Wu Chong-shi a0a1a2a3 a4 a5 a6 ( a7) a8Helmholtz a9a10a11a12a13a14a15a16a17a18a19a20a21a22a23a24a25a26 Legendrea9a10 1 sinθ d dθ parenleftbigg sinθdΘdθ parenrightbigg + bracketleftbigg λ? μsin2θ bracketrightbigg Θ = 0 a27a28a29a30a31a32a33a34 a21 Legendrea9a10 1 sinθ d dθ parenleftbigg sinθdΘdθ parenrightbigg + λΘ = 0, a35 a19a36x = cosθ, y(x) = Θ(θ)a21a37a38a22 a8 a29a39a40a41a42 d dx bracketleftbiggparenleftbig 1?x2parenrightbig dydx bracketrightbigg + bracketleftbigg λ? μ1?x2 bracketrightbigg y = 0 a43 d dx bracketleftbiggparenleftbig 1?x2parenrightbig dydx bracketrightbigg + λy = 0. a44a45a46a47a48a49a50 a9a10 a30a51 a21 a29a39a30a52a53a54a55a28a56 a11a17a18a19a20a57a58 a30a59a60a61 Wu Chong-shi §19.1 Legendrea62a63a64a65 a662a67 §19.1 Legendre a68a69a70a71 a11a72a73Legendrea9a10 a30a51a30a74a75a34a76a77a78 a21 a79a80a81a82 a17a9a10 a30a51a83a84 a47 ( a85a86a87a88)a21 a89a90a91 a22 a27a92 Legendre a9a10 a30a51a30a51a83a54 a35 a73a93a94 a61 star Legendrea95a96(a97a98a99z a100a101a102a103a104) d dz bracketleftbiggparenleftbig 1?z2parenrightbig dwdz bracketrightbigg + λw = 0. a105a106a107a108a109 a21z = ±1a110z = ∞a21 a111a112a113 a100a114a115 a108a109 a61 a116a117 a21 a118a119 a97 a106a107a109a120a121 a100 a108a109a122 a21Legendre a95a96a99a123a124a125a126a127a123a128 a61 star z = 0a109a100Legendrea95a96a99a129 a109 a21 a116a117 a21a95a96a99a123a124a130 z = 0a109a131a132a133a99a134a135 a132|z| < 1 a136a123 a128a21 a120 a130a137a138 a131 Taylor a139a140 a61a141a142a143a144a145a146a147a148 a119a149a107a150a151a152a153 a99a154a123a21a155a156a100 w1(z) = ∞summationdisplay n=0 22n (2n)! Γ parenleftBig n? ν2 parenrightBig Γ parenleftbigg n + ν + 12 parenrightbigg Γ parenleftBig ?ν2 parenrightBig Γ parenleftbiggν + 1 2 parenrightbigg z2n, w2(z) = ∞summationdisplay n=0 22n (2n + 1)! Γ parenleftbigg n? ν ?12 parenrightbigg Γ parenleftBig n + 1 + ν2 parenrightBig Γ parenleftbigg ?ν ?12 parenrightbigg Γ parenleftBig 1 + ν2 parenrightBig z2n+1, a157 a144 ν(ν + 1) = λ. a158 a97 a149a107 a154a123a159a123a128a160a161a21 a120 a130a162a163 Legendrea95a96a99a123a124 a157a164a165a166 a136a99a167a168a169 a61 a170 a100a21 a152a171a172a173 a21 a124a139a140a123a174a175 a132 a99 a132a176a177 a21 a178a179a180 a21a124 z = ±1a97 a149a109 a21a95a96a99a139a140a123a181a182a183a184a123a128 a61 a97a185 a177 a127a186 a148 a99a123a99a187a188a189a169 a120 a130a190 a148a61 a191a192w 1(z)a21a193na194a195a196a197a21 a157a198 a140 c2n = 2 2n (2n)! Γ parenleftBig n? ν2 parenrightBig Γ parenleftbigg n + ν + 12 parenrightbigg Γ parenleftBig ?