Wu Chong-shi a0a1a2a3a4 a5 a6 a7 ( a8) §21.1 a9a10 Legendre a11a12 a13a14a15a16a17a18 Legendre a19a20 d dx bracketleftbiggparenleftbig 1?x2parenrightbig dydx bracketrightbigg + parenleftBig λ? m 2 1?x2 parenrightBig y = 0 a21a22a23a24a25 y(±1)a22a23 a26a27a28a29 a19a30a31a32a33a34a35 a17a18 Legendre a19a20a36 Legendrea19a20a37a38 a27a39a40a29 a41a42a43a44a45a46 Legendre a47a48 d dz bracketleftbiggparenleftbig 1?z2parenrightbig dwdz bracketrightbigg + parenleftBig λ? m 2 1?z2 parenrightBig w = 0 a49a50a51a52a53a54a55 a29 a45a46 Legendre a47a48 a53a50a51a56 Legendre a47a48a57a58a59a60a61a62a63 z = ±1a56z = ∞a61 a64a65a66 a62a63a67a68 a50a51 a29 a49 z = ±1a52a53a69a70 a47a48a63 ρ(ρ?1) + ρ? m 2 4 = 0, a71a72 a61 a69a70a73 ρ = ±m2 . a74a75a76 a61 a45a46 Legendre a47a48 a53a77a78a72a79a80 w(z) = parenleftbig1?z2parenrightbigm/2 v(z) a53a81a82 a29a83a84 a47a48a61a85 a78a72a86a87 v(z)a71a88a89a53 a47a48parenleftbig 1?z2parenrightbigvprimeprime ?2(m + 1)zvprime + [λ?m(m + 1)]v = 0. (maltesecross) a74a90v(z)a49z = ±1a53a69a70a73 0 a91?ma29a69a70a73?ma53a77a49z = ±1a51a59a92a63a93a94 a53 a29 a95a96a97a98a99a100a78a72a101a76 a61a47a48 (maltesecross) a78a72a102a103 Legendre a47a48a104a105ma106 a64a86a87a29 star m = 0a90a107a108a67a109 a29 Wu Chong-shi §21.1 a110a111 Legendrea112a113 a1142a115 star a116m = ka90a80a117a61parenleftbig 1?z2parenrightbig parenleftBig v(k) parenrightBigprimeprime ?2(k + 1)z parenleftBig v(k) parenrightBigprime + [λ?k(k + 1)] parenleftBig v(k) parenrightBig = 0. a118 a104a105a59a106a61 parenleftbig1?z2parenrightbigparenleftBigv(k)parenrightBigprimeprimeprime ?2zparenleftBigv(k)parenrightBigprimeprime ?2(k + 1)zparenleftBigv(k)parenrightBigprimeprime ?2(k + 1)parenleftBigv(k)parenrightBigprime + [λ?k(k + 1)]parenleftBigv(k)parenrightBigprime = 0, a74 a60a85 a86a87 parenleftbig1?z2parenrightbigparenleftBigv(k+1)parenrightBigprimeprime ?2(k + 2)zparenleftBigv(k+1)parenrightBigprime + [λ?(k + 1)(k + 2)]v(k+1) = 0. a119a120a86a101 a29 square a121 a63a61 a45a46 Legendre a47a48 a49a122a123 0 < |z ?1| < 2 a124 a53a77 a85a63 w(z) = c1 parenleftbig1?z2parenrightbigm/2 P(m)ν (z) + c2 parenleftbig1?z2parenrightbigm/2 Q(m)ν (z), a125a126λ = ν(ν + 1) a29a127a128 a118a95a129a130a131a132 a92a133a134a135a136 a56 a134a135a137 a96 a29 a41a42a138a139a140x = 1 a141a142a143 a29 star Pν(x) a49x = 1a51a63 a129a130a53 a29 star Qν(x) a49x = 1a51a63a144 a96 a93a94 a53 a29 star parenleftbig1?x2parenrightbigm/2 P(m)ν (x) a49x = 1a51a66a63 a129a130a53 a61a145a63 a45a46 Legendre a47a48 a49x = 1a51a53a146a123 a124 a69a70 ρ = m/2a53a77a29 star Q(m)ν (x)a49x = 1a51a63 a72(x?1)?m a53 a47 a82 a93a94 a53 a61 a71a72 a61 