北京邮电大学：复变函数（2006年）：4HA

BDD3G1 BRGI
November 7,2006
1/54
FODRCBC6
1,D2B5BXGIE1BRGIG2BXBVF8GIE1BRGIARFGATCECVG2BHEF.
2,D2B5E8BRGIG2GAC3BRGIARB0DYG2GPFTBH.
3,G5F6GQB5G3F8GIFSD1EKAICEBRGIG2GAC3BRGIARE0DF.
2/54
FODRCBC6
1,D2B5BXGIE1BRGIG2BXBVF8GIE1BRGIARFGATCECVG2BHEF.
2,D2B5E8BRGIG2GAC3BRGIARB0DYG2GPFTBH.
3,G5F6GQB5G3F8GIFSD1EKAICEBRGIG2GAC3BRGIARE0DF.
2/54
FODRCBC6
1,D2B5BXGIE1BRGIG2BXBVF8GIE1BRGIARFGATCECVG2BHEF.
2,D2B5E8BRGIG2GAC3BRGIARB0DYG2GPFTBH.
3,G5F6GQB5G3F8GIFSD1EKAICEBRGIG2GAC3BRGIARE0DF.
2/54
FODRCBC6
1,D2B5BXGIE1BRGIG2BXBVF8GIE1BRGIARFGATCECVG2BHEF.
2,D2B5E8BRGIG2GAC3BRGIARB0DYG2GPFTBH.
3,G5F6GQB5G3F8GIFSD1EKAICEBRGIG2GAC3BRGIARE0DF.
2/54
BDE8AX BXGIE1BRGI
1,BXGIAFARBGDL
AN{αn}(n = 1,2,···)?E8BXGIAF,G1EL αn = an + ibn,D5
AN α = a + ib?E8B4F9ARBXGI,CGC2?ε > 0,?N(ε) > 0,EMAQA8 n > N
EE,CZ
|αn?α| < ε,
BSD3EH α F0BXGIAF{αn} A8 n → ∞ EEARBGDL,C8DU
limn→∞αn = α.
3/54
F9D2E8 BXGIAF{αn}(n = 1,2,···) FEGPD9 α ARFBDRCMDRF0
limn→∞an = a,limn→∞bn = b.
DD (FT)
4/54
DI 1 A7C7GIAF αn =
parenleftbigg1 + i
2
parenrightbiggn
ARFEGPBH.
B5 FGCZEPB5 αn = an + bni,B8A0EPC7DJA9 an G2 bn ARBGDL,BEB4
F9 αn ARGPFTBH.
FR?
parenleftbigg1 + i
2
parenrightbiggn
=
bracketleftBigg√
2
2
parenleftBig
cospi4 + isinpi4
parenrightBigbracketrightBiggn
= 12n/2
parenleftBig
cosnpi4 + isinnpi4
parenrightBig
.
FBDP
limn→∞ 12n/2cosnpi4 = 0,limn→∞ 12n/2sinnpi4 = 0.
EKEX αn =
parenleftbigg1 + i
2
parenrightbiggn
FEGPAKAY.
5/54
DI 1 A7C7GIAF αn =
parenleftbigg1 + i
2
parenrightbiggn
ARFEGPBH.
B5 FGCZEPB5 αn = an + bni,B8A0EPC7DJA9 an G2 bn ARBGDL,BEB4
F9 αn ARGPFTBH.
FR?
parenleftbigg1 + i
2
parenrightbiggn
=
bracketleftBigg√
2
2
parenleftBig
cospi4 + isinpi4
parenrightBigbracketrightBiggn
= 12n/2
parenleftBig
cosnpi4 + isinnpi4
parenrightBig
.
FBDP
limn→∞ 12n/2cosnpi4 = 0,limn→∞ 12n/2sinnpi4 = 0.
EKEX αn =
parenleftbigg1 + i
2
parenrightbiggn
FEGPAKAY.
5/54
DI 1 A7C7GIAF αn =
parenleftbigg1 + i
2
parenrightbiggn
ARFEGPBH.
B5 FGCZEPB5 αn = an + bni,B8A0EPC7DJA9 an G2 bn ARBGDL,BEB4
F9 αn ARGPFTBH.
FR?
parenleftbigg1 + i
2
parenrightbiggn
=
bracketleftBigg√
2
2
parenleftBig
cospi4 + isinpi4
parenrightBigbracketrightBiggn
= 12n/2
parenleftBig
cosnpi4 + isinnpi4
parenrightBig
.
FBDP
limn→∞ 12n/2cosnpi4 = 0,limn→∞ 12n/2sinnpi4 = 0.
EKEX αn =
parenleftbigg1 + i
2
parenrightbiggn
FEGPAKAY.
5/54
DI 1 A7C7GIAF αn =
parenleftbigg1 + i
2
parenrightbiggn
ARFEGPBH.
B5 FGCZEPB5 αn = an + bni,B8A0EPC7DJA9 an G2 bn ARBGDL,BEB4
F9 αn ARGPFTBH.
FR?
parenleftbigg1 + i
2
parenrightbiggn
=
bracketleftBigg√
2
2
parenleftBig
cospi4 + isinpi4
parenrightBigbracketrightBiggn
= 12n/2
parenleftBig
cosnpi4 + isinnpi4
parenrightBig
.
FBDP
limn→∞ 12n/2cosnpi4 = 0,limn→∞ 12n/2sinnpi4 = 0.
EKEX αn =
parenleftbigg1 + i
2
parenrightbiggn
FEGPAKAY.
5/54
DI 1 A7C7GIAF αn =
parenleftbigg1 + i
2
parenrightbiggn
ARFEGPBH.
B5 FGCZEPB5 αn = an + bni,B8A0EPC7DJA9 an G2 bn ARBGDL,BEB4
F9 αn ARGPFTBH.
FR?
parenleftbigg1 + i
2
parenrightbiggn
=
bracketleftBigg√
2
2
parenleftBig
cospi4 + isinpi4
parenrightBigbracketrightBiggn
= 12n/2
parenleftBig
cosnpi4 + isinnpi4
parenrightBig
.
FBDP
limn→∞ 12n/2cosnpi4 = 0,limn→∞ 12n/2sinnpi4 = 0.
EKEX αn =
parenleftbigg1 + i
2
parenrightbiggn
FEGPAKAY.
5/54
DI 2 A7C7GIAF αn = (?1)n + in + 1 ARFEGPBH.
B5 FR?
limn→∞(?1)n,
G0DJBF,AD
limn→∞ 1n + 1 = 0.
EKEX αn = (?1)n + in + 1 D9FT.
6/54
DI 2 A7C7GIAF αn = (?1)n + in + 1 ARFEGPBH.
B5 FR?
limn→∞(?1)n,
G0DJBF,AD
limn→∞ 1n + 1 = 0.
EKEX αn = (?1)n + in + 1 D9FT.
6/54
DI 2 A7C7GIAF αn = (?1)n + in + 1 ARFEGPBH.
B5 FR?
limn→∞(?1)n,
G0DJBF,AD
limn→∞ 1n + 1 = 0.
EKEX αn = (?1)n + in + 1 D9FT.
6/54
DI 2 A7C7GIAF αn = (?1)n + in + 1 ARFEGPBH.
B5 FR?
limn→∞(?1)n,
G0DJBF,AD
limn→∞ 1n + 1 = 0.
EKEX αn = (?1)n + in + 1 D9FT.
6/54
DI 3 DDEY:
limn→∞αn =


0,|α| > 1,
∞,|α| < 1,
1,α = 1,
G0DJBF,|α| = 1,α negationslash= 1.
B5 AN α = r(cosθ + isinθ),D9F0
(1) A8|α| < 1 EE,CZ r < 1,C3
0 lessorequalslant|αn| = |rn(cosnθ + sinnθ)| = rn
FR? limn→∞rn = 0,EKEX
limn→∞αn = 0.
(2) A8|α| > 1 EE,CZ
vextendsinglevextendsingle
vextendsinglevextendsingle1
α
vextendsinglevextendsingle
vextendsinglevextendsingle< 1,
C3DUGA (1) ARB4G0,CZ
limn→∞
parenleftbigg1
α
parenrightbiggn
= 0
EKEX limn→∞αn = ∞.
7/54
DI 3 DDEY:
limn→∞αn =


0,|α| > 1,
∞,|α| < 1,
1,α = 1,
G0DJBF,|α| = 1,α negationslash= 1.
B5 AN α = r(cosθ + isinθ),D9F0
(1) A8|α| < 1 EE,CZ r < 1,C3
0 lessorequalslant|αn| = |rn(cosnθ + sinnθ)| = rn
FR? limn→∞rn = 0,EKEX
limn→∞αn = 0.
(2) A8|α| > 1 EE,CZ
vextendsinglevextendsingle
vextendsinglevextendsingle1
α
vextendsinglevextendsingle
vextendsinglevextendsingle< 1,
C3DUGA (1) ARB4G0,CZ
limn→∞
parenleftbigg1
α
parenrightbiggn
= 0
EKEX limn→∞αn = ∞.
7/54
DI 3 DDEY:
limn→∞αn =


0,|α| > 1,
∞,|α| < 1,
1,α = 1,
G0DJBF,|α| = 1,α negationslash= 1.
B5 AN α = r(cosθ + isinθ),D9F0
(1) A8|α| < 1 EE,CZ r < 1,C3
0 lessorequalslant|αn| = |rn(cosnθ + sinnθ)| = rn
FR? limn→∞rn = 0,EKEX
limn→∞αn = 0.
(2) A8|α| > 1 EE,CZ
vextendsinglevextendsingle
vextendsinglevextendsingle1
α
vextendsinglevextendsingle
vextendsinglevextendsingle< 1,
C3DUGA (1) ARB4G0,CZ
limn→∞
parenleftbigg1
α
parenrightbiggn
= 0
EKEX limn→∞αn = ∞.
7/54
DI 3 DDEY:
limn→∞αn =


0,|α| > 1,
∞,|α| < 1,
1,α = 1,
G0DJBF,|α| = 1,α negationslash= 1.
B5 AN α = r(cosθ + isinθ),D9F0
(1) A8|α| < 1 EE,CZ r < 1,C3
0 lessorequalslant|αn| = |rn(cosnθ + sinnθ)| = rn
FR? limn→∞rn = 0,EKEX
limn→∞αn = 0.
(2) A8|α| > 1 EE,CZ
vextendsinglevextendsingle
vextendsinglevextendsingle1
α
vextendsinglevextendsingle
vextendsinglevextendsingle< 1,
C3DUGA (1) ARB4G0,CZ
limn→∞
parenleftbigg1
α
parenrightbiggn
= 0
EKEX limn→∞αn = ∞.
7/54
DI 3 DDEY:
limn→∞αn =


0,|α| > 1,
∞,|α| < 1,
1,α = 1,
G0DJBF,|α| = 1,α negationslash= 1.
B5 AN α = r(cosθ + isinθ),D9F0
(1) A8|α| < 1 EE,CZ r < 1,C3
0 lessorequalslant|αn| = |rn(cosnθ + sinnθ)| = rn
FR? limn→∞rn = 0,EKEX
limn→∞αn = 0.
(2) A8|α| > 1 EE,CZ
vextendsinglevextendsingle
vextendsinglevextendsingle1
α
vextendsinglevextendsingle
vextendsinglevextendsingle< 1,
C3DUGA (1) ARB4G0,CZ
limn→∞
parenleftbigg1
α
parenrightbiggn
= 0
EKEX limn→∞αn = ∞.
7/54
DI 3 DDEY:
limn→∞αn =


0,|α| > 1,
∞,|α| < 1,
1,α = 1,
G0DJBF,|α| = 1,α negationslash= 1.
B5 AN α = r(cosθ + isinθ),D9F0
(1) A8|α| < 1 EE,CZ r < 1,C3
0 lessorequalslant|αn| = |rn(cosnθ + sinnθ)| = rn
FR? limn→∞rn = 0,EKEX
limn→∞αn = 0.
(2) A8|α| > 1 EE,CZ
vextendsinglevextendsingle
vextendsinglevextendsingle1
α
vextendsinglevextendsingle
vextendsinglevextendsingle< 1,
C3DUGA (1) ARB4G0,CZ
limn→∞
parenleftbigg1
α
parenrightbiggn
= 0
EKEX limn→∞αn = ∞.
7/54
(3) A8 α = 1 EE,CZ αn = 1,C3
limn→∞αn = 1.
(4) A8|α| = 1,α negationslash= 1 EE,CZ α = cosθ + isinθ,EKEX
αn = cosnθ + isinnθ
FR? limn→∞cosnθ G2 limn→∞sinnθ H2G0DJBF,EKEX
limn→∞αn
G0DJBF.
8/54
(3) A8 α = 1 EE,CZ αn = 1,C3
limn→∞αn = 1.
(4) A8|α| = 1,α negationslash= 1 EE,CZ α = cosθ + isinθ,EKEX
αn = cosnθ + isinnθ
FR? limn→∞cosnθ G2 limn→∞sinnθ H2G0DJBF,EKEX
limn→∞αn
G0DJBF.
8/54
(3) A8 α = 1 EE,CZ αn = 1,C3
limn→∞αn = 1.
(4) A8|α| = 1,α negationslash= 1 EE,CZ α = cosθ + isinθ,EKEX
αn = cosnθ + isinnθ
FR? limn→∞cosnθ G2 limn→∞sinnθ H2G0DJBF,EKEX
limn→∞αn
G0DJBF.
8/54
(3) A8 α = 1 EE,CZ αn = 1,C3
limn→∞αn = 1.
(4) A8|α| = 1,α negationslash= 1 EE,CZ α = cosθ + isinθ,EKEX
αn = cosnθ + isinnθ
FR? limn→∞cosnθ G2 limn→∞sinnθ H2G0DJBF,EKEX
limn→∞αn
G0DJBF.
8/54
(3) A8 α = 1 EE,CZ αn = 1,C3
limn→∞αn = 1.
(4) A8|α| = 1,α negationslash= 1 EE,CZ α = cosθ + isinθ,EKEX
αn = cosnθ + isinnθ
FR? limn→∞cosnθ G2 limn→∞sinnθ H2G0DJBF,EKEX
limn→∞αn
G0DJBF.
8/54
2,BRGIARCECV
AN{αn} = {an + ibn}(n = 1,2,···)?E8BXGIAF,C4DSEQ
∞summationdisplay
n=1
αn = α1 + α2 +··· + αn +···
EH?FFEHBRGI,G1CREH n E1ARG2
sn = α1 + α2 +··· + αn
EH?BRGIARG4EPG2.
9/54
2,BRGIARCECV
AN{αn} = {an + ibn}(n = 1,2,···)?E8BXGIAF,C4DSEQ
∞summationdisplay
n=1
αn = α1 + α2 +··· + αn +···
EH?FFEHBRGI,G1CREH n E1ARG2
sn = α1 + α2 +··· + αn
EH?BRGIARG4EPG2.
9/54
BRGIFEGPDQD9FT
F9FECGC2G4EPG2GIAF{sn} FEGP,BSD3BRGI
∞summationdisplay
n=1
αn EH?FEGP,FBDPBG
DL limn→∞sn = s EH?BRGIARG2.
CGC2BRGIG0FEGP,BSD3BRGI
∞summationdisplay
n=1
αn EH?D9FT.
10/54
BRGIFEGPDQD9FT
F9FECGC2G4EPG2GIAF{sn} FEGP,BSD3BRGI
∞summationdisplay
n=1
αn EH?FEGP,FBDPBG
DL limn→∞sn = s EH?BRGIARG2.
CGC2BRGIG0FEGP,BSD3BRGI
∞summationdisplay
n=1
αn EH?D9FT.
10/54
F9D2AJ BRGI
∞summationdisplay
n=1
αn FEGPARFBDRCMDRF0BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGP.
DDFR
sn = α1 + α2 +··· + αn
= (a1 + a2 +··· + an) + i(b1 + b2 +··· + bn)
= σn + iτn,
CSF9D2E8,{sn} ARBGDLDJBFARFBDRCMDRF0{σn} G2{τn} ARBGDLDJBF,BP
BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGP.
11/54
F9D2AJ BRGI
∞summationdisplay
n=1
αn FEGPARFBDRCMDRF0BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGP.
