Wu Chong-shi a0a1a2a3 a4a5a6a7a8 ( a9) a10a11a12a13a14a15a16 a17a18a19a20a21a22a23a24a20a21a22a25a26 Lapalce a27a28 a29a30a31a32a33 (r,φ) a34a35a36 a32a33 (x,y) a37a38a39a40 x = r cosφ, y = r sinφ. a41a42a43a44a45a46 dr = cosφdx + sinφdy, dφ = ?sinφr dx + cosφr dy, a47 ?r ?x = cosφ, ?φ ?x = ? sinφ r , ?r ?y = sinφ, ?φ ?y = cosφ r . a48a49a50a51a52a53 a37 a45a54a55a56a57 ? ?x = ?r ?x ? ?r + ?φ ?x ? ?φ = cosφ ? ?r ? sinφ r ? ?φ, ? ?y = ?r ?y ? ?r + ?φ ?y ? ?φ = sinφ ? ?r + cosφ r ? ?φ. a58a59a60a61a62a63a64 ?2 ?x2 = parenleftbigg cosφ ??r ? sinφr ??φ parenrightbiggparenleftbigg cosφ ??r ? sinφr ??φ parenrightbigg = cos2φ ? 2 ?r2 ? 2sinφ cosφ r ?2 ?r?φ + sin2φ r2 ?2 ?φ2 + sin2 φ r ? ?r + 2sinφ cosφ r2 ? ?φ, ?2 ?y2 = parenleftbigg sinφ ??r + cosφr ??φ parenrightbiggparenleftbigg sinφ ??r + cosφr ??φ parenrightbigg = sin2φ ? 2 ?r2 + 2sinφ cosφ r ?2 ?r?φ + cos2φ r2 ?2 ?φ2 + cos2 φ r ? ?r ? 2sinφ cosφ r2 ? ?φ. a65a66a61a63a64a29a30a31a32a33 a39a67a37 Laplacea68a69 ?2 ≡ ? 2 ?r2 + 1 r ? ?r + 1 r2 ?2 ?φ2 ≡ 1r ??r parenleftbigg r ??r parenrightbigg + 1r2 ? 2 ?φ2. a70a42a71a72a73a57a74a75a76a63a64a77a32a33 a39a67a37 Laplacea68a69 ?2 ≡ ? 2 ?r2 + 1 r ? ?r + 1 r2 ?2 ?φ2 + ?2 ?z2 ≡ 1r ??r parenleftbigg r ??r parenrightbigg + 1r2 ? 2 ?φ2 + ?2 ?z2. Wu Chong-shi §18. a78a79a80a81a82a83 Lapalcea84a85 a862a87 a88a20a21a22a25a26 Lapalce a27a28 a89a32a33 (r,θ,φ) a34a35a36 a32a33 (x,y,z) a37a38a39a40 x = r sinθ cosφ, y = r sinθ sinφ, z = r cosθ. a41a42a75a76a90a46 dr = sinθ cosφdx + sinθ sinφdy + cosθdz, dθ = cosθ cosφr dx + cosθ sinφr dy ? sinθr dz, dφ = ? sinφrsinθdx + cosφrsinθdy. a91a42 ? ?x = ?r ?x ? ?r + ?θ ?x ? ?θ + ?φ ?x ? ?φ = sinθ cosφ ? ?r + cosθ cosφ r ? ?θ ? sinφ r sinθ ? ?φ, ? ?y = ?r ?y ? ?r + ?θ ?y ? ?θ + ?φ ?y ? ?φ = sinθ sinφ ? ?r + cosθ sinφ r ? ?θ + cosφ r sinθ ? ?φ, ? ?z = ?r ?z ? ?r + ?θ ?z ? ?θ = cosθ ? ?r ? sinθ r ? ?θ. a70a42a71a72a73a61a75a76a45a46 ?2 ?x2 = parenleftbigg sinθ cosφ ??r + cosθ cosφr ??θ ? sinφr sinθ ??φ parenrightbiggparenleftbigg sinθ cosφ ??r + cosθ cosφr ??θ ? sinφr sinθ ??φ parenrightbigg = sin2 θ cos2 φ ? 2 ?r2 + cos2θcos2φ r2 ?2 ?θ2 + sin2 φ r2 sin2 θ ?2 ?φ2 + 2sinθ cosθ cos2 φ r ?2 ?r?θ ? 2sinφ cosφr ? 2 ?r?φ ? 2cosθ sinφ cosφ r2 sinθ ?2 ?θ?φ + cos2 θ cos2 φ + sin2 φ r ? ?r + ?2sin 2 θ cosθ cos2 φ + cosθ sin2 φ r2 sinθ ? ?θ + 2sinφ cosφ r2 sin2 θ ? ?φ, ?2 ?y2 = parenleftbigg sinθ sinφ ??r + cosθ sinφr ??θ + cosφr sinθ ??φ parenrightbiggparenleftbigg sinθ sinφ ??r + cosθ sinφr ??θ + cosφr sinθ ??φ parenrightbigg = sin2θsin2φ ? 2 ?r2 + cos2θsin2φ r2 ?2 ?θ2 + cos2 φ r2 sin2 θ ?2 ?φ2 + 2sinθ cosθ sin2 φ r ?2 ?r?