北京邮电大学：复变函数（2006年）：5H

分类：数学格式：pdf 日期：2009年05月06日

BDFMG1 BFGI
December 5,2006
1/107
FODRCBC6
1,BRG0B5G3F8GIARAHG9G3BMAREPCP.
2,BFCSFGBMH2,ETBMBRG0BFGIF9D2BLG1BTCM.
BTCMBFGIF9D2DWEXD6C7DYGVBGETDMARFKEPBPDE?C7DYBF
AHG9G3BMBRARBFGI.
3,BTCMBFGIF9D2DODWEXC7DYE8AAF9FKEPG2BEFEFKEP.
2/107
FODRCBC6
1,BRG0B5G3F8GIARAHG9G3BMAREPCP.
2,BFCSFGBMH2,ETBMBRG0BFGIF9D2BLG1BTCM.
BTCMBFGIF9D2DWEXD6C7DYGVBGETDMARFKEPBPDE?C7DYBF
AHG9G3BMBRARBFGI.
3,BTCMBFGIF9D2DODWEXC7DYE8AAF9FKEPG2BEFEFKEP.
2/107
FODRCBC6
1,BRG0B5G3F8GIARAHG9G3BMAREPCP.
2,BFCSFGBMH2,ETBMBRG0BFGIF9D2BLG1BTCM.
BTCMBFGIF9D2DWEXD6C7DYGVBGETDMARFKEPBPDE?C7DYBF
AHG9G3BMBRARBFGI.
3,BTCMBFGIF9D2DODWEXC7DYE8AAF9FKEPG2BEFEFKEP.
2/107
FODRCBC6
1,BRG0B5G3F8GIARAHG9G3BMAREPCP.
2,BFCSFGBMH2,ETBMBRG0BFGIF9D2BLG1BTCM.
BTCMBFGIF9D2DWEXD6C7DYGVBGETDMARFKEPBPDE?C7DYBF
AHG9G3BMBRARBFGI.
3,BTCMBFGIF9D2DODWEXC7DYE8AAF9FKEPG2BEFEFKEP.
2/107
BDE8AX AHG9G3BM
ATAXARELB1CBC6:
1,CZGHB5G3F8GIARAHG9G3BMBRBDEPCP;
2,BEGHB5G3F8GIBFAHG9G3BMAPDXCBARBHEFBRBDGNFG.
3/107
BDE8AX AHG9G3BM
ATAXARELB1CBC6:
1,CZGHB5G3F8GIARAHG9G3BMBRBDEPCP;
2,BEGHB5G3F8GIBFAHG9G3BMAPDXCBARBHEFBRBDGNFG.
3/107
BDE8AX AHG9G3BM
ATAXARELB1CBC6:
1,CZGHB5G3F8GIARAHG9G3BMBRBDEPCP;
2,BEGHB5G3F8GIBFAHG9G3BMAPDXCBARBHEFBRBDGNFG.
3/107
AHG9G3BM:
CGC2F8GI f(z) BF z0 G0B5G3,A2BF z0 ARB8E8AMB1AP
DX 0 < |z?z0| < δ CBBRBRB5G3,BSD3 z0 EH? f(z) ARAH
G9G3BM.
DICG,F8GI
1
z,e
1
z
FTEX z = 0?AHG9G3BM.
4/107
AHG9G3BM:
CGC2F8GI f(z) BF z0 G0B5G3,A2BF z0 ARB8E8AMB1AP
DX 0 < |z?z0| < δ CBBRBRB5G3,BSD3 z0 EH? f(z) ARAH
G9G3BM.
DICG,F8GI
1
z,e
1
z
FTEX z = 0?AHG9G3BM.
4/107
AHG9G3BM:
CGC2F8GI f(z) BF z0 G0B5G3,A2BF z0 ARB8E8AMB1AP
DX 0 < |z?z0| < δ CBBRBRB5G3,BSD3 z0 EH? f(z) ARAH
G9G3BM.
DICG,F8GI
1
z,e
1
z
FTEX z = 0?AHG9G3BM.
4/107
G0AHG9ARG3BM:
DICG,F8GI f(z) = 1sin1
z
ARG3BMCZ:
z = 0,z = 1npi(n = 0,±1,±2,···)
FR?
limn→∞ 1npi = 0,
D9F0,BF z = 0 ARBRFBA2ARAMB1APDXCBD3CZ f(z) ARG3BM
DJBF.
EKEX z = 0 G0F0 f(z) ARAHG9G3BM.
5/107
G0AHG9ARG3BM:
DICG,F8GI f(z) = 1sin1
z
ARG3BMCZ:
z = 0,z = 1npi(n = 0,±1,±2,···)
FR?
limn→∞ 1npi = 0,
D9F0,BF z = 0 ARBRFBA2ARAMB1APDXCBD3CZ f(z) ARG3BM
DJBF.
EKEX z = 0 G0F0 f(z) ARAHG9G3BM.
5/107
G0AHG9ARG3BM:
DICG,F8GI f(z) = 1sin1
z
ARG3BMCZ:
z = 0,z = 1npi(n = 0,±1,±2,···)
FR?
limn→∞ 1npi = 0,
D9F0,BF z = 0 ARBRFBA2ARAMB1APDXCBD3CZ f(z) ARG3BM
DJBF.
EKEX z = 0 G0F0 f(z) ARAHG9G3BM.
5/107
G0AHG9ARG3BM:
DICG,F8GI f(z) = 1sin1
z
ARG3BMCZ:
z = 0,z = 1npi(n = 0,±1,±2,···)
FR?
limn→∞ 1npi = 0,
D9F0,BF z = 0 ARBRFBA2ARAMB1APDXCBD3CZ f(z) ARG3BM
DJBF.
EKEX z = 0 G0F0 f(z) ARAHG9G3BM.
5/107
G0AHG9ARG3BM:
DICG,F8GI f(z) = 1sin1
z
ARG3BMCZ:
z = 0,z = 1npi(n = 0,±1,±2,···)
FR?
limn→∞ 1npi = 0,
D9F0,BF z = 0 ARBRFBA2ARAMB1APDXCBD3CZ f(z) ARG3BM
DJBF.
EKEX z = 0 G0F0 f(z) ARAHG9G3BM.
5/107
AHG9G3BMAREPCPC1C7:
D6 f(z) BFENARAHG9G3BM z0 ARAMB1APDXCBFSD1EKGAC3
BRGI,DUGAFSD1EQARG0FVECF5GQAHG9G3BMEPCP.
6/107
AHG9G3BMAREPCPC1C7:
D6 f(z) BFENARAHG9G3BM z0 ARAMB1APDXCBFSD1EKGAC3
BRGI,DUGAFSD1EQARG0FVECF5GQAHG9G3BMEPCP.
6/107
1,DWAMG3BM
F9FE,CGC2BFGAC3BRGIELG0F5 z?z0 ARC3E8E1,BP
f(z) = c0+c1(z?z0)+···+cn(z?z0)n+···,0 < |z?z0| < δ,
BSD3AHG9G3BM z0 EH?DWAMG3BM.
DICG,z = 0 F0 f(z) = sinzz ARAHG9G3BM,DPCCF8GI
BF z = 0 ARAMB1APDXCBARGAC3FSD1EQ?
sinz
z =
1
z
parenleftbigg
z? 13!z3 + 15!z5?···
parenrightbigg
= 1? 13!z2 + 15!z4?···,0 < |z| < +∞
EQELG0F5 z ARC3E8E1,F0DWAMG3BM.
7/107
1,DWAMG3BM
F9FE,CGC2BFGAC3BRGIELG0F5 z?z0 ARC3E8E1,BP
f(z) = c0+c1(z?z0)+···+cn(z?z0)n+···,0 < |z?z0| < δ,
BSD3AHG9G3BM z0 EH?DWAMG3BM.
DICG,z = 0 F0 f(z) = sinzz ARAHG9G3BM,DPCCF8GI
BF z = 0 ARAMB1APDXCBARGAC3FSD1EQ?
sinz
z =
1
z
parenleftbigg
z? 13!z3 + 15!z5?···
parenrightbigg
= 1? 13!z2 + 15!z4?···,0 < |z| < +∞
EQELG0F5 z ARC3E8E1,F0DWAMG3BM.
7/107
1,DWAMG3BM
F9FE,CGC2BFGAC3BRGIELG0F5 z?z0 ARC3E8E1,BP
f(z) = c0+c1(z?z0)+···+cn(z?z0)n+···,0 < |z?z0| < δ,
BSD3AHG9G3BM z0 EH?DWAMG3BM.
DICG,z = 0 F0 f(z) = sinzz ARAHG9G3BM,DPCCF8GI
BF z = 0 ARAMB1APDXCBARGAC3FSD1EQ?
sinz
z =
1
z
parenleftbigg
z? 13!z3 + 15!z5?···
parenrightbigg
= 1? 13!z2 + 15!z4?···,0 < |z| < +∞
EQELG0F5 z ARC3E8E1,F0DWAMG3BM.
7/107
1,DWAMG3BM
F9FE,CGC2BFGAC3BRGIELG0F5 z?z0 ARC3E8E1,BP
f(z) = c0+c1(z?z0)+···+cn(z?z0)n+···,0 < |z?z0| < δ,
BSD3AHG9G3BM z0 EH?DWAMG3BM.
DICG,z = 0 F0 f(z) = sinzz ARAHG9G3BM,DPCCF8GI
BF z = 0 ARAMB1APDXCBARGAC3FSD1EQ?
sinz
z =
1
z
parenleftbigg
z? 13!z3 + 15!z5?···
parenrightbigg
= 1? 13!z2 + 15!z4?···,0 < |z| < +∞
EQELG0F5 z ARC3E8E1,F0DWAMG3BM.
7/107
1,DWAMG3BM
F9FE,CGC2BFGAC3BRGIELG0F5 z?z0 ARC3E8E1,BP
f(z) = c0+c1(z?z0)+···+cn(z?z0)n+···,0 < |z?z0| < δ,
BSD3AHG9G3BM z0 EH?DWAMG3BM.
DICG,z = 0 F0 f(z) = sinzz ARAHG9G3BM,DPCCF8GI
BF z = 0 ARAMB1APDXCBARGAC3FSD1EQ?
sinz
z =
1
z
parenleftbigg
z? 13!z3 + 15!z5?···
parenrightbigg
= 1? 13!z2 + 15!z4?···,0 < |z| < +∞
EQELG0F5 z ARC3E8E1,F0DWAMG3BM.
7/107
1,DWAMG3BM
F9FE,CGC2BFGAC3BRGIELG0F5 z?z0 ARC3E8E1,BP
f(z) = c0+c1(z?z0)+···+cn(z?z0)n+···,0 < |z?z0| < δ,
BSD3AHG9G3BM z0 EH?DWAMG3BM.
DICG,z = 0 F0 f(z) = sinzz ARAHG9G3BM,DPCCF8GI
BF z = 0 ARAMB1APDXCBARGAC3FSD1EQ?
sinz
z =
1
z
parenleftbigg
z? 13!z3 + 15!z5?···
parenrightbigg
= 1? 13!z2 + 15!z4?···,0 < |z| < +∞
EQELG0F5 z ARC3E8E1,F0DWAMG3BM.
7/107
1,DWAMG3BM
F9FE,CGC2BFGAC3BRGIELG0F5 z?z0 ARC3E8E1,BP
f(z) = c0+c1(z?z0)+···+cn(z?z0)n+···,0 < |z?z0| < δ,
BSD3AHG9G3BM z0 EH?DWAMG3BM.
DICG,z = 0 F0 f(z) = sinzz ARAHG9G3BM,DPCCF8GI
BF z = 0 ARAMB1APDXCBARGAC3FSD1EQ?
sinz
z =
1
z
parenleftbigg
z? 13!z3 + 15!z5?···
parenrightbigg
= 1? 13!z2 + 15!z4?···,0 < |z| < +∞
EQELG0F5 z ARC3E8E1,F0DWAMG3BM.
7/107
BH,DBB8 limz→0 sinzz = 1.
FR? f(z) = sinzz BF z = 0 BMFFF9FE,C3E0DRETAZF9FE
F8GIBF z = 0 BRARDU,EMAQ
f(0) = limz→0 sinzz = 1
G3BMFUDWAM,D9F0 f(z) = sinzz FUBF z = 0 B5G3AA.
8/107
BH,DBB8 limz→0 sinzz = 1.
FR? f(z) = sinzz BF z = 0 BMFFF9FE,C3E0DRETAZF9FE
F8GIBF z = 0 BRARDU,EMAQ
f(0) = limz→0 sinzz = 1
G3BMFUDWAM,D9F0 f(z) = sinzz FUBF z = 0 B5G3AA.
8/107
BH,DBB8 limz→0 sinzz = 1.
FR? f(z) = sinzz BF z = 0 BMFFF9FE,C3E0DRETAZF9FE
F8GIBF z = 0 BRARDU,EMAQ
f(0) = limz→0 sinzz = 1
G3BMFUDWAM,D9F0 f(z) = sinzz FUBF z = 0 B5G3AA.
8/107
BH,DBB8 limz→0 sinzz = 1.
FR? f(z) = sinzz BF z = 0 BMFFF9FE,C3E0DRETAZF9FE
F8GIBF z = 0 BRARDU,EMAQ
f(0) = limz→0 sinzz = 1
G3BMFUDWAM,D9F0 f(z) = sinzz FUBF z = 0 B5G3AA.
8/107
BH,DBB8 limz→0 sinzz = 1.
FR? f(z) = sinzz BF z = 0 BMFFF9FE,C3E0DRETAZF9FE
F8GIBF z = 0 BRARDU,EMAQ
f(0) = limz→0 sinzz = 1
G3BMFUDWAM,D9F0 f(z) = sinzz FUBF z = 0 B5G3AA.
8/107
DWAMG3BMARBHEF:
1,z = z0 F0 f(z) ARDWAMG3BM limz→z
0
f(z) DJBF.
DI 1,z = 0 F0 f(z) = sinzz ARAHG9G3BM,FR?
limz→0 sinzz = 1,
EKEX z = 0 F0DWAMG3BM.
9/107
DWAMG3BMARBHEF:
1,z = z0 F0 f(z) ARDWAMG3BM limz→z
0
f(z) DJBF.
DI 1,z = 0 F0 f(z) = sinzz ARAHG9G3BM,FR?
limz→0 sinzz = 1,
EKEX z = 0 F0DWAMG3BM.
9/107
DWAMG3BMARBHEF:
1,z = z0 F0 f(z) ARDWAMG3BM limz→z
0
f(z) DJBF.
DI 1,z = 0 F0 f(z) = sinzz ARAHG9G3BM,FR?
limz→0 sinzz = 1,
EKEX z = 0 F0DWAMG3BM.
9/107
DWAMG3BMARBHEF:
1,z = z0 F0 f(z) ARDWAMG3BM limz→z
0
f(z) DJBF.
DI 1,z = 0 F0 f(z) = sinzz ARAHG9G3BM,FR?
limz→0 sinzz = 1,
EKEX z = 0 F0DWAMG3BM.
9/107
2,BGBM
F9FE,CGC2BFGAC3BRGIELE0CZCZDLGTDR z?z0 ARC3E8
E1,DPG1ELB3D9 (z?z0)
1
ARDLD4E1? (z?z0)?m,BP
f(z) = c?m(z?z0)?m +···+ c?2(z?z0)?2 + c?1(z?z0)?1
+c0 + c1(z?z0) +···,(m greaterorequalslant 1,c?m negationslash= 0)
BSD3 z0 EH?F8GI f(z) AR m BRBRBGBGBM.
10/107
2,BGBM
F9FE,CGC2BFGAC3BRGIELE0CZCZDLGTDR z?z0 ARC3E8
E1,DPG1ELB3D9 (z?z0)
1
ARDLD4E1? (z?z0)?m,BP
f(z) = c?m(z?z0)?m +···+ c?2(z?z0)?2 + c?1(z?z0)?1
+c0 + c1(z?z0) +···,(m greaterorequalslant 1,c?m negationslash= 0)
BSD3 z0 EH?F8GI f(z) AR m BRBRBGBGBM.
10/107
2,BGBM
F9FE,CGC2BFGAC3BRGIELE0CZCZDLGTDR z?z0 ARC3E8
E1,DPG1ELB3D9 (z?z0)
1
ARDLD4E1? (z?z0)?m,BP
f(z) = c?m(z?z0)?m +···+ c?2(z?z0)?2 + c?1(z?z0)?1
+c0 + c1(z?z0) +···,(m greaterorequalslant 1,c?m negationslash= 0)
BSD3 z0 EH?F8GI f(z) AR m BRBRBGBGBM.
10/107
2,BGBM
F9FE,CGC2BFGAC3BRGIELE0CZCZDLGTDR z?z0 ARC3E8
E1,DPG1ELB3D9 (z?z0)
1
ARDLD4E1? (z?z0)?m,BP
f(z) = c?m(z?z0)?m +···+ c?2(z?z0)?2 + c?1(z?z0)?1
+c0 + c1(z?z0) +···,(m greaterorequalslant 1,c?m negationslash= 0)
BSD3 z0 EH?F8GI f(z) AR m BRBRBGBGBM.
10/107
2,BGBM
F9FE,CGC2BFGAC3BRGIELE0CZCZDLGTDR z?z0 ARC3E8
E1,DPG1ELB3D9 (z?z0)
1
ARDLD4E1? (z?z0)?m,BP
f(z) = c?m(z?z0)?m +···+ c?2(z?z0)?2 + c?1(z?z0)?1
+c0 + c1(z?z0) +···,(m greaterorequalslant 1,c?m negationslash= 0)
BSD3 z0 EH?F8GI f(z) AR m BRBRBGBGBM.
10/107
BGBMARBHEF
BHEF 1 DUGA m BRBGBMARGAC3FSD1EQ,CZ
f(z) = 1(z?z
0)m
g(z),(1)
G1EL,
g(z) = c?m + c?m+1(z?z0) + c?m+2(z?z0)2 + ···
BF |z?z0| < δ CBF0B5G3ARF8GI,DP g(z0) = c?m negationslash= 0.
DRDN,A8BRG3E8DRF8GI f(z) CDC4ER? (1) ARBBEQ,
DP g(z0) negationslash= 0,BSD3 z0 F0 f(z) AR m BRBGBM.
BHEF 2,z0 F0 f(z) ARBGBM limz→z
0
f(z) = ∞.
11/107
BGBMARBHEF
BHEF 1 DUGA m BRBGBMARGAC3FSD1EQ,CZ
f(z) = 1(z?z
0)m
g(z),(1)
G1EL,
g(z) = c?m + c?m+1(z?z0) + c?m+2(z?z0)2 + ···
BF |z?z0| < δ CBF0B5G3ARF8GI,DP g(z0) = c?m negationslash= 0.
DRDN,A8BRG3E8DRF8GI f(z) CDC4ER? (1) ARBBEQ,
DP g(z0) negationslash= 0,BSD3 z0 F0 f(z) AR m BRBGBM.
BHEF 2,z0 F0 f(z) ARBGBM limz→z
0
f(z) = ∞.
11/107
BGBMARBHEF
BHEF 1 DUGA m BRBGBMARGAC3FSD1EQ,CZ
f(z) = 1(z?z
0)m
g(z),(1)
G1EL,
g(z) = c?m + c?m+1(z?z0) + c?m+2(z?z0)2 + ···
BF |z?z0| < δ CBF0B5G3ARF8GI,DP g(z0) = c?m negationslash= 0.
DRDN,A8BRG3E8DRF8GI f(z) CDC4ER? (1) ARBBEQ,
DP g(z0) negationslash= 0,BSD3 z0 F0 f(z) AR m BRBGBM.
BHEF 2,z0 F0 f(z) ARBGBM limz→z
0
f(z) = ∞.
11/107
BGBMARBHEF
BHEF 1 DUGA m BRBGBMARGAC3FSD1EQ,CZ
f(z) = 1(z?z
0)m
g(z),(1)
G1EL,
g(z) = c?m + c?m+1(z?z0) + c?m+2(z?z0)2 + ···
BF |z?z0| < δ CBF0B5G3ARF8GI,DP g(z0) = c?m negationslash= 0.
DRDN,A8BRG3E8DRF8GI f(z) CDC4ER? (1) ARBBEQ,
DP g(z0) negationslash= 0,BSD3 z0 F0 f(z) AR m BRBGBM.
BHEF 2,z0 F0 f(z) ARBGBM limz→z
0
f(z) = ∞.
11/107
DI 1 F8GI f(z) = z?2(z2 + 1)(z?1)3 CZAAEFD3G3BM,CGC2
F0BGBM,DXFTENARBR.
B5 FR?
f(z) = 1(z?1)3 · z?2z2 + 1
FR? g(z) = z?2z2 + 1 BF z = 1 B5G3,DP g(1) =?12 negationslash= 0,
EKEX z = 1 F0ENARE8DRD2BRBGBM.
FVD2CZ
f(z) = 1z?i · z?2(z2 + 1)(z + i),f(z) = 1z + i · z? 2(z2 + 1)(z?i)
EKEX z = ±i F0ENARE8DRE8BRBGBM.
12/107
DI 1 F8GI f(z) = z?2(z2 + 1)(z?1)3 CZAAEFD3G3BM,CGC2
F0BGBM,DXFTENARBR.
B5 FR?
f(z) = 1(z?1)3 · z?2z2 + 1
FR? g(z) = z?2z2 + 1 BF z = 1 B5G3,DP g(1) =?12 negationslash= 0,
EKEX z = 1 F0ENARE8DRD2BRBGBM.
FVD2CZ
f(z) = 1z?i · z?2(z2 + 1)(z + i),f(z) = 1z + i · z? 2(z2 + 1)(z?i)
EKEX z = ±i F0ENARE8DRE8BRBGBM.
12/107
DI 1 F8GI f(z) = z?2(z2 + 1)(z?1)3 CZAAEFD3G3BM,CGC2
F0BGBM,DXFTENARBR.
B5 FR?
f(z) = 1(z?1)3 · z?2z2 + 1
FR? g(z) = z?2z2 + 1 BF z = 1 B5G3,DP g(1) =?12 negationslash= 0,
EKEX z = 1 F0ENARE8DRD2BRBGBM.
FVD2CZ
f(z) = 1z?i · z?2(z2 + 1)(z + i),f(z) = 1z + i · z? 2(z2 + 1)(z?i)
EKEX z = ±i F0ENARE8DRE8BRBGBM.
12/107
DI 1 F8GI f(z) = z?2(z2 + 1)(z?1)3 CZAAEFD3G3BM,CGC2
F0BGBM,DXFTENARBR.
B5 FR?
f(z) = 1(z?1)3 · z?2z2 + 1
FR? g(z) = z?2z2 + 1 BF z = 1 B5G3,DP g(1) =?12 negationslash= 0,
EKEX z = 1 F0ENARE8DRD2BRBGBM.
FVD2CZ
f(z) = 1z?i · z?2(z2 + 1)(z + i),f(z) = 1z + i · z? 2(z2 + 1)(z?i)
EKEX z = ±i F0ENARE8DRE8BRBGBM.
12/107
DI 1 F8GI f(z) = z?2(z2 + 1)(z?1)3 CZAAEFD3G3BM,CGC2
F0BGBM,DXFTENARBR.
B5 FR?
f(z) = 1(z?1)3 · z?2z2 + 1
FR? g(z) = z?2z2 + 1 BF z = 1 B5G3,DP g(1) =?12 negationslash= 0,
EKEX z = 1 F0ENARE8DRD2BRBGBM.
FVD2CZ
f(z) = 1z?i · z?2(z2 + 1)(z + i),f(z) = 1z + i · z? 2(z2 + 1)(z?i)
EKEX z = ±i F0ENARE8DRE8BRBGBM.
12/107
3,ATBHG3BM
CGC2BFGAC3BRGIELF5CZFFEHGTDR z?z0 ARC3E8E1,BSD3
AHG9G3BMEH? f(z) ARATBHG3BM,BP
f(z) = ··· + c?n(z?z0)?n + ··· + c?1(z?z0)?1
+c0 + c1(z?z0) +···,0 < |z?z0| < δ
DI 1 F8GI f(z) = e1z BF z = 0 ARAPDXCBARARGAC3BRGI
e1z = 1 + z?1 + 12!z?2 +··· + 1n!z?n + ···
EKEX z = 0 F0ENARATBHG3BM.
13/107
3,ATBHG3BM
CGC2BFGAC3BRGIELF5CZFFEHGTDR z?z0 ARC3E8E1,BSD3
AHG9G3BMEH? f(z) ARATBHG3BM,BP
f(z) = ··· + c?n(z?z0)?n + ··· + c?1(z?z0)?1
+c0 + c1(z?z0) +···,0 < |z?z0| < δ
DI 1 F8GI f(z) = e1z BF z = 0 ARAPDXCBARARGAC3BRGI
e1z = 1 + z?1 + 12!z?2 +··· + 1n!z?n + ···
EKEX z = 0 F0ENARATBHG3BM.
13/107
3,ATBHG3BM
CGC2BFGAC3BRGIELF5CZFFEHGTDR z?z0 ARC3E8E1,BSD3
AHG9G3BMEH? f(z) ARATBHG3BM,BP
f(z) = ··· + c?n(z?z0)?n + ··· + c?1(z?z0)?1
+c0 + c1(z?z0) +···,0 < |z?z0| < δ
DI 1 F8GI f(z) = e1z BF z = 0 ARAPDXCBARARGAC3BRGI
e1z = 1 + z?1 + 12!z?2 +··· + 1n!z?n + ···
EKEX z = 0 F0ENARATBHG3BM.
13/107
3,ATBHG3BM
CGC2BFGAC3BRGIELF5CZFFEHGTDR z?z0 ARC3E8E1,BSD3
AHG9G3BMEH? f(z) ARATBHG3BM,BP
f(z) = ··· + c?n(z?z0)?n + ··· + c?1(z?z0)?1
+c0 + c1(z?z0) +···,0 < |z?z0| < δ
DI 1 F8GI f(z) = e1z BF z = 0 ARAPDXCBARARGAC3BRGI
e1z = 1 + z?1 + 12!z?2 +··· + 1n!z?n + ···
EKEX z = 0 F0ENARATBHG3BM.
13/107
3,ATBHG3BM
CGC2BFGAC3BRGIELF5CZFFEHGTDR z?z0 ARC3E8E1,BSD3
AHG9G3BMEH? f(z) ARATBHG3BM,BP
f(z) = ··· + c?n(z?z0)?n + ··· + c?1(z?z0)?1
+c0 + c1(z?z0) +···,0 < |z?z0| < δ
DI 1 F8GI f(z) = e1z BF z = 0 ARAPDXCBARARGAC3BRGI
e1z = 1 + z?1 + 12!z?2 +··· + 1n!z?n + ···
EKEX z = 0 F0ENARATBHG3BM.
13/107
ATBHG3BMARBHEF:
BHEF 1,CGC2 z0? f(z) ARATBHG3BM limz→z
0
f(z) G0
DJBFDPG0? ∞.
DI 1 BM z = 0 F0 f(z) = e1z ARAHG9G3BM,DPA8 z GV x
F2AKE0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0+
e1x = +∞,
A8 z GV x F2C3E0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0?
e1x = 0,
EKEX limz→0e1z G0DJBF,D9F0 z = 0 F0ENARATBHG3BM.
14/107
ATBHG3BMARBHEF:
BHEF 1,CGC2 z0? f(z) ARATBHG3BM limz→z
0
f(z) G0
DJBFDPG0? ∞.
DI 1 BM z = 0 F0 f(z) = e1z ARAHG9G3BM,DPA8 z GV x
F2AKE0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0+
e1x = +∞,
A8 z GV x F2C3E0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0?
e1x = 0,
EKEX limz→0e1z G0DJBF,D9F0 z = 0 F0ENARATBHG3BM.
14/107
ATBHG3BMARBHEF:
BHEF 1,CGC2 z0? f(z) ARATBHG3BM limz→z
0
f(z) G0
DJBFDPG0? ∞.
DI 1 BM z = 0 F0 f(z) = e1z ARAHG9G3BM,DPA8 z GV x
F2AKE0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0+
e1x = +∞,
A8 z GV x F2C3E0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0?
e1x = 0,
EKEX limz→0e1z G0DJBF,D9F0 z = 0 F0ENARATBHG3BM.
14/107
ATBHG3BMARBHEF:
BHEF 1,CGC2 z0? f(z) ARATBHG3BM limz→z
0
f(z) G0
DJBFDPG0? ∞.
DI 1 BM z = 0 F0 f(z) = e1z ARAHG9G3BM,DPA8 z GV x
F2AKE0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0+
e1x = +∞,
A8 z GV x F2C3E0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0?
e1x = 0,
EKEX limz→0e1z G0DJBF,D9F0 z = 0 F0ENARATBHG3BM.
14/107
ATBHG3BMARBHEF:
BHEF 1,CGC2 z0? f(z) ARATBHG3BM limz→z
0
f(z) G0
DJBFDPG0? ∞.
DI 1 BM z = 0 F0 f(z) = e1z ARAHG9G3BM,DPA8 z GV x
F2AKE0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0+
e1x = +∞,
A8 z GV x F2C3E0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0?
e1x = 0,
EKEX limz→0e1z G0DJBF,D9F0 z = 0 F0ENARATBHG3BM.
14/107
ATBHG3BMARBHEF:
BHEF 1,CGC2 z0? f(z) ARATBHG3BM limz→z
0
f(z) G0
DJBFDPG0? ∞.
DI 1 BM z = 0 F0 f(z) = e1z ARAHG9G3BM,DPA8 z GV x
F2AKE0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0+
e1x = +∞,
A8 z GV x F2C3E0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0?
e1x = 0,
EKEX limz→0e1z G0DJBF,D9F0 z = 0 F0ENARATBHG3BM.
14/107
ATBHG3BMARBHEF:
BHEF 1,CGC2 z0? f(z) ARATBHG3BM limz→z
0
f(z) G0
DJBFDPG0? ∞.
DI 1 BM z = 0 F0 f(z) = e1z ARAHG9G3BM,DPA8 z GV x
F2AKE0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0+
e1x = +∞,
A8 z GV x F2C3E0E5EQD9 0 EE,CZ
limz→0e1z = lim
x→0?
e1x = 0,
EKEX limz→0e1z G0DJBF,D9F0 z = 0 F0ENARATBHG3BM.
14/107
BHEF 2,CGC2 z0?F8GI f(z) ARATBHG3BM,BSD3GHD9BR
FBDTF9ARBXGI A(E6GTDWCDG0AME8DRDU),DJBFGIAF {zn},
DP limn→∞zn = z0,GVDA
limn→∞f(zn) = A.
DI 1 EUDJ z = 0 F0F8GI f(z) = e1z ARATBHG3BM,GHD9
DEDRCZEHBXGI A negationslash= 0,CTAI
zn = 1lnA + 2npii → 0,
C3CZ
f(zn) = elnA+2npii = elnA = A.
DBB8,limn→∞f(zn) = A.
15/107
BHEF 2,CGC2 z0?F8GI f(z) ARATBHG3BM,BSD3GHD9BR
FBDTF9ARBXGI A(E6GTDWCDG0AME8DRDU),DJBFGIAF {zn},
DP limn→∞zn = z0,GVDA
limn→∞f(zn) = A.
DI 1 EUDJ z = 0 F0F8GI f(z) = e1z ARATBHG3BM,GHD9
DEDRCZEHBXGI A negationslash= 0,CTAI
zn = 1lnA + 2npii → 0,
C3CZ
f(zn) = elnA+2npii = elnA = A.
DBB8,limn→∞f(zn) = A.
15/107
BHEF 2,CGC2 z0?F8GI f(z) ARATBHG3BM,BSD3GHD9BR
FBDTF9ARBXGI A(E6GTDWCDG0AME8DRDU),DJBFGIAF {zn},
DP limn→∞zn = z0,GVDA
limn→∞f(zn) = A.
DI 1 EUDJ z = 0 F0F8GI f(z) = e1z ARATBHG3BM,GHD9
DEDRCZEHBXGI A negationslash= 0,CTAI
zn = 1lnA + 2npii → 0,
C3CZ
f(zn) = elnA+2npii = elnA = A.
DBB8,limn→∞f(zn) = A.
15/107
BHEF 2,CGC2 z0?F8GI f(z) ARATBHG3BM,BSD3GHD9BR
FBDTF9ARBXGI A(E6GTDWCDG0AME8DRDU),DJBFGIAF {zn},
DP limn→∞zn = z0,GVDA
limn→∞f(zn) = A.
DI 1 EUDJ z = 0 F0F8GI f(z) = e1z ARATBHG3BM,GHD9
DEDRCZEHBXGI A negationslash= 0,CTAI
zn = 1lnA + 2npii → 0,
C3CZ
f(zn) = elnA+2npii = elnA = A.
DBB8,limn→∞f(zn) = A.
15/107
BHEF 2,CGC2 z0?F8GI f(z) ARATBHG3BM,BSD3GHD9BR
FBDTF9ARBXGI A(E6GTDWCDG0AME8DRDU),DJBFGIAF {zn},
DP limn→∞zn = z0,GVDA
limn→∞f(zn) = A.
DI 1 EUDJ z = 0 F0F8GI f(z) = e1z ARATBHG3BM,GHD9
DEDRCZEHBXGI A negationslash= 0,CTAI
zn = 1lnA + 2npii → 0,
C3CZ
f(zn) = elnA+2npii = elnA = A.
DBB8,limn→∞f(zn) = A.
15/107
4,F8GIARAYBMDQBGBMARB3CG
m BRAYBM:
G0GOAVD9AYARB5G3F8GI f(z) CGC2CDC4EREK
f(z) = (z?z0)m?(z),
G1EL,?(z) BFBM z0 B5G3FBDP?(z0) negationslash= 0,m?B8E8AKAIGI,
BSD3 z0 EH? f(z) ARm BRAYBM.
16/107
4,F8GIARAYBMDQBGBMARB3CG
m BRAYBM:
G0GOAVD9AYARB5G3F8GI f(z) CGC2CDC4EREK
f(z) = (z?z0)m?(z),
G1EL,?(z) BFBM z0 B5G3FBDP?(z0) negationslash= 0,m?B8E8AKAIGI,
BSD3 z0 EH? f(z) ARm BRAYBM.
16/107
4,F8GIARAYBMDQBGBMARB3CG
m BRAYBM:
G0GOAVD9AYARB5G3F8GI f(z) CGC2CDC4EREK
f(z) = (z?z0)m?(z),
G1EL,?(z) BFBM z0 B5G3FBDP?(z0) negationslash= 0,m?B8E8AKAIGI,
BSD3 z0 EH? f(z) ARm BRAYBM.
16/107
m BRAYBMARBHEF
F9D2CGC2 f(z) BF z0 B5G3,BSD3 z0? f(z) AR m BRAY
BMARFBDRCMDRF0
f(n)(z0) = 0(n = 0,1,2,···,m?1),f(m)(z0) negationslash= 0.
DD BJDRBH AN?(z) BF z0 ARAICEFSD1EQ?
(z) = c0 + c1(z?z0) + c2(z?z0)2 + ···,
G1EL,c0 negationslash= 0.
