a0 a1 star a2a3a4a5a6a7a8a9a10 7 1 8 ￥ … ? 11 1 8 ￥ … ? Helmholtz §3￥ IXXee'lC § o Legendre § 1 sin d d sin d￡d ? + h ?? ?sin2 i ￡ = 0 –9§ Aˇ /§Legendre § 1 sin d d sin d￡d ? +?￡ = 0; C x = cos ; y(x) = ￡( )§Kq § U ? d dx ?? 1?x2 · dy dx ? + ? ?? ?1?x2 ? y = 0 d dx ?? 1?x2 · dy dx ? +?y = 0: ? ? § )§§ 5 9 3'lC {￥ AA^^' 16.1 Legendre § ) 12 16.1 Legendre § ) 3? Legendre § ) N/“ c§ ~ ' § ) n ( 18 )§flk –?Legendre § ) ) 5 ' F Legendre §(?p x·EC ) d dx ?? 1?x2 · dy dx ? +?y = 0: kn :§x = §1 x = 1§? · K :'ˇd§ ?n : U· : §Legendre § )3 ??) ' F x = 0:·Legendre § ~:§ˇd§ § )3–x = 0: % jxj < 1S ) § –—m Taylor??'18 ￥fi?? 5?’ A)§§ · y1(x) = 1X n=0 22n (2n)! Γ ? n? ”2 · Γ n+ ” +12 ? Γ ? ?”2 · Γ ” +1 2 ? x2n; y2(x) = 1X n=0 22n (2n+1)! Γ n? ” ?12 ? Γ ? n+1+ ”2 · Γ ?” ?12 ? Γ ? 1+ ”2 · x2n+1; ￥ ”(” +1) = ?: r? A) ) § – Legendre § )3 ?? S L “' ·§? X §3??)′? – §( ‘§3x = §1 ? :§ § ??)o ‰ ) '? l ? ) N/“ –w ' ?uy1(x)§ nv § X? c2n = 2 2n (2n)! Γ ? n? ”2 · Γ n+ ” +12 ? Γ ? ?”2 · Γ ” +1 2 ? ? 2 2n (2n+1)2n+1=2e?(2n+1)p2… ? n? ”2 ·n?(”+1)=2 e?n+”=2p2… Γ ? ?”2 · ￡ n+ ” +12 ?n+”=2 e?n?(”+1)=2p2… Γ ” +1 2 ? =~?￡ 1n: 16.1 Legendre § ) 13 ?‘?§ ~? §y1(x)3x = §1NC 1 § ln 11?x2 = 1X n=1 1 nx 2n 'ˇd§y1(x)3x = §1??u 'x = §1·y1(x) {:'XJrLegendre § 3x = 0 1 )y1(x)) ?? §§ ‰· ? …?' ?uy2(x)§ nv § k c2n+1 = 2 2n (2n+1)! Γ n? ” ?12 ? Γ ? n+1+ ”2 · Γ ?” ?12 ? Γ ? 1+ ”2 · ? 2 2n (2n+2)2n+3=2e?(2n+2)p2… ￡ n? ” ?12 ?n?”=2 e?n+(”?1)=2p2… Γ ?” ?12 ? ￡ ? n+1+ ”2 ·n+(”+1)=2 e?n?1?”=2p2… Γ ? 1+ ”2 · =~?￡ 12n+1: ?–§ ~? §y2(x)3x = §1NC 1 § ln 1+x1?x = 1X n=1 2 2n+1x 2n+1 'ˇd§y2(x)3x = §1 ??u 'x = §1 ·y2(x) {:'rLegendre § 3x = 0 1 )y2(x)) ?? §§ · ? …?' F –3x = 1(‰x = ?1): S?)Legendre §' dux = §1· § K :§ §3 0 < jx?1j < 2Sk K)§ y(x) = (x?1)‰ 1X n=0 cn(x?1)n; \Legendre §§ – 3x = 1: I § ‰(‰?1)+‰ = 0: ?–§‰1 = ‰2 = 0'?‘?Legendre §3x = 1: S 1 )￠S ·3 jx?1j < 2S) § 1 )K ‰?k?? §–x = 1( x = ?1) {:' U ~ ' §??){ IO ‰§ –? Legendre §3x = 1: S 1 ) P”(x) = 1X n=0 1 (n!)2 Γ(” +n+1) Γ(” ?n+1) x?1 2 ?n ; 16.1 Legendre § ) 14 ? ”g1 aLegendre…??1 ) Q”(x) = 12P”(x) ? ln x+1x?1 ?2 ?2?(” +1) ? + 1X n=0 1 (n!)2 Γ(” +n+1) Γ(” ?n+1) 1+ 12 +￠￠￠+ 1n ? x?1 2 ?n ; ? ”g1 aLegendre…?§ ￥ ·Euler?§?(z)·Γ…? ?? ?' d u…?P”(x)( ?? § §·–x = ?1 x = 1 {: ? … ?) Q”(x) ?? 5fik ‰5 5‰§?^ I AO5?' 16.2 Legendre? “ 15 16.2 Legendre? “ ￥/? Sx2 +y2 +z2 < a2 Laplace §> flK r2u = 0; uflfl§ = f(§); ￥§ L￥?x2 +y2 +z2 = a2 C:' ? y3?? m? N/G§g,? ^￥ IX5?)? ‰)flK§ ?r I : 3￥%'XJ>.^ k7, (ˇL￥% ) ‰?^= C ? ?5§@o§ , A r? ??? 4? ' ? J IX §? ? …?u , `?’§ u = u(r; ): N· ‰)flK3￥ IXe N/“' ·§I 5? F Laplace §3 = 0 = … ??§3? : ? 3u(r; )? ?' rLaplace §U ￥ IX § –‰)flK d5§7L ? u(r; )3 = 0 = … k.^ ' F Laplace §3 I :r = 0 ??§3T:? 3u(r; )?r ?' rLaplace §U ￥ IX § –‰)flK d5§ 7L ? u(r; )3 I :r = 0? k.