Wu Chong-shi a0a1a2a3 a4 a5 a6 (a1) §20.1 Legendre a7a8a9a10a11a12a13a14 Legendrea15a16a17a18a19a20a21a22a23a24a25a26a27a28a29a24a30 a31 a21a32a33a34 ra35a36a37a38a39a40a41a34a42a43a44 a45 a34 a42a43a46a21a34a24a41a47a48 z a49a50a51a44a52a53a34a42a43a21 (rprime, θ, φ)a34a24a42a22(a54a55a56φa57a58)a59a48 1√ r2 + rprime2 ?2rrprime cosθ = ? ???? ??? ? 1 r 1√ 1?2xt + t2, t = rprime r , 1 rprime 1√ 1?2xt + t2, t = r rprime, a60 a27x = cosθ a44a61a62a63a15a64a65a661/√1?2xt + t2 a24a40a64a67a68a48 1√ 1?2xt + t2 vextendsinglevextendsingle vextendsinglevextendsingle t=0 = 1 a30a21a52a69a24 a62a63a70a44a65a66 1/√1?2xt + t2 a21t = 0a34a71 a60a72a73a74 a18a75a76a24a44a77a78a79a80a81 Taylora82a83 1√ 1?2xt + t2 = ∞summationdisplay l=0 cltl, |t| < |x± radicalbig x2 ?1|. a70a84a85a86a82a83a87a66 cl a88 a18Legendrea15a16a17Pl(x)a44a59 1√ 1?2xt + t2 = ∞summationdisplay l=0 Pl(x)tl, |t| < |x± radicalbig x2 ?1|. a65a661/√1?2xt + t2 a59a89a48Legendrea15a16a17a24a90a91a65a66a30 a92 a93a94a95 a65a66 1/ √1?2xt + t2 a21t = 0a34a81Taylora82a83 1√ 1?2xt + t2 = 1radicalbig 1?2t + t2 ?2(x?1)t = 1 1?t bracketleftbigg 1? 2(x?1)t(1?t)2 bracketrightbigg?1/2 = 11?t ∞summationdisplay k=0 1 k! parenleftbigg ?12 parenrightbiggparenleftbigg ?32 parenrightbigg ··· parenleftbigg1 2 ?k parenrightbiggbracketleftbigg ?2(x?1)t(1?t)2 bracketrightbiggk = ∞summationdisplay k=0 (2k?1)!! k! (x?1) ktk(1?t)?(2k+1) = ∞summationdisplay k=0 (2k?1)!! k! (x?1) ktk ∞summationdisplay n=0 (2k + n)! n!(2k)! t n = ∞summationdisplay l=0 bracketleftBigg lsummationdisplay k=0 (l + k)! k!k!(l?k)! parenleftbiggx?1 2 parenrightbiggkbracketrightBigg tl. square a96a97Legendre a15a16a17a24a90a91a65a66a44a98a79a80a99a100a101a15a37 a97 a24a102a103a30a104a105a44a106 x = 1a44 a88 a99a100 1√ 1?2t + t2 = 1 1?t = ∞summationdisplay l=0 tl = ∞summationdisplay l=0 Pl(1)tl =? Pl(1) = 1. a107 a105a44 1√ 1?2xt + t2 = 1radicalbig 1?2(?x)(?t) + (?t)2, ∞summationdisplay l=0 Pl(x)tl = ∞summationdisplay l=0 Pl(?x)(?t)l, =? Pl(?x) = (?)lPl(x). Wu Chong-shi §20.2 Legendrea108a109a110a111a112a113a114a115 a1162a117 §20.2 Legendre a7a8a9a10a118a119a120a121 a122Legendre a15a16a17a24a90a91a65a66a123a124a44 a125a126a127a128 a123 a72a129Legendre a15a16a17a130a131a24a58a87a44a59Legendre a15a16a17a24a132a133a58a87a30 a134a135Legendre a15a16a17a24a90a91a65a66 1√ 1?2xt + t2 = ∞summationdisplay l=0 Pl(x)tl, a136a137a138t