D9F0 f(z) BF z0 ARAICEFSD1EQ?
f(z) = c0(z?z0)m +c1(z?z0)m+1 +c2(z?z0)m+2 +···,
C3f(n)(z0) = 0(n = 0,1,2,···,m?1),f
(m)(z0)
m! = c0 negationslash= 0.
17/107
m BRAYBMARBHEF
F9D2CGC2 f(z) BF z0 B5G3,BSD3 z0? f(z) AR m BRAY
BMARFBDRCMDRF0
f(n)(z0) = 0(n = 0,1,2,···,m?1),f(m)(z0) negationslash= 0.
DD BJDRBH AN?(z) BF z0 ARAICEFSD1EQ?
(z) = c0 + c1(z?z0) + c2(z?z0)2 + ···,
G1EL,c0 negationslash= 0.
D9F0 f(z) BF z0 ARAICEFSD1EQ?
f(z) = c0(z?z0)m +c1(z?z0)m+1 +c2(z?z0)m+2 +···,
C3f(n)(z0) = 0(n = 0,1,2,···,m?1),f
(m)(z0)
m! = c0 negationslash= 0.
17/107
m BRAYBMARBHEF
F9D2CGC2 f(z) BF z0 B5G3,BSD3 z0? f(z) AR m BRAY
BMARFBDRCMDRF0
f(n)(z0) = 0(n = 0,1,2,···,m?1),f(m)(z0) negationslash= 0.
DD BJDRBH AN?(z) BF z0 ARAICEFSD1EQ?
(z) = c0 + c1(z?z0) + c2(z?z0)2 + ···,
G1EL,c0 negationslash= 0.
D9F0 f(z) BF z0 ARAICEFSD1EQ?
f(z) = c0(z?z0)m +c1(z?z0)m+1 +c2(z?z0)m+2 +···,
C3f(n)(z0) = 0(n = 0,1,2,···,m?1),f
(m)(z0)
m! = c0 negationslash= 0.
17/107
m BRAYBMARBHEF
F9D2CGC2 f(z) BF z0 B5G3,BSD3 z0? f(z) AR m BRAY
BMARFBDRCMDRF0
f(n)(z0) = 0(n = 0,1,2,···,m?1),f(m)(z0) negationslash= 0.
DD BJDRBH AN?(z) BF z0 ARAICEFSD1EQ?
(z) = c0 + c1(z?z0) + c2(z?z0)2 + ···,
G1EL,c0 negationslash= 0.
D9F0 f(z) BF z0 ARAICEFSD1EQ?
f(z) = c0(z?z0)m +c1(z?z0)m+1 +c2(z?z0)m+2 +···,
C3f(n)(z0) = 0(n = 0,1,2,···,m?1),f
(m)(z0)
m! = c0 negationslash= 0.
17/107
m BRAYBMARBHEF
F9D2CGC2 f(z) BF z0 B5G3,BSD3 z0? f(z) AR m BRAY
BMARFBDRCMDRF0
f(n)(z0) = 0(n = 0,1,2,···,m?1),f(m)(z0) negationslash= 0.
DD BJDRBH AN?(z) BF z0 ARAICEFSD1EQ?
(z) = c0 + c1(z?z0) + c2(z?z0)2 + ···,
G1EL,c0 negationslash= 0.
D9F0 f(z) BF z0 ARAICEFSD1EQ?
f(z) = c0(z?z0)m +c1(z?z0)m+1 +c2(z?z0)m+2 +···,
C3f(n)(z0) = 0(n = 0,1,2,···,m?1),f
(m)(z0)
m! = c0 negationslash= 0.
17/107
DIGHD9F8GI f(z) = z3? 1,FR?
f(1) = 0,f′(1) = 3z2|z=1 = 3 negationslash= 0,
D0ADDJ z = 1 F0 f(z) ARE8BRAYBM.
18/107
BHEF,E8DRG0GO?AYARB5G3F8GIARAYBMF0AHG9AR.
DD,FR(z) BF z0 B5G3,DP?(z0) negationslash= 0,FRADENBF z0
ARB8DRA2APDXCBG0?AY.
EKEX f(z) = (z?z0)m?(z) BF z0 ARB8DRA2APDXCBG0?
AY.
19/107
BHEF,E8DRG0GO?AYARB5G3F8GIARAYBMF0AHG9AR.
DD,FR(z) BF z0 B5G3,DP?(z0) negationslash= 0,FRADENBF z0
ARB8DRA2APDXCBG0?AY.
EKEX f(z) = (z?z0)m?(z) BF z0 ARB8DRA2APDXCBG0?
AY.
19/107
BHEF,E8DRG0GO?AYARB5G3F8GIARAYBMF0AHG9AR.
DD,FR(z) BF z0 B5G3,DP?(z0) negationslash= 0,FRADENBF z0
ARB8DRA2APDXCBG0?AY.
EKEX f(z) = (z?z0)m?(z) BF z0 ARB8DRA2APDXCBG0?
AY.
19/107
DI 1 EMDD,CGC2 z0 F0B5G3F8GI f(z) AR m(m > 1) BR
AYBM,BSD3 z0 F0 f′(z) AR m? 1 BRAYBM.
DD z0 F0 f(z) AR m BRAYBM,C3
f(z) = (z?z0)m?(z),
G1EL,?(z) BF z0 B5G3DP?(z0) negationslash= 0.
AD
f′(z) = m(z?z0)m?1?(z) + (z?z0)m?′(z)
= (z?z0)m?1[m?(z) + (z?z0)?′(z)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright],
G1EL,g(z) = m?(z) + (z?z0)?′(z) BF z0 B5G3,
DP g(z0) = m?(z0) negationslash= 0,EKEX z0 F0 f′(z) AR m?1 BRAY
BM.
20/107
DI 1 EMDD,CGC2 z0 F0B5G3F8GI f(z) AR m(m > 1) BR
AYBM,BSD3 z0 F0 f′(z) AR m? 1 BRAYBM.
DD z0 F0 f(z) AR m BRAYBM,C3
f(z) = (z?z0)m?(z),
G1EL,?(z) BF z0 B5G3DP?(z0) negationslash= 0.
AD
f′(z) = m(z?z0)m?1?(z) + (z?z0)m?′(z)
= (z?z0)m?1[m?(z) + (z?z0)?′(z)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright],
G1EL,g(z) = m?(z) + (z?z0)?′(z) BF z0 B5G3,
DP g(z0) = m?(z0) negationslash= 0,EKEX z0 F0 f′(z) AR m?1 BRAY
BM.
20/107
DI 1 EMDD,CGC2 z0 F0B5G3F8GI f(z) AR m(m > 1) BR
AYBM,BSD3 z0 F0 f′(z) AR m? 1 BRAYBM.
DD z0 F0 f(z) AR m BRAYBM,C3
f(z) = (z?z0)m?(z),
G1EL,?(z) BF z0 B5G3DP?(z0) negationslash= 0.
AD
f′(z) = m(z?z0)m?1?(z) + (z?z0)m?′(z)
= (z?z0)m?1[m?(z) + (z?z0)?′(z)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright],
G1EL,g(z) = m?(z) + (z?z0)?′(z) BF z0 B5G3,
DP g(z0) = m?(z0) negationslash= 0,EKEX z0 F0 f′(z) AR m?1 BRAY
BM.
20/107
DI 1 EMDD,CGC2 z0 F0B5G3F8GI f(z) AR m(m > 1) BR
AYBM,BSD3 z0 F0 f′(z) AR m? 1 BRAYBM.
DD z0 F0 f(z) AR m BRAYBM,C3
f(z) = (z?z0)m?(z),
G1EL,?(z) BF z0 B5G3DP?(z0) negationslash= 0.
AD
f′(z) = m(z?z0)m?1?(z) + (z?z0)m?′(z)
= (z?z0)m?1[m?(z) + (z?z0)?′(z)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright],
G1EL,g(z) = m?(z) + (z?z0)?′(z) BF z0 B5G3,
DP g(z0) = m?(z0) negationslash= 0,EKEX z0 F0 f′(z) AR m?1 BRAY
BM.
20/107
DI 1 EMDD,CGC2 z0 F0B5G3F8GI f(z) AR m(m > 1) BR
AYBM,BSD3 z0 F0 f′(z) AR m? 1 BRAYBM.
DD z0 F0 f(z) AR m BRAYBM,C3
f(z) = (z?z0)m?(z),
G1EL,?(z) BF z0 B5G3DP?(z0) negationslash= 0.
AD
f′(z) = m(z?z0)m?1?(z) + (z?z0)m?′(z)
= (z?z0)m?1[m?(z) + (z?z0)?′(z)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright],
G1EL,g(z) = m?(z) + (z?z0)?′(z) BF z0 B5G3,
DP g(z0) = m?(z0) negationslash= 0,EKEX z0 F0 f′(z) AR m?1 BRAY
BM.
20/107
DI 2 CGC2 f(z) G2 g(z) F0EX z0?AYBMARH0DRG0GO?AY
ARB5G3F8GI,BSD3
limz→z
0
f(z)
g(z) = limz→z0
f′(z)
g′(z) (F9H0G8H2?∞)
DDAN z0 EPC7F0 f(z) G2 g(z) AR m BRAYBMG2 n BRAY
BM,C3
f(z) = (z?z0)m?1(z),g(z) = (z?z0)n?2(z),
G1EL,?1(z),?2(z) BF z0 B5G3,DP?1(z0) negationslash= 0,?2(z0) negationslash= 0.
21/107
DI 2 CGC2 f(z) G2 g(z) F0EX z0?AYBMARH0DRG0GO?AY
ARB5G3F8GI,BSD3
limz→z
0
f(z)
g(z) = limz→z0
f′(z)
g′(z) (F9H0G8H2?∞)
DDAN z0 EPC7F0 f(z) G2 g(z) AR m BRAYBMG2 n BRAY
BM,C3
f(z) = (z?z0)m?1(z),g(z) = (z?z0)n?2(z),
G1EL,?1(z),?2(z) BF z0 B5G3,DP?1(z0) negationslash= 0,?2(z0) negationslash= 0.
21/107
DI 2 CGC2 f(z) G2 g(z) F0EX z0?AYBMARH0DRG0GO?AY
ARB5G3F8GI,BSD3
limz→z
0
f(z)
g(z) = limz→z0
f′(z)
g′(z) (F9H0G8H2?∞)
DDAN z0 EPC7F0 f(z) G2 g(z) AR m BRAYBMG2 n BRAY
BM,C3
f(z) = (z?z0)m?1(z),g(z) = (z?z0)n?2(z),
G1EL,?1(z),?2(z) BF z0 B5G3,DP?1(z0) negationslash= 0,?2(z0) negationslash= 0.
21/107
DI 2 CGC2 f(z) G2 g(z) F0EX z0?AYBMARH0DRG0GO?AY
ARB5G3F8GI,BSD3
limz→z
0
f(z)
g(z) = limz→z0
f′(z)
g′(z) (F9H0G8H2?∞)
DDAN z0 EPC7F0 f(z) G2 g(z) AR m BRAYBMG2 n BRAY
BM,C3
f(z) = (z?z0)m?1(z),g(z) = (z?z0)n?2(z),
G1EL,?1(z),?2(z) BF z0 B5G3,DP?1(z0) negationslash= 0,?2(z0) negationslash= 0.
21/107
D9F0
f(z)
g(z) = (z?z0)
m?n?1(z)
2(z),
f′(z)
g′(z) = (z?z0)
m?nm?1(z) + (z?z0)?
′1(z)
n?2(z) + (z?z0)?′2(z),
DUGAH2EHH0EQ,CZ
A8 m > n EE,limz→z
0
f(z)
g(z) = 0 = limz→z0
f′(z)
g′(z);
A8 m = n EE,limz→z
0
f(z)
g(z) =
1(z0)
2(z0) = limz→z0
f′(z)
g′(z);
A8 m < n EE,limz→z
0
f(z)
g(z) = ∞ = limz→z0
f′(z)
g′(z).
22/107
D9F0
f(z)
g(z) = (z?z0)
m?n?1(z)
2(z),
f′(z)
g′(z) = (z?z0)
m?nm?1(z) + (z?z0)?
′1(z)
n?2(z) + (z?z0)?′2(z),
DUGAH2EHH0EQ,CZ
A8 m > n EE,limz→z
0
f(z)
g(z) = 0 = limz→z0
f′(z)
g′(z);
A8 m = n EE,limz→z
0
f(z)
g(z) =
1(z0)
2(z0) = limz→z0
f′(z)
g′(z);
A8 m < n EE,limz→z
0
f(z)
g(z) = ∞ = limz→z0
f′(z)
g′(z).
22/107
D9F0
f(z)
g(z) = (z?z0)
m?n?1(z)
2(z),
f′(z)
g′(z) = (z?z0)
m?nm?1(z) + (z?z0)?
′1(z)
n?2(z) + (z?z0)?′2(z),
DUGAH2EHH0EQ,CZ
A8 m > n EE,limz→z
0
f(z)
g(z) = 0 = limz→z0
f′(z)
g′(z);
A8 m = n EE,limz→z
0
f(z)
g(z) =
1(z0)
2(z0) = limz→z0
f′(z)
g′(z);
A8 m < n EE,limz→z
0
f(z)
g(z) = ∞ = limz→z0
f′(z)
g′(z).
22/107
D9F0
f(z)
g(z) = (z?z0)
m?n?1(z)
2(z),
f′(z)
g′(z) = (z?z0)
m?nm?1(z) + (z?z0)?
′1(z)
n?2(z) + (z?z0)?′2(z),
DUGAH2EHH0EQ,CZ
A8 m > n EE,limz→z
0
f(z)
g(z) = 0 = limz→z0
f′(z)
g′(z);
A8 m = n EE,limz→z
0
f(z)
g(z) =
1(z0)
2(z0) = limz→z0
f′(z)
g′(z);
A8 m < n EE,limz→z
0
f(z)
g(z) = ∞ = limz→z0
f′(z)
g′(z).
22/107
D9F0
f(z)
g(z) = (z?z0)
m?n?1(z)
2(z),
f′(z)
g′(z) = (z?z0)
m?nm?1(z) + (z?z0)?
′1(z)
n?2(z) + (z?z0)?′2(z),
DUGAH2EHH0EQ,CZ
A8 m > n EE,limz→z
0
f(z)
g(z) = 0 = limz→z0
f′(z)
g′(z);
A8 m = n EE,limz→z
0
f(z)
g(z) =
1(z0)
2(z0) = limz→z0
f′(z)
g′(z);
A8 m < n EE,limz→z
0
f(z)
g(z) = ∞ = limz→z0
f′(z)
g′(z).
22/107
D9F0
f(z)
g(z) = (z?z0)
m?n?1(z)
2(z),
f′(z)
g′(z) = (z?z0)
m?nm?1(z) + (z?z0)?
′1(z)
n?2(z) + (z?z0)?′2(z),
DUGAH2EHH0EQ,CZ
A8 m > n EE,limz→z
0
f(z)
g(z) = 0 = limz→z0
f′(z)
g′(z);
A8 m = n EE,limz→z
0
f(z)
g(z) =
1(z0)
2(z0) = limz→z0
f′(z)
g′(z);
A8 m < n EE,limz→z
0
f(z)
g(z) = ∞ = limz→z0
f′(z)
g′(z).
22/107
F8GIARAYBMDQBGBMARB3CG:
F9D2 CGC2 z0 F0 f(z) AR m BRBGBM,BSD3 z0 FUF0
1
f(z)
AR m BRAYBM.
DRDN,CGC2 z0 F0 f(z) AR m BRAYBM,BSD3 z0 FUF0
1
f(z)
AR m BRBGBM.
23/107
F8GIARAYBMDQBGBMARB3CG:
F9D2 CGC2 z0 F0 f(z) AR m BRBGBM,BSD3 z0 FUF0
1
f(z)
AR m BRAYBM.
DRDN,CGC2 z0 F0 f(z) AR m BRAYBM,BSD3 z0 FUF0
1
f(z)
AR m BRBGBM.
23/107
F8GIARAYBMDQBGBMARB3CG:
F9D2 CGC2 z0 F0 f(z) AR m BRBGBM,BSD3 z0 FUF0
1
f(z)
AR m BRAYBM.
DRDN,CGC2 z0 F0 f(z) AR m BRAYBM,BSD3 z0 FUF0
1
f(z)
AR m BRBGBM.
23/107
DDCGC2 z0 F0 f(z) AR m BRBGBM,C3
f(z) = 1(z?z
0)m
g(z),
G1EL g(z) BF z0 B5G3,DP g(z0) negationslash= 0.
EKEXA8 z negationslash= z0 EE,CZ
1
f(z) = (z?z0)
m · 1
g(z)
G1EL,h(z) defines 1g(z) BF z0 B5G3,DP h(z0) negationslash= 0.
EKEX z0 F0
1
f(z) AR m BRAYBM.
24/107
DDCGC2 z0 F0 f(z) AR m BRBGBM,C3
f(z) = 1(z?z
0)m
g(z),
G1EL g(z) BF z0 B5G3,DP g(z0) negationslash= 0.
EKEXA8 z negationslash= z0 EE,CZ
1
f(z) = (z?z0)
m · 1
g(z)
G1EL,h(z) defines 1g(z) BF z0 B5G3,DP h(z0) negationslash= 0.
EKEX z0 F0
1
f(z) AR m BRAYBM.
24/107
DDCGC2 z0 F0 f(z) AR m BRBGBM,C3
f(z) = 1(z?z
0)m
g(z),
G1EL g(z) BF z0 B5G3,DP g(z0) negationslash= 0.
EKEXA8 z negationslash= z0 EE,CZ
1
f(z) = (z?z0)
m · 1
g(z)
G1EL,h(z) defines 1g(z) BF z0 B5G3,DP h(z0) negationslash= 0.
EKEX z0 F0
1
f(z) AR m BRAYBM.
24/107
DDCGC2 z0 F0 f(z) AR m BRBGBM,C3
f(z) = 1(z?z
0)m
g(z),
G1EL g(z) BF z0 B5G3,DP g(z0) negationslash= 0.
EKEXA8 z negationslash= z0 EE,CZ
1
f(z) = (z?z0)
m · 1
g(z)
G1EL,h(z) defines 1g(z) BF z0 B5G3,DP h(z0) negationslash= 0.
EKEX z0 F0
1
f(z) AR m BRAYBM.
24/107
DI 1 F8GI
1
sinz CZAAEFD3G3BM? CGC2F0G3BM,DXFTENAR
BR.
B5 F8GI 1/sinz ARG3BMFUF0 sinz ARAYBM.
FR?DUGA sinz = 0 AQ z = kpi(k = 0,±1,±2,···),DB
B8ENDMF0AHG9G3BM.
D5CSD9
(sinz)′|z=kpi = cosz|z=kpi = (?1)k negationslash= 0,
EKEX z = kpi FTF0 sinz ARE8BRAYBM,?FUF0 1/sinz ARE8
BRBGBM.
25/107
DI 1 F8GI
1
sinz CZAAEFD3G3BM? CGC2F0G3BM,DXFTENAR
BR.
B5 F8GI 1/sinz ARG3BMFUF0 sinz ARAYBM.
FR?DUGA sinz = 0 AQ z = kpi(k = 0,±1,±2,···),DB
B8ENDMF0AHG9G3BM.
D5CSD9
(sinz)′|z=kpi = cosz|z=kpi = (?1)k negationslash= 0,
EKEX z = kpi FTF0 sinz ARE8BRAYBM,?FUF0 1/sinz ARE8
BRBGBM.
25/107
DI 1 F8GI
1
sinz CZAAEFD3G3BM? CGC2F0G3BM,DXFTENAR
BR.
B5 F8GI 1/sinz ARG3BMFUF0 sinz ARAYBM.
FR?DUGA sinz = 0 AQ z = kpi(k = 0,±1,±2,···),DB
B8ENDMF0AHG9G3BM.
D5CSD9
(sinz)′|z=kpi = cosz|z=kpi = (?1)k negationslash= 0,
EKEX z = kpi FTF0 sinz ARE8BRAYBM,?FUF0 1/sinz ARE8
BRBGBM.
25/107
DI 1 F8GI
1
sinz CZAAEFD3G3BM? CGC2F0G3BM,DXFTENAR
BR.
B5 F8GI 1/sinz ARG3BMFUF0 sinz ARAYBM.
FR?DUGA sinz = 0 AQ z = kpi(k = 0,±1,±2,···),DB
B8ENDMF0AHG9G3BM.
D5CSD9
(sinz)′|z=kpi = cosz|z=kpi = (?1)k negationslash= 0,
EKEX z = kpi FTF0 sinz ARE8BRAYBM,?FUF0 1/sinz ARE8
BRBGBM.
25/107
5,F8GIBFFFEHAFBMARBHAL
CGC2F8GI f(z) BFFFEHAFBM z = ∞ARAMB1AP
DX R < |z| < +∞ CBB5G3,BSD3EHBM ∞? f(z) ARAHG9
G3BM.
26/107
CTEHBRG0 f(z) BF z = ∞ APDXCBARBHEF
DUBVDQ t = 1z,GHD9CSBVDQCZ
t = 1z,z = ∞ mapsto?→ t = 0;
R < |z| < +∞ mapsto?→ 0 < |t| < 1R.
B9
f(z) = f(1t) =?(t).
GXAO,F3DMD6BFAMB1APDX R < |z| < +∞ CBGHF8GI f(z)
ARGNFGDE?AMB1APDX 0 < |t| < 1R CBGHF8GI?(t) ARGNFG.
BHF9,CGC2 t = 0 F0?(t) ARDWAMG3BM,m BRBGBMF9AT
BHG3BM,BSD3EHBM z = ∞F0 f(z) ARDWAMG3BM,m BRBGBM
F9ATBHG3BM.
27/107
CTEHBRG0 f(z) BF z = ∞ APDXCBARBHEF
DUBVDQ t = 1z,GHD9CSBVDQCZ
t = 1z,z = ∞ mapsto?→ t = 0;
R < |z| < +∞ mapsto?→ 0 < |t| < 1R.
B9
f(z) = f(1t) =?(t).
GXAO,F3DMD6BFAMB1APDX R < |z| < +∞ CBGHF8GI f(z)
ARGNFGDE?AMB1APDX 0 < |t| < 1R CBGHF8GI?(t) ARGNFG.
BHF9,CGC2 t = 0 F0?(t) ARDWAMG3BM,m BRBGBMF9AT
BHG3BM,BSD3EHBM z = ∞F0 f(z) ARDWAMG3BM,m BRBGBM
F9ATBHG3BM.
27/107
CTEHBRG0 f(z) BF z = ∞ APDXCBARBHEF
DUBVDQ t = 1z,GHD9CSBVDQCZ
t = 1z,z = ∞ mapsto?→ t = 0;
R < |z| < +∞ mapsto?→ 0 < |t| < 1R.
B9
f(z) = f(1t) =?(t).
GXAO,F3DMD6BFAMB1APDX R < |z| < +∞ CBGHF8GI f(z)
ARGNFGDE?AMB1APDX 0 < |t| < 1R CBGHF8GI?(t) ARGNFG.
BHF9,CGC2 t = 0 F0?(t) ARDWAMG3BM,m BRBGBMF9AT
BHG3BM,BSD3EHBM z = ∞F0 f(z) ARDWAMG3BM,m BRBGBM
F9ATBHG3BM.
27/107
CTEHBRG0 f(z) BF z = ∞ APDXCBARBHEF
DUBVDQ t = 1z,GHD9CSBVDQCZ
t = 1z,z = ∞ mapsto?→ t = 0;
R < |z| < +∞ mapsto?→ 0 < |t| < 1R.
B9
f(z) = f(1t) =?(t).
GXAO,F3DMD6BFAMB1APDX R < |z| < +∞ CBGHF8GI f(z)
ARGNFGDE?AMB1APDX 0 < |t| < 1R CBGHF8GI?(t) ARGNFG.
BHF9,CGC2 t = 0 F0?(t) ARDWAMG3BM,m BRBGBMF9AT
BHG3BM,BSD3EHBM z = ∞F0 f(z) ARDWAMG3BM,m BRBGBM
F9ATBHG3BM.
27/107
CTEHBRG0 f(z) BF z = ∞ APDXCBARBHEF
DUBVDQ t = 1z,GHD9CSBVDQCZ
t = 1z,z = ∞ mapsto?→ t = 0;
R < |z| < +∞ mapsto?→ 0 < |t| < 1R.
B9
f(z) = f(1t) =?(t).
GXAO,F3DMD6BFAMB1APDX R < |z| < +∞ CBGHF8GI f(z)
ARGNFGDE?AMB1APDX 0 < |t| < 1R CBGHF8GI?(t) ARGNFG.
BHF9,CGC2 t = 0 F0?(t) ARDWAMG3BM,m BRBGBMF9AT
BHG3BM,BSD3EHBM z = ∞F0 f(z) ARDWAMG3BM,m BRBGBM
F9ATBHG3BM.
27/107
BEDJA6?(t) BF 0 < |t| < 1R CBARGAC3BRGIFSD1EQ.
CSD9 f(z) BF R < |z| < +∞ CBB5G3,EKEXBFCSABDMDX
CBDWEXFSD1EKGAC3BRGI:
f(z) =
∞summationdisplay
n=1
c?nz?n + c0 +
∞summationdisplay
n=1
cnzn (2)
cn = 12pii
contintegraldisplay
C
f(ζ)
ζn+1dζ (n = 0,±1,±2,···),
FRCS,?(t) BFABDMDX 0 < |t| < 1R CBARGAC3BRGI
(t) =
∞summationdisplay
n=1
c?ntn + c0 +
∞summationdisplay
n=1
cnt?n.
28/107
BEDJA6?(t) BF 0 < |t| < 1R CBARGAC3BRGIFSD1EQ.
CSD9 f(z) BF R < |z| < +∞ CBB5G3,EKEXBFCSABDMDX
CBDWEXFSD1EKGAC3BRGI:
f(z) =
∞summationdisplay
n=1
c?nz?n + c0 +
∞summationdisplay
n=1
cnzn (2)
cn = 12pii
contintegraldisplay
C
f(ζ)
ζn+1dζ (n = 0,±1,±2,···),
FRCS,?(t) BFABDMDX 0 < |t| < 1R CBARGAC3BRGI
(t) =
∞summationdisplay
n=1
c?ntn + c0 +
∞summationdisplay
n=1
cnt?n.
28/107
BEDJA6?(t) BF 0 < |t| < 1R CBARGAC3BRGIFSD1EQ.
CSD9 f(z) BF R < |z| < +∞ CBB5G3,EKEXBFCSABDMDX
CBDWEXFSD1EKGAC3BRGI:
f(z) =
∞summationdisplay
n=1
c?nz?n + c0 +
∞summationdisplay
n=1
cnzn (2)
cn = 12pii
contintegraldisplay
C
f(ζ)
ζn+1dζ (n = 0,±1,±2,···),
FRCS,?(t) BFABDMDX 0 < |t| < 1R CBARGAC3BRGI
(t) =
∞summationdisplay
n=1
c?ntn + c0 +
∞summationdisplay
n=1
cnt?n.
28/107
D2H2EKGB,B3D9 z = ∞ AREPCP,CZCTEHARB4G0:
F9D2 1 GHD9 f(z) BF R < |z| < +∞CBARGAC3FSD1
EQ (2),
1) G0F5AKE8E1 =?z = ∞F0 f(z) ARDWAMG3BM;
2) F5CZCZDLGTARAKE8E1,DP zm?DLD4E1E8 =?z = ∞
F0 f(z) AR m BRBGBM;
3) F5CZFFEHGTARAKE8E1 =?z = ∞ F0 f(z) ARATBHG3
BM.
29/107
D2H2EKGB,B3D9 z = ∞ AREPCP,CZCTEHARB4G0:
F9D2 1 GHD9 f(z) BF R < |z| < +∞CBARGAC3FSD1
EQ (2),
1) G0F5AKE8E1 =?z = ∞F0 f(z) ARDWAMG3BM;
2) F5CZCZDLGTARAKE8E1,DP zm?DLD4E1E8 =?z = ∞
F0 f(z) AR m BRBGBM;
3) F5CZFFEHGTARAKE8E1 =?z = ∞ F0 f(z) ARATBHG3
BM.
29/107
D2H2EKGB,B3D9 z = ∞ AREPCP,CZCTEHARB4G0:
F9D2 1 GHD9 f(z) BF R < |z| < +∞CBARGAC3FSD1
EQ (2),
1) G0F5AKE8E1 =?z = ∞F0 f(z) ARDWAMG3BM;
2) F5CZCZDLGTARAKE8E1,DP zm?DLD4E1E8 =?z = ∞
F0 f(z) AR m BRBGBM;
3) F5CZFFEHGTARAKE8E1 =?z = ∞ F0 f(z) ARATBHG3
BM.
29/107
D2H2EKGB,B3D9 z = ∞ AREPCP,CZCTEHARB4G0:
F9D2 1 GHD9 f(z) BF R < |z| < +∞CBARGAC3FSD1
EQ (2),
1) G0F5AKE8E1 =?z = ∞F0 f(z) ARDWAMG3BM;
2) F5CZCZDLGTARAKE8E1,DP zm?DLD4E1E8 =?z = ∞
F0 f(z) AR m BRBGBM;
3) F5CZFFEHGTARAKE8E1 =?z = ∞ F0 f(z) ARATBHG3
BM.
29/107
D2H2EKGB,B3D9 z = ∞ AREPCP,CZCTEHARB4G0:
F9D2 1 GHD9 f(z) BF R < |z| < +∞CBARGAC3FSD1
EQ (2),
1) G0F5AKE8E1 =?z = ∞F0 f(z) ARDWAMG3BM;
2) F5CZCZDLGTARAKE8E1,DP zm?DLD4E1E8 =?z = ∞
F0 f(z) AR m BRBGBM;
3) F5CZFFEHGTARAKE8E1 =?z = ∞ F0 f(z) ARATBHG3
BM.
29/107
DUGABGDL limz→∞f(z) ARG0FVECF5,CZCTEHARB4G0
F9D2 2
1) limz→∞f(z) DJBFDPCZDLz = ∞F0 f(z) ARDWAMG3
BM;
2) limz→∞f(z) = ∞z = ∞ F0 f(z) ARBGBM;
3) limz→∞f(z) G0DJBFD5G0?FFEHDWz = ∞ F0 f(z)
ARATBHG3BM.
30/107
DUGABGDL limz→∞f(z) ARG0FVECF5,CZCTEHARB4G0
F9D2 2
1) limz→∞f(z) DJBFDPCZDLz = ∞F0 f(z) ARDWAMG3
BM;
2) limz→∞f(z) = ∞z = ∞ F0 f(z) ARBGBM;
3) limz→∞f(z) G0DJBFD5G0?FFEHDWz = ∞ F0 f(z)
ARATBHG3BM.
30/107
DUGABGDL limz→∞f(z) ARG0FVECF5,CZCTEHARB4G0
F9D2 2
1) limz→∞f(z) DJBFDPCZDLz = ∞F0 f(z) ARDWAMG3
BM;
2) limz→∞f(z) = ∞z = ∞ F0 f(z) ARBGBM;
3) limz→∞f(z) G0DJBFD5G0?FFEHDWz = ∞ F0 f(z)
ARATBHG3BM.
30/107
DUGABGDL limz→∞f(z) ARG0FVECF5,CZCTEHARB4G0
F9D2 2
1) limz→∞f(z) DJBFDPCZDLz = ∞F0 f(z) ARDWAMG3
BM;
2) limz→∞f(z) = ∞z = ∞ F0 f(z) ARBGBM;
3) limz→∞f(z) G0DJBFD5G0?FFEHDWz = ∞ F0 f(z)
ARATBHG3BM.
30/107
DI 1 A7GC z = ∞F0F8GI f(z) = zz + 1 AREFD3G3BM?
B5 E0DFE8DUGAGAC3BRGIFSD1EQA7GC
f(z) BFABDMDX 1 < |z| < +∞CBARGAC3BRGI
f(z) = 1
1 + 1z
= 1? 1z + 1z2? 1z3 + ··· + (?1)n 1zn + ···
ENG0F5AKE8E1,EKEX ∞ F0 f(z) ARDWAMG3BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞f(z) = limz→∞ zz + 1 = 1.
EKEX ∞ F0 f(z) ARDWAMG3BM.
31/107
DI 1 A7GC z = ∞F0F8GI f(z) = zz + 1 AREFD3G3BM?
B5 E0DFE8DUGAGAC3BRGIFSD1EQA7GC
f(z) BFABDMDX 1 < |z| < +∞CBARGAC3BRGI
f(z) = 1
1 + 1z
= 1? 1z + 1z2? 1z3 + ··· + (?1)n 1zn + ···
ENG0F5AKE8E1,EKEX ∞ F0 f(z) ARDWAMG3BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞f(z) = limz→∞ zz + 1 = 1.
EKEX ∞ F0 f(z) ARDWAMG3BM.
31/107
DI 1 A7GC z = ∞F0F8GI f(z) = zz + 1 AREFD3G3BM?
B5 E0DFE8DUGAGAC3BRGIFSD1EQA7GC
f(z) BFABDMDX 1 < |z| < +∞CBARGAC3BRGI
f(z) = 1
1 + 1z
= 1? 1z + 1z2? 1z3 + ··· + (?1)n 1zn + ···
ENG0F5AKE8E1,EKEX ∞ F0 f(z) ARDWAMG3BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞f(z) = limz→∞ zz + 1 = 1.
EKEX ∞ F0 f(z) ARDWAMG3BM.
31/107
DI 1 A7GC z = ∞F0F8GI f(z) = zz + 1 AREFD3G3BM?
B5 E0DFE8DUGAGAC3BRGIFSD1EQA7GC
f(z) BFABDMDX 1 < |z| < +∞CBARGAC3BRGI
f(z) = 1
1 + 1z
= 1? 1z + 1z2? 1z3 + ··· + (?1)n 1zn + ···
ENG0F5AKE8E1,EKEX ∞ F0 f(z) ARDWAMG3BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞f(z) = limz→∞ zz + 1 = 1.
EKEX ∞ F0 f(z) ARDWAMG3BM.
31/107
DI 1 A7GC z = ∞F0F8GI f(z) = zz + 1 AREFD3G3BM?
B5 E0DFE8DUGAGAC3BRGIFSD1EQA7GC
f(z) BFABDMDX 1 < |z| < +∞CBARGAC3BRGI
f(z) = 1
1 + 1z
= 1? 1z + 1z2? 1z3 + ··· + (?1)n 1zn + ···
ENG0F5AKE8E1,EKEX ∞ F0 f(z) ARDWAMG3BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞f(z) = limz→∞ zz + 1 = 1.
EKEX ∞ F0 f(z) ARDWAMG3BM.
31/107
DI 1 A7GC z = ∞F0F8GI f(z) = zz + 1 AREFD3G3BM?
B5 E0DFE8DUGAGAC3BRGIFSD1EQA7GC
f(z) BFABDMDX 1 < |z| < +∞CBARGAC3BRGI
f(z) = 1
1 + 1z
= 1? 1z + 1z2? 1z3 + ··· + (?1)n 1zn + ···
ENG0F5AKE8E1,EKEX ∞ F0 f(z) ARDWAMG3BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞f(z) = limz→∞ zz + 1 = 1.
