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WuChong-shi 1 §3.1 C ! # f(z)$C %&’ (! C)* +,n- .,z0=A,z1,z2,···,zn=B,ζk zk−1→zk- )*/.01 nXk=1f(ζk)(zk−zk−1)=nXk=1f(ζk)Δzk,23n→∞45max|Δzk|→0671 89:$;ζk =?@AB789C,# f(z)D! C E,ZCf(z)dz=limmax|Δzk|→0nXk=1f(ζk)Δzk.F3.1/G HI JGH %KLMZCf(z)dz=ZC(u+iv)(dx+idy)=ZC(udx−vdy)+iZC(vdx+udy).N7OPC -QR! f(z) C ST# A /&:$ UVWXY1.OP ZCf1(z)dz,ZCf2(z)dz,···,ZCfn(z)dzZ:$AZChf1(z)+f2(z)+···+fn(z)idz=ZCf1(z)dz+ZCf2(z)dz+···+ZCfn(z)dz;2.2C=C1+C2+···+CnAZC1f(z)dz+ZC2f(z)dz+···+ZCnf(z)dz=ZCf(z)dz;3.ZC−f(z)dz=−ZCf(z)dz[\C−]^C _‘a4.ZCaf(z)dz=aZCf(z)dz[\a,b a5. ZCf(z)dz ≤ZC|f(z)||dz|a6. ZCf(z)dz ≤Ml[\M, f(z) $C cl,C de WuChong-shi§3.1 2 fg Chij•k # •l.mno p q9r• st ujv& /Gk # 3l.w&6ujxy st C/z xy {3.1|ZCRezdzC,(i)DH}~0→1 j }1→1+ia(ii)D }~0→i jH}i→1+ia(iii)D 0→1+i uj(i)ZCRezdz=Z10xdx+Z10idy=12+i;uj(ii)ZCRezdz=Z10xdx=12;uj(iii)ZCRezdz=Z10(1+i)tdt=12(1+i).WuChong-shi 3 §3.2 Cauchy Cauchy& C st @ %@ J Y• Y ¡¢£⁄¥ƒ§¤'¤'“«‹›fifl a•– †‡· ¶ F3.2•‚„”»…‰‚„”» Cauchy¿ OP# f(z)$`S´ G\ˆ˜ADG\)¯/G -QR ˘M˙¨C( ˚3.3)%ICf(z)dz=0,¸ C˝˛ˇ G —c F3.3•‚„”» Cauchy WuChong-shi§3.2 Cauchy 4 , ` q $ Æ qª ¸G& Æ f0(z)$G\ST $7Æ q˛ˇŁØGreenŒºICP(x,y)dx+Q(x,y)dy=ZZS∂Q∂x−∂P∂ydxdyjICf(z)dz=ICudx−vdy+iICvdx+udy, ˘M˙¨ , IC udx−vdy=−ZZS∂v∂x+∂u∂ydxdy,IC vdx+udy=ZZS∂u∂x−∂v∂ydxdy. Cauchy-Riemannæ lJG \ k # ,0ı%ICf(z)dz=0.~jGreenŒº |¸ ł `S´ øœ /G%c oxœ ß ∞.$ (?c) f(z) ∞‹ ∞‹ « 0 Cauchy& /G ˆ˜# /GUV WY « “ ‹« « •Cauchy-Riemannæ ¸ @ º•Cauchy& A º ~Cauchy& !o˛ˇ5q # Y$%2f(z)$`S´ G\ˆ˜A ZCf(z)dzst?@ ( &’( ))¿*+,g$`S´ \ˆ˜# st?@N7OPw& .z0 -..z, .A0, 9 # Zzz0f(z)dz=F(z) `S´ G `C# B,f(z) x& ¿ 3.1OP# f(z)$`S´ G ˆ˜AF(z)=Zzz0f(z)dz /0f(z)1G23456f0(z)7158f00(z)9715z∈G5:;f0(z)‚56=?@A∂u/∂x,∂u/∂y,∂v/∂xB∂v/∂y‚CD =EWuChong-shi 5 ˝$G ˆ˜F;F0(z)=ddzZzz0f(z)dz=f(z). ø G|HF(z) I o˛ F3.5,7z G /.z+Δz J.O˚3.5AF(z)=Zzz0f(ζ)dζ,F(z+Δz)=Zz+Δzz0f(ζ)dζ.N, st?@ ˇΔFΔz=F(z+Δz)−F(z)Δz=1ΔzZz+Δzzf(ζ)dζ.~7˛5 ΔFΔz−f(z) = 1ΔzZz+Δzzf(ζ)dζ−f(z) = 1ΔzZz+Δzzf(ζ)−f(z)dζ ≤1|Δz|Zz+Δzz f(ζ)−f(z) · dζ .~jf(z) ST ıuj)v ε0:$δ043|ζ−z|δ6|f(ζ)−f(z)|ε ˇ ΔFΔz−f(z) ≤1|Δz|·ε·|Δz|=ε,o5F0(z)=limΔz→0ΔFΔz=f(z).