a139a140a44a37 ?12 ?2x+ 2t (1?2xt + t2)3/2 = ∞summationdisplay l=0 lPl(x)tl?1, a59 x?t (1?2xt + t2)1/2 = parenleftbig1?2xt + t2parenrightbig ∞summationdisplay l=0 lPl(x)tl?1 = (x?t) ∞summationdisplay l=0 Pl(x)tl. a141a142tl a16a24a87a66a44a37 xPl(x)?Pl?1(x) = (l + 1)Pl+1(x)?2xlPl(x) + (l?1)Pl?1(x), a143a144 a59a99 (2l + 1)xPl(x) = (l + 1)Pl+1(x) + lPl?1(x). (maltesecross) a52a69 a88 a99a100 Legendrea15a16a17a24a38a39a132a133a58a87a44a145a146a147a148a149a150a151a152 Legendrea153a154a155a156a157a158a159a160a30 a161a162a96a97 a52a39a132a133a58a87a44 a88 a79a80a163a164a165 a129 a24Legendrea15a16a17 a97a166a129Legendre a15a16a17P0(x) = 1a167 a38 a129 Legendre a15a16a17P1(x) = xa168a169a123a170a30 a95Legendre a15a16a17a24a90a91a65a66 1√ 1?2xt + t2 = ∞summationdisplay l=0 Pl(x)tl, a138x a171 a128 a44 a107a172 a99a100 ?12 ?2t (1?2xt + t2)3/2 = ∞summationdisplay l=0 Pprimel(x)tl, a173 a18 t ∞summationdisplay l=0 Pl(x)tl = parenleftbig1?2xt + t2parenrightbig ∞summationdisplay l=0 Pprimel(x)tl. a141a142tl+1 a16a24a87a66a44a99 Pl(x) = Pprimel+1(x)?2xPprimel(x) + Pprimel?1(x). (#) a52a39a132a133a58a87a27a44a123a174a24a18a175a39 a72a129 Legendre a15a16a17a71 a60a128 a66a30 Wu Chong-shi a176a177a178a179 a180 a181 a182 (a177) a1163a117 a163(maltesecross)a17 a138x a171 a128 a44a183a79a80a99a100 (2l + 1)Pl(x) + (2l + 1)xPprimel(x) = (l + 1)Pprimel+1(x) + lPprimel?1(x), a167(#)a17a184a185a44a186a187 Pprimel?1(x)a188Pprimel+1(x)a44 a107 a79a80a99a100a132a133a58a87 Pprimel+1(x) = xPprimel(x) + (l + 1)Pl(x), Pprimel?1(x) = xPprimel(x)?lPl(x). a52 a136 a39a132a133a58a87a44a189a18a163 Pprimel±1(x)a97Pl(x)a71 a60a128 a66a168a169a123a170a30 a163a52a190a132a133a58a87a191a192a193a194a44a183a79a80a29a38a195a99a100 a60a196a197 a17a24a132a133a58a87a30 a132a133a58a87a24a38a39 a97a198 a18a199a200a201a190a202a203a24a204a67a44a104a105integraldisplay 1 ?1 xPk(x)Pl(x)dx. a134a135 a132a133a58a87 (maltesecross)a44 a88 a172a205 a199a200a123integraldisplay 1 ?1 xPk(x)Pl(x)dx = l + 12l + 1 integraldisplay 1 ?1 Pk(x)Pl+1(x)dx + l2l + 1 integraldisplay 1 ?1 PkPl?1(x)dx = l + 12l + 1 22l + 3δl+1,k + l2l + 1 22l?1δl?1,k. Wu Chong-shi §20.3 Legendrea108a109a110a206a207a208a209 a1164a117 §20.3 Legendre a7a8a9a210a211a212a213 a214 20.1 a215a216a42a217a27a24 a128a218a219 a30 a31 a21a42a217a220a221a48E0 a24a215a216a42a217a27a36a29a38a39 a94a222a128a218a219 