DDFR
sn = α1 + α2 +··· + αn
= (a1 + a2 +··· + an) + i(b1 + b2 +··· + bn)
= σn + iτn,
CSF9D2E8,{sn} ARBGDLDJBFARFBDRCMDRF0{σn} G2{τn} ARBGDLDJBF,BP
BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGP.
11/54
F9D2AJ BRGI
∞summationdisplay
n=1
αn FEGPARFBDRCMDRF0BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGP.
DDFR
sn = α1 + α2 +··· + αn
= (a1 + a2 +··· + an) + i(b1 + b2 +··· + bn)
= σn + iτn,
CSF9D2E8,{sn} ARBGDLDJBFARFBDRCMDRF0{σn} G2{τn} ARBGDLDJBF,BP
BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGP.
11/54
F9D2AJ BRGI
∞summationdisplay
n=1
αn FEGPARFBDRCMDRF0BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGP.
DDFR
sn = α1 + α2 +··· + αn
= (a1 + a2 +··· + an) + i(b1 + b2 +··· + bn)
= σn + iτn,
CSF9D2E8,{sn} ARBGDLDJBFARFBDRCMDRF0{σn} G2{τn} ARBGDLDJBF,BP
BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGP.
11/54
F9D2AJ BRGI
∞summationdisplay
n=1
αn FEGPARFBDRCMDRF0BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGP.
DDFR
sn = α1 + α2 +··· + αn
= (a1 + a2 +··· + an) + i(b1 + b2 +··· + bn)
= σn + iτn,
CSF9D2E8,{sn} ARBGDLDJBFARFBDRCMDRF0{σn} G2{τn} ARBGDLDJBF,BP
BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGP.
11/54
F9D2AJ BRGI
∞summationdisplay
n=1
αn FEGPARFBDRCMDRF0BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGP.
DDFR
sn = α1 + α2 +··· + αn
= (a1 + a2 +··· + an) + i(b1 + b2 +··· + bn)
= σn + iτn,
CSF9D2E8,{sn} ARBGDLDJBFARFBDRCMDRF0{σn} G2{τn} ARBGDLDJBF,BP
BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGP.
11/54
BXGIE1BRGI
∞summationdisplay
n=1
αn FEGPARBJDRCMDR,limn→∞αn = 0.
DDCSEIGIE1BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGPARBJDRCMDRF0
limn→∞an = 0,limn→∞bn = 0,
DWAQ limn→∞αn = 0.
12/54
BXGIE1BRGI
∞summationdisplay
n=1
αn FEGPARBJDRCMDR,limn→∞αn = 0.
DDCSEIGIE1BRGI
∞summationdisplay
n=1
an G2
∞summationdisplay
n=1
bn FEGPARBJDRCMDRF0
limn→∞an = 0,limn→∞bn = 0,
DWAQ limn→∞αn = 0.
12/54
F9D2D2CGC2
∞summationdisplay
n=1
|αn|FEGP,BSD3
∞summationdisplay
n=1
αn?FEGP,DPG0AVEQ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
∞summationdisplay
n=1
αn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
∞summationdisplay
n=1
|αn|
EKG9.
DDCSD9
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
radicalbig
a2n + b2n FEGP,FBDP
|an|lessorequalslant
radicalbig
a2n + b2n,|bn|lessorequalslant
radicalbig
a2n + b2n.
DUGAEIGIE1BRGIARB3AMC7C3,DWDJBRGI
∞summationdisplay
n=1
|an|BL
∞summationdisplay
n=1
|bn| FTFEGP,FR
AD
∞summationdisplay
n=1
an BL
∞summationdisplay
n=1
bn FTH1GHFEGP,D9F0
∞summationdisplay
n=1
αn FEGP.
13/54
F9D2D2CGC2
∞summationdisplay
n=1
|αn|FEGP,BSD3
∞summationdisplay
n=1
αn?FEGP,DPG0AVEQ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
∞summationdisplay
n=1
αn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
∞summationdisplay
n=1
|αn|
EKG9.
DDCSD9
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
radicalbig
a2n + b2n FEGP,FBDP
|an|lessorequalslant
radicalbig
a2n + b2n,|bn|lessorequalslant
radicalbig
a2n + b2n.
DUGAEIGIE1BRGIARB3AMC7C3,DWDJBRGI
∞summationdisplay
n=1
|an|BL
∞summationdisplay
n=1
|bn| FTFEGP,FR
AD
∞summationdisplay
n=1
an BL
∞summationdisplay
n=1
bn FTH1GHFEGP,D9F0
∞summationdisplay
n=1
αn FEGP.
13/54
F9D2D2CGC2
∞summationdisplay
n=1
|αn|FEGP,BSD3
∞summationdisplay
n=1
αn?FEGP,DPG0AVEQ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
∞summationdisplay
n=1
αn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
∞summationdisplay
n=1
|αn|
EKG9.
DDCSD9
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
radicalbig
a2n + b2n FEGP,FBDP
|an|lessorequalslant
radicalbig
a2n + b2n,|bn|lessorequalslant
radicalbig
a2n + b2n.
DUGAEIGIE1BRGIARB3AMC7C3,DWDJBRGI
∞summationdisplay
n=1
|an|BL
∞summationdisplay
n=1
|bn| FTFEGP,FR
AD
∞summationdisplay
n=1
an BL
∞summationdisplay
n=1
bn FTH1GHFEGP,D9F0
∞summationdisplay
n=1
αn FEGP.
13/54
F9D2D2CGC2
∞summationdisplay
n=1
|αn|FEGP,BSD3
∞summationdisplay
n=1
αn?FEGP,DPG0AVEQ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
∞summationdisplay
n=1
αn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
∞summationdisplay
n=1
|αn|
EKG9.
DDCSD9
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
radicalbig
a2n + b2n FEGP,FBDP
|an|lessorequalslant
radicalbig
a2n + b2n,|bn|lessorequalslant
radicalbig
a2n + b2n.
DUGAEIGIE1BRGIARB3AMC7C3,DWDJBRGI
∞summationdisplay
n=1
|an|BL
∞summationdisplay
n=1
|bn| FTFEGP,FR
AD
∞summationdisplay
n=1
an BL
∞summationdisplay
n=1
bn FTH1GHFEGP,D9F0
∞summationdisplay
n=1
αn FEGP.
13/54
F9D2D2CGC2
∞summationdisplay
n=1
|αn|FEGP,BSD3
∞summationdisplay
n=1
αn?FEGP,DPG0AVEQ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
∞summationdisplay
n=1
αn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
∞summationdisplay
n=1
|αn|
EKG9.
DDCSD9
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
radicalbig
a2n + b2n FEGP,FBDP
|an|lessorequalslant
radicalbig
a2n + b2n,|bn|lessorequalslant
radicalbig
a2n + b2n.
DUGAEIGIE1BRGIARB3AMC7C3,DWDJBRGI
∞summationdisplay
n=1
|an|BL
∞summationdisplay
n=1
|bn| FTFEGP,FR
AD
∞summationdisplay
n=1
an BL
∞summationdisplay
n=1
bn FTH1GHFEGP,D9F0
∞summationdisplay
n=1
αn FEGP.
13/54
CSD9BRGI
∞summationdisplay
n=1
αn DQ
∞summationdisplay
n=1
|αn| ARG4EPG2GVDAG0AVEQ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
αk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
nsummationdisplay
k=1
|αk|
H0BQAIBGDLDWAQ,limn→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
αk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant limn→∞
nsummationdisplay
k=1
|αk|,BP
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
∞summationdisplay
n=1
αn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
∞summationdisplay
n=1
|αn|
14/54
CSD9BRGI
∞summationdisplay
n=1
αn DQ
∞summationdisplay
n=1
|αn| ARG4EPG2GVDAG0AVEQ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
αk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
nsummationdisplay
k=1
|αk|
H0BQAIBGDLDWAQ,limn→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
αk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant limn→∞
nsummationdisplay
k=1
|αk|,BP
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
∞summationdisplay
n=1
αn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
∞summationdisplay
n=1
|αn|
14/54
CSD9BRGI
∞summationdisplay
n=1
αn DQ
∞summationdisplay
n=1
|αn| ARG4EPG2GVDAG0AVEQ
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
αk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
nsummationdisplay
k=1
|αk|
H0BQAIBGDLDWAQ,limn→∞
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
nsummationdisplay
k=1
αk
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant limn→∞
nsummationdisplay
k=1
|αk|,BP
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
∞summationdisplay
n=1
αn
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglelessorequalslant
∞summationdisplay
n=1
|αn|
14/54
H1GHFEGPDQCMDRFEGP
F9FE CGC2
∞summationdisplay
n=1
|αn| FEGP,BSD3EHBRGI
∞summationdisplay
n=1
αn?H1GHFEGP.
EAH1GHFEGPARFEGPBRGIEH?CMDRFEGPBRGI.
15/54
H1GHFEGPDQCMDRFEGP
F9FE CGC2
∞summationdisplay
n=1
|αn| FEGP,BSD3EHBRGI
∞summationdisplay
n=1
αn?H1GHFEGP.
EAH1GHFEGPARFEGPBRGIEH?CMDRFEGPBRGI.
15/54
H1GHFEGPDQCMDRFEGP
F9FE CGC2
∞summationdisplay
n=1
|αn| FEGP,BSD3EHBRGI
∞summationdisplay
n=1
αn?H1GHFEGP.
EAH1GHFEGPARFEGPBRGIEH?CMDRFEGPBRGI.
15/54
F9D2D3 BRGI
∞summationdisplay
n=1
αn H1GHFEGPARFBDRCMDRF0BRGI
∞summationdisplay
n=1
an DQ
∞summationdisplay
n=1
bn H1
GHFEGP.
DD,FBEPBH
FR? |αn| =
radicalbig
a2n + b2n lessorequalslant|an|+|bn|,
EKEX
∞summationdisplay
n=1
|αn|lessorequalslant
∞summationdisplay
n=1
|an|+
∞summationdisplay
n=1
|bn|,
EKEXA8BRGI
∞summationdisplay
n=1
an DQ
∞summationdisplay
n=1
bn H1GHFEGPEE,BRGI
∞summationdisplay
n=1
αn?H1GHFEGP.
BJDRBH
DUGAF9D2D2ARDDEYC4ENBPAQ.
16/54
F9D2D3 BRGI
∞summationdisplay
n=1
αn H1GHFEGPARFBDRCMDRF0BRGI
∞summationdisplay
n=1
an DQ
∞summationdisplay
n=1
bn H1
GHFEGP.
DD,FBEPBH
FR? |αn| =
radicalbig
a2n + b2n lessorequalslant|an|+|bn|,
EKEX
∞summationdisplay
n=1
|αn|lessorequalslant
∞summationdisplay
n=1
|an|+
∞summationdisplay
n=1
|bn|,
EKEXA8BRGI
∞summationdisplay
n=1
an DQ
∞summationdisplay
n=1
bn H1GHFEGPEE,BRGI
∞summationdisplay
n=1
αn?H1GHFEGP.
BJDRBH
DUGAF9D2D2ARDDEYC4ENBPAQ.
16/54
F9D2D3 BRGI
∞summationdisplay
n=1
αn H1GHFEGPARFBDRCMDRF0BRGI
∞summationdisplay
n=1
an DQ
∞summationdisplay
n=1
bn H1
GHFEGP.
DD,FBEPBH
FR? |αn| =
radicalbig
a2n + b2n lessorequalslant|an|+|bn|,
EKEX
∞summationdisplay
n=1
|αn|lessorequalslant
∞summationdisplay
n=1
|an|+
∞summationdisplay
n=1
|bn|,
EKEXA8BRGI
∞summationdisplay
n=1
an DQ
∞summationdisplay
n=1
bn H1GHFEGPEE,BRGI
∞summationdisplay
n=1
αn?H1GHFEGP.
BJDRBH
DUGAF9D2D2ARDDEYC4ENBPAQ.
16/54
F9D2D3 BRGI
∞summationdisplay
n=1
αn H1GHFEGPARFBDRCMDRF0BRGI
∞summationdisplay
n=1
an DQ
∞summationdisplay
n=1
bn H1
GHFEGP.
DD,FBEPBH
FR? |αn| =
radicalbig
a2n + b2n lessorequalslant|an|+|bn|,
EKEX
∞summationdisplay
n=1
|αn|lessorequalslant
∞summationdisplay
n=1
|an|+
∞summationdisplay
n=1
|bn|,
EKEXA8BRGI
∞summationdisplay
n=1
an DQ
∞summationdisplay
n=1
bn H1GHFEGPEE,BRGI
∞summationdisplay
n=1
αn?H1GHFEGP.
BJDRBH
DUGAF9D2D2ARDDEYC4ENBPAQ.
16/54
F9D2D3 BRGI
∞summationdisplay
n=1
αn H1GHFEGPARFBDRCMDRF0BRGI
∞summationdisplay
n=1
an DQ
∞summationdisplay
n=1
bn H1
GHFEGP.
DD,FBEPBH
FR? |αn| =
radicalbig
a2n + b2n lessorequalslant|an|+|bn|,
EKEX
∞summationdisplay
n=1
|αn|lessorequalslant
∞summationdisplay
n=1
|an|+
∞summationdisplay
n=1
|bn|,
EKEXA8BRGI
∞summationdisplay
n=1
an DQ
∞summationdisplay
n=1
bn H1GHFEGPEE,BRGI
∞summationdisplay
n=1
αn?H1GHFEGP.
BJDRBH
DUGAF9D2D2ARDDEYC4ENBPAQ.
16/54
F9D2D3 BRGI
∞summationdisplay
n=1
αn H1GHFEGPARFBDRCMDRF0BRGI
∞summationdisplay
n=1
an DQ
∞summationdisplay
n=1
bn H1
GHFEGP.
DD,FBEPBH
FR? |αn| =
radicalbig
a2n + b2n lessorequalslant|an|+|bn|,
EKEX
∞summationdisplay
n=1
|αn|lessorequalslant
∞summationdisplay
n=1
|an|+
∞summationdisplay
n=1
|bn|,
EKEXA8BRGI
∞summationdisplay
n=1
an DQ
∞summationdisplay
n=1
bn H1GHFEGPEE,BRGI
∞summationdisplay
n=1
αn?H1GHFEGP.
BJDRBH
DUGAF9D2D2ARDDEYC4ENBPAQ.
16/54
DI 1 BRGI
∞summationdisplay
n=1
(?1)n + i√
n + 1 F0FGFEGP? F0FGH1GHFEGP?
B5FR?
∞summationdisplay
n=1
an =
∞summationdisplay
n=1
(?1)n√
n + 1
CMDRFEGP; ADBRGI
∞summationdisplay
n=1
bn =
∞summationdisplay
n=1
1√
n + 1
D9FT,APDUGAF9D2AJ,DJBRGID9FT.
17/54
DI 1 BRGI
∞summationdisplay
n=1
(?1)n + i√
n + 1 F0FGFEGP? F0FGH1GHFEGP?
B5FR?
∞summationdisplay
n=1
an =
∞summationdisplay
n=1
(?1)n√
n + 1
CMDRFEGP; ADBRGI
∞summationdisplay
n=1
bn =
∞summationdisplay
n=1
1√
n + 1
D9FT,APDUGAF9D2AJ,DJBRGID9FT.
17/54
DI 1 BRGI
∞summationdisplay
n=1
(?1)n + i√
n + 1 F0FGFEGP? F0FGH1GHFEGP?
B5FR?
∞summationdisplay
n=1
an =
∞summationdisplay
n=1
(?1)n√
n + 1
CMDRFEGP; ADBRGI
∞summationdisplay
n=1
bn =
∞summationdisplay
n=1
1√
n + 1
D9FT,APDUGAF9D2AJ,DJBRGID9FT.
17/54
DI 1 BRGI
∞summationdisplay
n=1
(?1)n + i√
n + 1 F0FGFEGP? F0FGH1GHFEGP?
B5FR?
∞summationdisplay
n=1
an =
∞summationdisplay
n=1
(?1)n√
n + 1
CMDRFEGP; ADBRGI
∞summationdisplay
n=1
bn =
∞summationdisplay
n=1
1√
n + 1
D9FT,APDUGAF9D2AJ,DJBRGID9FT.
17/54
DI 2 BRGI
∞summationdisplay
n=1
n
2n(1 + i)
n
F0FGFEGP? F0FGH1GHFEGP?
B5FR?
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsingle n2n(1 + i)n
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=1
n
parenleftbigg 1
√2
parenrightbiggn
FEGP,EKEXA6BRGIH1GHFEGP.
