCZDQBM?i
ARGDD0? 2 ARBMARBNB8,BPELB1i,DWF7? 2 ARAB.
i 2
O
y
x
D0E3GIH2DB,AN z = x + iy,E0ENBV?
|x + (y + 1)i| = 2.
FUF0
radicalbigx2 + (y + 1)2 = 2.
F9 x2 + (y + 1)2 = 4.
20/89
(2) D0BTG3H2DB,E0EN |z? 2i| = |z + 2| C4EREKCZAK
BM 2i G2?2 ARGDD0DNAVARBMARBNB8,BPGJAQBM 2i G2?2
ARDMGBARCADQEYEPDM.
y
x
O
y=?x
2
2i
D0E3GIH2DB,AN z = x + iy,E0ENBV?
|x + (y?2)i| = |(x + 2) + yi|.
FUF0 radicalbig
x2 + (y? 2)2 =
radicalbig
(x + 2)2 + y2
F9
y =?x.
21/89
(2) D0BTG3H2DB,E0EN |z? 2i| = |z + 2| C4EREKCZAK
BM 2i G2?2 ARGDD0DNAVARBMARBNB8,BPGJAQBM 2i G2?2
ARDMGBARCADQEYEPDM.
y
x
O
y=?x
2
2i
D0E3GIH2DB,AN z = x + iy,E0ENBV?
|x + (y?2)i| = |(x + 2) + yi|.
FUF0 radicalbig
x2 + (y? 2)2 =
radicalbig
(x + 2)2 + y2
F9
y =?x.
21/89
(2) D0BTG3H2DB,E0EN |z? 2i| = |z + 2| C4EREKCZAK
BM 2i G2?2 ARGDD0DNAVARBMARBNB8,BPGJAQBM 2i G2?2
ARDMGBARCADQEYEPDM.
y
x
O
y=?x
2
2i
D0E3GIH2DB,AN z = x + iy,E0ENBV?
|x + (y?2)i| = |(x + 2) + yi|.
FUF0 radicalbig
x2 + (y? 2)2 =
radicalbig
(x + 2)2 + y2
F9
y =?x.
21/89
(2) D0BTG3H2DB,E0EN |z? 2i| = |z + 2| C4EREKCZAK
BM 2i G2?2 ARGDD0DNAVARBMARBNB8,BPGJAQBM 2i G2?2
ARDMGBARCADQEYEPDM.
y
x
O
y=?x
2
2i
D0E3GIH2DB,AN z = x + iy,E0ENBV?
|x + (y?2)i| = |(x + 2) + yi|.
FUF0 radicalbig
x2 + (y? 2)2 =
radicalbig
(x + 2)2 + y2
F9
y =?x.
21/89
(3) |z+1|+|z?2| = 5; (4) ||z?3|?|z +2|| = 1;
B5 (3) D0BTG3H2DB,AVEQC4ERAKBM z1 =?1 ARGDD0DQAK
BM z2 = 2 ARGDD0DNG2? 5 ARFGBM z ARBNB8,GXF0E8DR
EX z1 =?1 DQ z2 = 2?A0BMARH1AB.
D0E3GIH2DB,AN z = x + iy,E0ENBV? ······.
(4) D0BTG3H2DB,AVEQC4ERAKBM z1 = 3 ARGDD0DQAK
BM z2 =?2 ARGDD0DNAB? 1 ARFGBM z ARBNB8,GXF0E8DR
EX z1 = 3 DQ z2 =?2?A0BMARCEETDM.
D0E3GIH2DB,AN z = x + iy,E0ENBV? ······
22/89
(3) |z+1|+|z?2| = 5; (4) ||z?3|?|z +2|| = 1;
B5 (3) D0BTG3H2DB,AVEQC4ERAKBM z1 =?1 ARGDD0DQAK
BM z2 = 2 ARGDD0DNG2? 5 ARFGBM z ARBNB8,GXF0E8DR
EX z1 =?1 DQ z2 = 2?A0BMARH1AB.
D0E3GIH2DB,AN z = x + iy,E0ENBV? ······.
(4) D0BTG3H2DB,AVEQC4ERAKBM z1 = 3 ARGDD0DQAK
BM z2 =?2 ARGDD0DNAB? 1 ARFGBM z ARBNB8,GXF0E8DR
EX z1 = 3 DQ z2 =?2?A0BMARCEETDM.
D0E3GIH2DB,AN z = x + iy,E0ENBV? ······
22/89
(3) |z+1|+|z?2| = 5; (4) ||z?3|?|z +2|| = 1;
B5 (3) D0BTG3H2DB,AVEQC4ERAKBM z1 =?1 ARGDD0DQAK
BM z2 = 2 ARGDD0DNG2? 5 ARFGBM z ARBNB8,GXF0E8DR
EX z1 =?1 DQ z2 = 2?A0BMARH1AB.
D0E3GIH2DB,AN z = x + iy,E0ENBV? ······.
(4) D0BTG3H2DB,AVEQC4ERAKBM z1 = 3 ARGDD0DQAK
BM z2 =?2 ARGDD0DNAB? 1 ARFGBM z ARBNB8,GXF0E8DR
EX z1 = 3 DQ z2 =?2?A0BMARCEETDM.
D0E3GIH2DB,AN z = x + iy,E0ENBV? ······
22/89
(3) |z+1|+|z?2| = 5; (4) ||z?3|?|z +2|| = 1;
B5 (3) D0BTG3H2DB,AVEQC4ERAKBM z1 =?1 ARGDD0DQAK
BM z2 = 2 ARGDD0DNG2? 5 ARFGBM z ARBNB8,GXF0E8DR
EX z1 =?1 DQ z2 = 2?A0BMARH1AB.
D0E3GIH2DB,AN z = x + iy,E0ENBV? ······.
(4) D0BTG3H2DB,AVEQC4ERAKBM z1 = 3 ARGDD0DQAK
BM z2 =?2 ARGDD0DNAB? 1 ARFGBM z ARBNB8,GXF0E8DR
EX z1 = 3 DQ z2 =?2?A0BMARCEETDM.
D0E3GIH2DB,AN z = x + iy,E0ENBV? ······
22/89
(3) |z+1|+|z?2| = 5; (4) ||z?3|?|z +2|| = 1;
B5 (3) D0BTG3H2DB,AVEQC4ERAKBM z1 =?1 ARGDD0DQAK
BM z2 = 2 ARGDD0DNG2? 5 ARFGBM z ARBNB8,GXF0E8DR
EX z1 =?1 DQ z2 = 2?A0BMARH1AB.
D0E3GIH2DB,AN z = x + iy,E0ENBV? ······.
(4) D0BTG3H2DB,AVEQC4ERAKBM z1 = 3 ARGDD0DQAK
BM z2 =?2 ARGDD0DNAB? 1 ARFGBM z ARBNB8,GXF0E8DR
EX z1 = 3 DQ z2 =?2?A0BMARCEETDM.
D0E3GIH2DB,AN z = x + iy,E0ENBV? ······
22/89
2,BXELEH
BKAR,CMELEHH2ARBMC4ERBXGI.
P
z
S O
y
x
z
N
CDGKGBELARE0EQ,ELEHH2ARBM,G0AMAGBG N A9,DQBXEY
EHH2ARBMDJBFE8E8GHBTARB3CG,D9F0F3DMFUDWEXCMELEHH2
ARBMC4ERBXGI.
B8A9,GHD9ELEHH2ARAGBG N,F3DMBHF9,BXEYEHH2CZE8
DRDXE8AR,FFEHAFBM”,DQAGBG N GHBT.
F3DMD5BHF9,BXGIELCZE8DRDXE8AR”FFEHDW” DQBXEYEH
H2FFEHAFBMGHBT,FBC8DU ∞,GXAOARELEHEH?BXELEH.
23/89
2,BXELEH
BKAR,CMELEHH2ARBMC4ERBXGI.
P
z
S O
y
x
z
N
CDGKGBELARE0EQ,ELEHH2ARBM,G0AMAGBG N A9,DQBXEY
EHH2ARBMDJBFE8E8GHBTARB3CG,D9F0F3DMFUDWEXCMELEHH2
ARBMC4ERBXGI.
B8A9,GHD9ELEHH2ARAGBG N,F3DMBHF9,BXEYEHH2CZE8
DRDXE8AR,FFEHAFBM”,DQAGBG N GHBT.
F3DMD5BHF9,BXGIELCZE8DRDXE8AR”FFEHDW” DQBXEYEH
H2FFEHAFBMGHBT,FBC8DU ∞,GXAOARELEHEH?BXELEH.
23/89
2,BXELEH
BKAR,CMELEHH2ARBMC4ERBXGI.
P
z
S O
y
x
z
N
CDGKGBELARE0EQ,ELEHH2ARBM,G0AMAGBG N A9,DQBXEY
EHH2ARBMDJBFE8E8GHBTARB3CG,D9F0F3DMFUDWEXCMELEHH2
ARBMC4ERBXGI.
B8A9,GHD9ELEHH2ARAGBG N,F3DMBHF9,BXEYEHH2CZE8
DRDXE8AR,FFEHAFBM”,DQAGBG N GHBT.
F3DMD5BHF9,BXGIELCZE8DRDXE8AR”FFEHDW” DQBXEYEH
H2FFEHAFBMGHBT,FBC8DU ∞,GXAOARELEHEH?BXELEH.
23/89
2,BXELEH
BKAR,CMELEHH2ARBMC4ERBXGI.
P
z
S O
y
x
z
N
CDGKGBELARE0EQ,ELEHH2ARBM,G0AMAGBG N A9,DQBXEY
EHH2ARBMDJBFE8E8GHBTARB3CG,D9F0F3DMFUDWEXCMELEHH2
ARBMC4ERBXGI.
B8A9,GHD9ELEHH2ARAGBG N,F3DMBHF9,BXEYEHH2CZE8
DRDXE8AR,FFEHAFBM”,DQAGBG N GHBT.
F3DMD5BHF9,BXGIELCZE8DRDXE8AR”FFEHDW” DQBXEYEH
H2FFEHAFBMGHBT,FBC8DU ∞,GXAOARELEHEH?BXELEH.
23/89
2,BXELEH
BKAR,CMELEHH2ARBMC4ERBXGI.
P
z
S O
y
x
z
N
CDGKGBELARE0EQ,ELEHH2ARBM,G0AMAGBG N A9,DQBXEY
EHH2ARBMDJBFE8E8GHBTARB3CG,D9F0F3DMFUDWEXCMELEHH2
ARBMC4ERBXGI.
B8A9,GHD9ELEHH2ARAGBG N,F3DMBHF9,BXEYEHH2CZE8
DRDXE8AR,FFEHAFBM”,DQAGBG N GHBT.
F3DMD5BHF9,BXGIELCZE8DRDXE8AR”FFEHDW” DQBXEYEH
H2FFEHAFBMGHBT,FBC8DU ∞,GXAOARELEHEH?BXELEH.
23/89
EHEDB5FFEHAFBM ∞BFCBARBXEYEHEH?B6FBBXEYEH,G0
EDB5FFEHAFBMBFCBARBXEYEHEH?CZDLEYEH,F9EHBXEYEH.
BXGI ∞ AREIG4,C2G4DQFN?H2FFFBFE,ENARAS:
|z| = +∞.
24/89
EHEDB5FFEHAFBM ∞BFCBARBXEYEHEH?B6FBBXEYEH,G0
EDB5FFEHAFBMBFCBARBXEYEHEH?CZDLEYEH,F9EHBXEYEH.
BXGI ∞ AREIG4,C2G4DQFN?H2FFFBFE,ENARAS:
|z| = +∞.
24/89
B3D9 ∞ ARD3C3B0DYDTCGCTBHF9:
CKDF,a +∞ = ∞+ a = ∞ (a negationslash= ∞)
DIDF,a?∞ = ∞?a = ∞ (a negationslash= ∞)
EMDF,a·∞ = ∞·a = ∞ (a negationslash= 0)
G0DF:a
∞ = 0,
∞
a = ∞(a negationslash= ∞),
a
0 = ∞(a negationslash= 0,A2DW?∞).
G1ENB0DY:∞±∞,0·∞,∞∞ G0BHF9G1FBFE.
25/89
B3D9 ∞ ARD3C3B0DYDTCGCTBHF9:
CKDF,a +∞ = ∞+ a = ∞ (a negationslash= ∞)
DIDF,a?∞ = ∞?a = ∞ (a negationslash= ∞)
EMDF,a·∞ = ∞·a = ∞ (a negationslash= 0)
G0DF:a
∞ = 0,
∞
a = ∞(a negationslash= ∞),
a
0 = ∞(a negationslash= 0,A2DW?∞).
G1ENB0DY:∞±∞,0·∞,∞∞ G0BHF9G1FBFE.
25/89
B3D9 ∞ ARD3C3B0DYDTCGCTBHF9:
CKDF,a +∞ = ∞+ a = ∞ (a negationslash= ∞)
DIDF,a?∞ = ∞?a = ∞ (a negationslash= ∞)
EMDF,a·∞ = ∞·a = ∞ (a negationslash= 0)
G0DF:a
∞ = 0,
∞
a = ∞(a negationslash= ∞),
a
0 = ∞(a negationslash= 0,A2DW?∞).
G1ENB0DY:∞±∞,0·∞,∞∞ G0BHF9G1FBFE.
25/89
B3D9 ∞ ARD3C3B0DYDTCGCTBHF9:
CKDF,a +∞ = ∞+ a = ∞ (a negationslash= ∞)
DIDF,a?∞ = ∞?a = ∞ (a negationslash= ∞)
EMDF,a·∞ = ∞·a = ∞ (a negationslash= 0)
G0DF:a
∞ = 0,
∞
a = ∞(a negationslash= ∞),
a
0 = ∞(a negationslash= 0,A2DW?∞).
G1ENB0DY:∞±∞,0·∞,∞∞ G0BHF9G1FBFE.
25/89
B3D9 ∞ ARD3C3B0DYDTCGCTBHF9:
CKDF,a +∞ = ∞+ a = ∞ (a negationslash= ∞)
DIDF,a?∞ = ∞?a = ∞ (a negationslash= ∞)
EMDF,a·∞ = ∞·a = ∞ (a negationslash= 0)
G0DF:a
∞ = 0,
∞
a = ∞(a negationslash= ∞),
a
0 = ∞(a negationslash= 0,A2DW?∞).
G1ENB0DY:∞±∞,0·∞,∞∞ G0BHF9G1FBFE.
25/89
B3D9 ∞ ARD3C3B0DYDTCGCTBHF9:
CKDF,a +∞ = ∞+ a = ∞ (a negationslash= ∞)
DIDF,a?∞ = ∞?a = ∞ (a negationslash= ∞)
EMDF,a·∞ = ∞·a = ∞ (a negationslash= 0)
G0DF:a
∞ = 0,
∞
a = ∞(a negationslash= ∞),
a
0 = ∞(a negationslash= 0,A2DW?∞).
G1ENB0DY:∞±∞,0·∞,∞∞ G0BHF9G1FBFE.
25/89
BDD3AX BXGIAREME8DQE0DU
1,EMFKDQGZ
ANCZH0DRBXGI
z1 = r1(cosθ1 + isinθ1),z2 = r2(cosθ2 + isinθ2).
BSD3 z1z2 = r1r2(cosθ1 + isinθ1)(cosθ2 + isinθ2)
= r1r2[cos(θ1 + θ2) + isin(θ1 + θ2)].
D9F0 |z1z2| = |z1||z2|
Arg(z1z2) = Arg(z1) + Arg(z2).
D0ADCZCTEHARF9D2.
F9D2E8H0DRBXGIEMFKARASAVD9ENDMARASAREMFK; H0DRBX
GIEMFKARFN?AVD9ENDMARFN?ARG2.
26/89
BDD3AX BXGIAREME8DQE0DU
1,EMFKDQGZ
ANCZH0DRBXGI
z1 = r1(cosθ1 + isinθ1),z2 = r2(cosθ2 + isinθ2).
BSD3 z1z2 = r1r2(cosθ1 + isinθ1)(cosθ2 + isinθ2)
= r1r2[cos(θ1 + θ2) + isin(θ1 + θ2)].
D9F0 |z1z2| = |z1||z2|
Arg(z1z2) = Arg(z1) + Arg(z2).
D0ADCZCTEHARF9D2.
F9D2E8H0DRBXGIEMFKARASAVD9ENDMARASAREMFK; H0DRBX
GIEMFKARFN?AVD9ENDMARFN?ARG2.
26/89
BDD3AX BXGIAREME8DQE0DU
1,EMFKDQGZ
ANCZH0DRBXGI
z1 = r1(cosθ1 + isinθ1),z2 = r2(cosθ2 + isinθ2).
BSD3 z1z2 = r1r2(cosθ1 + isinθ1)(cosθ2 + isinθ2)
= r1r2[cos(θ1 + θ2) + isin(θ1 + θ2)].
D9F0 |z1z2| = |z1||z2|
Arg(z1z2) = Arg(z1) + Arg(z2).
D0ADCZCTEHARF9D2.
F9D2E8H0DRBXGIEMFKARASAVD9ENDMARASAREMFK; H0DRBX
GIEMFKARFN?AVD9ENDMARFN?ARG2.
26/89
BDD3AX BXGIAREME8DQE0DU
1,EMFKDQGZ
ANCZH0DRBXGI
z1 = r1(cosθ1 + isinθ1),z2 = r2(cosθ2 + isinθ2).
BSD3 z1z2 = r1r2(cosθ1 + isinθ1)(cosθ2 + isinθ2)
= r1r2[cos(θ1 + θ2) + isin(θ1 + θ2)].
D9F0 |z1z2| = |z1||z2|
Arg(z1z2) = Arg(z1) + Arg(z2).
D0ADCZCTEHARF9D2.
F9D2E8H0DRBXGIEMFKARASAVD9ENDMARASAREMFK; H0DRBX
GIEMFKARFN?AVD9ENDMARFN?ARG2.
26/89
BDD3AX BXGIAREME8DQE0DU
1,EMFKDQGZ
ANCZH0DRBXGI
z1 = r1(cosθ1 + isinθ1),z2 = r2(cosθ2 + isinθ2).
BSD3 z1z2 = r1r2(cosθ1 + isinθ1)(cosθ2 + isinθ2)
= r1r2[cos(θ1 + θ2) + isin(θ1 + θ2)].
D9F0 |z1z2| = |z1||z2|
Arg(z1z2) = Arg(z1) + Arg(z2).
D0ADCZCTEHARF9D2.
F9D2E8H0DRBXGIEMFKARASAVD9ENDMARASAREMFK; H0DRBX
GIEMFKARFN?AVD9ENDMARFN?ARG2.
26/89
BDD3AX BXGIAREME8DQE0DU
1,EMFKDQGZ
ANCZH0DRBXGI
z1 = r1(cosθ1 + isinθ1),z2 = r2(cosθ2 + isinθ2).
BSD3 z1z2 = r1r2(cosθ1 + isinθ1)(cosθ2 + isinθ2)
= r1r2[cos(θ1 + θ2) + isin(θ1 + θ2)].
D9F0 |z1z2| = |z1||z2|
Arg(z1z2) = Arg(z1) + Arg(z2).
D0ADCZCTEHARF9D2.
F9D2E8H0DRBXGIEMFKARASAVD9ENDMARASAREMFK; H0DRBX
GIEMFKARFN?AVD9ENDMARFN?ARG2.
26/89
D0BTG3H2DB,EMFK z1z2 ARE5H2F0D0C4ER z1 ARE5H2CMBP
E8DR?G5 Argz2,FBARDJ (EGG9) AK |z2| ALAQAK.
θ2
θ2
z2
z1
z1z2
O
y
x
θ1
DICG,GHD9 iz,FR?
iz = z ×i = zeipi2,
EKEX iz DNA8D9GQ z CNEEA2CMBP 90?;
GHD9?z,FR?
z = z × (?1) = zeipi,
EKEX?z DNA8D9GQ z CNEEA2CMBP 180o.
27/89
D0BTG3H2DB,EMFK z1z2 ARE5H2F0D0C4ER z1 ARE5H2CMBP
E8DR?G5 Argz2,FBARDJ (EGG9) AK |z2| ALAQAK.
θ2
θ2
z2
z1
z1z2
O
y
x
θ1
DICG,GHD9 iz,FR?
iz = z ×i = zeipi2,
EKEX iz DNA8D9GQ z CNEEA2CMBP 90?;
GHD9?z,FR?
z = z × (?1) = zeipi,
EKEX?z DNA8D9GQ z CNEEA2CMBP 180o.