a44 a219 a24a223a224a48aa30a171 a219a225 a164a165a38a34a24a42 a22a30a226 a36a29 a128a218a219a227 a44a228 a173a229 a42a230a231a44a21 a128a218a219 a24 a219 a84a232 a88a233 a197 a91a38a63a24a230a90a84a42a43a67a234a44a78 a235a219a218 a91a48a236a22 a218 a30 star a219a225a164a165a38a34a24a237a42a22 a88 a18a33a37a24a215a216a42a217a24a42a22a167a230a90a42a43a24a42a22a24a238a239a30 star a219a218a94a222a44a165a240a241 a219a218 a24a42a22a48 0a30 star a77a48a21 a219a225 a35a35a242a37a42a43a44a46a80a21 a219a225 a24a42a22a243a244 Laplacea50a245a30 star a246 a97a219a247a248 a87a44 a247a248 a33a34a56 a219a249 a191a194a44a250a49a251a33a170a42a217a24a50a51a30 star a252a253a100a215a216a42a217a80a71 a219a218 a24 a138 a89a254a44a21 a219 a84a232a24a230a90a42a43a38a63a18a255a250a49a0a1a2a3a24a44a77a78a44 a138 a173a219a225 a164a165a38a34a44a57a23a18a230a90a42a43a4a90a24a24a42a22a44a188a18a237a42a22a44a98a5a18a255a250a49a0a1a2a3a24a30 a31u(r,θ) a18 a219a225 a38a34 (r,θ,φ)a24a237a42a22a44u1(r,θ)a167 u2(r,θ)a67a6a18a215a216a42a217a167a230a90a42a43a24a42a22a44 u1(r,θ) = ?E0z + u0 = ?E0rcosθ + u0, a7 a66u0 a59a48 a247a248 a33a34a35a24a42a22a30 u2(r,θ)a189a228a63a75a8a9 1 r2 ? ?r parenleftbigg r2?u2?r parenrightbigg + 1r2 sinθ ??θ parenleftbigg sinθ?u2?θ parenrightbigg = 0, u2vextendsinglevextendsingleθ=0 a37a10a44 u2vextendsinglevextendsingleθ=pia37a10, u2vextendsinglevextendsingler=a = E0acosθ?u0, u2vextendsinglevextendsingler→∞ → 0. a11 a63a30 u2(r,θ) a130a46a80a243a244 Laplace a50a245a44 a122a12a144 a232a13a44a18a228 a173 a230a90a42a43a14a18a67a234a21 a219 a84a232a44 a78 a219a225 a35a35a15a57a230a90a42a43a16a21a30 a122 a66a17a232a13a44a77a48 u(r,θ) = u1(r,θ) + u2(r,θ) a167a40a18a24 u1(r,θ) a5a243a244 Laplacea50a245a30a19a69a228 a173 a230a90a42a43a14a18a67a234a21 a219 a84a232a44a46a80a20 r → ∞a53 u2(r,θ)a231a20a21 a1730 a30 a171a75a63a75a8a9a30 a95 a50a245a167a37a10a22a23a67a24a3a25 a227 a44a79a80a99a100 1 sinθ d dθ bracketleftBig sinθdΘ(θ)dθ bracketrightBig + λΘ(θ) = 0, Θ(0)a37a10a44 Θ(pi)a37a10, d dr bracketleftBig r2dR(r)dr bracketrightBig ?λR(r) = 0, Wu Chong-shi a176a177a178a179 a180 a181 a182 (a177) a1165a117 a60 a27λa18a67a24a3a25a53a28a29a24a26a63a27a66a30a21 18a28a292a30a27a31a32a33a23a34a52a39a35a36a64a8a9a44 a60 a75a18 a35a36a64 λl = l(l + 1), l = 0,1,2,3,···, a35a36a65a66 Θl(θ) = Pl(cosθ). a48a37a171a75a58 a173 R(r) a24a50a245a44a38a55a79a80a81a3a39 t = lnr a44 a95 a50a245a3a48 d2Rl dt2 + dRl dt ?l(l + 1)Rl = 0. a173 a18 Rl(r) = Alelt + Ble?