ν2 parenrightBig Γ parenleftbiggν + 1 2 parenrightbigg ～ 2 2n (2n + 1)2n+1/2e?(2n+1)√2pi parenleftBig n? ν2 parenrightBign?(ν+1)/2 e?n+ν/2√2pi Γ parenleftBig ?ν2 parenrightBig parenleftbigg n + ν + 12 parenrightbiggn+ν/2 e?n?(ν+1)/2√2pi Γ parenleftbiggν + 1 2 parenrightbigg =a129a140× 1n. Wu Chong-shi a199a200a201a202 a203 a204 a205 ( a206) a663a67 a97 a180a207 a21 a118a119 a182 a107 a129a140a208 a122 a21 w1(z)a124z = ±1a209a210a99a211 a131 a21a110 ln 11?z2 = ∞summationdisplay n=1 1 nz 2n a212 a125a213a214 a61 a116a117 a21 w1(z)a124 z = ±1 a191 a140a215a216 a61 z = ±1 a100 w1(z)a99a217 a109 a61 a172a218a158 Legendre a95a96a124 z = 0a99 a141 a182a123w1(z)a123a128a160a161a163a125a126a127 a177 a21a155a182a183a100a182 a107a219a220a221 a140 a61 a191a192w 2(z)a21a193na194a195a196a197a21a222 a105 c2n+1 = 2 2n (2n + 1)! Γ parenleftbigg n? ν ?12 parenrightbigg Γ parenleftBig n + 1 + ν2 parenrightBig Γ parenleftbigg ?ν ?12 parenrightbigg Γ parenleftBig 1 + ν2 parenrightBig ～ 2 2n (2n + 2)2n+3/2e?(2n+2)√2pi × parenleftbigg n? ν ?12 parenrightbiggn?ν/2 e?n+(ν?1)/2√2pi Γ parenleftbigg ?ν ?12 parenrightbigg parenleftBig n + 1 + ν2 parenrightBign+(ν+1)/2 e?n?1?ν/2√2pi Γ parenleftBig 1 + ν2 parenrightBig =a129a140× 12n + 1. a223 a130a21 a118a119 a182 a107 a129a140a208 a122 a21 w2(z)a124z = ±1a209a210a99a211 a131 a21a110 ln 1 + z1?z = ∞summationdisplay n=1 2 2n + 1z 2n+1 a212 a125a213a214 a61 a116a117 a21 w2(z)a124 z = ±1 a222 a191 a140a215a216 a61 z = ±1 a222a100 w2(z)a99a217 a109 a61 a158 Legendre a95a96a124 z = 0a99 a141a224 a123w2(z)a123a128a160a161a163a125a126a127 a177 a21a155a222a100a182 a107a219a220a221 a140 a61 star a225 a120 a130a124 z = 1(a226z = ?1)a109a99a227a166a136a147a123 Legendrea95a96 a61 a228a192z = ±1 a100a95a96a99a114a115 a108a109 a21a95a96a124a229 a166 0 < |z ?1| < 2 a136 a105a149a107 a114a115a123a21a230 a120a231 w(z) = (z ?1)ρ ∞summationdisplay n=0 cn(z ?1)n, a232a233Legendre a95a96a21a234 a120 a130a162a163a124 z = 1a109a99a235a236a95a96 ρ(ρ?1) + ρ = 0. a223 a130a21ρ1 = ρ2 = 0a61a97 a180a207Legendre a95a96a124z = 1 a109a227 a166 a136a99 a141 a182a123a237a238 a177 a100a124 a132a166|z ?1| < 2 a136a123a128a99a21a239 a141a224 a123a115a182a183a240 a105a191 a140a241a21a130 z = 1(a110z = ?1)a131a217 a109 a61 a242a243 a129a244a245a95a96a139a140a123a246a99a236a247a248a249a21 a120 a130 a147a148 Legendre a95a96a124z = 1a109a227 a166 a136a99 a141 a182a123 Pν(z) = ∞summationdisplay n=0 1 (n!)2 Γ(ν + n + 1) Γ(ν ?n + 1) parenleftbiggz ?1 2 parenrightbiggn , Wu Chong-shi §19.1 Legendrea62a63a64a65 