27/89
D0BTG3H2DB,EMFK z1z2 ARE5H2F0D0C4ER z1 ARE5H2CMBP
E8DR?G5 Argz2,FBARDJ (EGG9) AK |z2| ALAQAK.
θ2
θ2
z2
z1
z1z2
O
y
x
θ1
DICG,GHD9 iz,FR?
iz = z ×i = zeipi2,
EKEX iz DNA8D9GQ z CNEEA2CMBP 90?;
GHD9?z,FR?
z = z × (?1) = zeipi,
EKEX?z DNA8D9GQ z CNEEA2CMBP 180o.
27/89
D0BTG3H2DB,EMFK z1z2 ARE5H2F0D0C4ER z1 ARE5H2CMBP
E8DR?G5 Argz2,FBARDJ (EGG9) AK |z2| ALAQAK.
θ2
θ2
z2
z1
z1z2
O
y
x
θ1
DICG,GHD9 iz,FR?
iz = z ×i = zeipi2,
EKEX iz DNA8D9GQ z CNEEA2CMBP 90?;
GHD9?z,FR?
z = z × (?1) = zeipi,
EKEX?z DNA8D9GQ z CNEEA2CMBP 180o.
27/89
BH,AKB4D2B5AVEQ
Arg(z1z2) = Arg(z1) + Arg(z2).
CSFN?ARGTDUBH,CCAVEQH0G8FTF0CSFFEHGTDRGIA8EKARH0
DRGIBK,AVEQC4ERH0G8DWCDAIAQARDUARASC6F0DNFVAR.
28/89
DI AN z1 =?1,z2 = i,C3 z1z2 =?i,
Argz1 = pi + 2npi,(n = 0,±1,±2,···),
Argz2 = pi2 + 2mpi,(m = 0,±1,±2,···),
Arg(z1z2) =?pi2 + 2kpi,(k = 0,±1,±2,···),
E3CKAK Arg(z1z2) = Arg(z1) + Arg(z2).
AQ 3pi
2 + 2(m + n)pi =?
pi
2 + 2kpi.
DREMH2EQEKG9,BJC4DPE0C1 k = m + n + 1.
E8E0EH,GHD9AVEQDQBQBRFBE8GH m,n,BFAVEQD1BQD3DW
FCAIE8DR k ARDUEMAQ k = m + n + 1;
B8E8E0EH,GHD9AVEQD1BQBRFBE8DR k,BFAVEQDQBQD3DW
FCAIE8GH m,n ARDUEMAQ m + n + 1 = k.
29/89
DI AN z1 =?1,z2 = i,C3 z1z2 =?i,
Argz1 = pi + 2npi,(n = 0,±1,±2,···),
Argz2 = pi2 + 2mpi,(m = 0,±1,±2,···),
Arg(z1z2) =?pi2 + 2kpi,(k = 0,±1,±2,···),
E3CKAK Arg(z1z2) = Arg(z1) + Arg(z2).
AQ 3pi
2 + 2(m + n)pi =?
pi
2 + 2kpi.
DREMH2EQEKG9,BJC4DPE0C1 k = m + n + 1.
E8E0EH,GHD9AVEQDQBQBRFBE8GH m,n,BFAVEQD1BQD3DW
FCAIE8DR k ARDUEMAQ k = m + n + 1;
B8E8E0EH,GHD9AVEQD1BQBRFBE8DR k,BFAVEQDQBQD3DW
FCAIE8GH m,n ARDUEMAQ m + n + 1 = k.
29/89
DI AN z1 =?1,z2 = i,C3 z1z2 =?i,
Argz1 = pi + 2npi,(n = 0,±1,±2,···),
Argz2 = pi2 + 2mpi,(m = 0,±1,±2,···),
Arg(z1z2) =?pi2 + 2kpi,(k = 0,±1,±2,···),
E3CKAK Arg(z1z2) = Arg(z1) + Arg(z2).
AQ 3pi
2 + 2(m + n)pi =?
pi
2 + 2kpi.
DREMH2EQEKG9,BJC4DPE0C1 k = m + n + 1.
E8E0EH,GHD9AVEQDQBQBRFBE8GH m,n,BFAVEQD1BQD3DW
FCAIE8DR k ARDUEMAQ k = m + n + 1;
B8E8E0EH,GHD9AVEQD1BQBRFBE8DR k,BFAVEQDQBQD3DW
FCAIE8GH m,n ARDUEMAQ m + n + 1 = k.
29/89
DI AN z1 =?1,z2 = i,C3 z1z2 =?i,
Argz1 = pi + 2npi,(n = 0,±1,±2,···),
Argz2 = pi2 + 2mpi,(m = 0,±1,±2,···),
Arg(z1z2) =?pi2 + 2kpi,(k = 0,±1,±2,···),
E3CKAK Arg(z1z2) = Arg(z1) + Arg(z2).
AQ 3pi
2 + 2(m + n)pi =?
pi
2 + 2kpi.
DREMH2EQEKG9,BJC4DPE0C1 k = m + n + 1.
E8E0EH,GHD9AVEQDQBQBRFBE8GH m,n,BFAVEQD1BQD3DW
FCAIE8DR k ARDUEMAQ k = m + n + 1;
B8E8E0EH,GHD9AVEQD1BQBRFBE8DR k,BFAVEQDQBQD3DW
FCAIE8GH m,n ARDUEMAQ m + n + 1 = k.
29/89
DI AN z1 =?1,z2 = i,C3 z1z2 =?i,
Argz1 = pi + 2npi,(n = 0,±1,±2,···),
Argz2 = pi2 + 2mpi,(m = 0,±1,±2,···),
Arg(z1z2) =?pi2 + 2kpi,(k = 0,±1,±2,···),
E3CKAK Arg(z1z2) = Arg(z1) + Arg(z2).
AQ 3pi
2 + 2(m + n)pi =?
pi
2 + 2kpi.
DREMH2EQEKG9,BJC4DPE0C1 k = m + n + 1.
E8E0EH,GHD9AVEQDQBQBRFBE8GH m,n,BFAVEQD1BQD3DW
FCAIE8DR k ARDUEMAQ k = m + n + 1;
B8E8E0EH,GHD9AVEQD1BQBRFBE8DR k,BFAVEQDQBQD3DW
FCAIE8GH m,n ARDUEMAQ m + n + 1 = k.
29/89
DI AN z1 =?1,z2 = i,C3 z1z2 =?i,
Argz1 = pi + 2npi,(n = 0,±1,±2,···),
Argz2 = pi2 + 2mpi,(m = 0,±1,±2,···),
Arg(z1z2) =?pi2 + 2kpi,(k = 0,±1,±2,···),
E3CKAK Arg(z1z2) = Arg(z1) + Arg(z2).
AQ 3pi
2 + 2(m + n)pi =?
pi
2 + 2kpi.
DREMH2EQEKG9,BJC4DPE0C1 k = m + n + 1.
E8E0EH,GHD9AVEQDQBQBRFBE8GH m,n,BFAVEQD1BQD3DW
FCAIE8DR k ARDUEMAQ k = m + n + 1;
B8E8E0EH,GHD9AVEQD1BQBRFBE8DR k,BFAVEQDQBQD3DW
FCAIE8GH m,n ARDUEMAQ m + n + 1 = k.
29/89
DI AN z1 =?1,z2 = i,C3 z1z2 =?i,
Argz1 = pi + 2npi,(n = 0,±1,±2,···),
Argz2 = pi2 + 2mpi,(m = 0,±1,±2,···),
Arg(z1z2) =?pi2 + 2kpi,(k = 0,±1,±2,···),
E3CKAK Arg(z1z2) = Arg(z1) + Arg(z2).
AQ 3pi
2 + 2(m + n)pi =?
pi
2 + 2kpi.
DREMH2EQEKG9,BJC4DPE0C1 k = m + n + 1.
E8E0EH,GHD9AVEQDQBQBRFBE8GH m,n,BFAVEQD1BQD3DW
FCAIE8DR k ARDUEMAQ k = m + n + 1;
B8E8E0EH,GHD9AVEQD1BQBRFBE8DR k,BFAVEQDQBQD3DW
FCAIE8GH m,n ARDUEMAQ m + n + 1 = k.
29/89
DI AN z1 =?1,z2 = i,C3 z1z2 =?i,
Argz1 = pi + 2npi,(n = 0,±1,±2,···),
Argz2 = pi2 + 2mpi,(m = 0,±1,±2,···),
Arg(z1z2) =?pi2 + 2kpi,(k = 0,±1,±2,···),
E3CKAK Arg(z1z2) = Arg(z1) + Arg(z2).
AQ 3pi
2 + 2(m + n)pi =?
pi
2 + 2kpi.
DREMH2EQEKG9,BJC4DPE0C1 k = m + n + 1.
E8E0EH,GHD9AVEQDQBQBRFBE8GH m,n,BFAVEQD1BQD3DW
FCAIE8DR k ARDUEMAQ k = m + n + 1;
B8E8E0EH,GHD9AVEQD1BQBRFBE8DR k,BFAVEQDQBQD3DW
FCAIE8GH m,n ARDUEMAQ m + n + 1 = k.
29/89
DI AN z1 =?1,z2 = i,C3 z1z2 =?i,
Argz1 = pi + 2npi,(n = 0,±1,±2,···),
Argz2 = pi2 + 2mpi,(m = 0,±1,±2,···),
Arg(z1z2) =?pi2 + 2kpi,(k = 0,±1,±2,···),
E3CKAK Arg(z1z2) = Arg(z1) + Arg(z2).
AQ 3pi
2 + 2(m + n)pi =?
pi
2 + 2kpi.
DREMH2EQEKG9,BJC4DPE0C1 k = m + n + 1.
E8E0EH,GHD9AVEQDQBQBRFBE8GH m,n,BFAVEQD1BQD3DW
FCAIE8DR k ARDUEMAQ k = m + n + 1;
B8E8E0EH,GHD9AVEQD1BQBRFBE8DR k,BFAVEQDQBQD3DW
FCAIE8GH m,n ARDUEMAQ m + n + 1 = k.
29/89
BXGIEMFKARDXGIC4ERBBEQ,CGC2CMDXGIBBEQC4ERBXGI
z1 = r1eiθ1,z2 = r2eiθ2,
BSD3F9D2E8DWEXC4ER?
z1z2 = r1r2ei(θ1+θ2).
CSCSFHG2DWDD,CGC2
zk = rkeiθk = rk(cosθk + isinθk),(k = 1,2,···,n).
BSD3
z1z2 ···zn = r1r2···rn[cos(θ1 + θ2 + ···+ θn)
+isin(θ1 + θ2 +···+ θn))]
= r1r2···rnei(θ1+θ2+···+θn).
30/89
BXGIEMFKARDXGIC4ERBBEQ,CGC2CMDXGIBBEQC4ERBXGI
z1 = r1eiθ1,z2 = r2eiθ2,
BSD3F9D2E8DWEXC4ER?
z1z2 = r1r2ei(θ1+θ2).
CSCSFHG2DWDD,CGC2
zk = rkeiθk = rk(cosθk + isinθk),(k = 1,2,···,n).
BSD3
z1z2 ···zn = r1r2···rn[cos(θ1 + θ2 + ···+ θn)
+isin(θ1 + θ2 +···+ θn))]
= r1r2···rnei(θ1+θ2+···+θn).
30/89
BXGIEMFKARDXGIC4ERBBEQ,CGC2CMDXGIBBEQC4ERBXGI
z1 = r1eiθ1,z2 = r2eiθ2,
BSD3F9D2E8DWEXC4ER?
z1z2 = r1r2ei(θ1+θ2).
CSCSFHG2DWDD,CGC2
zk = rkeiθk = rk(cosθk + isinθk),(k = 1,2,···,n).
BSD3
z1z2 ···zn = r1r2···rn[cos(θ1 + θ2 + ···+ θn)
+isin(θ1 + θ2 +···+ θn))]
= r1r2···rnei(θ1+θ2+···+θn).
30/89
BXGIEMFKARDXGIC4ERBBEQ,CGC2CMDXGIBBEQC4ERBXGI
z1 = r1eiθ1,z2 = r2eiθ2,
BSD3F9D2E8DWEXC4ER?
z1z2 = r1r2ei(θ1+θ2).
CSCSFHG2DWDD,CGC2
zk = rkeiθk = rk(cosθk + isinθk),(k = 1,2,···,n).
BSD3
z1z2 ···zn = r1r2···rn[cos(θ1 + θ2 + ···+ θn)
+isin(θ1 + θ2 +···+ θn))]
= r1r2···rnei(θ1+θ2+···+θn).
30/89
BXGIEMFKARDXGIC4ERBBEQ,CGC2CMDXGIBBEQC4ERBXGI
z1 = r1eiθ1,z2 = r2eiθ2,
BSD3F9D2E8DWEXC4ER?
z1z2 = r1r2ei(θ1+θ2).
CSCSFHG2DWDD,CGC2
zk = rkeiθk = rk(cosθk + isinθk),(k = 1,2,···,n).
BSD3
z1z2 ···zn = r1r2···rn[cos(θ1 + θ2 + ···+ θn)
+isin(θ1 + θ2 +···+ θn))]
= r1r2···rnei(θ1+θ2+···+θn).
30/89
B3D9GZ,A8 z1 negationslash= 0 EE,CZ
z2 = z2z
1
z1.
DUGAF9D2E8,CZ
|z2| = |z2z
1
||z1| DQ Argz2 = Arg(z2z
1
) + Argz1,
D9F0
|z2z
1
| = |z2||z
1|
,Arg(z2z
1
) = Argz2? Argz1,
CSCSAQ:
F9D2AJ H0DRBXGIARGZARASAVD9ENDMARASARGZ; H0DRBX
GIARGZARFN?AVD9AQG0GIDQG0GIARFN?DNAB.
31/89
B3D9GZ,A8 z1 negationslash= 0 EE,CZ
z2 = z2z
1
z1.
DUGAF9D2E8,CZ
|z2| = |z2z
1
||z1| DQ Argz2 = Arg(z2z
1
) + Argz1,
D9F0
|z2z
1
| = |z2||z
1|
,Arg(z2z
1
) = Argz2? Argz1,
CSCSAQ:
F9D2AJ H0DRBXGIARGZARASAVD9ENDMARASARGZ; H0DRBX
GIARGZARFN?AVD9AQG0GIDQG0GIARFN?DNAB.
31/89
B3D9GZ,A8 z1 negationslash= 0 EE,CZ
z2 = z2z
1
z1.
DUGAF9D2E8,CZ
|z2| = |z2z
1
||z1| DQ Argz2 = Arg(z2z
1
) + Argz1,
D9F0
|z2z
1
| = |z2||z
1|
,Arg(z2z
1
) = Argz2? Argz1,
CSCSAQ:
F9D2AJ H0DRBXGIARGZARASAVD9ENDMARASARGZ; H0DRBX
GIARGZARFN?AVD9AQG0GIDQG0GIARFN?DNAB.
31/89
B3D9GZ,A8 z1 negationslash= 0 EE,CZ
z2 = z2z
1
z1.
DUGAF9D2E8,CZ
|z2| = |z2z
1
||z1| DQ Argz2 = Arg(z2z
1
) + Argz1,
D9F0
|z2z
1
| = |z2||z
1|
,Arg(z2z
1
) = Argz2? Argz1,
CSCSAQ:
F9D2AJ H0DRBXGIARGZARASAVD9ENDMARASARGZ; H0DRBX
GIARGZARFN?AVD9AQG0GIDQG0GIARFN?DNAB.
31/89
B3D9GZ,A8 z1 negationslash= 0 EE,CZ
z2 = z2z
1
z1.
DUGAF9D2E8,CZ
|z2| = |z2z
1
||z1| DQ Argz2 = Arg(z2z
1
) + Argz1,
D9F0
|z2z
1
| = |z2||z
1|
,Arg(z2z
1
) = Argz2? Argz1,
CSCSAQ:
F9D2AJ H0DRBXGIARGZARASAVD9ENDMARASARGZ; H0DRBX
GIARGZARFN?AVD9AQG0GIDQG0GIARFN?DNAB.
31/89
BXGIGZARDXGIC4ERBBEQ,CGC2CMDXGIBBEQC4ERBXGI
z1 = r1eiθ1,z2 = r2eiθ2,
C3CZ z
2
z1 =
r2
r1e
i(θ2?θ1),r1 negationslash= 0.
32/89
BXGIGZARDXGIC4ERBBEQ,CGC2CMDXGIBBEQC4ERBXGI
z1 = r1eiθ1,z2 = r2eiθ2,
C3CZ z
2
z1 =
r2
r1e
i(θ2?θ1),r1 negationslash= 0.
32/89
BXGIGZARDXGIC4ERBBEQ,CGC2CMDXGIBBEQC4ERBXGI
z1 = r1eiθ1,z2 = r2eiθ2,
C3CZ z
2
z1 =
r2
r1e
i(θ2?θ1),r1 negationslash= 0.
32/89
DI 1 EUDJAKD2?BBARH0DRF6BM? z1 = 1 DQ z2 = 2 + i,
EMENARB8E8DRF6BM.
B5CGGB,GQC4ER z2?z1 ARE5H2BJ z1 CMBP
pi
3(F9?
pi
3) FU
AQAKB8E8DRE5H2,ENAREQBMFUF0EKEMARF6BM.
O
y
x z
1 =1
z2=2+i
z3
z3’
pi/3
pi/3
DUGABXGIAREMDF,CZ
z3?z1 = epi3i(z2?z1) = (12 +
√3
2 i)(1 + i)
= (12?
√3
2 ) + (
1
2 +
√3
2 )i.
EKEX z3 =
3?√3
2 +
1 +√3
2 i.33/89
DI 1 EUDJAKD2?BBARH0DRF6BM? z1 = 1 DQ z2 = 2 + i,
EMENARB8E8DRF6BM.
B5CGGB,GQC4ER z2?z1 ARE5H2BJ z1 CMBP
pi
3(F9?
pi
3) FU
AQAKB8E8DRE5H2,ENAREQBMFUF0EKEMARF6BM.
O
y
x z
1 =1
z2=2+i
z3
z3’
pi/3
pi/3
DUGABXGIAREMDF,CZ
z3?z1 = epi3i(z2?z1) = (12 +
√3
2 i)(1 + i)
= (12?
√3
2 ) + (
1
2 +
√3
2 )i.
EKEX z3 =
3?√3
2 +
1 +√3
2 i.33/89
DI 1 EUDJAKD2?BBARH0DRF6BM? z1 = 1 DQ z2 = 2 + i,
EMENARB8E8DRF6BM.
B5CGGB,GQC4ER z2?z1 ARE5H2BJ z1 CMBP
pi
3(F9?
pi
3) FU
AQAKB8E8DRE5H2,ENAREQBMFUF0EKEMARF6BM.
O
y
x z
1 =1
z2=2+i
z3
z3’
pi/3
pi/3
DUGABXGIAREMDF,CZ
z3?z1 = epi3i(z2?z1) = (12 +
√3
2 i)(1 + i)
= (12?
√3
2 ) + (
1
2 +
√3
2 )i.
EKEX z3 =
3?√3
2 +
1 +√3
2 i.33/89
DI 1 EUDJAKD2?BBARH0DRF6BM? z1 = 1 DQ z2 = 2 + i,
EMENARB8E8DRF6BM.
B5CGGB,GQC4ER z2?z1 ARE5H2BJ z1 CMBP
pi
3(F9?
pi
3) FU
AQAKB8E8DRE5H2,ENAREQBMFUF0EKEMARF6BM.
O
y
x z
1 =1
z2=2+i
z3
z3’
pi/3
pi/3
DUGABXGIAREMDF,CZ
z3?z1 = epi3i(z2?z1) = (12 +
√3
2 i)(1 + i)
= (12?
√3
2 ) + (
1
2 +
√3
2 )i.
EKEX z3 =
3?√3
2 +
1 +√3
2 i.33/89
DI 1 EUDJAKD2?BBARH0DRF6BM? z1 = 1 DQ z2 = 2 + i,
EMENARB8E8DRF6BM.
B5CGGB,GQC4ER z2?z1 ARE5H2BJ z1 CMBP
pi
3(F9?
pi
3) FU
AQAKB8E8DRE5H2,ENAREQBMFUF0EKEMARF6BM.
O
y
x z
1 =1
z2=2+i
z3
z3’
pi/3
pi/3
DUGABXGIAREMDF,CZ
z3?z1 = epi3i(z2?z1) = (12 +
√3
2 i)(1 + i)
= (12?
√3
2 ) + (
1
2 +
√3
2 )i.
