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WuChong-shi§2.1 20 §2.1 F znF ezF sinz,cosz,···F sinhz,coshz,···F······ !#$%&’ () *+,-./•012$%&’ -.3) *•45 ,678•45 9:)’ ;=,8WuChong-shi?@A BCD 21 FEFGzn•Hn=0,1,2,···IJzn(KLM67JNHn=1,2,···IJz=∞#OP/•Hn=−1,−2,−3,···IJznQz=0RSS67J(z=∞T67J(zn)0=nzn−1.•U V WXYZ[(n\)]^_( )Pn(z)=anzn+an−1zn−1+···+a1z+a0‘abFGR(z)=Pn(z)Qm(z),c+Pn(z)‘Qm(z)de#n\‘m\fgh/WuChong-shi§2.1 22 FiGFGezez=ex+iy=ex(cosy+isiny).j2.1(a)klmlno(b)klmlpo•q $rst $uv4wxyz{J|t) }~ /ez1·ez2=ex1+iy1·ex2+iy2=ex1·ex2·eiy1·eiy2=ex1+x2·ei(y1+y2)=e(x1+x2)+i(y1+y2)=ez1+z2.•ez(KLM67J(ez)0=ez.•ez( P Z[/ 0JHz & & t∞IJez , /;Jz=∞# ez,OP/•) ,=8 8Jc :2πiJez+2πi=ex+i(y+2π)=ex[cos(y+2π)+isin(y+2π)]=ex[cosy+isiny]=ex+iy=ez.WuChong-shi?@A BCD 23 F FGsinz,cosz,···) sinz,cosz ) Z[Jsinz=eiz−e−iz2i,cosz=eiz+e−iz2.j2.2(a) mlno(b) mlpoj2.3(a) mlno(b) mlpo•sinz,cosz(KLM67J(sinz)0=cosz,(cosz)0=−sinz.•z=∞# , XOP/•sinz‘cosz # J :2π/•sinz‘cosz, t1/isini=e−1−e12=−1.1752012···,cosi=e−1+e12=1.5430806···.WuChong-shi§2.1 24 c Jtanz,cotz,secz,cscz sinz‘coszZ[J h‘& IX Jtanz=sinzcosz,cotz=coszsinz,secz=1cosz,cscz=1sinz. ¡45Z[J¢£⁄¥J& ,ƒ§¤sh|t) }~ /F'“FGsinhz,coshz,··· sinhz,coshzT#«‹) Z[,/sinhz=ez−e−z2,coshz=ez+e−z2,tanhz=sinhzcoshz,cothz=coshzsinhz,sechz=1coshz,cschz=1sinhz.• ‘ ›fisinhz=−isiniz,coshz=cosiz,tanhz=−itaniz.fl J ,8–K U -†/• 8J sinhz,coshz, #2πi‡•· h(sinhz)0=coshz,(coshz)0=sinhz,(tanhz)0=sech2z.WuChong-shi?@A BCD 25 §2.2¶• ‚„”»…‰ ¿ `´”»ˆ˜¯˘˙¨ ˚¸/ ˝”»J˛»”»Jˇ— ”»J ‚„”»/ ˝”»…˛»”»J ”» ‚„”» Æ ª / ‚„”» ¿ ‚„”»ŁØ/FŒ_FG√z−aŒ_FGw=√zº ZXw ’ zJæ# shw2=z,w J # h √z, J ı #z,L / # z2=włw=z2,ø / ,f 8œß( ˛ „zJ ˝”»√z „/: !†f ,8Jß( d7X w=√z−a. œ hw=ρeiφ,z−a=reiθ, {=ρ=√r,φ=θ2+nπ,n=0,±1,±2,···.fl J|t Z,Xwz J= ww |% w1(z)=√reiθ/2$Ht M,n=0,±2,···w2(z)=√rei(π+θ/2)=−√reiθ/2$Htn=±1,±3,··· • w=√z−a,f 8 t ,f 8J J t z−a( ’ z) ,f 8/•f 8,œß{# w, /: Z J! w=√z−a¥ œ#|w|=p|z−a|,argw=12arg(z−a).: WXY$#f w=√z−a,8Jß( %&Z’zLM (XParg(z−a), J !)*z XZ +,-’fiIJ$%,w ,,-’fi/Hz XZ./01 +(łWuChong-shi§2.2CD 26 2 $3,01 +)’fiX 435SIJ 6†ß § 7/01 +8 9:aPz 01 +’fiX 435SJarg(z−a)TV5Jfl |%, ’/0 ;+,C1/j2.4a01 +8:=aPz’fiX 435SJarg(z−a)u2πJargw= uπJfl w V5/0 ;+,C2j2.4bfl JaP(f w=√z−a+?=@, A •HzBaPCXD435SIJ|%, V5‡• Hz BaPCXD435SIJ V5/aPE:f w=√z−a,FP/z=∞T#f w=√z−a,FP/4 ! J: –K Zf w=√z−a, ’ z G,|%HIJJ §Kz/LMNOºPQRS TUz−aºV WXYZ/H z−a, [\((w 8IJw=√z−a, T X Z Jfl w T X Z/ 0J&Z0≤arg(z−a)2π 2π≤arg(z−a)4πJss/WuChong-shi?