^ ' ‰)flK3￥ IXe L /“AT· 1 r2 @ @r r2@u@r ? + 1r2 sin @@ sin @u@ ? = 0; uflfl =0k.§ uflfl =…k.§ uflflr=0k.§ uflflr=a = f( ). 'lC '- u(r; ) = R(r)￡( ); \ § k.^ § U 'lC 1 sin d d sin d￡( )d ? +?￡( ) = 0; ￡(0)k.§ ￡(…)k.; d dr r2dR(r)dr ? ??R(r) = 0; 16.2 Legendre? “ 16 ￥?·'lC ? ‰o?' Legendre §§ k.^ § ? flK'ˇ~ C x = cos ; y(x) = ￡( )§? r ‰o?? ?”(” +1)§ flK C d dx ?? 1?x2 · dy dx ? +”(” +1)y = 0; y(§1)k.: ? …? F –lLegendre §3x = 0: S 5?’) u5?)' !fi? ? 5?’) /“§ y ?u ?(‰”) §? ) 3x = §1 ·??u ' ? § )3x = §1 k.§ ??(‰”) , Aˇ ' FlLegendre §3x = 1: S 5?’)P”(x) Q”(x) u5? ' P”(x) = 1X n=0 1 (n!)2 Γ(” +n+1) Γ(” ?n+1) x?1 2 ?n ; P”(x)3x = 1:·) § , ·k. ? Q”(x) = 12P”(x) ? ln x+1x?1 ?2 ?2?(” +1) ? + 1X n=0 1 (n!)2 Γ(” +n+1) Γ(” ?n+1) 1+ 12 +￠￠￠+ 1n ? x?1 2 ?n ; Q”(x)3x = 1:·??u ' rLegendre § ˇ) ? y(x) = c1P”(x)+c2Q”(x); du ?)3x = 1k.§7Lkc2 = 0§ c1 = 1' ?)3x = ?1: k.§ –‰ ? = ”(” +1)§l ? A …?' 3x = ?1:§P”(x) ? P”(?1) = 1X n=0 (?)n (n!)2 Γ(” +n+1) Γ(” ?n+1): 16.2 Legendre? “ 17 N·w § n > ”– §?? ? §ˇd? ??· ?? '§ ’ un un+1 = ? h(n+1)! n! i2Γ(” +n+1) Γ(” +n+2) Γ(” ?n) Γ(” ?n+1) = (n+1) 2 (n+” +1)(n?”) = 1+ 1 n +O 1 n2 ? ; Gauss O{ § –w F?u ” §P”(x)3x = ?1:u ' F P”(x)·????§§ U3x = ?1:k.? F ? flKk( "))§7L ?P”(x) ·????§= ? “' lP”(x) N/“w§? Uu)3” K ? '?–§ flK ) · ?l = l(l +1), l = 0;1;2;3;￠￠￠ ; …? yl(x) = Pl(x): Pl(x)· lg? “§? lgLegendre? “§ Pl(x) = lX n=0 1 (n!)2 (l +n)! (l?n)! x?1 2 ?n : N· Legendre? “3x = 1: ? Pl(1) = 1: Legendre? “· flK ) y §· Legendre §3k.^ e …? y ' $ A Legendre? “ L “ P0(x) = 1; P1(x) = x; P2(x) = 12 ? 3x2 ?1 · ; P3(x) = 12 ? 5x3 ?3x · ; P4(x) = 18 ? 35x4 ?30x2 +3 · : ?k y?”(” +1) ? 0§ˇ ” ? 0' Gauss O{ e?? 1P n=0 un￥ ’ – ? un un+1 = 1+ ? n +O ? n?? · ; ? = fi+ifl; ? > 1; K fi > 1 §?? ?′?? fi ? 1 §?? U ?′?' 16.2 Legendre? “ 18 § a/ a16.1' a16.1 Legendre? “ 16.3 Legendre? “ 'L? 19 16.3 Legendre? “ 'L? Legendre? “ 'L?· Pl(x) = 12ll! d l dxl ? x2 ?1 ·l : ? L “ ? Rodrigues?“' y ˇ ? x2 ?1 ·l = (x?1)l[2+(x?1)]l = lX n=0 l! n!(l?n)!2 l?n(x?1)l+n; ?– 1 2ll! dl dxl ? x2 ?1 ·l = d l dxl lX n=0 1 n!(l?n)!2 ?n(x?1)l+n = lX n=0 1 n!(l?n)! (l +n)! n! x?1 2 ?n : ? y? Legendre? “ 'L?' lLegendre? “ 'L?§?= –w Legendre? “ 5 l ? Pl(x)· …??l ? Pl(x)· …?§= Pl(?x) = (?)lPl(x): 2( Pl(x)3x = 1: ? §q – Pl(x)3x = ?1: ? § Pl(?1) = (?1)l: lLegendre? “ 'L? – ? Legendre? “￥?k X?§l Legendre? “ , w?L “' d§ ?x2 ?1￠l—m§ ? x2 ?1 ·l = lX r=0 (?)r l!r!(l?r)!x2l?2r; , ˉ ?lg§ dl dxl ? x2 ?1 ·l = d l dxl lX r=0 (?)r l!r!(l?r)!x2l?2r = [l=2]X r=0 (?)r l!r!(l?r)! (2l?2r)!(l?2r)! xl?2r; du ?lg §? “ g? $lg§?–?p “ d ?c lC ? [l=2]'?’ eLegendre? “ 'L?§ Pl(x) = [l=2]X r=0 (?)r (2l?2r)!2lr!(l?r)!(l?2r)!xl?2r: 16.3 Legendre? “ 'L? 110 l? L “?N·? Legendre? “Pl(x)3x = 0: ? P2l(0) = (?)l (2l)!22ll!l!; P2l+1(0) = 0: lLegendre? “ 'L?, – N?k?