EKEX z3 =
3?√3
2 +
1 +√3
2 i.33/89
CPD5DWAQ
z′3 = 3 +
√3
2 +
1?√3
2 i.
34/89
DI 2 ANBXGI z1,z2,z3 GHBTD9AVBQD2?BBARD2DRF6BM,FD
DD
z21 + z22 + z23?z1z2?z2z3?z1z3 = 0
y
O x
C
A B
z3
z1
z2
35/89
y
O x
C
A B
z3
z1
z2
DD CGGB,E5H2
→AB
CMBP
pi
3 AQAK
→AC,
E5H2
→BC
CMBP
pi
3
AQAK
→BA.
EKEX z3?z1 = e
pi
3i(z2?z1)
z1?z3 = epi3i(z3?z2)
CSCSAQ
(z3?z1)(z3?z2) = (z2?z1)(z1?z2),
DEDF,AQ
z21 + z22 + z23?z1z2?z2z3?z1z3 = 0
36/89
y
O x
C
A B
z3
z1
z2
DD CGGB,E5H2
→AB
CMBP
pi
3 AQAK
→AC,
E5H2
→BC
CMBP
pi
3
AQAK
→BA.
EKEX z3?z1 = e
pi
3i(z2?z1)
z1?z3 = epi3i(z3?z2)
CSCSAQ
(z3?z1)(z3?z2) = (z2?z1)(z1?z2),
DEDF,AQ
z21 + z22 + z23?z1z2?z2z3?z1z3 = 0
36/89
y
O x
C
A B
z3
z1
z2
DD CGGB,E5H2
→AB
CMBP
pi
3 AQAK
→AC,
E5H2
→BC
CMBP
pi
3
AQAK
→BA.
EKEX z3?z1 = e
pi
3i(z2?z1)
z1?z3 = epi3i(z3?z2)
CSCSAQ
(z3?z1)(z3?z2) = (z2?z1)(z1?z2),
DEDF,AQ
z21 + z22 + z23?z1z2?z2z3?z1z3 = 0
36/89
y
O x
C
A B
z3
z1
z2
DD CGGB,E5H2
→AB
CMBP
pi
3 AQAK
→AC,
E5H2
→BC
CMBP
pi
3
AQAK
→BA.
EKEX z3?z1 = e
pi
3i(z2?z1)
z1?z3 = epi3i(z3?z2)
CSCSAQ
(z3?z1)(z3?z2) = (z2?z1)(z1?z2),
DEDF,AQ
z21 + z22 + z23?z1z2?z2z3?z1z3 = 0
36/89
2,E8DQDU
n DRDNFVBXGI z AREMFKEH? z ARn CVE8,C8DU zn,BP
zn = z ·z···z.
DUGABXGIEMDFARCZB3B4C2,F3DMCZ
zn = rn(cosnθ + isinnθ).
CGC2F9FE z?n = 1zn,BSD3A8 n?C3AIGIEEH2EQ?F0EK
G9AR.
BTC7,A8 z ARAS r = 1 EE,CZ
(cosθ + isinθ)n = cosnθ + isinnθ.
CCEQEH?E5AZBIGYEQ.
37/89
2,E8DQDU
n DRDNFVBXGI z AREMFKEH? z ARn CVE8,C8DU zn,BP
zn = z ·z···z.
DUGABXGIEMDFARCZB3B4C2,F3DMCZ
zn = rn(cosnθ + isinnθ).
CGC2F9FE z?n = 1zn,BSD3A8 n?C3AIGIEEH2EQ?F0EK
G9AR.
BTC7,A8 z ARAS r = 1 EE,CZ
(cosθ + isinθ)n = cosnθ + isinnθ.
CCEQEH?E5AZBIGYEQ.
37/89
2,E8DQDU
n DRDNFVBXGI z AREMFKEH? z ARn CVE8,C8DU zn,BP
zn = z ·z···z.
DUGABXGIEMDFARCZB3B4C2,F3DMCZ
zn = rn(cosnθ + isinnθ).
CGC2F9FE z?n = 1zn,BSD3A8 n?C3AIGIEEH2EQ?F0EK
G9AR.
BTC7,A8 z ARAS r = 1 EE,CZ
(cosθ + isinθ)n = cosnθ + isinnθ.
CCEQEH?E5AZBIGYEQ.
37/89
2,E8DQDU
n DRDNFVBXGI z AREMFKEH? z ARn CVE8,C8DU zn,BP
zn = z ·z···z.
DUGABXGIEMDFARCZB3B4C2,F3DMCZ
zn = rn(cosnθ + isinnθ).
CGC2F9FE z?n = 1zn,BSD3A8 n?C3AIGIEEH2EQ?F0EK
G9AR.
BTC7,A8 z ARAS r = 1 EE,CZ
(cosθ + isinθ)n = cosnθ + isinnθ.
CCEQEH?E5AZBIGYEQ.
37/89
DI 1 C7DY
parenleftBigg
1 +√3i
1?√3i
parenrightBigg10
ARDU.
B5 FR?
1 +√3i = 2
parenleftBig
cospi3 + isinpi3
parenrightBig
1?√3i = 2
bracketleftBig
cos(?pi3) + isin(?pi3)
bracketrightBig
EKEX
parenleftBigg
1 +√3i
1?√3i
parenrightBigg10
=
2
parenleftBig
cospi3 + isinpi3
parenrightBig
2
bracketleftBig
cos(?pi3) + isin(?pi3)
bracketrightBig
10
=
parenleftbigg
cos2pi3 + isin2pi3
parenrightbigg10
= cos20pi3 + isin20pi3 =?12 +
√3
2 i.
38/89
DI 1 C7DY
parenleftBigg
1 +√3i
1?√3i
parenrightBigg10
ARDU.
B5 FR?
1 +√3i = 2
parenleftBig
cospi3 + isinpi3
parenrightBig
1?√3i = 2
bracketleftBig
cos(?pi3) + isin(?pi3)
bracketrightBig
EKEX
parenleftBigg
1 +√3i
1?√3i
parenrightBigg10
=
2
parenleftBig
cospi3 + isinpi3
parenrightBig
2
bracketleftBig
cos(?pi3) + isin(?pi3)
bracketrightBig
10
=
parenleftbigg
cos2pi3 + isin2pi3
parenrightbigg10
= cos20pi3 + isin20pi3 =?12 +
√3
2 i.
38/89
DI 1 C7DY
parenleftBigg
1 +√3i
1?√3i
parenrightBigg10
ARDU.
B5 FR?
1 +√3i = 2
parenleftBig
cospi3 + isinpi3
parenrightBig
1?√3i = 2
bracketleftBig
cos(?pi3) + isin(?pi3)
bracketrightBig
EKEX
parenleftBigg
1 +√3i
1?√3i
parenrightBigg10
=
2
parenleftBig
cospi3 + isinpi3
parenrightBig
2
bracketleftBig
cos(?pi3) + isin(?pi3)
bracketrightBig
10
=
parenleftbigg
cos2pi3 + isin2pi3
parenrightbigg10
= cos20pi3 + isin20pi3 =?12 +
√3
2 i.
38/89
DI 1 C7DY
parenleftBigg
1 +√3i
1?√3i
parenrightBigg10
ARDU.
B5 FR?
1 +√3i = 2
parenleftBig
cospi3 + isinpi3
parenrightBig
1?√3i = 2
bracketleftBig
cos(?pi3) + isin(?pi3)
bracketrightBig
EKEX
parenleftBigg
1 +√3i
1?√3i
parenrightBigg10
=
2
parenleftBig
cospi3 + isinpi3
parenrightBig
2
bracketleftBig
cos(?pi3) + isin(?pi3)
bracketrightBig
10
=
parenleftbigg
cos2pi3 + isin2pi3
parenrightbigg10
= cos20pi3 + isin20pi3 =?12 +
√3
2 i.
38/89
DI 1 C7DY
parenleftBigg
1 +√3i
1?√3i
parenrightBigg10
ARDU.
B5 FR?
1 +√3i = 2
parenleftBig
cospi3 + isinpi3
parenrightBig
1?√3i = 2
bracketleftBig
cos(?pi3) + isin(?pi3)
bracketrightBig
EKEX
parenleftBigg
1 +√3i
1?√3i
parenrightBigg10
=
2
parenleftBig
cospi3 + isinpi3
parenrightBig
2
bracketleftBig
cos(?pi3) + isin(?pi3)
bracketrightBig
10
=
parenleftbigg
cos2pi3 + isin2pi3
parenrightbigg10
= cos20pi3 + isin20pi3 =?12 +
√3
2 i.
38/89
BXGIARDU:
EVC3,DJFGEME0EN wn = z ARDUw,G1EL,z?EUDJBXGI.
B5 B9
z = r(cosθ + isinθ),w = ρ(cos? + isin?),
DUGAE5AZBIGYEQCZ
ρn(cosn? + isinn?) = r(cosθ + isinθ),
D9F0
ρn = r,cosn? = cosθ,sinn? = sinθ.
A0H0EQEKG9ARFBEPBJDRCMDRF0
n? = θ + 2kpi,(k = 0,±1,±2,···).
C3CZ ρ = r
1
n,? =
θ + 2kpi
n,
G1EL,r1/n F0DYGADU,EKEX
w = n√z = r1/n(cosθ + 2kpin + isinθ + 2kpin ).
A8 k = 0,1,2,···,n?1 EE,AQAK n DRDNFLARDU.
39/89
BXGIARDU:
EVC3,DJFGEME0EN wn = z ARDUw,G1EL,z?EUDJBXGI.
B5 B9
z = r(cosθ + isinθ),w = ρ(cos? + isin?),
DUGAE5AZBIGYEQCZ
ρn(cosn? + isinn?) = r(cosθ + isinθ),
D9F0
ρn = r,cosn? = cosθ,sinn? = sinθ.
A0H0EQEKG9ARFBEPBJDRCMDRF0
n? = θ + 2kpi,(k = 0,±1,±2,···).
C3CZ ρ = r
1
n,? =
θ + 2kpi
n,
G1EL,r1/n F0DYGADU,EKEX
w = n√z = r1/n(cosθ + 2kpin + isinθ + 2kpin ).
A8 k = 0,1,2,···,n?1 EE,AQAK n DRDNFLARDU.
39/89
BXGIARDU:
EVC3,DJFGEME0EN wn = z ARDUw,G1EL,z?EUDJBXGI.
B5 B9
z = r(cosθ + isinθ),w = ρ(cos? + isin?),
DUGAE5AZBIGYEQCZ
ρn(cosn? + isinn?) = r(cosθ + isinθ),
D9F0
ρn = r,cosn? = cosθ,sinn? = sinθ.
A0H0EQEKG9ARFBEPBJDRCMDRF0
n? = θ + 2kpi,(k = 0,±1,±2,···).
C3CZ ρ = r
1
n,? =
θ + 2kpi
n,
G1EL,r1/n F0DYGADU,EKEX
w = n√z = r1/n(cosθ + 2kpin + isinθ + 2kpin ).
A8 k = 0,1,2,···,n?1 EE,AQAK n DRDNFLARDU.
39/89
BXGIARDU:
EVC3,DJFGEME0EN wn = z ARDUw,G1EL,z?EUDJBXGI.
B5 B9
z = r(cosθ + isinθ),w = ρ(cos? + isin?),
DUGAE5AZBIGYEQCZ
ρn(cosn? + isinn?) = r(cosθ + isinθ),
D9F0
ρn = r,cosn? = cosθ,sinn? = sinθ.
A0H0EQEKG9ARFBEPBJDRCMDRF0
n? = θ + 2kpi,(k = 0,±1,±2,···).
C3CZ ρ = r
1
n,? =
θ + 2kpi
n,
G1EL,r1/n F0DYGADU,EKEX
w = n√z = r1/n(cosθ + 2kpin + isinθ + 2kpin ).
A8 k = 0,1,2,···,n?1 EE,AQAK n DRDNFLARDU.
39/89
BXGIARDU:
EVC3,DJFGEME0EN wn = z ARDUw,G1EL,z?EUDJBXGI.
B5 B9
z = r(cosθ + isinθ),w = ρ(cos? + isin?),
DUGAE5AZBIGYEQCZ
ρn(cosn? + isinn?) = r(cosθ + isinθ),
D9F0
ρn = r,cosn? = cosθ,sinn? = sinθ.
A0H0EQEKG9ARFBEPBJDRCMDRF0
n? = θ + 2kpi,(k = 0,±1,±2,···).
C3CZ ρ = r
1
n,? =
θ + 2kpi
n,
G1EL,r1/n F0DYGADU,EKEX
w = n√z = r1/n(cosθ + 2kpin + isinθ + 2kpin ).
A8 k = 0,1,2,···,n?1 EE,AQAK n DRDNFLARDU.
39/89
BXGIARDU:
EVC3,DJFGEME0EN wn = z ARDUw,G1EL,z?EUDJBXGI.
B5 B9
z = r(cosθ + isinθ),w = ρ(cos? + isin?),
DUGAE5AZBIGYEQCZ
ρn(cosn? + isinn?) = r(cosθ + isinθ),
D9F0
ρn = r,cosn? = cosθ,sinn? = sinθ.
A0H0EQEKG9ARFBEPBJDRCMDRF0
n? = θ + 2kpi,(k = 0,±1,±2,···).
C3CZ ρ = r
1
n,? =
θ + 2kpi
n,
G1EL,r1/n F0DYGADU,EKEX
w = n√z = r1/n(cosθ + 2kpin + isinθ + 2kpin ).
A8 k = 0,1,2,···,n?1 EE,AQAK n DRDNFLARDU.
39/89
BXGIARDU:
EVC3,DJFGEME0EN wn = z ARDUw,G1EL,z?EUDJBXGI.
B5 B9
z = r(cosθ + isinθ),w = ρ(cos? + isin?),
DUGAE5AZBIGYEQCZ
ρn(cosn? + isinn?) = r(cosθ + isinθ),
D9F0
ρn = r,cosn? = cosθ,sinn? = sinθ.
A0H0EQEKG9ARFBEPBJDRCMDRF0
n? = θ + 2kpi,(k = 0,±1,±2,···).
C3CZ ρ = r
1
n,? =
θ + 2kpi
n,
G1EL,r1/n F0DYGADU,EKEX
w = n√z = r1/n(cosθ + 2kpin + isinθ + 2kpin ).
A8 k = 0,1,2,···,n?1 EE,AQAK n DRDNFLARDU.
39/89
BXGIARDU:
EVC3,DJFGEME0EN wn = z ARDUw,G1EL,z?EUDJBXGI.
B5 B9
z = r(cosθ + isinθ),w = ρ(cos? + isin?),
DUGAE5AZBIGYEQCZ
ρn(cosn? + isinn?) = r(cosθ + isinθ),
D9F0
ρn = r,cosn? = cosθ,sinn? = sinθ.
A0H0EQEKG9ARFBEPBJDRCMDRF0
n? = θ + 2kpi,(k = 0,±1,±2,···).
C3CZ ρ = r
1
n,? =
θ + 2kpi
n,
G1EL,r1/n F0DYGADU,EKEX
w = n√z = r1/n(cosθ + 2kpin + isinθ + 2kpin ).
A8 k = 0,1,2,···,n?1 EE,AQAK n DRDNFLARDU.
39/89
BXGIARDU:
EVC3,DJFGEME0EN wn = z ARDUw,G1EL,z?EUDJBXGI.
B5 B9
z = r(cosθ + isinθ),w = ρ(cos? + isin?),
DUGAE5AZBIGYEQCZ
ρn(cosn? + isinn?) = r(cosθ + isinθ),
D9F0
ρn = r,cosn? = cosθ,sinn? = sinθ.
A0H0EQEKG9ARFBEPBJDRCMDRF0
n? = θ + 2kpi,(k = 0,±1,±2,···).
C3CZ ρ = r
1
n,? =
θ + 2kpi
n,
G1EL,r1/n F0DYGADU,EKEX
w = n√z = r1/n(cosθ + 2kpin + isinθ + 2kpin ).
A8 k = 0,1,2,···,n?1 EE,AQAK n DRDNFLARDU.
39/89
D0BTG3H2DB,n√z AR n DRDUFUF0EXA6BM?ELB1,r1/n?DW
F7ARABARCBAQAK n BQBBAR n DRF6BM.
40/89
DI 2 EM
4√1 + i.
B5FR?
1 + i = √2(cospi4 + isinpi4),
EKEX
4√1 + i = 8√2
cos
pi
4 + 2kpi
4 + isin
pi
4 + 2kpi
4
,k = 0,1,2,3.
BP w
0 = 8
√2parenleftBigcos pi
16 + isin
pi
16
parenrightBig;
w1 = 8√2
parenleftbigg
cos 916pi + isin 916pi
parenrightbigg;
w2 = 8√2
parenleftbigg
cos1716pi + isin1716pi
parenrightbigg;
w3 = 8√2
parenleftbigg
cos2516pi + isin2516pi
parenrightbigg
.
41/89
DI 2 EM
4√1 + i.
B5FR?
1 + i = √2(cospi4 + isinpi4),
EKEX
4√1 + i = 8√2
cos
pi
4 + 2kpi
4 + isin
pi
4 + 2kpi
4
,k = 0,1,2,3.
BP w
0 = 8
√2parenleftBigcos pi
16 + isin
pi
16
parenrightBig;
w1 = 8√2
parenleftbigg
cos 916pi + isin 916pi
parenrightbigg;
w2 = 8√2
parenleftbigg
cos1716pi + isin1716pi
parenrightbigg;
w3 = 8√2
parenleftbigg
cos2516pi + isin2516pi
parenrightbigg
.
41/89
DI 2 EM
4√1 + i.
B5FR?
1 + i = √2(cospi4 + isinpi4),
EKEX
4√1 + i = 8√2
cos
pi
4 + 2kpi
4 + isin
pi
4 + 2kpi
4
,k = 0,1,2,3.
BP w
0 = 8
√2parenleftBigcos pi
16 + isin
pi
16
parenrightBig;
w1 = 8√2
parenleftbigg
cos 916pi + isin 916pi
parenrightbigg;
w2 = 8√2
parenleftbigg
cos1716pi + isin1716pi
parenrightbigg;
w3 = 8√2
parenleftbigg
cos2516pi + isin2516pi
parenrightbigg
.
41/89
DI 2 EM
4√1 + i.
B5FR?
1 + i = √2(cospi4 + isinpi4),
EKEX
4√1 + i = 8√2
cos
pi
4 + 2kpi
4 + isin
pi
4 + 2kpi
4
,k = 0,1,2,3.
BP w
0 = 8
√2parenleftBigcos pi
16 + isin
pi
16
parenrightBig;
w1 = 8√2
parenleftbigg
cos 916pi + isin 916pi
parenrightbigg;
w2 = 8√2
parenleftbigg
cos1716pi + isin1716pi
parenrightbigg;
w3 = 8√2
parenleftbigg
cos2516pi + isin2516pi
parenrightbigg
.
41/89
DI 2 EM
4√1 + i.
B5FR?
1 + i = √2(cospi4 + isinpi4),
EKEX
4√1 + i = 8√2
cos
pi
4 + 2kpi
4 + isin
pi
4 + 2kpi
4
,k = 0,1,2,3.
BP w
0 = 8
√2parenleftBigcos pi
16 + isin
pi
16
parenrightBig;
w1 = 8√2
parenleftbigg
cos 916pi + isin 916pi
parenrightbigg;
w2 = 8√2
parenleftbigg
cos1716pi + isin1716pi
parenrightbigg;
w3 = 8√2
parenleftbigg
cos2516pi + isin2516pi
parenrightbigg
.
41/89
DI 2 EM
4√1 + i.
B5FR?
1 + i = √2(cospi4 + isinpi4),
EKEX
4√1 + i = 8√2
cos
pi
4 + 2kpi
4 + isin
pi
4 + 2kpi
4
,k = 0,1,2,3.
BP w
0 = 8
√2parenleftBigcos pi
16 + isin
pi
16
parenrightBig;
w1 = 8√2
parenleftbigg
cos 916pi + isin 916pi
parenrightbigg;
w2 = 8√2
parenleftbigg
cos1716pi + isin1716pi
parenrightbigg;
w3 = 8√2
parenleftbigg
cos2516pi + isin2516pi
parenrightbigg
.
41/89
O
y
x
1+i
w0
w1
w2
w3
42/89
D2G4G6DI
1,BXGIARE3GIB0DY
BXGIARE3GIB0DYEDB5G2AXABAXFKAXGZAXEME8G2E0DU,?
AAEMB0DYDXCKDFB0G2C7B4,BTCCFCC2G4F3ARC4EREQBHC4ER
BXGI,CZEEDODWEXCSBTG3FBFEBHBRG0BXGIARE3GIB0DY.
