北大数学物理方法(A)-复变函数教案07解析函数的Taylor展开

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Outline1Ôù)Û¼êTaylorÐmÔnÆêÆÔn{‘§|2007cSC.S.Wu1Ôù)Û¼êTaylorÐmOutlineùLJ:1TaylorÐmÐm½n?ØļêÐmª2TaylorÐmÞ~?ê¦{†–½Xê{õŠ¼êTaylorÐm3á:TaylorÐm3)ۼꍘ5)Û¼ê:á5)ۼꍘ5C.S.Wu1Ôù)Û¼êTaylorÐmOutlineùLJ:1TaylorÐmÐm½n?ØļêÐmª2TaylorÐmÞ~?ê¦{†–½Xê{õŠ¼êTaylorÐm3á:TaylorÐm3)ۼꍘ5)Û¼ê:á5)ۼꍘ5C.S.Wu1Ôù)Û¼êTaylorÐmOutlineùLJ:1TaylorÐmÐm½n?ØļêÐmª2TaylorÐmÞ~?ê¦{†–½Xê{õŠ¼êTaylorÐm3á:TaylorÐm3)ۼꍘ5)Û¼ê:á5)ۼꍘ5C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsReferencesÇÂÁ§5êÆÔn{6§§5.1—5.3ù&œ§5êÆÔn{6§§3.3nÎ!X1Á§5êÆÔn{6§§3.3,3.5C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsReferencesÇÂÁ§5êÆÔn{6§§5.1—5.3ù&œ§5êÆÔn{6§§3.3nÎ!X1Á§5êÆÔn{6§§3.3,3.5C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsReferencesÇÂÁ§5êÆÔn{6§§5.1—5.3ù&œ§5êÆÔn{6§§3.3nÎ!X1Á§5êÆÔn{6§§3.3,3.5C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctions)Û¼êTaylorÐm˜‡˜?ê3§Âñ S“L˜‡)Û¼êXÛr˜‡)Û¼êL«¤˜?êºC.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctions)Û¼êTaylorÐm˜‡˜?ê3§Âñ S“L˜‡)Û¼êXÛr˜‡)Û¼êL«¤˜?êºC.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctions)Û¼êTaylorÐm˜‡˜?ê3§Âñ S“L˜‡)Û¼êXÛr˜‡)Û¼êL«¤˜?êºC.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamplesùLJ:1TaylorÐmÐm½n?ØļêÐmª2TaylorÐmÞ~?ê¦{†–½Xê{õŠ¼êTaylorÐm3á:TaylorÐm3)ۼꍘ5)Û¼ê:á5)ۼꍘ5C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamplesÐm½n(Taylor)¼êf(z)3±a % CS9Cþ)Û§Kéu S?Ûz:§f(z)Œ^˜?êÐm(½ö`§f(z)Œ3a:Ðm˜?ê)f(z)=∞Xn=0an(z−a)nÙ¥an=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!C_ž•aa±ŒÈ©§ØAO`²± §þ_ž•C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamplesÐm½n(Taylor)(‡:)f(z)=∞Xn=0an(z−a)nan=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!ŠâCauchyÈ©úª§éu CS?¿˜:zf(z)=12πiICf(ζ)ζ−zdζ∵1ζ−z=1(ζ−a)−(z−a)=1ζ−a∞Xn=0z−aζ−an3 z−aζ−a ≤r1«¥˜—ÂñC.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamplesÐm½n(Taylor)(‡:)f(z)=∞Xn=0an(z−a)nan=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!ŠâCauchyÈ©úª§éu CS?¿˜:zf(z)=12πiICf(ζ)ζ−zdζ∵1ζ−z=1(ζ−a)−(z−a)=1ζ−a∞Xn=0z−aζ−an3 z−aζ−a ≤r1«¥˜—ÂñC.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamplesÐm½n(Taylor)(‡:)f(z)=∞Xn=0an(z−a)nan=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!ŠâCauchyÈ©úª§éu CS?¿˜:zf(z)=12πiICf(ζ)ζ−zdζ∵1ζ−z=1(ζ−a)−(z−a)=1ζ−a∞Xn=0z−aζ−an3 z−aζ−a ≤r1«¥˜—ÂñC.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamplesÐm½n(Taylor)(‡:)f(z)=∞Xn=0an(z−a)nan=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!őȩ∴f(z)=12πiIC∞Xn=0(z−a)n(ζ−a)n+1#f(ζ)dζ=∞Xn=012πiICf(ζ)(ζ−a)n+1dζ(z−a)nC.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamplesÐm½n(Taylor)(‡:)f(z)=∞Xn=0an(z−a)nan=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!őȩ∴f(z)=12πiIC∞Xn=0(z−a)n(ζ−a)n+1#f(ζ)dζ=∞Xn=012πiICf(ζ)(ζ−a)n+1dζ(z−a)nC.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamplesÐm½n(Taylor)(‡:)f(z)=∞Xn=0an(z−a)nan=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!∴f(z)=∞Xn=0an(z−a)nan=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamplesÐm½n(Taylor)(‡:)f(z)=∞Xn=0an(z−a)nan=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!∴f(z)=∞Xn=0an(z−a)nan=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamplesÐm½n(Taylor)(‡:)f(z)=∞Xn=0an(z−a)nan=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!∴f(z)=∞Xn=0an(z−a)nan=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamples~7.1¦ez3z=0:TaylorÐmªÏez3²¡)Û§ŒÐmez=∞Xn=0anzn|z|∞qÏez(n)=ez§¤±an=1n!ez(n) z=0=1n!∴ez=∞Xn=01n!zn|z|∞C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamples~7.1¦ez3z=0:TaylorÐmªÏez3²¡)Û§ŒÐmez=∞Xn=0anzn|z|∞qÏez(n)=ez§¤±an=1n!ez(n) z=0=1n!∴ez=∞Xn=01n!zn|z|∞C.S.Wu1Ôù)Û¼êTaylorÐmExpansi

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