@A BCD 27 ]w=√z−1J&Z0≤arg(z−1)2πJ^w(2),w(i),w(0),w(−i)/_argw=12arg(z−1)/fl:0≤arg(z−1)2πJ;arg(z−1) z=2=0,w(2)=1,arg(z−1) z=i=34π,w(i)=4√2e3πi/8,arg(z−1) z=0=π,w(0)=eπi/2=i,arg(z−1) z=−i=54π,w(−i)=4√2e5πi/8.•( &Z0≤arg(z−a)2π Jw, XZ[\(0≤argwπJł‘[\( aLM/(4 ,[\ Jw=√z−a ’ z Gb(XX|%,HI/j2.5aj2.5b•07&Z2π≤arg(z−a)4πJ{π≤argw2πJw2[\( aLM/w ’ z c=d,XX|%HI/•(4π≤arg(z−a)6π,6π≤arg(z−a)8π,··· ı−2π≤arg(z−a)0,−4π≤arg(z−a)−2π,···,&Z JVef)†ß45 7/fl J•ghiH&Z , ’fijkJ 2f / fi/• ’fi,ƒw J †f ,ƒw/ dF/•lw/ dF #/ Jmwf # ,ƒw/ dF,n‘/( M,o +Jf w=√z−a= w/ dFJde#w, aLM‘ aLM 0≤arg(z−a)2π †/ dFp 0≤argwπ,2π≤arg(z−a)4π †/ dFq π≤argw2π.2f rd:stw(uv w)/ dFJc& #[\z,’fi h/ 0( M, w+J #[\z xBz=aP ∞PCD/4§&Z y1 z zfi œß† (;2.5)/(zLM L{t& |z=aP}~9X +JX -3∞P/07&Z(WuChong-shi§2.2CD 28 + arg(z−a)=0J †/ dFp‡07&Z( + arg(z−a)=2πJ †/ dFq/4 w/ dF1 J x3Xw–m,wLMJłmwf w/ +,9 J #[\z,’fi h/Ut +, f , wFPJz=a‘∞Jfl Jz 6 BXwdFPCXD (4IJ IkB wdFPCXDV# ,)// dF,rdJ ı J ’fijk,&Z # X,/ 0JT &Z−π≤arg(z−a)π‘π≤arg(z−a)3π, −3π/2≤arg(z−a)π/2‘π/2≤arg(z−a)5π/2. +,9zf§f Juv # +/gh +, f ,dFPJ IiH&Z +X ( 0 ), ( ıs J&Z((XP, )ł /2f rd:/ dFJc P#Jlw/ dF #/ Jfl «,/ o ,678// ,dFP#OPJ |%t Xw/ dF/(FP JT b(Xwg|%tXw/ dF, */4§rd, P#=XZ, [8/fl: [\ , ’fijkJ 6 o X5 ) , /: 4w PJ X§–K Z ’ |%HI,Kz# S FGw z0º J ¡¢£¡zº⁄¥WXƒ§/Hz 4 +,-’fiIJ wT= ,-’fi/} w=√z−1: /&Zw(2)=1Jo z C1 C2,-’fi35PIJ w /C1‘C2#z=1:¤' 1:a“, a¤ ‘ a¤ /«~JHz C1‹›3z=0IJΔarg(z−1)=πJ;Δargw=12Δarg(z−1)=π2,w(0)=eiπ/2=i.Hz C2‹›3z=0IJΔarg(z−1)=−πJ;Δargw=12Δarg(z−1)=−π2,w(0)=e−iπ/2=−i. 4§KzJz,’fifi+ fl[\Jfl |Xw/ dFx›3 Xw/ dF/(y1; J4$Ht2 w ,zLM–† J‡XwM, + (arg(z−1)=2π)‘‡·wM, + (arg(z−1)=2π)$,J‡XwM, + (arg(z−1)=0)‘‡·wM, + (arg(z−1)=4π)$,/4 ·¶RiemannM(;2.6)/|t w=√z−1 √z−a J·¶RiemannM ,zP‘wLM ,P#XX|%,/WuChong-shi?@A BCD 29 j2.6•‚mlw=√z−a„Riemann”|t ) X5, h J 0w=3√z−a 3p(z−a)(z−b)JssJT »… o /g#‰h ¿ †f ,K`FPJN Z +,9z/(X´ˆ˜ J + 6 ¯X˘JT XZ‰h X˘ +K`FP ,† /2.˙GFGlnz| w=lnz,Z[#ew=zJT # J Z ’ z,Xw Jæ# ew=z,;=w ¨E:| w=lnz, / # w=ez,ø / w=u+ivJz=reiθJ x3eu·eiv=reiθ/;u=lnr=ln|z|,v=θ+2nπ(n=0,±1,±2,···).!J | w=lnz¥ œ#:w=lnz=ln|z|+i(θ+2nπ)=ln|z|+iargz.j2.7•‚mlw=lnzw=lnz˚¸ ˝˛z‚ˇ— •˛w‚fl J| w=lnzT#f ,Jcf 8, # z ,f 8Jf 8,œß{# w, `/|%lXwz J= fww J ,&`$ J `$ 2π,m /;2.7 † | w=lnz,#¿;/WuChong-shi§2.2CD 30 w=lnz,FP#z=0‘∞/9 +,†0 ∞J&Z +X ,argz Jł x3w=lnz,/ dF/w=lnz= fw