θ + 2sinφ cosφr ? 2 ?r?φ + 2cosθ sinφ cosφ r2 sinθ ?2 ?θ?φ + cos2 θ sin2 φ + cos2 φ r ? ?r + ?2sin 2 θ cosθ sin2 φ + cosθ cos2 φ r2 sinθ ? ?θ ? 2sinφ cosφ r2 sin2 θ ? ?φ, ?2 ?z2 = parenleftbigg cosθ ??r ? sinθr ??θ parenrightbiggparenleftbigg cosθ ??r ? sinθr ??θ parenrightbigg = cos2 θ ? 2 ?r2 + sin2 θ r2 ?2 ?θ2 ? 2sinθ cosθ r ?2 ?r?θ + 2sinθ cosθ r2 ? ?θ + sin2 θ r ? ?r. Wu Chong-shi a92a93a94a95 a96a97a98a99a100 ( a101) a102a103a104 a105 a79a80a81 a863a87 a65a66a61a63a64a89a32a33 a39a67a37 Laplacea68a69 ?2 ≡ ? 2 ?r2 + 2 r ? ?r + 1 r2 ?2 ?θ2 + cosθ r2 sinθ ? ?θ + 1 r2 sin2 θ ?2 ?φ2 ≡ 1r2 ??r parenleftbigg r2 ??r parenrightbigg + 1r2 sinθ ??θ parenleftbigg sinθ ??θ parenrightbigg + 1r2 sin2 θ ? 2 ?φ2. Wu Chong-shi §18.1 a106a107a108a109 a864a87 §18.1 a110 a111 a112 a113 a114a115a116a117a118a119a120a121a122a123a124a125a90a126a127a128 ?2u ?x2 + ?2u ?y2 = 0, x 2 + y2 < a2, uvextendsinglevextendsinglex2+y2=a2 = f. a70 a35a36 a32a33 a39a67 a57a129a130( a131a132Laplacea129a130)a133a134 a75a76a135a136a137a138a124a139a140a141a142a143a144 a134a145 a62a124a41a146a140a141 a37 a147a148 a40a149 a147a57a150a151 a134a152a153a154a155a156 a29a30a31a32a33 a39 a124 a70a29a30a31a32a33 a39a157 a57a158a159 a37 a125a90a126a127 a153a154 a75a76a160a128 1 r ? ?r parenleftbigg r?u?r parenrightbigg + 1r2 ? 2u ?φ2 = 0, 0 < r < a, uvextendsinglevextendsingler=a = f(φ). a161u(r,φ) = R(r)Φ(φ) a57a162a163a129a130a57a164 1 r d dr parenleftbigg rdRdr parenrightbigg Φ + Rr2 d 2Φ dφ2 = 0, =? r R d dr parenleftbigg rdRdr parenrightbigg = ?1Φ d 2Φ dφ2 = λ. a91a42a57a75a76a135a136a137a138a57 r ddr parenleftbigg rdRdr parenrightbigg ?λR = 0, d2Φ dφ2 + λΦ = 0. a139 a40 a140a141a142a143 R(a)Φ(φ) = f(φ) a165 a134a145 a62a135a136a137a138a57a91a128a140a141a142a143 a40a166a167a168a37 a124a169a170a171a172a62a173a174 a167a168 a129a130a135a136a137a138a57a63a64a175a176a177a164 a178a125a179a53 a37a167a168a180a181 a135a129a130a57a139 a40a182a183 a164a184 a153a37a167a168 a140a141a142a143a185a186a187a51a188a189a190a59a176a191a192a193a126a127a124 a194a195a196a197a198a199a200a201a202a203a204a205a206a207a208a57a209a210a211a212a213a119a214a215a119a216a217a218 a73a30a46a219 a37a220a221 a57a222a223 a40 a41a146a224a225 a157a37a226a227a228 a190 a37a229 a70 a149 a147a230a231 a37 a142a143 a67 a57a41a29a30 a35a36 a32a33 a39 a137a232a64a29a30a31a32a33 a39a233 a57a234a235 1 r ? ?r parenleftbigg r?u?r parenrightbigg + 1r2 ? 2u ?