43/89
D2G4G6DI
1,BXGIARE3GIB0DY
BXGIARE3GIB0DYEDB5G2AXABAXFKAXGZAXEME8G2E0DU,?
AAEMB0DYDXCKDFB0G2C7B4,BTCCFCC2G4F3ARC4EREQBHC4ER
BXGI,CZEEDODWEXCSBTG3FBFEBHBRG0BXGIARE3GIB0DY.
43/89
D2G4G6DI
1,BXGIARE3GIB0DY
BXGIARE3GIB0DYEDB5G2AXABAXFKAXGZAXEME8G2E0DU,?
AAEMB0DYDXCKDFB0G2C7B4,BTCCFCC2G4F3ARC4EREQBHC4ER
BXGI,CZEEDODWEXCSBTG3FBFEBHBRG0BXGIARE3GIB0DY.
43/89
DI 1 FRC4C7DY (5?i)4(1 + i),DDEY:
pi
4 = 4arctan
1
5? arctan
1
239
B5 FR?
(5?i)4(1 + i) = (24? 10i)2(1 + i)
= (476? 480i)(1 + i) = 4(239?i).
D5FR?
arg(5?i)4 =?4arctan15,arg(1 + i) = pi4,
arg[4(239?i)] = arctan 1239,
EKEX arctan 1239 =?4arctan15 + pi4.
44/89
DI 1 FRC4C7DY (5?i)4(1 + i),DDEY:
pi
4 = 4arctan
1
5? arctan
1
239
B5 FR?
(5?i)4(1 + i) = (24? 10i)2(1 + i)
= (476? 480i)(1 + i) = 4(239?i).
D5FR?
arg(5?i)4 =?4arctan15,arg(1 + i) = pi4,
arg[4(239?i)] = arctan 1239,
EKEX arctan 1239 =?4arctan15 + pi4.
44/89
DI 1 FRC4C7DY (5?i)4(1 + i),DDEY:
pi
4 = 4arctan
1
5? arctan
1
239
B5 FR?
(5?i)4(1 + i) = (24? 10i)2(1 + i)
= (476? 480i)(1 + i) = 4(239?i).
D5FR?
arg(5?i)4 =?4arctan15,arg(1 + i) = pi4,
arg[4(239?i)] = arctan 1239,
EKEX arctan 1239 =?4arctan15 + pi4.
44/89
DI 1 FRC4C7DY (5?i)4(1 + i),DDEY:
pi
4 = 4arctan
1
5? arctan
1
239
B5 FR?
(5?i)4(1 + i) = (24? 10i)2(1 + i)
= (476? 480i)(1 + i) = 4(239?i).
D5FR?
arg(5?i)4 =?4arctan15,arg(1 + i) = pi4,
arg[4(239?i)] = arctan 1239,
EKEX arctan 1239 =?4arctan15 + pi4.
44/89
DI 2 AN arg(z + 2) = pi3,arg(z?2) = 5pi6,EM z.
B5AN
z + 2 = r1
parenleftBig
cospi3 + isinpi3
parenrightBig
= 12r1 +
√3
2 ir1;
z?2 = r2
parenleftbigg
cos5pi6 + isin5pi6
parenrightbigg
=?
√3
2 r2 +
1
2ir2;
D9F0
z =
parenleftbigg1
2r1?2
parenrightbigg
+
√3
2 ir1 =
parenleftBigg
√3
2 r2 + 2
parenrightBigg
+ 12ir2.
B3AMEIG4DQC2G4,AQ
1
2r1?2 =?
√3
2 r2 + 2,
√3
2 r1 =
1
2ir2,
B5AQ r1 = 2,EKEX
z =?1 +√3i.
45/89
DI 2 AN arg(z + 2) = pi3,arg(z?2) = 5pi6,EM z.
B5AN
z + 2 = r1
parenleftBig
cospi3 + isinpi3
parenrightBig
= 12r1 +
√3
2 ir1;
z?2 = r2
parenleftbigg
cos5pi6 + isin5pi6
parenrightbigg
=?
√3
2 r2 +
1
2ir2;
D9F0
z =
parenleftbigg1
2r1?2
parenrightbigg
+
√3
2 ir1 =
parenleftBigg
√3
2 r2 + 2
parenrightBigg
+ 12ir2.
B3AMEIG4DQC2G4,AQ
1
2r1?2 =?
√3
2 r2 + 2,
√3
2 r1 =
1
2ir2,
B5AQ r1 = 2,EKEX
z =?1 +√3i.
45/89
DI 2 AN arg(z + 2) = pi3,arg(z?2) = 5pi6,EM z.
B5AN
z + 2 = r1
parenleftBig
cospi3 + isinpi3
parenrightBig
= 12r1 +
√3
2 ir1;
z?2 = r2
parenleftbigg
cos5pi6 + isin5pi6
parenrightbigg
=?
√3
2 r2 +
1
2ir2;
D9F0
z =
parenleftbigg1
2r1?2
parenrightbigg
+
√3
2 ir1 =
parenleftBigg
√3
2 r2 + 2
parenrightBigg
+ 12ir2.
B3AMEIG4DQC2G4,AQ
1
2r1?2 =?
√3
2 r2 + 2,
√3
2 r1 =
1
2ir2,
B5AQ r1 = 2,EKEX
z =?1 +√3i.
45/89
DI 2 AN arg(z + 2) = pi3,arg(z?2) = 5pi6,EM z.
B5AN
z + 2 = r1
parenleftBig
cospi3 + isinpi3
parenrightBig
= 12r1 +
√3
2 ir1;
z?2 = r2
parenleftbigg
cos5pi6 + isin5pi6
parenrightbigg
=?
√3
2 r2 +
1
2ir2;
D9F0
z =
parenleftbigg1
2r1?2
parenrightbigg
+
√3
2 ir1 =
parenleftBigg
√3
2 r2 + 2
parenrightBigg
+ 12ir2.
B3AMEIG4DQC2G4,AQ
1
2r1?2 =?
√3
2 r2 + 2,
√3
2 r1 =
1
2ir2,
B5AQ r1 = 2,EKEX
z =?1 +√3i.
45/89
DI 2 AN arg(z + 2) = pi3,arg(z?2) = 5pi6,EM z.
B5AN
z + 2 = r1
parenleftBig
cospi3 + isinpi3
parenrightBig
= 12r1 +
√3
2 ir1;
z?2 = r2
parenleftbigg
cos5pi6 + isin5pi6
parenrightbigg
=?
√3
2 r2 +
1
2ir2;
D9F0
z =
parenleftbigg1
2r1?2
parenrightbigg
+
√3
2 ir1 =
parenleftBigg
√3
2 r2 + 2
parenrightBigg
+ 12ir2.
B3AMEIG4DQC2G4,AQ
1
2r1?2 =?
√3
2 r2 + 2,
√3
2 r1 =
1
2ir2,
B5AQ r1 = 2,EKEX
z =?1 +√3i.
45/89
DI 3 B5E0EN (1 + z)5 = (1?z)5.
B5FR? z = 1 G0F0E0ENARDU,GQH2EQBVBB,AQ
w5 defines
parenleftbigg1 + z
1?z
parenrightbigg5
= 1 = ei0pi.
AD w5 = 1 ARDU?
eiα,G1ELα = 0,25pi,45pi,65pi,85pi,
CSw = 1 + z1?z,AQ
z = 1 + z1?z = e
iα? 1
eiα + 1 =
cosα? 1 + isinα
cosα + 1 + isinα
= itanα2.
46/89
DI 3 B5E0EN (1 + z)5 = (1?z)5.
B5FR? z = 1 G0F0E0ENARDU,GQH2EQBVBB,AQ
w5 defines
parenleftbigg1 + z
1?z
parenrightbigg5
= 1 = ei0pi.
AD w5 = 1 ARDU?
eiα,G1ELα = 0,25pi,45pi,65pi,85pi,
CSw = 1 + z1?z,AQ
z = 1 + z1?z = e
iα? 1
eiα + 1 =
cosα? 1 + isinα
cosα + 1 + isinα
= itanα2.
46/89
DI 3 B5E0EN (1 + z)5 = (1?z)5.
B5FR? z = 1 G0F0E0ENARDU,GQH2EQBVBB,AQ
w5 defines
parenleftbigg1 + z
1?z
parenrightbigg5
= 1 = ei0pi.
AD w5 = 1 ARDU?
eiα,G1ELα = 0,25pi,45pi,65pi,85pi,
CSw = 1 + z1?z,AQ
z = 1 + z1?z = e
iα? 1
eiα + 1 =
cosα? 1 + isinα
cosα + 1 + isinα
= itanα2.
46/89
DI 3 B5E0EN (1 + z)5 = (1?z)5.
B5FR? z = 1 G0F0E0ENARDU,GQH2EQBVBB,AQ
w5 defines
parenleftbigg1 + z
1?z
parenrightbigg5
= 1 = ei0pi.
AD w5 = 1 ARDU?
eiα,G1ELα = 0,25pi,45pi,65pi,85pi,
CSw = 1 + z1?z,AQ
z = 1 + z1?z = e
iα? 1
eiα + 1 =
cosα? 1 + isinα
cosα + 1 + isinα
= itanα2.
46/89
DI 3 B5E0EN (1 + z)5 = (1?z)5.
B5FR? z = 1 G0F0E0ENARDU,GQH2EQBVBB,AQ
w5 defines
parenleftbigg1 + z
1?z
parenrightbigg5
= 1 = ei0pi.
AD w5 = 1 ARDU?
eiα,G1ELα = 0,25pi,45pi,65pi,85pi,
CSw = 1 + z1?z,AQ
z = 1 + z1?z = e
iα? 1
eiα + 1 =
cosα? 1 + isinα
cosα + 1 + isinα
= itanα2.
46/89
DI 3 B5E0EN (1 + z)5 = (1?z)5.
B5FR? z = 1 G0F0E0ENARDU,GQH2EQBVBB,AQ
w5 defines
parenleftbigg1 + z
1?z
parenrightbigg5
= 1 = ei0pi.
AD w5 = 1 ARDU?
eiα,G1ELα = 0,25pi,45pi,65pi,85pi,
CSw = 1 + z1?z,AQ
z = 1 + z1?z = e
iα? 1
eiα + 1 =
cosα? 1 + isinα
cosα + 1 + isinα
= itanα2.
46/89
DI 4 AN |z0| < 1,DDEY,CT |z| < 1,C3
vextendsinglevextendsingle
vextendsinglevextendsingle z?z0
1?z0z
vextendsinglevextendsingle
vextendsinglevextendsingle< 1.
B5BPDRDD
|z?z0| < |1?z0z|.
FR?
|z?z0|2 = |z|2 +|z0|2? 2Re(zz0),
|1?z0z|2 = 1 +|z|2|z0|2? 2Re(zz0),
APDRDD
|z|2 +|z0|2 < 1 +|z|2|z0|2.
FR? |z| < 1,EKEX|z|2(1?|z0|2) < 1?|z0|2,APH2EQEKG9,AQ
DD.
47/89
DI 4 AN |z0| < 1,DDEY,CT |z| < 1,C3
vextendsinglevextendsingle
vextendsinglevextendsingle z?z0
1?z0z
vextendsinglevextendsingle
vextendsinglevextendsingle< 1.
B5BPDRDD
|z?z0| < |1?z0z|.
FR?
|z?z0|2 = |z|2 +|z0|2? 2Re(zz0),
|1?z0z|2 = 1 +|z|2|z0|2? 2Re(zz0),
APDRDD
|z|2 +|z0|2 < 1 +|z|2|z0|2.
FR? |z| < 1,EKEX|z|2(1?|z0|2) < 1?|z0|2,APH2EQEKG9,AQ
DD.
47/89
DI 4 AN |z0| < 1,DDEY,CT |z| < 1,C3
vextendsinglevextendsingle
vextendsinglevextendsingle z?z0
1?z0z
vextendsinglevextendsingle
vextendsinglevextendsingle< 1.
B5BPDRDD
|z?z0| < |1?z0z|.
FR?
|z?z0|2 = |z|2 +|z0|2? 2Re(zz0),
|1?z0z|2 = 1 +|z|2|z0|2? 2Re(zz0),
APDRDD
|z|2 +|z0|2 < 1 +|z|2|z0|2.
FR? |z| < 1,EKEX|z|2(1?|z0|2) < 1?|z0|2,APH2EQEKG9,AQ
DD.
47/89
DI 4 AN |z0| < 1,DDEY,CT |z| < 1,C3
vextendsinglevextendsingle
vextendsinglevextendsingle z?z0
1?z0z
vextendsinglevextendsingle
vextendsinglevextendsingle< 1.
B5BPDRDD
|z?z0| < |1?z0z|.
FR?
|z?z0|2 = |z|2 +|z0|2? 2Re(zz0),
|1?z0z|2 = 1 +|z|2|z0|2? 2Re(zz0),
APDRDD
|z|2 +|z0|2 < 1 +|z|2|z0|2.
FR? |z| < 1,EKEX|z|2(1?|z0|2) < 1?|z0|2,APH2EQEKG9,AQ
DD.
47/89
DI 4 AN |z0| < 1,DDEY,CT |z| < 1,C3
vextendsinglevextendsingle
vextendsinglevextendsingle z?z0
1?z0z
vextendsinglevextendsingle
vextendsinglevextendsingle< 1.
B5BPDRDD
|z?z0| < |1?z0z|.
FR?
|z?z0|2 = |z|2 +|z0|2? 2Re(zz0),
|1?z0z|2 = 1 +|z|2|z0|2? 2Re(zz0),
APDRDD
|z|2 +|z0|2 < 1 +|z|2|z0|2.
FR? |z| < 1,EKEX|z|2(1?|z0|2) < 1?|z0|2,APH2EQEKG9,AQ
DD.
47/89
2,EYEHBTG3EVC3ARBXGIE0DF
DUGABXGIDQEYEHH2ARBM (E5H2) DND1ARB3CG,CMBXGIE0
DFBHB5GZEYEHBTG3EVC3.
48/89
DI 1 DDEY,BXGI α DQ β EKC4ERARE5H2D4DNCADQARFBEP
BJDRCMDR? Re(αβ) = 0.
B
β
y
x O
|β?α|
α
A
|β|
|α|
49/89
B
β
y
x O
|β?α|
α
A
|β|
|α|
DD CGGB,E5H2
→OA ⊥→OB
ARFBEPBJDRCMDRF0
|?→AB|2 = |?→OA|2 + |→OB|2
BP |β?α|2 = |α|2 + |β|2.
D5FR?
|β?α|2 = (β?α)(β?α)
= ββ + αα? (αβ + αβ)
= |β|2 +|α|2? 2Re(αβ)
EKEX
→OA ⊥→OB
ARFBEPBJDRCMDRF0 Re(αβ) = 0.
50/89
B
β
y
x O
|β?α|
α
A
|β|
|α|
DD CGGB,E5H2
→OA ⊥→OB
ARFBEPBJDRCMDRF0
|?→AB|2 = |?→OA|2 + |→OB|2
BP |β?α|2 = |α|2 + |β|2.
D5FR?
|β?α|2 = (β?α)(β?α)
= ββ + αα? (αβ + αβ)
= |β|2 +|α|2? 2Re(αβ)
EKEX
→OA ⊥→OB
ARFBEPBJDRCMDRF0 Re(αβ) = 0.
50/89
B
β
y
x O
|β?α|
α
A
|β|
|α|
DD CGGB,E5H2
→OA ⊥→OB
ARFBEPBJDRCMDRF0
|?→AB|2 = |?→OA|2 + |→OB|2
BP |β?α|2 = |α|2 + |β|2.
D5FR?
|β?α|2 = (β?α)(β?α)
= ββ + αα? (αβ + αβ)
= |β|2 +|α|2? 2Re(αβ)
EKEX
→OA ⊥→OB
ARFBEPBJDRCMDRF0 Re(αβ) = 0.
50/89
B
β
y
x O
|β?α|
α
A
|β|
|α|
DD CGGB,E5H2
→OA ⊥→OB
ARFBEPBJDRCMDRF0
|?→AB|2 = |?→OA|2 + |→OB|2
BP |β?α|2 = |α|2 + |β|2.
D5FR?
|β?α|2 = (β?α)(β?α)
= ββ + αα? (αβ + αβ)
= |β|2 +|α|2? 2Re(αβ)
EKEX
→OA ⊥→OB
ARFBEPBJDRCMDRF0 Re(αβ) = 0.
50/89
B
β
y
x O
|β?α|
α
A
|β|
|α|
DD CGGB,E5H2
→OA ⊥→OB
ARFBEPBJDRCMDRF0
|?→AB|2 = |?→OA|2 + |→OB|2
BP |β?α|2 = |α|2 + |β|2.
D5FR?
|β?α|2 = (β?α)(β?α)
= ββ + αα? (αβ + αβ)
= |β|2 +|α|2? 2Re(αβ)
EKEX
→OA ⊥→OB
ARFBEPBJDRCMDRF0 Re(αβ) = 0.
50/89
B
β
y
x O
|β?α|
α
A
|β|
|α|
DD CGGB,E5H2
→OA ⊥→OB
ARFBEPBJDRCMDRF0
|?→AB|2 = |?→OA|2 + |→OB|2
BP |β?α|2 = |α|2 + |β|2.
D5FR?
|β?α|2 = (β?α)(β?α)
= ββ + αα? (αβ + αβ)
= |β|2 +|α|2? 2Re(αβ)
EKEX
→OA ⊥→OB
ARFBEPBJDRCMDRF0 Re(αβ) = 0.
50/89
DI 2 DDEY,EX z1,z2,z3 G2 w1,w2,w3?F6BMARH0DRD2
BBDND5ARFBDRCMDRF0
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
1 1 1
z1 z2 z3
w1 w2 w3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle= 0.
51/89
B5CSBXGIARBTG3FBFEBLBTG3ELH0D2?BBDND5ARA7F9F9
D2DJ,H0D2?BBDND5CZ
vextendsinglevextendsingle
vextendsinglevextendsinglez3?z1
z2?z1
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsinglew3?w1
w2?w1
vextendsinglevextendsingle
vextendsinglevextendsingle,
arg(z3?z1)?arg(z2?z1) = arg(w3?w1)? arg(w2?w1)
=? arg
parenleftbiggz
3?z1
z2?z1
parenrightbigg
= arg
parenleftbiggw
3?w1
w2?w1
parenrightbigg
,
BP
z3?z1
z2?z1 =
w3?w1
w2?w1.
BECSBDAFEQC7DY,AQ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
1 1 1
z1 z2 z3
w1 w2 w3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle= (z2?z1)(w3?w1)? (z3?z1)(w2?w1) = 0.
52/89
B5CSBXGIARBTG3FBFEBLBTG3ELH0D2?BBDND5ARA7F9F9
D2DJ,H0D2?BBDND5CZ
vextendsinglevextendsingle
vextendsinglevextendsinglez3?z1
z2?z1
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsinglew3?w1
w2?w1
vextendsinglevextendsingle
vextendsinglevextendsingle,
arg(z3?z1)?arg(z2?z1) = arg(w3?w1)? arg(w2?w1)
=? arg
parenleftbiggz
3?z1
z2?z1
parenrightbigg
= arg
parenleftbiggw
3?w1
w2?w1
parenrightbigg
,
BP
z3?z1
z2?z1 =
w3?w1
w2?w1.
BECSBDAFEQC7DY,AQ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
1 1 1
z1 z2 z3
w1 w2 w3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle= (z2?z1)(w3?w1)? (z3?z1)(w2?w1) = 0.
52/89
B5CSBXGIARBTG3FBFEBLBTG3ELH0D2?BBDND5ARA7F9F9
D2DJ,H0D2?BBDND5CZ
vextendsinglevextendsingle
vextendsinglevextendsinglez3?z1
z2?z1
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsinglew3?w1
w2?w1
vextendsinglevextendsingle
vextendsinglevextendsingle,
arg(z3?z1)?arg(z2?z1) = arg(w3?w1)? arg(w2?w1)
=? arg
parenleftbiggz
3?z1
z2?z1
parenrightbigg
= arg
parenleftbiggw
3?w1
w2?w1
parenrightbigg
,
BP
z3?z1
z2?z1 =
w3?w1
w2?w1.
BECSBDAFEQC7DY,AQ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
1 1 1
z1 z2 z3
w1 w2 w3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle= (z2?z1)(w3?w1)? (z3?z1)(w2?w1) = 0.