18/54
DI 2 BRGI
∞summationdisplay
n=1
n
2n(1 + i)
n
F0FGFEGP? F0FGH1GHFEGP?
B5FR?
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsingle n2n(1 + i)n
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=1
n
parenleftbigg 1
√2
parenrightbiggn
FEGP,EKEXA6BRGIH1GHFEGP.
18/54
DI 2 BRGI
∞summationdisplay
n=1
n
2n(1 + i)
n
F0FGFEGP? F0FGH1GHFEGP?
B5FR?
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsingle n2n(1 + i)n
vextendsinglevextendsingle
vextendsingle=
∞summationdisplay
n=1
n
parenleftbigg 1
√2
parenrightbiggn
FEGP,EKEXA6BRGIH1GHFEGP.
18/54
DI 3 BRGI
∞summationdisplay
n=1
(3 + 4i)n
n! F0FGFEGP? F0FGH1GHFEGP?
B5FR?
|αn| =
vextendsinglevextendsingle
vextendsinglevextendsingle(3 + 4i)n
n!
vextendsinglevextendsingle
vextendsinglevextendsingle= 5n
n!,
DUGAAKE1BRGIARB3DUA7C7DFDJ
∞summationdisplay
n=1
5n
n! FEGP.
APBRGI
∞summationdisplay
n=1
(3 + 4i)n
n! H1GHFEGP.
19/54
DI 3 BRGI
∞summationdisplay
n=1
(3 + 4i)n
n! F0FGFEGP? F0FGH1GHFEGP?
B5FR?
|αn| =
vextendsinglevextendsingle
vextendsinglevextendsingle(3 + 4i)n
n!
vextendsinglevextendsingle
vextendsinglevextendsingle= 5n
n!,
DUGAAKE1BRGIARB3DUA7C7DFDJ
∞summationdisplay
n=1
5n
n! FEGP.
APBRGI
∞summationdisplay
n=1
(3 + 4i)n
n! H1GHFEGP.
19/54
DI 3 BRGI
∞summationdisplay
n=1
(3 + 4i)n
n! F0FGFEGP? F0FGH1GHFEGP?
B5FR?
|αn| =
vextendsinglevextendsingle
vextendsinglevextendsingle(3 + 4i)n
n!
vextendsinglevextendsingle
vextendsinglevextendsingle= 5n
n!,
DUGAAKE1BRGIARB3DUA7C7DFDJ
∞summationdisplay
n=1
5n
n! FEGP.
APBRGI
∞summationdisplay
n=1
(3 + 4i)n
n! H1GHFEGP.
19/54
DI 3 BRGI
∞summationdisplay
n=1
(3 + 4i)n
n! F0FGFEGP? F0FGH1GHFEGP?
B5FR?
|αn| =
vextendsinglevextendsingle
vextendsinglevextendsingle(3 + 4i)n
n!
vextendsinglevextendsingle
vextendsinglevextendsingle= 5n
n!,
DUGAAKE1BRGIARB3DUA7C7DFDJ
∞summationdisplay
n=1
5n
n! FEGP.
APBRGI
∞summationdisplay
n=1
(3 + 4i)n
n! H1GHFEGP.
19/54
DI 4 BRGI
∞summationdisplay
n=1
in
lnn F0FGFEGP? F0FGH1GHFEGP?
B5FR?
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle in
lnn
vextendsinglevextendsingle
vextendsinglevextendsingle=
∞summationdisplay
n=1
1
lnn
FR?
∞summationdisplay
n=1
1
lnn D9FT,EKEXBRGIG0H1GHFEGP.
A8 n = 2k EE,BRGIBP
∞summationdisplay
n=1
in
lnn =
∞summationdisplay
k=1
(?1)k
ln2k
FEGP; A8 n = 2k + 1 EE,BRGIBP
∞summationdisplay
n=1
in
lnn =
∞summationdisplay
k=1
(?1)ki
ln(2k + 1)
FEGP.
EKEXA6BRGICMDRFEGP.
20/54
DI 4 BRGI
∞summationdisplay
n=1
in
lnn F0FGFEGP? F0FGH1GHFEGP?
B5FR?
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle in
lnn
vextendsinglevextendsingle
vextendsinglevextendsingle=
∞summationdisplay
n=1
1
lnn
FR?
∞summationdisplay
n=1
1
lnn D9FT,EKEXBRGIG0H1GHFEGP.
A8 n = 2k EE,BRGIBP
∞summationdisplay
n=1
in
lnn =
∞summationdisplay
k=1
(?1)k
ln2k
FEGP; A8 n = 2k + 1 EE,BRGIBP
∞summationdisplay
n=1
in
lnn =
∞summationdisplay
k=1
(?1)ki
ln(2k + 1)
FEGP.
EKEXA6BRGICMDRFEGP.
20/54
DI 4 BRGI
∞summationdisplay
n=1
in
lnn F0FGFEGP? F0FGH1GHFEGP?
B5FR?
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle in
lnn
vextendsinglevextendsingle
vextendsinglevextendsingle=
∞summationdisplay
n=1
1
lnn
FR?
∞summationdisplay
n=1
1
lnn D9FT,EKEXBRGIG0H1GHFEGP.
A8 n = 2k EE,BRGIBP
∞summationdisplay
n=1
in
lnn =
∞summationdisplay
k=1
(?1)k
ln2k
FEGP; A8 n = 2k + 1 EE,BRGIBP
∞summationdisplay
n=1
in
lnn =
∞summationdisplay
k=1
(?1)ki
ln(2k + 1)
FEGP.
EKEXA6BRGICMDRFEGP.
20/54
DI 4 BRGI
∞summationdisplay
n=1
in
lnn F0FGFEGP? F0FGH1GHFEGP?
B5FR?
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle in
lnn
vextendsinglevextendsingle
vextendsinglevextendsingle=
∞summationdisplay
n=1
1
lnn
FR?
∞summationdisplay
n=1
1
lnn D9FT,EKEXBRGIG0H1GHFEGP.
A8 n = 2k EE,BRGIBP
∞summationdisplay
n=1
in
lnn =
∞summationdisplay
k=1
(?1)k
ln2k
FEGP; A8 n = 2k + 1 EE,BRGIBP
∞summationdisplay
n=1
in
lnn =
∞summationdisplay
k=1
(?1)ki
ln(2k + 1)
FEGP.
EKEXA6BRGICMDRFEGP.
20/54
DI 4 BRGI
∞summationdisplay
n=1
in
lnn F0FGFEGP? F0FGH1GHFEGP?
B5FR?
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle in
lnn
vextendsinglevextendsingle
vextendsinglevextendsingle=
∞summationdisplay
n=1
1
lnn
FR?
∞summationdisplay
n=1
1
lnn D9FT,EKEXBRGIG0H1GHFEGP.
A8 n = 2k EE,BRGIBP
∞summationdisplay
n=1
in
lnn =
∞summationdisplay
k=1
(?1)k
ln2k
FEGP; A8 n = 2k + 1 EE,BRGIBP
∞summationdisplay
n=1
in
lnn =
∞summationdisplay
k=1
(?1)ki
ln(2k + 1)
FEGP.
EKEXA6BRGICMDRFEGP.
20/54
DI 4 BRGI
∞summationdisplay
n=1
in
lnn F0FGFEGP? F0FGH1GHFEGP?
B5FR?
∞summationdisplay
n=1
|αn| =
∞summationdisplay
n=1
vextendsinglevextendsingle
vextendsinglevextendsingle in
lnn
vextendsinglevextendsingle
vextendsinglevextendsingle=
∞summationdisplay
n=1
1
lnn
FR?
∞summationdisplay
n=1
1
lnn D9FT,EKEXBRGIG0H1GHFEGP.
A8 n = 2k EE,BRGIBP
∞summationdisplay
n=1
in
lnn =
∞summationdisplay
k=1
(?1)k
ln2k
FEGP; A8 n = 2k + 1 EE,BRGIBP
∞summationdisplay
n=1
in
lnn =
∞summationdisplay
k=1
(?1)ki
ln(2k + 1)
FEGP.
EKEXA6BRGICMDRFEGP.
20/54
DI 5 BRGI
∞summationdisplay
n=1
(?1)n + ni
n + 4 F0FGFEGP? F0FGH1GHFEGP?
B5 FR?
limn→∞αn = limn→∞ (?1)
n + ni
n + 4 = i negationslash= 0,
DUGABRGIFEGPARBJDRCMDR,CCBRGID9FT.
21/54
DI 5 BRGI
∞summationdisplay
n=1
(?1)n + ni
n + 4 F0FGFEGP? F0FGH1GHFEGP?
B5 FR?
limn→∞αn = limn→∞ (?1)
n + ni
n + 4 = i negationslash= 0,
DUGABRGIFEGPARBJDRCMDR,CCBRGID9FT.
21/54
DI 5 BRGI
∞summationdisplay
n=1
(?1)n + ni
n + 4 F0FGFEGP? F0FGH1GHFEGP?
B5 FR?
limn→∞αn = limn→∞ (?1)
n + ni
n + 4 = i negationslash= 0,
DUGABRGIFEGPARBJDRCMDR,CCBRGID9FT.
21/54
BDAJAX E8BRGI
1,E8BRGICECV
F9FE AN{fn(z)}(n = 1,2,···)?E8BXBVF8GICBAF,G1ELDSE1BFER
DX D CBCZF9FE,C4DSEQ
∞summationdisplay
n=1
fn(z) = f1(z) + f2(z) +··· + fn(z) +···
EH?BXBVF8GIE1BRGI.
BRGIARCR n E1G2
sn(z) = f1(z) + f2(z) +··· + fn(z)
EH?GXBRGIARG4EPG2.
22/54
BDAJAX E8BRGI
1,E8BRGICECV
F9FE AN{fn(z)}(n = 1,2,···)?E8BXBVF8GICBAF,G1ELDSE1BFER
DX D CBCZF9FE,C4DSEQ
∞summationdisplay
n=1
fn(z) = f1(z) + f2(z) +··· + fn(z) +···
EH?BXBVF8GIE1BRGI.
BRGIARCR n E1G2
sn(z) = f1(z) + f2(z) +··· + fn(z)
EH?GXBRGIARG4EPG2.
22/54
BDAJAX E8BRGI
1,E8BRGICECV
F9FE AN{fn(z)}(n = 1,2,···)?E8BXBVF8GICBAF,G1ELDSE1BFER
DX D CBCZF9FE,C4DSEQ
∞summationdisplay
n=1
fn(z) = f1(z) + f2(z) +··· + fn(z) +···
EH?BXBVF8GIE1BRGI.
BRGIARCR n E1G2
sn(z) = f1(z) + f2(z) +··· + fn(z)
EH?GXBRGIARG4EPG2.
22/54
F9FE CGC2GHD9 D CBARB8E8BM z0,BGDL
limn→∞sn(z0) = s(z0)
DJBF,BSD3EHBXBVF8GIE1BRGIBF z0 FEGP,AD s(z0) EH?ENARG2.
CGC2BRGIBF D CBBRBRFEGP,BSD3ENARG2F0 z ARE8DRF8GI s(z):
s(z) = f1(z) + f2(z) +··· + fn(z) +···
s(z) EH?BRGIARG2F8GI.
23/54
F9FE CGC2GHD9 D CBARB8E8BM z0,BGDL
limn→∞sn(z0) = s(z0)
DJBF,BSD3EHBXBVF8GIE1BRGIBF z0 FEGP,AD s(z0) EH?ENARG2.
CGC2BRGIBF D CBBRBRFEGP,BSD3ENARG2F0 z ARE8DRF8GI s(z):
s(z) = f1(z) + f2(z) +··· + fn(z) +···
s(z) EH?BRGIARG2F8GI.
23/54
F9FE A8 fn(z) = cn?1(z?a)n?1 F9 fn(z) = cn?1zn?1 EE,FUAQAKF8
GIE1BRGIARBTFRECBB
∞summationdisplay
n=1
cn(z?a)n = c0 + c1(z?a) + c2(z?a)2 +··· + cn(z?a)n +···
F9 ∞
summationdisplay
n=1
cnzn = c0 + c1z + c2z2 +··· + cnzn +···
GXERBRGIEH?E8BRGI.
24/54
CTEHBRG0E8BRGIARFEGPDXAREVC3.
F9D2E8 (BVAJAGF9D2) CGC2BRGI
∞summationdisplay
n=1
cnzn BF z = z0(negationslash= 0) FEGP,BSD3
GHGVDA|z| < |z0| AR z,BRGIBJH1GHFEGP.
CGC2BF z = z0 BRGID9FT,BSD3GHGVDA|z| > |z0|AR z,BRGIBJD9FT.
y
x O
z
z0
O
z0
y
z
x
25/54
CTEHBRG0E8BRGIARFEGPDXAREVC3.
F9D2E8 (BVAJAGF9D2) CGC2BRGI
∞summationdisplay
n=1
cnzn BF z = z0(negationslash= 0) FEGP,BSD3
GHGVDA|z| < |z0| AR z,BRGIBJH1GHFEGP.
CGC2BF z = z0 BRGID9FT,BSD3GHGVDA|z| > |z0|AR z,BRGIBJD9FT.
y
x O
z
z0
O
z0
y
z
x
25/54
DDFR?
∞summationdisplay
n=1
cnzn FEGP,EKEX limn→∞cnzn0 = 0.
EKEX,?M > 0,EMAQ?n,CZ
|cnzn0| < M.
CGC2|z| < |z0|,C3 q =
vextendsinglevextendsingle
vextendsinglevextendsinglez
z0
vextendsinglevextendsingle
vextendsinglevextendsingle< 1,
D9F0
|cnzn| = |cnzn0|·
vextendsinglevextendsingle
vextendsinglevextendsinglez
z0
vextendsinglevextendsingle
vextendsinglevextendsingle
n
< Mqn
FR?
∞summationdisplay
n=1
Mqn FEGP,EKEX
∞summationdisplay
n=1
|cnzn|FEGP,D0ADBRGI
∞summationdisplay
n=1
cnzn H1GHFE
GP.
26/54
DDFR?
∞summationdisplay
n=1
cnzn FEGP,EKEX limn→∞cnzn0 = 0.
EKEX,?M > 0,EMAQ?n,CZ
|cnzn0| < M.
CGC2|z| < |z0|,C3 q =
vextendsinglevextendsingle
vextendsinglevextendsinglez
z0
vextendsinglevextendsingle
vextendsinglevextendsingle< 1,
D9F0
|cnzn| = |cnzn0|·
vextendsinglevextendsingle
vextendsinglevextendsinglez
z0
vextendsinglevextendsingle
vextendsinglevextendsingle
n
< Mqn
FR?
∞summationdisplay
n=1
Mqn FEGP,EKEX
∞summationdisplay
n=1
|cnzn|FEGP,D0ADBRGI
∞summationdisplay
n=1
cnzn H1GHFE
GP.
26/54
DDFR?
∞summationdisplay
n=1
cnzn FEGP,EKEX limn→∞cnzn0 = 0.
EKEX,?M > 0,EMAQ?n,CZ
|cnzn0| < M.
CGC2|z| < |z0|,C3 q =
vextendsinglevextendsingle
vextendsinglevextendsinglez
z0
vextendsinglevextendsingle
vextendsinglevextendsingle< 1,
D9F0
|cnzn| = |cnzn0|·
vextendsinglevextendsingle
vextendsinglevextendsinglez
z0
vextendsinglevextendsingle
vextendsinglevextendsingle
n
< Mqn
FR?
∞summationdisplay
n=1
Mqn FEGP,EKEX
∞summationdisplay
n=1
|cnzn|FEGP,D0ADBRGI
∞summationdisplay
n=1
cnzn H1GHFE
GP.
26/54
DDFR?
∞summationdisplay
n=1
cnzn FEGP,EKEX limn→∞cnzn0 = 0.
EKEX,?M > 0,EMAQ?n,CZ
|cnzn0| < M.
CGC2|z| < |z0|,C3 q =
vextendsinglevextendsingle
vextendsinglevextendsinglez
z0
vextendsinglevextendsingle
vextendsinglevextendsingle< 1,
D9F0
|cnzn| = |cnzn0|·
vextendsinglevextendsingle
vextendsinglevextendsinglez
z0
vextendsinglevextendsingle
vextendsinglevextendsingle
n
< Mqn
FR?
∞summationdisplay
n=1
Mqn FEGP,EKEX
∞summationdisplay
n=1
|cnzn|FEGP,D0ADBRGI
∞summationdisplay
n=1
cnzn H1GHFE
GP.
