北大数学物理方法(A)-复变函数教案02解析函数

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Outline1ù)Û¼êÔnÆêÆÔn{‘§|2007cSC.S.Wu1ù)Û¼êOutlineùLJ:1)ۼꌆŒ‡¼ê)Û52м꘼êê¼ên¼êV­¼êC.S.Wu1ù)Û¼êOutlineùLJ:1)ۼꌆŒ‡¼ê)Û52м꘼êê¼ên¼êV­¼êC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsReferencesÇÂÁ§5êÆÔn{6§§2.1—2.3ù&œ§5êÆÔn{6§§1.4nÎ!X1Á§5êÆÔn{6§§1.2,1.3C.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsReferencesÇÂÁ§5êÆÔn{6§§2.1—2.3ù&œ§5êÆÔn{6§§1.4nÎ!X1Á§5êÆÔn{6§§1.2,1.3C.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsReferencesÇÂÁ§5êÆÔn{6§§2.1—2.3ù&œ§5êÆÔn{6§§1.4nÎ!X1Á§5êÆÔn{6§§1.2,1.3C.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityùLJ:1)ۼꌆŒ‡¼ê)Û52м꘼êê¼ên¼êV­¼êC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityêµ½Âw=f(z)´«GSüŠ¼ê§XJ3GS,:zlimΔz→0ΔwΔz=limΔz→0f(z+Δz)−f(z)Δz3§K¡¼êf(z)3z:Œd4Š§Pf0(z)§=¡f(z)3z:êC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityêµ½Âw=f(z)´«GSüŠ¼ê§XJ3GS,:zlimΔz→0ΔwΔz=limΔz→0f(z+Δz)−f(z)Δz3§K¡¼êf(z)3z:Œd4Š§Pf0(z)§=¡f(z)3z:êC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticity‡©µ½Âe¼êw=f(z)3z:UCþΔw=f(z+Δz)−f(z)Œ±¤Δw=A(z)Δz+ρ(Δz)Ù¥limΔz→0ρ(Δz)Δz=0K¡w=f(z)3z:Œ‡§Δw‚5Ü©A(z)Δz¡¼êw3z:‡©§PŠdw=A(z)dz½dz=ΔzC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticity‡©µ½Âe¼êw=f(z)3z:UCþΔw=f(z+Δz)−f(z)Œ±¤Δw=A(z)Δz+ρ(Δz)Ù¥limΔz→0ρ(Δz)Δz=0K¡w=f(z)3z:Œ‡§Δw‚5Ü©A(z)Δz¡¼êw3z:‡©§PŠdw=A(z)dz½dz=ΔzC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticity‡ûŒ±y²§e¼êw=f(z)3z:Œ§K˜½3T:Œ‡§‡ƒ½,¶¿…A(z)=f0(z)§=dw=f0(z)dz½dwdz=f0(z)ÏdꏡŠ‡ûC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticity‡ûŒ±y²§e¼êw=f(z)3z:Œ§K˜½3T:Œ‡§‡ƒ½,¶¿…A(z)=f0(z)§=dw=f0(z)dz½dwdz=f0(z)ÏdꏡŠ‡ûC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticity‡ûŒ±y²§e¼êw=f(z)3z:Œ§K˜½3T:Œ‡§‡ƒ½,¶¿…A(z)=f0(z)§=dw=f0(z)dz½dwdz=f0(z)ÏdꏡŠ‡ûC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticity‡ûŒ±y²§e¼êw=f(z)3z:Œ§K˜½3T:Œ‡§‡ƒ½,¶¿…A(z)=f0(z)§=dw=f0(z)dz½dwdz=f0(z)ÏdꏡŠ‡ûC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityµãf0(z)=limΔz→0f(z+Δz)−f(z)Δz¤¢limΔz→0(Δw/Δz)3§¿›XΔz±?¿ªªu0ž§Δw/ΔzѪuÓkŠ‡ƒ§eΔz±ØӐªªu0§Δw/ΔzªuØÓŠ§KlimΔz→0(Δw/Δz)Ø3AO´§ÄΔz→0ü«Õᐪ§ÒŒ±¼êŒ7‡^‡C.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityµãf0(z)=limΔz→0f(z+Δz)−f(z)Δz¤¢limΔz→0(Δw/Δz)3§¿›XΔz±?¿ªªu0ž§Δw/ΔzѪuÓkŠ‡ƒ§eΔz±ØӐªªu0§Δw/ΔzªuØÓŠ§KlimΔz→0(Δw/Δz)Ø3AO´§ÄΔz→0ü«Õᐪ§ÒŒ±¼êŒ7‡^‡C.