EKEX ∞ F0 f(z) ARDWAMG3BM.
31/107
DI 1 A7GC z = ∞F0F8GI f(z) = zz + 1 AREFD3G3BM?
B5 E0DFE8DUGAGAC3BRGIFSD1EQA7GC
f(z) BFABDMDX 1 < |z| < +∞CBARGAC3BRGI
f(z) = 1
1 + 1z
= 1? 1z + 1z2? 1z3 + ··· + (?1)n 1zn + ···
ENG0F5AKE8E1,EKEX ∞ F0 f(z) ARDWAMG3BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞f(z) = limz→∞ zz + 1 = 1.
EKEX ∞ F0 f(z) ARDWAMG3BM.
31/107
DI 2 A7GC z = ∞F0F8GI f(z) = sinz AREFD3G3BM?
B5 E0DFE8,DUGAGAC3BRGIFSD1EQA7GC
f(z) ARFSD1EQ
sinz = z? z
3
3! +
z5
5!?···+ (?1)
n z
2n+1
(2n + 1)! +···
F5CZFFEHGTDRAKE8E1,EKEX ∞F0 f(z) = sinz ARATBHG3
BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞sinz G0DJBFDPG0?FFEHDW,
EKEX ∞ F0 f(z) = sinz ARATBHG3BM.
32/107
DI 2 A7GC z = ∞F0F8GI f(z) = sinz AREFD3G3BM?
B5 E0DFE8,DUGAGAC3BRGIFSD1EQA7GC
f(z) ARFSD1EQ
sinz = z? z
3
3! +
z5
5!?···+ (?1)
n z
2n+1
(2n + 1)! +···
F5CZFFEHGTDRAKE8E1,EKEX ∞F0 f(z) = sinz ARATBHG3
BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞sinz G0DJBFDPG0?FFEHDW,
EKEX ∞ F0 f(z) = sinz ARATBHG3BM.
32/107
DI 2 A7GC z = ∞F0F8GI f(z) = sinz AREFD3G3BM?
B5 E0DFE8,DUGAGAC3BRGIFSD1EQA7GC
f(z) ARFSD1EQ
sinz = z? z
3
3! +
z5
5!?···+ (?1)
n z
2n+1
(2n + 1)! +···
F5CZFFEHGTDRAKE8E1,EKEX ∞F0 f(z) = sinz ARATBHG3
BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞sinz G0DJBFDPG0?FFEHDW,
EKEX ∞ F0 f(z) = sinz ARATBHG3BM.
32/107
DI 2 A7GC z = ∞F0F8GI f(z) = sinz AREFD3G3BM?
B5 E0DFE8,DUGAGAC3BRGIFSD1EQA7GC
f(z) ARFSD1EQ
sinz = z? z
3
3! +
z5
5!?···+ (?1)
n z
2n+1
(2n + 1)! +···
F5CZFFEHGTDRAKE8E1,EKEX ∞F0 f(z) = sinz ARATBHG3
BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞sinz G0DJBFDPG0?FFEHDW,
EKEX ∞ F0 f(z) = sinz ARATBHG3BM.
32/107
DI 2 A7GC z = ∞F0F8GI f(z) = sinz AREFD3G3BM?
B5 E0DFE8,DUGAGAC3BRGIFSD1EQA7GC
f(z) ARFSD1EQ
sinz = z? z
3
3! +
z5
5!?···+ (?1)
n z
2n+1
(2n + 1)! +···
F5CZFFEHGTDRAKE8E1,EKEX ∞F0 f(z) = sinz ARATBHG3
BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞sinz G0DJBFDPG0?FFEHDW,
EKEX ∞ F0 f(z) = sinz ARATBHG3BM.
32/107
DI 2 A7GC z = ∞F0F8GI f(z) = sinz AREFD3G3BM?
B5 E0DFE8,DUGAGAC3BRGIFSD1EQA7GC
f(z) ARFSD1EQ
sinz = z? z
3
3! +
z5
5!?···+ (?1)
n z
2n+1
(2n + 1)! +···
F5CZFFEHGTDRAKE8E1,EKEX ∞F0 f(z) = sinz ARATBHG3
BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞sinz G0DJBFDPG0?FFEHDW,
EKEX ∞ F0 f(z) = sinz ARATBHG3BM.
32/107
DI 2 A7GC z = ∞F0F8GI f(z) = sinz AREFD3G3BM?
B5 E0DFE8,DUGAGAC3BRGIFSD1EQA7GC
f(z) ARFSD1EQ
sinz = z? z
3
3! +
z5
5!?···+ (?1)
n z
2n+1
(2n + 1)! +···
F5CZFFEHGTDRAKE8E1,EKEX ∞F0 f(z) = sinz ARATBHG3
BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞sinz G0DJBFDPG0?FFEHDW,
EKEX ∞ F0 f(z) = sinz ARATBHG3BM.
32/107
DI 2 A7GC z = ∞F0F8GI f(z) = sinz AREFD3G3BM?
B5 E0DFE8,DUGAGAC3BRGIFSD1EQA7GC
f(z) ARFSD1EQ
sinz = z? z
3
3! +
z5
5!?···+ (?1)
n z
2n+1
(2n + 1)! +···
F5CZFFEHGTDRAKE8E1,EKEX ∞F0 f(z) = sinz ARATBHG3
BM.
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→∞sinz G0DJBFDPG0?FFEHDW,
EKEX ∞ F0 f(z) = sinz ARATBHG3BM.
32/107
DI 3 A7GCF8GI f(z) = sinz?zz3 BFB6FBBXEYEHH2ARG3
BM.
B5 E0DFE8,DUGAGAC3BRGIFSD1EQA7GC
FR? f(z) BF 0 < |z| < +∞CBARGAC3BRGIFSD1EQ
f(z) = sinz?zz3
= 1z3
bracketleftbiggparenleftbigg
z? z
3
3! +
z5
5!?
z7
7! + ···
parenrightbigg
z
bracketrightbigg
=?13! + z
2
5!?
z4
7! +···
DUGAF9FE,z = 0 F0 f(z) ARDWAMG3BM,z = ∞ F0 f(z) AR
ATBHG3BM.
33/107
DI 3 A7GCF8GI f(z) = sinz?zz3 BFB6FBBXEYEHH2ARG3
BM.
B5 E0DFE8,DUGAGAC3BRGIFSD1EQA7GC
FR? f(z) BF 0 < |z| < +∞CBARGAC3BRGIFSD1EQ
f(z) = sinz?zz3
= 1z3
bracketleftbiggparenleftbigg
z? z
3
3! +
z5
5!?
z7
7! + ···
parenrightbigg
z
bracketrightbigg
=?13! + z
2
5!?
z4
7! +···
DUGAF9FE,z = 0 F0 f(z) ARDWAMG3BM,z = ∞ F0 f(z) AR
ATBHG3BM.
33/107
DI 3 A7GCF8GI f(z) = sinz?zz3 BFB6FBBXEYEHH2ARG3
BM.
B5 E0DFE8,DUGAGAC3BRGIFSD1EQA7GC
FR? f(z) BF 0 < |z| < +∞CBARGAC3BRGIFSD1EQ
f(z) = sinz?zz3
= 1z3
bracketleftbiggparenleftbigg
z? z
3
3! +
z5
5!?
z7
7! + ···
parenrightbigg
z
bracketrightbigg
=?13! + z
2
5!?
z4
7! +···
DUGAF9FE,z = 0 F0 f(z) ARDWAMG3BM,z = ∞ F0 f(z) AR
ATBHG3BM.
33/107
DI 3 A7GCF8GI f(z) = sinz?zz3 BFB6FBBXEYEHH2ARG3
BM.
B5 E0DFE8,DUGAGAC3BRGIFSD1EQA7GC
FR? f(z) BF 0 < |z| < +∞CBARGAC3BRGIFSD1EQ
f(z) = sinz?zz3
= 1z3
bracketleftbiggparenleftbigg
z? z
3
3! +
z5
5!?
z7
7! + ···
parenrightbigg
z
bracketrightbigg
=?13! + z
2
5!?
z4
7! +···
DUGAF9FE,z = 0 F0 f(z) ARDWAMG3BM,z = ∞ F0 f(z) AR
ATBHG3BM.
33/107
DI 3 A7GCF8GI f(z) = sinz?zz3 BFB6FBBXEYEHH2ARG3
BM.
B5 E0DFE8,DUGAGAC3BRGIFSD1EQA7GC
FR? f(z) BF 0 < |z| < +∞CBARGAC3BRGIFSD1EQ
f(z) = sinz?zz3
= 1z3
bracketleftbiggparenleftbigg
z? z
3
3! +
z5
5!?
z7
7! + ···
parenrightbigg
z
bracketrightbigg
=?13! + z
2
5!?
z4
7! +···
DUGAF9FE,z = 0 F0 f(z) ARDWAMG3BM,z = ∞ F0 f(z) AR
ATBHG3BM.
33/107
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→0 sinz?zz3 = limz→0 cosz? 13z2 =?16
EKEX z = 0? f(z) ARDWAMG3BM.
FR?
limz→∞ sinz?zz3 G0DJBFDPG0?FFEHDW,
EKEX ∞ F0 f(z) ARATBHG3BM.
34/107
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→0 sinz?zz3 = limz→0 cosz? 13z2 =?16
EKEX z = 0? f(z) ARDWAMG3BM.
FR?
limz→∞ sinz?zz3 G0DJBFDPG0?FFEHDW,
EKEX ∞ F0 f(z) ARATBHG3BM.
34/107
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→0 sinz?zz3 = limz→0 cosz? 13z2 =?16
EKEX z = 0? f(z) ARDWAMG3BM.
FR?
limz→∞ sinz?zz3 G0DJBFDPG0?FFEHDW,
EKEX ∞ F0 f(z) ARATBHG3BM.
34/107
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→0 sinz?zz3 = limz→0 cosz? 13z2 =?16
EKEX z = 0? f(z) ARDWAMG3BM.
FR?
limz→∞ sinz?zz3 G0DJBFDPG0?FFEHDW,
EKEX ∞ F0 f(z) ARATBHG3BM.
34/107
E0DFAJ,DUGABGDLARBHEFA7GC
FR?
limz→0 sinz?zz3 = limz→0 cosz? 13z2 =?16
EKEX z = 0? f(z) ARDWAMG3BM.
FR?
limz→∞ sinz?zz3 G0DJBFDPG0?FFEHDW,
EKEX ∞ F0 f(z) ARATBHG3BM.
34/107
DI 4 F8GI
f(z) = (z
2? 1)(z?2)3
(sinpiz)3
BFB6FBBXEYEHCBCZAAEFD3CPBAARG3BM? CGC2F0BGBM,DXFT
ENARBR.
B5 F8GI f(z) EPBD?AYARBM z = 0,±1,±2,···.
FR?
(sinpiz)′ = picospiz
BF z = 0,±1,±2,··· BRG0?AY,FRCSGXAABMFTF0 sinpiz
ARE8BRAYBM,D0ADF0EPBD (sinpiz)3 ARD2BRAYBM.
D5FR?GXAABMELG0 1,?1,2 A9FTEMAQF8GI f(z) AREP
CU (z2? 1)(z?2)3 G0?AY,EKEXGXAABMELG0 1,?1,2 A9
F0 f(z) ARD2BRBGBM.
FR? 1 G2?1 F0EPCU (z2? 1)(z?2)3 ARE8BRAYBM,EK
EX 1 DQ?1 F0 f(z) AR 2 BRBGBM.
35/107
DI 4 F8GI
f(z) = (z
2? 1)(z?2)3
(sinpiz)3
BFB6FBBXEYEHCBCZAAEFD3CPBAARG3BM? CGC2F0BGBM,DXFT
ENARBR.
B5 F8GI f(z) EPBD?AYARBM z = 0,±1,±2,···.
FR?
(sinpiz)′ = picospiz
BF z = 0,±1,±2,··· BRG0?AY,FRCSGXAABMFTF0 sinpiz
ARE8BRAYBM,D0ADF0EPBD (sinpiz)3 ARD2BRAYBM.
D5FR?GXAABMELG0 1,?1,2 A9FTEMAQF8GI f(z) AREP
CU (z2? 1)(z?2)3 G0?AY,EKEXGXAABMELG0 1,?1,2 A9
F0 f(z) ARD2BRBGBM.
FR? 1 G2?1 F0EPCU (z2? 1)(z?2)3 ARE8BRAYBM,EK
EX 1 DQ?1 F0 f(z) AR 2 BRBGBM.
35/107
DI 4 F8GI
f(z) = (z
2? 1)(z?2)3
(sinpiz)3
BFB6FBBXEYEHCBCZAAEFD3CPBAARG3BM? CGC2F0BGBM,DXFT
ENARBR.
B5 F8GI f(z) EPBD?AYARBM z = 0,±1,±2,···.
FR?
(sinpiz)′ = picospiz
BF z = 0,±1,±2,··· BRG0?AY,FRCSGXAABMFTF0 sinpiz
ARE8BRAYBM,D0ADF0EPBD (sinpiz)3 ARD2BRAYBM.
D5FR?GXAABMELG0 1,?1,2 A9FTEMAQF8GI f(z) AREP
CU (z2? 1)(z?2)3 G0?AY,EKEXGXAABMELG0 1,?1,2 A9
F0 f(z) ARD2BRBGBM.
FR? 1 G2?1 F0EPCU (z2? 1)(z?2)3 ARE8BRAYBM,EK
EX 1 DQ?1 F0 f(z) AR 2 BRBGBM.
35/107
DI 4 F8GI
f(z) = (z
2? 1)(z?2)3
(sinpiz)3
BFB6FBBXEYEHCBCZAAEFD3CPBAARG3BM? CGC2F0BGBM,DXFT
ENARBR.
B5 F8GI f(z) EPBD?AYARBM z = 0,±1,±2,···.
FR?
(sinpiz)′ = picospiz
BF z = 0,±1,±2,··· BRG0?AY,FRCSGXAABMFTF0 sinpiz
ARE8BRAYBM,D0ADF0EPBD (sinpiz)3 ARD2BRAYBM.
D5FR?GXAABMELG0 1,?1,2 A9FTEMAQF8GI f(z) AREP
CU (z2? 1)(z?2)3 G0?AY,EKEXGXAABMELG0 1,?1,2 A9
F0 f(z) ARD2BRBGBM.
FR? 1 G2?1 F0EPCU (z2? 1)(z?2)3 ARE8BRAYBM,EK
EX 1 DQ?1 F0 f(z) AR 2 BRBGBM.
35/107
DI 4 F8GI
f(z) = (z
2? 1)(z?2)3
(sinpiz)3
BFB6FBBXEYEHCBCZAAEFD3CPBAARG3BM? CGC2F0BGBM,DXFT
ENARBR.
B5 F8GI f(z) EPBD?AYARBM z = 0,±1,±2,···.
FR?
(sinpiz)′ = picospiz
BF z = 0,±1,±2,··· BRG0?AY,FRCSGXAABMFTF0 sinpiz
ARE8BRAYBM,D0ADF0EPBD (sinpiz)3 ARD2BRAYBM.
D5FR?GXAABMELG0 1,?1,2 A9FTEMAQF8GI f(z) AREP
CU (z2? 1)(z?2)3 G0?AY,EKEXGXAABMELG0 1,?1,2 A9
F0 f(z) ARD2BRBGBM.
FR? 1 G2?1 F0EPCU (z2? 1)(z?2)3 ARE8BRAYBM,EK
EX 1 DQ?1 F0 f(z) AR 2 BRBGBM.
35/107
DI 4 F8GI
f(z) = (z
2? 1)(z?2)3
(sinpiz)3
BFB6FBBXEYEHCBCZAAEFD3CPBAARG3BM? CGC2F0BGBM,DXFT
ENARBR.
B5 F8GI f(z) EPBD?AYARBM z = 0,±1,±2,···.
FR?
(sinpiz)′ = picospiz
BF z = 0,±1,±2,··· BRG0?AY,FRCSGXAABMFTF0 sinpiz
ARE8BRAYBM,D0ADF0EPBD (sinpiz)3 ARD2BRAYBM.
D5FR?GXAABMELG0 1,?1,2 A9FTEMAQF8GI f(z) AREP
CU (z2? 1)(z?2)3 G0?AY,EKEXGXAABMELG0 1,?1,2 A9
F0 f(z) ARD2BRBGBM.
FR? 1 G2?1 F0EPCU (z2? 1)(z?2)3 ARE8BRAYBM,EK
EX 1 DQ?1 F0 f(z) AR 2 BRBGBM.
35/107
B3D9 z = 2,FR?
limz→2f(z) = limz→2 (z
2?1)(z? 2)3
(sinpiz)3 =
3
pi3,
EKEX z = 2 F0 f(z) ARDWAMG3BM.
B3D9 z = ∞,DJFG
f(1ξ) = (1?ξ
2)(1? 2ξ)3
ξ5sin3pi/ξ,
FR? ξ = 0,ξn = 1n FTF0ENARG3BM,A2A8 n → ∞ EE,
ξn → 0,EKEX ξ = 0 G0F0ENARAHG9G3BM,EKEX z = ∞G0
F0 f(z) ARAHG9G3BM.
36/107
B3D9 z = 2,FR?
limz→2f(z) = limz→2 (z
2?1)(z? 2)3
(sinpiz)3 =
3
pi3,
EKEX z = 2 F0 f(z) ARDWAMG3BM.
B3D9 z = ∞,DJFG
f(1ξ) = (1?ξ
2)(1? 2ξ)3
ξ5sin3pi/ξ,
FR? ξ = 0,ξn = 1n FTF0ENARG3BM,A2A8 n → ∞ EE,
ξn → 0,EKEX ξ = 0 G0F0ENARAHG9G3BM,EKEX z = ∞G0
F0 f(z) ARAHG9G3BM.
36/107
B3D9 z = 2,FR?
limz→2f(z) = limz→2 (z
2?1)(z? 2)3
(sinpiz)3 =
3
pi3,
EKEX z = 2 F0 f(z) ARDWAMG3BM.
B3D9 z = ∞,DJFG
f(1ξ) = (1?ξ
2)(1? 2ξ)3
ξ5sin3pi/ξ,
FR? ξ = 0,ξn = 1n FTF0ENARG3BM,A2A8 n → ∞ EE,
ξn → 0,EKEX ξ = 0 G0F0ENARAHG9G3BM,EKEX z = ∞G0
F0 f(z) ARAHG9G3BM.
36/107
B3D9 z = 2,FR?
limz→2f(z) = limz→2 (z
2?1)(z? 2)3
(sinpiz)3 =
3
pi3,
EKEX z = 2 F0 f(z) ARDWAMG3BM.
B3D9 z = ∞,DJFG
f(1ξ) = (1?ξ
2)(1? 2ξ)3
ξ5sin3pi/ξ,
FR? ξ = 0,ξn = 1n FTF0ENARG3BM,A2A8 n → ∞ EE,
ξn → 0,EKEX ξ = 0 G0F0ENARAHG9G3BM,EKEX z = ∞G0
F0 f(z) ARAHG9G3BM.
36/107
B3D9 z = 2,FR?
limz→2f(z) = limz→2 (z
2?1)(z? 2)3
(sinpiz)3 =
3
pi3,
EKEX z = 2 F0 f(z) ARDWAMG3BM.
B3D9 z = ∞,DJFG
f(1ξ) = (1?ξ
2)(1? 2ξ)3
ξ5sin3pi/ξ,
FR? ξ = 0,ξn = 1n FTF0ENARG3BM,A2A8 n → ∞ EE,
ξn → 0,EKEX ξ = 0 G0F0ENARAHG9G3BM,EKEX z = ∞G0
F0 f(z) ARAHG9G3BM.
36/107
BDAJAX BFGI
1,BFGIARF9FEBLBFGIF9D2
EVC3 1,CGC2F8GI f(z) BF z0 ARAPDXCBB5G3,BSD3DU
GADPG4 - AKCXF9D2,contintegraldisplay
C
f(z)dz = 0,
G1EL,C F0 z0 APDXCBARBRFBE8CMDFH0BGETDM.
EVC3 2,CGC2 z0? f(z) ARE8DRAHG9G3BM,BSD3GV
BF z0 ARB8DRAMB1APDX 0 < |z?z0| < R CBEDF5 z0 ARBRFB
E8CMAKE5DFH0BGETDM C ARFKEPcontintegraldisplay
C
f(z)dz
E8DOG0AVD9AY.
37/107
BDAJAX BFGI
1,BFGIARF9FEBLBFGIF9D2
EVC3 1,CGC2F8GI f(z) BF z0 ARAPDXCBB5G3,BSD3DU
GADPG4 - AKCXF9D2,contintegraldisplay
C
f(z)dz = 0,
G1EL,C F0 z0 APDXCBARBRFBE8CMDFH0BGETDM.
EVC3 2,CGC2 z0? f(z) ARE8DRAHG9G3BM,BSD3GV
BF z0 ARB8DRAMB1APDX 0 < |z?z0| < R CBEDF5 z0 ARBRFB
E8CMAKE5DFH0BGETDM C ARFKEPcontintegraldisplay
C
f(z)dz
E8DOG0AVD9AY.
37/107
BDAJAX BFGI
1,BFGIARF9FEBLBFGIF9D2
EVC3 1,CGC2F8GI f(z) BF z0 ARAPDXCBB5G3,BSD3DU
GADPG4 - AKCXF9D2,contintegraldisplay
C
f(z)dz = 0,
G1EL,C F0 z0 APDXCBARBRFBE8CMDFH0BGETDM.
EVC3 2,CGC2 z0? f(z) ARE8DRAHG9G3BM,BSD3GV
BF z0 ARB8DRAMB1APDX 0 < |z?z0| < R CBEDF5 z0 ARBRFB
E8CMAKE5DFH0BGETDM C ARFKEPcontintegraldisplay
C
f(z)dz
E8DOG0AVD9AY.
37/107
E0DF,GQF8GI f(z) BFCSAPDXCBFSD1EKGAC3BRGI
f(z) =
+∞summationdisplay
n=?∞
cn(z?z0)n
B8A0GHCSFSD1EQH0G8GV C FHE1FKEP,AQ
contintegraldisplay
C
f(z)dz =
+∞summationdisplay
n=?∞
cn
contintegraldisplay
C
(z?z0)ndz
FR? contintegraldisplay
C
dz
(z?z0)n+1 =
braceleftBig 2pii,n = 0,
0,n negationslash= 0,
EKEX contintegraldisplay
C
f(z)dz = 2piic?1.
38/107
E0DF,GQF8GI f(z) BFCSAPDXCBFSD1EKGAC3BRGI
f(z) =
+∞summationdisplay
n=?∞
cn(z?z0)n
B8A0GHCSFSD1EQH0G8GV C FHE1FKEP,AQ
contintegraldisplay
C
f(z)dz =
+∞summationdisplay
n=?∞
cn
contintegraldisplay
C
(z?z0)ndz
FR? contintegraldisplay
C
dz
(z?z0)n+1 =
braceleftBig 2pii,n = 0,
0,n negationslash= 0,
EKEX contintegraldisplay
C
f(z)dz = 2piic?1.
38/107
E0DF,GQF8GI f(z) BFCSAPDXCBFSD1EKGAC3BRGI
f(z) =
+∞summationdisplay
n=?∞
cn(z?z0)n
B8A0GHCSFSD1EQH0G8GV C FHE1FKEP,AQ
contintegraldisplay
C
f(z)dz =
+∞summationdisplay
n=?∞
cn
contintegraldisplay
C
(z?z0)ndz
FR? contintegraldisplay
C
dz
(z?z0)n+1 =
braceleftBig 2pii,n = 0,
0,n negationslash= 0,
EKEX contintegraldisplay
C
f(z)dz = 2piic?1.
38/107
E0DF,GQF8GI f(z) BFCSAPDXCBFSD1EKGAC3BRGI
f(z) =
+∞summationdisplay
n=?∞
cn(z?z0)n
B8A0GHCSFSD1EQH0G8GV C FHE1FKEP,AQ
contintegraldisplay
C
f(z)dz =
+∞summationdisplay
n=?∞
cn
contintegraldisplay
C
(z?z0)ndz
FR? contintegraldisplay
C
dz
(z?z0)n+1 =
braceleftBig 2pii,n = 0,
0,n negationslash= 0,
EKEX contintegraldisplay
C
f(z)dz = 2piic?1.
38/107
EH
1
2pii
contintegraldisplay
C
f(z)dz?BFGI,C8DU Res[f(z),z0],BP:
Res[f(z),z0] = 12pii
contintegraldisplay
C
f(z)dz.
D0ADCZ
Res[f(z),z0] = c?1.
C3 contintegraldisplay
C
f(z)dz = 2pii· Res[f(z),z0].
39/107
EH
1
2pii
contintegraldisplay
C
f(z)dz?BFGI,C8DU Res[f(z),z0],BP:
Res[f(z),z0] = 12pii
contintegraldisplay
C
f(z)dz.
D0ADCZ
Res[f(z),z0] = c?1.
C3 contintegraldisplay
C
f(z)dz = 2pii· Res[f(z),z0].
39/107
EH
1
2pii
contintegraldisplay
C
f(z)dz?BFGI,C8DU Res[f(z),z0],BP:
Res[f(z),z0] = 12pii
contintegraldisplay
C
f(z)dz.
D0ADCZ
Res[f(z),z0] = c?1.
C3 contintegraldisplay
C
f(z)dz = 2pii· Res[f(z),z0].
39/107
F9D2E8 (BFGIF9D2) ANF8GI f(z) BFERDX D CBG0CZDLDR
AHG9G3BM z1,z2,···,zn A9BRBRB5G3,C F0 D CBEDDWDSG3
BMARE8CMAKE5DFH0BGETDM,BSD3
contintegraldisplay
C
f(z)dz = 2pii
nsummationdisplay
k=1
Res[f(z),zk].
z1 C1
C2
C4
C3
C5
z2 z3
z4 z5
C D
40/107
z1 C1
C2
C4
C3
C5
z2 z3
z4 z5
C D
DDD6BF C CBARAHG9G3BM zk(k = 1,2,···,n) CMD4G0ED
F5ARAKE5DFH0BGETDM Ck DWBJGDBH,DUGABXG4BGF0F9D2CZ
contintegraldisplay
C
f(z)dz =
contintegraldisplay
C1
f(z)dz +
contintegraldisplay
C2
f(z)dz +···+
contintegraldisplay
Cn
f(z)dz
H0BQG0EX 2pii,AQ
1
2pii
contintegraldisplay
C
f(z)dz = Res[f(z),z1] + Res[f(z),z2] +···+ Res[f(z),zn]
BP
contintegraldisplay
C
f(z)dz = 2pii
nsummationdisplay
k=1
Res[f(z),zk].
41/107
z1 C1
C2
C4
C3
C5
z2 z3
z4 z5
C D
DDD6BF C CBARAHG9G3BM zk(k = 1,2,···,n) CMD4G0ED
F5ARAKE5DFH0BGETDM Ck DWBJGDBH,DUGABXG4BGF0F9D2CZ
contintegraldisplay
C
f(z)dz =
contintegraldisplay
C1
f(z)dz +
contintegraldisplay
C2
f(z)dz +···+
contintegraldisplay
Cn
f(z)dz
H0BQG0EX 2pii,AQ
1
2pii
contintegraldisplay
C
f(z)dz = Res[f(z),z1] + Res[f(z),z2] +···+ Res[f(z),zn]
BP
contintegraldisplay
C
f(z)dz = 2pii
nsummationdisplay
k=1
Res[f(z),zk].
41/107
z1 C1
C2
C4
C3
C5
z2 z3
z4 z5
C D
DDD6BF C CBARAHG9G3BM zk(k = 1,2,···,n) CMD4G0ED
F5ARAKE5DFH0BGETDM Ck DWBJGDBH,DUGABXG4BGF0F9D2CZ
contintegraldisplay
C
f(z)dz =
contintegraldisplay
C1
f(z)dz +
contintegraldisplay
C2
f(z)dz +···+
contintegraldisplay
Cn
f(z)dz
H0BQG0EX 2pii,AQ
1
2pii
contintegraldisplay
C
f(z)dz = Res[f(z),z1] + Res[f(z),z2] +···+ Res[f(z),zn]
BP
contintegraldisplay
C
f(z)dz = 2pii
nsummationdisplay
k=1
Res[f(z),zk].
41/107
C7DYBFGI Res[f(z),z0] ARE0DF:
1,CGC2 z0 F0 f(z) ARDWAMG3BM,BSD3 Res[f(z),z0] = 0.
2,CGC2 z0 F0 f(z) ARATBHG3BM,BSD3E0CDCMD6 f(z)
BF z0 FSD1EKGAC3BRGIARE0DFEM c?1.
3,CGC2 z0 F0 f(z) ARBGBM,BSD3CZCTEHBTDRC7DYBHC3.
42/107
C7DYBFGI Res[f(z),z0] ARE0DF:
1,CGC2 z0 F0 f(z) ARDWAMG3BM,BSD3 Res[f(z),z0] = 0.
2,CGC2 z0 F0 f(z) ARATBHG3BM,BSD3E0CDCMD6 f(z)
BF z0 FSD1EKGAC3BRGIARE0DFEM c?1.
3,CGC2 z0 F0 f(z) ARBGBM,BSD3CZCTEHBTDRC7DYBHC3.
42/107
C7DYBFGI Res[f(z),z0] ARE0DF:
1,CGC2 z0 F0 f(z) ARDWAMG3BM,BSD3 Res[f(z),z0] = 0.
2,CGC2 z0 F0 f(z) ARATBHG3BM,BSD3E0CDCMD6 f(z)
BF z0 FSD1EKGAC3BRGIARE0DFEM c?1.
3,CGC2 z0 F0 f(z) ARBGBM,BSD3CZCTEHBTDRC7DYBHC3.
42/107
C7DYBFGI Res[f(z),z0] ARE0DF:
1,CGC2 z0 F0 f(z) ARDWAMG3BM,BSD3 Res[f(z),z0] = 0.
2,CGC2 z0 F0 f(z) ARATBHG3BM,BSD3E0CDCMD6 f(z)
BF z0 FSD1EKGAC3BRGIARE0DFEM c?1.
3,CGC2 z0 F0 f(z) ARBGBM,BSD3CZCTEHBTDRC7DYBHC3.
42/107
2,BFGIARC7DYBHC3
BHC3 1 CGC2 z0 F0 f(z) ARE8BRBGBM,BSD3
Res[f(z),z0] = limz→z
0
(z?z0)f(z).
43/107
2,BFGIARC7DYBHC3
BHC3 1 CGC2 z0 F0 f(z) ARE8BRBGBM,BSD3
Res[f(z),z0] = limz→z
0
(z?z0)f(z).
43/107
BHC3 2 CGC2 z0 F0 f(z) AR m BRBGBM,BSD3
Res[f(z),z0] = 1(m? 1)! limz→z
0
dm?1
dzm?1{(z?z0)
mf(z)}.
DD FR? z0 F0 f(z) AR m BRBGBM,EKEX
f(z) = c?m(z?z0)?m +···+ c?2(z?z0)?2
+c?1(z?z0)?1 + c0 + c1(z?z0) +···
EX (z?z0)m EMH2EQARH0G8,AQ
(z?z0)mf(z) = c?m + c?m+1(z?z0) +···
+c?1(z?z0)m?1 + c0(z?z0)m +···
H0BQEM m?1 AUAJGI,AQ
dm?1
dzm?1{(z?z0)
mf(z)} = (m? 1)!c?1 +{
F5CZz?z0AKE8ARE1},
B9 z → z0,H0G8EMBGDL,BPAQ.
44/107
BHC3 2 CGC2 z0 F0 f(z) AR m BRBGBM,BSD3
Res[f(z),z0] = 1(m? 1)! limz→z
0
dm?1
dzm?1{(z?z0)
mf(z)}.
DD FR? z0 F0 f(z) AR m BRBGBM,EKEX
f(z) = c?m(z?z0)?m +···+ c?2(z?z0)?2
+c?1(z?z0)?1 + c0 + c1(z?z0) +···
EX (z?z0)m EMH2EQARH0G8,AQ
(z?z0)mf(z) = c?m + c?m+1(z?z0) +···
+c?1(z?z0)m?1 + c0(z?z0)m +···
H0BQEM m?1 AUAJGI,AQ
dm?1
dzm?1{(z?z0)
mf(z)} = (m? 1)!c?1 +{
F5CZz?z0AKE8ARE1},
B9 z → z0,H0G8EMBGDL,BPAQ.
44/107
BHC3 2 CGC2 z0 F0 f(z) AR m BRBGBM,BSD3
Res[f(z),z0] = 1(m? 1)! limz→z
0
dm?1
dzm?1{(z?z0)
mf(z)}.
DD FR? z0 F0 f(z) AR m BRBGBM,EKEX
f(z) = c?m(z?z0)?m +···+ c?2(z?z0)?2
+c?1(z?z0)?1 + c0 + c1(z?z0) +···
EX (z?z0)m EMH2EQARH0G8,AQ
(z?z0)mf(z) = c?m + c?m+1(z?z0) +···
+c?1(z?z0)m?1 + c0(z?z0)m +···
H0BQEM m?1 AUAJGI,AQ
dm?1
dzm?1{(z?z0)
mf(z)} = (m? 1)!c?1 +{
F5CZz?z0AKE8ARE1},
B9 z → z0,H0G8EMBGDL,BPAQ.
44/107
BHC3 2 CGC2 z0 F0 f(z) AR m BRBGBM,BSD3
Res[f(z),z0] = 1(m? 1)! limz→z
0
dm?1
dzm?1{(z?z0)
mf(z)}.
DD FR? z0 F0 f(z) AR m BRBGBM,EKEX
f(z) = c?m(z?z0)?m +···+ c?2(z?z0)?2
+c?1(z?z0)?1 + c0 + c1(z?z0) +···
EX (z?z0)m EMH2EQARH0G8,AQ
(z?z0)mf(z) = c?m + c?m+1(z?z0) +···
+c?1(z?z0)m?1 + c0(z?z0)m +···
H0BQEM m?1 AUAJGI,AQ
dm?1
dzm?1{(z?z0)
mf(z)} = (m? 1)!c?1 +{
F5CZz?z0AKE8ARE1},
B9 z → z0,H0G8EMBGDL,BPAQ.
44/107
BHC3 2 CGC2 z0 F0 f(z) AR m BRBGBM,BSD3
Res[f(z),z0] = 1(m? 1)! limz→z
0
dm?1
dzm?1{(z?z0)
mf(z)}.
DD FR? z0 F0 f(z) AR m BRBGBM,EKEX
f(z) = c?m(z?z0)?m +···+ c?2(z?z0)?2
+c?1(z?z0)?1 + c0 + c1(z?z0) +···
EX (z?z0)m EMH2EQARH0G8,AQ
(z?z0)mf(z) = c?m + c?m+1(z?z0) +···
+c?1(z?z0)m?1 + c0(z?z0)m +···
H0BQEM m?1 AUAJGI,AQ
dm?1
dzm?1{(z?z0)
mf(z)} = (m? 1)!c?1 +{
F5CZz?z0AKE8ARE1},
B9 z → z0,H0G8EMBGDL,BPAQ.