26/54
DDFR?
∞summationdisplay
n=1
cnzn FEGP,EKEX limn→∞cnzn0 = 0.
EKEX,?M > 0,EMAQ?n,CZ
|cnzn0| < M.
CGC2|z| < |z0|,C3 q =
vextendsinglevextendsingle
vextendsinglevextendsinglez
z0
vextendsinglevextendsingle
vextendsinglevextendsingle< 1,
D9F0
|cnzn| = |cnzn0|·
vextendsinglevextendsingle
vextendsinglevextendsinglez
z0
vextendsinglevextendsingle
vextendsinglevextendsingle
n
< Mqn
FR?
∞summationdisplay
n=1
Mqn FEGP,EKEX
∞summationdisplay
n=1
|cnzn|FEGP,D0ADBRGI
∞summationdisplay
n=1
cnzn H1GHFE
GP.
26/54
BVAJAGF9D2ARDXE8DOARC4DSBBEQ
F9D2 CGC2BRGI
∞summationdisplay
n=1
cn(z?α)n BF z = z0(z0 negationslash= α) FEGP,BSD3GHGV
DA|z?α| < |z0?α| AR z,BRGIBJH1GHFEGP.
CGC2BF z = z0 BRGID9FT,BSD3GHGVDA|z?α| > |z0?α| AR z,BRGIBJ
D9FT.
y
O x
z0 z
α
y
x O
α
z0
z
27/54
BVAJAGF9D2ARDXE8DOARC4DSBBEQ
F9D2 CGC2BRGI
∞summationdisplay
n=1
cn(z?α)n BF z = z0(z0 negationslash= α) FEGP,BSD3GHGV
DA|z?α| < |z0?α| AR z,BRGIBJH1GHFEGP.
CGC2BF z = z0 BRGID9FT,BSD3GHGVDA|z?α| > |z0?α| AR z,BRGIBJ
D9FT.
y
O x
z0 z
α
y
x O
α
z0
z
27/54
DI 1 E8BRGI
∞summationdisplay
n=0
cn(z?2)n CDFGBF z = 0 FEGPADBF z = 3 D9FT?
y
O
2 3
x
B5 G0CD,DUGABVAJAGF9D2,BF z = 0 FEGPARE8BRGI
∞summationdisplay
n=0
cn(z?2)n BJ
BFABDX
|z?2| < |0?2| = 2
CBBRBRFEGP.
AD z = 3 BFABDX|z?2| < 2 CB,EKEXE8BRGI
∞summationdisplay
n=0
cn(z?2)n G0CD
BF z = 0 FEGPADBF z = 3 D9FT.
28/54
DI 1 E8BRGI
∞summationdisplay
n=0
cn(z?2)n CDFGBF z = 0 FEGPADBF z = 3 D9FT?
y
O
2 3
x
B5 G0CD,DUGABVAJAGF9D2,BF z = 0 FEGPARE8BRGI
∞summationdisplay
n=0
cn(z?2)n BJ
BFABDX
|z?2| < |0?2| = 2
CBBRBRFEGP.
AD z = 3 BFABDX|z?2| < 2 CB,EKEXE8BRGI
∞summationdisplay
n=0
cn(z?2)n G0CD
BF z = 0 FEGPADBF z = 3 D9FT.
28/54
DI 1 E8BRGI
∞summationdisplay
n=0
cn(z?2)n CDFGBF z = 0 FEGPADBF z = 3 D9FT?
y
O
2 3
x
B5 G0CD,DUGABVAJAGF9D2,BF z = 0 FEGPARE8BRGI
∞summationdisplay
n=0
cn(z?2)n BJ
BFABDX
|z?2| < |0?2| = 2
CBBRBRFEGP.
AD z = 3 BFABDX|z?2| < 2 CB,EKEXE8BRGI
∞summationdisplay
n=0
cn(z?2)n G0CD
BF z = 0 FEGPADBF z = 3 D9FT.
28/54
DI 2 CTE8BRGI
∞summationdisplay
n=0
cn(z + i)n BF z = i BRD9FT,BSD3CCBRGIBF z = 2
BRFEGPDOF0D9FT?
y
x O
i
i
2
B5DUGABVAJAGF9D2,BF z = i D9FTARE8BRGI
∞summationdisplay
n=0
cn(z + i)n BJBFABEX
|z?(?i)| = |i?(?i)| = 2
A9D9FT.
AD z = 2 BFABDX|z?(?i)| = 2 A9,EKEXE8BRGI
∞summationdisplay
n=0
cn(z + i)n
BF z = 2 D9FT.
29/54
DI 2 CTE8BRGI
∞summationdisplay
n=0
cn(z + i)n BF z = i BRD9FT,BSD3CCBRGIBF z = 2
BRFEGPDOF0D9FT?
y
x O
i
i
2
B5DUGABVAJAGF9D2,BF z = i D9FTARE8BRGI
∞summationdisplay
n=0
cn(z + i)n BJBFABEX
|z?(?i)| = |i?(?i)| = 2
A9D9FT.
AD z = 2 BFABDX|z?(?i)| = 2 A9,EKEXE8BRGI
∞summationdisplay
n=0
cn(z + i)n
BF z = 2 D9FT.
29/54
DI 2 CTE8BRGI
∞summationdisplay
n=0
cn(z + i)n BF z = i BRD9FT,BSD3CCBRGIBF z = 2
BRFEGPDOF0D9FT?
y
x O
i
i
2
B5DUGABVAJAGF9D2,BF z = i D9FTARE8BRGI
∞summationdisplay
n=0
cn(z + i)n BJBFABEX
|z?(?i)| = |i?(?i)| = 2
A9D9FT.
AD z = 2 BFABDX|z?(?i)| = 2 A9,EKEXE8BRGI
∞summationdisplay
n=0
cn(z + i)n
BF z = 2 D9FT.
29/54
2,FEGPABDQFEGPDWF7
DUGABVAJAGF9D2,GHE8DRE8BRGI
nsummationdisplay
n=0
czzn BHCO,ENARFEGPECF5CZCTGB
D2ER:
(1) E8BRGIBFBXEYEHCBBRBRH1GHFEGP.
(2) E8BRGIBPBFA6BMFEGP.
(3) E8BRGIBFEXA6BM?ELB1ARABEX CR CBG4H1GHFEGP; BF CR A9D9FT.
FEGPABARDWF7 R EH?FEGPDWF7.
FEGPABEXH2BRGIARGPFTBH,DUGAGCC6ECF5EPG3B4F9.
y
x O
R
CR
30/54
2,FEGPABDQFEGPDWF7
DUGABVAJAGF9D2,GHE8DRE8BRGI
nsummationdisplay
n=0
czzn BHCO,ENARFEGPECF5CZCTGB
D2ER:
(1) E8BRGIBFBXEYEHCBBRBRH1GHFEGP.
(2) E8BRGIBPBFA6BMFEGP.
(3) E8BRGIBFEXA6BM?ELB1ARABEX CR CBG4H1GHFEGP; BF CR A9D9FT.
FEGPABARDWF7 R EH?FEGPDWF7.
FEGPABEXH2BRGIARGPFTBH,DUGAGCC6ECF5EPG3B4F9.
y
x O
R
CR
30/54
2,FEGPABDQFEGPDWF7
DUGABVAJAGF9D2,GHE8DRE8BRGI
nsummationdisplay
n=0
czzn BHCO,ENARFEGPECF5CZCTGB
D2ER:
(1) E8BRGIBFBXEYEHCBBRBRH1GHFEGP.
(2) E8BRGIBPBFA6BMFEGP.
(3) E8BRGIBFEXA6BM?ELB1ARABEX CR CBG4H1GHFEGP; BF CR A9D9FT.
FEGPABARDWF7 R EH?FEGPDWF7.
FEGPABEXH2BRGIARGPFTBH,DUGAGCC6ECF5EPG3B4F9.
y
x O
R
CR
30/54
2,FEGPABDQFEGPDWF7
DUGABVAJAGF9D2,GHE8DRE8BRGI
nsummationdisplay
n=0
czzn BHCO,ENARFEGPECF5CZCTGB
D2ER:
(1) E8BRGIBFBXEYEHCBBRBRH1GHFEGP.
(2) E8BRGIBPBFA6BMFEGP.
(3) E8BRGIBFEXA6BM?ELB1ARABEX CR CBG4H1GHFEGP; BF CR A9D9FT.
FEGPABARDWF7 R EH?FEGPDWF7.
FEGPABEXH2BRGIARGPFTBH,DUGAGCC6ECF5EPG3B4F9.
y
x O
R
CR
30/54
2,FEGPABDQFEGPDWF7
DUGABVAJAGF9D2,GHE8DRE8BRGI
nsummationdisplay
n=0
czzn BHCO,ENARFEGPECF5CZCTGB
D2ER:
(1) E8BRGIBFBXEYEHCBBRBRH1GHFEGP.
(2) E8BRGIBPBFA6BMFEGP.
(3) E8BRGIBFEXA6BM?ELB1ARABEX CR CBG4H1GHFEGP; BF CR A9D9FT.
FEGPABARDWF7 R EH?FEGPDWF7.
FEGPABEXH2BRGIARGPFTBH,DUGAGCC6ECF5EPG3B4F9.
y
x O
R
CR
30/54
DI 1 EME8BRGI
∞summationdisplay
n=0
zn = 1 + z + z2 +··· + zn +···
ARFEGPDTDWDQG2F8GI.
B5 BRGIARG4EPG2?
sn = 1 + z + z2 +··· + zn = 1?z
n
1?z,(z negationslash= 1).
D9F0,A8|z| < 1 EE,CZ limn→∞sn = 11?z,BRGIFEGP,DPG2F8GI
1
1?z;
A8|z|greaterorequalslant 1,E8DOE1 zn G0EQD9AY,APBRGID9FT.
31/54
DI 1 EME8BRGI
∞summationdisplay
n=0
zn = 1 + z + z2 +··· + zn +···
ARFEGPDTDWDQG2F8GI.
B5 BRGIARG4EPG2?
sn = 1 + z + z2 +··· + zn = 1?z
n
1?z,(z negationslash= 1).
D9F0,A8|z| < 1 EE,CZ limn→∞sn = 11?z,BRGIFEGP,DPG2F8GI
1
1?z;
A8|z|greaterorequalslant 1,E8DOE1 zn G0EQD9AY,APBRGID9FT.
31/54
DI 1 EME8BRGI
∞summationdisplay
n=0
zn = 1 + z + z2 +··· + zn +···
ARFEGPDTDWDQG2F8GI.
B5 BRGIARG4EPG2?
sn = 1 + z + z2 +··· + zn = 1?z
n
1?z,(z negationslash= 1).
D9F0,A8|z| < 1 EE,CZ limn→∞sn = 11?z,BRGIFEGP,DPG2F8GI
1
1?z;
A8|z|greaterorequalslant 1,E8DOE1 zn G0EQD9AY,APBRGID9FT.
31/54
DI 1 EME8BRGI
∞summationdisplay
n=0
zn = 1 + z + z2 +··· + zn +···
ARFEGPDTDWDQG2F8GI.
B5 BRGIARG4EPG2?
sn = 1 + z + z2 +··· + zn = 1?z
n
1?z,(z negationslash= 1).
D9F0,A8|z| < 1 EE,CZ limn→∞sn = 11?z,BRGIFEGP,DPG2F8GI
1
1?z;
A8|z|greaterorequalslant 1,E8DOE1 zn G0EQD9AY,APBRGID9FT.
31/54
3,FEGPDWF7AREMDF
F9D2AJ(B3DUDF) CGC2 limn→∞
vextendsinglevextendsingle
vextendsinglevextendsinglecn+1
cn
vextendsinglevextendsingle
vextendsinglevextendsingle= λ,
BSD3CZ
(1) A8 λ = 0 EE,R = ∞;
(2) A8 λ = +∞EE,R = 0.
(3) A8 0 < λ < +∞EE,R = 1λ.
32/54
3,FEGPDWF7AREMDF
F9D2AJ(B3DUDF) CGC2 limn→∞
vextendsinglevextendsingle
vextendsinglevextendsinglecn+1
cn
vextendsinglevextendsingle
vextendsinglevextendsingle= λ,
BSD3CZ
(1) A8 λ = 0 EE,R = ∞;
(2) A8 λ = +∞EE,R = 0.
(3) A8 0 < λ < +∞EE,R = 1λ.
32/54
F9D2AJ(DUDUDF) CGC2 limn→∞ n
radicalbig
|cn| = μ,BSD3CZ
(1) A8 μ = 0 EE,R = ∞;
(2) A8 μ = +∞EE,R = 0.
(3) A8 0 < μ < +∞EE,R = 1μ.
33/54
F9D2AJ(DUDUDF) CGC2 limn→∞ n
radicalbig
|cn| = μ,BSD3CZ
(1) A8 μ = 0 EE,R = ∞;
(2) A8 μ = +∞EE,R = 0.
(3) A8 0 < μ < +∞EE,R = 1μ.
33/54
DD (1) A8 μ = 0 EE,CZ limn→∞ n
radicalbig
|cn| = 0.
D9F0GHD9BRFB z,A8 n FBEPDWEE,CZ
nradicalbig|cn| < 1
2|z| =? |cn||z|
n < 1
2n
D9F0
∞summationdisplay
n=0
|cn||z|n FEGP,EKEX
∞summationdisplay
n=0
cnzn BFBRFBBM z BRFEGP.
EKEXE8BRGI
∞summationdisplay
n=0
cnzn ARFEGPDWF7F0,R = ∞.
34/54
DD (1) A8 μ = 0 EE,CZ limn→∞ n
radicalbig
|cn| = 0.
D9F0GHD9BRFB z,A8 n FBEPDWEE,CZ
nradicalbig|cn| < 1
2|z| =? |cn||z|
n < 1
2n
D9F0
∞summationdisplay
n=0
|cn||z|n FEGP,EKEX
∞summationdisplay
n=0
cnzn BFBRFBBM z BRFEGP.
EKEXE8BRGI
∞summationdisplay
n=0
cnzn ARFEGPDWF7F0,R = ∞.
34/54
DD (1) A8 μ = 0 EE,CZ limn→∞ n
radicalbig
|cn| = 0.
D9F0GHD9BRFB z,A8 n FBEPDWEE,CZ
nradicalbig|cn| < 1
2|z| =? |cn||z|
n < 1
2n
D9F0
∞summationdisplay
n=0
|cn||z|n FEGP,EKEX
∞summationdisplay
n=0
cnzn BFBRFBBM z BRFEGP.
EKEXE8BRGI
∞summationdisplay
n=0
cnzn ARFEGPDWF7F0,R = ∞.
34/54
DD (1) A8 μ = 0 EE,CZ limn→∞ n
radicalbig
|cn| = 0.
D9F0GHD9BRFB z,A8 n FBEPDWEE,CZ
nradicalbig|cn| < 1
2|z| =? |cn||z|
n < 1
2n
D9F0
∞summationdisplay
n=0
|cn||z|n FEGP,EKEX
∞summationdisplay
n=0
cnzn BFBRFBBM z BRFEGP.
EKEXE8BRGI
∞summationdisplay
n=0
cnzn ARFEGPDWF7F0,R = ∞.
34/54
(2) A8 μ = +∞EE,CZ limn→∞ n
radicalbig
|cn| = +∞.
D9F0GHD9BRFB z(z negationslash= 0),A8 n FBEPDWEE,CZ
nradicalbig|cn| > 1
|z| =? |cn||z|
n > 1
EKEX limn→∞cnzn negationslash= 0,D9F0
∞summationdisplay
n=0
cnzn BFBRFBBM z(z negationslash= 0) BRD9FT.
EKEXE8BRGI
∞summationdisplay
n=0
cnzn ARFEGPDWF7F0,R = 0.
35/54
(2) A8 μ = +∞EE,CZ limn→∞ n
radicalbig
|cn| = +∞.
D9F0GHD9BRFB z(z negationslash= 0),A8 n FBEPDWEE,CZ
nradicalbig|cn| > 1
|z| =? |cn||z|
n > 1
EKEX limn→∞cnzn negationslash= 0,D9F0
∞summationdisplay
n=0
cnzn BFBRFBBM z(z negationslash= 0) BRD9FT.
EKEXE8BRGI
∞summationdisplay
n=0
cnzn ARFEGPDWF7F0,R = 0.