a664a67 a250a131ν a251 a141 a182a252Legendrea221a140a253 a141a224 a123 a120a254a131 Qν(z) = 12Pν(z) bracketleftbigg ln z + 1z ?1 ?2γ ?2ψ(ν + 1) bracketrightbigg + ∞summationdisplay n=0 1 (n!)2 Γ(ν + n + 1) Γ(ν ?n + 1) parenleftbigg 1 + 12 +···+ 1n parenrightbiggparenleftbiggz ?1 2 parenrightbiggn , a250a131ν a251 a141a224 a252Legendrea221a140a21 a157 a144γ a100 Eulera140a21ψ(z)a100Γa221a140a99 a191 a140a244a255 a61 a0a1a2a3P ν(z)(a4a5a24a6a7a8 a9 a21 a29a10a27z = ?1a43z = ∞ a11a12a13 a30a14 a15a2a3) a43Q ν(z) a30a14 a15 a54a16 a17a18a19 a54a30a20 a19 a21a21 a60a22a23a53a31a24a25a26a61 Wu Chong-shi a199a200a201a202 a203 a204 a205 ( a206) a665a67 §19.2 Legendre a27a28a29 a30 a189 a165a166 a136x2 + y2 + z2 < a2 a99Laplacea95a96a31 a220a32a33 ?2u = 0, uvextendsinglevextendsingleΣ = f(Σ), a157 a144Σ a232 a167 a30 a127x 2 + y2 + z2 = a2 a177 a99a102 a109 a61 a34a35 a163a36a124 a223a37a171 a99a38a39 a165a166 a99a187a188a189a40a21a41 a42a43a44a45a46a47a48a49a50 a147 a123a97 a107 a183a123 a32a33 a21a239 a112 a43a158a47a48a51a52a53a54 a124 a30a133 a61 a172a218 a31a55a56a57a187 a105a58a59 a182 a107 ( a60a61 a30a133 a99) a62a183a63a64a65a184a102a99 a191a250a151 a21 a66a67 a21a193 a42 a222a234a68a193 a158 a97 a107a191a250 a63a99a95a69 a254a131a70a71 a99a95a69 a61 a97a72a73a74 a119a75 a236 a198a76 a21 a223a77 a147 a99a78a79 a221 a140 ua193 a42 a234a80φa152a153a21 u = u(r,θ). a81a82 a186 a148 a183a123 a32a33 a124 a30a75 a236 a198a83 a99a187a188a189a169 a61 a170 a100a21a84 a77a85a86a87 star Laplace a95a96a124θ = 0 a110θ = pia95a69 a177 a184a88a89a21a124a97a90 a109a177a91a157 a103a92a93a124u(r,θ)a191θ a99a134a94a95 a140 a61 a158 Laplace a95a96a96a186a163 a30a75 a236 a198 a197a21 a131a119a97a98 a183a123 a32a33 a99a99a100 a151 a21a101a102a103 a91a177 u(r,θ) a124θ = 0 a110θ =pia95a69 a177 a99 a105 a55a56a57 a61 star Laplacea95a96a124 a75 a236a104 a109 r = 0 a222a184a88a89a21a124a105 a109a91a157 a103a92a93a124 u(r,θ)a191r a99a134a94a95a140 a61 a158Laplace a95a96a96a186a163 a30a75 a236 a198 a197a21 a131a119a97a98 a183a123 a32a33 a99a99a100 a151 a21a225a101a102a103 a91a177u(r,θ) a124 a75 a236 a104 a109r = 0 a106a99 a105 a55a56a57 a61 a183a123 a32a33 a124 a30a75 a236 a198a83 a99 a212a107 a167a168a189a169a68a105a100 1 r2 ? ?r parenleftbigg r2?u?r parenrightbigg + 1r2 sinθ ??θ parenleftbigg sinθ?u?