parenleftbig1?x2parenrightbigm/2 Q(m) ν (x) a49x = 1a51a66 a59 a92a63a93a94 a53 a61a145a67a63 a45a46 Legendre a47a48 a49x = 1a51a53a146a123 a124 a69a70 ρ = ?m/2a53a77 a29 star a129a130a131a132a147a148a77a49 x = 1a51a129a130a61a71a72a61 c2 = 0a29 a118a138a139a140x = ?1 a141a142a143 a29 star a144 a121 a59a149 a53 ν a136a61a150 a147 P ν(x)a63a151a152a153 a96 a61a145 a49 x = ?1a51 a85a63a144 a96 a93a94 a53 a29 star x = ?1a51a85a63 P(m)ν (x)a53ma154a155 a51 a61 star a71a72a61parenleftbig1?x2parenrightbigm/2 P(m)ν (x) a49x = ?1a51a66a156a63a93a94a53a29 star a73a157a88a89a49 x = ?1a51a129a130a53a147a148a61a158a59 a53a78a159 a63 Pν(x) a160 a80a161a162a82 a61a163 ν a73a164a165a166a96a29 star a167 a121a49a77a126 a133a168 a53 a63 P(m)ν (x) a61 a71a72a169a170a129ν ≥ ma29 a171a172a173 a128 a53a174a175 a61a85 a148 a133 a157a45a46 Legendre a47a48 a49a129a130a131a132 a127 a53a77 a134a135a136 λl = l(l + 1), l = m,m + 1,m + 2,··· a134a135a137 a96 y l(x) = c1 parenleftbig1?x2parenrightbigm/2 P(m) l (x). a102a176a177c 1 = (?)m a61 a64a178 a134a135a137 a96a179a73 Pm l (x) a61 Pml (x) = (?)m parenleftbig1?x2parenrightbigm/2 P(m)l (x), Wu Chong-shi a180a181a182a183a184 a185 a112 a113 (a186) a1143a115 a187a73m a154la106 a45a46 Legendre a137 a96 a29 a45a46Legendre a137 a96 a61 a66 a63a188 a73 a134a135a136a189 a120a53a77a190 a163 a45a46 Legendre a47a48 a49a129a130a131a132 a127 a53 a134a135a137 a96a191 a84 a53 a61 a192a193 a61 a45a46 Legendre a137 a96a66a194a195a196a129 a67a197 a54a198a199a200a201a202a203a200a204a205a206a207 Legendre a208a209 a140a210a211 [?1,1] a212a213a214a61 integraldisplay 1 ?1 Pml (x)Pmk (x)dx = 0, k negationslash= l. a215a216a217a218a27 a31a61 a219a220a17a18 Legendre a19a20a221a222a61ma31a223a224 a27a225a226a227a228 a61a229a230a61 a21a231a232 a27 a233a234a39a40a235 a61 a17a18 Legendre a236 a228a27a237a228m a238a239a31a240a241 a27a29 a242a243a244 a19a20a35a245a61 a246a247a248a22a23a24a25 a61a221a249a250 a233a234a39a40a29 a215 a31a249a250 a13a251 a236 a228a233a234a252a27a253a254 a19a30 a29 a26 a232a255a0a1a2 a30a61a3 a248 a36a249a250Legendre a4a5a6 a27a233a234a252a7a8a27a9 a30 a29 a10 a167 a121k negationslash= l a61 a11a12a13 a116k < la29a121a63a61a83a84a45a46 Legendrea137 a96a53 a92a14a61 a15a43a16a17a43 a61a163 a86 integraldisplay 1 ?1 Pml (x)Pmk (x)dx = integraldisplay 1 ?1 parenleftbig1?x2parenrightbigm dmPk(x) dxm dmPl(x) dxm dx = parenleftbig1?x2parenrightbigm d mPk(x) dxm dm?1Pl(x) dxm?1 vextendsinglevextendsingle vextendsinglevextendsingle 1 ?1 ? integraldisplay 1 ?1 d dx bracketleftbiggparenleftbig 1?x2parenrightbigm d mPk(x) dxm bracketrightbigg dm?1P l(x) dxm?1 dx = ? integraldisplay 1 ?1 d dx