44/107
BHC3 3 AN f(z) = P(z)Q(z),P(z) BL Q(z) BF z0 FTB5G3,
CGC2 P(z0) negationslash= 0,Q(z0) = 0,Q′(z0) negationslash= 0,BSD3 z0? f(z)
ARE8BRBGBM,AD
Res[f(z),z0] = P(z0)Q′(z
0)
.
DD FGCZ,DDEY z0 F0 f(z) ARE8BRBGBM.
FR? Q(z0) = 0,Q′(z0) negationslash= 0,EKEX z0? Q(z) ARE8BRAY
BM,D0AD z0?
1
Q(z) ARE8BRBGBM,FRCS
1
Q(z) =
1
z?z0 ·?(z),
G1EL,?(z) BF z0 B5G3,DP?(z0) negationslash= 0.
45/107
BHC3 3 AN f(z) = P(z)Q(z),P(z) BL Q(z) BF z0 FTB5G3,
CGC2 P(z0) negationslash= 0,Q(z0) = 0,Q′(z0) negationslash= 0,BSD3 z0? f(z)
ARE8BRBGBM,AD
Res[f(z),z0] = P(z0)Q′(z
0)
.
DD FGCZ,DDEY z0 F0 f(z) ARE8BRBGBM.
FR? Q(z0) = 0,Q′(z0) negationslash= 0,EKEX z0? Q(z) ARE8BRAY
BM,D0AD z0?
1
Q(z) ARE8BRBGBM,FRCS
1
Q(z) =
1
z?z0 ·?(z),
G1EL,?(z) BF z0 B5G3,DP?(z0) negationslash= 0.
45/107
BHC3 3 AN f(z) = P(z)Q(z),P(z) BL Q(z) BF z0 FTB5G3,
CGC2 P(z0) negationslash= 0,Q(z0) = 0,Q′(z0) negationslash= 0,BSD3 z0? f(z)
ARE8BRBGBM,AD
Res[f(z),z0] = P(z0)Q′(z
0)
.
DD FGCZ,DDEY z0 F0 f(z) ARE8BRBGBM.
FR? Q(z0) = 0,Q′(z0) negationslash= 0,EKEX z0? Q(z) ARE8BRAY
BM,D0AD z0?
1
Q(z) ARE8BRBGBM,FRCS
1
Q(z) =
1
z?z0 ·?(z),
G1EL,?(z) BF z0 B5G3,DP?(z0) negationslash= 0.
45/107
FRCSAQ
f(z) = 1z?z
0
·?(z)P(z)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright,
G1EL,g(z) defines?(z)P(z)BF z0 B5G3,DPg(z0) =?(z0)P(z0) negationslash= 0.
AP z0? f(z) ARE8BRBGBM.
B8A0,DUGABHC3 1,CZ
Res[f(z),z0] = limz→z
0
(z?z0)f(z).
DGCM Q(z0) = 0,C3CZ
Res[f(z),z0] = limz→z
0
(z?z0)P(x)Q(z) = limz→z
0
P(z)
Q(z)?Q(z0)
z?z0
= P(z0)Q′(z
0)
46/107
FRCSAQ
f(z) = 1z?z
0
·?(z)P(z)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright,
G1EL,g(z) defines?(z)P(z)BF z0 B5G3,DPg(z0) =?(z0)P(z0) negationslash= 0.
AP z0? f(z) ARE8BRBGBM.
B8A0,DUGABHC3 1,CZ
Res[f(z),z0] = limz→z
0
(z?z0)f(z).
DGCM Q(z0) = 0,C3CZ
Res[f(z),z0] = limz→z
0
(z?z0)P(x)Q(z) = limz→z
0
P(z)
Q(z)?Q(z0)
z?z0
= P(z0)Q′(z
0)
46/107
FRCSAQ
f(z) = 1z?z
0
·?(z)P(z)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright,
G1EL,g(z) defines?(z)P(z)BF z0 B5G3,DPg(z0) =?(z0)P(z0) negationslash= 0.
AP z0? f(z) ARE8BRBGBM.
B8A0,DUGABHC3 1,CZ
Res[f(z),z0] = limz→z
0
(z?z0)f(z).
DGCM Q(z0) = 0,C3CZ
Res[f(z),z0] = limz→z
0
(z?z0)P(x)Q(z) = limz→z
0
P(z)
Q(z)?Q(z0)
z?z0
= P(z0)Q′(z
0)
46/107
FRCSAQ
f(z) = 1z?z
0
·?(z)P(z)bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright,
G1EL,g(z) defines?(z)P(z)BF z0 B5G3,DPg(z0) =?(z0)P(z0) negationslash= 0.
AP z0? f(z) ARE8BRBGBM.
B8A0,DUGABHC3 1,CZ
Res[f(z),z0] = limz→z
0
(z?z0)f(z).
DGCM Q(z0) = 0,C3CZ
Res[f(z),z0] = limz→z
0
(z?z0)P(x)Q(z) = limz→z
0
P(z)
Q(z)?Q(z0)
z?z0
= P(z0)Q′(z
0)
46/107
DI 1 CMGTERE0DFEM f(z) = 5z?2z(z? 1) BF z = 0
G2 z = 1 BRARBFGI.
B5 E0DFE8DGCMGAC3FSD1DF,
FR? f(z) BF 0 < |z| < 1 CBARGAC3FSD1EQF0
f(z) = 5z? 2z(z?1)
=?
parenleftbigg
5? 2z
parenrightbiggparenleftBigg ∞summationdisplay
n=0
zn
parenrightBigg
=?5(1 + z + z2 +···) + 2
parenleftbigg1
z + 1 + z + z
2 +···
parenrightbigg
,
0 < |z| < 1
EKEX,Res[f(z),0] = c?1 = 2.
47/107
DI 1 CMGTERE0DFEM f(z) = 5z?2z(z? 1) BF z = 0
G2 z = 1 BRARBFGI.
B5 E0DFE8DGCMGAC3FSD1DF,
FR? f(z) BF 0 < |z| < 1 CBARGAC3FSD1EQF0
f(z) = 5z? 2z(z?1)
=?
parenleftbigg
5? 2z
parenrightbiggparenleftBigg ∞summationdisplay
n=0
zn
parenrightBigg
=?5(1 + z + z2 +···) + 2
parenleftbigg1
z + 1 + z + z
2 +···
parenrightbigg
,
0 < |z| < 1
EKEX,Res[f(z),0] = c?1 = 2.
47/107
DI 1 CMGTERE0DFEM f(z) = 5z?2z(z? 1) BF z = 0
G2 z = 1 BRARBFGI.
B5 E0DFE8DGCMGAC3FSD1DF,
FR? f(z) BF 0 < |z| < 1 CBARGAC3FSD1EQF0
f(z) = 5z? 2z(z?1)
=?
parenleftbigg
5? 2z
parenrightbiggparenleftBigg ∞summationdisplay
n=0
zn
parenrightBigg
=?5(1 + z + z2 +···) + 2
parenleftbigg1
z + 1 + z + z
2 +···
parenrightbigg
,
0 < |z| < 1
EKEX,Res[f(z),0] = c?1 = 2.
47/107
DI 1 CMGTERE0DFEM f(z) = 5z?2z(z? 1) BF z = 0
G2 z = 1 BRARBFGI.
B5 E0DFE8DGCMGAC3FSD1DF,
FR? f(z) BF 0 < |z| < 1 CBARGAC3FSD1EQF0
f(z) = 5z? 2z(z?1)
=?
parenleftbigg
5? 2z
parenrightbiggparenleftBigg ∞summationdisplay
n=0
zn
parenrightBigg
=?5(1 + z + z2 +···) + 2
parenleftbigg1
z + 1 + z + z
2 +···
parenrightbigg
,
0 < |z| < 1
EKEX,Res[f(z),0] = c?1 = 2.
47/107
DI 1 CMGTERE0DFEM f(z) = 5z?2z(z? 1) BF z = 0
G2 z = 1 BRARBFGI.
B5 E0DFE8DGCMGAC3FSD1DF,
FR? f(z) BF 0 < |z| < 1 CBARGAC3FSD1EQF0
f(z) = 5z? 2z(z?1)
=?
parenleftbigg
5? 2z
parenrightbiggparenleftBigg ∞summationdisplay
n=0
zn
parenrightBigg
=?5(1 + z + z2 +···) + 2
parenleftbigg1
z + 1 + z + z
2 +···
parenrightbigg
,
0 < |z| < 1
EKEX,Res[f(z),0] = c?1 = 2.
47/107
FR? f(z) BF 0 < |z?1| < 1 CBARGAC3FSD1EQF0
f(z) = 5z?2z(z? 1) = 5(z? 1) + 3z? 1 · 11 + (z? 1)
=
parenleftbigg
5 + 3z?1
parenrightbiggparenleftBigg ∞summationdisplay
n=0
(?1)n(z?1)n
parenrightBigg
=?5[1? (z? 1) + (z?1)2 +···)
+3
bracketleftbigg 1
z? 1?1 + (z? 1)?···
bracketrightbigg
,
0 < |z? 1| < 1
EKEX Res[f(z),1] = c?1 = 3.
48/107
FR? f(z) BF 0 < |z?1| < 1 CBARGAC3FSD1EQF0
f(z) = 5z?2z(z? 1) = 5(z? 1) + 3z? 1 · 11 + (z? 1)
=
parenleftbigg
5 + 3z?1
parenrightbiggparenleftBigg ∞summationdisplay
n=0
(?1)n(z?1)n
parenrightBigg
=?5[1? (z? 1) + (z?1)2 +···)
+3
bracketleftbigg 1
z? 1?1 + (z? 1)?···
bracketrightbigg
,
0 < |z? 1| < 1
EKEX Res[f(z),1] = c?1 = 3.
48/107
FR? f(z) BF 0 < |z?1| < 1 CBARGAC3FSD1EQF0
f(z) = 5z?2z(z? 1) = 5(z? 1) + 3z? 1 · 11 + (z? 1)
=
parenleftbigg
5 + 3z?1
parenrightbiggparenleftBigg ∞summationdisplay
n=0
(?1)n(z?1)n
parenrightBigg
=?5[1? (z? 1) + (z?1)2 +···)
+3
bracketleftbigg 1
z? 1?1 + (z? 1)?···
bracketrightbigg
,
0 < |z? 1| < 1
EKEX Res[f(z),1] = c?1 = 3.
48/107
FR? f(z) BF 0 < |z?1| < 1 CBARGAC3FSD1EQF0
f(z) = 5z?2z(z? 1) = 5(z? 1) + 3z? 1 · 11 + (z? 1)
=
parenleftbigg
5 + 3z?1
parenrightbiggparenleftBigg ∞summationdisplay
n=0
(?1)n(z?1)n
parenrightBigg
=?5[1? (z? 1) + (z?1)2 +···)
+3
bracketleftbigg 1
z? 1?1 + (z? 1)?···
bracketrightbigg
,
0 < |z? 1| < 1
EKEX Res[f(z),1] = c?1 = 3.
48/107
FR? f(z) BF 0 < |z?1| < 1 CBARGAC3FSD1EQF0
f(z) = 5z?2z(z? 1) = 5(z? 1) + 3z? 1 · 11 + (z? 1)
=
parenleftbigg
5 + 3z?1
parenrightbiggparenleftBigg ∞summationdisplay
n=0
(?1)n(z?1)n
parenrightBigg
=?5[1? (z? 1) + (z?1)2 +···)
+3
bracketleftbigg 1
z? 1?1 + (z? 1)?···
bracketrightbigg
,
0 < |z? 1| < 1
EKEX Res[f(z),1] = c?1 = 3.
48/107
E0DFAJ DUGABHC3 1
Res[f(z),0] = limz→0z · 5z? 2z(z?1) = 2,
Res[f(z),1] = limz→1(z? 1)· 5z? 2z(z?1) = 3.
E0DFD2 DUGABHC3 3
Res[f(z),0] = 5z?2[z(z?1)]′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 2,
Res[f(z),1] = 5z?2[z(z?1)]′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
= 3.
49/107
E0DFAJ DUGABHC3 1
Res[f(z),0] = limz→0z · 5z? 2z(z?1) = 2,
Res[f(z),1] = limz→1(z? 1)· 5z? 2z(z?1) = 3.
E0DFD2 DUGABHC3 3
Res[f(z),0] = 5z?2[z(z?1)]′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 2,
Res[f(z),1] = 5z?2[z(z?1)]′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
= 3.
49/107
E0DFAJ DUGABHC3 1
Res[f(z),0] = limz→0z · 5z? 2z(z?1) = 2,
Res[f(z),1] = limz→1(z? 1)· 5z? 2z(z?1) = 3.
E0DFD2 DUGABHC3 3
Res[f(z),0] = 5z?2[z(z?1)]′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 2,
Res[f(z),1] = 5z?2[z(z?1)]′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
= 3.
49/107
E0DFAJ DUGABHC3 1
Res[f(z),0] = limz→0z · 5z? 2z(z?1) = 2,
Res[f(z),1] = limz→1(z? 1)· 5z? 2z(z?1) = 3.
E0DFD2 DUGABHC3 3
Res[f(z),0] = 5z?2[z(z?1)]′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 2,
Res[f(z),1] = 5z?2[z(z?1)]′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
= 3.
49/107
E0DFD3 DUGADPG4FKEPGYEQ,FR?
Res[f(z),z0] = 12pii
contintegraldisplay
C
f(z)dz = c?1.
G1EL C F0 z0 ARB8AMB1APDXCBE8CMDFH0BGETDM,D9F0DWEX
FRC4C7DYFKEP
1
2pii
contintegraldisplay
C
f(z)dz ARDUEMBFGI.
GHD9ATC3,CZ
Res[f(z),0] = 12pii
contintegraldisplay
|z|=12
5z?2
z(z? 1)dz
= 12pii
contintegraldisplay
|z|=12
5z? 2
z?1 ·
dz
z =
parenleftbigg5z? 2
z? 1
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 2,
Res[f(z),1] = 12pii
contintegraldisplay
|z?1|=12
5z?2
z(z? 1)dz
= 12pii
contintegraldisplay
|z?1|=12
5z? 2
z ·
dz
z? 1 =
parenleftbigg5z? 2
z
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 3.
50/107
E0DFD3 DUGADPG4FKEPGYEQ,FR?
Res[f(z),z0] = 12pii
contintegraldisplay
C
f(z)dz = c?1.
G1EL C F0 z0 ARB8AMB1APDXCBE8CMDFH0BGETDM,D9F0DWEX
FRC4C7DYFKEP
1
2pii
contintegraldisplay
C
f(z)dz ARDUEMBFGI.
GHD9ATC3,CZ
Res[f(z),0] = 12pii
contintegraldisplay
|z|=12
5z?2
z(z? 1)dz
= 12pii
contintegraldisplay
|z|=12
5z? 2
z?1 ·
dz
z =
parenleftbigg5z? 2
z? 1
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 2,
Res[f(z),1] = 12pii
contintegraldisplay
|z?1|=12
5z?2
z(z? 1)dz
= 12pii
contintegraldisplay
|z?1|=12
5z? 2
z ·
dz
z? 1 =
parenleftbigg5z? 2
z
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 3.
50/107
E0DFD3 DUGADPG4FKEPGYEQ,FR?
Res[f(z),z0] = 12pii
contintegraldisplay
C
f(z)dz = c?1.
G1EL C F0 z0 ARB8AMB1APDXCBE8CMDFH0BGETDM,D9F0DWEX
FRC4C7DYFKEP
1
2pii
contintegraldisplay
C
f(z)dz ARDUEMBFGI.
GHD9ATC3,CZ
Res[f(z),0] = 12pii
contintegraldisplay
|z|=12
5z?2
z(z? 1)dz
= 12pii
contintegraldisplay
|z|=12
5z? 2
z?1 ·
dz
z =
parenleftbigg5z? 2
z? 1
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 2,
Res[f(z),1] = 12pii
contintegraldisplay
|z?1|=12
5z?2
z(z? 1)dz
= 12pii
contintegraldisplay
|z?1|=12
5z? 2
z ·
dz
z? 1 =
parenleftbigg5z? 2
z
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 3.
50/107
E0DFD3 DUGADPG4FKEPGYEQ,FR?
Res[f(z),z0] = 12pii
contintegraldisplay
C
f(z)dz = c?1.
G1EL C F0 z0 ARB8AMB1APDXCBE8CMDFH0BGETDM,D9F0DWEX
FRC4C7DYFKEP
1
2pii
contintegraldisplay
C
f(z)dz ARDUEMBFGI.
GHD9ATC3,CZ
Res[f(z),0] = 12pii
contintegraldisplay
|z|=12
5z?2
z(z? 1)dz
= 12pii
contintegraldisplay
|z|=12
5z? 2
z?1 ·
dz
z =
parenleftbigg5z? 2
z? 1
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 2,
Res[f(z),1] = 12pii
contintegraldisplay
|z?1|=12
5z?2
z(z? 1)dz
= 12pii
contintegraldisplay
|z?1|=12
5z? 2
z ·
dz
z? 1 =
parenleftbigg5z? 2
z
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
z=0
= 3.
50/107
DI 2 EMF8GI f(z) = z? sinzz6 BF z = 0 BRARBFGI.
B5E0DFE8 DGCMGAC3BRGIFSD1EQ
z? sinz
z6 =
1
z6
bracketleftbigg
z?
parenleftbigg
z? 13!z3 + 15!z5?···
parenrightbiggbracketrightbigg
= 13!z3? 15!z + ···
EKEX
Res
bracketleftbiggz? sin
z6,0
bracketrightbigg
=?15!.
51/107
DI 2 EMF8GI f(z) = z? sinzz6 BF z = 0 BRARBFGI.
B5E0DFE8 DGCMGAC3BRGIFSD1EQ
z? sinz
z6 =
1
z6
bracketleftbigg
z?
parenleftbigg
z? 13!z3 + 15!z5?···
parenrightbiggbracketrightbigg
= 13!z3? 15!z + ···
EKEX
Res
bracketleftbiggz? sin
z6,0
bracketrightbigg
=?15!.
51/107
DI 2 EMF8GI f(z) = z? sinzz6 BF z = 0 BRARBFGI.
B5E0DFE8 DGCMGAC3BRGIFSD1EQ
z? sinz
z6 =
1
z6
bracketleftbigg
z?
parenleftbigg
z? 13!z3 + 15!z5?···
parenrightbiggbracketrightbigg
= 13!z3? 15!z + ···
EKEX
Res
bracketleftbiggz? sin
z6,0
bracketrightbigg
=?15!.
51/107
DI 2 EMF8GI f(z) = z? sinzz6 BF z = 0 BRARBFGI.
B5E0DFE8 DGCMGAC3BRGIFSD1EQ
z? sinz
z6 =
1
z6
bracketleftbigg
z?
parenleftbigg
z? 13!z3 + 15!z5?···
parenrightbiggbracketrightbigg
= 13!z3? 15!z + ···
EKEX
Res
bracketleftbiggz? sin
z6,0
bracketrightbigg
=?15!.
51/107
E0DFAJ DGCMBHC3 2,
FGCZB4F9BGBM z = 0 ARBRGI.
B9 P(x) = z? sinz,CSD9
P(0) = (z? sinz)|z=0 = 0,P′(0) = (1? cosz)|z=0 = 0,
P′′(0) = sinz|z=0 = 0,P′′′(0) = cosz|z=0 = 1 negationslash= 0,
FRCS z = 0 F0 z?sinz ARD2BRAYBM,EKEX z = 0 F0 f(z)
ARD2BRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbiggz? sinz
z6,0
bracketrightbigg
= 1(3?1)! limz→0 d
2
dz2
bracketleftbigg
z3 · z? sinzz6
bracketrightbigg
= 12 limz→0 d
2
dz2
bracketleftbiggz? sinz
z3
bracketrightbigg
=? 15!.
52/107
E0DFAJ DGCMBHC3 2,
FGCZB4F9BGBM z = 0 ARBRGI.
B9 P(x) = z? sinz,CSD9
P(0) = (z? sinz)|z=0 = 0,P′(0) = (1? cosz)|z=0 = 0,
P′′(0) = sinz|z=0 = 0,P′′′(0) = cosz|z=0 = 1 negationslash= 0,
FRCS z = 0 F0 z?sinz ARD2BRAYBM,EKEX z = 0 F0 f(z)
ARD2BRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbiggz? sinz
z6,0
bracketrightbigg
= 1(3?1)! limz→0 d
2
dz2
bracketleftbigg
z3 · z? sinzz6
bracketrightbigg
= 12 limz→0 d
2
dz2
bracketleftbiggz? sinz
z3
bracketrightbigg
=? 15!.
52/107
E0DFAJ DGCMBHC3 2,
FGCZB4F9BGBM z = 0 ARBRGI.
B9 P(x) = z? sinz,CSD9
P(0) = (z? sinz)|z=0 = 0,P′(0) = (1? cosz)|z=0 = 0,
P′′(0) = sinz|z=0 = 0,P′′′(0) = cosz|z=0 = 1 negationslash= 0,
FRCS z = 0 F0 z?sinz ARD2BRAYBM,EKEX z = 0 F0 f(z)
ARD2BRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbiggz? sinz
z6,0
bracketrightbigg
= 1(3?1)! limz→0 d
2
dz2
bracketleftbigg
z3 · z? sinzz6
bracketrightbigg
= 12 limz→0 d
2
dz2
bracketleftbiggz? sinz
z3
bracketrightbigg
=? 15!.
52/107
E0DFAJ DGCMBHC3 2,
FGCZB4F9BGBM z = 0 ARBRGI.
B9 P(x) = z? sinz,CSD9
P(0) = (z? sinz)|z=0 = 0,P′(0) = (1? cosz)|z=0 = 0,
P′′(0) = sinz|z=0 = 0,P′′′(0) = cosz|z=0 = 1 negationslash= 0,
FRCS z = 0 F0 z?sinz ARD2BRAYBM,EKEX z = 0 F0 f(z)
ARD2BRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbiggz? sinz
z6,0
bracketrightbigg
= 1(3?1)! limz→0 d
2
dz2
bracketleftbigg
z3 · z? sinzz6
bracketrightbigg
= 12 limz→0 d
2
dz2
bracketleftbiggz? sinz
z3
bracketrightbigg
=? 15!.
52/107
E0DFAJ DGCMBHC3 2,
FGCZB4F9BGBM z = 0 ARBRGI.
B9 P(x) = z? sinz,CSD9
P(0) = (z? sinz)|z=0 = 0,P′(0) = (1? cosz)|z=0 = 0,
P′′(0) = sinz|z=0 = 0,P′′′(0) = cosz|z=0 = 1 negationslash= 0,
FRCS z = 0 F0 z?sinz ARD2BRAYBM,EKEX z = 0 F0 f(z)
ARD2BRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbiggz? sinz
z6,0
bracketrightbigg
= 1(3?1)! limz→0 d
2
dz2
bracketleftbigg
z3 · z? sinzz6
bracketrightbigg
= 12 limz→0 d
2
dz2
bracketleftbiggz? sinz
z3
bracketrightbigg
=? 15!.
52/107
E0DFAJ DGCMBHC3 2,
FGCZB4F9BGBM z = 0 ARBRGI.
B9 P(x) = z? sinz,CSD9
P(0) = (z? sinz)|z=0 = 0,P′(0) = (1? cosz)|z=0 = 0,
P′′(0) = sinz|z=0 = 0,P′′′(0) = cosz|z=0 = 1 negationslash= 0,
FRCS z = 0 F0 z?sinz ARD2BRAYBM,EKEX z = 0 F0 f(z)
ARD2BRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbiggz? sinz
z6,0
bracketrightbigg
= 1(3?1)! limz→0 d
2
dz2
bracketleftbigg
z3 · z? sinzz6
bracketrightbigg
= 12 limz→0 d
2
dz2
bracketleftbiggz? sinz
z3
bracketrightbigg
=? 15!.
52/107
E0DFAJ DGCMBHC3 2,
FGCZB4F9BGBM z = 0 ARBRGI.
B9 P(x) = z? sinz,CSD9
P(0) = (z? sinz)|z=0 = 0,P′(0) = (1? cosz)|z=0 = 0,
P′′(0) = sinz|z=0 = 0,P′′′(0) = cosz|z=0 = 1 negationslash= 0,
FRCS z = 0 F0 z?sinz ARD2BRAYBM,EKEX z = 0 F0 f(z)
ARD2BRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbiggz? sinz
z6,0
bracketrightbigg
= 1(3?1)! limz→0 d
2
dz2
bracketleftbigg
z3 · z? sinzz6
bracketrightbigg
= 12 limz→0 d
2
dz2
bracketleftbiggz? sinz
z3
bracketrightbigg
=? 15!.
52/107
BH B3D9BHC3 2,
Res[f(z),z0] = 1(m? 1)! limz→z
0
dm?1
dzm?1{(z?z0)
mf(z)},
CGC2F8GI f(z) ARBGBM z0 AREICBBRGIB3 m B0,CCGYEQBX
B8EKG9,CZEE?AAC7DYE0BU,GQ m AIAQB3EICBARBRGID4.
E0DFD2 CGC2BFBHC3 2 ELAI m = 6,C3CZ
Res
bracketleftbiggz?sinz
z6,0
bracketrightbigg
= 1(6? 1)! limz→0 d
5
dz5
bracketleftbigg
z6 · z? sinzz6
bracketrightbigg
= 15! limz→0 d
5
dz5 [z? sinz] =?
1
5!.
53/107
BH B3D9BHC3 2,
Res[f(z),z0] = 1(m? 1)! limz→z
0
dm?1
dzm?1{(z?z0)
mf(z)},
CGC2F8GI f(z) ARBGBM z0 AREICBBRGIB3 m B0,CCGYEQBX
B8EKG9,CZEE?AAC7DYE0BU,GQ m AIAQB3EICBARBRGID4.
E0DFD2 CGC2BFBHC3 2 ELAI m = 6,C3CZ
Res
bracketleftbiggz?sinz
z6,0
bracketrightbigg
= 1(6? 1)! limz→0 d
5
dz5
bracketleftbigg
z6 · z? sinzz6
bracketrightbigg
= 15! limz→0 d
5
dz5 [z? sinz] =?
1
5!.
53/107
BH B3D9BHC3 2,
Res[f(z),z0] = 1(m? 1)! limz→z
0
dm?1
dzm?1{(z?z0)
mf(z)},
CGC2F8GI f(z) ARBGBM z0 AREICBBRGIB3 m B0,CCGYEQBX
B8EKG9,CZEE?AAC7DYE0BU,GQ m AIAQB3EICBARBRGID4.
E0DFD2 CGC2BFBHC3 2 ELAI m = 6,C3CZ
Res
bracketleftbiggz?sinz
z6,0
bracketrightbigg
= 1(6? 1)! limz→0 d
5
dz5
bracketleftbigg
z6 · z? sinzz6
bracketrightbigg
= 15! limz→0 d
5
dz5 [z? sinz] =?
1
5!.
53/107
BH B3D9BHC3 2,
Res[f(z),z0] = 1(m? 1)! limz→z
0
dm?1
dzm?1{(z?z0)
mf(z)},
CGC2F8GI f(z) ARBGBM z0 AREICBBRGIB3 m B0,CCGYEQBX
B8EKG9,CZEE?AAC7DYE0BU,GQ m AIAQB3EICBARBRGID4.
E0DFD2 CGC2BFBHC3 2 ELAI m = 6,C3CZ
Res
bracketleftbiggz?sinz
z6,0
bracketrightbigg
= 1(6? 1)! limz→0 d
5
dz5
bracketleftbigg
z6 · z? sinzz6
bracketrightbigg
= 15! limz→0 d
5
dz5 [z? sinz] =?
1
5!.
53/107
DI 3 EMF8GI f(z) = cos 11?z BFCZDLG3BMBRARBFGI.
B5 f(z) = cos 11?z CZG3BM z = 1.
DUGAGAC3BRGIFSD1EQ
cos 11?z = cos 1z?1 = 1? 12! 1(z?1)2 + 14! 1(z? 1)4?···
EKEX z = 1 F0 f(z) ARATBHBGBM,DP
Res
bracketleftbigg
cos 11?z,1
bracketrightbigg
= c?1 = 0.
54/107
DI 3 EMF8GI f(z) = cos 11?z BFCZDLG3BMBRARBFGI.
B5 f(z) = cos 11?z CZG3BM z = 1.
DUGAGAC3BRGIFSD1EQ
cos 11?z = cos 1z?1 = 1? 12! 1(z?1)2 + 14! 1(z? 1)4?···
EKEX z = 1 F0 f(z) ARATBHBGBM,DP
Res
bracketleftbigg
cos 11?z,1
bracketrightbigg
= c?1 = 0.
54/107
DI 3 EMF8GI f(z) = cos 11?z BFCZDLG3BMBRARBFGI.
B5 f(z) = cos 11?z CZG3BM z = 1.
DUGAGAC3BRGIFSD1EQ
cos 11?z = cos 1z?1 = 1? 12! 1(z?1)2 + 14! 1(z? 1)4?···
EKEX z = 1 F0 f(z) ARATBHBGBM,DP
Res
bracketleftbigg
cos 11?z,1
bracketrightbigg
= c?1 = 0.
54/107
DI 3 EMF8GI f(z) = cos 11?z BFCZDLG3BMBRARBFGI.
B5 f(z) = cos 11?z CZG3BM z = 1.
DUGAGAC3BRGIFSD1EQ
cos 11?z = cos 1z?1 = 1? 12! 1(z?1)2 + 14! 1(z? 1)4?···
EKEX z = 1 F0 f(z) ARATBHBGBM,DP
Res
bracketleftbigg
cos 11?z,1
bracketrightbigg
= c?1 = 0.
54/107
DI 4 EMF8GI f(z) = z2sin1z BFCZDLG3BMBRARBFGI.
B5BM z = 0 F0 f(z) ARG3BM.
BF z = 0 ARAMB1APDX 0 < |z| < +∞ CB,DUGAGAC3BRGI
FSD1EQ
z2sin1z = z2
parenleftbigg1
z?
1
3!
1
z3 +
1
5!
1
z5?···
parenrightbigg
= z? 13!1z + 15! 1z3?···
DWDJ z = 0 F0 z2sin1z ARATBHG3BM,DP
Res
bracketleftbigg
z2sin1z,0
bracketrightbigg
=?13! =?16.
55/107
DI 4 EMF8GI f(z) = z2sin1z BFCZDLG3BMBRARBFGI.
B5BM z = 0 F0 f(z) ARG3BM.
BF z = 0 ARAMB1APDX 0 < |z| < +∞ CB,DUGAGAC3BRGI
FSD1EQ
z2sin1z = z2
parenleftbigg1
z?
1
3!
1
z3 +
1
5!
1
z5?···
parenrightbigg
= z? 13!1z + 15! 1z3?···
DWDJ z = 0 F0 z2sin1z ARATBHG3BM,DP
Res
bracketleftbigg
z2sin1z,0
bracketrightbigg
=?13! =?16.
55/107
DI 4 EMF8GI f(z) = z2sin1z BFCZDLG3BMBRARBFGI.
B5BM z = 0 F0 f(z) ARG3BM.
BF z = 0 ARAMB1APDX 0 < |z| < +∞ CB,DUGAGAC3BRGI
FSD1EQ
z2sin1z = z2
parenleftbigg1
z?
1
3!
1
z3 +
1
5!
1
z5?···
parenrightbigg
= z? 13!1z + 15! 1z3?···
DWDJ z = 0 F0 z2sin1z ARATBHG3BM,DP
Res
bracketleftbigg
z2sin1z,0
bracketrightbigg
=?13! =?16.
55/107
DI 4 EMF8GI f(z) = z2sin1z BFCZDLG3BMBRARBFGI.
B5BM z = 0 F0 f(z) ARG3BM.
BF z = 0 ARAMB1APDX 0 < |z| < +∞ CB,DUGAGAC3BRGI
FSD1EQ
z2sin1z = z2
parenleftbigg1
z?
1
3!
1
z3 +
1
5!
1
z5?···
parenrightbigg
= z? 13!1z + 15! 1z3?···
DWDJ z = 0 F0 z2sin1z ARATBHG3BM,DP
Res
bracketleftbigg
z2sin1z,0
bracketrightbigg
=?13! =?16.
55/107
DI 4 EMF8GI f(z) = z2sin1z BFCZDLG3BMBRARBFGI.
B5BM z = 0 F0 f(z) ARG3BM.
BF z = 0 ARAMB1APDX 0 < |z| < +∞ CB,DUGAGAC3BRGI
FSD1EQ
z2sin1z = z2
parenleftbigg1
z?
1
3!
1
z3 +
1
5!
1
z5?···
parenrightbigg
= z? 13!1z + 15! 1z3?···
DWDJ z = 0 F0 z2sin1z ARATBHG3BM,DP
Res
bracketleftbigg
z2sin1z,0
bracketrightbigg
=?13! =?16.
55/107
DI 5 EMF8GI f(z) = 1zsinz BFCZDLG3BMBRARBFGI.
B5 z = 0 F0F8GI zsinz ARAJBRAYBM,EKEX z = 0 F0F8
GI
1
zsinz ARAJBRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbigg 1
zsinz,0
bracketrightbigg
= limz→0 ddz
parenleftbigg
z2 · 1zsinz
parenrightbigg
= limz→0 sinz?zcoszz2 ·
parenleftBig z
sinz
parenrightBig2
= 0.
B8A9,zk = kpi (k = ±1,±2,···)?F8GIARE8BRAYBM,
EKEX zk = kpi (k = ±1,±2,···) F0F8GIARE8BRBGBM.
DUGABHC3 3,CZ
Res
bracketleftbigg 1
zsinz,kpi
bracketrightbigg
=
1
z
(sinz)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=kpi
= 1kpicoskpi = (?1)
k
kpi,(k = ±1,±2,···)
56/107
DI 5 EMF8GI f(z) = 1zsinz BFCZDLG3BMBRARBFGI.
B5 z = 0 F0F8GI zsinz ARAJBRAYBM,EKEX z = 0 F0F8
GI
1
zsinz ARAJBRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbigg 1
zsinz,0
bracketrightbigg
= limz→0 ddz
parenleftbigg
z2 · 1zsinz
parenrightbigg
= limz→0 sinz?zcoszz2 ·
parenleftBig z
sinz
parenrightBig2
= 0.
B8A9,zk = kpi (k = ±1,±2,···)?F8GIARE8BRAYBM,
EKEX zk = kpi (k = ±1,±2,···) F0F8GIARE8BRBGBM.
DUGABHC3 3,CZ
Res
bracketleftbigg 1
zsinz,kpi
bracketrightbigg
=
1
z
(sinz)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=kpi
= 1kpicoskpi = (?1)
k
kpi,(k = ±1,±2,···)
56/107
DI 5 EMF8GI f(z) = 1zsinz BFCZDLG3BMBRARBFGI.