′ (J, ~X, rLegendre ? “ 'L? Rolle n( 5, Uy?l gLegendre ? “ l " : ‰‰ ··￠￠￠??,? u?m(?1; 1)S' 16.4 Legendre? “ 5 111 16.4 Legendre? “ 5 Legendre? “· flK …? y §ˇd§l flK u§ –y?Legendre? “ 5§= g? Legendre? “3? m[?1; 1] § Z 1 ?1 Pl(x)Pk(x)dx = 0; k 6= l: –l § u5y?' y3^, ? {y?? (J' kO ¨' Z 1 ?1 xkPl(x)dx; ￥k l · K ?' F?u? ¨'§l ¨…? 5 – Z 1 ?1 xkPl(x)dx = 0; k§l = ?: F k§l ? § Pl(x)^§ 'L? \§u·k Z 1 ?1 xkPl(x)dx = 12ll! Z 1 ?1 xk d l dxl ? x2 ?1 ·l dx = 12ll! ? xk d l?1 dxl?1 ? x2 ?1 ·lflfl fl 1 ?1 ? Z 1 ?1 dxk dx dl?1 dxl?1 ? x2 ?1 ·l dx ? : du d l?1 dxl?1 ?x2 ?1￠l￥ ‰?kˇf?x2 ?1￠§?–3 \ e x = §1 §' ¨' 5 ‰ 0§u· k Z 1 ?1 xkPl(x)dx = 12ll! Z 1 ?1 (?)1dx k dx dl?1 dxl?1 ? x2 ?1 ·l dx: ? §' ¨' g§ J Ly3n ? (1) UC g K ? (2) ?…??x2 ?1￠l ?~ g? (3) ?…?xk ?O\ g' ? §' ¨'lg § ?$ =￡ …?xk §(J C Z 1 ?1 xkPl(x)dx = 12ll! Z 1 ?1 (?)ld lxk dxl ? x2 ?1 ·l dx: ? k ? U§ ·k < l§…?xk ?lg ‰ 0§u· Z 1 ?1 xkPl(x)dx = 0; k < l: 16.4 Legendre? “ 5 112 , ? U·k > l§ -k = l +2n§u· Z 1 ?1 xl+2nPl(x)dx = 12ll! Z 1 ?1 (?)ld lxl+2n dxl ? x2 ?1 ·l dx = 12ll! (l +2n)!(2n)! Z 1 ?1 x2n ? 1?x2 ·l dx: C x2 = t§?|^B…? – ¨' Z 1 ?1 xl+2nPl(x)dx = 12ll! (l +2n)!(2n)! Z 1 0 tn?1=2 (1?t)l dt = 12ll! (l +2n)!(2n)! Γ n+ 12 ? Γ(l +1) Γ n+l + 32 ? = (l +2n)!2l+2nn! p… Γ n+l + 32 ? = 2l+1(l +2n)!(l +n)!n!(2l +2n+1)!: AO·k = l§=n = 0 § Z 1 ?1 xlPl(x)dx = l!2l p… Γ l + 32 ? = 2l+1 l!l!(2l +1)!: ? (J‘?§XJ …?xk gk uLegendre? “ g?l§@o§ … ?xk lgLegendre? “ ?¨3?m[?1; 1] ¨' ‰ 0' y3r ? (JA^u¨' Z 1 ?1 Pl(x)Pk(x)dx: F k 6= l § b k < l' ? Pk(x)·kg? “§ l ?k = ?‰ ?§? ? “￥ ?–Pl(x) ¨' ·0§?– y? g? Legendre? “3? m[?1; 1] ' Fe?? k = l /'? E, – Pl(x) §, ˉ ¨'§ Z 1 ?1 Pl(x)Pl(x)dx = Z 1 ?1 h clxl +cl?2xl?2 +cl?4xl?4 +￠￠￠ i Pl(x)dx: @o, 1 lgLegendre? “ ?¨ ¨' 0 , { lgLegendre? “ ?¨ ¨' 0. u·, k Z 1 ?1 Pl(x)Pl(x)dx = cl Z 1 ?1 xlPl(x)dx = cl ￡2l+1 l!l!(2l +1)!; 16.4 Legendre? “ 5 113 cl·lgLegendre? “￥xl X?§ cl = (2l)!2l(l!)2; ?–§Legendre? “ · Z 1 ?1 Pl(x)Pl(x)dx = 22l +1: FrLegendre? “ 5 ? 5§ – ? Z 1 ?1 Pk(x)Pl(x)dx = 22l +1–kl: F’uLegendre? “ 5 ? § –^ gC La' Z … 0 Pk(cos )Pl(cos )sin d = 22l +1–kl: ? ·‘§kgLegendre? “Pk(cos ) lgLegendre? “Pl(cos )3?m[0; …] – …?sin '?p …?sin — · ' § d d ? sin d￡d ? +?sin ￡ = 0 ￥ ? …?sin ' …? Legendre? “§ k 5 ?? 3?m[?1; 1]￥'aoY … ?f(x)§(3? ′? ?′e) –—m ?? f(x) = 1X l=0 clPl(x); ￥ —mX?cl – Legendre? “ 5? § cl = 2l +12 Z 1 ?1 f(x)Pl(x)dx: ~1 …?f(x) = x3ULegendre? “—m' ){1 x3 = 1P l=0 clPl(x)§K cl = 2l +12 Z 1 ?1 x3Pl(x)dx: XJ lim N!1 Z 1 ?1 flfl flf(x)? NP l=0 clPl(x) flfl fl2dx = 0; K??? 1P l=0 clPl(x)? ′? f(x)' 16.4 Legendre? “ 5 114 ? ? § – § l = 1 3 §cl 0' x3 = c1P1(x)+c3P3(x): —mX?c1 c3'O c1 = 32 Z 1 ?1 x4dx = 35; c3 = 72 Z 1 ?1 x3P3(x)dx = 25: (J · x3 = 35P1(x)+ 25P3(x): fl￠ §3? c1 § – { /? c3§?·ˇ 3—m“￥ \x = 1§ATkc1 +c3 = 1' ){2 ˇ x3 = c1P1(x)+c3P3(x) = c1x+c3 5 2x 3 ? 3 2x ? = 52c3x3 + c1 ? 