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WuChong-shi ( ) 1 ( )§5.1 ∞Xn=0cn(z−a)n=c0+c1(z−a)+c2(z−a)2+···+cn(z−a)n+···.! #$%&’ ( )*+,) - # ./05.1Abel(12)/034 ∞Pn=0cn(z−a)n567z089:5;a7 =|z0−a| ? @AB89C5|z−a|≤r(r|z0−a|)DE89.FG ∞Pn=0cn(z−a)n5z089HIJKLMNOlimn→∞cn(z0−a)n=0.GPQ5R qS|cn(z0−a)n|q.T;|cn(z−a)n|=|cn(z0−a)n|· z−az0−a nq z−az0−a n.G z−az0−a 1U|z−a||z0−a|V∞Pn=0 z−az0−a n89H∞Xn=0cn(z−a)n5|z−a||z0−a|@AB89.CW|z−a|≤r|z0−a|V|cn(z−a)n|≤qrn|z0−a|n, ∞Pn=0rn|z0−a|n89H∞Xn=0cn(z−a)n5|z−a|≤r(r|z0−a|)DE89.XYZ ∞Pn=0cn(z−a)n567z1[\:5|z−a|=|z1−a|]^^[\.F-_‘a.Z ∞Pn=0cn(z−a)n5|z−a|=|z1−a|]67z289:bAbelIc Ld5|z−a|=|z2−a|(|z2−a||z1−a|)@89efghi.H ∞Pn=0cn(z−a)n5|z−a|=|z1−a|]^^[\.WuChong-shi§5.1j 2 klmnklopqrs 5ztuv wx7y Mz89Mz[\.GPBr {|}~ ! 5ztuv 7 895 ] 7 [\.! 897e[\7 Q5s .F AbelIc!s I .!s} klm.F89 = z=a7.F89 ? klop.89? ; 0.!V89 s7. z=a7] 5 tu^^[\.89?( ; ∞.!V89} tu. 5 tu89 ∞7 I 7. 89? a - s 1. Cauchy aWlimn→∞|cn(z−a)n|1/n1U|z−a|1limn→∞|cn|1/nV AB89 Wlimn→∞|cn(z−a)n|1/n1U|z−a|1limn→∞|cn|1/nV [\.GP ∞Pn=0cn(z−a)n 89? R=1limn→∞|cn|1/n=limn→∞ 1cn 1/n.2. d’Alembert a34limn→∞ cn+1(z−a)n+1cn(z−a)n =|z−a|limn→∞ cn+1cn Q5:Wlimn→∞ cn+1(z−a)n+1cn(z−a)n 1U|z−a|limn→∞ cncn+1 V AB89 Wlimn→∞ cn+1(z−a)n+1cn(z−a)n 1U|z−a|limn→∞ cncn+1 V [\.GP ∞Pn=0cn(z−a)n 89? R=limn→∞ cncn+1 . ¡¢£⁄¥ƒ§¤'“«‹›fi.Cauchy¤'fl –†‡§·d’Alembert¤' fl«¶•§(‚¢„”limn→∞|cn/cn+1|»…).‰ ¿ `´ˆ˜¯¯˘˙¨ ˚¸.WuChong-shi ( ) 3 qr ∞Pn=0cn(z−a)n ˝ z ˛ˇ AbelIc— 5 89@w DE89GP 4.2 589@ s˛ˇ ( | 589@˛ˇ) ;B Zzz0∞Xn=0cn(z−a)ndz=∞Xn=0cnZzz0(z−a)ndz=∞Xn=0cnn+1(z−a)n+1−(z0−a)n+1,ddz∞Xn=0cn(z−a)n=∞Xn=0cnd(z−a)ndz=∞Xn=0cn+1(n+1)(z−a)n.F 589v 89Æ • ;^^89• ;^^[\•( ;5 7895 7[\.1+z+z2+···+zn+···5|z|=1v^^[\ z1+z22+z33+···+znn+···5|z|=1v z=1]ª89C5z=17[\ z21·2+z32·3+z43·4+···+znn(n−1)+···5|z|=1v^^89. # 89vy I 7. US5 7 d Ł 89 (U ØI Œ).g ∞Xn=0cn(z−a)n589@89ºf(z)34 589 v67z0(89 S(z0): 1 /0( ‘)— Wzq89@ærz0V M 5;z0 ı7, 2φπ łø@(œß5.1)f(z)}IærS(z0). 5.1 WuChong-shi§5.2 4 §5.2 §5.3D ˛ˇÆ ( ;-{ _ ˛ˇÆ./05.2g1.f(t,z)flt z§!#taz∈G2.