B5 z = 0 F0F8GI zsinz ARAJBRAYBM,EKEX z = 0 F0F8
GI
1
zsinz ARAJBRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbigg 1
zsinz,0
bracketrightbigg
= limz→0 ddz
parenleftbigg
z2 · 1zsinz
parenrightbigg
= limz→0 sinz?zcoszz2 ·
parenleftBig z
sinz
parenrightBig2
= 0.
B8A9,zk = kpi (k = ±1,±2,···)?F8GIARE8BRAYBM,
EKEX zk = kpi (k = ±1,±2,···) F0F8GIARE8BRBGBM.
DUGABHC3 3,CZ
Res
bracketleftbigg 1
zsinz,kpi
bracketrightbigg
=
1
z
(sinz)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=kpi
= 1kpicoskpi = (?1)
k
kpi,(k = ±1,±2,···)
56/107
DI 5 EMF8GI f(z) = 1zsinz BFCZDLG3BMBRARBFGI.
B5 z = 0 F0F8GI zsinz ARAJBRAYBM,EKEX z = 0 F0F8
GI
1
zsinz ARAJBRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbigg 1
zsinz,0
bracketrightbigg
= limz→0 ddz
parenleftbigg
z2 · 1zsinz
parenrightbigg
= limz→0 sinz?zcoszz2 ·
parenleftBig z
sinz
parenrightBig2
= 0.
B8A9,zk = kpi (k = ±1,±2,···)?F8GIARE8BRAYBM,
EKEX zk = kpi (k = ±1,±2,···) F0F8GIARE8BRBGBM.
DUGABHC3 3,CZ
Res
bracketleftbigg 1
zsinz,kpi
bracketrightbigg
=
1
z
(sinz)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=kpi
= 1kpicoskpi = (?1)
k
kpi,(k = ±1,±2,···)
56/107
DI 5 EMF8GI f(z) = 1zsinz BFCZDLG3BMBRARBFGI.
B5 z = 0 F0F8GI zsinz ARAJBRAYBM,EKEX z = 0 F0F8
GI
1
zsinz ARAJBRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbigg 1
zsinz,0
bracketrightbigg
= limz→0 ddz
parenleftbigg
z2 · 1zsinz
parenrightbigg
= limz→0 sinz?zcoszz2 ·
parenleftBig z
sinz
parenrightBig2
= 0.
B8A9,zk = kpi (k = ±1,±2,···)?F8GIARE8BRAYBM,
EKEX zk = kpi (k = ±1,±2,···) F0F8GIARE8BRBGBM.
DUGABHC3 3,CZ
Res
bracketleftbigg 1
zsinz,kpi
bracketrightbigg
=
1
z
(sinz)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=kpi
= 1kpicoskpi = (?1)
k
kpi,(k = ±1,±2,···)
56/107
DI 5 EMF8GI f(z) = 1zsinz BFCZDLG3BMBRARBFGI.
B5 z = 0 F0F8GI zsinz ARAJBRAYBM,EKEX z = 0 F0F8
GI
1
zsinz ARAJBRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbigg 1
zsinz,0
bracketrightbigg
= limz→0 ddz
parenleftbigg
z2 · 1zsinz
parenrightbigg
= limz→0 sinz?zcoszz2 ·
parenleftBig z
sinz
parenrightBig2
= 0.
B8A9,zk = kpi (k = ±1,±2,···)?F8GIARE8BRAYBM,
EKEX zk = kpi (k = ±1,±2,···) F0F8GIARE8BRBGBM.
DUGABHC3 3,CZ
Res
bracketleftbigg 1
zsinz,kpi
bracketrightbigg
=
1
z
(sinz)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=kpi
= 1kpicoskpi = (?1)
k
kpi,(k = ±1,±2,···)
56/107
DI 5 EMF8GI f(z) = 1zsinz BFCZDLG3BMBRARBFGI.
B5 z = 0 F0F8GI zsinz ARAJBRAYBM,EKEX z = 0 F0F8
GI
1
zsinz ARAJBRBGBM.
DUGABHC3 2,CZ
Res
bracketleftbigg 1
zsinz,0
bracketrightbigg
= limz→0 ddz
parenleftbigg
z2 · 1zsinz
parenrightbigg
= limz→0 sinz?zcoszz2 ·
parenleftBig z
sinz
parenrightBig2
= 0.
B8A9,zk = kpi (k = ±1,±2,···)?F8GIARE8BRAYBM,
EKEX zk = kpi (k = ±1,±2,···) F0F8GIARE8BRBGBM.
DUGABHC3 3,CZ
Res
bracketleftbigg 1
zsinz,kpi
bracketrightbigg
=
1
z
(sinz)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=kpi
= 1kpicoskpi = (?1)
k
kpi,(k = ±1,±2,···)
56/107
DGCMBFGIC7DYFKEP
DI 1 C7DYFKEP
contintegraldisplay
C
zez
z2? 1dz,C F0AKE5ABEX,|z| = 2.
B5 CSD9 f(z) = ze
z
z2? 1 CZH0DRE8BRBGBM +1,?1,ADGX
H0DRBGBMFTBFABEX|z| = 2 CB,EKEX
contintegraldisplay
C
zez
z2?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]}.
DUGABHC3 1,AQ
Res[f(z),1] = limz→1(z?1) ze
z
z2? 1 = limz→1
zez
z + 1 =
e
2,
Res[f(z),?1] = limz→?1(z + 1) ze
z
z2?1 = limz→1
zez
z?1 =
e?1
2,
FRCS
contintegraldisplay zez
z2? 1dz = 2pii
parenleftbigge
2 +
e?1
2
parenrightbigg
.
57/107
DGCMBFGIC7DYFKEP
DI 1 C7DYFKEP
contintegraldisplay
C
zez
z2? 1dz,C F0AKE5ABEX,|z| = 2.
B5 CSD9 f(z) = ze
z
z2? 1 CZH0DRE8BRBGBM +1,?1,ADGX
H0DRBGBMFTBFABEX|z| = 2 CB,EKEX
contintegraldisplay
C
zez
z2?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]}.
DUGABHC3 1,AQ
Res[f(z),1] = limz→1(z?1) ze
z
z2? 1 = limz→1
zez
z + 1 =
e
2,
Res[f(z),?1] = limz→?1(z + 1) ze
z
z2?1 = limz→1
zez
z?1 =
e?1
2,
FRCS
contintegraldisplay zez
z2? 1dz = 2pii
parenleftbigge
2 +
e?1
2
parenrightbigg
.
57/107
DGCMBFGIC7DYFKEP
DI 1 C7DYFKEP
contintegraldisplay
C
zez
z2? 1dz,C F0AKE5ABEX,|z| = 2.
B5 CSD9 f(z) = ze
z
z2? 1 CZH0DRE8BRBGBM +1,?1,ADGX
H0DRBGBMFTBFABEX|z| = 2 CB,EKEX
contintegraldisplay
C
zez
z2?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]}.
DUGABHC3 1,AQ
Res[f(z),1] = limz→1(z?1) ze
z
z2? 1 = limz→1
zez
z + 1 =
e
2,
Res[f(z),?1] = limz→?1(z + 1) ze
z
z2?1 = limz→1
zez
z?1 =
e?1
2,
FRCS
contintegraldisplay zez
z2? 1dz = 2pii
parenleftbigge
2 +
e?1
2
parenrightbigg
.
57/107
DGCMBFGIC7DYFKEP
DI 1 C7DYFKEP
contintegraldisplay
C
zez
z2? 1dz,C F0AKE5ABEX,|z| = 2.
B5 CSD9 f(z) = ze
z
z2? 1 CZH0DRE8BRBGBM +1,?1,ADGX
H0DRBGBMFTBFABEX|z| = 2 CB,EKEX
contintegraldisplay
C
zez
z2?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]}.
DUGABHC3 1,AQ
Res[f(z),1] = limz→1(z?1) ze
z
z2? 1 = limz→1
zez
z + 1 =
e
2,
Res[f(z),?1] = limz→?1(z + 1) ze
z
z2?1 = limz→1
zez
z?1 =
e?1
2,
FRCS
contintegraldisplay zez
z2? 1dz = 2pii
parenleftbigge
2 +
e?1
2
parenrightbigg
.
57/107
DGCMBFGIC7DYFKEP
DI 1 C7DYFKEP
contintegraldisplay
C
zez
z2? 1dz,C F0AKE5ABEX,|z| = 2.
B5 CSD9 f(z) = ze
z
z2? 1 CZH0DRE8BRBGBM +1,?1,ADGX
H0DRBGBMFTBFABEX|z| = 2 CB,EKEX
contintegraldisplay
C
zez
z2?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]}.
DUGABHC3 1,AQ
Res[f(z),1] = limz→1(z?1) ze
z
z2? 1 = limz→1
zez
z + 1 =
e
2,
Res[f(z),?1] = limz→?1(z + 1) ze
z
z2?1 = limz→1
zez
z?1 =
e?1
2,
FRCS
contintegraldisplay zez
z2? 1dz = 2pii
parenleftbigge
2 +
e?1
2
parenrightbigg
.
57/107
DGCMBFGIC7DYFKEP
DI 1 C7DYFKEP
contintegraldisplay
C
zez
z2? 1dz,C F0AKE5ABEX,|z| = 2.
B5 CSD9 f(z) = ze
z
z2? 1 CZH0DRE8BRBGBM +1,?1,ADGX
H0DRBGBMFTBFABEX|z| = 2 CB,EKEX
contintegraldisplay
C
zez
z2?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]}.
DUGABHC3 1,AQ
Res[f(z),1] = limz→1(z?1) ze
z
z2? 1 = limz→1
zez
z + 1 =
e
2,
Res[f(z),?1] = limz→?1(z + 1) ze
z
z2?1 = limz→1
zez
z?1 =
e?1
2,
FRCS
contintegraldisplay zez
z2? 1dz = 2pii
parenleftbigge
2 +
e?1
2
parenrightbigg
.
57/107
DGCMBFGIC7DYFKEP
DI 1 C7DYFKEP
contintegraldisplay
C
zez
z2? 1dz,C F0AKE5ABEX,|z| = 2.
B5 CSD9 f(z) = ze
z
z2? 1 CZH0DRE8BRBGBM +1,?1,ADGX
H0DRBGBMFTBFABEX|z| = 2 CB,EKEX
contintegraldisplay
C
zez
z2?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]}.
DUGABHC3 1,AQ
Res[f(z),1] = limz→1(z?1) ze
z
z2? 1 = limz→1
zez
z + 1 =
e
2,
Res[f(z),?1] = limz→?1(z + 1) ze
z
z2?1 = limz→1
zez
z?1 =
e?1
2,
FRCS
contintegraldisplay zez
z2? 1dz = 2pii
parenleftbigge
2 +
e?1
2
parenrightbigg
.
57/107
CSC3?DWEXCMBHC3 3BHEMBFGI:
Res[f(z),1] = ze
z
2z
vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
= e2;
Res[f(z),?1] = ze
z
2z
vextendsinglevextendsingle
vextendsinglevextendsingle
z=?1
= e
1
2,
58/107
CSC3?DWEXCMBHC3 3BHEMBFGI:
Res[f(z),1] = ze
z
2z
vextendsinglevextendsingle
vextendsinglevextendsingle
z=1
= e2;
Res[f(z),?1] = ze
z
2z
vextendsinglevextendsingle
vextendsinglevextendsingle
z=?1
= e
1
2,
58/107
DI 2 C7DYFKEP I =
contintegraldisplay
|z|=3
tgpizdz.
B5F8GI tgpiz = sinpizcospiz CZE8BRBGBM zk = k + 12 (k = 0,
±1,±2,···).
DUGABHC3 3,CZ
Res[tgpiz,zk] = sinpiz(cospiz)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=zk
=?1pi.
G1ELCZ 6 DRBGBM
z0 = 12,z1 = 32,z?1 =?12,z2 = 52,z?2 =?32,z3 =?52
BF |z| = 3 CB,EKEXDUGABFGIF9D2contintegraldisplay
|z|=3
tgpizdz = 2pii{Res[tgpiz,z0] + Res[tgpiz,z1] + Res[tgpiz,z?1]
+ Res[tgpiz,z2] + Res[tgpiz,z?2] + Res[tgpiz,z3]} =?12i.
59/107
DI 2 C7DYFKEP I =
contintegraldisplay
|z|=3
tgpizdz.
B5F8GI tgpiz = sinpizcospiz CZE8BRBGBM zk = k + 12 (k = 0,
±1,±2,···).
DUGABHC3 3,CZ
Res[tgpiz,zk] = sinpiz(cospiz)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=zk
=?1pi.
G1ELCZ 6 DRBGBM
z0 = 12,z1 = 32,z?1 =?12,z2 = 52,z?2 =?32,z3 =?52
BF |z| = 3 CB,EKEXDUGABFGIF9D2contintegraldisplay
|z|=3
tgpizdz = 2pii{Res[tgpiz,z0] + Res[tgpiz,z1] + Res[tgpiz,z?1]
+ Res[tgpiz,z2] + Res[tgpiz,z?2] + Res[tgpiz,z3]} =?12i.
59/107
DI 2 C7DYFKEP I =
contintegraldisplay
|z|=3
tgpizdz.
B5F8GI tgpiz = sinpizcospiz CZE8BRBGBM zk = k + 12 (k = 0,
±1,±2,···).
DUGABHC3 3,CZ
Res[tgpiz,zk] = sinpiz(cospiz)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=zk
=?1pi.
G1ELCZ 6 DRBGBM
z0 = 12,z1 = 32,z?1 =?12,z2 = 52,z?2 =?32,z3 =?52
BF |z| = 3 CB,EKEXDUGABFGIF9D2contintegraldisplay
|z|=3
tgpizdz = 2pii{Res[tgpiz,z0] + Res[tgpiz,z1] + Res[tgpiz,z?1]
+ Res[tgpiz,z2] + Res[tgpiz,z?2] + Res[tgpiz,z3]} =?12i.
59/107
DI 2 C7DYFKEP I =
contintegraldisplay
|z|=3
tgpizdz.
B5F8GI tgpiz = sinpizcospiz CZE8BRBGBM zk = k + 12 (k = 0,
±1,±2,···).
DUGABHC3 3,CZ
Res[tgpiz,zk] = sinpiz(cospiz)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=zk
=?1pi.
G1ELCZ 6 DRBGBM
z0 = 12,z1 = 32,z?1 =?12,z2 = 52,z?2 =?32,z3 =?52
BF |z| = 3 CB,EKEXDUGABFGIF9D2contintegraldisplay
|z|=3
tgpizdz = 2pii{Res[tgpiz,z0] + Res[tgpiz,z1] + Res[tgpiz,z?1]
+ Res[tgpiz,z2] + Res[tgpiz,z?2] + Res[tgpiz,z3]} =?12i.
59/107
DI 2 C7DYFKEP I =
contintegraldisplay
|z|=3
tgpizdz.
B5F8GI tgpiz = sinpizcospiz CZE8BRBGBM zk = k + 12 (k = 0,
±1,±2,···).
DUGABHC3 3,CZ
Res[tgpiz,zk] = sinpiz(cospiz)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=zk
=?1pi.
G1ELCZ 6 DRBGBM
z0 = 12,z1 = 32,z?1 =?12,z2 = 52,z?2 =?32,z3 =?52
BF |z| = 3 CB,EKEXDUGABFGIF9D2contintegraldisplay
|z|=3
tgpizdz = 2pii{Res[tgpiz,z0] + Res[tgpiz,z1] + Res[tgpiz,z?1]
+ Res[tgpiz,z2] + Res[tgpiz,z?2] + Res[tgpiz,z3]} =?12i.
59/107
3,BFFFEHAFBMARBFGI
F9FE ANF8GI f(z) BFABDMDX R < |z| < +∞CBB5G3,
C F0ABDMDXCBBJA6BMARBRG3E8CMAKE5DFH0BGETDM,BSD3FK
EP 1
2pii
contintegraldisplay
C?
f(z)dz
ARDUDQ C FFB3,EHCSDU?f(z) BF ∞ BMARBFGI,C8DU
Res[f(z),∞] = 12pii
contintegraldisplay
C?
f(z)dz
D5FR? c?1 = 12pii
contintegraldisplay
C
f(z)dz,EKEX
Res[f(z),∞] =?c?1.
F9D2,f(z) BF ∞BMARBFGIAVD9ENBF ∞BMARAMB1AP
DX R < |z| < +∞ CBGAC3FSD1EQEL z?1 ARCGGIBVFU.
60/107
3,BFFFEHAFBMARBFGI
F9FE ANF8GI f(z) BFABDMDX R < |z| < +∞CBB5G3,
C F0ABDMDXCBBJA6BMARBRG3E8CMAKE5DFH0BGETDM,BSD3FK
EP 1
2pii
contintegraldisplay
C?
f(z)dz
ARDUDQ C FFB3,EHCSDU?f(z) BF ∞ BMARBFGI,C8DU
Res[f(z),∞] = 12pii
contintegraldisplay
C?
f(z)dz
D5FR? c?1 = 12pii
contintegraldisplay
C
f(z)dz,EKEX
Res[f(z),∞] =?c?1.
F9D2,f(z) BF ∞BMARBFGIAVD9ENBF ∞BMARAMB1AP
DX R < |z| < +∞ CBGAC3FSD1EQEL z?1 ARCGGIBVFU.
60/107
3,BFFFEHAFBMARBFGI
F9FE ANF8GI f(z) BFABDMDX R < |z| < +∞CBB5G3,
C F0ABDMDXCBBJA6BMARBRG3E8CMAKE5DFH0BGETDM,BSD3FK
EP 1
2pii
contintegraldisplay
C?
f(z)dz
ARDUDQ C FFB3,EHCSDU?f(z) BF ∞ BMARBFGI,C8DU
Res[f(z),∞] = 12pii
contintegraldisplay
C?
f(z)dz
D5FR? c?1 = 12pii
contintegraldisplay
C
f(z)dz,EKEX
Res[f(z),∞] =?c?1.
F9D2,f(z) BF ∞BMARBFGIAVD9ENBF ∞BMARAMB1AP
DX R < |z| < +∞ CBGAC3FSD1EQEL z?1 ARCGGIBVFU.
60/107
3,BFFFEHAFBMARBFGI
F9FE ANF8GI f(z) BFABDMDX R < |z| < +∞CBB5G3,
C F0ABDMDXCBBJA6BMARBRG3E8CMAKE5DFH0BGETDM,BSD3FK
EP 1
2pii
contintegraldisplay
C?
f(z)dz
ARDUDQ C FFB3,EHCSDU?f(z) BF ∞ BMARBFGI,C8DU
Res[f(z),∞] = 12pii
contintegraldisplay
C?
f(z)dz
D5FR? c?1 = 12pii
contintegraldisplay
C
f(z)dz,EKEX
Res[f(z),∞] =?c?1.
F9D2,f(z) BF ∞BMARBFGIAVD9ENBF ∞BMARAMB1AP
DX R < |z| < +∞ CBGAC3FSD1EQEL z?1 ARCGGIBVFU.
60/107
3,BFFFEHAFBMARBFGI
F9FE ANF8GI f(z) BFABDMDX R < |z| < +∞CBB5G3,
C F0ABDMDXCBBJA6BMARBRG3E8CMAKE5DFH0BGETDM,BSD3FK
EP 1
2pii
contintegraldisplay
C?
f(z)dz
ARDUDQ C FFB3,EHCSDU?f(z) BF ∞ BMARBFGI,C8DU
Res[f(z),∞] = 12pii
contintegraldisplay
C?
f(z)dz
D5FR? c?1 = 12pii
contintegraldisplay
C
f(z)dz,EKEX
Res[f(z),∞] =?c?1.
F9D2,f(z) BF ∞BMARBFGIAVD9ENBF ∞BMARAMB1AP
DX R < |z| < +∞ CBGAC3FSD1EQEL z?1 ARCGGIBVFU.
60/107
3,BFFFEHAFBMARBFGI
F9FE ANF8GI f(z) BFABDMDX R < |z| < +∞CBB5G3,
C F0ABDMDXCBBJA6BMARBRG3E8CMAKE5DFH0BGETDM,BSD3FK
EP 1
2pii
contintegraldisplay
C?
f(z)dz
ARDUDQ C FFB3,EHCSDU?f(z) BF ∞ BMARBFGI,C8DU
Res[f(z),∞] = 12pii
contintegraldisplay
C?
f(z)dz
D5FR? c?1 = 12pii
contintegraldisplay
C
f(z)dz,EKEX
Res[f(z),∞] =?c?1.
F9D2,f(z) BF ∞BMARBFGIAVD9ENBF ∞BMARAMB1AP
DX R < |z| < +∞ CBGAC3FSD1EQEL z?1 ARCGGIBVFU.
60/107
F9D2AJ CGC2F8GI f(z) BFB6FBBXEYEHCBE0CZCZDLDRAH
G9G3BM,BSD3 f(z) BFEKCZDSG3BM (EDB5∞ BM) ARBFGIAR
D3G2BJAVD9AY.
DD G0 ∞ BMA9,AN f(z) ARCZDLDRG3BM?
z1,z2,···,zn,
D5AN C?E8CMBJA6BMARFBGQ zk(k = 1,2,···,n) EDF5BF
ENARCBG4ARAKE5DFH0BGETDM,BSD3DUGABFGIF9D2DQBFFFEH
AFBMARBFGIARF9FE,CZ
Res[f(z),∞] +
nsummationdisplay
k=1
Res[f(z),zk]
= 12pii
contintegraldisplay
C?
f(z)dz + 12pii
contintegraldisplay
C
f(z)dz = 0.
61/107
F9D2AJ CGC2F8GI f(z) BFB6FBBXEYEHCBE0CZCZDLDRAH
G9G3BM,BSD3 f(z) BFEKCZDSG3BM (EDB5∞ BM) ARBFGIAR
D3G2BJAVD9AY.
DD G0 ∞ BMA9,AN f(z) ARCZDLDRG3BM?
z1,z2,···,zn,
D5AN C?E8CMBJA6BMARFBGQ zk(k = 1,2,···,n) EDF5BF
ENARCBG4ARAKE5DFH0BGETDM,BSD3DUGABFGIF9D2DQBFFFEH
AFBMARBFGIARF9FE,CZ
Res[f(z),∞] +
nsummationdisplay
k=1
Res[f(z),zk]
= 12pii
contintegraldisplay
C?
f(z)dz + 12pii
contintegraldisplay
C
f(z)dz = 0.
61/107
F9D2AJ CGC2F8GI f(z) BFB6FBBXEYEHCBE0CZCZDLDRAH
G9G3BM,BSD3 f(z) BFEKCZDSG3BM (EDB5∞ BM) ARBFGIAR
D3G2BJAVD9AY.
DD G0 ∞ BMA9,AN f(z) ARCZDLDRG3BM?
z1,z2,···,zn,
D5AN C?E8CMBJA6BMARFBGQ zk(k = 1,2,···,n) EDF5BF
ENARCBG4ARAKE5DFH0BGETDM,BSD3DUGABFGIF9D2DQBFFFEH
AFBMARBFGIARF9FE,CZ
Res[f(z),∞] +
nsummationdisplay
k=1
Res[f(z),zk]
= 12pii
contintegraldisplay
C?
f(z)dz + 12pii
contintegraldisplay
C
f(z)dz = 0.
61/107
F9D2AJ CGC2F8GI f(z) BFB6FBBXEYEHCBE0CZCZDLDRAH
G9G3BM,BSD3 f(z) BFEKCZDSG3BM (EDB5∞ BM) ARBFGIAR
D3G2BJAVD9AY.
DD G0 ∞ BMA9,AN f(z) ARCZDLDRG3BM?
z1,z2,···,zn,
D5AN C?E8CMBJA6BMARFBGQ zk(k = 1,2,···,n) EDF5BF
ENARCBG4ARAKE5DFH0BGETDM,BSD3DUGABFGIF9D2DQBFFFEH
AFBMARBFGIARF9FE,CZ
Res[f(z),∞] +
nsummationdisplay
k=1
Res[f(z),zk]
= 12pii
contintegraldisplay
C?
f(z)dz + 12pii
contintegraldisplay
C
f(z)dz = 0.
61/107
BHC3 4 Res[f(z),∞] =?Res
bracketleftbigg
f(1z) · 1z2,0
bracketrightbigg
DD BFFFEHAFBMARBFGIF9FEEL,AIAKE5DFH0BGETDM C?
DWF7DAAADWARAKE5ABEX,|z| = ρ,B9 z =
1
ξ,FB
AN z = ρe
iθ,ξ = rei?,
BSD3 ρ = 1r,θ =,D9F0CZ
Res[f(z),∞] = 12pii
contintegraldisplay
C?
f(z)dz
= 12pii
integraldisplay?2pi
0
f(ρeiθ)ρieiθdθ
=? 12pii
integraldisplay 2pi
0
f( 1rei?) 1(rei?)2d(rei?)
=? 12pii
contintegraldisplay
|ξ|=1ρ
f(1ξ) 1ξ2dξ (|ξ| = 1ρ?AKE5).
62/107
BHC3 4 Res[f(z),∞] =?Res
bracketleftbigg
f(1z) · 1z2,0
bracketrightbigg
DD BFFFEHAFBMARBFGIF9FEEL,AIAKE5DFH0BGETDM C?
DWF7DAAADWARAKE5ABEX,|z| = ρ,B9 z =
1
ξ,FB
AN z = ρe
iθ,ξ = rei?,
BSD3 ρ = 1r,θ =,D9F0CZ
Res[f(z),∞] = 12pii
contintegraldisplay
C?
f(z)dz
= 12pii
integraldisplay?2pi
0
f(ρeiθ)ρieiθdθ
=? 12pii
integraldisplay 2pi
0
f( 1rei?) 1(rei?)2d(rei?)
=? 12pii
contintegraldisplay
|ξ|=1ρ
f(1ξ) 1ξ2dξ (|ξ| = 1ρ?AKE5).
62/107
BHC3 4 Res[f(z),∞] =?Res
bracketleftbigg
f(1z) · 1z2,0
bracketrightbigg
DD BFFFEHAFBMARBFGIF9FEEL,AIAKE5DFH0BGETDM C?
DWF7DAAADWARAKE5ABEX,|z| = ρ,B9 z =
1
ξ,FB
AN z = ρe
iθ,ξ = rei?,
BSD3 ρ = 1r,θ =,D9F0CZ
Res[f(z),∞] = 12pii
contintegraldisplay
C?
f(z)dz
= 12pii
integraldisplay?2pi
0
f(ρeiθ)ρieiθdθ
=? 12pii
integraldisplay 2pi
0
f( 1rei?) 1(rei?)2d(rei?)
=? 12pii
contintegraldisplay
|ξ|=1ρ
f(1ξ) 1ξ2dξ (|ξ| = 1ρ?AKE5).
62/107
BHC3 4 Res[f(z),∞] =?Res
bracketleftbigg
f(1z) · 1z2,0
bracketrightbigg
DD BFFFEHAFBMARBFGIF9FEEL,AIAKE5DFH0BGETDM C?
DWF7DAAADWARAKE5ABEX,|z| = ρ,B9 z =
1
ξ,FB
AN z = ρe
iθ,ξ = rei?,
BSD3 ρ = 1r,θ =,D9F0CZ
Res[f(z),∞] = 12pii
contintegraldisplay
C?
f(z)dz
= 12pii
integraldisplay?2pi
0
f(ρeiθ)ρieiθdθ
=? 12pii
integraldisplay 2pi
0
f( 1rei?) 1(rei?)2d(rei?)
=? 12pii
contintegraldisplay
|ξ|=1ρ
f(1ξ) 1ξ2dξ (|ξ| = 1ρ?AKE5).
62/107
BHC3 4 Res[f(z),∞] =?Res
bracketleftbigg
f(1z) · 1z2,0
bracketrightbigg
DD BFFFEHAFBMARBFGIF9FEEL,AIAKE5DFH0BGETDM C?
DWF7DAAADWARAKE5ABEX,|z| = ρ,B9 z =
1
ξ,FB
AN z = ρe
iθ,ξ = rei?,
BSD3 ρ = 1r,θ =,D9F0CZ
Res[f(z),∞] = 12pii
contintegraldisplay
C?
f(z)dz
= 12pii
integraldisplay?2pi
0
f(ρeiθ)ρieiθdθ
=? 12pii
integraldisplay 2pi
0
f( 1rei?) 1(rei?)2d(rei?)
=? 12pii
contintegraldisplay
|ξ|=1ρ
f(1ξ) 1ξ2dξ (|ξ| = 1ρ?AKE5).
62/107
BHC3 4 Res[f(z),∞] =?Res
bracketleftbigg
f(1z) · 1z2,0
bracketrightbigg
DD BFFFEHAFBMARBFGIF9FEEL,AIAKE5DFH0BGETDM C?
DWF7DAAADWARAKE5ABEX,|z| = ρ,B9 z =
1
ξ,FB
AN z = ρe
iθ,ξ = rei?,
BSD3 ρ = 1r,θ =,D9F0CZ
Res[f(z),∞] = 12pii
contintegraldisplay
C?
f(z)dz
= 12pii
integraldisplay?2pi
0
f(ρeiθ)ρieiθdθ
=? 12pii
integraldisplay 2pi
0
f( 1rei?) 1(rei?)2d(rei?)
=? 12pii
contintegraldisplay
|ξ|=1ρ
f(1ξ) 1ξ2dξ (|ξ| = 1ρ?AKE5).
62/107
BHC3 4 Res[f(z),∞] =?Res
bracketleftbigg
f(1z) · 1z2,0
bracketrightbigg
DD BFFFEHAFBMARBFGIF9FEEL,AIAKE5DFH0BGETDM C?
DWF7DAAADWARAKE5ABEX,|z| = ρ,B9 z =
1
ξ,FB
AN z = ρe
iθ,ξ = rei?,
BSD3 ρ = 1r,θ =,D9F0CZ
Res[f(z),∞] = 12pii
contintegraldisplay
C?
f(z)dz
= 12pii
integraldisplay?2pi
0
f(ρeiθ)ρieiθdθ
=? 12pii
integraldisplay 2pi
0
f( 1rei?) 1(rei?)2d(rei?)
=? 12pii
contintegraldisplay
|ξ|=1ρ
f(1ξ) 1ξ2dξ (|ξ| = 1ρ?AKE5).
62/107
CSD9 f(z) BF ρ < |z| < +∞CBB5G3,D0AD f(1ξ)
BF 0 < |ξ| < 1ρ CBB5G3,FRCS f(1ξ) 1ξ2 BF |ξ| < 1ρ G0 ξ = 0
A9DACZG1EMG3BM.
CSBFGIF9D2,AQ
1
2pii
contintegraldisplay
|ξ|=1ρ
f(1ξ) 1ξ2dξ = Res
bracketleftbigg
f(1ξ)· 1ξ2,0
bracketrightbigg
.
DDBA.
63/107
CSD9 f(z) BF ρ < |z| < +∞CBB5G3,D0AD f(1ξ)
BF 0 < |ξ| < 1ρ CBB5G3,FRCS f(1ξ) 1ξ2 BF |ξ| < 1ρ G0 ξ = 0
A9DACZG1EMG3BM.
CSBFGIF9D2,AQ
1
2pii
contintegraldisplay
|ξ|=1ρ
f(1ξ) 1ξ2dξ = Res
bracketleftbigg
f(1ξ)· 1ξ2,0
bracketrightbigg
.
DDBA.
63/107
C7DYBFGIAREVC3
DI 1 EM Res[f(z),∞] ARDU,G1ELf(z) =
1
z(z + 1)4(z?4).
B5,DGCMBHC3 4 C7DY.
DUGAGYEQ
Res[f(z),∞] =?Res
bracketleftbigg 1
z2f(
1
z),0
bracketrightbigg
=?Res
bracketleftbigg z4
(1 + z)4(1?4z),0
bracketrightbigg
,
AD z = 0 F0
z4
(1 + z)4(1? 4z)ARDWAMG3BM,EKEX
Res[f(z),∞] =?Res
bracketleftbigg z4
(1 + z)4(1? 4z),0
bracketrightbigg
= 0.
64/107
C7DYBFGIAREVC3
DI 1 EM Res[f(z),∞] ARDU,G1ELf(z) =
1
z(z + 1)4(z?4).
B5,DGCMBHC3 4 C7DY.
DUGAGYEQ
Res[f(z),∞] =?Res
bracketleftbigg 1
z2f(
1
z),0
bracketrightbigg
=?Res
bracketleftbigg z4
(1 + z)4(1?4z),0
bracketrightbigg
,
AD z = 0 F0
z4
(1 + z)4(1? 4z)ARDWAMG3BM,EKEX
Res[f(z),∞] =?Res
bracketleftbigg z4
(1 + z)4(1? 4z),0
bracketrightbigg
= 0.
64/107
C7DYBFGIAREVC3
DI 1 EM Res[f(z),∞] ARDU,G1ELf(z) =
1
z(z + 1)4(z?4).
B5,DGCMBHC3 4 C7DY.
DUGAGYEQ
Res[f(z),∞] =?Res
bracketleftbigg 1
z2f(
1
z),0
bracketrightbigg
=?Res
bracketleftbigg z4
(1 + z)4(1?4z),0
bracketrightbigg
,
AD z = 0 F0
z4
(1 + z)4(1? 4z)ARDWAMG3BM,EKEX
Res[f(z),∞] =?Res
bracketleftbigg z4
(1 + z)4(1? 4z),0
bracketrightbigg
= 0.
64/107
C7DYBFGIAREVC3
DI 1 EM Res[f(z),∞] ARDU,G1ELf(z) =
1
z(z + 1)4(z?4).
B5,DGCMBHC3 4 C7DY.
DUGAGYEQ
Res[f(z),∞] =?Res
bracketleftbigg 1
z2f(
1
z),0
bracketrightbigg
=?Res
bracketleftbigg z4
(1 + z)4(1?4z),0
bracketrightbigg
,
AD z = 0 F0
z4
(1 + z)4(1? 4z)ARDWAMG3BM,EKEX
Res[f(z),∞] =?Res
bracketleftbigg z4
(1 + z)4(1? 4z),0
bracketrightbigg
= 0.
64/107
DI 2 EM Res[f(z),∞] ARDU,G1EL f(z) = e
z
z2?1.
B5,DUGAB6FBBXEYEHCBARBFGIDNG2?AYC7DY.
F8GI f(z) = e
z
z2?1 BFB6FBBXEYEHH2CZG3BM,±1,∞.
FR? ±1? f(z) ARE8BRBGBM,DP
Res[f(z),1] = limz→1(z?1)f(z) = limz→1 e
z
z + 1 =
1
2e;
Res[f(z),?1] = limz→?1(z + 1)f(z) = limz→?1 e
z
z? 1 =?
1
2e
1.
DUGAF9D2AJ,CZ
Res[f(z),∞] + Res[f(z),1] + Res[f(z),?1] = 0,
EKEX
Res[f(z),∞] =?{Res[f(z),1] + Res[f(z),?1]} = 12(e?1? e).
65/107
DI 2 EM Res[f(z),∞] ARDU,G1EL f(z) = e
z
z2?1.
B5,DUGAB6FBBXEYEHCBARBFGIDNG2?AYC7DY.
F8GI f(z) = e
z
z2?1 BFB6FBBXEYEHH2CZG3BM,±1,∞.