35/54
(2) A8 μ = +∞EE,CZ limn→∞ n
radicalbig
|cn| = +∞.
D9F0GHD9BRFB z(z negationslash= 0),A8 n FBEPDWEE,CZ
nradicalbig|cn| > 1
|z| =? |cn||z|
n > 1
EKEX limn→∞cnzn negationslash= 0,D9F0
∞summationdisplay
n=0
cnzn BFBRFBBM z(z negationslash= 0) BRD9FT.
EKEXE8BRGI
∞summationdisplay
n=0
cnzn ARFEGPDWF7F0,R = 0.
35/54
(2) A8 μ = +∞EE,CZ limn→∞ n
radicalbig
|cn| = +∞.
D9F0GHD9BRFB z(z negationslash= 0),A8 n FBEPDWEE,CZ
nradicalbig|cn| > 1
|z| =? |cn||z|
n > 1
EKEX limn→∞cnzn negationslash= 0,D9F0
∞summationdisplay
n=0
cnzn BFBRFBBM z(z negationslash= 0) BRD9FT.
EKEXE8BRGI
∞summationdisplay
n=0
cnzn ARFEGPDWF7F0,R = 0.
35/54
(3) A8 0 < μ < +∞,FR?
limn→∞ n
radicalbig
|cn||z|n = limn→∞ n
radicalbig
|cn|·|z| = μ|z|
EKEXA8|z| < 1μ EE,BRGI
∞summationdisplay
n=0
|cn||z|n FEGP.
DUGAH2AXARF9D2D2,BRGI
∞summationdisplay
n=0
cnzn BFABDX|z| < 1μ CBFEGP.
BEDDA8|z| > 1μ EE,BRGI
∞summationdisplay
n=0
cnzn D9FT.
36/54
(3) A8 0 < μ < +∞,FR?
limn→∞ n
radicalbig
|cn||z|n = limn→∞ n
radicalbig
|cn|·|z| = μ|z|
EKEXA8|z| < 1μ EE,BRGI
∞summationdisplay
n=0
|cn||z|n FEGP.
DUGAH2AXARF9D2D2,BRGI
∞summationdisplay
n=0
cnzn BFABDX|z| < 1μ CBFEGP.
BEDDA8|z| > 1μ EE,BRGI
∞summationdisplay
n=0
cnzn D9FT.
36/54
(3) A8 0 < μ < +∞,FR?
limn→∞ n
radicalbig
|cn||z|n = limn→∞ n
radicalbig
|cn|·|z| = μ|z|
EKEXA8|z| < 1μ EE,BRGI
∞summationdisplay
n=0
|cn||z|n FEGP.
DUGAH2AXARF9D2D2,BRGI
∞summationdisplay
n=0
cnzn BFABDX|z| < 1μ CBFEGP.
BEDDA8|z| > 1μ EE,BRGI
∞summationdisplay
n=0
cnzn D9FT.
36/54
DRDDDF,CQANBFAB|z| > 1μ A9DJBFE8BM z0,EMAQBRGI
∞summationdisplay
n=0
cnzn0 FEGP.
AIE8BM z1,EMAQ
1
λ < |z1| < |z0|
DUGABVAJAGF9D2,BRGI
∞summationdisplay
n=0
|cn||z1|n FEGP.
A2F0FR?|z1| > 1μ,EKEX
limn→∞ n
radicalbig
|cn||z1|n = μ|z1| > 1.
DQBRGI
∞summationdisplay
n=0
|cn||z1|n FEGPCVGP.
FRCS,
∞summationdisplay
n=0
cnzn BFAB|z1| > 1μ A9D9FT.
D2H2EKGB,BRGIARFEGPDWF7F0,R = 1μ.
37/54
DRDDDF,CQANBFAB|z| > 1μ A9DJBFE8BM z0,EMAQBRGI
∞summationdisplay
n=0
cnzn0 FEGP.
AIE8BM z1,EMAQ
1
λ < |z1| < |z0|
DUGABVAJAGF9D2,BRGI
∞summationdisplay
n=0
|cn||z1|n FEGP.
A2F0FR?|z1| > 1μ,EKEX
limn→∞ n
radicalbig
|cn||z1|n = μ|z1| > 1.
DQBRGI
∞summationdisplay
n=0
|cn||z1|n FEGPCVGP.
FRCS,
∞summationdisplay
n=0
cnzn BFAB|z1| > 1μ A9D9FT.
D2H2EKGB,BRGIARFEGPDWF7F0,R = 1μ.
37/54
DRDDDF,CQANBFAB|z| > 1μ A9DJBFE8BM z0,EMAQBRGI
∞summationdisplay
n=0
cnzn0 FEGP.
AIE8BM z1,EMAQ
1
λ < |z1| < |z0|
DUGABVAJAGF9D2,BRGI
∞summationdisplay
n=0
|cn||z1|n FEGP.
A2F0FR?|z1| > 1μ,EKEX
limn→∞ n
radicalbig
|cn||z1|n = μ|z1| > 1.
DQBRGI
∞summationdisplay
n=0
|cn||z1|n FEGPCVGP.
FRCS,
∞summationdisplay
n=0
cnzn BFAB|z1| > 1μ A9D9FT.
D2H2EKGB,BRGIARFEGPDWF7F0,R = 1μ.
37/54
DRDDDF,CQANBFAB|z| > 1μ A9DJBFE8BM z0,EMAQBRGI
∞summationdisplay
n=0
cnzn0 FEGP.
AIE8BM z1,EMAQ
1
λ < |z1| < |z0|
DUGABVAJAGF9D2,BRGI
∞summationdisplay
n=0
|cn||z1|n FEGP.
A2F0FR?|z1| > 1μ,EKEX
limn→∞ n
radicalbig
|cn||z1|n = μ|z1| > 1.
DQBRGI
∞summationdisplay
n=0
|cn||z1|n FEGPCVGP.
FRCS,
∞summationdisplay
n=0
cnzn BFAB|z1| > 1μ A9D9FT.
D2H2EKGB,BRGIARFEGPDWF7F0,R = 1μ.
37/54
DRDDDF,CQANBFAB|z| > 1μ A9DJBFE8BM z0,EMAQBRGI
∞summationdisplay
n=0
cnzn0 FEGP.
AIE8BM z1,EMAQ
1
λ < |z1| < |z0|
DUGABVAJAGF9D2,BRGI
∞summationdisplay
n=0
|cn||z1|n FEGP.
A2F0FR?|z1| > 1μ,EKEX
limn→∞ n
radicalbig
|cn||z1|n = μ|z1| > 1.
DQBRGI
∞summationdisplay
n=0
|cn||z1|n FEGPCVGP.
FRCS,
∞summationdisplay
n=0
cnzn BFAB|z1| > 1μ A9D9FT.
D2H2EKGB,BRGIARFEGPDWF7F0,R = 1μ.
37/54
DRDDDF,CQANBFAB|z| > 1μ A9DJBFE8BM z0,EMAQBRGI
∞summationdisplay
n=0
cnzn0 FEGP.
AIE8BM z1,EMAQ
1
λ < |z1| < |z0|
DUGABVAJAGF9D2,BRGI
∞summationdisplay
n=0
|cn||z1|n FEGP.
A2F0FR?|z1| > 1μ,EKEX
limn→∞ n
radicalbig
|cn||z1|n = μ|z1| > 1.
DQBRGI
∞summationdisplay
n=0
|cn||z1|n FEGPCVGP.
FRCS,
∞summationdisplay
n=0
cnzn BFAB|z1| > 1μ A9D9FT.
D2H2EKGB,BRGIARFEGPDWF7F0,R = 1μ.
37/54
DI 1 EME8BRGI
∞summationdisplay
n=0
(cosin)zn ARFEGPDWF7.
B5 FR? cn = cosin = 12(en + e?n),EKEX
limn→∞
vextendsinglevextendsingle
vextendsinglevextendsinglecn+1
cn
vextendsinglevextendsingle
vextendsinglevextendsingle= lim
n→∞
en+1 + e?n?1
en + e?n = limn→∞
e + e?2n?1
1 + e?2n = e,
APFEGPDWF7 R = 1e.
38/54
DI 1 EME8BRGI
∞summationdisplay
n=0
(cosin)zn ARFEGPDWF7.
B5 FR? cn = cosin = 12(en + e?n),EKEX
limn→∞
vextendsinglevextendsingle
vextendsinglevextendsinglecn+1
cn
vextendsinglevextendsingle
vextendsinglevextendsingle= lim
n→∞
en+1 + e?n?1
en + e?n = limn→∞
e + e?2n?1
1 + e?2n = e,
APFEGPDWF7 R = 1e.
38/54
DI 1 EME8BRGI
∞summationdisplay
n=0
(cosin)zn ARFEGPDWF7.
B5 FR? cn = cosin = 12(en + e?n),EKEX
limn→∞
vextendsinglevextendsingle
vextendsinglevextendsinglecn+1
cn
vextendsinglevextendsingle
vextendsinglevextendsingle= lim
n→∞
en+1 + e?n?1
en + e?n = limn→∞
e + e?2n?1
1 + e?2n = e,
APFEGPDWF7 R = 1e.
38/54
DI 2 EME8BRGI
∞summationdisplay
n=1
parenleftBig z
lnin
parenrightBign
ARFEGPDWF7.
B5 FR? cn =
parenleftbigg 1
lnin
parenrightbiggn
,EKEX
limn→∞ n
radicalbig
|cn| = limn→∞ 1|lnin| = limn→∞ 1vextendsinglevextendsingle
vextendsinglelnn + ipi2
vextendsinglevextendsingle
vextendsingle
= limn→∞ 1radicalBig
(lnn)2 + pi24
= 0
APFEGPDWF7 R = ∞.
39/54
DI 2 EME8BRGI
∞summationdisplay
n=1
parenleftBig z
lnin
parenrightBign
ARFEGPDWF7.
B5 FR? cn =
parenleftbigg 1
lnin
parenrightbiggn
,EKEX
limn→∞ n
radicalbig
|cn| = limn→∞ 1|lnin| = limn→∞ 1vextendsinglevextendsingle
vextendsinglelnn + ipi2
vextendsinglevextendsingle
vextendsingle
= limn→∞ 1radicalBig
(lnn)2 + pi24
= 0
APFEGPDWF7 R = ∞.
39/54
DI 2 EME8BRGI
∞summationdisplay
n=1
parenleftBig z
lnin
parenrightBign
ARFEGPDWF7.
B5 FR? cn =
parenleftbigg 1
lnin
parenrightbiggn
,EKEX
limn→∞ n
radicalbig
|cn| = limn→∞ 1|lnin| = limn→∞ 1vextendsinglevextendsingle
vextendsinglelnn + ipi2
vextendsinglevextendsingle
vextendsingle
= limn→∞ 1radicalBig
(lnn)2 + pi24
= 0
APFEGPDWF7 R = ∞.
39/54
DI 2 EME8BRGI
∞summationdisplay
n=1
parenleftBig z
lnin
parenrightBign
ARFEGPDWF7.
B5 FR? cn =
parenleftbigg 1
lnin
parenrightbiggn
,EKEX
limn→∞ n
radicalbig
|cn| = limn→∞ 1|lnin| = limn→∞ 1vextendsinglevextendsingle
vextendsinglelnn + ipi2
vextendsinglevextendsingle
vextendsingle
= limn→∞ 1radicalBig
(lnn)2 + pi24
= 0
APFEGPDWF7 R = ∞.
39/54
DI 3 EME8BRGI
∞summationdisplay
n=1
nlnnzn ARFEGPDWF7.
B5 FR? cn = nlnn,EKEX
limn→∞ n
radicalbig
|cn| = limn→∞nlnnn
= limn→∞elnnn ·lnn = limn→∞e(
lnn)2
n,
DUGAGABJDSDFC3,FR?
limx→∞ (lnx)
2
x = limx→∞
2lnx
x = limx→∞
2
x = 0
EKEX
limn→∞ n
radicalbig
|cn| = limn→∞e
(lnn)2
n = e0 = 1.
APFEGPDWF7 R = 1.
40/54
DI 3 EME8BRGI
∞summationdisplay
n=1
nlnnzn ARFEGPDWF7.
B5 FR? cn = nlnn,EKEX
limn→∞ n
radicalbig
|cn| = limn→∞nlnnn
= limn→∞elnnn ·lnn = limn→∞e(
lnn)2
n,
DUGAGABJDSDFC3,FR?
limx→∞ (lnx)
2
x = limx→∞
2lnx
x = limx→∞
2
x = 0
EKEX
limn→∞ n
radicalbig
|cn| = limn→∞e
(lnn)2
n = e0 = 1.
APFEGPDWF7 R = 1.
40/54
DI 3 EME8BRGI
∞summationdisplay
n=1
nlnnzn ARFEGPDWF7.
B5 FR? cn = nlnn,EKEX
limn→∞ n
radicalbig
|cn| = limn→∞nlnnn
= limn→∞elnnn ·lnn = limn→∞e(
lnn)2
n,
DUGAGABJDSDFC3,FR?
limx→∞ (lnx)
2
x = limx→∞
2lnx
x = limx→∞
2
x = 0
EKEX
limn→∞ n
radicalbig
|cn| = limn→∞e
(lnn)2
n = e0 = 1.
APFEGPDWF7 R = 1.
40/54
DI 3 EME8BRGI
∞summationdisplay
n=1
nlnnzn ARFEGPDWF7.
B5 FR? cn = nlnn,EKEX
limn→∞ n
radicalbig
|cn| = limn→∞nlnnn
= limn→∞elnnn ·lnn = limn→∞e(
lnn)2
n,
DUGAGABJDSDFC3,FR?
limx→∞ (lnx)
2
x = limx→∞
2lnx
x = limx→∞
2
x = 0
EKEX
limn→∞ n
radicalbig
|cn| = limn→∞e
(lnn)2
n = e0 = 1.
APFEGPDWF7 R = 1.
40/54
DI 3 EME8BRGI
∞summationdisplay
n=1
nlnnzn ARFEGPDWF7.
B5 FR? cn = nlnn,EKEX
limn→∞ n
radicalbig
|cn| = limn→∞nlnnn
= limn→∞elnnn ·lnn = limn→∞e(
lnn)2
n,
DUGAGABJDSDFC3,FR?
limx→∞ (lnx)
2
x = limx→∞
2lnx
x = limx→∞
2
x = 0
EKEX
limn→∞ n
radicalbig
|cn| = limn→∞e
(lnn)2
n = e0 = 1.
APFEGPDWF7 R = 1.
40/54
DI 4 EME8BRGI
∞summationdisplay
n=1
1
an + ibnz
n(a > 0,b > 0)
ARFEGPDWF7.
B5 FR? cn = 1an + ibn,DUGADUDUDF,CZ
limn→∞ n
radicalbig
|cn| = limn→∞ n
radicalBiggvextendsingle
vextendsinglevextendsingle
vextendsingle
1
an + ibn
vextendsinglevextendsingle
vextendsinglevextendsingle= lim
n→∞
1
2n√a2n + b2n
FR? max{a,b}lessorequalslant 2n
radicalbig
a2n + b2n lessorequalslant 2n√2max{a,b}
DUGACHB1F9D2,CZ
limn→∞ n
radicalbig
|cn| = 1max{a,b}
APFEGPDWF7 R = max{a,b}.
41/54
DI 4 EME8BRGI
∞summationdisplay
n=1
1
an + ibnz
n(a > 0,b > 0)
ARFEGPDWF7.
B5 FR? cn = 1an + ibn,DUGADUDUDF,CZ
limn→∞ n
radicalbig
|cn| = limn→∞ n
radicalBiggvextendsingle
vextendsinglevextendsingle
vextendsingle
1
an + ibn
vextendsinglevextendsingle
vextendsinglevextendsingle= lim
n→∞
1
2n√a2n + b2n
FR? max{a,b}lessorequalslant 2n
radicalbig
a2n + b2n lessorequalslant 2n√2max{a,b}
DUGACHB1F9D2,CZ
limn→∞ n
radicalbig
|cn| = 1max{a,b}
APFEGPDWF7 R = max{a,b}.
41/54
DI 4 EME8BRGI
∞summationdisplay
n=1
1
an + ibnz
n(a > 0,b > 0)
ARFEGPDWF7.
