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1、ComplexAnalysisCourseNotes—HarvardUniversity—Math213aFall2000,2006,2010C.McMullenJanuary7,2011Contents1Basiccomplexanalysis......................12Thesimply-connectedRiemannsurfaces............273Entireandmeromorphicfunctions...............394Conformalmapping.......................565Elliptic
2、functionsandellipticcurves..............79ForwardComplexanalysisisanexusformanymathematicalfields,including:1.Algebra(theoryoffieldsandequations);2.Algebraicgeometryandcomplexmanifolds;3.Geometry(Platonicsolids;flattori;hyperbolicmanifoldsofdimen-sionstwoandthree);4.Liegroups,discretesubgroupsan
3、dhomogeneousspaces(e.g.H/SL2(Z);5.Dynamics(iteratedrationalmaps);6.Numbertheoryandautomorphicforms(ellipticfunctions,zetafunc-tions);7.TheoryofRiemannsurfaces(Teichm¨ullertheory,curvesandtheirJa-cobians);8.Severalcomplexvariablesandcomplexmanifolds;9.RealanalysisandPDE(harmonicfunctions,ellip
4、ticequationsanddistributions).Thiscoursecoverssomebasicmaterialonboththegeometricandanalyticaspectsofcomplexanalysisinonevariable.Prerequisites:Backgroundinrealanalysisandbasicdifferentialtopology(suchascoveringspacesanddifferentialforms),andafirstcourseincomplexanalysis.1BasiccomplexanalysisWeb
5、eginwithaquickreviewofelementaryfactsaboutthecomplexplaneandanalyticfunctions.Somenotation.ThecomplexnumberswillbedenotedC.Welet∆,HandCbdenotetheunitdisk
6、z
7、<1,theupperhalfplaneIm(z)>0,andtheRiemannsphereC∪{∞}.WewriteS1(r)forthecircle
8、z
9、=r,andS1fortheunitcircle,eachorientedcounter-clockwise.We
10、alsoset∆∗=∆−{0}andC∗=C−{0}.Algebraofcomplexnumbers.Thecomplexnumbersareformallyde-finedasthefieldC=R[i],wherei2=−1.Theyarerepresentedinthe1Euclideanplanebyz=(x,y)=x+iy.Therearetwosquare-rootsof−1inC;thenumberiistheonewithpositiveimaginarypart.AnimportantroleisplayedbytheGaloisinvolutionz7→z.Wed
11、efine
12、z
13、2=N(z)=zz=x2+y2.(Comparethecaseofarealquadraticfield,√whereN(a+bd)=a2−db2givesanindefiniteform.)Compatibilityof
14、z
15、withtheEuclideanmetricjustifiestheidentificationofCandR2.Wealsoseethatzisafield:1/z=z/
16、z
17、.Itisalsoconvenienttodescribecomplexn