32c3 ? x; ?– 5 2c3 = 1; c1 ? 3 2c3 = 0: dd – c3 = 25; c1 = 32c3 = 35: ){3 ˇ x3 = c1P1(x)+c3P3(x); \P3(x) "": x = r 3 5; k c1 = x 3 P1(x) flfl flfl x= p 3=5 = x2flflx=p3=5 = 35; c3 = 1?c1 = 25: ’uLegendre? “ 5§ –U^– gC La'? §XJ … ?f( )ULegendre? “Pl(cos )—m§ f( ) = 1X l=0 clPl(cos ); 16.4 Legendre? “ 5 115 K—mX? cl = 2l +12 Z … 0 f( )Pl(cos )sin d : 16.5 Legendre? “ )?…? 116 16.5 Legendre? “ )?…? Legendre? “·?k3? ?￥ ? ' 3 :r? k :> § :> ?3: z? §? :> 3(r0; ; `): >?(w, `?’)= 1p r2 +r02 ?2rr0cos = 8> >>< >>>: 1 r 1p 1?2xt+t2; t = r0 r ; 1 r0 1p 1?2xt+t2; t = r r0; ￥x = cos §?5‰? …?1=p?2xt+t2 '{ 1p 1?2xt+t2 flfl flfl t=0 = 1: 3? 5‰e§…?1=p1?2xt+t23t = 0:9 S·) §ˇ – Taylor—m 1p 1?2xt+t2 = 1X l=0 cltl; jtj < jx§ p x2 ?1j: e?y?—mX?cl ·Legendre? “Pl(x)§= 1p 1?2xt+t2 = 1X l=0 Pl(x)tl; jtj < jx§ p x2 ?1j: …?1=p1?2xt+t2=? Legendre? “ )?…?' y …?1=p1?2xt+t23t = 0: Taylor—m 1p 1?2xt+t2 = 1p 1?2t+t2 ?2(x?1)t = 11?t ? 1? 2(x?1)t(1?t)2 ??1=2 = 11?t 1X k=0 1 k! ?12 ? ?32 ? ￠￠￠ 1 2 ?k ?? ?2(x?1)t(1?t)2 ?k = 1X k=0 (2k?1)!! k! (x?1) ktk(1?t)?(2k+1) = 1X k=0 (2k?1)!! k! (x?1) ktk 1X n=0 (2k +n)! n!(2k)! t n = 1X l=0 " lX k=0 (l +k)! k!k!(l?k)! x?1 2 ?k# tl: –w §?p —mX? ·lgLegendre? “'? ? ’uLegendre? “)? …? y?'?? ′? § –d)?…?1=p1?2xt+t2 :(‰' |^Legendre? “ )?…?§ – N?k^ (J'~X§-x = 1§ 1p 1?2t+t2 = 1 1?t = 1X l=0 tl = 1X l=0 Pl(1)tl; 16.5 Legendre? “ )?…? 117 ?–§Pl(1) = 1' qX§ 1p 1?2xt+t2 = 1p 1?2(?x)(?t)+(?t)2; 1X l=0 Pl(x)tl = 1X l=0 Pl(?x)(?t)l; –y?Legendre? “ 5Pl(?x) = (?)lPl(x)' 16.6 Legendre? “ 4 ’X 118 16.6 Legendre? “ 4 ’X lLegendre? “ )?…? u§?N· gLegendre? “ m ’X§ =Legendre? “ 4 ’X' Legendre? “ )?…? 1p 1?2xt+t2 = 1X l=0 Pl(x)tl; ?t ?§k ?12 ?2x+2t(1?2xt+t2)3=2 = 1X l=0 lPl(x)tl?1; = x?t (1?2xt+t2)1=2 = ? 1?2xt+t2 · 1X l=0 lPl(x)tl?1 = (x?t) 1X l=0 Pl(x)tl: ’ tl X?§k xPl(x)?Pl?1(x) = (l +1)Pl+1(x)?2xlPl(x)+(l?1)Pl?1(x); n= (2l +1)xPl(x) = (l +1)Pl+1(x)+lPl?1(x): (z) ? Legendre? “ 4 ’X§§ n gLegendre? “ m ?X' E|^? 4 ’X§ –r??g Legendre? “^"gLegendre? “P0(x) = 1 gLegendre? “P1(x) = xL? 5' Legendre? “ )?…? 1p 1?2xt+t2 = 1X l=0 Pl(x)tl; ?x? §qU ?12 ?2t(1?2xt+t2)3=2 = 1X l=0 P0l(x)tl; u· t 1X l=0 Pl(x)tl = ? 1?2xt+t2 · 1X l=0 P0l(x)tl: ’ tl+1 X?§ Pl(x) = P0l+1(x)?2xP0l(x)+P0l?1(x): (#) ? 4 ’X￥§ y ·n gLegendre? “9 ?' 16.6 Legendre? “ 4 ’X 119 r(z)“?x? § – (2l +1)Pl(x)+(2l +1)xP0l(x) = (l +1)P0l+1(x)+lP0l?1(x); (#)“??§ P0l?1(x)‰P0l+1(x)§q – 4 ’X P0l+1(x) = xP0l(x)+(l +1)Pl(x); P0l?1(x) = xP0l(x)?lPl(x): ? 4 ’X§K·rP0l§1(x)^Pl(x)9 ?L? 5' r? 4 ’X?#| § –? ?/“ 4 ’X' 4 ’X ^ ·O , a. ¨'§~X Z 1 ?1 xPk(x)Pl(x)dx: 4 ’X(z)§ U O Z 1 ?1 xPk(x)Pl(x)dx = l +12l +1 Z 1 ?1 Pk(x)Pl+1(x)dx+ l2l +1 Z 1 ?1 PkPl?1(x)dx = l +12l +1 22l +3–l+1;k + l2l +1 22l?1–l?1;k: 16.7 Legendre? “A^ ~ 120 16.7 Legendre? “A^ ~ ~2 !>|￥ N￥' 3>|r E0 !>|￥ ? / N￥§￥ ? a'?