complex-analysis

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1、ComplexAnalysisAntonDeitmarContents1Thecomplexnumbers32Holomorphy73PowerSeries94PathIntegrals145Cauchy'sTheorem176Homotopy197Cauchy'sIntegralFormula258Singularities319TheResidueTheorem3410Constructionoffunctions3811Gamma&Zeta451COMPLEXANALYSIS212Theupperhalfplane4713Con

2、formalmappings5014Simpleconnectedness53COMPLEXANALYSIS31ThecomplexnumbersProposition1.1Thecomplexconjugationhasthefollowingproperties:(a)z+w=z+w,(b)zw=zw,
(c)z1=z1,orz=z,ww(d)z=z,(e)z+z=2Re(z),andzz=2iIm(z).COMPLEXANALYSIS4Proposition1.2Theabsolutevaluesatises:(a)j

3、zj=0,z=0,(b)jzwj=jzjjwj,(c)jzj=jzj,(d)jz1j=jzj1,(e)jz+wjjzj+jwj,(triangleinequality).Proposition1.3AsubsetACisclosediforeverysequence(an)inAthatconvergesinCthelimita=limn!1analsobelongstoA.WesaythatAcontainsallitslimitpoints.COMPLEXANALYSIS5Proposition1.4LetOdenote

4、thesystemofallopensetsinC.Then(a);2O,C2O,(b)A;B2O)AB2O,S(c)Ai2Oforeveryi2Iimpliesi2IAi2O.Proposition1.5ForasubsetKCthefollowingareequivalent:(a)Kiscompact.(b)Everysequence(zn)inKhasaconvergentsubsequencewithlimitinK.COMPLEXANALYSIS6Theorem1.6LetSCbecompactandf:S!Cbec

5、ontinuous.Then(a)f(S)iscompact,and(b)therearez1;z22Ssuchthatforeveryz2S,jf(z1)jjf(z)jjf(z2)j:COMPLEXANALYSIS72HolomorphyProposition2.1LetDCbeopen.Iff;gareholomorphicinD,thensoareffor2C,f+g,andfg.Wehave00000(f)=f;(f+g)=f+g;000(fg)=fg+fg:LetfbeholomorphiconDandgbeh

6、olomorphiconE,wheref(D)E.ThengfisholomorphiconDand000(gf)(z)=g(f(z))f(z):Finally,iffisholomorphiconDandf(z)6=0forevery1z2D,thenisholomorphiconDwithf1f0(z)0()(z)=:ff(z)2COMPLEXANALYSIS8Theorem2.2(Cauchy-RiemannEquations)Letf=u+ivbecomplexdierentiableatz=x+iy.Thenthe

7、partialderivativesux;uy;vx;vyallexistandsatisfyux=vy;uy=vx:Proposition2.3SupposefisholomorphiconadiskD.(a)Iff0=0inD,thenfisconstant.(b)Ifjfjisconstant,thenfisconstant.COMPLEXANALYSIS93PowerSeriesProposition3.1Let(an)beasequenceofcomplexnumbers.P(a)Supposethatanconverge

8、s.Thenthesequence(an)tendstozero.Inparticular,thesequence(an)isbounded.PP(b)Ifjanjconverges,thenanconverges.In