bracketleftbiggparenleftbig 1?x2parenrightbigm d mPk(x) dxm bracketrightbigg dm?1P l(x) dxm?1 dx. a43a16a17a43 a59a106 a53a172a18a19a157a49a17a43a20a21a22a23 a59a24 a165a20a25 a61a85a150 a11a103 a63 a178a26a17 a137 a96a126P l(x) a53 a104a105a27 a28 a59a106 a87a125a29a53a192a30a173a29a78a72a31a32 a61 a49a43a16a17a43 m a106a33a61a85 a194a195a86a87 integraldisplay 1 ?1 Pml (x)Pmk (x)dx = (?)m integraldisplay 1 ?1 dm dxm bracketleftbiggparenleftbig 1?x2parenrightbigm d mPk(x) dxm bracketrightbigg Pl(x)dx. a34a35a173a82a36 a47 a53a26a17 a137 a96 a63 l a106Legendrea161a162a82a56a37a59a24a161a162a82 dm dxm bracketleftbiggparenleftbig 1?x2parenrightbigm d mPk(x) dxm bracketrightbigg a53a38a17 a29 a39a40a148 a133 a74 a24 a161a162a82a53 a106 a96a73 k?m+2m?m = k a29 a167 a121k < l a61 a117 a163a85 a101a86a45a46Legendre a137 a96a53 a67a197 a54 a29 square a188a41a42x = cosθ a61 a156a78a72a86a87a45a46 Legendre a137 a96 a67a197 a54a53a37 a59a43a44a45 a81a82 a61a163 integraldisplay pi 0 Pml (cosθ)Pmk (cosθ)sinθdθ = 0, k negationslash= l. a34a35 a61 a74a46 a133a168 a157 a67a197a47a48 sinθ a29 a57a58a49a50 a21 a128 a53a51a100 a61 a156a159a148a86a45a46 Legendre a137 a96a53 a49a47 a29 a74 a150 a147a49a72a173a101a76a103 a48 a53a52a82a126 Wu Chong-shi §21.1 a110a111 Legendrea112a113 a1144a115 a177k = l a163 a78 a29a121 a63a61integraldisplay 1 ?1 Pml (x)Pml (x)dx = (?)m integraldisplay 1 ?1 dm dxm bracketleftbiggparenleftbig 1?x2parenrightbigm d mPl(x) dxm bracketrightbigg Pl(x)dx. a133a168 a49a53a82a36a54a53a26a17 a137 a96 a63 la106Legendrea161a162a82a56a37a59a24 la106 a161a162a82 dm dxm bracketleftbiggparenleftbig 1?x2parenrightbigm d mPl(x) dxm bracketrightbigg = 12ll! d m dxm bracketleftbiggparenleftbig 1?x2parenrightbigm d l+m dxl+m parenleftbigx2 ?1parenrightbiglbracketrightbigg a53a38a17 a29 a16718a55a564a57 a53a174a175a78a58 a61a144 a17a43 a136 a53 a158a59a59a60a85a150a61a62 a74 a24 a161a162a82a53a63a64a65 a106 a162a29a39 a40a148 a133 a74 a24 a63a64a65 a106 a162a53a66a96 a63 (?)m 12ll! (2l)!(l ?m)! (l + m)!l! , a71a72 a61a85 a86a87 integraldisplay 1 ?1 Pml (x)Pml (x)dx = (2l)!2l(l!)2 (l + m)!(l ?m)! integraldisplay 1 ?1 xlPl(x)dx = (l + m)!(l ?m)! 22l + 1, a67a68a69 a59a70a188a41a42 x = cosθ a61 integraldisplay 1 ?1 Pml (cosθ)Pml (cosθ)sinθdθ = (l + m)!(l ?m)! 22l + 1. a244a71a72 a231 a222a61 a17a18 Legendre a236 a228a27a73 a4 a252a74a75a242a76Legendre a4a5a6 a27 a240 a247 a252a74a77a78a29 a244a79a29 Wu Chong-shi a180a181a182a183a184 a185 a112 a113 (a186) a1145a115 §21.2 a80a81a82a83a11a12 a84a21a85a78Laplace a19a20 a21a86a87 a253a40a26a27a88a89a90a91a29a92a93a94 a224a95a96a61 a97a98a99 a15a16a86a100Laplace a19a20 a27a101 a0 a7a102a103a104a105a29 a49a106a107a70a66 a127 a61a92 a77 a189 a120 a63 1 r2 ? ?r parenleftbigg r2?u?r parenrightbigg + 1r2 sinθ ??