¸Kª F(z)$G ˛IF;F0(z)=f(z) L’(OP# Φ(z) I Φ0(z)=f(z)AΦ(z)B,f(z) M# &’ f(z) x& K f(z) /GM# ujv& /G# f(z)NłM# x O/ )*JGM# øPQ/Gb ¸ N,OPΦ1(z)Φ2(z)Z f(z) M# AΦ01(z)=f(z),Φ02(z)=f(z).WuChong-shi§3.2 Cauchy 6 ˇΦ1(z)−Φ2(z)0=0Φ1(z)−Φ2(z)=C.R¨ k # M# ˛4 STU, Φ(z),f(z) /GM# Af(z) x& F(z)=Zzz0f(z)dz=Φ(z)+C.V fg%F(z0)=Φ(z0)+C=0,C=−Φ(z0). ˇZzz0f(z)dz=Φ(z)−Φ(z0).{3.2ST Zbazndzn,W 3n,Xg 6zn$Y ˆ˜1n+1zn+1 /GM# N7ujz )*/Æ s %Zbazndz=1n+1bn+1−an+1.3n=−2,−3,−4,···6zn$xß z=0.$ )*/G`S´ ˆ˜[M# Z˛,1n+1zn+1 N7Z%Zbazndz=1n+1bn+1−an+1.V~jq/[\3 MN7]Pujxß z=0.$ )* ^! 3n=−16z−1˝ $xß z=0$ )/ ˆ˜V[M# Ł,lnz N7$xß z=0 )/`S´ Zbadzz=lnb−lna._ ‘*$/G`S´ 3gst?@ V afl b«¥ ¶ bc«d‹ef‹ ghij b« ST k l¸ M# mC# N7 C~a b æº%@ 39n$x z=0 /G`S´ 6K (lnz9n$o/G`C p ı Clnb−lna O/q& ujxy `S´ K˛œuŁjlnz xy`C pN C˝K˛œxy WuChong-shirstuvwxy7z§3.3 Cauchy – Cauchy¿ OPf(z) S´ G\ `Cˆ˜# AIC0f(z)dz=nXi=1ICif(z)dz,[\C0,C1,C2,···,Cn {^ S´ G —c |G -QR˘M! C1,C2,···,CnZß $C0 } ; % st~‘Py F3.6‰‚„”» Cauchy O˚3.6x C0,C1,C2,···,Cn ,_6 æ‘ 0 3 + (C1,C2,···,Cn1C0S] N 5/G`S´ G0f(z)$`S´ G0 ˆ˜ N ˛ˇŁØ`S´ Cauchy& IC0f(z)dz+Zb1a1f(z)dz+IC−1f(z)dz+Za1b1f(z)dz+Zb2a2f(z)dz+IC−2f(z)dz+Za2b2f(z)dz+···+Zbnanf(z)dz+IC−nf(z)dz+Zanbnf(z)dz=0.~jf(z)$G0 `CıDy/+ J C P Zbiaif(z)dz+Zaibif(z)dz=0. ˇIC0f(z)dz+nXi=1IC−if(z)dz=0,(3.1)IC0f(z)dz=−nXi=1IC−if(z)dz=nXi=1ICif(z)dz.(3.2)WuChong-shi§3.3 Cauchy 8 {3.3STICzndzCn,W C ~‘,_6 æ‘ 3n,Xg 6fg `S´ Cauchy& ICzndz=0.3n, W 6OPC x z=0A˝%ICzndz=0.OPC %z=0A S´ Cauchy& %ICzndz=I|z|=1zndz=Z2π0 eiθneiθidθ=Z2π0ei(n+1)θidθ=2πi,n=−1;0,n=−2,−3,−4,···. ] ]PK%ICzndz=2πi,n=−1,;C %z=0;0,[ . /z IC(z−a)ndz=2πi,n=−1,;C %z=a;0,[ .WuChong-shi 9 §3.4 3.1OP# f(z)$z=a. J STF;3θ1≤arg(z−a)≤θ2,|z−a|→06(z−a)f(z)/ jkAlimδ→0ZCδf(z)dz=ik(θ2−θ1),[\Cδ ˇz=a, δ, t ,θ2−θ1 |z−a|=δ,θ1≤arg(z−a)≤θ2 ˚3.7 N,ZCδdzz−a=i(θ2−θ1),F3.7 ˇ ZCδf(z)dz−ik(θ2−θ1) = ZCδf(z)−kz−adz ≤ZCδ|(z−a)f(z)−k||dz||z−a|.~j3θ1≤arg(z−a)≤θ2z−a→06(z−a)f(z)/ jk¸* ∀ε0∃(arg(z−a)?@ )r(ε)043|(z−a)|=δr6|(z−a)f(z)−k|ε ˇ ZCδf(z)dz−ik(θ2−θ1) ≤ε(θ2−θ1),olimδ→0ZCδf(z)dz=ik(θ2−θ1).WuChong-shi§3.4 ¡¢£ ⁄ 10 3.2f(z)$∞. J ST3θ1≤argz≤θ2z→∞6zf(z)/ jKAlimR→∞ZCRf(z)dz=iK(θ2−θ1),[\CR ˇM., R, t¥ ,θ2−θ1 |z|=R,θ1≤argz≤θ2( ˚3.8) 7ƒ ª 1 3.1 ª P§ N,ZCRdzz=i θ2−θ1 ˇF3.8 ZCRf(z)dz−iK(θ2−θ1) = ZCRf(z)−Kzdz = ZCRzf(z)−Kdzz ≤ZCR zf(z)−K ·|dz||z|.~j3θ1≤argz≤θ2z→∞6zf(z)/ jK¸* ∀ε0∃(argz?@ )M(ε)043|z|=RM6|zf(z)−K|ε ˇ ZCRf(z)dz−iK θ2−θ1 ≤ε(θ2−θ1),olimR→∞ZCRf(z)dz=iK θ2−θ1.