复变函数复习资料 葵花宝典

1373841Rouch¶e454752,(complexanalysis),(analysecomplexe)(theoriedefonctionsvariablescomplexes)(Funktionentheorie).,Borel1952:Thesweepingdevelopmentofmathematicsduringthelasttwocenturiesisdueinlargepartduetotheintroductionofcomplexnumbers;paradoxi-cally,thisisbasedontheseeminglyabsurdnotionthattherearenumberswhosesquaresarenegatives(:;,).()1),.2)²Goldbach:6.(1+1).(1+2)..²Riemann:Riemannzeta-³(z)=1Xn=11nz;z1:³(z)C,³(z)z=11.³(¡2n)=0;n=1;2;:::;³.Riemann:³z=12..3):.4):().15).6).7).²Joukovskii,.f(z)=12µz+1z¶Joukovskii.²,.253.:,,:,Cauchy,,,,,,,,:,,,@@z,@@z:,(Riemann),1,:1)...:(),,,()Cauchy-Riemann:C-R,,,C-R@@z,@@z.:Jacobi(1),:,Cauchy,(WeierstrassM),Abel,,,,Taylor:,,,1-41)5.:.3.2:.2).CauchyCauchyCauchyCauchylifeblood.:Cauchy:Cauchy,Cauchy,,Morera,:1,(2),,,,,Cauchy:Cauchy,Cauchy,(2),Liouville,Liouville,,,,Cauchy,.,,.:D'(u;v)=u2+v2D(),D().,.LaurentLaurent:Laurent,Laurent,Laurent,1Laurent:,,,,,1:,Mittag-Le²er,C,C453.1)z0f,z0g=1f.,.2)Laurent.3)C.C().CC.4).:(1),,,1.Rouch¶e:,,Rouch¶e,Rouch¶e:1-6151)1,z0Laurent1=(z¡z0),1Laurent1=z.,f(z)»a(z¡z0)m(z!z0);m1,a6=0,Res(f;z0)=1(m¡1)!limz!z0(f(z)(z¡z0)m)(m¡1):,m=1,Res(f;z0)=a:f(z)»azm(z!1);553m1,a6=0,Res(f;1)=8:¡a;m=1;0;m1:2)2..3)Rouch¶e.4).,..1):,C-R.2).Laurent.3)..6531.:Bz+Bz+C=0z1;z2Bz1+Bz2+C=0:(1).BBz+Bz+C=0.w1,w2w1w2+w1w2=0.=)Bz+Bz+C=0z1;z2,Bz1+z22+Bz1+z22+C=0;(2)i(z1¡z2)B,i(z1¡z2)B+i(z1¡z2)B=0:(3)Bz1¡z22+Bz2¡z12=0:(4)(2)(4)(1).(=(1),Bz2+Bz1+C=0:(5)(1)(5)2,(2).z1+z22Bz+Bz+C=0.(1)(5)B(z1¡z2)+B(z2¡z1)=0:(3),i(z1¡z2)B.z1¡z2Bz+Bz+C=0.,Bz+Bz+C=0z1;z2.2.S½C.S0S,S.S0,S=S[S0.7.S0,C¡S0.,z02C¡S0,,±0,D0(z0;±)\S=;.z2D0(z0;±),0,D(z;)½D0(z0;±),D(z;)\S=;,D0(z;)\S=;.z=2S0,D(z0;±)\S0=;.D(z0;±)½C¡S0.C¡S0.S=S[S0.S½S[S0.,z2S,z=2S[S0,,0,D(z;)\S=;,S½C¡D(z;).,S½C¡D(z;),z2S.S½S[S0.S[S0½S.,z2S[S0,z=2S,F,F¾S,z=2F,z2C¡F.C¡F,0,D(z;)½C¡F½C¡S.z2S[S0,D(z;)\S6=;..S½S[S0.,S=S[S0.3.F,F..fD(z;1):z2FgF.F,z1;:::;zn2F,F½nSk=1D(zk;1).F.F.F,z02F0¡F.Uk=½z2C:jz¡z0j1k¾;k=1;2;::::fUk:k=1;2;:::gF.F,n1,F½n[k=1Uk=Un=½z2C:jz¡z0j1n¾=C¡Dµz0;1n¶½C¡Dµz0;1n¶:F\Dµz0;1n¶=;:,z02F0,D¡z0;1n¢\F6=;..,F.4.S½C,diamS=diamS..S½S,diamS6diamS.diamS=+1,diamS=diamS=+1.diamS+1.0,853,z1;z22S,jz1¡z2jdiamS¡:z1;z22S,w1;w22S,jw1¡z1j;jw2¡z2j:jz1¡z2j6jz1¡w1j+jw1¡w2j+jw2¡z2jdiamS+2:diamSdiamS+3:0,diamS6diamS.,diamS=diamS..,diamS+1(S),S(),z1;z22S,jz1¡z2j=diamS:5.­½C,:­..­.,z02­,L(z0)z0,L(z0)=[nU:U;z02U½­o:z2L(z0)½­,­,0,D(z;)½­.D(z;)½­(),L(z0)z,D(z;)½L(z0).L(z0).