θ parenleftbigg sinθ?u?θ parenrightbigg + 1r2sin2θ ? 2u ?φ2 = 0, uvextendsinglevextendsingleθ=0 a129a130a61 uvextendsinglevextendsingleθ=pia129a130, uvextendsinglevextendsingleφ=0 = uvextendsinglevextendsingleφ=2pi, ?u?φ vextendsinglevextendsingle vextendsingle φ=0 = ?u?φ vextendsinglevextendsingle vextendsingle φ=2pi , uvextendsinglevextendsingler=0 a129a130a61 uvextendsinglevextendsingler=a = f(θ,φ). a48a10818a55a563a57 a53 a70a109a61 a110u(r,θ,φ) = R(r)S(θ,φ) a61 a178a173a128a53 a47a48 a56a111 a106a112 a130a131a132a43a113 a41a114a61 a86 d dr bracketleftbigg r2 dR(r)dr bracketrightbigg ?λR(r) = 0, uvextendsinglevextendsingler=0a129a130, a56 1 sinθ ? ?θ bracketleftbigg sinθ?S(θ,φ)?θ bracketrightbigg + 1sin2θ ? 2S(θ,φ) ?φ2 + λS(θ,φ) = 0, Svextendsinglevextendsingleθ=0 a129a130a61 Svextendsinglevextendsingleθ=pia129a130, Svextendsinglevextendsingleφ=0 = Svextendsinglevextendsingleφ=2pi, ?S?φ vextendsinglevextendsingle vextendsingle φ=0 = ?S?φ vextendsinglevextendsingle vextendsingle φ=2pi . a74a66 a63a59a24a134a135a136a189 a120 a61a115a116a117a118a119 a205a120a121a122a123a124 a29 a73a157a148 a133a134a135a136λ a56a125a194a53 a134a135a137 a96 a61 a78a72a118a110S(θ,φ) = Θ(θ)Φ(φ) a61 a69 a59a70 a43a113 a41a114a61a85 a129 1 sinθ d dθ bracketleftbigg sinθdΘ(θ)dθ bracketrightbigg + bracketleftbigg λ? μsin2θ bracketrightbigg Θ(θ) = 0, Θ(0)a129a130a61 Θ(pi)a129a130 a56 Φ primeprime + μΦ = 0, Φ(0) = Φ(2pi), Φprime(0) = Φprime(2pi). a74a126 a24 a176 a104 a43 a47a48 a53 a134a135a136a189 a120 a62a127a128 a174a175a103 a61 a43a129a130a173 a59a57 a56 a56 18a55a561a57 a29 a74 a60a61a144 a121a131 a104 a43 a47a48 a53 a134a135a136a189 a120 a61 a75 a61a134a135a136a85a63 λl = l(l + 1), l = 0,1,2,3,···, a64 a144 a194a121 a59a24a134a135a136 λl a61 a1292l + 1 a24a134a135a137 a96 Slm1(θ,φ) = Pml (cosθ)cosmφ, m = 0,1,2,···,l, Slm2(θ,φ) = Pml (cosθ)sinmφ, m = 1,2,···,l. a74a132 a134a135a137 a96 a61a133 a187a73a134a135a136a137 a208a209a61 a67a134a135a138 a208a209 a29 a134a135a136a189 a120a53a139a15a140 a63 2l + 1a61a141 a121a176 a104 a43 a47a48a134a135a136a189 a120a71a142a78a53a139a15a140 2 a29 a143a121Ra53a176 a104 a43 a47a48a61 a49 a56 20a55a563a57 a126 a127a128 a174a175a103 a29 a145 a49a129a130a131a132 a127 a53a77 a63 Rl(r) = rl a29 a74 a60a61 a131 a104 a43 a47a48a92 a77 a189 a120a53a144a77 a85a63 ulm1(r,θ,φ) = rlPml (cosθ)cosmφ, l = 0,1,2,···, m = 0,1,2,···,l Wu Chong-shi §21.2 a185a145a146a147a112a113 a1146a115 a56 ulm2(r,θ,φ) = rlPml (cosθ)sinmφ, l = 0,1,2,···, m = 1,2,···,l. a64 a59a149 