B5 FR? cn = 1an + ibn,DUGADUDUDF,CZ
limn→∞ n
radicalbig
|cn| = limn→∞ n
radicalBiggvextendsingle
vextendsinglevextendsingle
vextendsingle
1
an + ibn
vextendsinglevextendsingle
vextendsinglevextendsingle= lim
n→∞
1
2n√a2n + b2n
FR? max{a,b}lessorequalslant 2n
radicalbig
a2n + b2n lessorequalslant 2n√2max{a,b}
DUGACHB1F9D2,CZ
limn→∞ n
radicalbig
|cn| = 1max{a,b}
APFEGPDWF7 R = max{a,b}.
41/54
DI 4 EME8BRGI
∞summationdisplay
n=1
1
an + ibnz
n(a > 0,b > 0)
ARFEGPDWF7.
B5 FR? cn = 1an + ibn,DUGADUDUDF,CZ
limn→∞ n
radicalbig
|cn| = limn→∞ n
radicalBiggvextendsingle
vextendsinglevextendsingle
vextendsingle
1
an + ibn
vextendsinglevextendsingle
vextendsinglevextendsingle= lim
n→∞
1
2n√a2n + b2n
FR? max{a,b}lessorequalslant 2n
radicalbig
a2n + b2n lessorequalslant 2n√2max{a,b}
DUGACHB1F9D2,CZ
limn→∞ n
radicalbig
|cn| = 1max{a,b}
APFEGPDWF7 R = max{a,b}.
41/54
BH,E8BRGIBFFEGPABEXH2ARGPFTBHCZD2ERECF5:
(1) E8BRGIBFFEGPABEXH2FTD9FT.
DICG,E8BRGI
∞summationdisplay
n=0
zn
BFFEGPABEX C,|z| = 1 FTD9FT.
42/54
BH,E8BRGIBFFEGPABEXH2ARGPFTBHCZD2ERECF5:
(1) E8BRGIBFFEGPABEXH2FTD9FT.
DICG,E8BRGI
∞summationdisplay
n=0
zn
BFFEGPABEX C,|z| = 1 FTD9FT.
42/54
BH,E8BRGIBFFEGPABEXH2ARGPFTBHCZD2ERECF5:
(1) E8BRGIBFFEGPABEXH2FTD9FT.
DICG,E8BRGI
∞summationdisplay
n=0
zn
BFFEGPABEX C,|z| = 1 FTD9FT.
42/54
(2) E8BRGIBFFEGPABEXH2FTFEGP.
DICG,E8BRGI
∞summationdisplay
n=0
zn
n2BFFEGPABEX C,|z| = 1 H2CZ,
∞summationdisplay
n=0
vextendsinglevextendsingle
vextendsinglevextendsinglezn
n2
vextendsinglevextendsingle
vextendsinglevextendsingle=
∞summationdisplay
n=0
1
n2
EKEXE8BRGI
∞summationdisplay
n=0
zn
n2BFFEGPABEX C,|z| = 1 H2FTH1GHFEGP.
43/54
(2) E8BRGIBFFEGPABEXH2FTFEGP.
DICG,E8BRGI
∞summationdisplay
n=0
zn
n2BFFEGPABEX C,|z| = 1 H2CZ,
∞summationdisplay
n=0
vextendsinglevextendsingle
vextendsinglevextendsinglezn
n2
vextendsinglevextendsingle
vextendsinglevextendsingle=
∞summationdisplay
n=0
1
n2
EKEXE8BRGI
∞summationdisplay
n=0
zn
n2BFFEGPABEX C,|z| = 1 H2FTH1GHFEGP.
43/54
(2) E8BRGIBFFEGPABEXH2FTFEGP.
DICG,E8BRGI
∞summationdisplay
n=0
zn
n2BFFEGPABEX C,|z| = 1 H2CZ,
∞summationdisplay
n=0
vextendsinglevextendsingle
vextendsinglevextendsinglezn
n2
vextendsinglevextendsingle
vextendsinglevextendsingle=
∞summationdisplay
n=0
1
n2
EKEXE8BRGI
∞summationdisplay
n=0
zn
n2BFFEGPABEX C,|z| = 1 H2FTH1GHFEGP.
43/54
(3) E8BRGIBFFEGPABEXH2BPCZFEGPBM?CZD9FTBM.
DICG,E8BRGI
∞summationdisplay
n=0
zn
n,FR?A8 z = 1 EE,BRGI
∞summationdisplay
n=0
1
n
D9FT; A8 z =?1 EE,BRGI
∞summationdisplay
n=0
(?1)n
n
FEGP,EKEXE8BRGI
∞summationdisplay
n=0
zn
n BFFEGPABEXH2C9CZFEGPBM?CZD9FTBM.
44/54
(3) E8BRGIBFFEGPABEXH2BPCZFEGPBM?CZD9FTBM.
DICG,E8BRGI
∞summationdisplay
n=0
zn
n,FR?A8 z = 1 EE,BRGI
∞summationdisplay
n=0
1
n
D9FT; A8 z =?1 EE,BRGI
∞summationdisplay
n=0
(?1)n
n
FEGP,EKEXE8BRGI
∞summationdisplay
n=0
zn
n BFFEGPABEXH2C9CZFEGPBM?CZD9FTBM.
44/54
(3) E8BRGIBFFEGPABEXH2BPCZFEGPBM?CZD9FTBM.
DICG,E8BRGI
∞summationdisplay
n=0
zn
n,FR?A8 z = 1 EE,BRGI
∞summationdisplay
n=0
1
n
D9FT; A8 z =?1 EE,BRGI
∞summationdisplay
n=0
(?1)n
n
FEGP,EKEXE8BRGI
∞summationdisplay
n=0
zn
n BFFEGPABEXH2C9CZFEGPBM?CZD9FTBM.
44/54
(3) E8BRGIBFFEGPABEXH2BPCZFEGPBM?CZD9FTBM.
DICG,E8BRGI
∞summationdisplay
n=0
zn
n,FR?A8 z = 1 EE,BRGI
∞summationdisplay
n=0
1
n
D9FT; A8 z =?1 EE,BRGI
∞summationdisplay
n=0
(?1)n
n
FEGP,EKEXE8BRGI
∞summationdisplay
n=0
zn
n BFFEGPABEXH2C9CZFEGPBM?CZD9FTBM.
44/54
DI 1 CGC2BRGI
∞summationdisplay
n=0
Cnzn BFENARFEGPABARABEXH2E8BM z0 BRH1GHFEGP,
DDEY,ENBFFEGPABEKDWEKARBGERDXH2H1GHFEGP.
DD,GHD9FEGPABEXH2BRFBE8BM z(z negationslash= z0),CZ
|Cnzn| = |Cn||zn| = |Cn||z0|n,
FR?BRGI
∞summationdisplay
n=0
Cnzn0 H1GHFEGP,EKEXBRGI
∞summationdisplay
n=0
|Cn||zn0|FEGP.
D9F0BRGI
∞summationdisplay
n=0
|Cn||zn|FEGP,APBRGI
∞summationdisplay
n=0
Cnzn BFABEXH2BRFBE8BMz BR
H1GHFEGP.
BEDUGABVAJAGF9D2,BRGIBFFEGPABEKDWEKARBGERDXH2H1GHFEGP.
45/54
DI 1 CGC2BRGI
∞summationdisplay
n=0
Cnzn BFENARFEGPABARABEXH2E8BM z0 BRH1GHFEGP,
DDEY,ENBFFEGPABEKDWEKARBGERDXH2H1GHFEGP.
DD,GHD9FEGPABEXH2BRFBE8BM z(z negationslash= z0),CZ
|Cnzn| = |Cn||zn| = |Cn||z0|n,
FR?BRGI
∞summationdisplay
n=0
Cnzn0 H1GHFEGP,EKEXBRGI
∞summationdisplay
n=0
|Cn||zn0|FEGP.
D9F0BRGI
∞summationdisplay
n=0
|Cn||zn|FEGP,APBRGI
∞summationdisplay
n=0
Cnzn BFABEXH2BRFBE8BMz BR
H1GHFEGP.
BEDUGABVAJAGF9D2,BRGIBFFEGPABEKDWEKARBGERDXH2H1GHFEGP.
45/54
DI 1 CGC2BRGI
∞summationdisplay
n=0
Cnzn BFENARFEGPABARABEXH2E8BM z0 BRH1GHFEGP,
DDEY,ENBFFEGPABEKDWEKARBGERDXH2H1GHFEGP.
DD,GHD9FEGPABEXH2BRFBE8BM z(z negationslash= z0),CZ
|Cnzn| = |Cn||zn| = |Cn||z0|n,
FR?BRGI
∞summationdisplay
n=0
Cnzn0 H1GHFEGP,EKEXBRGI
∞summationdisplay
n=0
|Cn||zn0|FEGP.
D9F0BRGI
∞summationdisplay
n=0
|Cn||zn|FEGP,APBRGI
∞summationdisplay
n=0
Cnzn BFABEXH2BRFBE8BMz BR
H1GHFEGP.
BEDUGABVAJAGF9D2,BRGIBFFEGPABEKDWEKARBGERDXH2H1GHFEGP.
45/54
DI 1 CGC2BRGI
∞summationdisplay
n=0
Cnzn BFENARFEGPABARABEXH2E8BM z0 BRH1GHFEGP,
DDEY,ENBFFEGPABEKDWEKARBGERDXH2H1GHFEGP.
DD,GHD9FEGPABEXH2BRFBE8BM z(z negationslash= z0),CZ
|Cnzn| = |Cn||zn| = |Cn||z0|n,
FR?BRGI
∞summationdisplay
n=0
Cnzn0 H1GHFEGP,EKEXBRGI
∞summationdisplay
n=0
|Cn||zn0|FEGP.
D9F0BRGI
∞summationdisplay
n=0
|Cn||zn|FEGP,APBRGI
∞summationdisplay
n=0
Cnzn BFABEXH2BRFBE8BMz BR
H1GHFEGP.
BEDUGABVAJAGF9D2,BRGIBFFEGPABEKDWEKARBGERDXH2H1GHFEGP.
45/54
DI 1 CGC2BRGI
∞summationdisplay
n=0
Cnzn BFENARFEGPABARABEXH2E8BM z0 BRH1GHFEGP,
DDEY,ENBFFEGPABEKDWEKARBGERDXH2H1GHFEGP.
DD,GHD9FEGPABEXH2BRFBE8BM z(z negationslash= z0),CZ
|Cnzn| = |Cn||zn| = |Cn||z0|n,
FR?BRGI
∞summationdisplay
n=0
Cnzn0 H1GHFEGP,EKEXBRGI
∞summationdisplay
n=0
|Cn||zn0|FEGP.
D9F0BRGI
∞summationdisplay
n=0
|Cn||zn|FEGP,APBRGI
∞summationdisplay
n=0
Cnzn BFABEXH2BRFBE8BMz BR
H1GHFEGP.
BEDUGABVAJAGF9D2,BRGIBFFEGPABEKDWEKARBGERDXH2H1GHFEGP.
45/54
DI 2 ANBRGI
∞summationdisplay
n=0
Cn FEGP,AD
∞summationdisplay
n=0
|Cn| D9FT,DDEY
∞summationdisplay
n=0
Cnzn ARFEGP
DWF7? 1.
DD (1) BRGI
∞summationdisplay
n=0
Cn FEGP,C3E8BRGI
∞summationdisplay
n=0
Cnzn BF z = 1 FEGP.
D9F0CSBVAJAGF9D2,GHD9EKCZGVDA|z| < 1 AR z,E8BRGI
∞summationdisplay
n=0
Cnzn H1
GHFEGP,EKEXE8BRGIARFEGPDWF7? R greaterorequalslant 1.
(2) A2F0CGC2 R > 1,C3E8BRGI
∞summationdisplay
n=0
Cnzn BFFEGPAB|z| < R CBH1GHFE
GP.
D9F0BF z = 1 BR?H1GHFEGP,BP
∞summationdisplay
n=0
|Cn| FEGP,CVGP.
AP
∞summationdisplay
n=0
Cnzn ARFEGPDWF7? 1.
46/54
DI 2 ANBRGI
∞summationdisplay
n=0
Cn FEGP,AD
∞summationdisplay
n=0
|Cn| D9FT,DDEY
∞summationdisplay
n=0
Cnzn ARFEGP
DWF7? 1.
DD (1) BRGI
∞summationdisplay
n=0
Cn FEGP,C3E8BRGI
∞summationdisplay
n=0
Cnzn BF z = 1 FEGP.
D9F0CSBVAJAGF9D2,GHD9EKCZGVDA|z| < 1 AR z,E8BRGI
∞summationdisplay
n=0
Cnzn H1
GHFEGP,EKEXE8BRGIARFEGPDWF7? R greaterorequalslant 1.
(2) A2F0CGC2 R > 1,C3E8BRGI
∞summationdisplay
n=0
Cnzn BFFEGPAB|z| < R CBH1GHFE
GP.
D9F0BF z = 1 BR?H1GHFEGP,BP
∞summationdisplay
n=0
|Cn| FEGP,CVGP.
AP
∞summationdisplay
n=0
Cnzn ARFEGPDWF7? 1.
46/54
DI 2 ANBRGI
∞summationdisplay
n=0
Cn FEGP,AD
∞summationdisplay
n=0
|Cn| D9FT,DDEY
∞summationdisplay
n=0
Cnzn ARFEGP
DWF7? 1.
DD (1) BRGI
∞summationdisplay
n=0
Cn FEGP,C3E8BRGI
∞summationdisplay
n=0
Cnzn BF z = 1 FEGP.
D9F0CSBVAJAGF9D2,GHD9EKCZGVDA|z| < 1 AR z,E8BRGI
∞summationdisplay
n=0
Cnzn H1
GHFEGP,EKEXE8BRGIARFEGPDWF7? R greaterorequalslant 1.
(2) A2F0CGC2 R > 1,C3E8BRGI
∞summationdisplay
n=0
Cnzn BFFEGPAB|z| < R CBH1GHFE
GP.
D9F0BF z = 1 BR?H1GHFEGP,BP
∞summationdisplay
n=0
|Cn| FEGP,CVGP.
AP
∞summationdisplay
n=0
Cnzn ARFEGPDWF7? 1.
46/54
DI 2 ANBRGI
∞summationdisplay
n=0
Cn FEGP,AD
∞summationdisplay
n=0
|Cn| D9FT,DDEY
∞summationdisplay
n=0
Cnzn ARFEGP
DWF7? 1.
DD (1) BRGI
∞summationdisplay
n=0
Cn FEGP,C3E8BRGI
∞summationdisplay
n=0
Cnzn BF z = 1 FEGP.
D9F0CSBVAJAGF9D2,GHD9EKCZGVDA|z| < 1 AR z,E8BRGI
∞summationdisplay
n=0
Cnzn H1
GHFEGP,EKEXE8BRGIARFEGPDWF7? R greaterorequalslant 1.
(2) A2F0CGC2 R > 1,C3E8BRGI
∞summationdisplay
n=0
Cnzn BFFEGPAB|z| < R CBH1GHFE
GP.
D9F0BF z = 1 BR?H1GHFEGP,BP
∞summationdisplay
n=0
|Cn| FEGP,CVGP.
AP
∞summationdisplay
n=0
Cnzn ARFEGPDWF7? 1.
46/54
DI 2 ANBRGI
∞summationdisplay
n=0
Cn FEGP,AD
∞summationdisplay
n=0
|Cn| D9FT,DDEY
∞summationdisplay
n=0
Cnzn ARFEGP
DWF7? 1.
DD (1) BRGI
∞summationdisplay
n=0
Cn FEGP,C3E8BRGI
∞summationdisplay
n=0
Cnzn BF z = 1 FEGP.
D9F0CSBVAJAGF9D2,GHD9EKCZGVDA|z| < 1 AR z,E8BRGI
∞summationdisplay
n=0
Cnzn H1
GHFEGP,EKEXE8BRGIARFEGPDWF7? R greaterorequalslant 1.
(2) A2F0CGC2 R > 1,C3E8BRGI
∞summationdisplay
n=0
Cnzn BFFEGPAB|z| < R CBH1GHFE
GP.
D9F0BF z = 1 BR?H1GHFEGP,BP
∞summationdisplay
n=0
|Cn| FEGP,CVGP.
AP
∞summationdisplay
n=0
Cnzn ARFEGPDWF7? 1.
46/54
DI 2 ANBRGI
∞summationdisplay
n=0
Cn FEGP,AD
∞summationdisplay
n=0
|Cn| D9FT,DDEY
∞summationdisplay
n=0
Cnzn ARFEGP
DWF7? 1.