­,.,­.C.,,­C,­.9536.S,f:S!C.z02S,r0,f(D(z0;r)\S)=ff(z0)g,fS..fS.z02S,S1=fz2S:f(z)=f(z0)g;S2=fz2S:f(z)6=f(z0)g:z02S1,S16=;.z2S1,,r(z)0,f(D(z;r(z))\S)=ff(z)g=ff(z0)g:O1=[z2S1D(z;r(z));O1,z02S1½O1.z2S2,f(z)6=f(z0),,(z)0,f(z0)=2f(D(z;(z)\S):O2=[z2S2D(z;(z));O2,S2½O2.S=S1[S2=(S\O1)[(S\O1)½O1[O2;S\O16=;;(S\O1)\(S\O2)=;:S,S\O2=;,S\O1=S.S=S1,S2=;.fS.7.1)KC,FC.K\P=;,dist(K;F)0.2)D,S½D,dist(S;@D)0..1)z2K,FK\F=;,0D(z;)\F=;.F½C¡D(z;),dist(z;F)0.KnD(z;dist(z;F)2):z2Ko.K,z1;:::;zn2K,K½n[k=1Dµzk;dist(zk;F)2¶:1053=min16k6ndist(zk;F)0;z2K;w2F,zk2fz1;:::;zng,jz¡zkjdist(zk;F)2,jz¡wjjw¡zkj¡jzk¡zjdist(zk;F)¡dist(zk;F)22;dist(K;F)20.2)SzD,D,S\@D=;.S,,@D,1)dist(S;@D)0.8.S.T½S,TST.T½SS,S¡TS.:SS()..T½SSCFT=F\S.=)T½S,,T0\S½T.T=T\S½T\S=(T0[T)\S=(T0\S)[(T\S)½T[T=T;T\S=T.F=T,FC,F\S=T.(=CF,T=F\S,T\S=F\S\S½(F\S)\S=(F\S)\S=F\S=T:T0\S½(T0[T)\S=T\S½T:T½SS.T½SSCGT=1153G\S.,,TS()S¡TS()CFS¡T=F\S()CFT=S¡(S¡T)=(C¡F)\S()CGT=G\S:S()O1;O2S½O1[O2;O1\S6=;;O2\S6=;;(O1\S)\(O2\S)=;()O1;O2S=(O1\S)[(O2\S);O1\S6=;;O2\S6=;;(O1\S)\(O2\S)=;()SS1;S2S=S1[S2:,S1;S2S,S=S1[S2,S1;S2S,,S()SS1;S2S=S1[S2:9.z1;z2;z3,:z1;z2;z3z1+z2+z3=0..=)z1;z2;z3,,0,z1+z2+z33=0.z1+z2+z3=0.(=jz1j=jz2j=jz3j=1,z1+z2+z3=0,1=jz1j2=jz2+z3j2=jz2j2+2(z2z3)+jz3j2=2+2(z2z3):(z2z3)=¡12.jz2¡z3j2=jz2j2¡2(z2z3)+jz3j2=3:jz2¡z3j=p3.jz3¡z1j=jz1¡z2j=p3.z1;z2;z3.125310.f(z)C,f(z)C.1)f(z)=u(x;y)+iv(x;y),f(z)=u(x;¡y)¡iv(x;¡y)=u1(x;y)+iv1(x;y):u1;v1R2.fC-R,@u1@x=ux(x;¡y)=vy(x;¡y)=@v1@y;@v1@x=¡vx(x;¡y)=uy(x;¡y)=¡u1@y:f(z)C.2)f,z2C,f(z+¢z)¡f(z)=f0(z)¢z+o(¢z)(¢z!0):f(z+¢z)¡f(z)=f0(z)¢z+o(¢z)(¢z!0):,f(z+¢z)¡f(z)=f0(z)¢z+o(¢z)(¢z!0):(f(z))0(f(z))0=f0(z);z2C:f(z)C.11.1)f(z)f(z)­,f(z)­.2)f(z)­,jf(z)j,f(z)­.1)f(z),z2­,f(z+h)¡f(z)=f0(z)h+o(h)(h!0);f(z+h)¡f(z)=f0(z)¢h+o(h)(h!0):f(z),limh!0f(z+h)¡f(z)h=limh!0f0(z)hh1353.limh!0hh,f0(z)=0,f0(z)=0.f­.2),C0,z2­,f(z)f(z)=C2.f(z)=u(x;y)+iv(x;y),u2(x;y)+v2(x;y)=C2;8z2­:C=0,.C0,,ux(x;y)u(x;y)+vx(x;y)v(x;y)=0;uy(x;y)u(x;y)+vy(x;y)v(x;y)=0;8z2­:u2(x;y)+v2(x;y)´C20,ux(x;y)vy(x;y)¡uy(x;y)vx(x;y)´0;z2­:fC-R,u2x(x;y)+u2y(x;y)=v2y(x;y)+v2x(x;y)´0;z2­:ux(x;y)=uy(x;y)=vx(x;y)=vy(x;y)´0:u;v­.f(z)­.12.f(z)=u+iv­,u=sinv,f(z)­.C-R,z2­,ux=cosvvx=¡cosvuy=¡cos2vvy=¡cos2vux;uy=cosvvy=cosvux=cos2vvx=¡cos2vuy:ux´0;uy´0;z2­:,C-R,vx´0;vy´0;z2­:u;v­.f­.145313.f­C1.:f­,k2N,f(k)­..,k=0,f(k)=f­.k=n2N,f(n)­,k=n+1f(k)­.,f(n)C-R,z2­,@f(n)(z)@z=0;@f(n)(z)@z=f(n+1)(z):@f(n+1)(z)@z=@@zÃ@f(n)(z)@z!=@@zÃ@f(n)(z)@z!=0;8z2­:f(n+1)­C-R,f(n+1)­.,k2N,f(k)­.14.f­,z02­,f0(z0)6=0.:z0Uf(z0)V,f:U!V.C-Rf¡1:V!U.f(z)=u(x;y)+iv(x;y),fT:(x;y)7¡!(u(x;y);v(x;y)).Tz0JacobiJT(z0)=jf0(z0)j20:,0,T:D(z0;)!T(D(z0;)).U=D(z0;),V=f(D(z0;))=T(D(z0;)),U;Vz0f(z0),f:U!V.C-R