FR? ±1? f(z) ARE8BRBGBM,DP
Res[f(z),1] = limz→1(z?1)f(z) = limz→1 e
z
z + 1 =
1
2e;
Res[f(z),?1] = limz→?1(z + 1)f(z) = limz→?1 e
z
z? 1 =?
1
2e
1.
DUGAF9D2AJ,CZ
Res[f(z),∞] + Res[f(z),1] + Res[f(z),?1] = 0,
EKEX
Res[f(z),∞] =?{Res[f(z),1] + Res[f(z),?1]} = 12(e?1? e).
65/107
DI 2 EM Res[f(z),∞] ARDU,G1EL f(z) = e
z
z2?1.
B5,DUGAB6FBBXEYEHCBARBFGIDNG2?AYC7DY.
F8GI f(z) = e
z
z2?1 BFB6FBBXEYEHH2CZG3BM,±1,∞.
FR? ±1? f(z) ARE8BRBGBM,DP
Res[f(z),1] = limz→1(z?1)f(z) = limz→1 e
z
z + 1 =
1
2e;
Res[f(z),?1] = limz→?1(z + 1)f(z) = limz→?1 e
z
z? 1 =?
1
2e
1.
DUGAF9D2AJ,CZ
Res[f(z),∞] + Res[f(z),1] + Res[f(z),?1] = 0,
EKEX
Res[f(z),∞] =?{Res[f(z),1] + Res[f(z),?1]} = 12(e?1? e).
65/107
DI 2 EM Res[f(z),∞] ARDU,G1EL f(z) = e
z
z2?1.
B5,DUGAB6FBBXEYEHCBARBFGIDNG2?AYC7DY.
F8GI f(z) = e
z
z2?1 BFB6FBBXEYEHH2CZG3BM,±1,∞.
FR? ±1? f(z) ARE8BRBGBM,DP
Res[f(z),1] = limz→1(z?1)f(z) = limz→1 e
z
z + 1 =
1
2e;
Res[f(z),?1] = limz→?1(z + 1)f(z) = limz→?1 e
z
z? 1 =?
1
2e
1.
DUGAF9D2AJ,CZ
Res[f(z),∞] + Res[f(z),1] + Res[f(z),?1] = 0,
EKEX
Res[f(z),∞] =?{Res[f(z),1] + Res[f(z),?1]} = 12(e?1? e).
65/107
DI 2 EM Res[f(z),∞] ARDU,G1EL f(z) = e
z
z2?1.
B5,DUGAB6FBBXEYEHCBARBFGIDNG2?AYC7DY.
F8GI f(z) = e
z
z2?1 BFB6FBBXEYEHH2CZG3BM,±1,∞.
FR? ±1? f(z) ARE8BRBGBM,DP
Res[f(z),1] = limz→1(z?1)f(z) = limz→1 e
z
z + 1 =
1
2e;
Res[f(z),?1] = limz→?1(z + 1)f(z) = limz→?1 e
z
z? 1 =?
1
2e
1.
DUGAF9D2AJ,CZ
Res[f(z),∞] + Res[f(z),1] + Res[f(z),?1] = 0,
EKEX
Res[f(z),∞] =?{Res[f(z),1] + Res[f(z),?1]} = 12(e?1? e).
65/107
DI 2 EM Res[f(z),∞] ARDU,G1EL f(z) = e
z
z2?1.
B5,DUGAB6FBBXEYEHCBARBFGIDNG2?AYC7DY.
F8GI f(z) = e
z
z2?1 BFB6FBBXEYEHH2CZG3BM,±1,∞.
FR? ±1? f(z) ARE8BRBGBM,DP
Res[f(z),1] = limz→1(z?1)f(z) = limz→1 e
z
z + 1 =
1
2e;
Res[f(z),?1] = limz→?1(z + 1)f(z) = limz→?1 e
z
z? 1 =?
1
2e
1.
DUGAF9D2AJ,CZ
Res[f(z),∞] + Res[f(z),1] + Res[f(z),?1] = 0,
EKEX
Res[f(z),∞] =?{Res[f(z),1] + Res[f(z),?1]} = 12(e?1? e).
65/107
DI 3 EM Res[f(z),∞],G1EL f(z) = e 1z2.
B5,DGCMGAC3BRGIFSD1EQ.
BF ∞ ARAMB1APDX 0 < |z| < +∞CB,
e 1z2 = 1 + 1z2 + 12! 1z4 + ··· + 1n! 1z2n + ···
EKEX
Res[e 1z2,∞] =?c?1 = 0.
66/107
DI 3 EM Res[f(z),∞],G1EL f(z) = e 1z2.
B5,DGCMGAC3BRGIFSD1EQ.
BF ∞ ARAMB1APDX 0 < |z| < +∞CB,
e 1z2 = 1 + 1z2 + 12! 1z4 + ··· + 1n! 1z2n + ···
EKEX
Res[e 1z2,∞] =?c?1 = 0.
66/107
DI 3 EM Res[f(z),∞],G1EL f(z) = e 1z2.
B5,DGCMGAC3BRGIFSD1EQ.
BF ∞ ARAMB1APDX 0 < |z| < +∞CB,
e 1z2 = 1 + 1z2 + 12! 1z4 + ··· + 1n! 1z2n + ···
EKEX
Res[e 1z2,∞] =?c?1 = 0.
66/107
DI 3 EM Res[f(z),∞],G1EL f(z) = e 1z2.
B5,DGCMGAC3BRGIFSD1EQ.
BF ∞ ARAMB1APDX 0 < |z| < +∞CB,
e 1z2 = 1 + 1z2 + 12! 1z4 + ··· + 1n! 1z2n + ···
EKEX
Res[e 1z2,∞] =?c?1 = 0.
66/107
DI 3 C7DYF8GI f(z) = 11 + z2eimz(m negationslash= 0 F0EIDIGI) BF
B6FBBXEYEHCBARBFGI.
B5 z = ±i F0 f(z) ARE8BRBGBM,z = ∞F0 f(z) ARAT
BHG3BM.
DUGABHC3 1,CZ
Res[f(z),i] = limz→i(z?i)· 11 + z2eimz
= limz→i 1z + ieimz = e
m
2i =?
i
2e
m;
Res[f(z),?i] = limz→?i(z + i) · 11 + z2eimz
= limz→?i 1z?ieimz = i2em;
DUGAF9D2AJ,CZ
Res[f(z),∞] =?Res[f(z),i]? Res[f(z),?i] = i2e?m? i2em.
67/107
DI 3 C7DYF8GI f(z) = 11 + z2eimz(m negationslash= 0 F0EIDIGI) BF
B6FBBXEYEHCBARBFGI.
B5 z = ±i F0 f(z) ARE8BRBGBM,z = ∞F0 f(z) ARAT
BHG3BM.
DUGABHC3 1,CZ
Res[f(z),i] = limz→i(z?i)· 11 + z2eimz
= limz→i 1z + ieimz = e
m
2i =?
i
2e
m;
Res[f(z),?i] = limz→?i(z + i) · 11 + z2eimz
= limz→?i 1z?ieimz = i2em;
DUGAF9D2AJ,CZ
Res[f(z),∞] =?Res[f(z),i]? Res[f(z),?i] = i2e?m? i2em.
67/107
DI 3 C7DYF8GI f(z) = 11 + z2eimz(m negationslash= 0 F0EIDIGI) BF
B6FBBXEYEHCBARBFGI.
B5 z = ±i F0 f(z) ARE8BRBGBM,z = ∞F0 f(z) ARAT
BHG3BM.
DUGABHC3 1,CZ
Res[f(z),i] = limz→i(z?i)· 11 + z2eimz
= limz→i 1z + ieimz = e
m
2i =?
i
2e
m;
Res[f(z),?i] = limz→?i(z + i) · 11 + z2eimz
= limz→?i 1z?ieimz = i2em;
DUGAF9D2AJ,CZ
Res[f(z),∞] =?Res[f(z),i]? Res[f(z),?i] = i2e?m? i2em.
67/107
DI 3 C7DYF8GI f(z) = 11 + z2eimz(m negationslash= 0 F0EIDIGI) BF
B6FBBXEYEHCBARBFGI.
B5 z = ±i F0 f(z) ARE8BRBGBM,z = ∞F0 f(z) ARAT
BHG3BM.
DUGABHC3 1,CZ
Res[f(z),i] = limz→i(z?i)· 11 + z2eimz
= limz→i 1z + ieimz = e
m
2i =?
i
2e
m;
Res[f(z),?i] = limz→?i(z + i) · 11 + z2eimz
= limz→?i 1z?ieimz = i2em;
DUGAF9D2AJ,CZ
Res[f(z),∞] =?Res[f(z),i]? Res[f(z),?i] = i2e?m? i2em.
67/107
DI 3 C7DYF8GI f(z) = 11 + z2eimz(m negationslash= 0 F0EIDIGI) BF
B6FBBXEYEHCBARBFGI.
B5 z = ±i F0 f(z) ARE8BRBGBM,z = ∞F0 f(z) ARAT
BHG3BM.
DUGABHC3 1,CZ
Res[f(z),i] = limz→i(z?i)· 11 + z2eimz
= limz→i 1z + ieimz = e
m
2i =?
i
2e
m;
Res[f(z),?i] = limz→?i(z + i) · 11 + z2eimz
= limz→?i 1z?ieimz = i2em;
DUGAF9D2AJ,CZ
Res[f(z),∞] =?Res[f(z),i]? Res[f(z),?i] = i2e?m? i2em.
67/107
DI 4 EMF8GI f(z) = sin2z(z + 1)3 BFB6FBBXEYEHCBARBFGI.
B5 z =?1 F0 f(z) AR 3 BRBGBM,z = ∞ F0 f(z) ARAT
BHG3BM.
DUGABHC3 2,CZ
Res[f(z),?1] = 12! limz→?1 d
2
dz2[(z + 1)
3 · sin2z
(z + 1)3]
= 12! limz→?1 d
2
dz2sin2z =?2 sin(?2) = 2sin2.
DUGAF9D2AJ,
Res[f(z),∞] =?Res[f(z),?1] =?2sin2..
68/107
DI 4 EMF8GI f(z) = sin2z(z + 1)3 BFB6FBBXEYEHCBARBFGI.
B5 z =?1 F0 f(z) AR 3 BRBGBM,z = ∞ F0 f(z) ARAT
BHG3BM.
DUGABHC3 2,CZ
Res[f(z),?1] = 12! limz→?1 d
2
dz2[(z + 1)
3 · sin2z
(z + 1)3]
= 12! limz→?1 d
2
dz2sin2z =?2 sin(?2) = 2sin2.
DUGAF9D2AJ,
Res[f(z),∞] =?Res[f(z),?1] =?2sin2..
68/107
DI 4 EMF8GI f(z) = sin2z(z + 1)3 BFB6FBBXEYEHCBARBFGI.
B5 z =?1 F0 f(z) AR 3 BRBGBM,z = ∞ F0 f(z) ARAT
BHG3BM.
DUGABHC3 2,CZ
Res[f(z),?1] = 12! limz→?1 d
2
dz2[(z + 1)
3 · sin2z
(z + 1)3]
= 12! limz→?1 d
2
dz2sin2z =?2 sin(?2) = 2sin2.
DUGAF9D2AJ,
Res[f(z),∞] =?Res[f(z),?1] =?2sin2..
68/107
DI 5 EMF8GI f(z) = e
1
z
1?z BFB6FBBXEYEHCBARBFGI.
B5 z = 1 F0 f(z) ARE8BRBGBM,z = 0 F0 f(z) ARATBH
G3BM,z = ∞F0 f(z) ARE8BRAYBM.
(1) DUGABHC3 1,CZ
Res[f(z),1] = limz→1(z?1)· e
1
z
1?z
=?limz→1 e1z =?e.
69/107
DI 5 EMF8GI f(z) = e
1
z
1?z BFB6FBBXEYEHCBARBFGI.
B5 z = 1 F0 f(z) ARE8BRBGBM,z = 0 F0 f(z) ARATBH
G3BM,z = ∞F0 f(z) ARE8BRAYBM.
(1) DUGABHC3 1,CZ
Res[f(z),1] = limz→1(z?1)· e
1
z
1?z
=?limz→1 e1z =?e.
69/107
DI 5 EMF8GI f(z) = e
1
z
1?z BFB6FBBXEYEHCBARBFGI.
B5 z = 1 F0 f(z) ARE8BRBGBM,z = 0 F0 f(z) ARATBH
G3BM,z = ∞F0 f(z) ARE8BRAYBM.
(1) DUGABHC3 1,CZ
Res[f(z),1] = limz→1(z?1)· e
1
z
1?z
=?limz→1 e1z =?e.
69/107
(2) DUGAGAC3BRGIFSD1EQ,A8|z| FBEPDWEE,
f(z) =?1z 11? 1
z
e1z
=?1z
parenleftbigg
1 + 1z +···
parenrightbiggparenleftbigg
1 + 1z + 12! 1z2 + ···
parenrightbigg
= ···? 1z + ···
EKEX Res[f(z),∞] =?c?1 =?(?1) = 1.
(3) DUGAF9D2AJ,CZ
Res[f(z),0] =?Res[f(z),∞]? Res[f(z),1] = e? 1.
70/107
(2) DUGAGAC3BRGIFSD1EQ,A8|z| FBEPDWEE,
f(z) =?1z 11? 1
z
e1z
=?1z
parenleftbigg
1 + 1z +···
parenrightbiggparenleftbigg
1 + 1z + 12! 1z2 + ···
parenrightbigg
= ···? 1z + ···
EKEX Res[f(z),∞] =?c?1 =?(?1) = 1.
(3) DUGAF9D2AJ,CZ
Res[f(z),0] =?Res[f(z),∞]? Res[f(z),1] = e? 1.
70/107
DGCMBFGIC7DYFKEP
DI 1 C7DYFKEP
contintegraldisplay
C
z
z4? 1dz,C?AKE5ABEX,|z| = 2.
E0DFE8 AQFKF8GI f(z) = zz4? 1 CZD3DRE8BRBG
BM ±1,±i FTBFABEXCB,EKEXcontintegraldisplay
C
z
z4?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]
+Res[f(z),i] + Res[f(z),?i]}.
CSBHC3 3,P(z)Q′(z) = z4z3 = 14z2,AP
contintegraldisplay
C
z
z4?1dz = 2pii
braceleftbigg1
4 +
1
4?
1
4?
1
4
bracerightbigg
= 0.
71/107
DGCMBFGIC7DYFKEP
DI 1 C7DYFKEP
contintegraldisplay
C
z
z4? 1dz,C?AKE5ABEX,|z| = 2.
E0DFE8 AQFKF8GI f(z) = zz4? 1 CZD3DRE8BRBG
BM ±1,±i FTBFABEXCB,EKEXcontintegraldisplay
C
z
z4?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]
+Res[f(z),i] + Res[f(z),?i]}.
CSBHC3 3,P(z)Q′(z) = z4z3 = 14z2,AP
contintegraldisplay
C
z
z4?1dz = 2pii
braceleftbigg1
4 +
1
4?
1
4?
1
4
bracerightbigg
= 0.
71/107
DGCMBFGIC7DYFKEP
DI 1 C7DYFKEP
contintegraldisplay
C
z
z4? 1dz,C?AKE5ABEX,|z| = 2.
E0DFE8 AQFKF8GI f(z) = zz4? 1 CZD3DRE8BRBG
BM ±1,±i FTBFABEXCB,EKEXcontintegraldisplay
C
z
z4?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]
+Res[f(z),i] + Res[f(z),?i]}.
CSBHC3 3,P(z)Q′(z) = z4z3 = 14z2,AP
contintegraldisplay
C
z
z4?1dz = 2pii
braceleftbigg1
4 +
1
4?
1
4?
1
4
bracerightbigg
= 0.
71/107
E0DFAJ AQFKF8GI f(z) = zz4? 1 BF |z| = 2 ARA9G4,
G0 ∞ BMA9DACZG1ENG3BM.
DUGABFGIDNG2?AYBLBFGIF9D2,CZ
contintegraldisplay
C
z
z4?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]
+Res[f(z),i] + Res[f(z),?i]}
=?2piiRes[f(z),∞]
= 2piiRes
bracketleftbigg
f(1z)· 1z2,0
bracketrightbigg
= 2piiRes
bracketleftbigg z
1?z2,0
bracketrightbigg
= 0.
72/107
E0DFAJ AQFKF8GI f(z) = zz4? 1 BF |z| = 2 ARA9G4,
G0 ∞ BMA9DACZG1ENG3BM.
DUGABFGIDNG2?AYBLBFGIF9D2,CZ
contintegraldisplay
C
z
z4?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]
+Res[f(z),i] + Res[f(z),?i]}
=?2piiRes[f(z),∞]
= 2piiRes
bracketleftbigg
f(1z)· 1z2,0
bracketrightbigg
= 2piiRes
bracketleftbigg z
1?z2,0
bracketrightbigg
= 0.
72/107
E0DFAJ AQFKF8GI f(z) = zz4? 1 BF |z| = 2 ARA9G4,
G0 ∞ BMA9DACZG1ENG3BM.
DUGABFGIDNG2?AYBLBFGIF9D2,CZ
contintegraldisplay
C
z
z4?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]
+Res[f(z),i] + Res[f(z),?i]}
=?2piiRes[f(z),∞]
= 2piiRes
bracketleftbigg
f(1z)· 1z2,0
bracketrightbigg
= 2piiRes
bracketleftbigg z
1?z2,0
bracketrightbigg
= 0.
72/107
E0DFAJ AQFKF8GI f(z) = zz4? 1 BF |z| = 2 ARA9G4,
G0 ∞ BMA9DACZG1ENG3BM.
DUGABFGIDNG2?AYBLBFGIF9D2,CZ
contintegraldisplay
C
z
z4?1dz = 2pii{Res[f(z),1] + Res[f(z),?1]
+Res[f(z),i] + Res[f(z),?i]}
=?2piiRes[f(z),∞]
= 2piiRes
bracketleftbigg
f(1z)· 1z2,0
bracketrightbigg
= 2piiRes
bracketleftbigg z
1?z2,0
bracketrightbigg
= 0.
72/107
DI 2 C7DYFKEP
contintegraldisplay
C
z3
1 + ze
1
zdz,C,|z| = 2
AKE5.
B5 F8GI
z3
1 + ze
1
z
CZH0DRG3BM,z = 0,z =?1,DPFT
BF C CB,ADBF C A9BPCZAHG9G3BM∞.
DUGABFGIDNG2?AYBLBFGIF9D2DWDJcontintegraldisplay
C
z3
1 + ze
1
z dz = 2pii{Res[f(z),0] + Res[f(z),?1]}
=?2piiRes[f(z),∞]
= 2piiRes
bracketleftbigg 1
z2f(
1
z),0
bracketrightbigg
= 2piiRes
bracketleftbigg ez
z4(1 + z),0
bracketrightbigg
= 2pii· 1(4? 1)! limz→0
bracketleftbigg
z4 · e
z
z4(1 + z)
bracketrightbigg(3)
=?2pii3,
73/107
DI 2 C7DYFKEP
contintegraldisplay
C
z3
1 + ze
1
zdz,C,|z| = 2
AKE5.
B5 F8GI
z3
1 + ze
1
z
CZH0DRG3BM,z = 0,z =?1,DPFT
BF C CB,ADBF C A9BPCZAHG9G3BM∞.
DUGABFGIDNG2?AYBLBFGIF9D2DWDJcontintegraldisplay
C
z3
1 + ze
1
z dz = 2pii{Res[f(z),0] + Res[f(z),?1]}
=?2piiRes[f(z),∞]
= 2piiRes
bracketleftbigg 1
z2f(
1
z),0
bracketrightbigg
= 2piiRes
bracketleftbigg ez
z4(1 + z),0
bracketrightbigg
= 2pii· 1(4? 1)! limz→0
bracketleftbigg
z4 · e
z
z4(1 + z)
bracketrightbigg(3)
=?2pii3,
73/107
DI 2 C7DYFKEP
contintegraldisplay
C
z3
1 + ze
1
zdz,C,|z| = 2
AKE5.
B5 F8GI
z3
1 + ze
1
z
CZH0DRG3BM,z = 0,z =?1,DPFT
BF C CB,ADBF C A9BPCZAHG9G3BM∞.
DUGABFGIDNG2?AYBLBFGIF9D2DWDJcontintegraldisplay
C
z3
1 + ze
1
z dz = 2pii{Res[f(z),0] + Res[f(z),?1]}
=?2piiRes[f(z),∞]
= 2piiRes
bracketleftbigg 1
z2f(
1
z),0
bracketrightbigg
= 2piiRes
bracketleftbigg ez
z4(1 + z),0
bracketrightbigg
= 2pii· 1(4? 1)! limz→0
bracketleftbigg
z4 · e
z
z4(1 + z)
bracketrightbigg(3)
=?2pii3,
73/107
DI 2 C7DYFKEP
contintegraldisplay
C
z3
1 + ze
1
zdz,C,|z| = 2
AKE5.
B5 F8GI
z3
1 + ze
1
z
CZH0DRG3BM,z = 0,z =?1,DPFT
BF C CB,ADBF C A9BPCZAHG9G3BM∞.
DUGABFGIDNG2?AYBLBFGIF9D2DWDJcontintegraldisplay
C
z3
1 + ze
1
z dz = 2pii{Res[f(z),0] + Res[f(z),?1]}
=?2piiRes[f(z),∞]
= 2piiRes
bracketleftbigg 1
z2f(
1
z),0
bracketrightbigg
= 2piiRes
bracketleftbigg ez
z4(1 + z),0
bracketrightbigg
= 2pii· 1(4? 1)! limz→0
bracketleftbigg
z4 · e
z
z4(1 + z)
bracketrightbigg(3)
=?2pii3,
73/107
DI 2 C7DYFKEP
contintegraldisplay
C
z3
1 + ze
1
zdz,C,|z| = 2
AKE5.
B5 F8GI
z3
1 + ze
1
z
CZH0DRG3BM,z = 0,z =?1,DPFT
BF C CB,ADBF C A9BPCZAHG9G3BM∞.
DUGABFGIDNG2?AYBLBFGIF9D2DWDJcontintegraldisplay
C
z3
1 + ze
1
z dz = 2pii{Res[f(z),0] + Res[f(z),?1]}
=?2piiRes[f(z),∞]
= 2piiRes
bracketleftbigg 1
z2f(
1
z),0
bracketrightbigg
= 2piiRes
bracketleftbigg ez
z4(1 + z),0
bracketrightbigg
= 2pii· 1(4? 1)! limz→0
bracketleftbigg
z4 · e
z
z4(1 + z)
bracketrightbigg(3)
=?2pii3,
73/107
DI 2 C7DYFKEP
contintegraldisplay
C
z3
1 + ze
1
zdz,C,|z| = 2
AKE5.
B5 F8GI
z3
1 + ze
1
z
CZH0DRG3BM,z = 0,z =?1,DPFT
BF C CB,ADBF C A9BPCZAHG9G3BM∞.
DUGABFGIDNG2?AYBLBFGIF9D2DWDJcontintegraldisplay
C
z3
1 + ze
1
z dz = 2pii{Res[f(z),0] + Res[f(z),?1]}
=?2piiRes[f(z),∞]
= 2piiRes
bracketleftbigg 1
z2f(
1
z),0
bracketrightbigg
= 2piiRes
bracketleftbigg ez
z4(1 + z),0
bracketrightbigg
= 2pii· 1(4? 1)! limz→0
bracketleftbigg
z4 · e
z
z4(1 + z)
bracketrightbigg(3)
=?2pii3,
73/107
DGCMBFGIDQBFGIF9D2DDEYDNB3F3C3
DI 1 AN a F0 f(z) ARAHG9G3BM,FDDD,CT f(z) F0G3F8
GI,C3
Res[f(z),a] = Res[f(z),?a]
CT f(z) F0GRF8GI,C3
Res[f(z),a] =?Res[f(z),?a]
B5 FR? a F0 f(z) ARAHG9G3BM,EKEXBF a ARB8DRA2AP
DXCB f(z) DWEXFSD1EKGAC3BRGI.
f(z) =
2summationdisplay
n=?∞
cn(z?a)n + c?1z?a +
+∞summationdisplay
n=0
cn(z?a)n
FRf(?z) = f(z),EKEX
f(?z) =
2summationdisplay
n=?∞
cn(z?a)n + c?1z?a +
+∞summationdisplay
n=0
cn(z?a)n
74/107
DGCMBFGIDQBFGIF9D2DDEYDNB3F3C3
DI 1 AN a F0 f(z) ARAHG9G3BM,FDDD,CT f(z) F0G3F8
GI,C3
Res[f(z),a] = Res[f(z),?a]
CT f(z) F0GRF8GI,C3
Res[f(z),a] =?Res[f(z),?a]
B5 FR? a F0 f(z) ARAHG9G3BM,EKEXBF a ARB8DRA2AP
DXCB f(z) DWEXFSD1EKGAC3BRGI.
f(z) =
2summationdisplay
n=?∞
cn(z?a)n + c?1z?a +
+∞summationdisplay
n=0
cn(z?a)n
FRf(?z) = f(z),EKEX
f(?z) =
2summationdisplay
n=?∞
cn(z?a)n + c?1z?a +
+∞summationdisplay
n=0
cn(z?a)n
74/107
DGCMBFGIDQBFGIF9D2DDEYDNB3F3C3
DI 1 AN a F0 f(z) ARAHG9G3BM,FDDD,CT f(z) F0G3F8
GI,C3
Res[f(z),a] = Res[f(z),?a]
CT f(z) F0GRF8GI,C3
Res[f(z),a] =?Res[f(z),?a]
B5 FR? a F0 f(z) ARAHG9G3BM,EKEXBF a ARB8DRA2AP
DXCB f(z) DWEXFSD1EKGAC3BRGI.
f(z) =
2summationdisplay
n=?∞
cn(z?a)n + c?1z?a +
+∞summationdisplay
n=0
cn(z?a)n
FRf(?z) = f(z),EKEX
f(?z) =
2summationdisplay
n=?∞
cn(z?a)n + c?1z?a +
+∞summationdisplay
n=0
cn(z?a)n
74/107
DGCMBFGIDQBFGIF9D2DDEYDNB3F3C3
DI 1 AN a F0 f(z) ARAHG9G3BM,FDDD,CT f(z) F0G3F8
GI,C3
Res[f(z),a] = Res[f(z),?a]
CT f(z) F0GRF8GI,C3
Res[f(z),a] =?Res[f(z),?a]
B5 FR? a F0 f(z) ARAHG9G3BM,EKEXBF a ARB8DRA2AP
DXCB f(z) DWEXFSD1EKGAC3BRGI.
f(z) =
2summationdisplay
n=?∞
cn(z?a)n + c?1z?a +
+∞summationdisplay
n=0
cn(z?a)n
FRf(?z) = f(z),EKEX
f(?z) =
2summationdisplay
n=?∞
cn(z?a)n + c?1z?a +
+∞summationdisplay
n=0
cn(z?a)n
74/107
BP
f(z) =?
bracketleftBigg?2summationdisplay
n=?∞
cn(?z?a)n + c?1?z?a +
+∞summationdisplay
n=0
cn(?z?a)n
bracketrightBigg
=
2summationdisplay
n=?∞
cn(?1)n+1(z + a)n + c?1z + a +
+∞summationdisplay
n=0
cn(?1)n+1(z + a)n
EKEX
Res[f(z),?a] = c?1 = Res[f(z),a]
FVD2,A8 f(z) F0GRF8GIEE,CZ
Res[f(z),?a] =?c?1 =?Res[f(z),a]
75/107
BP
f(z) =?
bracketleftBigg?2summationdisplay
n=?∞
cn(?z?a)n + c?1?z?a +
+∞summationdisplay
n=0
cn(?z?a)n
bracketrightBigg
=
2summationdisplay
n=?∞
cn(?1)n+1(z + a)n + c?1z + a +
+∞summationdisplay
n=0
cn(?1)n+1(z + a)n
EKEX
Res[f(z),?a] = c?1 = Res[f(z),a]
FVD2,A8 f(z) F0GRF8GIEE,CZ
Res[f(z),?a] =?c?1 =?Res[f(z),a]
75/107
BP
f(z) =?
bracketleftBigg?2summationdisplay
n=?∞
cn(?z?a)n + c?1?z?a +
+∞summationdisplay
n=0
cn(?z?a)n
bracketrightBigg
=
2summationdisplay
n=?∞
cn(?1)n+1(z + a)n + c?1z + a +
+∞summationdisplay
n=0
cn(?1)n+1(z + a)n
EKEX
Res[f(z),?a] = c?1 = Res[f(z),a]
FVD2,A8 f(z) F0GRF8GIEE,CZ
Res[f(z),?a] =?c?1 =?Res[f(z),a]
75/107
DI 2 AN C?ERDX D CBARE8CMAKE5DFH0BGETDM,z0
C CBE8BM,CGC2 f(z) BF D CBB5G3,DP f(z0) = 0,
f′(z0) negationslash= 0,BF C CB f(z) FFG1EMAYBM,FDDD
1
2pii
contintegraldisplay
C
zf′(z)
f(z) dz = z0.
DD FR? f(z0) = 0,f,(z0) negationslash= 0,EKEX z = z0 F0 f(z) AR
E8BRAYBM,D0ADF0
zf′(z)
f(z) ARE8BRBGBM.
D9F0
Res
bracketleftbiggzf′(z)
f(z),z0
bracketrightbigg
= limz→z
0
(z?z0)· zf
′(z)
f(z)
= z0f′(z0)· limz→z
0
z?z0
f(z)?f(z0)
= z0f′(z0)· 1f′(z
0)
= z0
AQDD.76/107
DI 2 AN C?ERDX D CBARE8CMAKE5DFH0BGETDM,z0
C CBE8BM,CGC2 f(z) BF D CBB5G3,DP f(z0) = 0,
f′(z0) negationslash= 0,BF C CB f(z) FFG1EMAYBM,FDDD
1
2pii
contintegraldisplay
C
zf′(z)
f(z) dz = z0.
DD FR? f(z0) = 0,f,(z0) negationslash= 0,EKEX z = z0 F0 f(z) AR
E8BRAYBM,D0ADF0
zf′(z)
f(z) ARE8BRBGBM.
D9F0
Res
bracketleftbiggzf′(z)
f(z),z0
bracketrightbigg
= limz→z
0
(z?z0)· zf
′(z)
f(z)
= z0f′(z0)· limz→z
0
z?z0
f(z)?f(z0)
= z0f′(z0)· 1f′(z
0)
= z0
AQDD.76/107
DI 2 AN C?ERDX D CBARE8CMAKE5DFH0BGETDM,z0
C CBE8BM,CGC2 f(z) BF D CBB5G3,DP f(z0) = 0,
f′(z0) negationslash= 0,BF C CB f(z) FFG1EMAYBM,FDDD
1
2pii
contintegraldisplay
C
zf′(z)
f(z) dz = z0.
DD FR? f(z0) = 0,f,(z0) negationslash= 0,EKEX z = z0 F0 f(z) AR
E8BRAYBM,D0ADF0
zf′(z)
f(z) ARE8BRBGBM.
D9F0
Res
bracketleftbiggzf′(z)
f(z),z0
bracketrightbigg
= limz→z
0
(z?z0)· zf
′(z)
f(z)
= z0f′(z0)· limz→z
0
z?z0
f(z)?f(z0)
= z0f′(z0)· 1f′(z
0)
= z0
AQDD.76/107
DI 3 AN a F0 f(z) ARE8BRBGBM,DPBF a BRARBFGIF0 A,
DDEY:
limz→a |f
′(z)|
1 + |f(z)|2 =
1
|A|
B5FR? a F0 f(z) ARE8BRBGBM,EKEX
f(z) = 1z?a?(z),
G1EL,?(z) BF a BMB5G3,DP?(a) negationslash= 0.
B8A9,DUGABHC3 2,CZ
A = Res[f(z),a] =?(z)(z?a)′
vextendsinglevextendsingle
vextendsinglevextendsingle
x=a
=?(a),
77/107
DI 3 AN a F0 f(z) ARE8BRBGBM,DPBF a BRARBFGIF0 A,
DDEY:
limz→a |f
′(z)|
1 + |f(z)|2 =
1
|A|
B5FR? a F0 f(z) ARE8BRBGBM,EKEX
f(z) = 1z?a?(z),
G1EL,?(z) BF a BMB5G3,DP?(a) negationslash= 0.
B8A9,DUGABHC3 2,CZ
A = Res[f(z),a] =?(z)(z?a)′
vextendsinglevextendsingle
vextendsinglevextendsingle
x=a
=?(a),
77/107
DI 3 AN a F0 f(z) ARE8BRBGBM,DPBF a BRARBFGIF0 A,
DDEY:
limz→a |f
′(z)|
1 + |f(z)|2 =
1
|A|
B5FR? a F0 f(z) ARE8BRBGBM,EKEX
f(z) = 1z?a?(z),
G1EL,?(z) BF a BMB5G3,DP?(a) negationslash= 0.
B8A9,DUGABHC3 2,CZ
A = Res[f(z),a] =?(z)(z?a)′
vextendsinglevextendsingle
vextendsinglevextendsingle
x=a
=?(a),
77/107
EKEX
limz→a |f
′(z)|
1 + |f(z)|2 = limz→a
vextendsinglevextendsingle
vextendsinglevextendsingle?′(z)(z?a)(z)
(z?a)2
vextendsinglevextendsingle
vextendsinglevextendsingle
1 +
vextendsinglevextendsingle
vextendsinglevextendsingle?(z)
z?a
vextendsinglevextendsingle
vextendsinglevextendsingle
2
= limz→a |?
′(z)(z?a)(z)|
|z?a|2 +|?(z)|2
= 1?(a) = 1|A|.
78/107
BDD2AX BFGIBFF9FKEPC7DYH2ARBTCM
DGCMBFGIC7DYEIFKEP,ALBLH0DREVC3:
1,GQF9FKEPARFKEPERD1BPDE?BGETDM;
2,GQF9FKEPARAQFKF8GI (EIF8GI) BPDE?BXF8GI.
79/107
BDD2AX BFGIBFF9FKEPC7DYH2ARBTCM
DGCMBFGIC7DYEIFKEP,ALBLH0DREVC3:
1,GQF9FKEPARFKEPERD1BPDE?BGETDM;
2,GQF9FKEPARAQFKF8GI (EIF8GI) BPDE?BXF8GI.
79/107
BDD2AX BFGIBFF9FKEPC7DYH2ARBTCM
DGCMBFGIC7DYEIFKEP,ALBLH0DREVC3:
1,GQF9FKEPARFKEPERD1BPDE?BGETDM;
2,GQF9FKEPARAQFKF8GI (EIF8GI) BPDE?BXF8GI.
79/107
1,BBCG
integraldisplay 2pi
0
R(cosθ,sinθ)dθ ARFKEP,G1
EL R(cosθ,sinθ)? cosθ DQ sinθ ARCZD2F8GI.