52/89
B5CSBXGIARBTG3FBFEBLBTG3ELH0D2?BBDND5ARA7F9F9
D2DJ,H0D2?BBDND5CZ
vextendsinglevextendsingle
vextendsinglevextendsinglez3?z1
z2?z1
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsinglew3?w1
w2?w1
vextendsinglevextendsingle
vextendsinglevextendsingle,
arg(z3?z1)?arg(z2?z1) = arg(w3?w1)? arg(w2?w1)
=? arg
parenleftbiggz
3?z1
z2?z1
parenrightbigg
= arg
parenleftbiggw
3?w1
w2?w1
parenrightbigg
,
BP
z3?z1
z2?z1 =
w3?w1
w2?w1.
BECSBDAFEQC7DY,AQ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
1 1 1
z1 z2 z3
w1 w2 w3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle= (z2?z1)(w3?w1)? (z3?z1)(w2?w1) = 0.
52/89
B5CSBXGIARBTG3FBFEBLBTG3ELH0D2?BBDND5ARA7F9F9
D2DJ,H0D2?BBDND5CZ
vextendsinglevextendsingle
vextendsinglevextendsinglez3?z1
z2?z1
vextendsinglevextendsingle
vextendsinglevextendsingle=
vextendsinglevextendsingle
vextendsinglevextendsinglew3?w1
w2?w1
vextendsinglevextendsingle
vextendsinglevextendsingle,
arg(z3?z1)?arg(z2?z1) = arg(w3?w1)? arg(w2?w1)
=? arg
parenleftbiggz
3?z1
z2?z1
parenrightbigg
= arg
parenleftbiggw
3?w1
w2?w1
parenrightbigg
,
BP
z3?z1
z2?z1 =
w3?w1
w2?w1.
BECSBDAFEQC7DY,AQ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle
1 1 1
z1 z2 z3
w1 w2 w3
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglevextendsingle= (z2?z1)(w3?w1)? (z3?z1)(w2?w1) = 0.
52/89
DI 3 AN z1,z2,z3 D2BMF3G4CMDR:z1 + z2 + z3 = 0,
|z1| = |z2| = |z3| = 1,DDEY:z1,z2,z3 F0CBAQD9H0EG
AB |z| = 1 ARE8DRAKD2?BBARF6BM.
B5DUGABDE8G1CBC3BD 11 C3ARB4G0,
|z1 + z2|2 + |z1?z2|2 = 2(|z1|2 + |z2|2) = 4.
DGCM z1 + z2 + z3 = 0 E3CKAKH2EQDWAQ
|?z3|2 + |z1?z2|2 = 4 =? |z1?z2|2 = 3.
FVD2
|z1?z3|2 = 3; |z2?z3|2 = 3.
EKEX z1,z2,z3 F0CBAQD9H0EGAB|z| = 1 ARE8DRAKD2?BBAR
F6BM.
53/89
DI 3 AN z1,z2,z3 D2BMF3G4CMDR:z1 + z2 + z3 = 0,
|z1| = |z2| = |z3| = 1,DDEY:z1,z2,z3 F0CBAQD9H0EG
AB |z| = 1 ARE8DRAKD2?BBARF6BM.
B5DUGABDE8G1CBC3BD 11 C3ARB4G0,
|z1 + z2|2 + |z1?z2|2 = 2(|z1|2 + |z2|2) = 4.
DGCM z1 + z2 + z3 = 0 E3CKAKH2EQDWAQ
|?z3|2 + |z1?z2|2 = 4 =? |z1?z2|2 = 3.
FVD2
|z1?z3|2 = 3; |z2?z3|2 = 3.
EKEX z1,z2,z3 F0CBAQD9H0EGAB|z| = 1 ARE8DRAKD2?BBAR
F6BM.
53/89
DI 3 AN z1,z2,z3 D2BMF3G4CMDR:z1 + z2 + z3 = 0,
|z1| = |z2| = |z3| = 1,DDEY:z1,z2,z3 F0CBAQD9H0EG
AB |z| = 1 ARE8DRAKD2?BBARF6BM.
B5DUGABDE8G1CBC3BD 11 C3ARB4G0,
|z1 + z2|2 + |z1?z2|2 = 2(|z1|2 + |z2|2) = 4.
DGCM z1 + z2 + z3 = 0 E3CKAKH2EQDWAQ
|?z3|2 + |z1?z2|2 = 4 =? |z1?z2|2 = 3.
FVD2
|z1?z3|2 = 3; |z2?z3|2 = 3.
EKEX z1,z2,z3 F0CBAQD9H0EGAB|z| = 1 ARE8DRAKD2?BBAR
F6BM.
53/89
DI 3 AN z1,z2,z3 D2BMF3G4CMDR:z1 + z2 + z3 = 0,
|z1| = |z2| = |z3| = 1,DDEY:z1,z2,z3 F0CBAQD9H0EG
AB |z| = 1 ARE8DRAKD2?BBARF6BM.
B5DUGABDE8G1CBC3BD 11 C3ARB4G0,
|z1 + z2|2 + |z1?z2|2 = 2(|z1|2 + |z2|2) = 4.
DGCM z1 + z2 + z3 = 0 E3CKAKH2EQDWAQ
|?z3|2 + |z1?z2|2 = 4 =? |z1?z2|2 = 3.
FVD2
|z1?z3|2 = 3; |z2?z3|2 = 3.
EKEX z1,z2,z3 F0CBAQD9H0EGAB|z| = 1 ARE8DRAKD2?BBAR
F6BM.
53/89
DI 3 AN z1,z2,z3 D2BMF3G4CMDR:z1 + z2 + z3 = 0,
|z1| = |z2| = |z3| = 1,DDEY:z1,z2,z3 F0CBAQD9H0EG
AB |z| = 1 ARE8DRAKD2?BBARF6BM.
B5DUGABDE8G1CBC3BD 11 C3ARB4G0,
|z1 + z2|2 + |z1?z2|2 = 2(|z1|2 + |z2|2) = 4.
DGCM z1 + z2 + z3 = 0 E3CKAKH2EQDWAQ
|?z3|2 + |z1?z2|2 = 4 =? |z1?z2|2 = 3.
FVD2
|z1?z3|2 = 3; |z2?z3|2 = 3.
EKEX z1,z2,z3 F0CBAQD9H0EGAB|z| = 1 ARE8DRAKD2?BBAR
F6BM.
53/89
DI 4 DDEYBXEYEHH2ARDQDME0ENDWALEK:
αz + αz = c,(α negationslash= 0?BXDIGI,c?EIDIGI).
DDAN z = x + iy,α = a + ib,E3CKAKE0ENEL,AQ
(a + ib)(x?iy) + (a?ib)(x + iy) = c,
DEDFAQ 2ax + 2by = c,GXF0DQDMARE8DOE0EN.
54/89
DI 4 DDEYBXEYEHH2ARDQDME0ENDWALEK:
αz + αz = c,(α negationslash= 0?BXDIGI,c?EIDIGI).
DDAN z = x + iy,α = a + ib,E3CKAKE0ENEL,AQ
(a + ib)(x?iy) + (a?ib)(x + iy) = c,
DEDFAQ 2ax + 2by = c,GXF0DQDMARE8DOE0EN.
54/89
DI 4 DDEYBXEYEHH2ARDQDME0ENDWALEK:
αz + αz = c,(α negationslash= 0?BXDIGI,c?EIDIGI).
DDAN z = x + iy,α = a + ib,E3CKAKE0ENEL,AQ
(a + ib)(x?iy) + (a?ib)(x + iy) = c,
DEDFAQ 2ax + 2by = c,GXF0DQDMARE8DOE0EN.
54/89
DRDN,GHD9BRFBDQDM,Ax+ By + C = 0(A,B G0FVEE?
AY),B9 z = x + iy,C3
x = z + z2,y = z?z2i,
E3CKAKE0EN,AQ
Az + z2 + Bz?z2i + C = 0
BP
parenleftbigg1
2A?
B
2 i
parenrightbigg
z +
parenleftbigg1
2A +
B
2 i
parenrightbigg
z + C = 0,
AI α = A + iB2,c =?C,C3E0ENDWAL?
αz + αz = c,(α negationslash= 0?BXDIGI,c?EIDIGI).
55/89
DRDN,GHD9BRFBDQDM,Ax+ By + C = 0(A,B G0FVEE?
AY),B9 z = x + iy,C3
x = z + z2,y = z?z2i,
E3CKAKE0EN,AQ
Az + z2 + Bz?z2i + C = 0
BP
parenleftbigg1
2A?
B
2 i
parenrightbigg
z +
parenleftbigg1
2A +
B
2 i
parenrightbigg
z + C = 0,
AI α = A + iB2,c =?C,C3E0ENDWAL?
αz + αz = c,(α negationslash= 0?BXDIGI,c?EIDIGI).
55/89
DRDN,GHD9BRFBDQDM,Ax+ By + C = 0(A,B G0FVEE?
AY),B9 z = x + iy,C3
x = z + z2,y = z?z2i,
E3CKAKE0EN,AQ
Az + z2 + Bz?z2i + C = 0
BP
parenleftbigg1
2A?
B
2 i
parenrightbigg
z +
parenleftbigg1
2A +
B
2 i
parenrightbigg
z + C = 0,
AI α = A + iB2,c =?C,C3E0ENDWAL?
αz + αz = c,(α negationslash= 0?BXDIGI,c?EIDIGI).
55/89
DRDN,GHD9BRFBDQDM,Ax+ By + C = 0(A,B G0FVEE?
AY),B9 z = x + iy,C3
x = z + z2,y = z?z2i,
E3CKAKE0EN,AQ
Az + z2 + Bz?z2i + C = 0
BP
parenleftbigg1
2A?
B
2 i
parenrightbigg
z +
parenleftbigg1
2A +
B
2 i
parenrightbigg
z + C = 0,
AI α = A + iB2,c =?C,C3E0ENDWAL?
αz + αz = c,(α negationslash= 0?BXDIGI,c?EIDIGI).
55/89
DRDN,GHD9BRFBDQDM,Ax+ By + C = 0(A,B G0FVEE?
AY),B9 z = x + iy,C3
x = z + z2,y = z?z2i,
E3CKAKE0EN,AQ
Az + z2 + Bz?z2i + C = 0
BP
parenleftbigg1
2A?
B
2 i
parenrightbigg
z +
parenleftbigg1
2A +
B
2 i
parenrightbigg
z + C = 0,
AI α = A + iB2,c =?C,C3E0ENDWAL?
αz + αz = c,(α negationslash= 0?BXDIGI,c?EIDIGI).
55/89
DI 5 DDEYBXEYEHH2ARABEXE0ENDWALEK
zz + αz + αz + c = 0,(α?BXDIGI,c?EIDIGI).
DDABEXARE8DOE0EN?
x2 + y2 + 2ax + 2by + c = 0.
B9 z = x + iy,C3 x = z + z2,y = z?z2i,E3CKAKE0EN,AQ
parenleftbiggz + z
2
parenrightbigg2
+
parenleftbiggz?z
2i
parenrightbigg2
+ 2az + z2 + 2bz?z2i + c = 0
BP zz + (a?bi)z + (a + bi)z + c = 0,
B9 α = a + bi,APABEXE0ENDWAL?
zz + αz + αz + c = 0,(α?BXDIGI,c?EIDIGI).
56/89
DI 5 DDEYBXEYEHH2ARABEXE0ENDWALEK
zz + αz + αz + c = 0,(α?BXDIGI,c?EIDIGI).
DDABEXARE8DOE0EN?
x2 + y2 + 2ax + 2by + c = 0.
B9 z = x + iy,C3 x = z + z2,y = z?z2i,E3CKAKE0EN,AQ
parenleftbiggz + z
2
parenrightbigg2
+
parenleftbiggz?z
2i
parenrightbigg2
+ 2az + z2 + 2bz?z2i + c = 0
BP zz + (a?bi)z + (a + bi)z + c = 0,
B9 α = a + bi,APABEXE0ENDWAL?
zz + αz + αz + c = 0,(α?BXDIGI,c?EIDIGI).
56/89
DI 5 DDEYBXEYEHH2ARABEXE0ENDWALEK
zz + αz + αz + c = 0,(α?BXDIGI,c?EIDIGI).
DDABEXARE8DOE0EN?
x2 + y2 + 2ax + 2by + c = 0.
B9 z = x + iy,C3 x = z + z2,y = z?z2i,E3CKAKE0EN,AQ
parenleftbiggz + z
2
parenrightbigg2
+
parenleftbiggz?z
2i
parenrightbigg2
+ 2az + z2 + 2bz?z2i + c = 0
BP zz + (a?bi)z + (a + bi)z + c = 0,
B9 α = a + bi,APABEXE0ENDWAL?
zz + αz + αz + c = 0,(α?BXDIGI,c?EIDIGI).
56/89
DI 5 DDEYBXEYEHH2ARABEXE0ENDWALEK
zz + αz + αz + c = 0,(α?BXDIGI,c?EIDIGI).
DDABEXARE8DOE0EN?
x2 + y2 + 2ax + 2by + c = 0.
B9 z = x + iy,C3 x = z + z2,y = z?z2i,E3CKAKE0EN,AQ
parenleftbiggz + z
2
parenrightbigg2
+
parenleftbiggz?z
2i
parenrightbigg2
+ 2az + z2 + 2bz?z2i + c = 0
BP zz + (a?bi)z + (a + bi)z + c = 0,
B9 α = a + bi,APABEXE0ENDWAL?
zz + αz + αz + c = 0,(α?BXDIGI,c?EIDIGI).
56/89
DI 5 DDEYBXEYEHH2ARABEXE0ENDWALEK
zz + αz + αz + c = 0,(α?BXDIGI,c?EIDIGI).
DDABEXARE8DOE0EN?
x2 + y2 + 2ax + 2by + c = 0.
B9 z = x + iy,C3 x = z + z2,y = z?z2i,E3CKAKE0EN,AQ
parenleftbiggz + z
2
parenrightbigg2
+
parenleftbiggz?z
2i
parenrightbigg2
+ 2az + z2 + 2bz?z2i + c = 0
BP zz + (a?bi)z + (a + bi)z + c = 0,
B9 α = a + bi,APABEXE0ENDWAL?
zz + αz + αz + c = 0,(α?BXDIGI,c?EIDIGI).
56/89
DI 5 DDEYBXEYEHH2ARABEXE0ENDWALEK
zz + αz + αz + c = 0,(α?BXDIGI,c?EIDIGI).
DDABEXARE8DOE0EN?
x2 + y2 + 2ax + 2by + c = 0.
B9 z = x + iy,C3 x = z + z2,y = z?z2i,E3CKAKE0EN,AQ
parenleftbiggz + z
2
parenrightbigg2
+
parenleftbiggz?z
2i
parenrightbigg2
+ 2az + z2 + 2bz?z2i + c = 0
BP zz + (a?bi)z + (a + bi)z + c = 0,
B9 α = a + bi,APABEXE0ENDWAL?
zz + αz + αz + c = 0,(α?BXDIGI,c?EIDIGI).
56/89
BDD3AX ERDX
1,ERDXARCECV
APDX,EYEHH2EX z0?ELB1,δ(BRFBARAKGI)?DWF7ARAB:
|z?z0| < δ
CBG4ARBMARBKG4EH? z0 ARAPDX,ADEHCSG0AVEQ
0 < |z?z0| < δ
EKB4F9ARBMBK? z0 ARAMB1APDX.
57/89
BDD3AX ERDX
1,ERDXARCECV
APDX,EYEHH2EX z0?ELB1,δ(BRFBARAKGI)?DWF7ARAB:
|z?z0| < δ
CBG4ARBMARBKG4EH? z0 ARAPDX,ADEHCSG0AVEQ
0 < |z?z0| < δ
EKB4F9ARBMBK? z0 ARAMB1APDX.
57/89
D1BK,AN G?E8EYEHBMBK,z0? G ELBRFBE8BM,CGC2DJ
BF z0 ARE8DRAPDX,CCAPDXCBAREKCZBMFTG9D9 G,BSD3
EH z0? G ARCBBM.
CGC2 G CBARDEDRBMFTF0ENARCBBM,BSD3EH G?D1BK.
Γ
z1
z2
z0
δ
C1
γ1
γ2
z
ERDX:EYEHBMBK D EH?E8DRERDX,CGC2ENGVDACTAFH0DR
CMDR:
1) D F0E8DRD1BK;
2) D F0GJFRAR,BP D ELBRG3H0BMFTDWEXCMAHASG9D9 D
ARE8CMGQDMGJAQGDBH.
58/89
D1BK,AN G?E8EYEHBMBK,z0? G ELBRFBE8BM,CGC2DJ
BF z0 ARE8DRAPDX,CCAPDXCBAREKCZBMFTG9D9 G,BSD3
EH z0? G ARCBBM.
CGC2 G CBARDEDRBMFTF0ENARCBBM,BSD3EH G?D1BK.
Γ
z1
z2
z0
δ
C1
γ1
γ2
z
ERDX:EYEHBMBK D EH?E8DRERDX,CGC2ENGVDACTAFH0DR
CMDR:
1) D F0E8DRD1BK;
2) D F0GJFRAR,BP D ELBRG3H0BMFTDWEXCMAHASG9D9 D
ARE8CMGQDMGJAQGDBH.
58/89
D1BK,AN G?E8EYEHBMBK,z0? G ELBRFBE8BM,CGC2DJ
BF z0 ARE8DRAPDX,CCAPDXCBAREKCZBMFTG9D9 G,BSD3
EH z0? G ARCBBM.
CGC2 G CBARDEDRBMFTF0ENARCBBM,BSD3EH G?D1BK.
Γ
z1
z2
z0
δ
C1
γ1
γ2
z
ERDX:EYEHBMBK D EH?E8DRERDX,CGC2ENGVDACTAFH0DR
CMDR:
1) D F0E8DRD1BK;
2) D F0GJFRAR,BP D ELBRG3H0BMFTDWEXCMAHASG9D9 D
ARE8CMGQDMGJAQGDBH.
58/89
BQBA,AN D?BXEYEHCBARE8DRERDX,CGC2BM P G0G9
D9 D,A2BF P ARBRFBA2ARAPDXCBD3EDF5CZ D ELARBM,EH
GXAOARBM? D ARBQBABM.
D AREKCZBQBABMDGEK D ARBQBA.
BGERDX,ERDX D DQENARBQBAE8GDA8EKBGERDX,C8DU D.
CZBA,CGC2E8DRERDX D DWEXAQEDF5BFE8DREXA6BM?EL
B1ARABCB,BP,?M > 0,EMAQ:
|z| < M,?z ∈ D,
BSD3 D EH?CZBAAR,FGC3EH?FFBAAR.
59/89
BQBA,AN D?BXEYEHCBARE8DRERDX,CGC2BM P G0G9
D9 D,A2BF P ARBRFBA2ARAPDXCBD3EDF5CZ D ELARBM,EH
GXAOARBM? D ARBQBABM.
D AREKCZBQBABMDGEK D ARBQBA.
BGERDX,ERDX D DQENARBQBAE8GDA8EKBGERDX,C8DU D.
CZBA,CGC2E8DRERDX D DWEXAQEDF5BFE8DREXA6BM?EL
B1ARABCB,BP,?M > 0,EMAQ:
|z| < M,?z ∈ D,
BSD3 D EH?CZBAAR,FGC3EH?FFBAAR.
59/89
BQBA,AN D?BXEYEHCBARE8DRERDX,CGC2BM P G0G9
D9 D,A2BF P ARBRFBA2ARAPDXCBD3EDF5CZ D ELARBM,EH
GXAOARBM? D ARBQBABM.
D AREKCZBQBABMDGEK D ARBQBA.
BGERDX,ERDX D DQENARBQBAE8GDA8EKBGERDX,C8DU D.
CZBA,CGC2E8DRERDX D DWEXAQEDF5BFE8DREXA6BM?EL
B1ARABCB,BP,?M > 0,EMAQ:
|z| < M,?z ∈ D,
BSD3 D EH?CZBAAR,FGC3EH?FFBAAR.
59/89
BQBA,AN D?BXEYEHCBARE8DRERDX,CGC2BM P G0G9
D9 D,A2BF P ARBRFBA2ARAPDXCBD3EDF5CZ D ELARBM,EH
GXAOARBM? D ARBQBABM.
D AREKCZBQBABMDGEK D ARBQBA.
BGERDX,ERDX D DQENARBQBAE8GDA8EKBGERDX,C8DU D.
CZBA,CGC2E8DRERDX D DWEXAQEDF5BFE8DREXA6BM?EL
B1ARABCB,BP,?M > 0,EMAQ:
|z| < M,?z ∈ D,
BSD3 D EH?CZBAAR,FGC3EH?FFBAAR.