DD (1) BRGI
∞summationdisplay
n=0
Cn FEGP,C3E8BRGI
∞summationdisplay
n=0
Cnzn BF z = 1 FEGP.
D9F0CSBVAJAGF9D2,GHD9EKCZGVDA|z| < 1 AR z,E8BRGI
∞summationdisplay
n=0
Cnzn H1
GHFEGP,EKEXE8BRGIARFEGPDWF7? R greaterorequalslant 1.
(2) A2F0CGC2 R > 1,C3E8BRGI
∞summationdisplay
n=0
Cnzn BFFEGPAB|z| < R CBH1GHFE
GP.
D9F0BF z = 1 BR?H1GHFEGP,BP
∞summationdisplay
n=0
|Cn| FEGP,CVGP.
AP
∞summationdisplay
n=0
Cnzn ARFEGPDWF7? 1.
46/54
DI 2 ANBRGI
∞summationdisplay
n=0
Cn FEGP,AD
∞summationdisplay
n=0
|Cn| D9FT,DDEY
∞summationdisplay
n=0
Cnzn ARFEGP
DWF7? 1.
DD (1) BRGI
∞summationdisplay
n=0
Cn FEGP,C3E8BRGI
∞summationdisplay
n=0
Cnzn BF z = 1 FEGP.
D9F0CSBVAJAGF9D2,GHD9EKCZGVDA|z| < 1 AR z,E8BRGI
∞summationdisplay
n=0
Cnzn H1
GHFEGP,EKEXE8BRGIARFEGPDWF7? R greaterorequalslant 1.
(2) A2F0CGC2 R > 1,C3E8BRGI
∞summationdisplay
n=0
Cnzn BFFEGPAB|z| < R CBH1GHFE
GP.
D9F0BF z = 1 BR?H1GHFEGP,BP
∞summationdisplay
n=0
|Cn| FEGP,CVGP.
AP
∞summationdisplay
n=0
Cnzn ARFEGPDWF7? 1.
46/54
4,E8BRGIARB0DYG2BHEF
F9D2E8 ANE8BRGI
f(z) =
∞summationdisplay
n=0
anzn,R = r1; g(z) =
∞summationdisplay
n=0
bnzn,R = r2,
BSD3BFEXA6BM?ELB1,r1,r2 ELAMA2ARE8DR?DWF7ARABCB,H0DRBRGIDWEX
BRBDCK,DI,EMB0DY,EKAQARE8BRGIARFEGPDWF7DWD9F9AVD9 r1,r2 ELAMA2
ARE8DR,FBDP
f(z) ±g(z) =
∞summationdisplay
n=0
anzn ±
∞summationdisplay
n=0
bnzn =
∞summationdisplay
n=0
(an ±bn)zn,|z| < R.
f(z)g(z) =
parenleftBigg ∞summationdisplay
n=0
anzn
parenrightBiggparenleftBigg∞summationdisplay
n=0
bnzn
parenrightBigg
=
∞summationdisplay
n=0
(anb0 + an?1b1 + an?2b2 +···+ a0bn)zn,|z| < R.
G1EL,R = min(r1,r2).
47/54
4,E8BRGIARB0DYG2BHEF
F9D2E8 ANE8BRGI
f(z) =
∞summationdisplay
n=0
anzn,R = r1; g(z) =
∞summationdisplay
n=0
bnzn,R = r2,
BSD3BFEXA6BM?ELB1,r1,r2 ELAMA2ARE8DR?DWF7ARABCB,H0DRBRGIDWEX
BRBDCK,DI,EMB0DY,EKAQARE8BRGIARFEGPDWF7DWD9F9AVD9 r1,r2 ELAMA2
ARE8DR,FBDP
f(z) ±g(z) =
∞summationdisplay
n=0
anzn ±
∞summationdisplay
n=0
bnzn =
∞summationdisplay
n=0
(an ±bn)zn,|z| < R.
f(z)g(z) =
parenleftBigg ∞summationdisplay
n=0
anzn
parenrightBiggparenleftBigg∞summationdisplay
n=0
bnzn
parenrightBigg
=
∞summationdisplay
n=0
(anb0 + an?1b1 + an?2b2 +···+ a0bn)zn,|z| < R.
G1EL,R = min(r1,r2).
47/54
4,E8BRGIARB0DYG2BHEF
F9D2E8 ANE8BRGI
f(z) =
∞summationdisplay
n=0
anzn,R = r1; g(z) =
∞summationdisplay
n=0
bnzn,R = r2,
BSD3BFEXA6BM?ELB1,r1,r2 ELAMA2ARE8DR?DWF7ARABCB,H0DRBRGIDWEX
BRBDCK,DI,EMB0DY,EKAQARE8BRGIARFEGPDWF7DWD9F9AVD9 r1,r2 ELAMA2
ARE8DR,FBDP
f(z) ±g(z) =
∞summationdisplay
n=0
anzn ±
∞summationdisplay
n=0
bnzn =
∞summationdisplay
n=0
(an ±bn)zn,|z| < R.
f(z)g(z) =
parenleftBigg ∞summationdisplay
n=0
anzn
parenrightBiggparenleftBigg∞summationdisplay
n=0
bnzn
parenrightBigg
=
∞summationdisplay
n=0
(anb0 + an?1b1 + an?2b2 +···+ a0bn)zn,|z| < R.
G1EL,R = min(r1,r2).
47/54
4,E8BRGIARB0DYG2BHEF
F9D2E8 ANE8BRGI
f(z) =
∞summationdisplay
n=0
anzn,R = r1; g(z) =
∞summationdisplay
n=0
bnzn,R = r2,
BSD3BFEXA6BM?ELB1,r1,r2 ELAMA2ARE8DR?DWF7ARABCB,H0DRBRGIDWEX
BRBDCK,DI,EMB0DY,EKAQARE8BRGIARFEGPDWF7DWD9F9AVD9 r1,r2 ELAMA2
ARE8DR,FBDP
f(z) ±g(z) =
∞summationdisplay
n=0
anzn ±
∞summationdisplay
n=0
bnzn =
∞summationdisplay
n=0
(an ±bn)zn,|z| < R.
f(z)g(z) =
parenleftBigg ∞summationdisplay
n=0
anzn
parenrightBiggparenleftBigg∞summationdisplay
n=0
bnzn
parenrightBigg
=
∞summationdisplay
n=0
(anb0 + an?1b1 + an?2b2 +···+ a0bn)zn,|z| < R.
G1EL,R = min(r1,r2).
47/54
4,E8BRGIARB0DYG2BHEF
F9D2E8 ANE8BRGI
f(z) =
∞summationdisplay
n=0
anzn,R = r1; g(z) =
∞summationdisplay
n=0
bnzn,R = r2,
BSD3BFEXA6BM?ELB1,r1,r2 ELAMA2ARE8DR?DWF7ARABCB,H0DRBRGIDWEX
BRBDCK,DI,EMB0DY,EKAQARE8BRGIARFEGPDWF7DWD9F9AVD9 r1,r2 ELAMA2
ARE8DR,FBDP
f(z) ±g(z) =
∞summationdisplay
n=0
anzn ±
∞summationdisplay
n=0
bnzn =
∞summationdisplay
n=0
(an ±bn)zn,|z| < R.
f(z)g(z) =
parenleftBigg ∞summationdisplay
n=0
anzn
parenrightBiggparenleftBigg∞summationdisplay
n=0
bnzn
parenrightBigg
=
∞summationdisplay
n=0
(anb0 + an?1b1 + an?2b2 +···+ a0bn)zn,|z| < R.
G1EL,R = min(r1,r2).
47/54
4,E8BRGIARB0DYG2BHEF
F9D2E8 ANE8BRGI
f(z) =
∞summationdisplay
n=0
anzn,R = r1; g(z) =
∞summationdisplay
n=0
bnzn,R = r2,
BSD3BFEXA6BM?ELB1,r1,r2 ELAMA2ARE8DR?DWF7ARABCB,H0DRBRGIDWEX
BRBDCK,DI,EMB0DY,EKAQARE8BRGIARFEGPDWF7DWD9F9AVD9 r1,r2 ELAMA2
ARE8DR,FBDP
f(z) ±g(z) =
∞summationdisplay
n=0
anzn ±
∞summationdisplay
n=0
bnzn =
∞summationdisplay
n=0
(an ±bn)zn,|z| < R.
f(z)g(z) =
parenleftBigg ∞summationdisplay
n=0
anzn
parenrightBiggparenleftBigg∞summationdisplay
n=0
bnzn
parenrightBigg
=
∞summationdisplay
n=0
(anb0 + an?1b1 + an?2b2 +···+ a0bn)zn,|z| < R.
G1EL,R = min(r1,r2).
47/54
4,E8BRGIARB0DYG2BHEF
F9D2E8 ANE8BRGI
f(z) =
∞summationdisplay
n=0
anzn,R = r1; g(z) =
∞summationdisplay
n=0
bnzn,R = r2,
BSD3BFEXA6BM?ELB1,r1,r2 ELAMA2ARE8DR?DWF7ARABCB,H0DRBRGIDWEX
BRBDCK,DI,EMB0DY,EKAQARE8BRGIARFEGPDWF7DWD9F9AVD9 r1,r2 ELAMA2
ARE8DR,FBDP
f(z) ±g(z) =
∞summationdisplay
n=0
anzn ±
∞summationdisplay
n=0
bnzn =
∞summationdisplay
n=0
(an ±bn)zn,|z| < R.
f(z)g(z) =
parenleftBigg ∞summationdisplay
n=0
anzn
parenrightBiggparenleftBigg∞summationdisplay
n=0
bnzn
parenrightBigg
=
∞summationdisplay
n=0
(anb0 + an?1b1 + an?2b2 +···+ a0bn)zn,|z| < R.
G1EL,R = min(r1,r2).
47/54
BH H0DRE8BRGIEYC4B0DYA0EKAQARE8BRGIARFEGPDWF7CZDWCDDW
D9 min(r1,r2).
DI 3 ANCZE8BRGI
∞summationdisplay
n=0
zn DQ
∞summationdisplay
n=0
1
1 + anz
n(0 < a < 1),
EM
∞summationdisplay
n=0
zn?
∞summationdisplay
n=0
1
1 + anz
n =
∞summationdisplay
n=0
an
1 + anz
n
ARFEGPDWF7.
B5 C6F0A8DD,BRGI
∞summationdisplay
n=0
zn DQ
∞summationdisplay
n=0
1
1 + anz
n
ARFEGPDWF7FTAVD9 1;
ADBRGI
∞summationdisplay
n=0
an
1 + anz
n
ARFEGPDWF7F0
1
a > 1.
A2F0BHFB,EMAVEQ
∞summationdisplay
n=0
zn?
∞summationdisplay
n=1
1
1 + anz
n =
∞summationdisplay
n=1
an
1 + anz
n
EKG9ARFEGPABDXBXF0|z| < 1,G0CDB6DW.
48/54
BH H0DRE8BRGIEYC4B0DYA0EKAQARE8BRGIARFEGPDWF7CZDWCDDW
D9 min(r1,r2).
DI 3 ANCZE8BRGI
∞summationdisplay
n=0
zn DQ
∞summationdisplay
n=0
1
1 + anz
n(0 < a < 1),
EM
∞summationdisplay
n=0
zn?
∞summationdisplay
n=0
1
1 + anz
n =
∞summationdisplay
n=0
an
1 + anz
n
ARFEGPDWF7.
B5 C6F0A8DD,BRGI
∞summationdisplay
n=0
zn DQ
∞summationdisplay
n=0
1
1 + anz
n
ARFEGPDWF7FTAVD9 1;
ADBRGI
∞summationdisplay
n=0
an
1 + anz
n
ARFEGPDWF7F0
1
a > 1.
A2F0BHFB,EMAVEQ
∞summationdisplay
n=0
zn?
∞summationdisplay
n=1
1
1 + anz
n =
∞summationdisplay
n=1
an
1 + anz
n
EKG9ARFEGPABDXBXF0|z| < 1,G0CDB6DW.
48/54
BH H0DRE8BRGIEYC4B0DYA0EKAQARE8BRGIARFEGPDWF7CZDWCDDW
D9 min(r1,r2).
DI 3 ANCZE8BRGI
∞summationdisplay
n=0
zn DQ
∞summationdisplay
n=0
1
1 + anz
n(0 < a < 1),
EM
∞summationdisplay
n=0
zn?
∞summationdisplay
n=0
1
1 + anz
n =
∞summationdisplay
n=0
an
1 + anz
n
ARFEGPDWF7.
B5 C6F0A8DD,BRGI
∞summationdisplay
n=0
zn DQ
∞summationdisplay
n=0
1
1 + anz
n
ARFEGPDWF7FTAVD9 1;
ADBRGI
∞summationdisplay
n=0
an
1 + anz
n
ARFEGPDWF7F0
1
a > 1.
A2F0BHFB,EMAVEQ
∞summationdisplay
n=0
zn?
∞summationdisplay
n=1
1
1 + anz
n =
∞summationdisplay
n=1
an
1 + anz
n
EKG9ARFEGPABDXBXF0|z| < 1,G0CDB6DW.
48/54
BH H0DRE8BRGIEYC4B0DYA0EKAQARE8BRGIARFEGPDWF7CZDWCDDW
D9 min(r1,r2).
DI 3 ANCZE8BRGI
∞summationdisplay
n=0
zn DQ
∞summationdisplay
n=0
1
1 + anz
n(0 < a < 1),
EM
∞summationdisplay
n=0
zn?
∞summationdisplay
n=0
1
1 + anz
n =
∞summationdisplay
n=0
an
1 + anz
n
ARFEGPDWF7.
B5 C6F0A8DD,BRGI
∞summationdisplay
n=0
zn DQ
∞summationdisplay
n=0
1
1 + anz
n
ARFEGPDWF7FTAVD9 1;
ADBRGI
∞summationdisplay
n=0
an
1 + anz
n
ARFEGPDWF7F0
1
a > 1.
A2F0BHFB,EMAVEQ
∞summationdisplay
n=0
zn?
∞summationdisplay
n=1
1
1 + anz
n =
∞summationdisplay
n=1
an
1 + anz
n
EKG9ARFEGPABDXBXF0|z| < 1,G0CDB6DW.
48/54
BH H0DRE8BRGIEYC4B0DYA0EKAQARE8BRGIARFEGPDWF7CZDWCDDW
D9 min(r1,r2).
DI 3 ANCZE8BRGI
∞summationdisplay
n=0
zn DQ
∞summationdisplay
n=0
1
1 + anz
n(0 < a < 1),
EM
∞summationdisplay
n=0
zn?
∞summationdisplay
n=0
1
1 + anz
n =
∞summationdisplay
n=0
an
1 + anz
n
ARFEGPDWF7.
B5 C6F0A8DD,BRGI
∞summationdisplay
n=0
zn DQ
∞summationdisplay
n=0
1
1 + anz
n
ARFEGPDWF7FTAVD9 1;
ADBRGI
∞summationdisplay
n=0
an
1 + anz
n
ARFEGPDWF7F0
1
a > 1.
A2F0BHFB,EMAVEQ
∞summationdisplay
n=0
zn?
∞summationdisplay
n=1
1
1 + anz
n =
∞summationdisplay
n=1
an
1 + anz
n
EKG9ARFEGPABDXBXF0|z| < 1,G0CDB6DW.
48/54
F9D2AJ (E3DQ (BXG4) B0DY) CGC2A8|z| < r EE,f(z) =
∞summationdisplay
n=0
anzn,D5
ANBF|z| < R CB g(z) B5G3DPGVDA|g(z)| < r,BSD3A8|z| < R EE,CZ
f[g(z)] =
∞summationdisplay
n=0
an[g(z)]n.
49/54
DI 1 D6F8GI
1
z?b C4EREKBBCG
∞summationdisplay
n=1
cn(z?a)n ARE8BRGI,G1EL,a negationslash= b.
B5D6F8GI
1
z?b ALEKCGCTARBBEQ
1
z?b =
1
(z?a)?(b?a) =?
1
b?a ·
1
1? z?ab?a
DUGADI 1
1
1?z = 1 + z + z
2 +··· + zn +··· |z| < 1.