(l+1)t = Alrl + Blr?l?1. a77a40a44a243a244Laplacea50a245a167a37a10a22a23a24a38a41a75 a88 a18 u2(r,θ) = ∞summationdisplay l=0 parenleftbigA lrl + Blr?l?1 parenrightbigP l(cosθ). a252a253a100a57a42a43a22a23 u2vextendsinglevextendsingler→∞ → 0a44a231a44a37 Al = 0. a45a46a47a219 a84 r = aa232a24a48a10a22a23a44 u2(r,θ)vextendsinglevextendsingler=a = ∞summationdisplay l=0 Bla?l?1Pl(cosθ) = E0acosθ?u0 = E0aP1(cosθ)?u0P0(cosθ), a46a80a37 B0 = ?u0a, B1 = E0a3, a167 Bl = 0, l ≥ 2. a52a69 a88 a171a99 u2(r,θ) = ?u0ar + E0a 3 r2 cosθ. a49a50a51a52a53u 2(r,θ) a54a55a56a57a58a59 a60a61a62a63a64a65a66a53a67a68a69a70 a30 a71a72a73a65a74a53a75a76a77 a44 a78a79a60a61a62a53a63a64a65a66a80 a54a81a82a81a83a84a85a86 a53 a86 a65a66a87a65a88a89a90a53a91a92 a30a86 a65a66a53a65 a93a94?4piε 0u0a a95 a65a88a89a90a53a88a89a96a94 4piε 0E0a3 a44a97a98a99 a72a73a65a74a53 a97a98 a80a100 a30 a95u 1(r,θ)a167u2(r,θ)a238a239a44a88 a99a100 a219a225 a164a165a38a34a24a237a42a22a101 u(r,θ) = u0 parenleftBig 1? ar parenrightBig ?E0 parenleftBig 1? a 3 r3 parenrightBig rcosθ. a10220.1 a103a123a37a34a250a49a24a164a165a38a39a104a84a232a42a217a105a24a67a234 a102 a30 Wu Chong-shi §20.3 Legendrea108a109a110a206a207a208a209 a1166a117 a102 20.1 a215a216a42a217a27a24 a128a218a219 a214 20.2 a215a216a106a42a107a108a109a24 a229 a42a22a30 a31 a37a38a215a216a107a108a109a44a223a224a48 aa44a237a42a43a48M a44a171a145a21a110a131a164a165a38a34a24 a229 a42a22a30a226 a38 a45a219a247a248 a87a44 a247a248 a33a34a36a21a109 a249 a44a78a108a109a189a35a21a111a112a84a232a30a52a53a44a110a131a164a165a38a34(r,θ,φ) a24 a229 a42a22a231a44a56 φa57a58a44u = u(r,θ)a30a79a80a113a123ua46a243a244a24a50a245a167a114a67a63a75a22a23 1 r2 ? ?r parenleftbigg r2?u?r parenrightbigg + 1r2 sinθ ??θ parenleftbigg sinθ?u?θ parenrightbigg = 0, (r,θ)negationslash= parenleftBig a, pi2 parenrightBig , uvextendsinglevextendsingleθ=0a37a10, uvextendsinglevextendsingleθ=pia37a10, uvextendsinglevextendsingler=0a37a10, uvextendsinglevextendsingler→∞ → 0. a49a115a116a117a118a119a120a121a53a122a123a124a125( a126a127a128a44 a49a119a122a123a124a125a116a117a129a122a53) a44a130 a94a131a132a133 a57a58 a134a135a64a136a65a137a53a138 (a65a66a67a68) a53a69a70 a30 a139 a113a123a215a216a107a108a109a232a24a42a43a140a221a44a22a141 a139a97 a100 δ a65a66a30a50a245 a88 a3a48 1 r2 ? ?r parenleftbigg r2?u?r parenrightbigg + 1r2 sinθ ??