59/89
DICG,GVDAG0AVEQ r1 < |z?z0| < r2 AREKCZBMA8EKE8DR
ERDX.
CCERDXF0CZBAAR,ERDXARBQBAF0H0DRABEX:
|z?z0| = r1,|z?z0| = r2.
r1
r2
z0
60/89
2,H0GJFRDXDQGTGJFRDX
GJCHETDM:CGC2 x(t) G2 y(t) F0H0DRGJCHAREIBVF8GI,BS
D3,E0ENDG
x = x(t),y = y(t),(a lessorequalslant t lessorequalslant b)
E3C4E8CMEYEHETDM,EH?GJCHETDM.
CGC2B9
z(t) = x(t) + iy(t),
BSD3GXCMETDMDWEXCME8DRBXGIE0EN
z = z(t),(alessorequalslant t lessorequalslant b)
BHE3C4,GXF0EYEHETDMARBXGIC4EREQ.
61/89
2,H0GJFRDXDQGTGJFRDX
GJCHETDM:CGC2 x(t) G2 y(t) F0H0DRGJCHAREIBVF8GI,BS
D3,E0ENDG
x = x(t),y = y(t),(a lessorequalslant t lessorequalslant b)
E3C4E8CMEYEHETDM,EH?GJCHETDM.
CGC2B9
z(t) = x(t) + iy(t),
BSD3GXCMETDMDWEXCME8DRBXGIE0EN
z = z(t),(alessorequalslant t lessorequalslant b)
BHE3C4,GXF0EYEHETDMARBXGIC4EREQ.
61/89
2,H0GJFRDXDQGTGJFRDX
GJCHETDM:CGC2 x(t) G2 y(t) F0H0DRGJCHAREIBVF8GI,BS
D3,E0ENDG
x = x(t),y = y(t),(a lessorequalslant t lessorequalslant b)
E3C4E8CMEYEHETDM,EH?GJCHETDM.
CGC2B9
z(t) = x(t) + iy(t),
BSD3GXCMETDMDWEXCME8DRBXGIE0EN
z = z(t),(alessorequalslant t lessorequalslant b)
BHE3C4,GXF0EYEHETDMARBXGIC4EREQ.
61/89
DICTAFE0ENC4EREFD3AOARETDM?
(1) z = 3ei2pit,0 lessorequalslant t lessorequalslant 1,
(2) z = 1 + t2 + it,0 lessorequalslant t lessorequalslant 1.
B5 (1) FR?
z = x(t) + iy(t) = 3ei2pit = 3cos2pit + i3sin2pit,
EKEX x(t) = 3cos2pit,y(t) = 3sin2pit(0 lessorequalslant t lessorequalslant 1),F0EX 0
ABB1,3?DWF7ARABEX.
(2) FR?
z = x(t) + iy(t) = 1 + t2 + it,
EKEX x(t) = 1 + t2,y(t) = t(0 lessorequalslant t lessorequalslant 1),F0?FW
DM y2 = x?1 H2ARE8GB.
62/89
DICTAFE0ENC4EREFD3AOARETDM?
(1) z = 3ei2pit,0 lessorequalslant t lessorequalslant 1,
(2) z = 1 + t2 + it,0 lessorequalslant t lessorequalslant 1.
B5 (1) FR?
z = x(t) + iy(t) = 3ei2pit = 3cos2pit + i3sin2pit,
EKEX x(t) = 3cos2pit,y(t) = 3sin2pit(0 lessorequalslant t lessorequalslant 1),F0EX 0
ABB1,3?DWF7ARABEX.
(2) FR?
z = x(t) + iy(t) = 1 + t2 + it,
EKEX x(t) = 1 + t2,y(t) = t(0 lessorequalslant t lessorequalslant 1),F0?FW
DM y2 = x?1 H2ARE8GB.
62/89
DICTAFE0ENC4EREFD3AOARETDM?
(1) z = 3ei2pit,0 lessorequalslant t lessorequalslant 1,
(2) z = 1 + t2 + it,0 lessorequalslant t lessorequalslant 1.
B5 (1) FR?
z = x(t) + iy(t) = 3ei2pit = 3cos2pit + i3sin2pit,
EKEX x(t) = 3cos2pit,y(t) = 3sin2pit(0 lessorequalslant t lessorequalslant 1),F0EX 0
ABB1,3?DWF7ARABEX.
(2) FR?
z = x(t) + iy(t) = 1 + t2 + it,
EKEX x(t) = 1 + t2,y(t) = t(0 lessorequalslant t lessorequalslant 1),F0?FW
DM y2 = x?1 H2ARE8GB.
62/89
DICTAFE0ENC4EREFD3AOARETDM?
(1) z = 3ei2pit,0 lessorequalslant t lessorequalslant 1,
(2) z = 1 + t2 + it,0 lessorequalslant t lessorequalslant 1.
B5 (1) FR?
z = x(t) + iy(t) = 3ei2pit = 3cos2pit + i3sin2pit,
EKEX x(t) = 3cos2pit,y(t) = 3sin2pit(0 lessorequalslant t lessorequalslant 1),F0EX 0
ABB1,3?DWF7ARABEX.
(2) FR?
z = x(t) + iy(t) = 1 + t2 + it,
EKEX x(t) = 1 + t2,y(t) = t(0 lessorequalslant t lessorequalslant 1),F0?FW
DM y2 = x?1 H2ARE8GB.
62/89
DICTAFE0ENC4EREFD3AOARETDM?
(1) z = 3ei2pit,0 lessorequalslant t lessorequalslant 1,
(2) z = 1 + t2 + it,0 lessorequalslant t lessorequalslant 1.
B5 (1) FR?
z = x(t) + iy(t) = 3ei2pit = 3cos2pit + i3sin2pit,
EKEX x(t) = 3cos2pit,y(t) = 3sin2pit(0 lessorequalslant t lessorequalslant 1),F0EX 0
ABB1,3?DWF7ARABEX.
(2) FR?
z = x(t) + iy(t) = 1 + t2 + it,
EKEX x(t) = 1 + t2,y(t) = t(0 lessorequalslant t lessorequalslant 1),F0?FW
DM y2 = x?1 H2ARE8GB.
62/89
DICTAFE0ENC4EREFD3AOARETDM?
(1) z = 3ei2pit,0 lessorequalslant t lessorequalslant 1,
(2) z = 1 + t2 + it,0 lessorequalslant t lessorequalslant 1.
B5 (1) FR?
z = x(t) + iy(t) = 3ei2pit = 3cos2pit + i3sin2pit,
EKEX x(t) = 3cos2pit,y(t) = 3sin2pit(0 lessorequalslant t lessorequalslant 1),F0EX 0
ABB1,3?DWF7ARABEX.
(2) FR?
z = x(t) + iy(t) = 1 + t2 + it,
EKEX x(t) = 1 + t2,y(t) = t(0 lessorequalslant t lessorequalslant 1),F0?FW
DM y2 = x?1 H2ARE8GB.
62/89
DICTAFE0ENC4EREFD3AOARETDM?
(1) z = 3ei2pit,0 lessorequalslant t lessorequalslant 1,
(2) z = 1 + t2 + it,0 lessorequalslant t lessorequalslant 1.
B5 (1) FR?
z = x(t) + iy(t) = 3ei2pit = 3cos2pit + i3sin2pit,
EKEX x(t) = 3cos2pit,y(t) = 3sin2pit(0 lessorequalslant t lessorequalslant 1),F0EX 0
ABB1,3?DWF7ARABEX.
(2) FR?
z = x(t) + iy(t) = 1 + t2 + it,
EKEX x(t) = 1 + t2,y(t) = t(0 lessorequalslant t lessorequalslant 1),F0?FW
DM y2 = x?1 H2ARE8GB.
62/89
BDDBETDM:
CGC2BFERD1 a lessorequalslant t lessorequalslant b H2 x′(t) G2 y′(t) FTF0GJCHAR,DP
GHD9 t ARDEE8DRDU,CZ
{x′(t)}2+{y′(t)}2 negationslash= 0,(AVCSD9z′(t) = x′(t)+iy′(t) negationslash= 0).
BSD3GXETDMEH?BDDBAR.
CSBTGBEDCVDNGJARBDDBETDMEKDGEKARETDMEH?EPGBBDDB
ETDM.
63/89
BDDBETDM:
CGC2BFERD1 a lessorequalslant t lessorequalslant b H2 x′(t) G2 y′(t) FTF0GJCHAR,DP
GHD9 t ARDEE8DRDU,CZ
{x′(t)}2+{y′(t)}2 negationslash= 0,(AVCSD9z′(t) = x′(t)+iy′(t) negationslash= 0).
BSD3GXETDMEH?BDDBAR.
CSBTGBEDCVDNGJARBDDBETDMEKDGEKARETDMEH?EPGBBDDB
ETDM.
63/89
BDDBETDM:
CGC2BFERD1 a lessorequalslant t lessorequalslant b H2 x′(t) G2 y′(t) FTF0GJCHAR,DP
GHD9 t ARDEE8DRDU,CZ
{x′(t)}2+{y′(t)}2 negationslash= 0,(AVCSD9z′(t) = x′(t)+iy′(t) negationslash= 0).
BSD3GXETDMEH?BDDBAR.
CSBTGBEDCVDNGJARBDDBETDMEKDGEKARETDMEH?EPGBBDDB
ETDM.
63/89
DFH0ETDM:
AN C,z = z(t)(a lessorequalslant t lessorequalslant b)?E8CMGJCHETDM,z(a)
DQ z(b) EPC7EH? C ARGDBMDQEQBM.
CGC2GHD9GVDA a < t1 < b,a lessorequalslant t2 lessorequalslant b AR t1 DQ t2,
A8 t1 negationslash= t2 ADCZ z(t1) = z(t2) EE,BM z(t1) EH?ETDM C
ARETBM,DACZETBMARGJCHETDM C,EH?DFH0ETDMF9CTA8ET
DM.
CGC2DFH0ETDMARGDBMG2EQBMETG4,BSD3ETDM C EH?DF
H0BGETDM.
64/89
DFH0ETDM:
AN C,z = z(t)(a lessorequalslant t lessorequalslant b)?E8CMGJCHETDM,z(a)
DQ z(b) EPC7EH? C ARGDBMDQEQBM.
CGC2GHD9GVDA a < t1 < b,a lessorequalslant t2 lessorequalslant b AR t1 DQ t2,
A8 t1 negationslash= t2 ADCZ z(t1) = z(t2) EE,BM z(t1) EH?ETDM C
ARETBM,DACZETBMARGJCHETDM C,EH?DFH0ETDMF9CTA8ET
DM.
CGC2DFH0ETDMARGDBMG2EQBMETG4,BSD3ETDM C EH?DF
H0BGETDM.
64/89
DFH0ETDM:
AN C,z = z(t)(a lessorequalslant t lessorequalslant b)?E8CMGJCHETDM,z(a)
DQ z(b) EPC7EH? C ARGDBMDQEQBM.
CGC2GHD9GVDA a < t1 < b,a lessorequalslant t2 lessorequalslant b AR t1 DQ t2,
A8 t1 negationslash= t2 ADCZ z(t1) = z(t2) EE,BM z(t1) EH?ETDM C
ARETBM,DACZETBMARGJCHETDM C,EH?DFH0ETDMF9CTA8ET
DM.
CGC2DFH0ETDMARGDBMG2EQBMETG4,BSD3ETDM C EH?DF
H0BGETDM.
64/89
DFH0ETDM:
AN C,z = z(t)(a lessorequalslant t lessorequalslant b)?E8CMGJCHETDM,z(a)
DQ z(b) EPC7EH? C ARGDBMDQEQBM.
CGC2GHD9GVDA a < t1 < b,a lessorequalslant t2 lessorequalslant b AR t1 DQ t2,
A8 t1 negationslash= t2 ADCZ z(t1) = z(t2) EE,BM z(t1) EH?ETDM C
ARETBM,DACZETBMARGJCHETDM C,EH?DFH0ETDMF9CTA8ET
DM.
CGC2DFH0ETDMARGDBMG2EQBMETG4,BSD3ETDM C EH?DF
H0BGETDM.
64/89
H0GJFRDX,BXEYEHH2ARE8DRERDX B,CGC2BFG1ELBRDUE8
CMDFH0BGETDM,ADETDMARCBG4D3G9D9 B,EH?H0GJFRDX.
E8DRERDXCGC2G0F0H0GJFR,FUEH?GTGJFRDX.
65/89
BDBKFMAX BXBVF8GI
1,BXBVF8GIARF9FE
F9FEAN G F0E8DRBXGI z = x + iy ARBKG4,CGC2CZE8DR
B4F9ARDFC3DJBF,GHD9 G ELDEE8DR z,CZE8DRF9BTDRBX
GI w = u + iv DQDNGHBT,BSD3EHBXBVGI w F0BXBVGI z AR
F8GI,C8DU
w = f(z).
CGC2 z ARE8DRDUGHBT w ARE8DRDU,BSD3EHF8GI f(z) F0H0
DUBXBVF8GI.
66/89
BDBKFMAX BXBVF8GI
1,BXBVF8GIARF9FE
F9FEAN G F0E8DRBXGI z = x + iy ARBKG4,CGC2CZE8DR
B4F9ARDFC3DJBF,GHD9 G ELDEE8DR z,CZE8DRF9BTDRBX
GI w = u + iv DQDNGHBT,BSD3EHBXBVGI w F0BXBVGI z AR
F8GI,C8DU
w = f(z).
CGC2 z ARE8DRDUGHBT w ARE8DRDU,BSD3EHF8GI f(z) F0H0
DUBXBVF8GI.
66/89
GHD9BXGI z = x + iy G2 w = u + iv,BXBVF8
GI w = f(z) DNA8D9H0DRAJA3EIBVF8GI:
u = u(x,y),v = v(x,y),
DICG,GHD9BXBVF8GI w = z2,B9 z = x + iy,
w = u + iv,C3CZ
u + iv = (x + iy)2 = x2?y2 + 2xyi,
D9F0BXBVF8GI w = z2 GHBTD9H0DRAJA3EIBVF8GI:
u = x2?y2,v = 2xy.
67/89
GHD9BXGI z = x + iy G2 w = u + iv,BXBVF8
GI w = f(z) DNA8D9H0DRAJA3EIBVF8GI:
u = u(x,y),v = v(x,y),
DICG,GHD9BXBVF8GI w = z2,B9 z = x + iy,
w = u + iv,C3CZ
u + iv = (x + iy)2 = x2?y2 + 2xyi,
D9F0BXBVF8GI w = z2 GHBTD9H0DRAJA3EIBVF8GI:
u = x2?y2,v = 2xy.
67/89
GHD9BXGI z = x + iy G2 w = u + iv,BXBVF8
GI w = f(z) DNA8D9H0DRAJA3EIBVF8GI:
u = u(x,y),v = v(x,y),
DICG,GHD9BXBVF8GI w = z2,B9 z = x + iy,
w = u + iv,C3CZ
u + iv = (x + iy)2 = x2?y2 + 2xyi,
D9F0BXBVF8GI w = z2 GHBTD9H0DRAJA3EIBVF8GI:
u = x2?y2,v = 2xy.
67/89
2,C6AJARCECV
GHD9E8DREIBVF8GI y = f(x),DICG:y = x2,ENARGBE4DW
EXBFE8DRDQ?DVC1CGELC4ERFTBH.
O
y
x
y=x2
ADE8DRBXBVF8GI w = f(z) GHBTD9H0DRAJA3EIBVF8
GI u = u(x,y),v = v(x,y),EKEXE8DRBXBVF8GIARGBE4G0
DWCDBFE8DRDQ?DVC1CGELC4ERFTBH.
68/89
2,C6AJARCECV
GHD9E8DREIBVF8GI y = f(x),DICG:y = x2,ENARGBE4DW
EXBFE8DRDQ?DVC1CGELC4ERFTBH.
O
y
x
y=x2
ADE8DRBXBVF8GI w = f(z) GHBTD9H0DRAJA3EIBVF8
GI u = u(x,y),v = v(x,y),EKEXE8DRBXBVF8GIARGBE4G0
DWCDBFE8DRDQ?DVC1CGELC4ERFTBH.
68/89
2,C6AJARCECV
GHD9E8DREIBVF8GI y = f(x),DICG:y = x2,ENARGBE4DW
EXBFE8DRDQ?DVC1CGELC4ERFTBH.
O
y
x
y=x2
ADE8DRBXBVF8GI w = f(z) GHBTD9H0DRAJA3EIBVF8
GI u = u(x,y),v = v(x,y),EKEXE8DRBXBVF8GIARGBE4G0
DWCDBFE8DRDQ?DVC1CGELC4ERFTBH.
68/89
CGC2CM z EYEHH2ARBMC4ERCVBVH2 z ARDU,ADCMB8E8DREY
EH w EYEHH2ARBMC4ERF8GI w ARDU,BSD3F8GI w = f(z)
BFBTG3H2FUDWEXDBDUF0D6 z EYEHH2ARE8DRBMBK G BVAK w
EYEHH2ARE8DRBMBK G? ARC6AJ(F9BVDQ).
DICG:DJFGF8GI w = z EKA8EKARC6AJ:
w,z = a + bi(zEYEH)?→ w = a?bi(wEYEH);
w,z1 = 2 + 3i(zEYEH)?→ w1 = 2? 3i(wEYEH);
w,△ABC(zEYEH)?→ △A′B′C′(wEYEH).
x
y
u
v
A
B
C
B’ A’
C’
(Z) (W)
69/89
CGC2CM z EYEHH2ARBMC4ERCVBVH2 z ARDU,ADCMB8E8DREY
EH w EYEHH2ARBMC4ERF8GI w ARDU,BSD3F8GI w = f(z)
BFBTG3H2FUDWEXDBDUF0D6 z EYEHH2ARE8DRBMBK G BVAK w
EYEHH2ARE8DRBMBK G? ARC6AJ(F9BVDQ).
DICG:DJFGF8GI w = z EKA8EKARC6AJ:
w,z = a + bi(zEYEH)?→ w = a?bi(wEYEH);
w,z1 = 2 + 3i(zEYEH)?→ w1 = 2? 3i(wEYEH);
w,△ABC(zEYEH)?→ △A′B′C′(wEYEH).
x
y
u
v
A
B
C
B’ A’
C’
(Z) (W)
69/89
F8GIG2C6AJARB3CG
F8GIG2C6AJFTF0E8ERGHBTB3CGARDRC6,ENDMF0FVE8CECV.
BFEPG3EL,CBBAD6BVH2DND1ARGHBTB3CGEH?F8GI,
CG w = f(z) FUDRC6BFB4F9ARDFC3CT,BKG4 G ELARGI z DQ
BKG4 G? ELARGI w DND1ARF8GIB3CG.
BFBTG3EL,CBBAD6H0DRBMBKDND1ARGHBTB3CGEH?C6AJ,
CG w = f(z) FUDRC6BF z EYEHH2E8DRBMBK G DQ w EYEHH2
E8DRBMBK G? DND1ARC6AJB3CG.
70/89
DICG,CTEHGNFGF8GI w = z2 EKA8EKARC6AJ.
FGCZ,DUGAB3D9EMDFARASDQFN?ARF9D2DWDJ,FRC4C6
AJ w = z2,z ARFN?C7DWE8AL.
FRCS,z EYEHH2DQAKEIF2A2 α AR?BBDXC6AJEK w
EYEHH2DQAKEIF2A2 2α AR?BBDX.
O
y
x
α
v
u O
2α
71/89
DICG,CTEHGNFGF8GI w = z2 EKA8EKARC6AJ.
FGCZ,DUGAB3D9EMDFARASDQFN?ARF9D2DWDJ,FRC4C6
AJ w = z2,z ARFN?C7DWE8AL.
FRCS,z EYEHH2DQAKEIF2A2 α AR?BBDXC6AJEK w
EYEHH2DQAKEIF2A2 2α AR?BBDX.
O
y
x
α
v
u O
2α
71/89
DICG,CTEHGNFGF8GI w = z2 EKA8EKARC6AJ.
FGCZ,DUGAB3D9EMDFARASDQFN?ARF9D2DWDJ,FRC4C6
AJ w = z2,z ARFN?C7DWE8AL.
FRCS,z EYEHH2DQAKEIF2A2 α AR?BBDXC6AJEK w
EYEHH2DQAKEIF2A2 2α AR?BBDX.
O
y
x
α
v
u O
2α
71/89
B8A9,CSD9 w = z2 GHBTH0DRAJA3EIBVF8GI
u = x2?y2,v = 2xy.
