北大数学物理方法(A)-复变函数教案08解析函数的Laurent展开

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Outline1lù)Û¼êLaurentÐmÔnÆêÆÔn{‘§|2007cSC.S.Wu1lù)Û¼êLaurentÐmOutlineùLJ:1LaurentÐmÐm½nÞ~õŠ¼êLaurentÐm2üŠ¼êáÛ:áÛ:áÛ:©a¼ê3á?ÛÉ53)Ûòÿ˜‡~f)ÛòÿVgC.S.Wu1lù)Û¼êLaurentÐmOutlineùLJ:1LaurentÐmÐm½nÞ~õŠ¼êLaurentÐm2üŠ¼êáÛ:áÛ:áÛ:©a¼ê3á?ÛÉ53)Ûòÿ˜‡~f)ÛòÿVgC.S.Wu1lù)Û¼êLaurentÐmOutlineùLJ:1LaurentÐmÐm½nÞ~õŠ¼êLaurentÐm2üŠ¼êáÛ:áÛ:áÛ:©a¼ê3á?ÛÉ53)Ûòÿ˜‡~f)ÛòÿVgC.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationReferencesÇÂÁ§5êÆÔn{6§§5.4—5.7ù&œ§5êÆÔn{6§§3.5,3.4nÎ!X1Á§5êÆÔn{6§§3.4,3.5C.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationReferencesÇÂÁ§5êÆÔn{6§§5.4—5.7ù&œ§5êÆÔn{6§§3.5,3.4nÎ!X1Á§5êÆÔn{6§§3.4,3.5C.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationReferencesÇÂÁ§5êÆÔn{6§§5.4—5.7ù&œ§5êÆÔn{6§§3.5,3.4nÎ!X1Á§5êÆÔn{6§§3.4,3.5C.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuation)Û¼êLaurentÐm˜‡¼ê،3)Û:ŠTaylorÐm §kž„I‡ò§3Û:NCÐm¤˜?êùÒ´LaurentÐmC.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuation)Û¼êLaurentÐm˜‡¼ê،3)Û:ŠTaylorÐm §kž„I‡ò§3Û:NCÐm¤˜?êùÒ´LaurentÐmC.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuation)Û¼êLaurentÐm˜‡¼ê،3)Û:ŠTaylorÐm §kž„I‡ò§3Û:NCÐm¤˜?êùÒ´LaurentÐmC.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationTheorem(Laurent)IllustrativeExamplesLaurentExpansion:MultivaluedFunctionsùLJ:1LaurentÐmÐm½nÞ~õŠ¼êLaurentÐm2üŠ¼êáÛ:áÛ:áÛ:©a¼ê3á?ÛÉ53)Ûòÿ˜‡~f)ÛòÿVgC.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationTheorem(Laurent)IllustrativeExamplesLaurentExpansion:MultivaluedFunctionsÐm½n(Laurent)¼êf(z)3±b %‚/«R1≤|z−b|≤R2þüŠ)Û§Kéu‚S?Ûz:§f(z)Œ±ÐmLaurent?êf(z)=∞Xn=−∞an(z−b)nR1|z−b|R2an=12πiICf(ζ)(ζ−b)n+1dζC´‚S7S ˜±?¿˜^4Ü­‚C.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationTheorem(Laurent)IllustrativeExamplesLaurentExpansion:MultivaluedFunctionsÐm½n(Laurent)(‡:)f(z)=∞Xn=−∞an(z−b)nR1|z−b|R2an=12πiICf(ζ)(ζ−b)n+1dζò‚S .©OPC1ÚC2§ŠâEëÏ«CauchyÈ©úª§éu‚/«S?¿˜:z§kf(z)=12πiIC2f(ζ)ζ−zdζ−12πiIC1f(ζ)ζ−zdζe¡©OOŽ÷C1ÚC2È©C.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationTheorem(Laurent)IllustrativeExamplesLaurentExpansion:MultivaluedFunctionsÐm½n(Laurent)(‡:)f(z)=∞Xn=−∞an(z−b)nR1|z−b|R2an=12πiICf(ζ)(ζ−b)n+1dζò‚S .