θ parenleftbigg sinθ?u?θ parenrightbigg = ? 1ε 0 f(r)δ(r?a)δ parenleftBig θ? pi2 parenrightBig , (#) a65a66f(r)a79a80a228 integraldisplayintegraldisplayintegraldisplay f(r)δ(r ?a)δ parenleftBig θ? pi2 parenrightBig r2 sinθdrdθdφ = 1 a63a123a44a78a142a228 a173 f(r)δ(r ?a) = f(a)δ(r?a) a44a46a80a37 f(r) = f(a) = 12pia2. a143 a226a144a226a145a146 a30a228δ a65a66a24a254a147a79a80a148a112a44a20r negationslash= aa53a44a50a245 (#) a88a149a150 a48Laplacea50a245a30a52 a69a44 a45 a102a194a37a10a22a23a167a57a42a43a22a23a44 a88 a99a100 u(r,θ) = ?? ??? ? ??? ?? ∞summationdisplay l=0 AlrlPl(cosθ), r < a, ∞summationdisplay l=0 Blr?l?1Pl(cosθ), r > a. a55 a227 a88 a231a44 a96a97 a108a109a24a147a25a67a234 (a59a50a245(#)a151 a137 a24a152a153 a129 a16) a63a123a87a66 Al a167Bl a30 Wu Chong-shi a176a177a178a179 a180 a181 a182 (a177) a1167a117 star a252a253a100δ a65a66a231a44a18a131a154a65a66a24 a128 a66a44a46a80 u(r,θ)a21 a219 a84r = aa232a38a63a18a155a156a24a44 u(r,θ)vextendsinglevextendsingler=a+0r=a?0 = 0, star ?u(r,θ)/?ra21a219a84r = aa232a38a63a18a2a155a156a24a44a145a21a219a84r = aa136a157a24a158a3a79a80a228a50a245(#) a138r a204a67a99a100a101 r2?u?r vextendsinglevextendsingle vextendsingle r=a+0 r=a?0 = ? 12piε 0 δ parenleftBig θ? pi2 parenrightBig a59 ?u ?r vextendsinglevextendsingle vextendsingle r=a+0 r=a?0 = ? 12piε 0a2 δ parenleftBig θ? pi2 parenrightBig . a95δ(θ?pi/2) a98a159Legendrea15a16a17a82a83 δ parenleftBig θ? pi2 parenrightBig = ∞summationtext l=0 clPl(cosθ), cl = 2l + 12 integraldisplay pi 0 δ parenleftBig θ? pi2 parenrightBig Pl(cosθ)sinθdθ = 2l + 12 Pl(0). a77a40 Alal = Bla?l?1, Allal+1 + Bl(l + 1)a?l = 2l + 14piε 0 Pl(0). a75a130a59a99 Al = 14piε 0 a?l?1Pl(0), Bl = 14piε 0 alPl(0). a46a80 u(r,θ) = ? ??? ? ??? ?? 1 4piε0a ∞summationdisplay l=0 parenleftBigr a parenrightBigl Pl(0)Pl(cosθ), r < a, 1 4piε0a ∞summationdisplay l=0 parenleftBiga r parenrightBigl+1 Pl(0)Pl(cosθ), r > a. a46a47P l(0)a64a44a59a99 u(r,θ) = ?? ??? ? ??? ?? 