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WuChong-shi ( ) 1 ( )§7.1 !Laurent#$%&’()*+,-./0Taylor123456789:;,/=12?@A(BC6DEFLaurent12BGH7.1(Laurent)I’(f(z),JbKLMNOPQRR1≤|z−b|≤R2STU-.4VWXORYNZ[z/4f(z)+J\@A(12Kf(z)=∞Xn=−∞an(z−b)n,R1|z−b|R2,]^an=12πiICf(ζ)(ζ−b)n+1dζ,C_ORY‘YL%aNZb%cdefg(hi7.1)Bj7.1Laurentklm:ORNY3nopqrKC1sC24VtuvwxQRNCauchyypz{45f(z)=12πiIC2f(ζ)ζ−zdζ−12πiIC1f(ζ)ζ−zdζ.WXC2SNyp4+J|}~\J N 412πiIC2f(ζ)ζ−zdζ=∞Xn=0an(z−b)n,|z−b|R2,an=12πiIC2f(ζ)(ζ−b)n+1dζ.WXC1SNyp−12πiIC1f(ζ)ζ−zdζ=12πiIC1f(ζ)(z−b)−(ζ−b)dζ=12πiIC1f(ζ)z−b∞Xk=0ζ−bz−bkdζ=∞Xk=0(z−b)−k−1·12πiIC1f(ζ)(ζ−b)kdζ. −k−1=n,k=−(n+1)4V−12πiIC1f(ζ)ζ−zdζ=−∞Xn=−1(z−b)n·12πiIC1f(ζ)(ζ−b)n+1dζ=−∞Xn=−1an(z−b)n,|z−b|R1,]^an=12πiIC1f(ζ)(ζ−b)n+1dζ.WuChong-shi§7.1 Laurent 2 pe 4D5f(z)=∞Xn=−∞an(z−b)n,R1|z−b|R2,an=12πiICf(ζ)(ζ−b)n+1dζ.C& K’(f(z),ORR1|z−b|R2YNLaurent124]^NA( KLaurentA(B an CB( ¡¢)£⁄¥FLaurent12Ncƒ§+J¤'Kf(z),R1|z−b|R2YTU-.BFWXLaurent12 “4«((‹›_fi@flN«()an6=1n!f(n)(b).Ff(z),YLC1^ -.B%–“ 4,C1S_5/NB†Xb/4+‡_f(z)N/4§+‡_f(z)N-./B•· b/_C2YN %/4VC1+J¶•‚„4”»…‰D ?0|z−b|RBC6DEFf(z),¿ /bN`RYNLaurent12B•3LC2N´ˆ§+JK∞4˜†,∞/§”»BFLaurent12¯5fi@fl4˘5˙@flB•fi@fl,LC2Y(|z−b|R2)¨W”»4,C2YNZb%&dQR^% ”»4 KLaurentA(NfiV p˚•˙@fl,LC13(|z−b|R1)¨W”»4,C13NZb%&dQR^% ”»4 KLaurentA(N¸9 pB• pe 4D ?LaurentA(4,ORR1|z−b|R2Y¨W”»4,ORYNZb%&dQR^% ”»B•˝R1=064LaurentA(N¸9 pD˛ˇ— *f(z),z=b/N BWuChong-shi ( ) 3 FLaurent If(z),ORR1|z−b|R2Y5 &LaurentA(4f(z)=∞Xn=−∞an(z−b)n=∞Xn=−∞a0n(z−b)n. J(z−b)−k−14 ORY‘YL%aNZ%‰ Cyp(C &A(,‰ S % ”»4Æ +Jªflyp)4V XIC(z−b)n−k−1dz=2πiδnk, 5ak=a0kBÆKkZb4 5ak=a0k,k=0,±1,±2,···.‹ ELaurent12N % B ŁØŒLaurentº æ 4ı łø (œß ø )BWuChong-shi§7.2Laurent 4 §7.2Laurent ! Laurent 4 ( 4 )B 4! Laurent #$%B&’( )*+,-. Laurent / 40 412 $%43 45 ) 6,-.785f(z) 9: 4;= + f(z) Laurent BTaylor ? $%4 @!A BC4D E F Laurent BG7.1H1z(z−1),0|z|1Ys|z|1YN12{BI1z(z−1),0|z|1YN12P{%J_∞Pn=−∞anznBKJ1z(z−1)=−1z11−z=−1z∞Xn=0zn=−∞Xn=0zn−1=−∞Xn=−1zn,0|z|1.§+J\ pp{NLM¥1z(z−1)=−1z−11−z=−1z−∞Xn=0zn=−∞Xn=−1zn.1z(z−1),|z|1YNLaurent12P{§_∞Pn=−∞anzn41z(z−1)=1z211−1z=1z2∞Xn=01zn=−∞Xn=−2zn,|z|1. NOP54Q 6( )1Q R-. Laurent S1TQ B1/z(z−1))0|z|1. Laurent 3! 6U9V4W)|z|1. Laurent !XYZ6U9V4[\!]9VBG7.2\^J«(MHcotz,z=0`RYNLaurent12BI^J«(M_‡\X5•&˙@fl(fi@fl)N‘PBcotz=∞Xn=−1b2n+1z2n+1.( 3! 