a77 a68 a73 u(r,θ,φ) = ∞summationdisplay l=0 lsummationdisplay m=0 rlPml (cosθ)[Alm cosmφ + Blm sinmφ]. a148a14919 a55a564a57 a56 a134a55a561a57 a53a174a175 a61 a78a72a150 a133a61la151ma203a200a205a134a135a136a137a208a209a140a152a153 4pia154 a155a156 a212a157a158a159a213a214 a205 a61 integraldisplay pi 0 Pml (cosθ)Pnk(cosθ)sinθdθ integraldisplay 2pi 0 cosmφcosnφdφ = 0, l negationslash= k, m negationslash= n, integraldisplay pi 0 Pml (cosθ)Pnk(cosθ)sinθdθ integraldisplay 2pi 0 sinmφsinnφdφ = 0, l negationslash= k, m negationslash= n, integraldisplay pi 0 Pml (cosθ)Pnk(cosθ)sinθdθ integraldisplay 2pi 0 cosmφsinnφdφ = 0, l negationslash= k, m negationslash= n. a160 a60a61 a156a78a72a79 a133 a134a135a136a137 a208a209 a205a161 a118 integraldisplay pi 0 bracketleftbigPm l (cosθ) bracketrightbig2 sinθdθintegraldisplay 2pi 0 cos2mφdφ = (l + m)!(l ?m)! 2pi2l + 1 (1 + δm0), integraldisplay pi 0 bracketleftbigPm l (cosθ) bracketrightbig2 sinθdθintegraldisplay 2pi 0 sin2mφdφ = (l + m)!(l ?m)! 2pi2l + 1. a49a162a163a173a176a95a53 a63 a37 a59a43 a81a82a53a106 a128a164 a56 a137 a96 a29 star a56a59a61a63 a178 a134a135a136a189 a120 Φprimeprime + μΦ = 0, Φ(0) = Φ(2pi), Φprime(0) = Φprime(2pi). a53a77a49a81a82a173a79a80 a134a135a136 μm = m 2, m = 0,±1,±2,±3···, a134a135a137 a96 Φ m(φ) = eimφ. a74 a60a61a144 a194a121 a59a24a134a135a136 λl = l(l + 1)a61l = 0,1,2,3,···a61 a131 a104 a43 a47a48a134a135a136a189 a120a53 a134a135a137 a96 a85a63 Slm(θ,φ) = P|m|l (cosθ)eimφ, m = 0,±1,±2,···,±l. a74 a60a92a14 a53a106 a128a164 a56 a137 a96 a61 a125 a67a197 a143a66a56 a49a47 a78a72a79a80a165a139a166a53a81a82 a61integraldisplay pi 0 integraldisplay 2pi 0 Slm(θ,φ)S?kn(θ,φ)sinθdθdφ= (l +|m|)!(l ?|m|)! 4pi2l + 1δlkδmn. a76 a220a84a21a27a13a251 a236 a228 a31a167a236 a228 a61a168 a243 a21 a233a234a39a40 a36a169a19 a27a170 a6 a235 a61a171a172a173 a235a27 a0a1a13 a251 a236 a228a174 a167a175a176 a29 a173a177a178 a71 a229a31 a92a93a179 a249 a13a251 a236 a228a27 a169a19a180 a92 a233a103a29 Wu Chong-shi a180a181a182a183a184 a185 a112 a113 (a186) a1147a115 star a56a181a61 a102a176a165 a63a182 a95a98 a59a183 a53a106 a128a164 a56 a137 a96 a29a184a185 a61 Yml (θ,φ) = radicalBigg (l ?|m|)! (l +|m|)! 2l + 1 4pi P |m| l (cosθ)e imφ, m = 0,±1,±2,···,±l. a74a90 a85 a129 a67a197 a98 a59 a143a66 integraldisplay pi 0 integraldisplay 2pi 0 Yml (θ,φ)Yn?k (θ,φ)sinθdθdφ = δlkδmn. a247a186a217a218 a27 a31a61 a21 a97 a241 a27a187a188a235 a61 Yml (θ,φ) a189a189 a22 a97 a241 a27 a224a190 a29 a21a191a248a192a193 a171a194a195a196 a219a29 Yml (θ,φ) a224a190 a235a27a197a219a103a198a199a200a242a243a201a202 a61 a215 a31a229 a92 P?ml (x) = (?)m(l ?m)!(l + m)!Pml (x).