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityµãf0(z)=limΔz→0f(z+Δz)−f(z)Δz¤¢limΔz→0(Δw/Δz)3§¿›XΔz±?¿ªªu0ž§Δw/ΔzѪuÓkŠ‡ƒ§eΔz±ØӐªªu0§Δw/ΔzªuØÓŠ§KlimΔz→0(Δw/Δz)Ø3AO´§ÄΔz→0ü«Õᐪ§ÒŒ±¼êŒ7‡^‡C.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityµãf0(z)=limΔz→0f(z+Δz)−f(z)Δz¤¢limΔz→0(Δw/Δz)3§¿›XΔz±?¿ªªu0ž§Δw/ΔzѪuÓkŠ‡ƒ§eΔz±ØӐªªu0§Δw/ΔzªuØÓŠ§KlimΔz→0(Δw/Δz)Ø3AO´§ÄΔz→0ü«Õᐪ§ÒŒ±¼êŒ7‡^‡C.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityΔx→0§Δy=0f0(z)=limΔz→0ΔwΔz=limΔx→0Δu+iΔvΔx=∂u∂x+i∂v∂xΔx=0§Δy→0f0(z)=limΔz→0ΔwΔz=limΔy→0Δu+iΔviΔy=∂v∂y−i∂u∂yCauchy-Riemann§∂u∂x=∂v∂y∂u∂y=−∂v∂xC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityΔx→0§Δy=0f0(z)=limΔz→0ΔwΔz=limΔx→0Δu+iΔvΔx=∂u∂x+i∂v∂xΔx=0§Δy→0f0(z)=limΔz→0ΔwΔz=limΔy→0Δu+iΔviΔy=∂v∂y−i∂u∂yCauchy-Riemann§∂u∂x=∂v∂y∂u∂y=−∂v∂xC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityΔx→0§Δy=0f0(z)=limΔz→0ΔwΔz=limΔx→0Δu+iΔvΔx=∂u∂x+i∂v∂xΔx=0§Δy→0f0(z)=limΔz→0ΔwΔz=limΔy→0Δu+iΔviΔy=∂v∂y−i∂u∂yCauchy-Riemann§∂u∂x=∂v∂y∂u∂y=−∂v∂xC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityCauchy-Riemann§∂u∂x=∂v∂y∂u∂y=−∂v∂xCauchy-Riemann§´¼êŒ7‡^‡Ø´¿©^‡Œ±y²§ef(z)=u(x,y)+iv(x,y)¢Üu(x,y)ÚJÜv(x,y)þŒ‡1§…÷vCauchy-Riemann§§K¼êf(z)Œ1=o‡ ê∂u/∂x,∂u/∂y,∂v/∂xÚ∂v/∂y3…ëYC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityCauchy-Riemann§∂u∂x=∂v∂y∂u∂y=−∂v∂xCauchy-Riemann§´¼êŒ7‡^‡Ø´¿©^‡Œ±y²§ef(z)=u(x,y)+iv(x,y)¢Üu(x,y)ÚJÜv(x,y)þŒ‡1§…÷vCauchy-Riemann§§K¼êf(z)Œ1=o‡ ê∂u/∂x,∂u/∂y,∂v/∂xÚ∂v/∂y3…ëYC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityCauchy-Riemann§∂u∂x=∂v∂y∂u∂y=−∂v∂xCauchy-Riemann§´¼êŒ7‡^‡Ø´¿©^‡Œ±y²§ef(z)=u(x,y)+iv(x,y)¢Üu(x,y)ÚJÜv(x,y)þŒ‡1§…÷vCauchy-Riemann§§K¼êf(z)Œ1=o‡ ê∂u/∂x,∂u/∂y,∂v/∂xÚ∂v/∂y3…ëYC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityCauchy-Riemann§∂u∂x=∂v∂y∂u∂y=−∂v∂xCauchy-Riemann§´¼êŒ7‡^‡Ø´¿©^‡Œ±y²§ef(z)=u(x,y)+iv(x,y)¢Üu(x,y)ÚJÜv(x,y)þŒ‡1§…÷vCauchy-Riemann§§K¼êf(z)Œ1=o‡ ê∂u/∂x,∂u/∂y,∂v/∂xÚ∂v/∂y3…ëYC.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityµãÚ¢êœ/˜XJ¼êf(z)3z:Œ§K3z:7ëY´¼ê3,:ëY§¿ØUíѼê3T:Œ$–kùœ¹µ¼ê3,«S??ëY§%??،C.S.Wu1ù)Û¼êAnalyticFunctionsElementaryFunctionsDifferentiabilityAnalyticityµãÚ¢êœ/˜XJ¼êf(z)3z:Œ§K3z:7ëY´¼ê3,:ëY§¿ØUíѼê3T:Œ$–kùœ¹µ¼ê3,«S??ëY§%??،C.S.Wu1ù)Û¼êAnalyticFunctionsElementar

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