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WuChong-shi ( )Cauchy 1 ( )Cauchy §4.1Cauchy Cauchy f(z)!G#$%&’()*+G$,-C./0123+a4G567+8f(a)=12πiICf(z)z−adz,9#:.;3C$=?@AG5BC|z−a|r(DE4.1+FGCH|z−a|=rAG5)+8IJKLM!$CauchyNO+PICf(z)z−adz=I|z−a|=rf(z)z−adz,Q4.1RSTUVCauchyWXYZ[\]^_r$‘abc+defr→0?g4limz→a(z−a)f(z)z−a=f(a),hiO3.1+jkl12πiICf(z)z−adz=f(a).WuChong-shi§4.1Cauchy 2 m Cauchy nob-!+pqr f(z)As%tuvwCxyCz({|b}~7)%&’(? + A 12πiICf(z)z−adz,9#a4Cz67+:.;3C$ + b}~7$=+ E4.2?ACz B6 74C +R4 $‘CCR+ +noC CR {v$KLM!+IJP-!$Cauchy:. +PQ4.2 STUVCauchyWXYZ12πiICRf(z)z−adz+12πiICf(z)z−adz=f(a), :.;3CR$ ? qR ‘+ \] ¡j_R$¢£‘abc+o+efR→∞+⁄l¥12πiICf(z)z−adz=f(a)−limR→∞12πiICRf(z)z−adz. ]f(z)ƒ §¤'iO3.2$q“+8e «‘CCR$:.$‹›&+limR→∞12πiICRf(z)z−adz=K,K=limz→∞z·f(z)z−a=f(∞).g[+12πiICf(z)z−adz=f(a)−K.fifl K=0 +jl¥ m Cauchy ]f(z)As%tuvwCxyCz’(+– z→∞ +f(z)6† ‡o0+8Cauchy:. f(a)=12πiICf(z)z−adz·¡ ¶+[•a4Cz67+:.;3C4 ?WuChong-shi ( )Cauchy 3 §4.2‚„”»…‰ ¿» Cauchy:. +e `´«6 ˆq\˜ ]f(z)AG#’(+8AG5f(z)$¯˘˙¨*f(n)(z) ˚A+¸–f(n)(z)=n!2πiICf(ζ)(ζ−z)n+1dζ,9#CG$=,-+z4G5¯ 67+ E4.3?@˝˛“f0(z)?g4f(z+h)−f(z)h=12πi1hICf(ζ)ζ−z−h−f(ζ)ζ−zdζ=12πiICf(ζ)(ζ−z−h)(ζ−z)dζ,Q4.3ˇ— YZ ‹›h→0+ 4f0(z)+⁄ :)*$‹›4f(ζ)/(ζ−z)2?4 k A:. “‹›u + ICf(ζ)dζ(ζ−z−h)(ζ−z)−ICf(ζ)dζ(ζ−z)2=hICf(ζ)dζ(ζ−z−h)(ζ−z)2.hof(ζ)ACxLÆ+dACxP|f(ζ)|≤M+ z¥C$ ª 4δ+l4C$ +8P ICf(ζ)(ζ−z−h)(ζ−z)dζ−ICf(ζ)(ζ−z)2dζ ≤|h|·Mlδ2(δ−|h|)→0,g[+:. “‹›u +f0(z)=12πiICf(ζ)(ζ−z)2dζ.Ł +e “«f00(z)=limh→0f0(z+h)−f0(z)h=limh→012πi1hICf(ζ)(ζ−z−h)2−f(ζ)(ζ−z)2dζ=limh→012πiIC2ζ−2z−h(ζ−z−h)2(ζ−z)2f(ζ)dζ=2!2πiICf(ζ)(ζ−z)3dζ. [ØÆ+ e“«f(n)(z)?F \]Œ +6 Kº)*+ qA6 !