￥ ?? : >?' ) ? N￥ §du?>aA§3 N￥ ￥? ?/? ‰ a)?> ' § ?￥N? ?N' F￥ ?? : o>? · k !>| >? a)> >? U\' F￥N /§? X￥N >? 0' Fˇ 3￥ ??vk> §?–3￥ >? vLaplace §' F ^￥ IX§ I : ￥%? §4? 5>| ' F ? !>|–9￥N ??5§3￥? a)> ‰·74?^= C §ˇ §?u￥ ?? :§? ·a)> ) >?§‰·o>?§ ·74?^ = C ' u(r; )·￥ :(r; ;`) o>?§u1(r; ) u2(r; )'O· !>| a)> >?§ u1(r; ) = ?E0z +u0 = ?E0rcos +u0; ~?u0= I :? >?'u2(r; )Kd‰)flK 1 r2 @ @r r2@u2@r ? + 1r2 sin @@ sin @u2@ ? = 0; u2flfl =0k.§ u2flfl =…k.; u2flflr=a = E0acos ?u0, u2flflr!1 ! 0: ?‰' u2(r; ) ?– vLaplace §§l n ‘§·dua)> ·' 3￥ ? § ￥ ?? ?a)> 3'l?? ‘§ˇ u(r; ) = u1(r; ) + u2(r; ) u1(r; ) vLaplace §' dua)> ·' 3￥? §?– r !1 u2(r; )A “u0' ?)‰)flK' § k.^ 'lC § – 1 sin d d h sin d￡( )d i +?￡( ) = 0; ￡(0)k.§ ￡(…)k.; d dr h r2dR(r)dr i ??R(r) = 0; 16.7 Legendre? “A^ ~ 121 ￥?·'lC ? ‰o?'316.2!￥fi?? L? flK§ )· ?l = l(l +1), l = 0;1;2;3;￠￠￠ ; …? ￡l( ) = Pl(cos ): ?)’uR(r) §§E, – C t = lnr§ §C d2Rl dt2 + dRl dt ?l(l +1)Rl = 0: u· Rl(r) = Alelt +Ble?(l+1)t = Alrl +Blr?l?1: ˇd§ vLaplace § k.^ ) · u2(r; ) = 1X l=0 ? Alrl +Blr?l?1 · Pl(cos ): ? ?? ^ u2flflr!1 ! 0§ATk Al = 0: 2 \￥?r = a >.^ § u2(r; )flflr=a = 1X l=0 Bla?l?1Pl(cos ) = E0acos ?u0 = E0aP1(cos )?u0P0(cos ); ?–k B0 = ?u0a; B1 = E0a3; Bl = 0; l ? 2: ? ? u2(r; ) = ?u0ar + E0a 3 r2 cos : ?p? u2(r; ) , N ￥? a)> ' ?'3 !>| ^ e§ /￥? a)> u u I : :> > 4f U\': > > ?4…"0u0a ?> 4f 4 4…"0E0a3§ !>| ' u1(r; ) u2(r; )U\§ ￥ ?? : o>? u(r; ) = u0 ? 1? ar · ?E0 ? 1? a 3 r3 · rcos : a16.2 L4? ?? ? >| ' a' 16.7 Legendre? “A^ ~ 122 a16.2 !>|￥ N￥ ~3 :> K e !0 ￥ >?' Xa16.3??§ k ? a !0 ￥(>N˙ ")§ ￥%b (b > a)? :> q§?0 ￥S ?? : >?' x y z a q(0; 0; b) (x; y; z) a16.3 :> K e 0 ￥ ) 3:> ^e§0 ￥u)4z' du0 ￥· ! §? 4z> 8￥3 ￥ L?'ˇd§0 ￥S! ?? : >?§ ·:> >? 4z?> >? U \' F ￥ IX§ I : 3￥%§4? :> §w,§? ·:> >?§‰· 4z> >?§ ·74?^= C § ·‘§ `?’' F (r; ;`): o>? u(r; )§4z> ) >? v(r; )§Kk u(r; ) = 14…" 0 qp r2 +b2 ?2rbcos +v(r; ): m 1 :> q ) >?§"0 >N˙' Fdu￥S >N˙ §?–§3?v(r; )‰u(r; ) §I ?'r < a(￥S) r > a(￥ )' F^v<(r; ) v>(r; )'OL?￥S(r < a) ￥ (r > a) v(r; ) § F^u<(r; ) u>(r; )'OL?￥S(r < a) ￥ (r > a) u(r; ) ' 16.7 Legendre? “A^ ~ 123 y3 5?v<(r; ) v>(r; )'du4z> ' 3￥? §?–§ ￥? : §v<(r; ) v>(r; )?? vLaplace §'2 ? k.^ ?? ^ § ATk 1 r2 @ @r r2@v<@r ? + 1r2 sin @@ sin @v<@ ? = 0; v<flfl =0k.§ v<flfl =…k.; v<flflr=0k.; 1 r2 @ @r r2@v>@r ? + 1r2 sin @@ sin @v>@ ? = 0; v>flfl =0k.§ v>flfl =…k.; v>flflr!1 ! 0: ~2￥ {§ –? v<(r; ) = 1X l=0 AlrlPl(cos ); v>(r; ) = 1X l=0 Blr?l?1Pl(cos ); ￥X?Al Bl ‰' /‰ v<(r; ) v>(r; )§ A § 3￥? 7L v o ^ ￥ S! o>?u<(r; ) u>(r; )§3￥? ‰ v>?oY > ￡￥ { ' o Y§ u<flflr=a = u>flflr=a; "@u<@r flfl fl r=a = @u>@r flfl fl r=a : ?–§?uv<(r; ) v>(r; )§k v<flflr=a = v>flflr=a; " ?@v < @r + 1 4…"0 @ @r qp r2 +b2 ?2rbcos ? r=a = ?