complex analysis

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1、COMPLEXANALYSIS1DouglasN.Arnold2References:JohnB.Conway,FunctionsofOneComplexVariable,Springer-Verlag,1978.LarsV.Ahlfors,ComplexAnalysis,McGraw-Hill,1966.RaghavanNarasimhan,ComplexAnalysisinOneVariable,Birkh¨auser,1985.CONTENTSI.TheComplexNumberSystem...................

2、..........................2II.ElementaryPropertiesandExamplesofAnalyticFns................3Differentiabilityandanalyticity...........................................4TheLogarithm............................................................6Conformality....................

3、..........................................6Cauchy–RiemannEquations...............................................7M¨obiustransformations...................................................9III.ComplexIntegrationandApplicationstoAnalyticFns.............11Localresultsandco

4、nsequences...........................................12HomotopyofpathsandCauchy’sTheorem...............................14WindingnumbersandCauchy’sIntegralFormula.........................15Zerocounting;OpenMappingTheorem..................................17Morera’sTheorem

5、andGoursat’sTheorem...............................18IV.SingularitiesofAnalyticFunctions....................................19Laurentseries............................................................20Residueintegrals......................................................

6、...23V.Furtherresultsonanalyticfunctions.................................26ThetheoremsofWeierstrass,Hurwitz,andMontel.......................26Schwarz’sLemma........................................................28TheRiemannMappingTheorem................................

7、.........29ComplementsonConformalMapping....................................31VI.HarmonicFunctions......................................................32ThePoissonkernel.......................................................33SubharmonicfunctionsandthesolutionoftheDiri

8、chletProblem.........36TheSchwarzReflectionPrinciple.........................................391Theselecturenot