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WuChong-shi ( ) Helmholtz Legendre 1sinθddθsinθdΘdθ+λ−μsin2θΘ=0 ! Legendre 1sinθddθsinθdΘdθ+λΘ=0,# $x=cosθ,y(x)=Θ(θ) %& ’()*ddx 1−x2dydx+λ−μ1−x2y=0+ddx 1−x2dydx+λy=0.,-./012 3 ’ 4567 8 9: ;=WuChong-shi§19.1Legendre?@AB2C§19.1LegendreDEFG HILegendre 3 JKLMN OPQR 3ST/(UVWX) YZ[ \Legendre 3 3S6#I]^=FLegendre_‘(abczdefgh)ddz 1−z2dwdz+λw=0.ijklm z=±1nz=∞ opqdrslm=tu vwajkmxydlmz Legendre_‘c{|}~ { =Fz=0mdLegendre_‘c m tu _‘c{| z=0m c |z|1 { x Taylor = w k c { dw1(z)=∞Xn=022n(2n)!Γn−ν2Γn+ν+12Γ−ν2Γν+12z2n,w2(z)=∞Xn=022n(2n+1)!Γn−ν−12Γn+1+ν2Γ−ν−12Γ1+ν2z2n+1, ν(ν+1)=λ. a k { { ¡ x ¢£Legendre_‘c{| ⁄¥ƒ c§¤'=“d «‹› | {fifl c – †‡· |z=±1a m _‘c { ¶•‚{ =a„– ” c{c»…‰'x =¿ w1(z) `n´ˆ˜¯ ˘ c2n=22n(2n)!Γn−ν2Γn+ν+12Γ−ν2Γν+12∼22n(2n+1)2n+1/2e−(2n+1)√2πn−ν2n−(ν+1)/2e−n+ν/2√2πΓ−ν2n+ν+12n+ν/2e−n−(ν+1)/2√2πΓν+12= ×1n.WuChong-shi˙¨ ˚¸ ˝(˛)B3Ca·ˇ vw¶k —z w1(z)|z=±1 c nln11−z2=∞Xn=11nz2n } =tu w1(z)|z=±1¿ =z=±1dw1(z)c m=‹ Legendre_‘|z=0c ¶{w1(z){ ¡£}~ – ¶•d¶k =¿ w2(z) `n´ˆ˜¯ ic2n+1=22n(2n+1)!Γn−ν−12Γn+1+ν2Γ−ν−12Γ1+ν2∼22n(2n+2)2n+3/2e−(2n+2)√2π×n−ν−12n−ν/2e−n+(ν−1)/2√2πΓ−ν−12n+1+ν2n+(ν+1)/2e−n−1−ν/2√2πΓ1+ν2= ×12n+1. vw¶k —z w2(z)|z=±1 c nln1+z1−z=∞Xn=122n+1z2n+1 } =tu w2(z)|z=±1 ¿ =z=±1 dw2(z)c m= Legendre_‘|z=0c {w2(z){ ¡£}~ – d¶k =FÆx |z=1( z=−1)mcªƒ {Legendre_‘= z=±1d_‘crslm _‘| ƒ0|z−1|2 i krs{ x w(z)=(z−1)ρ∞Xn=0cn(z−1)n,ŁØLegendre_‘ Œx ¢£|z=1mcº _‘ρ(ρ−1)+ρ=0. ρ1=ρ2=0=a·ˇLegendre_‘|z=1mªƒ c ¶{ –d| ƒ|z−1|2 { c {s¶• i¿ æ z=1(nz=−1) m= ı_‘ { c łø x Legendre_‘|z=1mªƒ c ¶{Pν(z)=∞Xn=01(n!)2Γ(ν+n+1)Γ(ν−n+1)z−12n,WuChong-shi§19.1Legendre?@AB4Cœ νß ¶ Legendre {x Qν(z)=12Pν(z)lnz+1z−1−2γ−2ψ(ν+1)+∞Xn=01(n!)2Γ(ν+n+1)Γ(ν−n+1)1+12+···+1nz−12n,œ νß Legendre γdEuler ψ(z)dΓ c¿ = Pν(z)( z=−1+z=∞ )+Qν(z) 6 6 5 =WuChong-shi˙¨ ˚¸ ˝(˛)B5C§19.2Legendre ‰¥ƒ x2+y2+z2a2cLaplace_‘ !∇2u=0,u Σ=f(Σ), ΣŁ§ x2+y2+z2=a2–cfm=#£$| %«c&’¥ƒc»…‰( )*+,-./012 {ak•{! p+ /03456| =‹ 789»i:;¶k(= c)•?@A‚fc¿œ BC `* ŒD` ak¿œ?c_E FGc_E=aHIJwK ˘L M cNO u`*ŒPφ u=u(r,θ).QR” •{!| K ˘Sc»…‰'=“d TMUVWFLaplace_‘|θ=0nθ=π_E–‚XY |aZm–[ g\]|u(r,θ)¿θc ^_ = Laplace_‘‘”£ K ˘¯ wab•{!ccd efg[–u(r,θ)|θ=0nθ=π_E–ci789=FLaplace_‘|K hmr=0 ‚XY |im[ g\]|u(r,θ)¿rc ^_ = Laplace_‘‘”£ K ˘¯ wab•{!ccd Æefg[–u(r,θ)|K hmr=0jci789=•{!| K ˘Sc k§¤‰'Did1r2∂∂rr2∂u∂r+1r2sinθ∂∂θsinθ∂u∂θ=0,u θ=0i7 u θ=πi7 u r=0i7 u r=a=f(θ).