@v > @r + 1 4…"0 @ @r qp r2 +b2 ?2rbcos ? r=a : d1 “§ Alal = Bla?l?1: 2 —m“ 1p a2 +b2 ?2abcos = 1 b 1X l=0 ?a b ·l Pl(cos ); 16.7 Legendre? “A^ ~ 124 –? @ @r 1p r2 +b2 ?2rbcos flfl fl r=a = 1b2 1X l=0 ?a b ·l?1 lPl(cos ); q "Allal?1 + ("?1)q4…" 0 lal?1 bl+1 = ?Bl(l +1)a ?l?2: )’uAl Bl?? §§ U ? Al = ?("?1)q4…" 0 l l +1+"l 1 bl+1; Bl = ?("?1)q4…" 0 l l +1+"l a2l+1 bl+1 : ￡ 5 ??￥§ )v<(r; ) v>(r; ). ￥S o>?u<(r; ) u>(r; )' ~4 ![ ?' k ![ § ? a§ M§?§3 m?? : ?' : m k :> m ?> § lCoulomb‰?§ˇd§ ? ?>? § vPoisson §'? §3 flK￥§ —33 : § ?AT???? vLaplace §' ) E ￥ IX§ I : 3 %§ K?3? ? '? § m?? :(r; ;`) ?AT `?’§u = u(r; )' – u? v § '‰)^ 1 r2 @ @r r2@u@r ? + 1r2 sin @@ sin @u@ ? = 0; (r; )6= ? a; …2 · ; uflfl =0k.; uflfl =…k.; uflflr=0k.; uflflr!1 ! 0: ? · ‰)flK(‰ ‘§? ‰)flK ·?‰ )§ˇ ?vk N ) ? ( ' ) ?' ![ ' N §?7 ^ –…?' § C 1 r2 @ @r r2@u@r ? + 1r2 sin @@ sin @u@ ? = ?4…GMf(r)–(r?a)– ? ? …2 · ; (#) ￥G· ~?§…?f(r) –d ZZZ f(r)–(r?a)– ? ? …2 · r2 sin drd d` = 1 ‰ § duf(r)–(r?a) = f(a)–(r?a)§?–k f(r) = f(a) = 12…a2: 16.7 Legendre? “A^ ~ 125 ?)‰)flK'd–…? 5 – § r 6= a § §(#) z Laplace §'? §2( k.^ ?? ^ § u(r; ) = 8> >>>< >>> >: 1X l=0 AlrlPl(cos ); r < a; 1X l=0 Blr?l?1Pl(cos ); r > a: , AT|^ ' (= §(#)m g )‰ X?Al Bl' F ? –…?AT·m …? ?§?–u(r; )3￥?r = a ‰·oY § u(r; )flflr=a+0r=a?0 = 0; F @u(r; )=@r3￥?r = a ‰· oY §§3￥?r = a C –d §(#)?r¨' r2@u@r flfl fl r=a+0 r=a?0 = ?2GM– ? ? …2 · ; = @u @r flfl fl r=a+0 r=a?0 = ?2GMa2 – ? ? …2 · : –( ?…=2) ULegendre? “—m – ? ? …2 · = 1P l=0 clPl(cos ); cl = 2l +12 Z … 0 – ? ? …2 · Pl(cos )sin d = 2l +12 Pl(0): ˇd Alal = Bla?l?1; Allal+1 +Bl(l +1)a?l = (2l +1)GMPl(0): ) = Al = GMa?l?1Pl(0); Bl = GMalPl(0): ?– u(r; ) = 8> >>>< >>> >: GM a 1X l=0 ?r a ·l Pl(0)Pl(cos ); r < a; GM a 1X l=0 ?a r ·l+1 Pl(0)Pl(cos ); r > a: \Pl(0) §= u(r; ) = 8 >>>> < >>>> : GM a 1X l=0 (?)l (2l)!22ll!l! ?r a ·2l P2l(cos ); r < a; GM a 1X l=0 (?)l (2l)!22ll!l! ?a r ·2l+1 P2l(cos ); r > a: 16.7 Legendre? “A^ ~ 126 – 0 ·dflK IO){ l ‰)flK u§ )§, …? 5‰ U\X?' 3? flK￥§ § g kAˇ5 3 r = a; = …=2 0§ § g 33 ? 1(? U yo k )' A/§3?) ^ Aˇ {§= g §(#)=z g § ￥?r = a o ^ ' ?p￠S ·0 ? o ^ {{' K k ? IO ){§= g § ‰)^ (k.^ ?? ^ )§? )§, ? |^o ^ ! Legendre? “ 5‰X?§ ·r )w? ·u(r; )3r = 0‰r = 1: S Taylor—m§ {? u(r; )3, Aˇ ? § Taylor—m 5‰ U\X?' du ??5§ ?? : ? (r; ) = (r;0)‰(r;…) : l §ˇ –dCoulomb‰? U\ ? ?? :(r;0)‰(r;…) ?§ u(r; )flfl =0;… = I GM 2…a dlp a2 +r2 = GMp a2 +r2: Taylor—m u(r; )flfl =0;… = 8 >>>> < >>>> : GM a 1X l=0 (?)l (2l)!22l (l!)2 ?r a ·2l ; r < a; GM r 1X l=0 (?)l (2l)!22l (l!)2 ?a r ·2l ; r > a: , ?§d )q – u(r; ) flfl fl =0 = 8> >>> < >>> >: 1X l=0 Alrl; r < a; 1X l=0 Blr?l?1; r > a ‰ u(r; ) flfl fl =… = 8> >>>< >>>> : 1X l=0 (?)lAlrl; r < a; 1X l=0 (?)lBlr?l?1; r > a: ’ ( ·Taylor—m 5)§ –? A2l = (?)lGMa (2l)!22ll!l!a?2l; A2l+1 = 0; B2l = (?)lGMa (2l)!22ll!l!a2l+1; B2l+1 = 0: –w §? )“ c? ' 16.8 o Legendre…? 