#6˙¨*••˚A( æ ı )+8 $¯˘˙¨*ł˚A+¸–ł !5$’()*?FAøº)*#¸œ [?ß ¸ hf0(x)$˚A`´«f00(x)$˚A?FKº)*#f(z)A6!#••e¨( ’()6 $q“?øº)*#f0(x)$˚A { xA* x(6N! 5)º nf(x)$q“+⁄Kº)*#f0(z)$˚A8{ A !xnf(z)$q“?WuChong-shi§4.3Cauchy 4 §4.3Cauchy …‚„ Ax6 co’()* ˙¨* $k #+f(z)$’( £ A (1)f(z)e Cauchy:. (2)f(z)ACxLÆ?g[+ˆKx $ +je k A6!./01$(tu tu)23CxLÆ$)*φ(ζ) # $:.f(z)=12πiZCφ(ζ)ζ−zdζ($4Cauchy% )23z7z$’()*+f0(z)eM :. “¨⁄l¥+f(p)(z)=p!2πiZCφ(ζ)(ζ−z)p+1dζ.& :.f(z)=12πiI|ζ|=1ζ∗ζ−zdζ,|z|6=1.’ 6 Cauchy(:.?g4A|ζ|=1xζ∗=1/ζ+df(z)=12πiI|ζ|=11ζ(ζ−z)dζ. |z|1 +[:.e Cauchy:. +f(z)=12πiI|ζ|=11ζ1ζ−zdζ=−1z. 0|z|1 +f(z)=12πiI|ζ|=11z1ζ−z−1ζdζ=0.)*+«+[\]noz=0· ¶?,u x\]+jPf(z)=12πiI|ζ|=1ζ∗ζ−zdζ=−1z,|z|1,0,|z|1.h[eD+f(z)A|z|6=1•’(+-.ζ∗A/ ’(?WuChong-shi ( )Cauchy 5 0 Cauchy(:.+je `«123 ’45?674.2 1.f(t,z) t8z 9:ı +t∈[a,b]+z∈G+2.;[a,b]= ?t@+f(t,z) G= A@ ı +8F(z)=Zbaf(t,z)dtAG5’($+–F0(z)=Zba∂f(t,z)∂zdt.@g4f(t,z)AGx’(+dnoG5$¯˘67z+Cauchy:. ¶+f(t,z)=12πiICf(t,ζ)ζ−zdζ.BCF(z)$ND+¸EF:.GH(g4f(t,z)LÆ)+lF(z)=Zbadt2πiICf(t,ζ)ζ−zdζ=12πiIC1ζ−zZbaf(t,ζ)dt#dζ. 6 Cauchy(:.+Zbaf(t,z)dtLÆ+dF(z)4G5$’()*+–F0(z)=12πiIC1(ζ−z)2Zbaf(t,ζ)dt#dζ=Zba12πiICf(t,ζ)(ζ−z)2dζdt=Zba∂f(t,z)∂zdt.I¡+ \˜JK oZCf(t,z)dt? ^ q“C./0123+ tACxºL+z∈G +f(t,z)t z$LÆ)*?k $ _x MŁ?WuChong-shi§4.1NOPO 6 (Q)RSTUVWX +YZ [X + ı \]^ _‘abcd?efghı 8Yiı j k[X lm ?noı X pq8roı st uv8wx yz?§4.1{»|»6}K*~*u0+u1+u2+···+un+···=∞Xn=0un.fun$ø .fl4αn_βn,8∞Xn=0un=∞Xn=0αn+i∞Xn=0βn.6 K*~*Pun / o ø*~*Pαn Pβn+ ¡? ]~*$ . Sn=u0+u1+u2+···+un # $H {Sn} +8$~*Pun +H {Sn}$‹›S=limn→∞Sn+$4~*Pun$ ∞Xn=0un=limn→∞Sn. 8+~*Pun $?