θ parenrightbigg = 0, uvextendsinglevextendsingleθ=0 a105a55a21 uvextendsinglevextendsingleθ=pi a105a55a21 uvextendsinglevextendsingler=0 a105a55a21 uvextendsinglevextendsingler=a = f(θ). a245a108a102a103 a61a109 u(r,θ) = R(r)Θ(θ), a232a233 a95a96a110 a105 a55a56a57a21a234 a121 a195a245a108a102a103a239a162a163 1 sinθ d dθ parenleftbigg sinθdΘ(θ)dθ parenrightbigg + λΘ(θ) = 0 Θ(0)a105a55 Θ(pi)a105a55 a110 d dr parenleftbigg r2dR(r)dr parenrightbigg ?λR(r) = 0, a157 a144λ a100a245a108a102a103a197a110a111a99a112a183a113a140 a61 Legendrea95a96a21 a114a177a105 a55a56a57a21 a115 a88a116a117 a220a32a33 a61 a60a129a159a102a118 x = cosθ, y(x) = Θ(θ)a21 a111a112a158 Wu Chong-shi §19.2 Legendrea119a120a121 a666a67 a112a183a113a140λa186a88ν(ν + 1)a21a116a117 a220a32a33 a234a102 a131 d dx bracketleftbiggparenleftbig 1?x2parenrightbig dydx bracketrightbigg + ν(ν + 1)y = 0, y(±1)a105a55. a122a123a124a125a126a123a124a127a128 star a129a130a131 Legendre a132a133a134 x = 0 a52a135a136a137a138a139a140a141a142a143a144a145a146a147a122a144a61 a177 a182a148 a145a146a149a148 a119 a97 a149a107a150a151a152a153 a123a99a189a169a21a225 a171a150a119a191a192 a182a151a99 λ(a226 ν) a220a21a97 a149a107 a123a124 x = ±1a113a100a191a140a215a216a99a61 a131a119a152 a162a95a96a99a123a124 x = ±1a153 a105 a55a21a234 a77 a147 λ( a226ν)a254a59a90a154a154 a220 a61 star a185Legendrea95a96a124x = 1a109a227 a166 a136a99 a149a107a150a151a152a153 a123 Pν(x)a110Qν(x)a148a215 a50a37a171 a61 Pν(x) = ∞summationdisplay n=0 1 (n!)2 Γ(ν + n + 1) Γ(ν ?n + 1) parenleftbiggx?1 2 parenrightbiggn , Pν(x)a124x = 1a109a100a123a128a99a21a193 a42 a222a234a100 a105 a55a99a253 Qν(x) = 12Pν(x) bracketleftbigg ln x + 1x?1 ?2γ ?2ψ(ν + 1) bracketrightbigg + ∞summationdisplay n=0 1 (n!)2 Γ(ν + n + 1) Γ(ν ?n + 1) parenleftbigg 1 + 12 +···+ 1n parenrightbiggparenleftbiggx?1 2 parenrightbiggn , Qν(x)a124x = 1a109a100 a191 a140a215a216a99 a61 a158Legendre a95a96a99a60a123a186a88 y(x) = c1Pν(x) + c2Qν(x), a228a192a77 a147 a123a124 x = 1a105a55a21a101a102 a105c 2 = 0a21a239 a112 a184a155 a254 c 1 = 1 a61 a77 a147 a123a124 x = ?1 a109 a222 a105 a55a21a234 a120 a130a183 a148 a116a117 a220 λ = ν(ν + 1) a21a185a239 a147a148 a213a68a99a116a117 a221 a140 a61 a124x = ?1a109a21Pν(x)a99a140 a220a131 Pν(?1) = ∞summationdisplay n=0 (?)n (n!)2 Γ(ν + n + 1) Γ(ν ?n + 1) = ? sinνpi pi ∞summationdisplay n=0 Γ(n + ν + 1)Γ(n?