127 16.8 o Legendre…? !? o Legendre § d dx ?? 1?x2 · dy dx ? + ? ?? m 2 1?x2 · y = 0 3k.^ y(§1)k. e )' {{··``a? o Legendre § Legendre § m ’X' ?k' o Legendre §3 :? 5 'o Legendre § : Legendre § § ·x = §1 x = 1§ · K :'3x = §1? I §· ‰(‰?1)+‰? m 2 4 = 0; ?–§ I ‰ = §m2 : ?‘?§o Legendre § ) – ?y(x) = ?1?x2￠§m=2 v(x) /“'?–§b y(x) = ? 1?x2 ·m=2 v(x); \ §§ – v(x)? v § ? 1?x2 · v00 ?2(m+1)xv0 +[??m(m+1)]v = 0: (z) ? v(x)3x = §1 I 0 ?m' I ?m )3x = §1: ‰·u ' ^??8B{ –y?§ §(z) –ˇLLegendre § ?mg ' F m = 0 w, (' F m = k ??§ ? 1?x2 ·? v(k) ·00 ?2(k +1)x ? v(k) ·0 +[??k(k +1)] ? v(k) · = 0: 2 ? g§ ? 1?x2 ·? v(k) ·000 ?2x ? v(k) ·00 ?2(k +1)x ? v(k) ·00 ?2(k +1) ? v(k) ·0 +[??k(k +1)] ? v(k) ·0 = 0; ? ? 1?x2 ·? v(k+1) ·00 ?2(k +2)x ? v(k+1) ·0 +[??(k +1)(k +2)]v(k+1) = 0: ?K y' 16.8 o Legendre…? 128 u·§o Legendre §3 0 < jx?1j < 2S ) · y(x) = c1 ? 1?x2 ·m=2 P(m)” (x)+c2 ? 1?x2 ·m=2 Q(m)” (x); ￥? = ”(” +1)'e?2^k.^ ‰ …?' ?k ?3x = 1:k.' F P”(x)3x = 1:·k. ' F Q”(x)3x = 1:·??u ' F ?1?x2￠m=2 P(m)” (x)3x = 1: ·k. §§·o Legendre §3x = 1: S I‰ = m=2 )' F Q(m)” (x)3x = 1:·–(x?1)?m “u §?–§?1?x2￠m=2 Q(m)” (x)3x = 1: ‰·u §§ ·o Legendre §3x = 1: S I‰ = ?m=2 )' Fk.^ ?)3x = 1:k.§?–§c2 = 0' 2 ?3x = ?1:k.' F?u ” § P”(x)·????§§3x = ?1: ·??u ' F x = ?1: ·P(m)” (x) m 4:§ F?–§?1?x2￠m=2 P(m)” (x)3x = ?1: ·u ' F v3x = ?1:k. ?§ U·P”(x) ?? “§=” K ?' Fdu3)￥ y ·P(m)” (x)§?–7Lk” ? m' o( ? ? § ? o Legendre §3k.^ e ) ?l = l(l +1); l = m;m+1;m+2;￠￠￠ …? yl(x) = c1 ? 1?x2 ·m=2 P(m)l (x): ˇ~ c1 = (?)m§ …?P Pml (x)§ Pml (x) = (?)m ? 1?x2 ·m=2 P(m)l (x); ? m lgo Legendre…?' o Legendre…?§ · flK )!=o Legendre §3k.^ e …? \ §ˇd§o Legendre…? A k 5 g o Legendre… ?3?m[?1;1] § Z 1 ?1 Pml (x)Pmk (x)dx = 0; k 6= l: 16.8 o Legendre…? 129 ?p5? ·§?uo Legendre §5‘§m· ‰ fi o?§ˇd§3 ? ’X￥§o Legendre…? ?m7L· ' –l § u§?A^k.^ §5y? ’X'?·y? …? 5 IO {{' e? {§ ^ y?Legendre? “ 5aq {' y duk 6= l§ b k < l'u·§ \o Legendre…? ‰′§?' ¨'§= Z 1 ?1 Pml (x)Pmk (x)dx = Z 1 ?1 ? 1?x2 ·m dmPk(x) dxm dmPl(x) dxm dx = ? 1?x2 ·m dmPk(x) dxm dm?1Pl(x) dxm?1 flfl flfl 1 ?1 ? Z 1 ?1 d dx ?? 1?x2 ·m dmPk(x) dxm ? dm?1P l(x) dxm?1 dx = ? Z 1 ?1 d dx ?? 1?x2 ·m dmPk(x) dxm ? dm?1P l(x) dxm?1 dx: ' ¨' g (J 3¨' cO\ K § L· ¨…?￥Pl(x) ? =￡ g { ˇf ' – §3' ¨'mg § A Z 1 ?1 Pml (x)Pmk (x)dx = (?)m Z 1 ?1 dm dxm ?? 1?x2 ·m dmPk(x) dxm ? Pl(x)dx: 5? “m ¨…?·lgLegendre? “ , ? “ dm dxm ?? 1?x2 ·m dmPk(x) dxm ? ?¨'N·? ? ? “ g? k ? m + 2m ? m = k'duk < l§?= y o Legendre…? 5' C x = cos § – o Legendre…? 5 , ?L /“§= Z … 0 Pml (cos )Pmk (cos )sin d = 0; k 6= l: 5?§?p y ?sin ' c? {§ U? o Legendre…? '? 3– y?L§ “ ￥ k = l= 'u·§ Z 1 ?1 Pml (x)Pml (x)dx = (?)m Z 1 ?1 dm dxm ?? 1?x2 ·m dmPl(x) dxm ? Pl(x)dx: y3 “m ¨…?·lgLegendre? “ , lg? “ dm dxm ?? 1?x2 ·m dmPl(x) dxm ? = 12ll! d m dxm ?? 1?x2 ·m dl+m dxl+m ? x2 ?1 ·l? 16.8 o Legendre…? 130 ?