©OPC1ÚC2§ŠâEëÏ«CauchyÈ©úª§éu‚/«S?¿˜:z§kf(z)=12πiIC2f(ζ)ζ−zdζ−12πiIC1f(ζ)ζ−zdζe¡©OOŽ÷C1ÚC2È©C.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationTheorem(Laurent)IllustrativeExamplesLaurentExpansion:MultivaluedFunctionsÐm½n(Laurent)(‡:)f(z)=∞Xn=−∞an(z−b)nR1|z−b|R2an=12πiICf(ζ)(ζ−b)n+1dζò‚S .©OPC1ÚC2§ŠâEëÏ«CauchyÈ©úª§éu‚/«S?¿˜:z§kf(z)=12πiIC2f(ζ)ζ−zdζ−12πiIC1f(ζ)ζ−zdζe¡©OOŽ÷C1ÚC2È©C.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationTheorem(Laurent)IllustrativeExamplesLaurentExpansion:MultivaluedFunctionsÐm½n(Laurent)(‡:)f(z)=∞Xn=−∞an(z−b)nR1|z−b|R2an=12πiICf(ζ)(ζ−b)n+1dζéu÷C1È©−12πiIC1f(ζ)ζ−zdζ=12πiIC1f(ζ)(z−b)−(ζ−b)dζC.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationTheorem(Laurent)IllustrativeExamplesLaurentExpansion:MultivaluedFunctionsÐm½n(Laurent)(‡:)f(z)=∞Xn=−∞an(z−b)nR1|z−b|R2an=12πiICf(ζ)(ζ−b)n+1dζéu÷C1È©−12πiIC1f(ζ)ζ−zdζ=12πiIC1f(ζ)z−b∞Xk=0ζ−bz−bkdζC.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationTheorem(Laurent)IllustrativeExamplesLaurentExpansion:MultivaluedFunctionsÐm½n(Laurent)(‡:)f(z)=∞Xn=−∞an(z−b)nR1|z−b|R2an=12πiICf(ζ)(ζ−b)n+1dζéu÷C1È©−12πiIC1f(ζ)ζ−zdζ=12πiIC1f(ζ)z−b∞Xk=0ζ−bz−bkdζ=∞Xk=012πiIC1f(ζ)(ζ−b)kdζ(z−b)−k−1(|z−b|R1)C.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationTheorem(Laurent)IllustrativeExamplesLaurentExpansion:MultivaluedFunctionsÐm½n(Laurent)(‡:)f(z)=∞Xn=−∞an(z−b)nR1|z−b|R2an=12πiICf(ζ)(ζ−b)n+1dζéu÷C1È©−12πiIC1f(ζ)ζ−zdζ=12πiIC1f(ζ)z−b∞Xk=0ζ−bz−bkdζ=−∞Xn=−112πiIC1f(ζ)(ζ−b)n+1dζ(z−b)−n(|z−b|R1)C.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationTheorem(Laurent)IllustrativeExamplesLaurentExpansion:MultivaluedFunctionsÐm½n(Laurent)(‡:)f(z)=∞Xn=−∞an(z−b)nR1|z−b|R2an=12πiICf(ζ)(ζ−b)n+1dζéu÷C2È©§Œ†Ú^TaylorÐm(J12πiIC2f(ζ)ζ−zdζ=12πiIC2f(ζ)ζ−b∞Xn=0z−bζ−bndζ=∞Xn=012πiIC2f(ζ)(ζ−b)n+1dζ(z−b)n(|z−b|R2)C.S.Wu1lù)Û¼êLaurentÐmExpansioninLaurentSeriesIsolatedSingularitiesofUniformFunctionAnalyticContinuationTheorem(Laurent)IllustrativeExamplesLaurentExpansion:MultivaluedFunctionsÐm½n(Laurent)(‡:)f(z)=∞Xn=−∞an(z−b)nR1|z−b|R2an=12πiICf(ζ)(ζ−b)n+1dζò

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