E0DF B9 z = eiθ,BSD3 dz = ieiθdθ,
sinθ = 12i(eiθ?e?iθ) = z
2?1
2iz,
cosθ = 12i(eiθ + e?iθ) = z
2 + 1
2z,
D0AD,EKANFKEPDE?GVAKE5H0EGABEXARFKEP:
contintegraldisplay
|z|=1
R
braceleftbiggz2 + 1
2z,
z2? 1
2iz
bracerightbiggdz
iz =
contintegraldisplay
|z|=1
f(z)dz,
G1EL,f(z)? z ARCZD2F8GI.
80/107
1,BBCG
integraldisplay 2pi
0
R(cosθ,sinθ)dθ ARFKEP,G1
EL R(cosθ,sinθ)? cosθ DQ sinθ ARCZD2F8GI.
E0DF B9 z = eiθ,BSD3 dz = ieiθdθ,
sinθ = 12i(eiθ?e?iθ) = z
2?1
2iz,
cosθ = 12i(eiθ + e?iθ) = z
2 + 1
2z,
D0AD,EKANFKEPDE?GVAKE5H0EGABEXARFKEP:
contintegraldisplay
|z|=1
R
braceleftbiggz2 + 1
2z,
z2? 1
2iz
bracerightbiggdz
iz =
contintegraldisplay
|z|=1
f(z)dz,
G1EL,f(z)? z ARCZD2F8GI.
80/107
1,BBCG
integraldisplay 2pi
0
R(cosθ,sinθ)dθ ARFKEP,G1
EL R(cosθ,sinθ)? cosθ DQ sinθ ARCZD2F8GI.
E0DF B9 z = eiθ,BSD3 dz = ieiθdθ,
sinθ = 12i(eiθ?e?iθ) = z
2?1
2iz,
cosθ = 12i(eiθ + e?iθ) = z
2 + 1
2z,
D0AD,EKANFKEPDE?GVAKE5H0EGABEXARFKEP:
contintegraldisplay
|z|=1
R
braceleftbiggz2 + 1
2z,
z2? 1
2iz
bracerightbiggdz
iz =
contintegraldisplay
|z|=1
f(z)dz,
G1EL,f(z)? z ARCZD2F8GI.
80/107
CGC2 f(z) BFH0EGABEX |z| = 1 H2AREPBDG0?AY,C3GVDA
BFGIF9D2ARCMDR,DUGABFGIF9D2,CZ
contintegraldisplay
|z|=1
f(z)dz = 2pii
nsummationdisplay
k=1
Res[f(z),zk],
G1EL,zk(k = 1,2,···,n)?EDF5BFH0EGABEX|z| = 1 CB
AR f(z) ARAHG9G3BM.
81/107
DI 1 C7DY
integraldisplay 2pi
0
sin2θ
a + bcosθdθ (a > b > 0)
B5CSD9 a > b > 0,AQFKF8GIAREPBD a + bcosθ
BF 0 lessorequalslant θ lessorequalslant 2pi CBG0?AY,FRCSFKEPF0CZFBFEAR.
B9 z = eiθ,BSD3
dz = ieiθdθ,sinθ = z
2? 1
2iz,cosθ =
z2 + 1
2z,
D9F0
integraldisplay 2pi
0
sin2θ
a + bcosθdθ =
i
2
contintegraldisplay
|z|=1
(z2? 1)2
z2(bz2 + 2az + b)dz.
82/107
DI 1 C7DY
integraldisplay 2pi
0
sin2θ
a + bcosθdθ (a > b > 0)
B5CSD9 a > b > 0,AQFKF8GIAREPBD a + bcosθ
BF 0 lessorequalslant θ lessorequalslant 2pi CBG0?AY,FRCSFKEPF0CZFBFEAR.
B9 z = eiθ,BSD3
dz = ieiθdθ,sinθ = z
2? 1
2iz,cosθ =
z2 + 1
2z,
D9F0
integraldisplay 2pi
0
sin2θ
a + bcosθdθ =
i
2
contintegraldisplay
|z|=1
(z2? 1)2
z2(bz2 + 2az + b)dz.
82/107
DI 1 C7DY
integraldisplay 2pi
0
sin2θ
a + bcosθdθ (a > b > 0)
B5CSD9 a > b > 0,AQFKF8GIAREPBD a + bcosθ
BF 0 lessorequalslant θ lessorequalslant 2pi CBG0?AY,FRCSFKEPF0CZFBFEAR.
B9 z = eiθ,BSD3
dz = ieiθdθ,sinθ = z
2? 1
2iz,cosθ =
z2 + 1
2z,
D9F0
integraldisplay 2pi
0
sin2θ
a + bcosθdθ =
i
2
contintegraldisplay
|z|=1
(z2? 1)2
z2(bz2 + 2az + b)dz.
82/107
DI 1 C7DY
integraldisplay 2pi
0
sin2θ
a + bcosθdθ (a > b > 0)
B5CSD9 a > b > 0,AQFKF8GIAREPBD a + bcosθ
BF 0 lessorequalslant θ lessorequalslant 2pi CBG0?AY,FRCSFKEPF0CZFBFEAR.
B9 z = eiθ,BSD3
dz = ieiθdθ,sinθ = z
2? 1
2iz,cosθ =
z2 + 1
2z,
D9F0
integraldisplay 2pi
0
sin2θ
a + bcosθdθ =
i
2
contintegraldisplay
|z|=1
(z2? 1)2
z2(bz2 + 2az + b)dz.
82/107
F8GI f(z) = (z
2? 1)2
z2(bz2 + 2az + b) BF |z| = 1 CBCZH0DRBG
BM,0(AJBRBGBM) G2
a + √a2?b2
b (E8BRBGBM),G1BFGIEP
C7?
Res[f(z),0] = limz→0(z2f(z))′ = limz→0
bracketleftbigg (z2? 1)2
bz2 + 2az + b
bracketrightbigg′
=?2ab2,
Res
bracketleftBigg
f(z),?a +
√a2?b2
b
bracketrightBigg
=
(z2?1)2
z2
(bz2 + 2az + b)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=?a+
√
a2?b2
b
= 2
√a2?b2
b2,
83/107
F8GI f(z) = (z
2? 1)2
z2(bz2 + 2az + b) BF |z| = 1 CBCZH0DRBG
BM,0(AJBRBGBM) G2
a + √a2?b2
b (E8BRBGBM),G1BFGIEP
C7?
Res[f(z),0] = limz→0(z2f(z))′ = limz→0
bracketleftbigg (z2? 1)2
bz2 + 2az + b
bracketrightbigg′
=?2ab2,
Res
bracketleftBigg
f(z),?a +
√a2?b2
b
bracketrightBigg
=
(z2?1)2
z2
(bz2 + 2az + b)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=?a+
√
a2?b2
b
= 2
√a2?b2
b2,
83/107
F8GI f(z) = (z
2? 1)2
z2(bz2 + 2az + b) BF |z| = 1 CBCZH0DRBG
BM,0(AJBRBGBM) G2
a + √a2?b2
b (E8BRBGBM),G1BFGIEP
C7?
Res[f(z),0] = limz→0(z2f(z))′ = limz→0
bracketleftbigg (z2? 1)2
bz2 + 2az + b
bracketrightbigg′
=?2ab2,
Res
bracketleftBigg
f(z),?a +
√a2?b2
b
bracketrightBigg
=
(z2?1)2
z2
(bz2 + 2az + b)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=?a+
√
a2?b2
b
= 2
√a2?b2
b2,
83/107
DUGABFGIF9D2,CZ
integraldisplay 2pi
0
sin2θ
a + bcosθdθ
= i2 · 2pii
braceleftBigg
Res[f(z),0] + Res
bracketleftBigg
f(z),?a +
√a2?b2
b
bracketrightBiggbracerightBigg
= 2pib2 (a?
radicalbig
a2?b2).
84/107
2,BBCG
integraldisplay +∞
∞
R(x)dx ARFKEP,A8AQFKF8GI R(x) F0 x
ARCZD2F8GI,ADEPBDARCVGIE6A8B3EPCUARCVGID4H0CV,FB
DP R(z) BFEIF2H2DACZAHG9G3BMEE.
E0DF AN
R(z) = z
n + a1zn?1 + ··· + an
zm + b1zm?1 + ··· + bm,m?n greaterorequalslant 2
E8EUALEPEQ.
85/107
2,BBCG
integraldisplay +∞
∞
R(x)dx ARFKEP,A8AQFKF8GI R(x) F0 x
ARCZD2F8GI,ADEPBDARCVGIE6A8B3EPCUARCVGID4H0CV,FB
DP R(z) BFEIF2H2DACZAHG9G3BMEE.
E0DF AN
R(z) = z
n + a1zn?1 + ··· + an
zm + b1zm?1 + ··· + bm,m?n greaterorequalslant 2
E8EUALEPEQ.
85/107
y
x O?R R
CR
z1
z2
z3
AIFKEPF0DMCGGB,AI R F3A8DW,EM R(z) EKCZARBFH2DW
EYEHCBARBGBM zk FTEDF5BFGXFKEPF0DMCB.
DUGABFGIF9D2,AQ
integraldisplay R
R
R(x)dx +
integraldisplay
CR
R(z)dz = 2pii
summationdisplay
Res[R(z),zk].
CTEHDDEY,lim
R→+∞
integraldisplay
CR
R(z)dz = 0:
86/107
y
x O?R R
CR
z1
z2
z3
AIFKEPF0DMCGGB,AI R F3A8DW,EM R(z) EKCZARBFH2DW
EYEHCBARBGBM zk FTEDF5BFGXFKEPF0DMCB.
DUGABFGIF9D2,AQ
integraldisplay R
R
R(x)dx +
integraldisplay
CR
R(z)dz = 2pii
summationdisplay
Res[R(z),zk].
CTEHDDEY,lim
R→+∞
integraldisplay
CR
R(z)dz = 0:
86/107
y
x O?R R
CR
z1
z2
z3
AIFKEPF0DMCGGB,AI R F3A8DW,EM R(z) EKCZARBFH2DW
EYEHCBARBGBM zk FTEDF5BFGXFKEPF0DMCB.
DUGABFGIF9D2,AQ
integraldisplay R
R
R(x)dx +
integraldisplay
CR
R(z)dz = 2pii
summationdisplay
Res[R(z),zk].
CTEHDDEY,lim
R→+∞
integraldisplay
CR
R(z)dz = 0:
86/107
FR?
|R(z)| = 1|z|m?n · |1 + a1z
1 + ··· + anz?n|
|1 + b1z?1 + ··· + bmz?m|
lessorequalslant 1|z|m?n · 1 + |a1z
1 + ··· + anz?n|
1?|b1z?1 +··· + bmz?m|,
A8 |z| FBEPDWEE,D3DWEM
|a1z?1 + ··· + anz?n| < 110,|b1z?1 + ··· + bmz?m| < 110
CSD9 m?n greaterorequalslant 2,APCZ
|R(z)| < 1|z|m?n · 1 +
1
10
1? 110 <
2
|z|2.
87/107
FR?
|R(z)| = 1|z|m?n · |1 + a1z
1 + ··· + anz?n|
|1 + b1z?1 + ··· + bmz?m|
lessorequalslant 1|z|m?n · 1 + |a1z
1 + ··· + anz?n|
1?|b1z?1 +··· + bmz?m|,
A8 |z| FBEPDWEE,D3DWEM
|a1z?1 + ··· + anz?n| < 110,|b1z?1 + ··· + bmz?m| < 110
CSD9 m?n greaterorequalslant 2,APCZ
|R(z)| < 1|z|m?n · 1 +
1
10
1? 110 <
2
|z|2.
87/107
FR?
|R(z)| = 1|z|m?n · |1 + a1z
1 + ··· + anz?n|
|1 + b1z?1 + ··· + bmz?m|
lessorequalslant 1|z|m?n · 1 + |a1z
1 + ··· + anz?n|
1?|b1z?1 +··· + bmz?m|,
A8 |z| FBEPDWEE,D3DWEM
|a1z?1 + ··· + anz?n| < 110,|b1z?1 + ··· + bmz?m| < 110
CSD9 m?n greaterorequalslant 2,APCZ
|R(z)| < 1|z|m?n · 1 +
1
10
1? 110 <
2
|z|2.
87/107
FR?
|R(z)| = 1|z|m?n · |1 + a1z
1 + ··· + anz?n|
|1 + b1z?1 + ··· + bmz?m|
lessorequalslant 1|z|m?n · 1 + |a1z
1 + ··· + anz?n|
1?|b1z?1 +··· + bmz?m|,
A8 |z| FBEPDWEE,D3DWEM
|a1z?1 + ··· + anz?n| < 110,|b1z?1 + ··· + bmz?m| < 110
CSD9 m?n greaterorequalslant 2,APCZ
|R(z)| < 1|z|m?n · 1 +
1
10
1? 110 <
2
|z|2.
87/107
FRCS,BFDWF7 R FBEPDWAR CR H2,CZ
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
CR
R(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
CR
|R(z)|ds lessorequalslant 2R2·piR = 2piR → 0 (R → +
EKEX,
integraldisplay +∞
∞
R(x)dx = 2pii
summationdisplay
Res[R(z),zk].
88/107
FRCS,BFDWF7 R FBEPDWAR CR H2,CZ
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay
CR
R(z)dz
vextendsinglevextendsingle
vextendsinglevextendsinglelessorequalslant
integraldisplay
CR
|R(z)|ds lessorequalslant 2R2·piR = 2piR → 0 (R → +
EKEX,
integraldisplay +∞
∞
R(x)dx = 2pii
summationdisplay
Res[R(z),zk].
88/107
DI 1 C7DYFKEP
integraldisplay +∞
0
x2
1 + x4dx,C AKE5.
B5F8GI f(z) = z
2
1 + z4 CZD3DRE8BRBGBM,G1ELH0DRBG
BM:
z0 = epi4i,z1 = e3pi4 i
EGD9H2DWEYEH,G1BFGIEPC7?
Res[f(z),z0] = z
2
(1 + z4)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=z0
= 14e?pi4i,
Res[f(z),z1] = z
2
(1 + z4)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=z1
= 14e?3pi4 i.
D9F0
integraldisplay +∞
∞
x2
1 + x4dx = 2pii{Res[f(z),z0] + Res[f(z),z1]} =
√2
2 pi.
89/107
DI 1 C7DYFKEP
integraldisplay +∞
0
x2
1 + x4dx,C AKE5.
B5F8GI f(z) = z
2
1 + z4 CZD3DRE8BRBGBM,G1ELH0DRBG
BM:
z0 = epi4i,z1 = e3pi4 i
EGD9H2DWEYEH,G1BFGIEPC7?
Res[f(z),z0] = z
2
(1 + z4)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=z0
= 14e?pi4i,
Res[f(z),z1] = z
2
(1 + z4)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=z1
= 14e?3pi4 i.
D9F0
integraldisplay +∞
∞
x2
1 + x4dx = 2pii{Res[f(z),z0] + Res[f(z),z1]} =
√2
2 pi.
89/107
DI 1 C7DYFKEP
integraldisplay +∞
0
x2
1 + x4dx,C AKE5.
B5F8GI f(z) = z
2
1 + z4 CZD3DRE8BRBGBM,G1ELH0DRBG
BM:
z0 = epi4i,z1 = e3pi4 i
EGD9H2DWEYEH,G1BFGIEPC7?
Res[f(z),z0] = z
2
(1 + z4)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=z0
= 14e?pi4i,
Res[f(z),z1] = z
2
(1 + z4)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=z1
= 14e?3pi4 i.
D9F0
integraldisplay +∞
∞
x2
1 + x4dx = 2pii{Res[f(z),z0] + Res[f(z),z1]} =
√2
2 pi.
89/107
DI 1 C7DYFKEP
integraldisplay +∞
0
x2
1 + x4dx,C AKE5.
B5F8GI f(z) = z
2
1 + z4 CZD3DRE8BRBGBM,G1ELH0DRBG
BM:
z0 = epi4i,z1 = e3pi4 i
EGD9H2DWEYEH,G1BFGIEPC7?
Res[f(z),z0] = z
2
(1 + z4)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=z0
= 14e?pi4i,
Res[f(z),z1] = z
2
(1 + z4)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=z1
= 14e?3pi4 i.
D9F0
integraldisplay +∞
∞
x2
1 + x4dx = 2pii{Res[f(z),z0] + Res[f(z),z1]} =
√2
2 pi.
89/107
DI 1 C7DYFKEP
integraldisplay +∞
0
x2
1 + x4dx,C AKE5.
B5F8GI f(z) = z
2
1 + z4 CZD3DRE8BRBGBM,G1ELH0DRBG
BM:
z0 = epi4i,z1 = e3pi4 i
EGD9H2DWEYEH,G1BFGIEPC7?
Res[f(z),z0] = z
2
(1 + z4)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=z0
= 14e?pi4i,
Res[f(z),z1] = z
2
(1 + z4)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=z1
= 14e?3pi4 i.
D9F0
integraldisplay +∞
∞
x2
1 + x4dx = 2pii{Res[f(z),z0] + Res[f(z),z1]} =
√2
2 pi.
89/107
DI 1 C7DYFKEP
integraldisplay +∞
0
x2
1 + x4dx,C AKE5.
B5F8GI f(z) = z
2
1 + z4 CZD3DRE8BRBGBM,G1ELH0DRBG
BM:
z0 = epi4i,z1 = e3pi4 i
EGD9H2DWEYEH,G1BFGIEPC7?
Res[f(z),z0] = z
2
(1 + z4)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=z0
= 14e?pi4i,
Res[f(z),z1] = z
2
(1 + z4)′
vextendsinglevextendsingle
vextendsinglevextendsingle
z=z1
= 14e?3pi4 i.
D9F0
integraldisplay +∞
∞
x2
1 + x4dx = 2pii{Res[f(z),z0] + Res[f(z),z1]} =
√2
2 pi.
89/107
FR?AQFKF8GIF0GRF8GI,EKEX
integraldisplay +∞
0
x2
1 + x4dx =
√2
4 pi.
90/107
DI 2 C7DYFKEPI =
integraldisplay +∞
∞
x2
(x2 + a2)(x2 + b2)dx(a > 0,b > 0).
B5 GXD4 m = 4,n = 2,FBDPEIF2H2 R(z) DACZAHG9G3
BM,FRCSFKEPDJBF.
F8GI
z2
(z2 + a2)(z2 + b2) CZD3DRE8BRBGBM±ai,±bi,G1
EL ai DQ bi BFH2DWEYEHCB,ENDMARBFGIEPC7?
Res[R(z),ai] = limz→ai
bracketleftbigg
(z?ai) z
2
(z2 + a2)(z2 + b2)
bracketrightbigg
= a2i(a2?b2).
Res[R(z),bi] = lim
z→bi
bracketleftbigg
(z?bi) z
2
(z2 + a2)(z2 + b2)
bracketrightbigg
= b2i(b2?a2).
EKEXI = 2pii{Res[R(z),ai] + Res[R(z),bi]} = pia + b.
91/107
DI 2 C7DYFKEPI =
integraldisplay +∞
∞
x2
(x2 + a2)(x2 + b2)dx(a > 0,b > 0).
B5 GXD4 m = 4,n = 2,FBDPEIF2H2 R(z) DACZAHG9G3
BM,FRCSFKEPDJBF.
F8GI
z2
(z2 + a2)(z2 + b2) CZD3DRE8BRBGBM±ai,±bi,G1
EL ai DQ bi BFH2DWEYEHCB,ENDMARBFGIEPC7?
Res[R(z),ai] = limz→ai
bracketleftbigg
(z?ai) z
2
(z2 + a2)(z2 + b2)
bracketrightbigg
= a2i(a2?b2).
Res[R(z),bi] = lim
z→bi
bracketleftbigg
(z?bi) z
2
(z2 + a2)(z2 + b2)
bracketrightbigg
= b2i(b2?a2).
EKEXI = 2pii{Res[R(z),ai] + Res[R(z),bi]} = pia + b.
91/107
DI 2 C7DYFKEPI =
integraldisplay +∞
∞
x2
(x2 + a2)(x2 + b2)dx(a > 0,b > 0).
B5 GXD4 m = 4,n = 2,FBDPEIF2H2 R(z) DACZAHG9G3
BM,FRCSFKEPDJBF.
F8GI
z2
(z2 + a2)(z2 + b2) CZD3DRE8BRBGBM±ai,±bi,G1
EL ai DQ bi BFH2DWEYEHCB,ENDMARBFGIEPC7?
Res[R(z),ai] = limz→ai
bracketleftbigg
(z?ai) z
2
(z2 + a2)(z2 + b2)
bracketrightbigg
= a2i(a2?b2).
Res[R(z),bi] = lim
z→bi
bracketleftbigg
(z?bi) z
2
(z2 + a2)(z2 + b2)
bracketrightbigg
= b2i(b2?a2).
EKEXI = 2pii{Res[R(z),ai] + Res[R(z),bi]} = pia + b.
91/107
DI 2 C7DYFKEPI =
integraldisplay +∞
∞
x2
(x2 + a2)(x2 + b2)dx(a > 0,b > 0).
B5 GXD4 m = 4,n = 2,FBDPEIF2H2 R(z) DACZAHG9G3
BM,FRCSFKEPDJBF.
F8GI
z2
(z2 + a2)(z2 + b2) CZD3DRE8BRBGBM±ai,±bi,G1
EL ai DQ bi BFH2DWEYEHCB,ENDMARBFGIEPC7?
Res[R(z),ai] = limz→ai
bracketleftbigg
(z?ai) z
2
(z2 + a2)(z2 + b2)
bracketrightbigg
= a2i(a2?b2).
Res[R(z),bi] = lim
z→bi
bracketleftbigg
(z?bi) z
2
(z2 + a2)(z2 + b2)
bracketrightbigg
= b2i(b2?a2).
EKEXI = 2pii{Res[R(z),ai] + Res[R(z),bi]} = pia + b.
91/107
DI 2 C7DYFKEPI =
integraldisplay +∞
∞
x2
(x2 + a2)(x2 + b2)dx(a > 0,b > 0).
B5 GXD4 m = 4,n = 2,FBDPEIF2H2 R(z) DACZAHG9G3
BM,FRCSFKEPDJBF.
F8GI
z2
(z2 + a2)(z2 + b2) CZD3DRE8BRBGBM±ai,±bi,G1
EL ai DQ bi BFH2DWEYEHCB,ENDMARBFGIEPC7?
Res[R(z),ai] = limz→ai
bracketleftbigg
(z?ai) z
2
(z2 + a2)(z2 + b2)
bracketrightbigg
= a2i(a2?b2).
Res[R(z),bi] = lim
z→bi
bracketleftbigg
(z?bi) z
2
(z2 + a2)(z2 + b2)
bracketrightbigg
= b2i(b2?a2).
EKEXI = 2pii{Res[R(z),ai] + Res[R(z),bi]} = pia + b.
91/107
DI 3 C7DYFKEPI =
integraldisplay +∞
∞
dx
(1 + x2)n+1.
B5 AN f(z) = 1(1 + z2)n+1,ENBFH2DWEYEHCZDXE8ARAH
G9G3BM z = i,ENF0 f(z) AR n + 1 BRBGBM,G1BFGIF0
Res[f(z),i] = 1n! d
n
dzn
braceleftbigg 1
(z + i)n+1
bracerightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
z=i
= 1n! (?1)
n(n + 1)(n + 2)···(2n)
(2i)2n+1
= 12i (2n?1)!!(2n)!!
EKEX I = 2piiRes[f(z),i] = 12i(2n? 1)!!(2n)!! pi.
92/107
DI 3 C7DYFKEPI =
integraldisplay +∞
∞
dx
(1 + x2)n+1.
B5 AN f(z) = 1(1 + z2)n+1,ENBFH2DWEYEHCZDXE8ARAH
G9G3BM z = i,ENF0 f(z) AR n + 1 BRBGBM,G1BFGIF0
Res[f(z),i] = 1n! d
n
dzn
braceleftbigg 1
(z + i)n+1
bracerightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
z=i
= 1n! (?1)
n(n + 1)(n + 2)···(2n)
(2i)2n+1
= 12i (2n?1)!!(2n)!!
EKEX I = 2piiRes[f(z),i] = 12i(2n? 1)!!(2n)!! pi.
92/107
DI 3 C7DYFKEPI =
integraldisplay +∞
∞
dx
(1 + x2)n+1.
B5 AN f(z) = 1(1 + z2)n+1,ENBFH2DWEYEHCZDXE8ARAH
G9G3BM z = i,ENF0 f(z) AR n + 1 BRBGBM,G1BFGIF0
Res[f(z),i] = 1n! d
n
dzn
braceleftbigg 1
(z + i)n+1
bracerightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
z=i
= 1n! (?1)
n(n + 1)(n + 2)···(2n)
(2i)2n+1
= 12i (2n?1)!!(2n)!!
EKEX I = 2piiRes[f(z),i] = 12i(2n? 1)!!(2n)!! pi.
92/107
DI 3 C7DYFKEPI =
integraldisplay +∞
∞
dx
(1 + x2)n+1.
B5 AN f(z) = 1(1 + z2)n+1,ENBFH2DWEYEHCZDXE8ARAH
G9G3BM z = i,ENF0 f(z) AR n + 1 BRBGBM,G1BFGIF0
Res[f(z),i] = 1n! d
n
dzn
braceleftbigg 1
(z + i)n+1
bracerightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
z=i
= 1n! (?1)
n(n + 1)(n + 2)···(2n)
(2i)2n+1
= 12i (2n?1)!!(2n)!!
EKEX I = 2piiRes[f(z),i] = 12i(2n? 1)!!(2n)!! pi.
92/107
DI 3 C7DYFKEPI =
integraldisplay +∞
∞
dx
(1 + x2)n+1.
B5 AN f(z) = 1(1 + z2)n+1,ENBFH2DWEYEHCZDXE8ARAH
G9G3BM z = i,ENF0 f(z) AR n + 1 BRBGBM,G1BFGIF0
Res[f(z),i] = 1n! d
n
dzn
braceleftbigg 1
(z + i)n+1
bracerightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
z=i
= 1n! (?1)
n(n + 1)(n + 2)···(2n)
(2i)2n+1
= 12i (2n?1)!!(2n)!!
EKEX I = 2piiRes[f(z),i] = 12i(2n? 1)!!(2n)!! pi.
92/107
3,BBCG
integraldisplay +∞
∞
R(x)eaixdx (a > 0) ARFKEP,A8 R(x)
F0 x ARCZD2F8GI,ADEPBDARCVGIE6A8B3EPCUARCVGID4E8
CV,FBDP R(z) BFEIF2H2DACZAHG9G3BMEE.
E0DF CPD5 2 ELARBRD2,CSD9 m?n greaterorequalslant 1,APGHFBEPDW
AR |z|,CZ
|R(z)| < 2|z|.
FRCS,BFDWF7 R FBDWAR CR H2,CZvextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
CR
R(z)eaizdz
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
integraldisplay
CR
|R(z)||eaiz|dz < 2R
integraldisplay
CR
e?ayds
= 2
integraldisplay pi
0
e?aRsinθdθ = 4
integraldisplay pi2
0
e?aRsinθdθ
lessorequalslant
integraldisplay pi2
0
e?aR(2θ/pi)dθ = 2piaR(1? e?aR) → 0,
(R → +∞).
93/107
3,BBCG
integraldisplay +∞
∞
R(x)eaixdx (a > 0) ARFKEP,A8 R(x)
F0 x ARCZD2F8GI,ADEPBDARCVGIE6A8B3EPCUARCVGID4E8
CV,FBDP R(z) BFEIF2H2DACZAHG9G3BMEE.
E0DF CPD5 2 ELARBRD2,CSD9 m?n greaterorequalslant 1,APGHFBEPDW
AR |z|,CZ
|R(z)| < 2|z|.
FRCS,BFDWF7 R FBDWAR CR H2,CZvextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
CR
R(z)eaizdz
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
integraldisplay
CR
|R(z)||eaiz|dz < 2R
integraldisplay
CR
e?ayds
= 2
integraldisplay pi
0
e?aRsinθdθ = 4
integraldisplay pi2
0
e?aRsinθdθ
lessorequalslant
integraldisplay pi2
0
e?aR(2θ/pi)dθ = 2piaR(1? e?aR) → 0,
(R → +∞).
93/107
3,BBCG
integraldisplay +∞
∞
R(x)eaixdx (a > 0) ARFKEP,A8 R(x)
F0 x ARCZD2F8GI,ADEPBDARCVGIE6A8B3EPCUARCVGID4E8
CV,FBDP R(z) BFEIF2H2DACZAHG9G3BMEE.
E0DF CPD5 2 ELARBRD2,CSD9 m?n greaterorequalslant 1,APGHFBEPDW
AR |z|,CZ
|R(z)| < 2|z|.
FRCS,BFDWF7 R FBDWAR CR H2,CZvextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
CR
R(z)eaizdz
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
integraldisplay
CR
|R(z)||eaiz|dz < 2R
integraldisplay
CR
e?ayds
= 2
integraldisplay pi
0
e?aRsinθdθ = 4
integraldisplay pi2
0
e?aRsinθdθ
lessorequalslant
integraldisplay pi2
0
e?aR(2θ/pi)dθ = 2piaR(1? e?aR) → 0,
(R → +∞).
93/107
3,BBCG
integraldisplay +∞
∞
R(x)eaixdx (a > 0) ARFKEP,A8 R(x)
F0 x ARCZD2F8GI,ADEPBDARCVGIE6A8B3EPCUARCVGID4E8
CV,FBDP R(z) BFEIF2H2DACZAHG9G3BMEE.
E0DF CPD5 2 ELARBRD2,CSD9 m?n greaterorequalslant 1,APGHFBEPDW
AR |z|,CZ
|R(z)| < 2|z|.
FRCS,BFDWF7 R FBDWAR CR H2,CZvextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
CR
R(z)eaizdz
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
integraldisplay
CR
|R(z)||eaiz|dz < 2R
integraldisplay
CR
e?ayds
= 2
integraldisplay pi
0
e?aRsinθdθ = 4
integraldisplay pi2
0
e?aRsinθdθ
lessorequalslant
integraldisplay pi2
0
e?aR(2θ/pi)dθ = 2piaR(1? e?aR) → 0,
(R → +∞).
93/107
3,BBCG
integraldisplay +∞
∞
R(x)eaixdx (a > 0) ARFKEP,A8 R(x)
F0 x ARCZD2F8GI,ADEPBDARCVGIE6A8B3EPCUARCVGID4E8
CV,FBDP R(z) BFEIF2H2DACZAHG9G3BMEE.
E0DF CPD5 2 ELARBRD2,CSD9 m?n greaterorequalslant 1,APGHFBEPDW
AR |z|,CZ
|R(z)| < 2|z|.
FRCS,BFDWF7 R FBDWAR CR H2,CZvextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
CR
R(z)eaizdz
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
integraldisplay
CR
|R(z)||eaiz|dz < 2R
integraldisplay
CR
e?ayds
= 2
integraldisplay pi
0
e?aRsinθdθ = 4
integraldisplay pi2
0
e?aRsinθdθ
lessorequalslant
integraldisplay pi2
0
e?aR(2θ/pi)dθ = 2piaR(1? e?aR) → 0,
(R → +∞).
93/107
3,BBCG
integraldisplay +∞
∞
R(x)eaixdx (a > 0) ARFKEP,A8 R(x)
F0 x ARCZD2F8GI,ADEPBDARCVGIE6A8B3EPCUARCVGID4E8
CV,FBDP R(z) BFEIF2H2DACZAHG9G3BMEE.
E0DF CPD5 2 ELARBRD2,CSD9 m?n greaterorequalslant 1,APGHFBEPDW
AR |z|,CZ
|R(z)| < 2|z|.
FRCS,BFDWF7 R FBDWAR CR H2,CZvextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
CR
R(z)eaizdz
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
integraldisplay
CR
|R(z)||eaiz|dz < 2R
integraldisplay
CR
e?ayds
= 2
integraldisplay pi
0
e?aRsinθdθ = 4
integraldisplay pi2
0
e?aRsinθdθ
lessorequalslant
integraldisplay pi2
0
e?aR(2θ/pi)dθ = 2piaR(1? e?aR) → 0,
(R → +∞).
93/107
3,BBCG
integraldisplay +∞
∞
R(x)eaixdx (a > 0) ARFKEP,A8 R(x)
F0 x ARCZD2F8GI,ADEPBDARCVGIE6A8B3EPCUARCVGID4E8
CV,FBDP R(z) BFEIF2H2DACZAHG9G3BMEE.
E0DF CPD5 2 ELARBRD2,CSD9 m?n greaterorequalslant 1,APGHFBEPDW
AR |z|,CZ
|R(z)| < 2|z|.
FRCS,BFDWF7 R FBDWAR CR H2,CZvextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay
CR
R(z)eaizdz
vextendsinglevextendsingle
vextendsinglevextendsingle lessorequalslant
integraldisplay
CR
|R(z)||eaiz|dz < 2R
integraldisplay
CR
e?ayds
= 2
integraldisplay pi
0
e?aRsinθdθ = 4
integraldisplay pi2
0
e?aRsinθdθ
lessorequalslant
integraldisplay pi2
0
e?aR(2θ/pi)dθ = 2piaR(1? e?aR) → 0,
(R → +∞).
93/107
FRCSAQ
integraldisplay +∞
∞
R(x)eaixdx = 2pii
summationdisplay
Res[R(z)eaiz,zk],
F9GU
integraldisplay +∞
∞
R(x)cosaxdx + i
integraldisplay +∞
∞
R(x)sinaxdx
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
= 2pii
summationdisplay
Res[R(z)eaiz,zk].
94/107
FRCSAQ
integraldisplay +∞
∞
R(x)eaixdx = 2pii
summationdisplay
Res[R(z)eaiz,zk],
F9GU
integraldisplay +∞
∞
R(x)cosaxdx + i
integraldisplay +∞
∞
R(x)sinaxdx
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
= 2pii
summationdisplay
Res[R(z)eaiz,zk].
94/107
FRCSAQ
integraldisplay +∞
∞
R(x)eaixdx = 2pii
summationdisplay
Res[R(z)eaiz,zk],
F9GU
integraldisplay +∞
∞
R(x)cosaxdx + i
integraldisplay +∞
∞
R(x)sinaxdx
bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright
= 2pii
summationdisplay
Res[R(z)eaiz,zk].
94/107
DI 1 C7DY I =
integraldisplay +∞
0
cosax
1 + x2dx (a > 0).
B5 AN R(z) = 11 + z2,R(z) BFH2DWEYEHCZDXE8ARAHG9
G3BM (E8BRBGBM)z = i,AP
integraldisplay +∞
∞
1
1 + x2e
iaxdx = 2pii·Res[R(z)eiaz,i] = 2pii· 1
2ie
a = pie?a.
FRCS,
integraldisplay +∞
∞
cosax
1 + x2dx = pie
a,
EKEX
integraldisplay +∞
0
cosax
1 + x2dx =
pi
2e
a.
95/107
DI 1 C7DY I =
integraldisplay +∞
0
cosax
1 + x2dx (a > 0).
B5 AN R(z) = 11 + z2,R(z) BFH2DWEYEHCZDXE8ARAHG9
G3BM (E8BRBGBM)z = i,AP
integraldisplay +∞
∞
1
1 + x2e
iaxdx = 2pii·Res[R(z)eiaz,i] = 2pii· 1
2ie
a = pie?a.