6U9V4 6 a)5.6bc2B)cosz=sinz∞Xn=−1b2n+1z2n+1,∞Xn=0(−)n(2n)!z2n=∞Xk=0(−)k(2k+1)!z2k+1∞Xl=0b2l−1z2l−1=∞Xk=0∞Xl=0(−)k(2k+1)!b2l−1z2(k+l)=∞Xn=0nXl=0(−)n−l(2n−2l+1)!b2l−1#z2n.WuChong-shi ( ) 5 dEFefg«nXl=0(−)l(2n−2l+1)!b2l−1=1(2n)!.ªhH-4‹En=0:b−1=1;n=1:13!b−1−11!b1=12!,b1=−13;n=2:15!b−1−13!b1+11!b3=14!,b3=−145;n=3:17!b−1−15!b1+13!b3−11!b5=16!,b5=−2945;...KJcotz=1z−13z−145z3−2945z5−···.tucotzN/pi4+jkdA(N”»…‰K0|z|πBlm7+Jn\A()MBcotz=1tanz=1z+13z3+215z5+17315z7+···=1z11+13z2+215z4+17315z6+···=1zh1−13z2+215z4+17315z6+···+13z2+215z4+17315z6+···2−13z2+215z4+17315z6+···3+13z2+215z4+17315z6+···4−+···i=1zh1−13z2+−215+19z4+−17315+2×13×215−127z6+···i=1z1−13z2−145z4−2945z6−···=1z−13z−145z3−2945z5+···.WuChong-shi§7.2Laurent 6 opq Laurent G7.3H’(lnz−2z−1,1|z|2r2|z|∞YN@A(12BIlm^sJN12QR_OPQR4KJ4· ‡0@A(12Nt4EFN%J_LaurentA(B’(lnz−2z−15 &u/¥z=1sz=24 ,OR1|z|2Y +‡0Laurent12B,OR2|z|∞Y4’(lnz−2z−1_TU-.N4vwxyz{|JTUpu}4L+0Laurent12B~·4 |J, gS arg(z−2)−arg(z−1)=π4Vlnz−2z−1 z=∞=0.X_5lnz−2z−1=ln1−2/z1−1/z=ln1−2z−ln1−1z=−2z−122z2−132z3−···#−−1z−121z2−131z3−···#=−1z−321z2−731z3−···−2n−1n1zn−···.G7.4Hexpz2t−1t,0|t|∞YNLaurent12BI\A( MBÆKezt/2=∞Xk=0z2ktkk!,|t|∞,e−z/2t=∞Xl=0z2l(−)ll!1tl, 1t ∞‹|t|0,KJexpz2t−1t=∞Xk=0z2ktkk!∞Xl=0z2l(−)ll!1tl=∞Xk=0∞Xl=0(−)lk!l!z2k+ltk−l=∞Xn=0h∞Xl=0(−)ll!(l+n)!z22l+nitn+−∞Xn=−1h∞Xl=−n(−)ll!(l+n)!z22l+nitn=∞Xn=−∞Jn(z)tn,WuChong-shi ( ) 7 ]^Jn(z)=∞Xl=0(−)ll!(l+n)!z22l+n,n=0,1,2,···;∞Xl=−n(−)ll!(l+n)!z22l+n,n=−1,−2,−3,··· Kn Bessel’(B· ¶ /_’(f(z)N/4 ,¶ /N`RYTU-.Nt4V+:f(z),∞/N`RY0Laurent12(56D T “?,∞/0Laurent12)BF f(z))∞ -.(∞ ) 4= t=1/z4( f(1/t))t=0 -.(t=0 ) 40Wf1t=∞Xn=−∞antn,0|t|r,f(z)=∞Xn=−∞anz−n,1r|z|∞. 78 P ∞ 6,-BFf(1/t) Laurent: ]9V( V) ] 4U9V B0 4 E¡4NO f(z))z=∞ -. Laurent: 4z U9V¢ ] 4W]9V¢ B]9V £⁄ ( f(z))∞ ¥ƒ§BS¤N~4s~6N'“«‘PJr~74§‹+J›?_,∞/`RYLaurent12BWuChong-shi§7.3fifl –†‡ 8 §7.3· !¶•‚„G”If(z)KTU’((»…U’(N%&TUpu)4b/_;N/B· ,b/‰,%&`R4, `RY()b/3)4f(z)¿¿+ 4V bKf(z)N¿ /B`¿ /N~´BWX’(1/sin(1/z)4 41/z=nπ4‹z=1/nπ4n=0,±1,±2,···_;N/BVz=0_Cˆ/N˜/(¯•/)¥,z=0NZb%&`R^4˘‰,¶ …&/4 z=0_`¿ /B· z=b_TU’(f(z)N¿ /4V%J‰,%&OR0|z−b|R4, ORY4f(z)+J12?LaurentA(4f(z)=∞Xn=−∞an(z−b)n.C6+‡˙¨ «‘˚¥FA(12{ ¸˙@fl¥b/ Kf(z)N+ /Bz=0D_’(sinzz=∞Xn=0(−)n(2n+1)!z2n,|z|∞s1z−cotz=13z+145z3+2945z5+···,|z|πN+ /BFA(12{_¸5•&˙@fl¥b/ Kf(z)N¯/BFA(12{¸5¶ …&˙@fl¥b/ Kf(z)Nl /B˝¤pq£⁄’(, «/¿N˛KBߡ— X,+ /¿4A(12{^ ¸˙@fl4 A( __,ORY”»4 ,ORN^M4‹+ /z=b¿§_”»NBwwC6N”»QR_%&L4LM,+ /z=b4A(,”»LYNZ%dQR^% ”»4wws’(w 4limz→bf(z)=limz→b∞Xn=0an(z−b)n=a0.WuChong-shi ( ) 9 ’(,+