~*$ + $ . H $ ND$?g[+IJH $ q! +e «~* $ q! Cauchy q! lε0+ n+ ; p+ |un+1+un+2+···+un+p|ε.fifl+fp=1+jl¥~* $¡q! limn→∞un=0.FA ¢º“ GH$£⁄ +e ¥ ~*¸ƒ?u1+u2+u3+u4+···=(u1+u2)+(u3+u4)+···.WuChong-shi (§)¤'PO 7 F ]~*∞Pn=0|un| +8$~*∞Pn=0un“n ?“n $~*6N $?|un+1+un+2+···un+p|≤|un+1|+|un+2|+···+|un+p|. +6 $~*+« 6N“n $?F“n ~*$‹fl ?F›fifl –†|un|vn+⁄∞Pn=0vn +8∞Pn=0|un| ∞Pn=0un“n ?†|un|vn,⁄∞Pn=0vn +8∞Pn=0|un| ?F›‡fl –†˚A_nbc$·*ρ+8 un+1un ρ1 +~*∞Pn=0un“n un+1un ρ1 +~*∞Pn=0|un| ?•s@ Zx ¶• ;ef‚kX +„b|un+1/un| ab””^sun ab»A…f+ ‰ks@ Zx ¿ `´ ˆP|un| ˜¯˘?•ρ ˘˙•¨w ˚¸ k ˝˛ab+ˇd’Alembert Zx?Fd’Alembertfl – ]limn→∞|un+1/un|=l1,8∞Pn=0un“n ]limn→∞|un+1/un|=l1,8∞Pn=0|un| ?•d’Alembert Zx ¶• d— + = ˝˛ ^s s@ Zx ρ …»A?•d’Alembert Zx • k z ZX ˜¯8 +ˇklimn→∞|un+1/un| ˆX ∞Pn=0|un| ˜¯+ klimn→∞|un+1/un| ˆX ? ;limn→∞|un+1/un|≥1 limn→∞|un+1/un|≤1 aÆ ª ˆ+ limn→∞|un+1/un|=limn→∞|un+1/un|=limn→∞|un+1/un|.•Cauchy Zx ¶•Æ Łzd Łlimn→∞|un|1/n ˆX ØŒ;˜¯?FCauchyfl – ]limn→∞|un|1/n1,8~*∞Pn=0|un| ]limn→∞|un|1/n1,8~*∞Pn=0un ?WuChong-shi§4.1NOPO 8 º 5 1.¢FGH? u0+u1+u2+u3+u4+···=u0+u1+u2+u4+u3+u6+u8+u5+···.2.e 6 “n ~* æ ~*+ ~*·“n ?∞Xn=0un=∞Xn=0u2n+∞Xn=0u2n+1.3. “n ~* :·¡“n +Xkuk·Xlvl=Xk,lukvl. $ :6 ˆ~*u0v0+u0v1+u0v2+u0v3+···+u1v0+u1v1+u1v2+u1v3+···+u2v0+u2v1+u2v2+u2v3+···+u3v0+u3v1+u3v2+u3v3+···+···.9“n ı e ł¯ H“ +9& º? e k+l=n$‘a Hø +∞Xk=0uk·∞Xl=0vl=∞Xn=0wn,wn=nXk=0ukvn−k. ]›o œ“ GH( œ“ GH¢Pfiß$ˆq )+8 $! e Puk,Pvlł +–9# 6“n Puk,Pvl Pwnł ?WuChong-shi (§)¤'PO 9 §4.2”»|» 5 uk(z)(k=1,2,···)A!G#PND? ]noG#67z0+~*∞Pk=1uk(z0) +8$~*∞Pk=1uk(z)Az07 ? + ]∞Pk=1vk(z0) +8$~*∞Pk=1vk(z)Az07 ? ]~*∞Pk=1uk(z)A!G5 67ł +8$~*AG5 ?9 )*S(z)G5$%&)*? 5 ]no¯ N$ε0,˚A6 _zbc$N(