DGCME3DQ (BXG4) B0DY,A8
vextendsinglevextendsingle
vextendsinglevextendsinglez?a
b?a
vextendsinglevextendsingle
vextendsinglevextendsingle< 1
EE,CZ
1
1? z?ab?a
= 1 +
parenleftbiggz?a
b?a
parenrightbigg
+
parenleftbiggz?a
b?a
parenrightbigg2
+···+
parenleftbiggz?a
b?a
parenrightbiggn
+···,
50/54
DI 1 D6F8GI
1
z?b C4EREKBBCG
∞summationdisplay
n=1
cn(z?a)n ARE8BRGI,G1EL,a negationslash= b.
B5D6F8GI
1
z?b ALEKCGCTARBBEQ
1
z?b =
1
(z?a)?(b?a) =?
1
b?a ·
1
1? z?ab?a
DUGADI 1
1
1?z = 1 + z + z
2 +··· + zn +··· |z| < 1.
DGCME3DQ (BXG4) B0DY,A8
vextendsinglevextendsingle
vextendsinglevextendsinglez?a
b?a
vextendsinglevextendsingle
vextendsinglevextendsingle< 1
EE,CZ
1
1? z?ab?a
= 1 +
parenleftbiggz?a
b?a
parenrightbigg
+
parenleftbiggz?a
b?a
parenrightbigg2
+···+
parenleftbiggz?a
b?a
parenrightbiggn
+···,
50/54
DI 1 D6F8GI
1
z?b C4EREKBBCG
∞summationdisplay
n=1
cn(z?a)n ARE8BRGI,G1EL,a negationslash= b.
B5D6F8GI
1
z?b ALEKCGCTARBBEQ
1
z?b =
1
(z?a)?(b?a) =?
1
b?a ·
1
1? z?ab?a
DUGADI 1
1
1?z = 1 + z + z
2 +··· + zn +··· |z| < 1.
DGCME3DQ (BXG4) B0DY,A8
vextendsinglevextendsingle
vextendsinglevextendsinglez?a
b?a
vextendsinglevextendsingle
vextendsinglevextendsingle< 1
EE,CZ
1
1? z?ab?a
= 1 +
parenleftbiggz?a
b?a
parenrightbigg
+
parenleftbiggz?a
b?a
parenrightbigg2
+···+
parenleftbiggz?a
b?a
parenrightbiggn
+···,
50/54
DI 1 D6F8GI
1
z?b C4EREKBBCG
∞summationdisplay
n=1
cn(z?a)n ARE8BRGI,G1EL,a negationslash= b.
B5D6F8GI
1
z?b ALEKCGCTARBBEQ
1
z?b =
1
(z?a)?(b?a) =?
1
b?a ·
1
1? z?ab?a
DUGADI 1
1
1?z = 1 + z + z
2 +··· + zn +··· |z| < 1.
DGCME3DQ (BXG4) B0DY,A8
vextendsinglevextendsingle
vextendsinglevextendsinglez?a
b?a
vextendsinglevextendsingle
vextendsinglevextendsingle< 1
EE,CZ
1
1? z?ab?a
= 1 +
parenleftbiggz?a
b?a
parenrightbigg
+
parenleftbiggz?a
b?a
parenrightbigg2
+···+
parenleftbiggz?a
b?a
parenrightbiggn
+···,
50/54
D0AD
1
z?b =?
1
b?a?
1
(b?a)2(z?a)?
1
(b?a)3(z?a)
2?···
1(b?a)n+1(z?a)n?···
AN R = |b?a|,BSD3A8|z?a| < R EE,H2EQD1G8BRGIFEGP;
A8|z?a| > R EE,BRGID9FT.
51/54
D0AD
1
z?b =?
1
b?a?
1
(b?a)2(z?a)?
1
(b?a)3(z?a)
2?···
1(b?a)n+1(z?a)n?···
AN R = |b?a|,BSD3A8|z?a| < R EE,H2EQD1G8BRGIFEGP;
A8|z?a| > R EE,BRGID9FT.
51/54
D0AD
1
z?b =?
1
b?a?
1
(b?a)2(z?a)?
1
(b?a)3(z?a)
2?···
1(b?a)n+1(z?a)n?···
AN R = |b?a|,BSD3A8|z?a| < R EE,H2EQD1G8BRGIFEGP;
A8|z?a| > R EE,BRGID9FT.
51/54
F9D2D3ANE8BRGI
∞summationdisplay
n=1
cn(z?a)n ARFEGPDWF7? R,BSD3
1) ENARG2F8GI f(z),BP
f(z) =
∞summationdisplay
n=0
cn(z?a)n
F0FEGPAB:|z?a| < R CBARB5G3F8GI.
2) f(z) BFFEGPABCBARAJGIDWGQG1E8BRGIFHE1EMAJAQAK,BP
f′(z) =
∞summationdisplay
n=1
ncn(z?a)n?1
3) f(z) BFFEGPABCBDWGQG1E8BRGIFHE1EMFKEP,BP
integraldisplay
C
f(z)dz =
∞summationdisplay
n=0
cn
integraldisplay
C
(z?a)ndz,C ∈ |z?a| < R.
F9 integraldisplay
z
a
f(ζ)dζ =
∞summationdisplay
n=0
cn
n + 1(z?a)
n+1.
52/54
F9D2D3ANE8BRGI
∞summationdisplay
n=1
cn(z?a)n ARFEGPDWF7? R,BSD3
1) ENARG2F8GI f(z),BP
f(z) =
∞summationdisplay
n=0
cn(z?a)n
F0FEGPAB:|z?a| < R CBARB5G3F8GI.
2) f(z) BFFEGPABCBARAJGIDWGQG1E8BRGIFHE1EMAJAQAK,BP
f′(z) =
∞summationdisplay
n=1
ncn(z?a)n?1
3) f(z) BFFEGPABCBDWGQG1E8BRGIFHE1EMFKEP,BP
integraldisplay
C
f(z)dz =
∞summationdisplay
n=0
cn
integraldisplay
C
(z?a)ndz,C ∈ |z?a| < R.
F9 integraldisplay
z
a
f(ζ)dζ =
∞summationdisplay
n=0
cn
n + 1(z?a)
n+1.
52/54
F9D2D3ANE8BRGI
∞summationdisplay
n=1
cn(z?a)n ARFEGPDWF7? R,BSD3
1) ENARG2F8GI f(z),BP
f(z) =
∞summationdisplay
n=0
cn(z?a)n
F0FEGPAB:|z?a| < R CBARB5G3F8GI.
2) f(z) BFFEGPABCBARAJGIDWGQG1E8BRGIFHE1EMAJAQAK,BP
f′(z) =
∞summationdisplay
n=1
ncn(z?a)n?1
3) f(z) BFFEGPABCBDWGQG1E8BRGIFHE1EMFKEP,BP
integraldisplay
C
f(z)dz =
∞summationdisplay
n=0
cn
integraldisplay
C
(z?a)ndz,C ∈ |z?a| < R.
F9 integraldisplay
z
a
f(ζ)dζ =
∞summationdisplay
n=0
cn
n + 1(z?a)
n+1.
52/54
F9D2D3ANE8BRGI
∞summationdisplay
n=1
cn(z?a)n ARFEGPDWF7? R,BSD3
1) ENARG2F8GI f(z),BP
f(z) =
∞summationdisplay
n=0
cn(z?a)n
F0FEGPAB:|z?a| < R CBARB5G3F8GI.
2) f(z) BFFEGPABCBARAJGIDWGQG1E8BRGIFHE1EMAJAQAK,BP
f′(z) =
∞summationdisplay
n=1
ncn(z?a)n?1
3) f(z) BFFEGPABCBDWGQG1E8BRGIFHE1EMFKEP,BP
integraldisplay
C
f(z)dz =
∞summationdisplay
n=0
cn
integraldisplay
C
(z?a)ndz,C ∈ |z?a| < R.
F9 integraldisplay
z
a
f(ζ)dζ =
∞summationdisplay
n=0
cn
n + 1(z?a)
n+1.
52/54
EME8BRGIARG2F8GIEVC3
EMG2F8GIARE8DOE0DF:
(1),DGCMGYEQEMFTFEGPDWF7;
(2),DGCME8BRGIBFFEGPABCBDWEXFHE1EMAJGIG2FHE1EMFKEPARBHEF,D6
CCBRGIDE?EUDJF8GIARE8BRGI,DICMGYEQ:
1
1?z = 1 + z + z
2 +···,|z| < 1;
(3) BRBDEXH2B0DYEMCNB0DY,C7DYFT f(z).
53/54
EME8BRGIARG2F8GIEVC3
EMG2F8GIARE8DOE0DF:
(1),DGCMGYEQEMFTFEGPDWF7;
(2),DGCME8BRGIBFFEGPABCBDWEXFHE1EMAJGIG2FHE1EMFKEPARBHEF,D6
CCBRGIDE?EUDJF8GIARE8BRGI,DICMGYEQ:
1
1?z = 1 + z + z
2 +···,|z| < 1;
(3) BRBDEXH2B0DYEMCNB0DY,C7DYFT f(z).
53/54
EME8BRGIARG2F8GIEVC3
EMG2F8GIARE8DOE0DF:
(1),DGCMGYEQEMFTFEGPDWF7;
(2),DGCME8BRGIBFFEGPABCBDWEXFHE1EMAJGIG2FHE1EMFKEPARBHEF,D6
CCBRGIDE?EUDJF8GIARE8BRGI,DICMGYEQ:
1
1?z = 1 + z + z
2 +···,|z| < 1;
(3) BRBDEXH2B0DYEMCNB0DY,C7DYFT f(z).
53/54
DI 1 EME8BRGI
∞summationdisplay
n=1
n2zn ARG2F8GI,FBC7DY
∞summationdisplay
n=0
n2
2n ARDU.
B5 F0DJE8BRGIARFEGPDWF7F0 R = 1,BFFEGPAB|z| < 1 CBANG1G2F8GI
f(z),BP
f(z) =
∞summationdisplay
n=0
n2zn,|z| < 1
DGCMFHE1EMAJ,AQ
f(z) = z ·
∞summationdisplay
n=1
n2zn?1 = z ·
parenleftBigg ∞summationdisplay
n=1
nzn
parenrightBigg′
= z ·
bracketleftBigg
z ·
parenleftBigg ∞summationdisplay
n=1
nzn?1
parenrightBiggbracketrightBigg′
= z ·
bracketleftBigg
z ·
parenleftBigg ∞summationdisplay
n=1
zn
parenrightBigg′bracketrightBigg′
= z ·
bracketleftbigg
z ·
parenleftbigg z
1?z
parenrightbigg′bracketrightbigg′
= z(1 + z)(1?z)3
EKEX
∞summationdisplay
n=0
n2
2n = f(
1
2) = 6.
54/54
DI 1 EME8BRGI
∞summationdisplay
n=1
n2zn ARG2F8GI,FBC7DY
∞summationdisplay
n=0
n2
2n ARDU.
B5 F0DJE8BRGIARFEGPDWF7F0 R = 1,BFFEGPAB|z| < 1 CBANG1G2F8GI
f(z),BP
f(z) =
∞summationdisplay
n=0
n2zn,|z| < 1
DGCMFHE1EMAJ,AQ
f(z) = z ·
∞summationdisplay
n=1
n2zn?1 = z ·
parenleftBigg ∞summationdisplay
n=1
nzn
parenrightBigg′
= z ·
bracketleftBigg
z ·
parenleftBigg ∞summationdisplay
n=1
nzn?1
parenrightBiggbracketrightBigg′
= z ·
bracketleftBigg
z ·
parenleftBigg ∞summationdisplay
n=1
zn
parenrightBigg′bracketrightBigg′
= z ·
bracketleftbigg
z ·
parenleftbigg z
1?z
parenrightbigg′bracketrightbigg′
= z(1 + z)(1?z)3
EKEX
∞summationdisplay
n=0
n2
2n = f(
1
2) = 6.
54/54
DI 1 EME8BRGI
∞summationdisplay
n=1
n2zn ARG2F8GI,FBC7DY
∞summationdisplay
n=0
n2
2n ARDU.
B5 F0DJE8BRGIARFEGPDWF7F0 R = 1,BFFEGPAB|z| < 1 CBANG1G2F8GI
f(z),BP
f(z) =
∞summationdisplay
n=0
n2zn,|z| < 1
DGCMFHE1EMAJ,AQ
f(z) = z ·
∞summationdisplay
n=1
n2zn?1 = z ·
parenleftBigg ∞summationdisplay
n=1
nzn
parenrightBigg′
= z ·
bracketleftBigg
z ·
parenleftBigg ∞summationdisplay
n=1
nzn?1
parenrightBiggbracketrightBigg′
= z ·
bracketleftBigg
z ·
parenleftBigg ∞summationdisplay
n=1
zn
parenrightBigg′bracketrightBigg′
= z ·
bracketleftbigg
z ·
parenleftbigg z
1?z
parenrightbigg′bracketrightbigg′
= z(1 + z)(1?z)3
EKEX
∞summationdisplay
n=0
n2
2n = f(
1
2) = 6.
54/54
DI 1 EME8BRGI
∞summationdisplay
n=1
n2zn ARG2F8GI,FBC7DY
∞summationdisplay
n=0
n2
2n ARDU.
B5 F0DJE8BRGIARFEGPDWF7F0 R = 1,BFFEGPAB|z| < 1 CBANG1G2F8GI
f(z),BP
f(z) =
∞summationdisplay
n=0
n2zn,|z| < 1
DGCMFHE1EMAJ,AQ
f(z) = z ·
∞summationdisplay
n=1
n2zn?1 = z ·
parenleftBigg ∞summationdisplay
n=1
nzn
parenrightBigg′
= z ·
bracketleftBigg
z ·
parenleftBigg ∞summationdisplay
n=1
nzn?1
parenrightBiggbracketrightBigg′
= z ·
bracketleftBigg
z ·
parenleftBigg ∞summationdisplay
n=1
zn
parenrightBigg′bracketrightBigg′
= z ·
bracketleftbigg
z ·
parenleftbigg z
1?z
parenrightbigg′bracketrightbigg′
= z(1 + z)(1?z)3
EKEX
∞summationdisplay
n=0
n2
2n = f(
1
2) = 6.
54/54
DI 1 EME8BRGI
∞summationdisplay
n=1
n2zn ARG2F8GI,FBC7DY
∞summationdisplay
n=0
n2
2n ARDU.
B5 F0DJE8BRGIARFEGPDWF7F0 R = 1,BFFEGPAB|z| < 1 CBANG1G2F8GI
f(z),BP
f(z) =
∞summationdisplay
n=0
n2zn,|z| < 1
DGCMFHE1EMAJ,AQ
f(z) = z ·
∞summationdisplay
n=1
n2zn?1 = z ·
parenleftBigg ∞summationdisplay
n=1
nzn
parenrightBigg′
= z ·
bracketleftBigg
z ·
parenleftBigg ∞summationdisplay
n=1
nzn?1
parenrightBiggbracketrightBigg′
= z ·
bracketleftBigg
z ·
parenleftBigg ∞summationdisplay
n=1
zn
parenrightBigg′bracketrightBigg′
= z ·
bracketleftbigg
z ·
parenleftbigg z
1?z
parenrightbigg′bracketrightbigg′
= z(1 + z)(1?z)3
EKEX
∞summationdisplay
n=0
n2
2n = f(
1
2) = 6.
54/54
DI 1 EME8BRGI
∞summationdisplay
n=1
n2zn ARG2F8GI,FBC7DY
∞summationdisplay
n=0
n2
2n ARDU.
B5 F0DJE8BRGIARFEGPDWF7F0 R = 1,BFFEGPAB|z| < 1 CBANG1G2F8GI
f(z),BP
f(z) =
∞summationdisplay
n=0
n2zn,|z| < 1
DGCMFHE1EMAJ,AQ
f(z) = z ·
∞summationdisplay
n=1
n2zn?1 = z ·
parenleftBigg ∞summationdisplay
n=1
nzn
parenrightBigg′
= z ·
bracketleftBigg
z ·
parenleftBigg ∞summationdisplay
n=1
nzn?1
parenrightBiggbracketrightBigg′
= z ·
bracketleftBigg
z ·
parenleftBigg ∞summationdisplay
n=1
zn
parenrightBigg′bracketrightBigg′
= z ·
bracketleftbigg
z ·
parenleftbigg z
1?z
parenrightbigg′bracketrightbigg′
= z(1 + z)(1?z)3
EKEX
∞summationdisplay
n=0
n2
2n = f(
1
2) = 6.
54/54