Outline1Ôù)Û¼êTaylorÐmÔnÆêÆÔn{§|2007cSC.S.Wu1Ôù)Û¼êTaylorÐmOutlineùÇ:1TaylorÐmÐm½n?ØļêÐmª2TaylorÐmÞ~?ê¦{½Xê{õ¼êTaylorÐm3á:TaylorÐm3)Û¼ê5)Û¼ê:á5)Û¼ê5C.S.Wu1Ôù)Û¼êTaylorÐmOutlineùÇ:1TaylorÐmÐm½n?ØļêÐmª2TaylorÐmÞ~?ê¦{½Xê{õ¼êTaylorÐm3á:TaylorÐm3)Û¼ê5)Û¼ê:á5)Û¼ê5C.S.Wu1Ôù)Û¼êTaylorÐmOutlineùÇ:1TaylorÐmÐm½n?ØļêÐmª2TaylorÐmÞ~?ê¦{½Xê{õ¼êTaylorÐm3á:TaylorÐm3)Û¼ê5)Û¼ê:á5)Û¼ê5C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsReferencesÇÂÁ§5êÆÔn{6§§5.1—5.3ù&§5êÆÔn{6§§3.3nÎ!X1Á§5êÆÔn{6§§3.3,3.5C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsReferencesÇÂÁ§5êÆÔn{6§§5.1—5.3ù&§5êÆÔn{6§§3.3nÎ!X1Á§5êÆÔn{6§§3.3,3.5C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsReferencesÇÂÁ§5êÆÔn{6§§5.1—5.3ù&§5êÆÔn{6§§3.3nÎ!X1Á§5êÆÔn{6§§3.3,3.5C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctions)Û¼êTaylorÐm?ê3§ÂñSL)Û¼êXÛr)Û¼êL«¤?êºC.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctions)Û¼êTaylorÐm?ê3§ÂñSL)Û¼êXÛr)Û¼êL«¤?êºC.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctions)Û¼êTaylorÐm?ê3§ÂñSL)Û¼êXÛr)Û¼êL«¤?êºC.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamplesùÇ:1TaylorÐmÐm½n?ØļêÐmª2TaylorÐmÞ~?ê¦{½Xê{õ¼êTaylorÐm3á:TaylorÐm3)Û¼ê5)Û¼ê:á5)Û¼ê5C.S.Wu1Ôù)Û¼êTaylorÐmExpansioninTaylorSeriesTaylorExpansion:ExamplesIdentityTheoremforAnalyticFunctionsTheorem(Taylor)Discussions&RemarksIllustrativeExamplesÐm½n(Taylor)¼êf(z)3±a%CS9Cþ)Û§KéuS?Ûz:§f(z)^?êÐm(½ö`§f(z)3a:Ðm?ê)f(z)=∞Xn=0an(z−a)nÙ¥an=12πiICf(ζ)(ζ−a)n+1dζ=f(n)(a)n!C_aa± È©§ØAO`²± §þ_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!âCauchyÈ©úª§éuCS?¿:zf(z)=12πiICf(ζ)ζ−zdζ∵1ζ−z=1(ζ−a)−(z−a)=1ζ−a∞Xn=0z−aζ−an3z−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!âCauchyÈ©úª§éuCS?¿:zf(z)=12πiICf(ζ)ζ−zdζ∵1ζ−z=1(ζ−a)−(z−a)=1ζ−a∞Xn=0z−aζ−an3z−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!âCauchyÈ©úª§éuCS?¿:zf(z)=12πiICf(ζ)ζ−zdζ∵1ζ−z=1(ζ−a)−(z−a)=1ζ−a∞Xn=0z−aζ−an3z−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!ÅÈ©∴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!ÅÈ©∴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:TaylorÐmªÏez3²¡)Û§Ðmez=∞Xn=0anzn|z|∞qÏ ez(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:TaylorÐmªÏez3²¡)Û§Ðmez=∞Xn=0anzn|z|∞qÏ ez(n)=ez§¤±an=1n! ez(n)z=0=1n!∴ez=∞Xn=01n!zn|z|∞C.S.Wu1Ôù)Û¼êTaylorÐmExpansi