ν) (n!)2 . a228Stirling a156a169 a120 a130a157a158a162 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 ～ 1 n a223 a130 Wu Chong-shi a199a200a201a202 a203 a204 a205 ( a206) a667a67 star a191a192a182a151a99 ν a220a21Pν(x)a124x = ?1a109a215a216 a61 star a92 a77P ν(x)a100 a152a159 a139a140a21a155a234a184 a120a121 a124 x = ?1a109a105a55a253 star a77a152a162a116a117 a220a32a33a105 ( a160a161)a123a21a101a102 a77 a147 P ν(x)a184a100 a152a159 a139a140a21a162a163a164 a131a219 a241a169 a61 a185Pν(x)a99a187a188a189a169a190a21a97a92 a121 a215a165a124 ν a131a160a166 a107 a140a197 a61 a223 a130a21a116a117 a220a32a33 a99a123a234a100 a116a117 a220 λ l = l(l + 1), l = 0,1,2,3,···, a116a117 a221 a140 yl(x) = Pl(x). Pl(x)a100a182a107l a251a219a241a169a21a250a131 la251Legendrea219a241a169a21 Pl(x) = lsummationdisplay n=0 1 (n!)2 (l + n)! (l?n)! parenleftbiggx?1 2 parenrightbiggn . a81a82 a162a163Legendrea219a241a169a124x = 1a109a99a140 a220a87 Pl(1) = 1. Legendre a14a167a76a10a35a11a44a168a15a169a170a30a51a73a171a30a21a10a35a11 Legendrea9a10a11 a17a172a173a174 a16 a30 a44a168a2a3 a73a171 a30a61 a175 a148a176a177 a99a178 a107 Legendrea219 a241a169a99a167a168a169 a87 P0(x) = 1, P1(x) = x, P2(x) = 12 parenleftbig3x2 ?1parenrightbig, P3(x) = 12 parenleftbig5x3 ?3xparenrightbig, P4(x) = 18 parenleftbig35x4 ?30x2 + 3parenrightbig. a155a156a99a179a189a180a179 19.1a61 Wu Chong-shi §19.2 Legendrea119a120a121 a668a67 a18119.1 Legendre a119a120a121 Wu Chong-shi a199a200a201a202 a203 a204 a205 ( a206) a669a67 §19.3 Legendre a27a28a29a70a182a183a184a185 Legendrea219a241a169a99a244a245a167a186a100 Pl(x) = 12ll! d l dxl parenleftbigx2 ?1parenrightbigl . a97 a107 a167a168a169a222 a250a131 Rodrigues a156a169 a61 a187 a116a131 parenleftbigx2 ?1parenrightbigl = (x?1)l[2 + (x?1)]l = lsummationdisplay n=0 l! n!(l?n)!2 l?n(x?1)l+n, a223 a130 1 2ll! dl dxl parenleftbigx2 ?1parenrightbigl = dl dxl lsummationdisplay n=0 1 n!(l?n)!2 ?n(x?1)l+n = lsummationdisplay n=0 1 n!(l?n)! (l + n)! n! parenleftbiggx?1 2 parenrightbiggn . a97a72a234 a150a207a119 Legendrea219 a241a169a99a244a245a167a186 a61 square a185 Legendrea219a241a169a99a244a245a167a186a21a89a162 a120 a130a190 a148Legendre a219 a241a169a99 a108a188a151a87 l a131a188 a140a197 Pl(x) a100 a188a221 a140a253 la131a108a140a197Pl(x)a100 a108a221 a140a21a162 Pl(?x) = (?)lPl(x). a189a190a191P l(x)a124x = 1 a109 a99a140 a220 a21a192 a120 a130a162a163 Pl(x)a124x = ?1a109a99a140 a220 a21 Pl(?1) = (?1)l. a185Legendrea219a241a169a99a244a245a167a186a225 a120 a130a193a194 a147a148Legendre a219 a241a169 a144 a223a105a195 a241a99 a198 a140a21a185a239a95 a148 Legendrea219a241a169a99a196a182 a107a197a207 a167a168a169 a61 a131a117 a21 a120a198 parenleftbigx2 ?1parenrightbigl a137a138a21 parenleftbigx2 ?1parenrightbigl = lsummationdisplay r=0 (?)r l!r!(l ?r)!x2l?2r, a42a76a199 a241a244a255 la251a21 dl dxl parenleftbigx2 ?1parenrightbigl = dl dxl lsummationdisplay r=0 (?)r l!r!