复变报告-解析函数的五种等价叙述

Fiveofanalyticfunctionsequivalentdescription解析函数的五种等价表达Analyticfunction：Definition1.Afunctionofthecomplexvariableisanalyticinanopensetifithasaderivativeateachpointinthatset。2.Ifweshouldspeakofafunctionthatisanalyticinasetwhichisnotopen,itistobeunderstoodthatisanalyticinanopensetcontaining。3.Inparticular,isanalyticatapointifitisanalyticthroughoutsomeneighborhoodof。Equivalentdescription:定理1函数在区域内解析的充要条件是：二元函数，在区域内可微并且函数，在区域内满足条件。Theorem1：AfunctionisanalyticinadomainifandonlyifandaredifferentiableandsatisfytheCauchy-Riemannequations，ateverypointof.Example:Toshowthatthefunctionisanalyticeverywhere.

Proof:Because.Thus.SoaredifferentiableandsatisfytheCauchy-Riemannequations,Henceisanalyticeverywhere.

定理2函数在区域内为解析函数的充要条件是：与具有连续的偏导数，且满足方程。Theorem2：Ifandhavecontinuousfirst-orderpartialderivativeswhichsatisfytheCauchy-Riemannequations，Thenisanalytic。ExampleToverifythefunctionnotanalyticateverypointofplane。Proof：Supposethat，thenSo，Because，sonotsatisfytheequationsateverypointofplane。Hencenotanalyticateverypointofplane。定理3函数在区域内解析的充要条件是：在区域内函数是的共轭调和函数。Theorem3：Afunctionsisanalyticinadomainifandonlyifisaharmonicconjugateof。ExampleToshowthatandbothareharmonicfunctions,butisnotaanalyticfunction。Proof：Because；。So.Henceisaharmonicfunctionattheplaneof;isaharmonicfunctionattheplaneof。But,soandnotsatisfytheequationsatplane,henceisnotaharmonicconjugateof。Soisnotaanalyticfunctionattheplane。定理4设是单连通区域，那么在区域内解析的充要条件是：在区域内连续且沿内任一围线有。Theorem4：(Morea)Ifisdefinedandcontinuousinadomainandifforallclosedcontourin,thenisanalyticthroughout。ExampleFindtheintegralunitcircle。Sol：Becausearethesingularpointsofthefunction,notinteriortoandontheunitcircle。Sothattheintegralisanalyticinteriortoandontheunitcircle。ByCauchyTheorem,。定理5函数在区域内解析的充要条件为：在内任一点的邻域内可展成为关于的幂级数，即泰勒级数。Theorem5：Letbeapowerserieswhoseradiusofconvergenceis。Thenisanalyticontheopendisc。ExampleToexpandthefunctionintoaseriesinvolvingpowerofandgiveitsradius。Sol：Becausehasonlyasingularpointatth