ılfg=mu(r,θ)=R(r)Θ(θ),ŁØ_‘ni789 Œyˆılfg ¢£1sinθddθsinθdΘ(θ)dθ+λΘ(θ)=0Θ(0)i7Θ(π)i7nddrr2dR(r)dr−λR(r)=0, λdılfg¯nocp•q =Legendre_‘ r–i789 sXtu != fvx=cosθ,y(x)=Θ(θ) op WuChong-shi§19.2LegendrewxyB6Cp•q λ”Xν(ν+1) tu !Œf ddx 1−x2dydx+ν(ν+1)y=0,y(±1)i7.z{|}~{| F Legendre x=04 z =–¶ wa k {c‰' Æ« w¿ ¶ cλ( ν) a k{|x=±1qd¿ c= w ¢_‘c{|x=±1 i7 ŒM λ( ν) ;Z =F„Legendre_‘|x=1mªƒ c k {Pν(x)nQν(x) 2%«=Pν(x)=∞Xn=01(n!)2Γ(ν+n+1)Γ(ν−n+1)x−12n,Pν(x)|x=1md{ c `* Œdi7c Qν(x)=12Pν(x)lnx+1x−1−2γ−2ψ(ν+1)+∞Xn=01(n!)2Γ(ν+n+1)Γ(ν−n+1)1+12+···+1nx−12n,Qν(x)|x=1md¿ c= Legendre_‘c{”Xy(x)=c1Pν(x)+c2Qν(x), M {|x=1i7 efic2=0 p‚ c1=1=M {|x=−1m i7 Œx • tu λ=ν(ν+1) „ Dctu =|x=−1m Pν(x)c Pν(−1)=∞Xn=0(−)n(n!)2Γ(ν+n+1)Γ(ν−n+1)=−sinνππ∞Xn=0Γ(n+ν+1)Γ(n−ν)(n!)2. Stirling 'x ¢Γ(n+ν+1)Γ(n−ν)(n!)2∼(n+ν+1)n+ν+1/2e−n−ν−1(n−ν)n−ν−1/2e−n+ν(n+1)n+1/2e−n−1(n+1)n+1/2e−n−1∼1n WuChong-shi˙¨ ˚¸ ˝(˛)B7CF¿ ¶ cν Pν(x)|x=−1m =F\MPν(x)d Œ‚xy|x=−1mi7 FM ¢tu !i( ¡){ efM Pν(x)‚d ¢£⁄ æ'=„Pν(x)c»…‰' a\y ¥|ν ƒk ¯= tu !c{Œdtu λl=l(l+1),l=0,1,2,3,···,tu yl(x)=Pl(x).Pl(x)d¶klß æ' œ lßLegendre æ' Pl(x)=lXn=01(n!)2(l+n)!(l−n)!x−12n.QR¢£Legendre æ'|x=1mc WPl(1)=1.Legendre §L # ,¤ '“ 3I« # Legendre ‹›fi ,¤ I« =fl –c†kLegendre æ'c§¤'WP0(x)=1,P1(x)=x,P2(x)=12 3x2−1,P3(x)=12 5x3−3x,P4(x)=18 35x4−30x2+3. c‡‰·‡19.1=WuChong-shi§19.2LegendrewxyB8C 19.1LegendrewxyWuChong-shi˙¨ ˚¸ ˝(˛)B9C§19.3Legendre F¶•‚„Legendre æ'c ı§”dPl(x)=12ll!dldxl x2−1l.ak§¤' œ Rodrigues '=»t x2−1l=(x−1)l[2+(x−1)]l=lXn=0l!n!(l−n)!2l−n(x−1)l+n, 12ll!dldxl x2−1l=dldxllXn=01n!(l−n)!2−n(x−1)l+n=lXn=01n!(l−n)!(l+n)!n!x−12n.aHŒ ˇwLegendre æ'c ı§”=„Legendre æ'c ı§” Y¢x Legendre æ'cl… Wl … ¯Pl(x)d… l l ¯Pl(x)dl ¢Pl(−x)=(−)lPl(x).‰ ¿Pl(x)|x=1mc x ¢£Pl(x)|x=−1mc Pl(−1)=(−1)l.„Legendre æ'c ı§”Æx `´ Legendre æ' iˆæc˘ „ _ Legendre æ'c˜¶k¯ˇ§¤'= u x˘ x2−1l x2−1l=lXr=0(−)rl!r!(l−r)!x2l−2r,*L˙æ lß dldxl x2−1l=dldxllXr=0(−)rl!r!(l−r)!x2l−2r=[l/2]Xr=0(−)rl!r!(l−r)!(2l−2r)!(l−2r)!xl−2r, lßL æ'cß M¨–lß abn'c– Œ ˚clf Lc[l/2]=¿¸¶SLegendre æ'c ı§” Œ¢£Pl(x)=[l/2]Xr=0(−)r(2l−2r)!2lr!(l−r)!(l−2r)!xl−2r.„ak§¤' QR Legendre æ'Pl(x)|x=0mc WP2l(0)=(−)l(2l)!22ll!l!,P2l+1(0)=0.WuChong-shi§19.4Legendrewxy@˝˛ˇ— B10C§19.4Legendre F Legendre §L # ,¤ '“ ,¤ I« ,¤ '“I Legendre §L 6 Æ ªLegendre Ł[−1,1]ØŒº Z1−1Pl(x)Pk(x)dx=0,k6=l.x „_‘ 2 ˇ=$|-˜¶ _ ˇak = ıZ1−1xkPl(x)dx, knlqd ƒk =F¿ ak ı „ cl… x æ