FRCS,
integraldisplay +∞
∞
cosax
1 + x2dx = pie
a,
EKEX
integraldisplay +∞
0
cosax
1 + x2dx =
pi
2e
a.
95/107
DI 1 C7DY I =
integraldisplay +∞
0
cosax
1 + x2dx (a > 0).
B5 AN R(z) = 11 + z2,R(z) BFH2DWEYEHCZDXE8ARAHG9
G3BM (E8BRBGBM)z = i,AP
integraldisplay +∞
∞
1
1 + x2e
iaxdx = 2pii·Res[R(z)eiaz,i] = 2pii· 1
2ie
a = pie?a.
FRCS,
integraldisplay +∞
∞
cosax
1 + x2dx = pie
a,
EKEX
integraldisplay +∞
0
cosax
1 + x2dx =
pi
2e
a.
95/107
DI 1 C7DY I =
integraldisplay +∞
0
cosax
1 + x2dx (a > 0).
B5 AN R(z) = 11 + z2,R(z) BFH2DWEYEHCZDXE8ARAHG9
G3BM (E8BRBGBM)z = i,AP
integraldisplay +∞
∞
1
1 + x2e
iaxdx = 2pii·Res[R(z)eiaz,i] = 2pii· 1
2ie
a = pie?a.
FRCS,
integraldisplay +∞
∞
cosax
1 + x2dx = pie
a,
EKEX
integraldisplay +∞
0
cosax
1 + x2dx =
pi
2e
a.
95/107
DI 1 C7DY I =
integraldisplay +∞
0
cosax
1 + x2dx (a > 0).
B5 AN R(z) = 11 + z2,R(z) BFH2DWEYEHCZDXE8ARAHG9
G3BM (E8BRBGBM)z = i,AP
integraldisplay +∞
∞
1
1 + x2e
iaxdx = 2pii·Res[R(z)eiaz,i] = 2pii· 1
2ie
a = pie?a.
FRCS,
integraldisplay +∞
∞
cosax
1 + x2dx = pie
a,
EKEX
integraldisplay +∞
0
cosax
1 + x2dx =
pi
2e
a.
95/107
DI 1 C7DY I =
integraldisplay +∞
0
cosax
1 + x2dx (a > 0).
B5 AN R(z) = 11 + z2,R(z) BFH2DWEYEHCZDXE8ARAHG9
G3BM (E8BRBGBM)z = i,AP
integraldisplay +∞
∞
1
1 + x2e
iaxdx = 2pii·Res[R(z)eiaz,i] = 2pii· 1
2ie
a = pie?a.
FRCS,
integraldisplay +∞
∞
cosax
1 + x2dx = pie
a,
EKEX
integraldisplay +∞
0
cosax
1 + x2dx =
pi
2e
a.
95/107
DI 2 C7DY I =
integraldisplay +∞
0
xsinx
x2 + a2dx (a > 0).
B5 GXD4GXD4 m = 2,n = 1,FBDPEIF2H2 R(z) DACZAH
G9G3BM,FRCSFKEPDJBF.
F8GI R(z) = zz2 + a2 BFH2DWEYEHCBCZE8DRE8BRBGBM ai,
AP
integraldisplay +∞
0
x
x2 + a2e
ixdx = 2pii·Res[R(z)eiz,ai] = 2pii· e
a
2,
FRCS,
integraldisplay +∞
0
xsinx
x2 + a2dx =
1
2pie
a.
96/107
DI 2 C7DY I =
integraldisplay +∞
0
xsinx
x2 + a2dx (a > 0).
B5 GXD4GXD4 m = 2,n = 1,FBDPEIF2H2 R(z) DACZAH
G9G3BM,FRCSFKEPDJBF.
F8GI R(z) = zz2 + a2 BFH2DWEYEHCBCZE8DRE8BRBGBM ai,
AP
integraldisplay +∞
0
x
x2 + a2e
ixdx = 2pii·Res[R(z)eiz,ai] = 2pii· e
a
2,
FRCS,
integraldisplay +∞
0
xsinx
x2 + a2dx =
1
2pie
a.
96/107
DI 2 C7DY I =
integraldisplay +∞
0
xsinx
x2 + a2dx (a > 0).
B5 GXD4GXD4 m = 2,n = 1,FBDPEIF2H2 R(z) DACZAH
G9G3BM,FRCSFKEPDJBF.
F8GI R(z) = zz2 + a2 BFH2DWEYEHCBCZE8DRE8BRBGBM ai,
AP
integraldisplay +∞
0
x
x2 + a2e
ixdx = 2pii·Res[R(z)eiz,ai] = 2pii· e
a
2,
FRCS,
integraldisplay +∞
0
xsinx
x2 + a2dx =
1
2pie
a.
96/107
DI 2 C7DY I =
integraldisplay +∞
0
xsinx
x2 + a2dx (a > 0).
B5 GXD4GXD4 m = 2,n = 1,FBDPEIF2H2 R(z) DACZAH
G9G3BM,FRCSFKEPDJBF.
F8GI R(z) = zz2 + a2 BFH2DWEYEHCBCZE8DRE8BRBGBM ai,
AP
integraldisplay +∞
0
x
x2 + a2e
ixdx = 2pii·Res[R(z)eiz,ai] = 2pii· e
a
2,
FRCS,
integraldisplay +∞
0
xsinx
x2 + a2dx =
1
2pie
a.
96/107
DI 2 C7DY I =
integraldisplay +∞
0
xsinx
x2 + a2dx (a > 0).
B5 GXD4GXD4 m = 2,n = 1,FBDPEIF2H2 R(z) DACZAH
G9G3BM,FRCSFKEPDJBF.
F8GI R(z) = zz2 + a2 BFH2DWEYEHCBCZE8DRE8BRBGBM ai,
AP
integraldisplay +∞
0
x
x2 + a2e
ixdx = 2pii·Res[R(z)eiz,ai] = 2pii· e
a
2,
FRCS,
integraldisplay +∞
0
xsinx
x2 + a2dx =
1
2pie
a.
96/107
DI 3 DDEY:
integraldisplay +∞
0
sinx2dx =
integraldisplay +∞
0
cosx2dx = 12
radicalbiggpi
2.
(EUDJ
integraldisplay +∞
0
e?x2dx =
√pi
2,)
B5DJFGF8GI eiz2,A8 z = x EE,CZ
eix2 = cosx2 + isinx2,
ENAREIG4DQC2G4EPC7F0EKEMFKEPARAQFKF8GI.
AIFKEPBGETDM C F0DWF7? R AR
pi
4 GVBBARBQBA.
CSD9 eiz2 BF D CBBLG1BQBAH2B5G3,DUGADPG4 - AKCXF9
D2 contintegraldisplay
C
eiz2dz = 0,
BP
contintegraldisplay
OA
eiz2dz +
contintegraldisplay
hatwidestAB
eiz2dz +
contintegraldisplay
BO
eiz2dz = 0.
97/107
DI 3 DDEY:
integraldisplay +∞
0
sinx2dx =
integraldisplay +∞
0
cosx2dx = 12
radicalbiggpi
2.
(EUDJ
integraldisplay +∞
0
e?x2dx =
√pi
2,)
B5DJFGF8GI eiz2,A8 z = x EE,CZ
eix2 = cosx2 + isinx2,
ENAREIG4DQC2G4EPC7F0EKEMFKEPARAQFKF8GI.
AIFKEPBGETDM C F0DWF7? R AR
pi
4 GVBBARBQBA.
CSD9 eiz2 BF D CBBLG1BQBAH2B5G3,DUGADPG4 - AKCXF9
D2 contintegraldisplay
C
eiz2dz = 0,
BP
contintegraldisplay
OA
eiz2dz +
contintegraldisplay
hatwidestAB
eiz2dz +
contintegraldisplay
BO
eiz2dz = 0.
97/107
DI 3 DDEY:
integraldisplay +∞
0
sinx2dx =
integraldisplay +∞
0
cosx2dx = 12
radicalbiggpi
2.
(EUDJ
integraldisplay +∞
0
e?x2dx =
√pi
2,)
B5DJFGF8GI eiz2,A8 z = x EE,CZ
eix2 = cosx2 + isinx2,
ENAREIG4DQC2G4EPC7F0EKEMFKEPARAQFKF8GI.
AIFKEPBGETDM C F0DWF7? R AR
pi
4 GVBBARBQBA.
CSD9 eiz2 BF D CBBLG1BQBAH2B5G3,DUGADPG4 - AKCXF9
D2 contintegraldisplay
C
eiz2dz = 0,
BP
contintegraldisplay
OA
eiz2dz +
contintegraldisplay
hatwidestAB
eiz2dz +
contintegraldisplay
BO
eiz2dz = 0.
97/107
DI 3 DDEY:
integraldisplay +∞
0
sinx2dx =
integraldisplay +∞
0
cosx2dx = 12
radicalbiggpi
2.
(EUDJ
integraldisplay +∞
0
e?x2dx =
√pi
2,)
B5DJFGF8GI eiz2,A8 z = x EE,CZ
eix2 = cosx2 + isinx2,
ENAREIG4DQC2G4EPC7F0EKEMFKEPARAQFKF8GI.
AIFKEPBGETDM C F0DWF7? R AR
pi
4 GVBBARBQBA.
CSD9 eiz2 BF D CBBLG1BQBAH2B5G3,DUGADPG4 - AKCXF9
D2 contintegraldisplay
C
eiz2dz = 0,
BP
contintegraldisplay
OA
eiz2dz +
contintegraldisplay
hatwidestAB
eiz2dz +
contintegraldisplay
BO
eiz2dz = 0.
97/107
DI 3 DDEY:
integraldisplay +∞
0
sinx2dx =
integraldisplay +∞
0
cosx2dx = 12
radicalbiggpi
2.
(EUDJ
integraldisplay +∞
0
e?x2dx =
√pi
2,)
B5DJFGF8GI eiz2,A8 z = x EE,CZ
eix2 = cosx2 + isinx2,
ENAREIG4DQC2G4EPC7F0EKEMFKEPARAQFKF8GI.
AIFKEPBGETDM C F0DWF7? R AR
pi
4 GVBBARBQBA.
CSD9 eiz2 BF D CBBLG1BQBAH2B5G3,DUGADPG4 - AKCXF9
D2 contintegraldisplay
C
eiz2dz = 0,
BP
contintegraldisplay
OA
eiz2dz +
contintegraldisplay
hatwidestAB
eiz2dz +
contintegraldisplay
BO
eiz2dz = 0.
97/107
O
y
x
B
D
A
R
R
G1ELDSETDMARGJGIE0ENEPC7F0
OA,z = x,0 lessorequalslant x lessorequalslant R;
hatwidestAB,z = Reiθ,0 lessorequalslant θ lessorequalslant pi
4;
BO,z = reipi4,r,R → 0;
EKEX
integraldisplay R
0
eix2dx +
integraldisplay pi4
0
eiR2ei2θdθ +
integraldisplay 0
R
eir2e
pi2i
epi4idx = 0.
98/107
O
y
x
B
D
A
R
R
G1ELDSETDMARGJGIE0ENEPC7F0
OA,z = x,0 lessorequalslant x lessorequalslant R;
hatwidestAB,z = Reiθ,0 lessorequalslant θ lessorequalslant pi
4;
BO,z = reipi4,r,R → 0;
EKEX
integraldisplay R
0
eix2dx +
integraldisplay pi4
0
eiR2ei2θdθ +
integraldisplay 0
R
eir2e
pi2i
epi4idx = 0.
98/107
BP
integraldisplay R
0
(cosx2 + isinx2)dx
= epi4i
integraldisplay R
0
e?r2dr?
integraldisplay pi4
0
eiR2cos2θ?R2sin2θiReiθdθ.
A8 R → ∞ EE,
epi4i
integraldisplay ∞
0
e?r2dr =
√pi
2 ·e
pi
4i = 12
radicalbiggpi
2 +
i
2
radicalbiggpi
2.
ADBDAJDRFKEP
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay pi4
0
eiR2cos2θ?R2sin2θiReiθdθ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lessorequalslant
integraldisplay pi4
0
|eiR2cos2θ?R2sin2θ|·|i|·|R|·|eiθ|dθ
lessorequalslant
integraldisplay pi4
0
e?R2sin2θRdθ lessorequalslant R
integraldisplay pi4
0
e?pi4R2θdθ
= pi4R(1?e?R2) → 0,(R → ∞).
99/107
BP
integraldisplay R
0
(cosx2 + isinx2)dx
= epi4i
integraldisplay R
0
e?r2dr?
integraldisplay pi4
0
eiR2cos2θ?R2sin2θiReiθdθ.
A8 R → ∞ EE,
epi4i
integraldisplay ∞
0
e?r2dr =
√pi
2 ·e
pi
4i = 12
radicalbiggpi
2 +
i
2
radicalbiggpi
2.
ADBDAJDRFKEP
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay pi4
0
eiR2cos2θ?R2sin2θiReiθdθ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lessorequalslant
integraldisplay pi4
0
|eiR2cos2θ?R2sin2θ|·|i|·|R|·|eiθ|dθ
lessorequalslant
integraldisplay pi4
0
e?R2sin2θRdθ lessorequalslant R
integraldisplay pi4
0
e?pi4R2θdθ
= pi4R(1?e?R2) → 0,(R → ∞).
99/107
BP
integraldisplay R
0
(cosx2 + isinx2)dx
= epi4i
integraldisplay R
0
e?r2dr?
integraldisplay pi4
0
eiR2cos2θ?R2sin2θiReiθdθ.
A8 R → ∞ EE,
epi4i
integraldisplay ∞
0
e?r2dr =
√pi
2 ·e
pi
4i = 12
radicalbiggpi
2 +
i
2
radicalbiggpi
2.
ADBDAJDRFKEP
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay pi4
0
eiR2cos2θ?R2sin2θiReiθdθ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lessorequalslant
integraldisplay pi4
0
|eiR2cos2θ?R2sin2θ|·|i|·|R|·|eiθ|dθ
lessorequalslant
integraldisplay pi4
0
e?R2sin2θRdθ lessorequalslant R
integraldisplay pi4
0
e?pi4R2θdθ
= pi4R(1?e?R2) → 0,(R → ∞).
99/107
BP
integraldisplay R
0
(cosx2 + isinx2)dx
= epi4i
integraldisplay R
0
e?r2dr?
integraldisplay pi4
0
eiR2cos2θ?R2sin2θiReiθdθ.
A8 R → ∞ EE,
epi4i
integraldisplay ∞
0
e?r2dr =
√pi
2 ·e
pi
4i = 12
radicalbiggpi
2 +
i
2
radicalbiggpi
2.
ADBDAJDRFKEP
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay pi4
0
eiR2cos2θ?R2sin2θiReiθdθ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lessorequalslant
integraldisplay pi4
0
|eiR2cos2θ?R2sin2θ|·|i|·|R|·|eiθ|dθ
lessorequalslant
integraldisplay pi4
0
e?R2sin2θRdθ lessorequalslant R
integraldisplay pi4
0
e?pi4R2θdθ
= pi4R(1?e?R2) → 0,(R → ∞).
99/107
BP
integraldisplay R
0
(cosx2 + isinx2)dx
= epi4i
integraldisplay R
0
e?r2dr?
integraldisplay pi4
0
eiR2cos2θ?R2sin2θiReiθdθ.
A8 R → ∞ EE,
epi4i
integraldisplay ∞
0
e?r2dr =
√pi
2 ·e
pi
4i = 12
radicalbiggpi
2 +
i
2
radicalbiggpi
2.
ADBDAJDRFKEP
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay pi4
0
eiR2cos2θ?R2sin2θiReiθdθ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lessorequalslant
integraldisplay pi4
0
|eiR2cos2θ?R2sin2θ|·|i|·|R|·|eiθ|dθ
lessorequalslant
integraldisplay pi4
0
e?R2sin2θRdθ lessorequalslant R
integraldisplay pi4
0
e?pi4R2θdθ
= pi4R(1?e?R2) → 0,(R → ∞).
99/107
BP
integraldisplay R
0
(cosx2 + isinx2)dx
= epi4i
integraldisplay R
0
e?r2dr?
integraldisplay pi4
0
eiR2cos2θ?R2sin2θiReiθdθ.
A8 R → ∞ EE,
epi4i
integraldisplay ∞
0
e?r2dr =
√pi
2 ·e
pi
4i = 12
radicalbiggpi
2 +
i
2
radicalbiggpi
2.
ADBDAJDRFKEP
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
integraldisplay pi4
0
eiR2cos2θ?R2sin2θiReiθdθ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
lessorequalslant
integraldisplay pi4
0
|eiR2cos2θ?R2sin2θ|·|i|·|R|·|eiθ|dθ
lessorequalslant
integraldisplay pi4
0
e?R2sin2θRdθ lessorequalslant R
integraldisplay pi4
0
e?pi4R2θdθ
= pi4R(1?e?R2) → 0,(R → ∞).
99/107
EKEXA8 R → ∞ EE,CZ
integraldisplay ∞
0
(cosx2 + isinx2)dx = 12
radicalbiggpi
2 +
i
2
radicalbiggpi
2.
B9H0G8AREIG4G2C2G4EPC7DNAV,AQ
integraldisplay +∞
0
sinx2dx =
integraldisplay +∞
0
cosx2dx = 12
radicalbiggpi
2.
100/107
EKEXA8 R → ∞ EE,CZ
integraldisplay ∞
0
(cosx2 + isinx2)dx = 12
radicalbiggpi
2 +
i
2
radicalbiggpi
2.
B9H0G8AREIG4G2C2G4EPC7DNAV,AQ
integraldisplay +∞
0
sinx2dx =
integraldisplay +∞
0
cosx2dx = 12
radicalbiggpi
2.
100/107
BDD3AX GHGIBFGIDQFN?A6D2
1,GHGIBFGI
CTAFBBEQARFKEP
1
2pii
contintegraldisplay
C
f′(z)
f(z) dz
EH? f(z) B3D9ETDM C ARGHGIBFGI.
101/107
BDD3AX GHGIBFGIDQFN?A6D2
1,GHGIBFGI
CTAFBBEQARFKEP
1
2pii
contintegraldisplay
C
f′(z)
f(z) dz
EH? f(z) B3D9ETDM C ARGHGIBFGI.
101/107
F9D2E8CGC2 f(z) BFDFH0BGETDM C H2B5G3DPG0?AY,
BF C ARCBG4G0AMCZDLDRBGBMEXA9?BRBRB5G3,BSD3
1
2pii
contintegraldisplay
C
f′(z)
f(z) dz = N?P,
G1EL,N? f(z) BF C CBAYBMARD3DRGI,P? f(z) BF C
CBBGBMARD3DRGI,DP C AIAKE5.
G1EL,BFC7DYAYBMDQBGBMARDRGIEE,m BRARAYBMF9BGBM
DYDU m DRAYBMF9BGBM.
102/107
F9D2E8CGC2 f(z) BFDFH0BGETDM C H2B5G3DPG0?AY,
BF C ARCBG4G0AMCZDLDRBGBMEXA9?BRBRB5G3,BSD3
1
2pii
contintegraldisplay
C
f′(z)
f(z) dz = N?P,
G1EL,N? f(z) BF C CBAYBMARD3DRGI,P? f(z) BF C
CBBGBMARD3DRGI,DP C AIAKE5.
G1EL,BFC7DYAYBMDQBGBMARDRGIEE,m BRARAYBMF9BGBM
DYDU m DRAYBMF9BGBM.
102/107
DD AN f(z) BF C CBCZE8DR nk BRARAYBM ak,BSD3BF ak
ARAPDX |z?zk| < δ CB,CZ
f(z) = (z?ak)nk?(z),
G1EL,?(z) BFCCAPDXCBB5G3,DP?(ak) negationslash= 0.
D9F0
f′(z) = nk(z?ak)nk?1?(z) + (z?zk)nk?′(z),
APBF 0 < |z?ak| < δ CB,CZ
f′(z)
f(z) =
nk
z?ak +
′(z)
(z)
CSD9BF 0 < |z?ak| < δ CB,?(z) B5G3,FRAD?′(z) B5
G3,DP?(z) negationslash= 0,EKEX
′(z)
(z) BFCCAPDXCBB5G3.
103/107
DD AN f(z) BF C CBCZE8DR nk BRARAYBM ak,BSD3BF ak
ARAPDX |z?zk| < δ CB,CZ
f(z) = (z?ak)nk?(z),
G1EL,?(z) BFCCAPDXCBB5G3,DP?(ak) negationslash= 0.
D9F0
f′(z) = nk(z?ak)nk?1?(z) + (z?zk)nk?′(z),
APBF 0 < |z?ak| < δ CB,CZ
f′(z)
f(z) =
nk
z?ak +
′(z)
(z)
CSD9BF 0 < |z?ak| < δ CB,?(z) B5G3,FRAD?′(z) B5
G3,DP?(z) negationslash= 0,EKEX
′(z)
(z) BFCCAPDXCBB5G3.
103/107
DD AN f(z) BF C CBCZE8DR nk BRARAYBM ak,BSD3BF ak
ARAPDX |z?zk| < δ CB,CZ
f(z) = (z?ak)nk?(z),
G1EL,?(z) BFCCAPDXCBB5G3,DP?(ak) negationslash= 0.
D9F0
f′(z) = nk(z?ak)nk?1?(z) + (z?zk)nk?′(z),
APBF 0 < |z?ak| < δ CB,CZ
f′(z)
f(z) =
nk
z?ak +
′(z)
(z)
CSD9BF 0 < |z?ak| < δ CB,?(z) B5G3,FRAD?′(z) B5
G3,DP?(z) negationslash= 0,EKEX
′(z)
(z) BFCCAPDXCBB5G3.
103/107
DD AN f(z) BF C CBCZE8DR nk BRARAYBM ak,BSD3BF ak
ARAPDX |z?zk| < δ CB,CZ
f(z) = (z?ak)nk?(z),
G1EL,?(z) BFCCAPDXCBB5G3,DP?(ak) negationslash= 0.
D9F0
f′(z) = nk(z?ak)nk?1?(z) + (z?zk)nk?′(z),
APBF 0 < |z?ak| < δ CB,CZ
f′(z)
f(z) =
nk
z?ak +
′(z)
(z)
CSD9BF 0 < |z?ak| < δ CB,?(z) B5G3,FRAD?′(z) B5
G3,DP?(z) negationslash= 0,EKEX
′(z)
(z) BFCCAPDXCBB5G3.
103/107
DD AN f(z) BF C CBCZE8DR nk BRARAYBM ak,BSD3BF ak
ARAPDX |z?zk| < δ CB,CZ
f(z) = (z?ak)nk?(z),
G1EL,?(z) BFCCAPDXCBB5G3,DP?(ak) negationslash= 0.
D9F0
f′(z) = nk(z?ak)nk?1?(z) + (z?zk)nk?′(z),
APBF 0 < |z?ak| < δ CB,CZ
f′(z)
f(z) =
nk
z?ak +
′(z)
(z)
CSD9BF 0 < |z?ak| < δ CB,?(z) B5G3,FRAD?′(z) B5
G3,DP?(z) negationslash= 0,EKEX
′(z)
(z) BFCCAPDXCBB5G3.
103/107
DD AN f(z) BF C CBCZE8DR nk BRARAYBM ak,BSD3BF ak
ARAPDX |z?zk| < δ CB,CZ
f(z) = (z?ak)nk?(z),
G1EL,?(z) BFCCAPDXCBB5G3,DP?(ak) negationslash= 0.
D9F0
f′(z) = nk(z?ak)nk?1?(z) + (z?zk)nk?′(z),
APBF 0 < |z?ak| < δ CB,CZ
f′(z)
f(z) =
nk
z?ak +
′(z)
(z)
CSD9BF 0 < |z?ak| < δ CB,?(z) B5G3,FRAD?′(z) B5
G3,DP?(z) negationslash= 0,EKEX
′(z)
(z) BFCCAPDXCBB5G3.
103/107
DD AN f(z) BF C CBCZE8DR nk BRARAYBM ak,BSD3BF ak
ARAPDX |z?zk| < δ CB,CZ
f(z) = (z?ak)nk?(z),
G1EL,?(z) BFCCAPDXCBB5G3,DP?(ak) negationslash= 0.
D9F0
f′(z) = nk(z?ak)nk?1?(z) + (z?zk)nk?′(z),
APBF 0 < |z?ak| < δ CB,CZ
f′(z)
f(z) =
nk
z?ak +
′(z)
(z)
CSD9BF 0 < |z?ak| < δ CB,?(z) B5G3,FRAD?′(z) B5
G3,DP?(z) negationslash= 0,EKEX
′(z)
(z) BFCCAPDXCBB5G3.
103/107
DD AN f(z) BF C CBCZE8DR nk BRARAYBM ak,BSD3BF ak
ARAPDX |z?zk| < δ CB,CZ
f(z) = (z?ak)nk?(z),
G1EL,?(z) BFCCAPDXCBB5G3,DP?(ak) negationslash= 0.
D9F0
f′(z) = nk(z?ak)nk?1?(z) + (z?zk)nk?′(z),
APBF 0 < |z?ak| < δ CB,CZ
f′(z)
f(z) =
nk
z?ak +
′(z)
(z)
CSD9BF 0 < |z?ak| < δ CB,?(z) B5G3,FRAD?′(z) B5
G3,DP?(z) negationslash= 0,EKEX
′(z)
(z) BFCCAPDXCBB5G3.
103/107
EKEX
Res
bracketleftbiggf′(z)
f(z),ak
bracketrightbigg
= 12pii
contintegraldisplay
Ck
f′(z)
f(z) dz
= 12pii
contintegraldisplay
Ck
nk
z?akdz +
1
2pii
contintegraldisplay
Ck
′(z)
(z) dz
= nk + 0.
G1EL,Ck F0BPEDB5 ak ARBGETDM.
104/107
EKEX
Res
bracketleftbiggf′(z)
f(z),ak
bracketrightbigg
= 12pii
contintegraldisplay
Ck
f′(z)
f(z) dz
= 12pii
contintegraldisplay
Ck
nk
z?akdz +
1
2pii
contintegraldisplay
Ck
′(z)
(z) dz
= nk + 0.
G1EL,Ck F0BPEDB5 ak ARBGETDM.
104/107
EKEX
Res
bracketleftbiggf′(z)
f(z),ak
bracketrightbigg
= 12pii
contintegraldisplay
Ck
f′(z)
f(z) dz
= 12pii
contintegraldisplay
Ck
nk
z?akdz +
1
2pii
contintegraldisplay
Ck
′(z)
(z) dz
= nk + 0.
G1EL,Ck F0BPEDB5 ak ARBGETDM.
104/107
FVD2,CGC2 f(z) BF C CBCZE8DR pk BRARBGBM bk,BSD3
DWEXDDEY:
Res
bracketleftbiggf′(z)
f(z),bk
bracketrightbigg
=?pk.
D2H2EKGB,CGC2 f(z) BF C CBCZ l DRBRGIEPC7
n1,n2,···,nl ARAYBM a1,a2,···,al G2 m DRBRGIEPC7
p1,p2,···,pm ARBGBM b1,b2,···,bm,BSD3DUGAEXH2EK
GBG2BFGIF9D2,AQ
1
2pii
contintegraldisplay
C
f′(z)
f(z) dz =
lsummationdisplay
k=1
Res
bracketleftbiggf′(z)
f(z),ak
bracketrightbigg
+
msummationdisplay
k=1
Res
bracketleftbiggf′(z)
f(z),bk
bracketrightbigg
= (n1 + n2 +···+ nl)? (p1 + p2 +···+ pm)
= N?P.
105/107
FVD2,CGC2 f(z) BF C CBCZE8DR pk BRARBGBM bk,BSD3
DWEXDDEY:
Res
bracketleftbiggf′(z)
f(z),bk
bracketrightbigg
=?pk.
D2H2EKGB,CGC2 f(z) BF C CBCZ l DRBRGIEPC7
n1,n2,···,nl ARAYBM a1,a2,···,al G2 m DRBRGIEPC7
p1,p2,···,pm ARBGBM b1,b2,···,bm,BSD3DUGAEXH2EK
GBG2BFGIF9D2,AQ
1
2pii
contintegraldisplay
C
f′(z)
f(z) dz =
lsummationdisplay
k=1
Res
bracketleftbiggf′(z)
f(z),ak
bracketrightbigg
+
msummationdisplay
k=1
Res
bracketleftbiggf′(z)
f(z),bk
bracketrightbigg
= (n1 + n2 +···+ nl)? (p1 + p2 +···+ pm)
= N?P.
105/107
FVD2,CGC2 f(z) BF C CBCZE8DR pk BRARBGBM bk,BSD3
DWEXDDEY:
Res
bracketleftbiggf′(z)
f(z),bk
bracketrightbigg
=?pk.
D2H2EKGB,CGC2 f(z) BF C CBCZ l DRBRGIEPC7
n1,n2,···,nl ARAYBM a1,a2,···,al G2 m DRBRGIEPC7
p1,p2,···,pm ARBGBM b1,b2,···,bm,BSD3DUGAEXH2EK
GBG2BFGIF9D2,AQ
1
2pii
contintegraldisplay
C
f′(z)
f(z) dz =
lsummationdisplay
k=1
Res
bracketleftbiggf′(z)
f(z),ak
bracketrightbigg
+
msummationdisplay
k=1
Res
bracketleftbiggf′(z)
f(z),bk
bracketrightbigg
= (n1 + n2 +···+ nl)? (p1 + p2 +···+ pm)
= N?P.
105/107
DI 1 C7DYFKEP
contintegraldisplay
|z|=3
z
z2? 1dz;
B5AN f(z) = z2?1,C3 f(z) BF |z| = 3 CBCZH0DRE8BR
AYBM ±1,FFBGBM,BP N = 2,P = 0.
EKEX
contintegraldisplay
|z|=3
z
z2? 1dz =
1
2
contintegraldisplay
|z|=3
2z
z2? 1dz
= 12
contintegraldisplay
|z|=3
f′(z)
f(z) dz
= 12 · 2pii(N?P) = 2pii.
106/107
DI 1 C7DYFKEP
contintegraldisplay
|z|=3
z
z2? 1dz;
B5AN f(z) = z2?1,C3 f(z) BF |z| = 3 CBCZH0DRE8BR
AYBM ±1,FFBGBM,BP N = 2,P = 0.
EKEX
contintegraldisplay
|z|=3
z
z2? 1dz =
1
2
contintegraldisplay
|z|=3
2z
z2? 1dz
= 12
contintegraldisplay
|z|=3
f′(z)
f(z) dz
= 12 · 2pii(N?P) = 2pii.
106/107
DI 1 C7DYFKEP
contintegraldisplay
|z|=3
z
z2? 1dz;
B5AN f(z) = z2?1,C3 f(z) BF |z| = 3 CBCZH0DRE8BR
AYBM ±1,FFBGBM,BP N = 2,P = 0.
EKEX
contintegraldisplay
|z|=3
z
z2? 1dz =
1
2
contintegraldisplay
|z|=3
2z
z2? 1dz
= 12
contintegraldisplay
|z|=3
f′(z)
f(z) dz
= 12 · 2pii(N?P) = 2pii.
106/107
DI 1 C7DYFKEP
contintegraldisplay
|z|=3
z
z2? 1dz;
B5AN f(z) = z2?1,C3 f(z) BF |z| = 3 CBCZH0DRE8BR
AYBM ±1,FFBGBM,BP N = 2,P = 0.
EKEX
contintegraldisplay
|z|=3
z
z2? 1dz =
1
2
contintegraldisplay
|z|=3
2z
z2? 1dz
= 12
contintegraldisplay
|z|=3
f′(z)
f(z) dz
= 12 · 2pii(N?P) = 2pii.
106/107
DI 1 C7DYFKEP
contintegraldisplay
|z|=3
z
z2? 1dz;
B5AN f(z) = z2?1,C3 f(z) BF |z| = 3 CBCZH0DRE8BR
AYBM ±1,FFBGBM,BP N = 2,P = 0.
EKEX
contintegraldisplay
|z|=3
z
z2? 1dz =
1
2
contintegraldisplay
|z|=3
2z
z2? 1dz
= 12
contintegraldisplay
|z|=3
f′(z)
f(z) dz
= 12 · 2pii(N?P) = 2pii.
106/107
DI 2 C7DYFKEP
contintegraldisplay
|z|=3
tanzdz;
B5AN f(z) = cosz,C3 f(z) BF |z| = 3 CBCZH0DRE8BRAY
BM ±pi2,FFBGBM,BP N = 2,P = 0.
EKEX
contintegraldisplay
|z|=3
tanzdz =?
contintegraldisplay
|z|=3
sinz
cosz dz
=?
contintegraldisplay
|z|=3
f′(z)
f(z) dz
=?2pii(N?P) =?4pii.
107/107
DI 2 C7DYFKEP
contintegraldisplay
|z|=3
tanzdz;
B5AN f(z) = cosz,C3 f(z) BF |z| = 3 CBCZH0DRE8BRAY
BM ±pi2,FFBGBM,BP N = 2,P = 0.
EKEX
contintegraldisplay
|z|=3
tanzdz =?
contintegraldisplay
|z|=3
sinz
cosz dz
=?
contintegraldisplay
|z|=3
f′(z)
f(z) dz
=?2pii(N?P) =?4pii.
107/107
DI 2 C7DYFKEP
contintegraldisplay
|z|=3
tanzdz;
B5AN f(z) = cosz,C3 f(z) BF |z| = 3 CBCZH0DRE8BRAY
BM ±pi2,FFBGBM,BP N = 2,P = 0.
EKEX
contintegraldisplay
|z|=3
tanzdz =?
contintegraldisplay
|z|=3
sinz
cosz dz
=?
contintegraldisplay
|z|=3
f′(z)
f(z) dz
=?2pii(N?P) =?4pii.
107/107
DI 2 C7DYFKEP
contintegraldisplay
|z|=3
tanzdz;
B5AN f(z) = cosz,C3 f(z) BF |z| = 3 CBCZH0DRE8BRAY
BM ±pi2,FFBGBM,BP N = 2,P = 0.
EKEX
contintegraldisplay
|z|=3
tanzdz =?
contintegraldisplay
|z|=3
sinz
cosz dz
=?
contintegraldisplay
|z|=3
f′(z)
f(z) dz
=?2pii(N?P) =?4pii.
107/107
DI 2 C7DYFKEP
contintegraldisplay
|z|=3
tanzdz;
B5AN f(z) = cosz,C3 f(z) BF |z| = 3 CBCZH0DRE8BRAY
BM ±pi2,FFBGBM,BP N = 2,P = 0.
EKEX
contintegraldisplay
|z|=3
tanzdz =?
contintegraldisplay
|z|=3
sinz
cosz dz
=?
contintegraldisplay
|z|=3
f′(z)
f(z) dz
=?2pii(N?P) =?4pii.
107/107

课件简介

课件名称：	北京邮电大学：复变函数（2006年）
课件分类：	数学
课件类型：	电子教案
文件大小：	8.31MB
下载次数：	55
评论次数：	10
用户评分：	8.6