FRCS,CZ
w = z2,x2?y2 = c1(zEYEH)?→ u = c1(wEYEH)
2xy = c2(zEYEH)?→ v = c2(wEYEH)
BEDJFGDQDM x = λ ARE6,D6 x = λ E3CKAK u = x2?y2,
v = 2xy AQ
u = λ2?y2,v = 2λy.
EEAMGJGI y AQ
v2 = 4λ2(λ2?u).
BPDQDM x = λ BF w EYEHARE6.
FVD2,DQDM y = μ ARE6ARE0EN?
v2 = 4μ2(ν2 + u).
72/89
B8A9,CSD9 w = z2 GHBTH0DRAJA3EIBVF8GI
u = x2?y2,v = 2xy.
FRCS,CZ
w = z2,x2?y2 = c1(zEYEH)?→ u = c1(wEYEH)
2xy = c2(zEYEH)?→ v = c2(wEYEH)
BEDJFGDQDM x = λ ARE6,D6 x = λ E3CKAK u = x2?y2,
v = 2xy AQ
u = λ2?y2,v = 2λy.
EEAMGJGI y AQ
v2 = 4λ2(λ2?u).
BPDQDM x = λ BF w EYEHARE6.
FVD2,DQDM y = μ ARE6ARE0EN?
v2 = 4μ2(ν2 + u).
72/89
B8A9,CSD9 w = z2 GHBTH0DRAJA3EIBVF8GI
u = x2?y2,v = 2xy.
FRCS,CZ
w = z2,x2?y2 = c1(zEYEH)?→ u = c1(wEYEH)
2xy = c2(zEYEH)?→ v = c2(wEYEH)
BEDJFGDQDM x = λ ARE6,D6 x = λ E3CKAK u = x2?y2,
v = 2xy AQ
u = λ2?y2,v = 2λy.
EEAMGJGI y AQ
v2 = 4λ2(λ2?u).
BPDQDM x = λ BF w EYEHARE6.
FVD2,DQDM y = μ ARE6ARE0EN?
v2 = 4μ2(ν2 + u).
72/89
B8A9,CSD9 w = z2 GHBTH0DRAJA3EIBVF8GI
u = x2?y2,v = 2xy.
FRCS,CZ
w = z2,x2?y2 = c1(zEYEH)?→ u = c1(wEYEH)
2xy = c2(zEYEH)?→ v = c2(wEYEH)
BEDJFGDQDM x = λ ARE6,D6 x = λ E3CKAK u = x2?y2,
v = 2xy AQ
u = λ2?y2,v = 2λy.
EEAMGJGI y AQ
v2 = 4λ2(λ2?u).
BPDQDM x = λ BF w EYEHARE6.
FVD2,DQDM y = μ ARE6ARE0EN?
v2 = 4μ2(ν2 + u).
72/89
B8A9,CSD9 w = z2 GHBTH0DRAJA3EIBVF8GI
u = x2?y2,v = 2xy.
FRCS,CZ
w = z2,x2?y2 = c1(zEYEH)?→ u = c1(wEYEH)
2xy = c2(zEYEH)?→ v = c2(wEYEH)
BEDJFGDQDM x = λ ARE6,D6 x = λ E3CKAK u = x2?y2,
v = 2xy AQ
u = λ2?y2,v = 2λy.
EEAMGJGI y AQ
v2 = 4λ2(λ2?u).
BPDQDM x = λ BF w EYEHARE6.
FVD2,DQDM y = μ ARE6ARE0EN?
v2 = 4μ2(ν2 + u).
72/89
B8A9,CSD9 w = z2 GHBTH0DRAJA3EIBVF8GI
u = x2?y2,v = 2xy.
FRCS,CZ
w = z2,x2?y2 = c1(zEYEH)?→ u = c1(wEYEH)
2xy = c2(zEYEH)?→ v = c2(wEYEH)
BEDJFGDQDM x = λ ARE6,D6 x = λ E3CKAK u = x2?y2,
v = 2xy AQ
u = λ2?y2,v = 2λy.
EEAMGJGI y AQ
v2 = 4λ2(λ2?u).
BPDQDM x = λ BF w EYEHARE6.
FVD2,DQDM y = μ ARE6ARE0EN?
v2 = 4μ2(ν2 + u).
72/89
DI 1 F8GIw = 1zD6CTAF z EYEHH2ARETDMC6AJEK w EYEH
H2C6AOARETDM.
1) x2 + y2 = 4; 3) x = 1.
B5B9 z = x+iy,w = u+iv,C3 w = 1z,BP z = 1w,DN
A8D9
x + iy = 1u + iv = uu2 + v2?i vu2 + v2.
BP
x = uu2 + v2,y =?vu2 + v2.
1) A8 x2 + y2 = 4 EE,CZ
x2 + y2 =
parenleftbigg u
u2 + v2
parenrightbigg2
+
parenleftbigg?v
u2 + v2
parenrightbigg2
= 4.
DEDF,BPAQ w EYEHH2ARE6ETDMF0,u2 + v2 = 14.
73/89
DI 1 F8GIw = 1zD6CTAF z EYEHH2ARETDMC6AJEK w EYEH
H2C6AOARETDM.
1) x2 + y2 = 4; 3) x = 1.
B5B9 z = x+iy,w = u+iv,C3 w = 1z,BP z = 1w,DN
A8D9
x + iy = 1u + iv = uu2 + v2?i vu2 + v2.
BP
x = uu2 + v2,y =?vu2 + v2.
1) A8 x2 + y2 = 4 EE,CZ
x2 + y2 =
parenleftbigg u
u2 + v2
parenrightbigg2
+
parenleftbigg?v
u2 + v2
parenrightbigg2
= 4.
DEDF,BPAQ w EYEHH2ARE6ETDMF0,u2 + v2 = 14.
73/89
DI 1 F8GIw = 1zD6CTAF z EYEHH2ARETDMC6AJEK w EYEH
H2C6AOARETDM.
1) x2 + y2 = 4; 3) x = 1.
B5B9 z = x+iy,w = u+iv,C3 w = 1z,BP z = 1w,DN
A8D9
x + iy = 1u + iv = uu2 + v2?i vu2 + v2.
BP
x = uu2 + v2,y =?vu2 + v2.
1) A8 x2 + y2 = 4 EE,CZ
x2 + y2 =
parenleftbigg u
u2 + v2
parenrightbigg2
+
parenleftbigg?v
u2 + v2
parenrightbigg2
= 4.
DEDF,BPAQ w EYEHH2ARE6ETDMF0,u2 + v2 = 14.
73/89
DI 1 F8GIw = 1zD6CTAF z EYEHH2ARETDMC6AJEK w EYEH
H2C6AOARETDM.
1) x2 + y2 = 4; 3) x = 1.
B5B9 z = x+iy,w = u+iv,C3 w = 1z,BP z = 1w,DN
A8D9
x + iy = 1u + iv = uu2 + v2?i vu2 + v2.
BP
x = uu2 + v2,y =?vu2 + v2.
1) A8 x2 + y2 = 4 EE,CZ
x2 + y2 =
parenleftbigg u
u2 + v2
parenrightbigg2
+
parenleftbigg?v
u2 + v2
parenrightbigg2
= 4.
DEDF,BPAQ w EYEHH2ARE6ETDMF0,u2 + v2 = 14.
73/89
DI 1 F8GIw = 1zD6CTAF z EYEHH2ARETDMC6AJEK w EYEH
H2C6AOARETDM.
1) x2 + y2 = 4; 3) x = 1.
B5B9 z = x+iy,w = u+iv,C3 w = 1z,BP z = 1w,DN
A8D9
x + iy = 1u + iv = uu2 + v2?i vu2 + v2.
BP
x = uu2 + v2,y =?vu2 + v2.
1) A8 x2 + y2 = 4 EE,CZ
x2 + y2 =
parenleftbigg u
u2 + v2
parenrightbigg2
+
parenleftbigg?v
u2 + v2
parenrightbigg2
= 4.
DEDF,BPAQ w EYEHH2ARE6ETDMF0,u2 + v2 = 14.
73/89
DI 1 F8GIw = 1zD6CTAF z EYEHH2ARETDMC6AJEK w EYEH
H2C6AOARETDM.
1) x2 + y2 = 4; 3) x = 1.
B5B9 z = x+iy,w = u+iv,C3 w = 1z,BP z = 1w,DN
A8D9
x + iy = 1u + iv = uu2 + v2?i vu2 + v2.
BP
x = uu2 + v2,y =?vu2 + v2.
1) A8 x2 + y2 = 4 EE,CZ
x2 + y2 =
parenleftbigg u
u2 + v2
parenrightbigg2
+
parenleftbigg?v
u2 + v2
parenrightbigg2
= 4.
DEDF,BPAQ w EYEHH2ARE6ETDMF0,u2 + v2 = 14.
73/89
2) A8 x = 1 EE,C3CZ
x = uu2 + v2 = 1,
DEDF,AQ u = u2 + v2,BP
parenleftbigg
u? 12
parenrightbigg2
+ v2 = 14.
BP x = 1 BF w EYEHH2ARE6ETDM.
74/89
2) A8 x = 1 EE,C3CZ
x = uu2 + v2 = 1,
DEDF,AQ u = u2 + v2,BP
parenleftbigg
u? 12
parenrightbigg2
+ v2 = 14.
BP x = 1 BF w EYEHH2ARE6ETDM.
74/89
2) A8 x = 1 EE,C3CZ
x = uu2 + v2 = 1,
DEDF,AQ u = u2 + v2,BP
parenleftbigg
u? 12
parenrightbigg2
+ v2 = 14.
BP x = 1 BF w EYEHH2ARE6ETDM.
74/89
2) A8 x = 1 EE,C3CZ
x = uu2 + v2 = 1,
DEDF,AQ u = u2 + v2,BP
parenleftbigg
u? 12
parenrightbigg2
+ v2 = 14.
BP x = 1 BF w EYEHH2ARE6ETDM.
74/89
BDBKAX BXBVF8GIARBGDLG2GJCHBH
1,F8GIARBGDL
F9FE ANF8GI w = f(z) F9FEBF z0 ARAMB1AP
DX 0 < |z?z0| < ρ CB,CGC2CZE8B4F9ARGI A DJBF,GHD9
BRFBDTF9AR ε > 0,DJBF δ(ε) > 0(0 < δ lessorequalslant ρ),EMAQ
A8 0 < |z?z0| < δ EECZ
|f(x)?A| < ε,
BSD3EHA? f(z) A8 z EQE5D9 z0 ARBGDL,C8DU
limz→z
0
f(z) = A.
y
x O
z0
z f(z)
A ε δ
v
u O
75/89
BH 1,F9FEEL z EQE5D9 z0 ARE0EQF0BRFBAR,?FUF0CO,
FFG0 z D0EFD3E0E5,EXG3ERE0EQEQE5D9 z0,f(z) FTDREQE5
D9 A.
76/89
CTEHDJFGBXBVF8GIBGDLARC7DY.
F9D2AJCGC2 limz→z
0
f(z) = A,limz→z
0
g(z) = B,BSD3
1) limz→z
0
[f(z)±g(z)] = A±B;
2) limz→z
0
f(z)g(z) = AB;
3) limz→z
0
f(z)
g(z) =
A
B (B negationslash= 0)
77/89
CTEHDJFGBXBVF8GIBGDLARC7DY.
F9D2AJCGC2 limz→z
0
f(z) = A,limz→z
0
g(z) = B,BSD3
1) limz→z
0
[f(z)±g(z)] = A±B;
2) limz→z
0
f(z)g(z) = AB;
3) limz→z
0
f(z)
g(z) =
A
B (B negationslash= 0)
77/89
CTEHDJFGBXBVF8GIBGDLARC7DY.
F9D2AJCGC2 limz→z
0
f(z) = A,limz→z
0
g(z) = B,BSD3
1) limz→z
0
[f(z)±g(z)] = A±B;
2) limz→z
0
f(z)g(z) = AB;
3) limz→z
0
f(z)
g(z) =
A
B (B negationslash= 0)
77/89
CTEHDJFGBXBVF8GIBGDLARC7DY.
F9D2AJCGC2 limz→z
0
f(z) = A,limz→z
0
g(z) = B,BSD3
1) limz→z
0
[f(z)±g(z)] = A±B;
2) limz→z
0
f(z)g(z) = AB;
3) limz→z
0
f(z)
g(z) =
A
B (B negationslash= 0)
77/89
DI 1 EMBGDL limz→i z?iz(1 + z2).
B5FR?
z?i
z(1 + z2) =
z?i
z(z?i)(z + i) =
1
z(z + i),
EKEX
lim
z→i
z?i
z(1 + z2) = limz→i
1
z(z + i) =?
1
2.
78/89
DI 1 EMBGDL limz→i z?iz(1 + z2).
B5FR?
z?i
z(1 + z2) =
z?i
z(z?i)(z + i) =
1
z(z + i),
EKEX
lim
z→i
z?i
z(1 + z2) = limz→i
1
z(z + i) =?
1
2.
78/89
DI 1 EMBGDL limz→i z?iz(1 + z2).
B5FR?
z?i
z(1 + z2) =
z?i
z(z?i)(z + i) =
1
z(z + i),
EKEX
lim
z→i
z?i
z(1 + z2) = limz→i
1
z(z + i) =?
1
2.
78/89
DI 2 EMBGDL limz→1 zz + 2z?z? 2z2? 1,
B5FR?
zz + 2z?z? 2
z2? 1 =
(z + 2)(z?1)
(z + 1)(z?1) =
z + 2
z + 1,
EKEX
limz→1 zz + 2z?z? 2z2? 1 = limz→1 z + 2z + 1 = 32.
79/89
DI 2 EMBGDL limz→1 zz + 2z?z? 2z2? 1,
B5FR?
zz + 2z?z? 2
z2? 1 =
(z + 2)(z?1)
(z + 1)(z?1) =
z + 2
z + 1,
EKEX
limz→1 zz + 2z?z? 2z2? 1 = limz→1 z + 2z + 1 = 32.
79/89
DI 2 EMBGDL limz→1 zz + 2z?z? 2z2? 1,
B5FR?
zz + 2z?z? 2
z2? 1 =
(z + 2)(z?1)
(z + 1)(z?1) =
z + 2
z + 1,
EKEX
limz→1 zz + 2z?z? 2z2? 1 = limz→1 z + 2z + 1 = 32.
79/89
F9D2E8AN f(z) = u(x,y) + iv(x,y),A = u0 + iv0,
z0 = x0 + iy0,BSD3 limz→z
0
f(z) = A ARFBDRCMDRF0
lim
x→x0
y →y0
u(x,y) = u0,lim
x→x0
y →y0
v(x,y) = v0.
80/89
DD,BJDRBH.
CGC2 limz→z
0
f(z) = A,BSD3DUGABGDLARF9FE,FU
CZ?ε > 0,?δ > 0,A8 0 < |(x + iy)?(x0 + iy0)| < δ EE,
BPA8 0 <radicalbig(x?x0)2 + (y?y0)2 < δ EE,CZ
|(u + iv)? (u0 + iv0)| < ε.
BP
|(u?u0) + i(v?v0)| < ε.
C3 |u?u0| < ε,|v?v0| < ε.
D9F0
lim
x→x0
y →y0
u(x,y) = u0,lim
x→x0
y →y0
v(x,y) = v0.
81/89
DD,BJDRBH.
CGC2 limz→z
0
f(z) = A,BSD3DUGABGDLARF9FE,FU
CZ?ε > 0,?δ > 0,A8 0 < |(x + iy)?(x0 + iy0)| < δ EE,
BPA8 0 <radicalbig(x?x0)2 + (y?y0)2 < δ EE,CZ
|(u + iv)? (u0 + iv0)| < ε.
BP
|(u?u0) + i(v?v0)| < ε.
C3 |u?u0| < ε,|v?v0| < ε.
D9F0
lim
x→x0
y →y0
u(x,y) = u0,lim
x→x0
y →y0
v(x,y) = v0.
81/89
DD,BJDRBH.
CGC2 limz→z
0
f(z) = A,BSD3DUGABGDLARF9FE,FU
CZ?ε > 0,?δ > 0,A8 0 < |(x + iy)?(x0 + iy0)| < δ EE,
BPA8 0 <radicalbig(x?x0)2 + (y?y0)2 < δ EE,CZ
|(u + iv)? (u0 + iv0)| < ε.
BP
|(u?u0) + i(v?v0)| < ε.
C3 |u?u0| < ε,|v?v0| < ε.
D9F0
lim
x→x0
y →y0
u(x,y) = u0,lim
x→x0
y →y0
v(x,y) = v0.
81/89
DD,BJDRBH.
CGC2 limz→z
0
f(z) = A,BSD3DUGABGDLARF9FE,FU
CZ?ε > 0,?δ > 0,A8 0 < |(x + iy)?(x0 + iy0)| < δ EE,
BPA8 0 <radicalbig(x?x0)2 + (y?y0)2 < δ EE,CZ
|(u + iv)? (u0 + iv0)| < ε.
BP
|(u?u0) + i(v?v0)| < ε.
C3 |u?u0| < ε,|v?v0| < ε.
D9F0
lim
x→x0
y →y0
u(x,y) = u0,lim
x→x0
y →y0
v(x,y) = v0.
81/89
DD,BJDRBH.
CGC2 limz→z
0
f(z) = A,BSD3DUGABGDLARF9FE,FU
CZ?ε > 0,?δ > 0,A8 0 < |(x + iy)?(x0 + iy0)| < δ EE,
BPA8 0 <radicalbig(x?x0)2 + (y?y0)2 < δ EE,CZ
|(u + iv)? (u0 + iv0)| < ε.
BP
|(u?u0) + i(v?v0)| < ε.
C3 |u?u0| < ε,|v?v0| < ε.
D9F0
lim
x→x0
y →y0
u(x,y) = u0,lim
x→x0
y →y0
v(x,y) = v0.
81/89
FBEPBH.
CGC2H2EHH0EQEKG9,BSD3
A8 0 <radicalbig(x?x0)2 + (y?y0)2 < δ EE,CZ
|u?u0| < ε2,|v?v0| < ε2.
C3CZ
|f(z)?A| = |(u?u0)+i(v?v0)|lessorequalslant|u?u0|+|v?v0| < ε.
EKEX limz→z
0
f(z) = A.
82/89
FBEPBH.
CGC2H2EHH0EQEKG9,BSD3
A8 0 <radicalbig(x?x0)2 + (y?y0)2 < δ EE,CZ
|u?u0| < ε2,|v?v0| < ε2.
C3CZ
|f(z)?A| = |(u?u0)+i(v?v0)|lessorequalslant|u?u0|+|v?v0| < ε.
EKEX limz→z
0
f(z) = A.
82/89
FBEPBH.
CGC2H2EHH0EQEKG9,BSD3
A8 0 <radicalbig(x?x0)2 + (y?y0)2 < δ EE,CZ
|u?u0| < ε2,|v?v0| < ε2.
C3CZ
|f(z)?A| = |(u?u0)+i(v?v0)|lessorequalslant|u?u0|+|v?v0| < ε.
EKEX limz→z
0
f(z) = A.
82/89
FBEPBH.
CGC2H2EHH0EQEKG9,BSD3
A8 0 <radicalbig(x?x0)2 + (y?y0)2 < δ EE,CZ
|u?u0| < ε2,|v?v0| < ε2.
C3CZ
|f(z)?A| = |(u?u0)+i(v?v0)|lessorequalslant|u?u0|+|v?v0| < ε.
EKEX limz→z
0
f(z) = A.
82/89
DI 3 AN
f(z) = 12i(zz? zz),(z negationslash= 0).
FDDDA8 z → 0 EE f(z) ARBGDLG0DJBF.
DD B9 z = x + iy,f(z) = u(x,y) + iv(x,y),C3
u(x,y) + iv(x,y) = 12i
parenleftbiggx + iy
x?iy?
x?iy
x + iy
parenrightbigg
= 2xyx2 + y2
CSCSDWAQ,u(x,y) = 2xyx2 + y2,v(x,y) = 0.DUGAF9D2E8,
limz→0f(z) DJBF lim
x→0
y →0
u(x,y)G2 lim
x→0
y →0
v(x,y) FTDJBF.
83/89
DI 3 AN
f(z) = 12i(zz? zz),(z negationslash= 0).
FDDDA8 z → 0 EE f(z) ARBGDLG0DJBF.
DD B9 z = x + iy,f(z) = u(x,y) + iv(x,y),C3
u(x,y) + iv(x,y) = 12i
parenleftbiggx + iy
x?iy?
x?iy
x + iy
parenrightbigg
= 2xyx2 + y2
CSCSDWAQ,u(x,y) = 2xyx2 + y2,v(x,y) = 0.DUGAF9D2E8,
limz→0f(z) DJBF lim
x→0
y →0
u(x,y)G2 lim
x→0
y →0
v(x,y) FTDJBF.