¨'d16.4! ? §?¨' z 5g? ? “ p g 'N· ? ? p g X?· (?)m 12ll! (2l)!(l?m)! (l +m)!l! ; ?–§ Z 1 ?1 Pml (x)Pml (x)dx = (2l)!2l(l!)2 (l +m)!(l?m)! Z 1 ?1 xlPl(x)dx = (l +m)!(l?m)! 22l +1; ‰ ? C x = cos § Z 1 ?1 Pml (cos )Pml (cos )sin d = (l +m)!(l?m)! 22l +1: l K ‘§o Legendre…? N?5 dLegendre? “ A5 'l ' 16.9 ￥?N …? 131 16.9 ￥?N …? y3￡ Laplace §3￥ IXe 'lC ' (‰ § k? ￥ SLaplace § 11 a> flK' 3￥ IXe§‰)flK· 1 r2 @ @r r2@u@r ? + 1r2 sin @@ sin @u@ ? + 1r2sin2 @ 2u @`2 = 0; uflfl =0k.§ uflfl =…k.; uflfl`=0 = uflfl`=2…, @u@` flfl fl `=0 = @u@` flfl fl `=2… ; uflflr=0k.§ uflflr=a = f( ;`): ?E15.6! ‰§-u(r; ;`) = R(r)S( ;`)§ ? § g>.^ 'lC §= d dr ? r2dR(r)dr ? ??R(r) = 0; uflflr=0k.; 1 sin @ @ ? sin @S( ;`)@ ? + 1sin2 @ 2S( ;`) @`2 +?S( ;`) = 0; Sflfl =0k.§ Sflfl =…k.; Sflfl`=0 = Sflfl`=2…, @S@` flfl fl `=0 = @S@` flfl fl `=2… : ? · flK§ ' § flK' ? ? A …?§ –2-S( ;`) = ￡( )'(`)§? 'lC § k 1 sin d d ? sin d￡( )d ? + h ?? ?sin2 i ￡( ) = 0; ￡(0)k.§ ￡(…)k.; '00 +?' = 0; '(0) = '(2…), '0(0) = '0(2…): ? ~ ' § flK fi?? L§'O 16.8! 15.4!'? §?u ' § flK5‘§ · ?l = l(l +1); l = 0;1;2;3;￠￠￠ ; 16.9 ￥?N …? 132 ?Au ?l§k2l +1 …? Slm1( ;`) = Pml (cos )cosm`; m = 0;1;2;￠￠￠ ;l; Slm2( ;`) = Pml (cos )sinm`; m = 1;2;￠￠￠ ;l: ? …?§ ? ￥?N …?§‰￥? …?' flK {? ·2l +1§ u~ '' § flK?N {? 2' ’uR ~ ' §§316.7!￥fi?? L'§3k.^ e )·Rl(r) = rl'? § ' §‰)flK A) · ulm1(r; ;`) = rlPml (cos )cosm`; l = 0;1;2;￠￠￠ ; m = 0;1;2;￠￠￠ ;l ulm2(r; ;`) = rlPml (cos )sinm`; l = 0;1;2;￠￠￠ ; m = 1;2;￠￠￠ ;l: )K u(r; ;`) = 1X l=0 lX m=0 rlPml (cos )[Alm cosm`+Blm sinm`]: ￡ e13.8!￥ ? '3T!￥ LLaplace §3 IX￥ ? “ )§§ ? A)· ' F ^ IX §ˇ L “ N/“ ' Fr ? A)U Ix; y; z …?§ –w §§ ·x; y; z l g? “' F ‰ l §? ? “k2l +1 ' ? ·13.8!￥J L ( ' n 16.4! 16.8! ? § –w §l‰m ￥?N …?3 4…?N ·* d § Z … 0 Pml (cos )Pnk(cos )sin d Z 2… 0 cosm`cosn`d` = 0; l 6= k; m 6= n; Z … 0 Pml (cos )Pnk(cos )sin d Z 2… 0 sinm`sinn`d` = 0; l 6= k; m 6= n; Z … 0 Pml (cos )Pnk(cos )sin d Z 2… 0 cosm`sinn`d` = 0; l 6= k; m 6= n: 16.9 ￥?N …? 133 § – ￥?N …? Z … 0 ￡Pm l (cos ) ?2 sin d Z 2… 0 cos2m`d` = (l +m)!(l?m)! 2…2l +1 (1+–m0); Z … 0 ￡Pm l (cos ) ?2 sin d Z 2… 0 sin2m`d` = (l +m)!(l?m)! 2…2l +1: 3 n ~^ ·, ?/“ ￥?N …?' F1 §· flK '00 +?' = 0; '(0) = '(2…), '0(0) = '0(2…): )3/“ ? ?m = m2; m = 0;§1;§2;§3￠￠￠ ; …? 'm(`) = eim`: ? §?Au ?l = l(l+1)§l = 0;1;2;3;￠￠￠§ ' § flK …? · Slm( ;`) = Pjmjl (cos )eim`; m = 0;§1;§2;￠￠￠ ;§l: ? ‰′ ￥?N …?§ ’X – ? { /“§ Z … 0 Z 2… 0 Slm( ;`)S?kn( ;`)sin d d`=(l +jmj)!(l?jmj)! 4…2l +1–lk–mn: duy3 …?·E…?§?–3 ’X ?“￥§ r ￥ …? E ' ˇ· y …? ' F1 §ˇ~ · ^8 z ￥?N …?'~X§ Yml ( ;`) = s (l?jmj)! (l +jmj)! 2l +1 4… P jmj l (cos )e im`; m = 0;§1;§2;￠￠￠ ;§l: ? k 8 ’X Z … 0 Z 2… 0 Yml ( ;`)Yn?k ( ;`)sin d d` = –lk–mn: A 5? ·§3 'z￥§Yml ( ;`)~~k ‰′'3?^ I @ ?' Yml ( ;`)‰′￥ ? ? – K§?·ˇ P?ml (x) = (?)m(l?m)!(l +m)!Pml (x):