φ2 = 0, 0 < r < a, uvextendsinglevextendsingler=a = f(φ). a182a145 a222a223a236a237a146a158a159 a37 a125a90a126a127a238a239a240a241a57a242 a182a145 a189a190a59a176a222a243 a37 a125a90a126a127a124 Wu Chong-shi a92a93a94a95 a96a97a98a99a100 ( a101) a102a103a104 a105 a79a80a81 a865a87 star a244 a59a57a70a53a245a73a57a158a159a125a90a126a127 a37a181 a135a129a130a70 a149a246a247a247 a190a248a238 a134 a188a137a232a64a29a30a31a32a33a66a57a129 a130a70a230a249 a37a250a251 φ = 0a34φ = 2pia182a145 a190a248a124a252a253a241a57a70a29a30a31a32a33 a157 a57a151a137a138 φ a37 a137a254a255a0 a40[0, 2pi] a57a91a128 u(r, φ)a70 a250a251φ = 0a34φ = 2pia247a37a1 a54a53 a183 a164a125a2a57a3a4a138a65a5a6a7a62a125a2 u(r, φ)a70a175a176a250a251a247a37a8a9a1 a54a53a124 a10a175a176 a250a251a11a12a40 a41a146 a155a156 a29a30a31a32a33 a39a13 a160 a149 a147a188a46a219 a37 a57 a182a166a14a15a37a16a17 a140a141a57a70a158a18 a37 a125a90a126a127 a157 a57a61a19 a145 a73a20a21a125a184 a153a37 a140a141a142a143a124a10a22a61a54a23a70a73a30 a37 a234a235 a157 a6 a183 a164a24a46 u(r,φ) a70φ = 0 a34φ = 2pia247a25a153a133a26a27a37 a140a141a142a143a124 a28a29a64a29a30a31a32a33 a39a37a30a251 a57 (r, φ = 0) a34(r, φ = 2pi)a162a31a37a40 a29a30a73 a37a32 a59 a251 a57 a25 a76a57a33a128 a222a243 a37 a125a90a126a127a57 a153a133a34 a3a73a35a36a142a143 u(r,φ)vextendsinglevextendsingleφ=0 = u(r,φ)vextendsinglevextendsingleφ=2pi a34 ?u(r,φ) ?φ vextendsinglevextendsingle vextendsingle φ=0 = ?u(r,φ)?φ vextendsinglevextendsingle vextendsingle φ=2pi . a10a22a57a73a30a37a64 a37 a41a146a38Laplacea129a130a39 a35a36 a32a33 a39a40 a232a64a31a32a33 a39a233 a188a41a42 a37a43a44 a57a75a76a45 a46a35a36a142a143a188a63a64 a34a47 a124 star a244a131 a57a158a159 a37 a129a130a70a32a33a158 a251(x, y) = (0, 0)a6a40a190a248a37a124a139a40 a57a137a232a64a29a30a31a32a33a66a57a129a130a70 r = 0a251a182a145 a190a248a124a91a128u(r,φ)a70r = 0 a251a37a1 a54a53a6 a182a183 a164a125a2a57a3a4a138a6a7a62a125a2 u(r, φ) a70r = 0 a251a37a8a9a1 a54a53a124 r = 0a251 a33a128a151a137a138r a37a250a251 a57a6 a11a12a40 a41 a155a156 a31a32a33 a39 a188a46a219 a37 a57a242 a182a145a40a149 a147a230a231 a37a16a17 a140a141a124a10a22a6a74a48a20 a34 a3a73 u(r, φ)a70r = 0 a251a25a153a133a26a27a37 a140a141a142a143a124 a28a29a64a158a159 a37 a129a130 a40a167a168a37 a57a70 a149a246(a49a50 a32a33a158 a251)a40a51a52a37 a57a91a42a57u(r, φ)a70a32a33a158 a251a153 a133a40 a164a141 a37 a57 a153a133 a20 a34 a3a73a164a141a142a143 u(r, φ)vextendsinglevextendsingler=0a164a141. a53a54 a229 a194a55a56a211a195a196a197a198a199a200a57a57a121a58a122a123a59a202a60a206a61 1 r ? ?r parenleftbigg r?u?r parenrightbigg + 1r2 ? 2u ?φ2 = 0, 0 < φ < 2pi, 0 < r < a, u(r,φ)vextendsinglevextendsingleφ=0 = u(r,φ)vextendsinglevextendsingleφ=2pi, 0 < r < a, ?u(r,φ) ?