(l?r)!x2l?2r = [l/2]summationdisplay r=0 (?)r l!r!(l ?r)! (2l?2r)!(l ?2r)! xl?2r, a228a192 a244a255 l a251 a76 a21 a219 a241a169a99a251a140 a77a200 a177 l a251a21 a223 a130a97a98a110a169a99 a177a201 a234 a228 a244a255a202a99 l a102 a131 a244a255 a76 a99 [l/2]a61a191a203a182 a83 Legendrea219 a241a169a99a244a245a167a186a21a234a162a163 Pl(x) = [l/2]summationdisplay r=0 (?)r (2l?2r)!2l r!(l?r)!(l ?2r)!xl?2r. a185a97 a107 a167a168a169a204 a81a82 a147a148 Legendre a219 a241a169Pl(x)a124x = 0a109a99a140 a220a87 P2l(0) = (?)l (2l)!22l l!l!, P2l+1(0) = 0. Wu Chong-shi §19.4 Legendrea119a120a121a64a205a206a207a208a209 a6610a67 §19.4 Legendre a27a28a29a70a210a211a212a213a214 Legendre a14a167a76a10a35a11 a44a168a15a169a170 a30 a44a168a2a3 a73a171 a30 a21a215a216a21a217 a44a168a15a169a170 a73a218a21a22 a27a219a220Legendre a14a167a76a30a221a222a54 a21a223a224a225a226 a128a227 Legendre a228a229a230a134a231a232 [?1, 1] a233a234 a235 a21 integraldisplay 1 ?1 Pl(x)Pk(x)dx = 0, k negationslash= l. a120 a130a185a95a96 a148 a215 a50a150a207 a61 a36a124 a45 a196a182a236a95a246 a150a207 a97 a107a190a218 a61 a237 a158a238a239a245 integraldisplay 1 ?1 xkPl(x)dx, a157 a144k a110la113a100a160a166 a107 a140 a61 star a191a192a97 a107 a239a245a21a185a240a239 a221 a140a99 a108a188a151a120 a130a241a164integraldisplay 1 ?1 xkPl(x)dx = 0, a193k ±l =a108a140. star a193k ±la131a188a140a197a21 a120a198 P l(x) a45 a155a99a244a245a167a186 a232a233 a21 a192 a100 a105 integraldisplay 1 ?1 xkPl(x)dx = 12l l! integraldisplay 1 ?1 xk d l dxl parenleftbigx2 ?1parenrightbigl dx = 12l l! bracketleftbigg xk d l?1 dxl?1 parenleftbigx2 ?1parenrightbigl vextendsinglevextendsinglevextendsingle1 ?1 ? integraldisplay 1 ?1 dxk dx dl?1 dxl?1 parenleftbigx2 ?1parenrightbigl dxbracketrightbigg. a0a1 d l?1 dxl?1 parenleftbigx2 ?1parenrightbigl a58a242 a19a243a17 a215a244 parenleftbigx2 ?1parenrightbig a21a245 a27 a11a246a247a248a16a249x = ±1a9a21a17a250a251 a17a73a252 a30a167 a242 a19 a11 0a21 a1a10a91a17 integraldisplay 1 ?1 xkPl(x)dx = 12l l! integraldisplay 1 ?1 (?)1dx k dx dl?1 dxl?1 parenleftbigx2 ?1parenrightbigl dx. a97a72a21a245a253a239a245a182a251a21 a157a254a218 a234a167a36a124 a106 a95a127 a87 (1) a40a19a242a255 a221a0a1 a253 (2) a92a2a3parenleftbigx2 ?1parenrightbigl a30a82a2a3a4a242a255a253 (3) a92a2a3xk a30a82a2a5a6a242a255a61 a97a72a21a245a253a239a245 la251 a76 a21a244a255a7a238a234a125a253a65a8a163 a221 a140 xk a177a9a190a10a11a12a13integraldisplay 1 ?1 xkPl(x)dx = 12l l! integraldisplay 1 ?1 (?)ld lxk dxl parenleftbigx2 ?1parenrightbigl dx. a14a15a16a17a18a19a20a9a21a22 k < l a9a23a24 xk a25a26 l a27 a21a28a13 0 a9a29a22 integraldisplay 1 ?1 xkPl(x)dx = 0, a30k < l. Wu Chong-shi a31a32a33a34 a35 a36 a37 ( a38) a3911a40 a41a21a18a19a20a22 k ≥ l a9a42a29 k ±l = a43 a24a9a44a45a46a47 k = l + 2n a9a29a22 integraldisplay 1 ?1 xl+2nPl(x)dx = 12l l! integraldisplay 1 ?1 (?)ld lxl+2n dxl parenleftbigx2 ?1parenrightbigl dx = 1 2l l! (l + 2n)! (2n)! integraldisplay 1 ?1 x2n parenleftbig1?x2parenrightbigl dx. a48a12a49 x2 = t a9a50a51a52 B a23a24a11a19a53a54a55a56a57 integraldisplay 1 ?1 xl+2nPl(x)dx = 12l l! (l + 2n)!(2n)! integraldisplay 1 0 tn?1/2 (1?t)l dt = 12l l! (l + 2n)!(2n)! Γ parenleftbigg n + 12 parenrightbigg Γ(l + 1) Γ parenleftbigg n + l + 32 parenrightbigg = (l + 2n)!2l+2n n! √pi Γ parenleftbigg n + l + 32 parenrightbigg = 2l+1(l + 2n)!(l + n)!n!(2l + 2n + 1)!. a58a59a22 k = l a9a60 n = 0 a15a9 integraldisplay 1 ?1 xlPl(x)dx = l!2l √pi Γ parenleftbigg l + 32 parenrightbigg = 2l+1 l!l!(2l + 1)!. a61a62a63a64a10a65a66a9a67a68a69a70a71 xk a72 a69a73 k a74a75 Legendre a76a77a78 a72 a73a71 l a9a79a80a9a69a70a71 xk a81 l a73 Legendre a76a77a78 a72a82a83a84a85a86 [?1, 1] a87 a72a83a88a89a90a91 0 a92 a93a94a95a61a62a63a64a10a96a52a29a56a57 integraldisplay 1 ?1 Pl(x)Pk(x)dx. star a30 k negationslash= l a15a9a45a46a97a98 k < l a92a99a100a101 Pk(x) a22 k a27a102a103a104 a9a45a105 l ?k = a106 a24a107 a43 a24a9a14a108 a102a103a104a109 a63a110 a103a111 a53 P l(x) a63a56a57a112a22 0 a9a113a53a11a114a66a115a116a117a73a71 a72 Legendre a76a77a78 a84a85a86 [?1, 1] a87a118a119a92 star a120 a62a121a105 k = l a63a122a123 a92 a14a15a124a125a19a53a126a55a21a108 P l(x) a63a110 a103 a9a125a127a128 a103 a56a57a9 integraldisplay 1 ?1 Pl(x)Pl(x)dx = integraldisplay 1 ?1 bracketleftbigc lxl + cl?2xl?2 + cl?4xl?4 +··· bracketrightbigP l(x)dx. a129a130a9a131a115a132a21 a103a133 l a27 Legendre a102a103a104 a63 a111 a56a63a56a57a45a13 0 a134 a9a135a136a110 a103a133 l a27 Legendre a102a103a104 a63 a111 a56a63a56a57a137a13 0 a92 a29a22a9a11a16 integraldisplay 1 ?1 Pl(x)Pl(x)dx = cl integraldisplay 1 ?1 xlPl(x)dx = cl ×2l+1 l!l!(2l + 1)!, cl a22 l a27 Legendre a102a103a104a109xl a103 a63a138a24a9 cl = (2l)!2l(l!)2, a113a53a9 Legendre a102a103a104 a63a139a140a22 integraldisplay 1 ?1 Pl(x)Pl(x)dx = 22l + 1. Wu Chong-shi §19.4 Legendre a141a142a143a144a145a146a147a148a149 a3912a40 star a95 Legendre a102a103a104 a63a150a151a152 a133 a139a140a153a50a154a155a9a156a19a53a126a157 integraldisplay 1 ?1 Pk(x)Pl(x)dx = 22l + 1δkl. star a158 a29 Legendre