1 4piε0a ∞summationdisplay l=0 (?)l (2l)!22l l!l! parenleftBigr a parenrightBig2l P2l(cosθ), r < a, 1 4piε0a ∞summationdisplay l=0 (?)l (2l)!22l l!l! parenleftBiga r parenrightBig2l+1 P2l(cosθ), r > a. a160 a62a161a162a53a117a163a124a125a53 a84a164 a123a165 a101 a166 a120a121a53a122a123a124a125a134a167 a44 a168a134a118a169a123 a44a55a170a171a172a173a174 a175a176a53a177a178a179a122a134a91a92a180a176 a30 a71a49a119a124a125a181 a44a97a182 a53a183a184a185a186a187a133a188a189a179 a101 a190a71a191a192 r = a, θ = pi/2a62a116a940 a44 a193a194 a44 a183a184a185a186a71a191a192a62a53a176a195a94∞(a49a196a197a198a199a200a201a201a65a66a93a94a133a202a195) a30 a80a203a79 a44 a71a51a123a204a205 a56a206 a76 a59 a188a189a53a207a165 a44 a208a209a183a184a185 a97a182(#)a210a211 a94a184a185 a97a182 a87a60a61 r = aa62a53a212a78a213a214a30 a49a50a215a216a62a117a161a162 a59 a118a217a168a212a78a213a214a53 a97 a165 a30 Wu Chong-shi §20.3 Legendrea108a109a110a206a207a208a209 a1168a117 a35a9a183a37a38a218a219a220a221a158 a226a222 a44a223a224a225a226a152a227a228a229 a144a226a230a231 (a232a233 a230a231 a229a234a235a236 a230a231 ) a44 a143 a147 a237a238 a226 a44a239a240a241a242a243a244a245a246 a230a231 a247 a224a225 Legendre a153a154a155a158a248a249a250 a144 a160a251a44a252a253a254 a237a238 a226a255a0 a253 u(r,θ) a1 r = 0 a2 r = ∞ a3a158a151a4a5a158 Taylor a6a7a8a9 a222a10a11 u(r,θ) a1a12a13a14a15 θ a16a17a18a19a20 a21 a8a22a23a24a25 Taylor a6a7a19a26a13a27a28a29a30a31a32a20a33 a34a35a36a37a38a39a40a41 a8 a36a37a42a43a44a45a46a47a48a49a42 (r,θ) = (r,0) a50(r,pi)a46a38a51a52a53a54a8 a55a56a57a58 a34Coulomb a59a60a61a62a63a64a65 a48a49a42a43a44a45a46 (r,0) a50(r,pi)a38a66a67a68a8 u(r,θ)vextendsinglevextendsingleθ=0,pi = contintegraldisplay 1 8pi2ε0a dl√ a2 + r2 = 1 4piε0 1√ a2 + r2. a69Taylor a70a71 u(r,θ)vextendsinglevextendsingleθ=0,pi = ? ??? ? ??? ? 1 4piε0a ∞summationdisplay l=0 (?)l (2l)!22l (l!)2 parenleftBigr a parenrightBig2l , r < a, 1 4piε0r ∞summationdisplay l=0 (?)l (2l)!22l (l!)2 parenleftBiga r parenrightBig2l , r > a. a72a45a73a74 a8 a34a45a75a76a77a57a58a78a47 u(r,θ) vextendsinglevextendsingle vextendsingle θ=0 = ? ??? ?? ??? ?? ∞summationdisplay l=0 Alrl, r < a, ∞summationdisplay l=0 Blr?l?1, r > a a50 u(r,θ) vextendsinglevextendsingle vextendsingle θ=pi = ?? ??? ? ??? ? ∞summationdisplay l=0 (?)lAlrl, r < a, ∞summationdisplay l=0 (?)lBlr?l?1, r > a. a79a80a81a82( a83a84a85Taylora70a71 a38a86a45a41) a8a87 a57a58a88a78 A2l = (?) l 4piε0a (2l)! 22l l!l!a ?2l, A2l+1 = 0, B2l = (?) l 4piε0a (2l)! 22l l!l!a 2l+1, B2l+1 = 0. a57a58a89 a65a8a90a91 a78a47a38a76a92a93a94a74a38a95a96a53a97 a33