$%&’t≥af(t,z)flG(§˚)*+#3.,-Z∞af(t,z)dt…G(./£⁄0∀ε0∃T(ε) T2T1T(ε)˜« ZT2T1f(t,z)dt ε,:F(z)=Z∞af(t,z)dt5G@ ˛ˇ 1F0(z)=Z∞a∂f(t,z)∂zdt.Fw2s3456{an}a0=aa1a2a3···anan+1···,limn→∞an=∞.7un(z)=Zan+1anf(t,z)dt: 3.7 r I ˛ˇÆ Ic 8un(z) G@ 9Œ˛ˇ .:G F(z)=∞Xn=0un(z)5GvE89H WeierstrassIc8F(z)=∞Xn=0un(z)=Z∞af(t,z)dt5G@˛ˇ1F0(z)=∞Xn=0u0n(z)=Z∞a∂f(t,z)∂zdt.Br ; ( ;=^c.5?-!sIcV@M ABC ( ; ) DE89. - a EF»…#φ(t)GH|f(t,z)|φ(t)z∈G·IZ∞aφ(t)dt£⁄ Z∞af(t,z)dt…G(J$·I./£⁄.K BC sLMNu F(z)=Z∞0e−t2cos2ztdt.(5.1)!s D O PdJKIc Q sNOC1G BrR z=x+iy |cos2zt|=qcosh22yt−cos22xt≤cosh2|yt|≤e2|yt|.WuChong-shi ( ) 5 T;Brztuv wxs v|Imz|y0r e−t2cos2zt e−t2+2y0t,C Z∞0e−t2+2y0tdt89T; BC (5.1)E89GP!s K z 5ztuv wxs @˛ˇ.STU} F0(z)=−Z∞0e−t22tsin2ztdt=e−t2sin2zt ∞0−2zZ∞0e−t2cos2ztdt=−2zF(z).˛!sV WX} ;YºF(z)=Ce−z2 D C C=F(0)=Z∞0e−t2dt=12√π,! )Z}YºZ∞0e−t2cos2ztdt=12√πe−z2.WuChong-shi§5.3 [ Taylor\] 6 §5.3 ^ Taylor_‘.¡a#…b§£⁄cdef.¡*+#.E’g.¡*+#fh†ai# /05.1(Taylor)g f(z)5;a = C@jCv˛ˇ:Br@ wkz7f(z) - lm ( |f(z) 5a7lm )f(z)=∞Xn=0an(z−a)n, Dan=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!,C2nVoWpq.F Cauchy r’BrC@wx7z f(z)=12πiICf(ζ)ζ−zdζ. 1ζ−z=1(ζ−a)−(z−a)=1ζ−a∞Xn=0z−aζ−an.P 5 z−aζ−a ≤r1 DE89GP ; f(z)=12πiIC∞Xn=0(z−a)n(ζ−a)n+1#f(ζ)dζ=∞Xn=012πiICf(ζ)(ζ−a)n+1dζ(z−a)n=∞Xn=0an(z−a)n,an=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!.|s 1.Ic NO ;tu Mf(z)5C@˛ˇU . ˜$%vw§zxyzza{c|}.cC0gz~ …cd.f(z)…C0d C0(fl*+§.q WuChong-shi ( ) 7 2.! Taylorlm &’ D Taylorr’ NO .F5 Df(x) wk Q5 K; ‘Taylorr’Q5( Taylorr’89).F5R D˛ˇ M ( Q5)}K; ‘Taylor 89.3.kl f(z) 7 ¡I Taylor 89?.gb f(z) ¢a7)£ 7:⁄|{89?R=|b−a|.f(z)…c|z−a||b−a|d¥¥*+f(z)yz…cdƒ§{Taylori#(¤ 'Taylori#…c|z−a||b−a|d£⁄). “fl'f(z)§Taylori#£⁄¥ƒ«‹%|b−a|.£⁄¥ƒ.›fi«`fl%|b−a|. bfi“~–…£⁄cd†·ai#…£⁄cd¥¥*+‡bfi{·fi§ ¶•‚(„”bfifly»·fi…5.5‰).11+z2=∞Xn=0(−)nz2n,|z|1. 7z=±i}¡I Taylor 89?R=|±i|=1.C5 łø@Taylor 89?e Æ ¿ }`; .11+x2=∞Xn=0(−)nx2n,−1x1,}`;c˛89? k 1G 1/(1+x2)5´s ˆv˝ ˜¯ ,˘1wk ˝ Q5 ˙4.Taylor¨ ˚¸2 ˝Is5C@˛ˇ :˛ Taylorlm ˇ Ulm an ØI .F—I sTaylor 5C@˝89º s˛ˇ f(z)f(z)=a0+a1(z−a)+a2(z−a)2+···+an(z−a)n+···=a00+a01(z−a)+a02(z−a)2+···+a0n(z−a)n+···.2 z→a:qr 5C@ w DE89H a0=a00. V 2 z→a:Ya1=a01.3P ¯U ‘Yan=a0n,n=0,1,2,···.Taylorlm ˇÆ— WuChong-shi§5.3 [ Taylor\] 8 F - zWaYº f(z)5 s@ Taylorlm ˇ .GP IM- aIlm .F345 7lm sTaylor : ; .•L 5 7lm sTaylor ; .• s 5 7lmYº sTaylor US r 89 ( Ł lm .