83/89
DI 3 AN
f(z) = 12i(zz? zz),(z negationslash= 0).
FDDDA8 z → 0 EE f(z) ARBGDLG0DJBF.
DD B9 z = x + iy,f(z) = u(x,y) + iv(x,y),C3
u(x,y) + iv(x,y) = 12i
parenleftbiggx + iy
x?iy?
x?iy
x + iy
parenrightbigg
= 2xyx2 + y2
CSCSDWAQ,u(x,y) = 2xyx2 + y2,v(x,y) = 0.DUGAF9D2E8,
limz→0f(z) DJBF lim
x→0
y →0
u(x,y)G2 lim
x→0
y →0
v(x,y) FTDJBF.
83/89
DI 3 AN
f(z) = 12i(zz? zz),(z negationslash= 0).
FDDDA8 z → 0 EE f(z) ARBGDLG0DJBF.
DD B9 z = x + iy,f(z) = u(x,y) + iv(x,y),C3
u(x,y) + iv(x,y) = 12i
parenleftbiggx + iy
x?iy?
x?iy
x + iy
parenrightbigg
= 2xyx2 + y2
CSCSDWAQ,u(x,y) = 2xyx2 + y2,v(x,y) = 0.DUGAF9D2E8,
limz→0f(z) DJBF lim
x→0
y →0
u(x,y)G2 lim
x→0
y →0
v(x,y) FTDJBF.
83/89
A2F0,GHD9 lim
x→0
y →0
u(x,y),A8 z GVDQDM y = kx EQD9 0
EE,CZ
lim
x→0
y →0
y = kx
u(x,y) = lim
x→0
y →0
y = kx
2xy
x2 + y2 = limx→0
2x2
x2 + k2x2 =
2
1 + k2
AP lim
x→0
y →0
u(x,y) G0DJBF,EKEX limz→0f(z) G0DJBF.
84/89
A2F0,GHD9 lim
x→0
y →0
u(x,y),A8 z GVDQDM y = kx EQD9 0
EE,CZ
lim
x→0
y →0
y = kx
u(x,y) = lim
x→0
y →0
y = kx
2xy
x2 + y2 = limx→0
2x2
x2 + k2x2 =
2
1 + k2
AP lim
x→0
y →0
u(x,y) G0DJBF,EKEX limz→0f(z) G0DJBF.
84/89
A2F0,GHD9 lim
x→0
y →0
u(x,y),A8 z GVDQDM y = kx EQD9 0
EE,CZ
lim
x→0
y →0
y = kx
u(x,y) = lim
x→0
y →0
y = kx
2xy
x2 + y2 = limx→0
2x2
x2 + k2x2 =
2
1 + k2
AP lim
x→0
y →0
u(x,y) G0DJBF,EKEX limz→0f(z) G0DJBF.
84/89
DI 4 DDEYF8GI f(z) = Re(z)|z| A8 z → 0 EEARBGDLG0DJ
BF.
DD B9 z = x + iy,C3
f(z) = xradicalbigx2 + y2.
CSCSDWAQ,u(x,y) = xradicalbigx2 + y2,v(x,y) = 0.
DUGAF9D2E8,CZ
limz→0f(z) DJBF lim
x→0
y →0
u(x,y)G2 lim
x→0
y →0
v(x,y) FTDJBF.
85/89
DI 4 DDEYF8GI f(z) = Re(z)|z| A8 z → 0 EEARBGDLG0DJ
BF.
DD B9 z = x + iy,C3
f(z) = xradicalbigx2 + y2.
CSCSDWAQ,u(x,y) = xradicalbigx2 + y2,v(x,y) = 0.
DUGAF9D2E8,CZ
limz→0f(z) DJBF lim
x→0
y →0
u(x,y)G2 lim
x→0
y →0
v(x,y) FTDJBF.
85/89
DI 4 DDEYF8GI f(z) = Re(z)|z| A8 z → 0 EEARBGDLG0DJ
BF.
DD B9 z = x + iy,C3
f(z) = xradicalbigx2 + y2.
CSCSDWAQ,u(x,y) = xradicalbigx2 + y2,v(x,y) = 0.
DUGAF9D2E8,CZ
limz→0f(z) DJBF lim
x→0
y →0
u(x,y)G2 lim
x→0
y →0
v(x,y) FTDJBF.
85/89
DI 4 DDEYF8GI f(z) = Re(z)|z| A8 z → 0 EEARBGDLG0DJ
BF.
DD B9 z = x + iy,C3
f(z) = xradicalbigx2 + y2.
CSCSDWAQ,u(x,y) = xradicalbigx2 + y2,v(x,y) = 0.
DUGAF9D2E8,CZ
limz→0f(z) DJBF lim
x→0
y →0
u(x,y)G2 lim
x→0
y →0
v(x,y) FTDJBF.
85/89
A2F0,GHD9 lim
x→0
y →0
u(x,y),A8 z GVDQDM y = kx EQD9 0
EE,CZ
lim
x→0
y →0
y = kx
u(x,y) = lim
x→0
y →0
y = kx
xradicalbig
x2 + y2
= limx→0 xradicalbig(1 + k2)x2 = ± 1√1 + k2
AP lim
x→0
y →0
u(x,y) G0DJBF,EKEX limz→0f(z) G0DJBF.
86/89
A2F0,GHD9 lim
x→0
y →0
u(x,y),A8 z GVDQDM y = kx EQD9 0
EE,CZ
lim
x→0
y →0
y = kx
u(x,y) = lim
x→0
y →0
y = kx
xradicalbig
x2 + y2
= limx→0 xradicalbig(1 + k2)x2 = ± 1√1 + k2
AP lim
x→0
y →0
u(x,y) G0DJBF,EKEX limz→0f(z) G0DJBF.
86/89
A2F0,GHD9 lim
x→0
y →0
u(x,y),A8 z GVDQDM y = kx EQD9 0
EE,CZ
lim
x→0
y →0
y = kx
u(x,y) = lim
x→0
y →0
y = kx
xradicalbig
x2 + y2
= limx→0 xradicalbig(1 + k2)x2 = ± 1√1 + k2
AP lim
x→0
y →0
u(x,y) G0DJBF,EKEX limz→0f(z) G0DJBF.
86/89
2,F8GIARGJCHBH
F9FE CGC2 limz→z
0
f(z) = f(z0),BSD3EHf(z) BF z0 BRGJ
CH.
CGC2 f(z) BFERDX D CBBRBRGJCH,C3EHf(z) BF D CBGJ
CH.
F9D2D2 F8GIf(z) = u(x,y)+iv(x,y)BFz0 = x0 +iy0
BRGJCHARFBDRCMDRF0,u(x,y) G2 v(x,y) BF (x0,y0) GJCH.
F9D2D3 1) BF z0 GJCHARH0DRF8GI f(z),g(z) ARG2,AB,
FK,GZ (EPBDBF z0 G0?AY) BF z0 BRBXGJCH.
2) CGC2F8GI h = g(z) BF z GJCH,F8GI w = f(h)
BF h0 = g(z0) GJCH,BSD3BXG4F8GI w = f[g(z)] BF z0 BR
GJCH.
87/89
2,F8GIARGJCHBH
F9FE CGC2 limz→z
0
f(z) = f(z0),BSD3EHf(z) BF z0 BRGJ
CH.
CGC2 f(z) BFERDX D CBBRBRGJCH,C3EHf(z) BF D CBGJ
CH.
F9D2D2 F8GIf(z) = u(x,y)+iv(x,y)BFz0 = x0 +iy0
BRGJCHARFBDRCMDRF0,u(x,y) G2 v(x,y) BF (x0,y0) GJCH.
F9D2D3 1) BF z0 GJCHARH0DRF8GI f(z),g(z) ARG2,AB,
FK,GZ (EPBDBF z0 G0?AY) BF z0 BRBXGJCH.
2) CGC2F8GI h = g(z) BF z GJCH,F8GI w = f(h)
BF h0 = g(z0) GJCH,BSD3BXG4F8GI w = f[g(z)] BF z0 BR
GJCH.
87/89
2,F8GIARGJCHBH
F9FE CGC2 limz→z
0
f(z) = f(z0),BSD3EHf(z) BF z0 BRGJ
CH.
CGC2 f(z) BFERDX D CBBRBRGJCH,C3EHf(z) BF D CBGJ
CH.
F9D2D2 F8GIf(z) = u(x,y)+iv(x,y)BFz0 = x0 +iy0
BRGJCHARFBDRCMDRF0,u(x,y) G2 v(x,y) BF (x0,y0) GJCH.
F9D2D3 1) BF z0 GJCHARH0DRF8GI f(z),g(z) ARG2,AB,
FK,GZ (EPBDBF z0 G0?AY) BF z0 BRBXGJCH.
2) CGC2F8GI h = g(z) BF z GJCH,F8GI w = f(h)
BF h0 = g(z0) GJCH,BSD3BXG4F8GI w = f[g(z)] BF z0 BR
GJCH.
87/89
2,F8GIARGJCHBH
F9FE CGC2 limz→z
0
f(z) = f(z0),BSD3EHf(z) BF z0 BRGJ
CH.
CGC2 f(z) BFERDX D CBBRBRGJCH,C3EHf(z) BF D CBGJ
CH.
F9D2D2 F8GIf(z) = u(x,y)+iv(x,y)BFz0 = x0 +iy0
BRGJCHARFBDRCMDRF0,u(x,y) G2 v(x,y) BF (x0,y0) GJCH.
F9D2D3 1) BF z0 GJCHARH0DRF8GI f(z),g(z) ARG2,AB,
FK,GZ (EPBDBF z0 G0?AY) BF z0 BRBXGJCH.
2) CGC2F8GI h = g(z) BF z GJCH,F8GI w = f(h)
BF h0 = g(z0) GJCH,BSD3BXG4F8GI w = f[g(z)] BF z0 BR
GJCH.
87/89
CZBABGBKH2GJCHF8GIARBHEF
AN G?BXEYEHH2ARCZBABGBK,F8GI w = f(z) BF G H2
GJCH,C3
(1) F8GI f(z) BFBGBK G H2CZBA,BPDJBFDIGI M,EM
AQ?z ∈ G,FTCZ
|f(z)|lessorequalslant M.
BFBGETDMF9EDF5ETDMG8BMBFCBARETDMGBH2GJCHF8
GI f(z) BFETDMH2F0CZBAAR,BPDJBFDIGI M,BFETDMH2CZ
|f(z)|lessorequalslant M.
(2) F8GI |f(z)|BFBGBK G H2DSAKDLDWDUG2DLA2DU.
(3) F8GI f(z) BFBGBK G H2E8E7GJCH.
88/89
CZBABGBKH2GJCHF8GIARBHEF
AN G?BXEYEHH2ARCZBABGBK,F8GI w = f(z) BF G H2
GJCH,C3
(1) F8GI f(z) BFBGBK G H2CZBA,BPDJBFDIGI M,EM
AQ?z ∈ G,FTCZ
|f(z)|lessorequalslant M.
BFBGETDMF9EDF5ETDMG8BMBFCBARETDMGBH2GJCHF8
GI f(z) BFETDMH2F0CZBAAR,BPDJBFDIGI M,BFETDMH2CZ
|f(z)|lessorequalslant M.
(2) F8GI |f(z)|BFBGBK G H2DSAKDLDWDUG2DLA2DU.
(3) F8GI f(z) BFBGBK G H2E8E7GJCH.
88/89
CZBABGBKH2GJCHF8GIARBHEF
AN G?BXEYEHH2ARCZBABGBK,F8GI w = f(z) BF G H2
GJCH,C3
(1) F8GI f(z) BFBGBK G H2CZBA,BPDJBFDIGI M,EM
AQ?z ∈ G,FTCZ
|f(z)|lessorequalslant M.
BFBGETDMF9EDF5ETDMG8BMBFCBARETDMGBH2GJCHF8
GI f(z) BFETDMH2F0CZBAAR,BPDJBFDIGI M,BFETDMH2CZ
|f(z)|lessorequalslant M.
(2) F8GI |f(z)|BFBGBK G H2DSAKDLDWDUG2DLA2DU.
(3) F8GI f(z) BFBGBK G H2E8E7GJCH.
88/89
CZBABGBKH2GJCHF8GIARBHEF
AN G?BXEYEHH2ARCZBABGBK,F8GI w = f(z) BF G H2
GJCH,C3
(1) F8GI f(z) BFBGBK G H2CZBA,BPDJBFDIGI M,EM
AQ?z ∈ G,FTCZ
|f(z)|lessorequalslant M.
BFBGETDMF9EDF5ETDMG8BMBFCBARETDMGBH2GJCHF8
GI f(z) BFETDMH2F0CZBAAR,BPDJBFDIGI M,BFETDMH2CZ
|f(z)|lessorequalslant M.
(2) F8GI |f(z)|BFBGBK G H2DSAKDLDWDUG2DLA2DU.
(3) F8GI f(z) BFBGBK G H2E8E7GJCH.
88/89
CZBABGBKH2GJCHF8GIARBHEF
AN G?BXEYEHH2ARCZBABGBK,F8GI w = f(z) BF G H2
GJCH,C3
(1) F8GI f(z) BFBGBK G H2CZBA,BPDJBFDIGI M,EM
AQ?z ∈ G,FTCZ
|f(z)|lessorequalslant M.
BFBGETDMF9EDF5ETDMG8BMBFCBARETDMGBH2GJCHF8
GI f(z) BFETDMH2F0CZBAAR,BPDJBFDIGI M,BFETDMH2CZ
|f(z)|lessorequalslant M.
(2) F8GI |f(z)|BFBGBK G H2DSAKDLDWDUG2DLA2DU.
(3) F8GI f(z) BFBGBK G H2E8E7GJCH.
88/89
CZBABGBKH2GJCHF8GIARBHEF
AN G?BXEYEHH2ARCZBABGBK,F8GI w = f(z) BF G H2
GJCH,C3
(1) F8GI f(z) BFBGBK G H2CZBA,BPDJBFDIGI M,EM
AQ?z ∈ G,FTCZ
|f(z)|lessorequalslant M.
BFBGETDMF9EDF5ETDMG8BMBFCBARETDMGBH2GJCHF8
GI f(z) BFETDMH2F0CZBAAR,BPDJBFDIGI M,BFETDMH2CZ
|f(z)|lessorequalslant M.
(2) F8GI |f(z)|BFBGBK G H2DSAKDLDWDUG2DLA2DU.
(3) F8GI f(z) BFBGBK G H2E8E7GJCH.
88/89
32,(pp,34) FDDD argz BFA6BMDQC3EIF2H2G0GJCH.
DDB9 f(z) = argz,CSD9 f(z) BF z = 0 FFF9FE,EK
EX f(z) = argz BFA6BMG0GJCH.
FRpi < argz lessorequalslant pi,D9F0CZCTEHARB4G0.
(1) AN z0 negationslash= 0 F0C3EIF2H2ARBRFBE8BM,C3
CZ f(z0) = argz0 = pi.
(2) A8 z D0H2DWEYEHEQE5 z0 EE,f(z) = argz → pi.
(3) A8 z D0CTDWEYEHEQE5 z0 EE,f(z) = argz →?pi.
O
y
x
z0
AP limz→z
0
f(z) G0DJBF,f(z) BF z0 BMG0GJCH,EKEX argz BF
C3EIF2H2G0GJCH.
89/89
32,(pp,34) FDDD argz BFA6BMDQC3EIF2H2G0GJCH.
DDB9 f(z) = argz,CSD9 f(z) BF z = 0 FFF9FE,EK
EX f(z) = argz BFA6BMG0GJCH.
FRpi < argz lessorequalslant pi,D9F0CZCTEHARB4G0.
(1) AN z0 negationslash= 0 F0C3EIF2H2ARBRFBE8BM,C3
CZ f(z0) = argz0 = pi.
(2) A8 z D0H2DWEYEHEQE5 z0 EE,f(z) = argz → pi.
(3) A8 z D0CTDWEYEHEQE5 z0 EE,f(z) = argz →?pi.
O
y
x
z0
AP limz→z
0
f(z) G0DJBF,f(z) BF z0 BMG0GJCH,EKEX argz BF
C3EIF2H2G0GJCH.
89/89
32,(pp,34) FDDD argz BFA6BMDQC3EIF2H2G0GJCH.
DDB9 f(z) = argz,CSD9 f(z) BF z = 0 FFF9FE,EK
EX f(z) = argz BFA6BMG0GJCH.
FRpi < argz lessorequalslant pi,D9F0CZCTEHARB4G0.
(1) AN z0 negationslash= 0 F0C3EIF2H2ARBRFBE8BM,C3
CZ f(z0) = argz0 = pi.
(2) A8 z D0H2DWEYEHEQE5 z0 EE,f(z) = argz → pi.
(3) A8 z D0CTDWEYEHEQE5 z0 EE,f(z) = argz →?pi.
O
y
x
z0
AP limz→z
0
f(z) G0DJBF,f(z) BF z0 BMG0GJCH,EKEX argz BF
C3EIF2H2G0GJCH.
89/89
32,(pp,34) FDDD argz BFA6BMDQC3EIF2H2G0GJCH.
DDB9 f(z) = argz,CSD9 f(z) BF z = 0 FFF9FE,EK
EX f(z) = argz BFA6BMG0GJCH.
FRpi < argz lessorequalslant pi,D9F0CZCTEHARB4G0.
(1) AN z0 negationslash= 0 F0C3EIF2H2ARBRFBE8BM,C3
CZ f(z0) = argz0 = pi.
(2) A8 z D0H2DWEYEHEQE5 z0 EE,f(z) = argz → pi.
(3) A8 z D0CTDWEYEHEQE5 z0 EE,f(z) = argz →?pi.
O
y
x
z0
AP limz→z
0
f(z) G0DJBF,f(z) BF z0 BMG0GJCH,EKEX argz BF
C3EIF2H2G0GJCH.
89/89
32,(pp,34) FDDD argz BFA6BMDQC3EIF2H2G0GJCH.
DDB9 f(z) = argz,CSD9 f(z) BF z = 0 FFF9FE,EK
EX f(z) = argz BFA6BMG0GJCH.
FRpi < argz lessorequalslant pi,D9F0CZCTEHARB4G0.
(1) AN z0 negationslash= 0 F0C3EIF2H2ARBRFBE8BM,C3
CZ f(z0) = argz0 = pi.
(2) A8 z D0H2DWEYEHEQE5 z0 EE,f(z) = argz → pi.
(3) A8 z D0CTDWEYEHEQE5 z0 EE,f(z) = argz →?pi.
O
y
x
z0
AP limz→z
0
f(z) G0DJBF,f(z) BF z0 BMG0GJCH,EKEX argz BF
C3EIF2H2G0GJCH.
89/89
32,(pp,34) FDDD argz BFA6BMDQC3EIF2H2G0GJCH.
DDB9 f(z) = argz,CSD9 f(z) BF z = 0 FFF9FE,EK
EX f(z) = argz BFA6BMG0GJCH.
FRpi < argz lessorequalslant pi,D9F0CZCTEHARB4G0.
(1) AN z0 negationslash= 0 F0C3EIF2H2ARBRFBE8BM,C3
CZ f(z0) = argz0 = pi.
(2) A8 z D0H2DWEYEHEQE5 z0 EE,f(z) = argz → pi.
(3) A8 z D0CTDWEYEHEQE5 z0 EE,f(z) = argz →?pi.
O
y
x
z0
AP limz→z
0
f(z) G0DJBF,f(z) BF z0 BMG0GJCH,EKEX argz BF
C3EIF2H2G0GJCH.
89/89
32,(pp,34) FDDD argz BFA6BMDQC3EIF2H2G0GJCH.
DDB9 f(z) = argz,CSD9 f(z) BF z = 0 FFF9FE,EK
EX f(z) = argz BFA6BMG0GJCH.
FRpi < argz lessorequalslant pi,D9F0CZCTEHARB4G0.
(1) AN z0 negationslash= 0 F0C3EIF2H2ARBRFBE8BM,C3
CZ f(z0) = argz0 = pi.
(2) A8 z D0H2DWEYEHEQE5 z0 EE,f(z) = argz → pi.
(3) A8 z D0CTDWEYEHEQE5 z0 EE,f(z) = argz →?pi.
O
y
x
z0
AP limz→z
0
f(z) G0DJBF,f(z) BF z0 BMG0GJCH,EKEX argz BF
C3EIF2H2G0GJCH.
89/89