姚国武复变作业答案

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z=rei,* 
f
z=pr1+jcosjjsinjrei,Ϗ
z!0,r!0,¤±=1ž,،,Ù{œ¹þŒöS2.ÁE¼êw=f(z),¦§3CþëY§3z=0Œ§30:?ۘ%Sþkf(z))Û:†Û:‰Y.~fµw=f(z)=8:z2;Imz0;z2;Imz0:öS3.n1½ê§ ¼êw=f(z)=8:znz;z6=0;0;z=0:3z=0?ëY59Œ5J«.Ó1˜öS¥{Œ3z=0?µn2žŒ,n=2žëY،,n=1žØëYöS4.ÖþP67:10(4)(5)Proof.(4).argf(z)2(;])eif(z)𢊧A^10(1)y²L(Øѧeif(z)const,lf(z)const:(5).a;b;c؏0)a2+b26=0,au+bv=c)aux+bvx0;auy+bvy0)(aux+bvx)2+(auy+bvy)20)(a2+b2)(u2x+u2y)=(a2+b2)(v2x+v2y)0)u2x+u2y0;v2x+v2y0)ux=vy0;uy=vx0)uconst;vconst:öS5.é?¿‰½z2C,¦y:limn!1nln(1+zn)=z:1J«.nln(1+zn)=nlnp(1+xn)2+(yn)2+inarctanyn1+xn!x+iy;n!1.öS6.Jordan4­‚CŒ¤«¡ÈS,Á¦eÈ©ŠµICzdz:‰Yµ2iS.öS7.Öþ1nÙSK:21.J«.µ5¿ICf0(z)zz0dzICf(z)(zz0)2dz=IC[f(z)zz0]0dz=0;Ϗf(z)zz0Š[f(z)zz0]03Dþ¼ê§ŠâÈ©´»Ã'5n‡¿‡^‡=öS8.0,a2C;jaj6=§¦y:I=Ijzj=jdzjjzaj2=8:22jaj2;jaj;2jaj22;jaj:Proof.-z=rei,3C:jzj=,þ,jzj2=zz=2:jdzj=ds=d=idzz.u´I=Ijzj=jdzjjzaj2=ICidz(za)(za)z=iICdz(za)(2az)Case1.jaj.I=iIC12azzadz=i
2i12azjz=a=22jaj2;Case2.jaj.aqŒöS9.f(z)3D:jzz0jRþ)Û§3DþëY§y²¡È²þŠúªµf(z0)=1R2ZZDf(z)dxdy:J«.f(z0)=12Z20f(z0+rei)d)ZR0f(z0)rdr=12ZR0Z20f(z0+rei)rddr)f(z0)=1R2ZZDf(z)dxdy:öS10.f(z)3k.«Dþ)Û§3DþëY§…??؏0§y²µeé8z2@Dkjf(z)jM(M,‡~ê¤,K3Dþ§f(z)Mei,é,‡2[0;2).J«.éf(z)91f(z)©OA^Œn,KØJwjf(z)jŒ†þ3.þ§Ïjf(z)jM,lf(z)const:2öS11.Pn(z)n(1)gõ‘ª§…p‘Xê1§¦yµM=maxjzj=1jPn(z)j1:Proof.y{1.|^Cauchyت,5¿n!=jP(n)(0)jn!M1n,=y{2.-w=1z;KPn(z)zn=zn+a1zn1+


+anzn=wn+a1wn1+


+an=Q(w),5¿M=maxjzj=1jPn(z)j=maxjwj=1jQ(w)j:ÏkMjQ(0)j=1.öS12.f(z)¼ê§…®f(0)=A,f0(0)=B,鉽r09ên§Á¦µ(1)I1=R20fn(rei)cos22d,(2)I2=R20fn(rei)sin22d.(d?ng)J«.z=rei;d=dziz;cos22=1+cos2=(1+z+z2r)=2=2rz+z2+r24rz;sin22=1cos2=(1z+z2r)=2=2rzz2r24rz,ù§I1=Z20fn(rei)cos22d=Ijzj=rfn(z)2rz+z2+r24rzdziz=Ijzj=rfn(z)2rz+r24riz2dz=2i[fn(0)2i+r4i[fn(z)]0z=0]=2i[An2i+rnAn1B4i]=[An+rnAn1B2]:aqŒµI2=Z20fn(rei)sin22d=[AnrnAn1B2]:öS13.f(z)3jzj2S)Û§®f(0)=A;f0(0)=B;f00(0)=C,Á¦È©I=12iIjzj=1(2z)f(z)dz:J«.kz¤ëêÈ©§z=ei;dz=ieid§,2=¤CauchyÈ©úª½pêúªOŽ,Œ¦Ijzj=1f(z)dz=2iB;Ijzj=1zf(z)dz=iC:öS14.f(z)3jzj1þ)ۅk.§‰½z0÷vjz0j1±9ên,¦yµIjzj=2f(z)dz(zz0)n+1=0:Proof.3jzj1þ,jf(z)jM.é¤k¿©Œr,dEÜ4´½nkIjzj=2f(z)dz(zz0)n+1=Ijzz0j=rf(z)dz(zz0)n+1:3jIjzj=2f(z)dz(zz0)n+1j=jIjzz0j=rf(z)dz(zz0)n+1jIjzz0j=rjf(z)jjdzjjzz0jn+1Ijzz0j=rMdsrn+1=2Mrn:-r!+1,K·‚I‡(ØN‘,ÿ䴚K‰Yµ(X)1!limn!+1zn=0…zn6=0(8n2N)§ef(z)3zn(8n2N)?،§Kf(z)3z=0?Ø)Û()2!limn!+1zn=0…zn6=0(8n2N)§ef(z)3zn(8n2N)?)Û§…f(z)3z=0:ëYŒ§Kf(z)3z=0?)Û()3!z0(2C)f(z)Û:§Kf(z)3z0?،()4!f(z)3üëÏ«D¥÷?ۘ^{ü4­‚È©0§Kf(z)3D¥)Û(X)5!f(z)3«D(C)þ)ۅarg(f2(z))ð2,Kf(z)3Dþð~ê4