a102a103a104 a150a151a152a63a121a105 a133 a139a140a9a159a19a53a52 θ a13a160a12a161a162a163 a92integraldisplay pi 0 Pk(cosθ)Pl(cosθ)sinθdθ = 22l + 1δkl. a14a11a22a65a9 k a73 Legendre a76a77a78 Pk(cosθ) a81 l a73 Legendre a76a77a78 Pl(cosθ) a84a85a86 [0, pi] a87a164 a165a70a71 sinθ a118a119a92 a14a166a63a165a70a71 sinθ a118a167a168a169a170 a88a171a172 d dθ bracketleftbigg sinθdΘdθ bracketrightbigg + λsinθΘ = 0 a173a174a175a176 λ a177 a72 a70a71 sinθ a92 a48a13a178a179a23a24a63Legendre a102a103a104 a9a159a180a16a181a182a152a183a184a185a21a108a94a186a187 [?1, 1] a109 a57a188a189a190a63a23a24 f(x) a9 (a94a191a137a192a193a194a185a195a120) a19a53a196a197a13a198a24 f(x) = ∞summationdisplay l=0 clPl(x), a135 a109 a63a196a197a138a24 c l a19a53a199a200 Legendre a102a103a104 a63a150a151a152a201a202a9 cl = 2l + 12 integraldisplay 1 ?1 f(x)Pl(x)dx. a203 19.1 a204 a23a24 f(x) = x3 a205 Legendre a102a103a104 a196a197 a92 a206a207 1 a98 x3 = ∞summationtext l=0 clPl(x) a9a208 cl = 2l + 12 integraldisplay 1 ?1 x3Pl(x)dx. a199a200a61a62a63a121a105a9a19a53a209a210a9a131a115 l = 1 a133 3 a134 a9 c l a137a13 0 a92 x3 = c1P1(x) + c3P3(x). a196a197a138a24 c 1 a133 c3 a57a59a13 c1 = 32 integraldisplay 1 ?1 x4dx = 35, c3 = 72 integraldisplay 1 ?1 x3P3(x)dx = 25. a211a127a63a64a10a11a22 x3 = 35P1(x) + 25P3(x). a194 a212a213 lim N→∞ integraldisplay 1 ?1 vextendsinglevextendsingle vextendsinglef(x)? Nsummationtext l=0 clPl(x) vextendsinglevextendsingle vextendsingle 2dx = 0, a214a215a216a217 ∞summationtext l=0 clPl(x)a218a219a220a221a222 f(x)a223 Wu Chong-shi a31a32a33a34 a35 a36 a37 ( a38) a3913a40 a224a225a226a9a227a228a229a230c 1 a231a232 a9a233a234a235a236a237a238a228a239 c 3 a9a240a241a242a243a227a244a245a246a247a248a249 x = 1 a9 a250a251a252 c 1 + c3 = 1 a92 a206a207 2 a253 a13 x3 = c1P1(x) + c3P3(x) = c1x + c3 parenleftbigg5 2x 3 ? 3 2x parenrightbigg = 52c3x3 + parenleftbigg c1 ? 32c3 parenrightbigg x, a113a53 5 2c3 = 1, c1 ? 3 2c3 = 0. a42a254a159a19a53a202 a101 c3 = 25, c1 = 32c3 = 35. a206a207 3 a253 a13 x3 = c1P1(x) + c3P3(x), a255a0 P 3(x) a63a1a2a2a3a4 x = radicalbigg 3 5, a11a16 c1 = x 3 P1(x) vextendsinglevextendsingle vextendsinglevextendsingle x= √ 3/5 = x2vextendsinglevextendsinglex=√3/5 = 35, c3 = 1?c1 = 25. a158 a29 Legendre a102a103a104 a63a181a182a152a9a159a19a53a5a52a53 θ a13a160a12a161a162a163 a92 a14a15a9a6a10a7 a204 a23a24 f(θ) a205 Legendre a102a103a104 Pl(cosθ) a196a197a9 f(θ) = ∞summationdisplay l=0 clPl(cosθ), a208a196a197a138a24a13 cl = 2l + 12 integraldisplay pi 0 f(θ)Pl(cosθ)sinθdθ.