北大数学物理方法(A)-数学物理方程教案06球函数1

Outline18ù¥¼ê(˜)ÔnÆ2007cSC.S.Wu18ù¥¼ê(˜)OutlineùLJ:1Legendreõ‘ªÚ\Legendre§)Legendreõ‘ª2Legendreõ‘ª5ŸLegendreõ‘ª‡©L«Legendreõ‘ª5Legendreõ‘ª5C.S.Wu18ù¥¼ê(˜)OutlineùLJ:1Legendreõ‘ªÚ\Legendre§)Legendreõ‘ª2Legendreõ‘ª5ŸLegendreõ‘ª‡©L«Legendreõ‘ª5Legendreõ‘ª5C.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsReferencesÇÂÁ§5êÆÔn{6§§16.1,16.2,16.3,16.4ù&œ§5êÆÔn{6§§10.1nÎ!X1Á§5êÆÔn{6§§12.3C.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsLegendreõ‘ªÚ\C.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsë‘Legendre§òHelmholtz§3¥‹IXe©lCþ§Œë‘Legendre§1sinθddθsinθdΘdθ+λ−μsin2θΘ=0ŠC†x=cosθ,y(x)=Θ(θ)§KŒU¤ddx1−x2
dydx+λ−μ1−x2y=0C.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsë‘Legendre§òHelmholtz§3¥‹IXe©lCþ§Œë‘Legendre§1sinθddθsinθdΘdθ+λ−μsin2θΘ=0ŠC†x=cosθ,y(x)=Θ(θ)§KŒU¤ddx1−x2
dydx+λ−μ1−x2y=0C.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsLegendre§ŠAϜ/§μ=0§Legendre§1sinθddθsinθdΘdθ+λΘ=0ŠC†x=cosθ,y(x)=Θ(θ)§KŒU¤ddx1−x2
dydx+λy=0ù9e˜ùò?Øùü‡§)§§‚̇5Ÿ9Ù3©lCþ{¥A^C.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsLegendre§ŠAϜ/§μ=0§Legendre§1sinθddθsinθdΘdθ+λΘ=0ŠC†x=cosθ,y(x)=Θ(θ)§KŒU¤ddx1−x2
dydx+λy=0ù9e˜ùò?Øùü‡§)§§‚̇5Ÿ9Ù3©lCþ{¥A^C.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsLegendre§ŠAϜ/§μ=0§Legendre§1sinθddθsinθdΘdθ+λΘ=0ŠC†x=cosθ,y(x)=Θ(θ)§KŒU¤ddx1−x2
dydx+λy=0ù9e˜ùò?Øùü‡§)§§‚̇5Ÿ9Ù3©lCþ{¥A^C.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsùLJ:1Legendreõ‘ªÚ\Legendre§)Legendreõ‘ª2Legendreõ‘ª5ŸLegendreõ‘ª‡©L«Legendreõ‘ª5Legendreõ‘ª5C.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomials'uLegendre§?ØLegendre§ddz1−z2
dwdz+λw=03¦ÑLegendre§)äN/ªƒc§Šâ~‡©§)Ûnا¯kҌ±éLegendre§))Û5ŠÑäC.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomials'uLegendre§?ØLegendre§ddz1−z2
dwdz+λw=0FLegendre§kn‡Û:§z=±1Ú∞§¿…Ñ´KÛ:©Ïd§Øùn‡:ŒU´)Û: §Legendre§)3²¡)ÛC.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomials'uLegendre§?ØLegendre§ddz1−z2
dwdz+λw=0FLegendre§kn‡Û:§z=±1Ú∞§¿…Ñ´KÛ:©Ïd§Øùn‡:ŒU´)Û: §Legendre§)3²¡)ÛFz=0:´Legendre§~:§Ïd§§)3±z=0:%ü |z|1S)Û§Œ±ÐmTaylor?êC.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomials'uLegendre§?ØLegendre§ddz(1−z2)dwdz+λw=0Legendre§3z=0Sü‡‚5Ã')w1(z)=∞Xn=022n(2n)!Γn−ν2Γn+ν+12Γ−ν2Γν+12z2nw2(z)=∞Xn=022n(2n+1)!Γn−ν−12Γn+1+ν2Γ−ν−12Γ1+ν2z2n+1C.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsw1(z)3z=±1Âñ5éuw1(z)§nvŒž§ÙXêc2n=22n(2n)!Γn−ν2Γn+ν+12Γ−ν2Γν+12∼22nΓ−ν2Γν+12n−ν2n−(ν+1)/2e−n+ν/2(2n+1)2n+1/2e−(2n+1)×n+ν+12n+ν/2e−n−(ν+1)/2√2πC.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsw1(z)3z=±1Âñ5Ïd§nvŒžc2n∼~ê×1nù`²§Ø˜‡~ê §w1(z)3z=±1NC1§Úln11−z2=∞Xn=11nz2nƒÓÏd§w1(z)3z=±1éêuÑ©z=±1´w1(z){:XJrLegendre§3z=01˜)w1(z))Ûòÿ²¡þ§§˜½´˜‡õŠ¼êC.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsw1(z)3z=±1Âñ5Ïd§nvŒžc2n∼~ê×1nù`²§Ø˜‡~ê §w1(z)3z=±1NC1§Úln11−z2=∞Xn=11nz2nƒÓÏd§w1(z)3z=±1éêuÑ©z=±1´w1(z){:XJrLegendre§3z=01˜)w1(z))Ûòÿ²¡þ§§˜½´˜‡õŠ¼êC.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsw1(z)3z=±1Âñ5Ïd§nvŒžc2n∼~ê×1nù`²§Ø˜‡~ê §w1(z)3z=±1NC1§Úln11−z2=∞Xn=11nz2nƒÓÏd§w1(z)3z=±1éêuÑ©z=±1´w1(z){:XJrLegendre§3z=01˜)w1(z))Ûòÿ²¡þ§§˜½´˜‡õŠ¼êC.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsw1(z)3z=±1Âñ5Ïd§nvŒžc2n∼~ê×1nù`²§Ø˜‡~ê §w1(z)3z=±1NC1§Úln11−z2=∞Xn=11nz2nƒÓÏd§w1(z)3z=±1éêuÑ©z=±1´w1(z){:XJrLegendre§3z=01˜)w1(z))Ûòÿ²¡þ§§˜½´˜‡õŠ¼êC.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsw2(z)3z=±1Âñ5Ó§éuw2(z)§nvŒž§ÙXêc2n+1=22n(2n+1)!Γn−ν−12Γn+1+ν2Γ−ν−12Γ1+ν2∼22nn−ν−12n−ν/2e−n+(ν−1)/2Γn−ν−12Γn+1+ν2(2n+2)2n+3/2e−(2n+2)×n+1+ν2n+(ν+1)/2e−n−1−ν/2√2πC.S.Wu18ù¥¼ê(˜)LegendrePolynomialsPropertiesofLegendrePolynomialsSolutionstotheLegendreEquationLegendrePolynomialsw2(z)3z=±1Âñ5Ïd§nvŒžc2n+1∼~ê×12n+1ù`²§Ø˜‡~ê §w2(z)3z=±1NC1§Úln1+z1−z=∞Xn=122n+1z2n+1ƒÓÏd§w2(z)3z=±1éêuÑ©z=±1´w1(z){:XJrLegendre§3z=01)w2(z))Ûòÿ²¡þ§§˜½´˜‡õŠ¼êC.S.Wu18ù