φ vextendsinglevextendsingle vextendsingle φ=0 = ?u(r,φ)?φ vextendsinglevextendsingle vextendsingle φ=2pi , 0 < r < a, u(r, φ)vextendsinglevextendsingler=0a164a141, 0 < φ < 2pi, uvextendsinglevextendsingler=a = f(φ), 0 < φ < 2pi. Wu Chong-shi §18.1 a106a107a108a109 a866a87 a219a70a57a62a159a63a50a135a136a137a138 a37 a60a64a57a61a75a76a65a64a57a66a67a68a30a69a70a63a64 a37 a175a176 a167a168a180a181 a135a129a130 r ddr parenleftbigg rdRdr parenrightbigg ?λR = 0, d2Φ dφ2 + λΦ = 0 a186a71a57a41a35a36a142a143a74a75a76a63a64 Φ(0) = Φ(2pi), Φprime(0) = Φprime(2pi). a10a22a57a72a63a64a67a59a176a73 a37 a191a192a193a126a127 d2Φ dφ2 + λΦ = 0, Φ(0) = Φ(2pi), Φprime(0) = Φprime(2pi). a74a75a76a77a78a79a80a81a82a83a84 a229 a85a84a86a87a88a89a90a91a92a81a93a94a95a96a97a98a99a100a101a102a103a104a105a106a81a124 a74a75a76a77a78a79a80a81a107a108a109a110a111a112a113a114a81a82a83a124 a133λ = 0a233 a57 a180a181 a135a129a130 a37 a45a90a128 Φ0(φ) = A0φ + B0. a162a163a35a36a142a143a57a164 B0 = A02pi + B0, A0 = A0. a91a42 A0 = 0, B0 a115a116 . a10a241a117λ = 0 a40 a191a192a193a57a184 a153a37 a191a192a52a53 a40 Φ0(φ) = 1. a133λ negationslash= 0a233 a57a129a130 a37 a45a90a128 Φ(φ) = Asin √ λφ + Bcos √ λφ. a162a163a35a36a142a143a57a63a64 B = Asin √ λ2pi+ Bcos √ λ2pi, A = Acos √ λ2pi?Bsin √ λ2pi. a10a75a76a65a190 a40a38 a146 a39 a53 A a34B a37a118a119a167a168 a162a53a129a130a120a57a164 a166a121 a90 a37 a3a135a122a20a142a143 a40vextendsingle vextendsinglevextendsingle vextendsinglevextendsingle sin √λ2pi cos√λ2pi?1 cos√λ2pi?1 ?sin√λ2pi vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = 0, a472(cos√λ2pi?1) = 0a124a10a22a72a75a76a45a63a191a192a193 λm = m2, m = 1,2,3,···, a184 a153a37a166a121 a90 a40 A a115a116 , B a115a116 . Wu Chong-shi a92a93a94a95 a96a97a98a99a100 ( a101) a102a103a104 a105 a79a80a81 a867a87 a10a61 a40 a241a57a123 a153 a146a59a176a191a192a193 λ m a57a164a175a176a191a192a52a53 Φm1(φ) = sinmφ, Φm2(φ) = cosmφ. a108a109a57a124a125a126a127a128 λ 0 = 0 a81a129a130a98λ 0 negationslash= 0 a81a129a130a131a132a133a134a57a135a99a136a106 Φm1(φ) = sinmφ, Φm2(φ) = cosmφ, a137 a128ma81a138a78a139a140a141a142a143 0,1,2,3,···a124 a48a49a135a136a137a138a55 a37 a33a144a60a64a57a62a159a45 a180a181 a135a129a130 r ddr parenleftbigg rdRdr parenrightbigg ?λR = 0 a37 a90a124a145 a116 a10a176 a180a181 a135a129a130 a40 a59a176 a30a146a37 a137 a39 a53a129a130a57a70a46a151a137a138 a37 a137a232 d dt = r d dr a47 t = lnr a66a57a61a75a76a137a128 a180a39 a53 a37a180a181 a135a129a130 a229 d2R dt2 ?λR = 0. a25 a76a57 a133λ0 = 0a233 a57a45a90a128 R0(r) = C0 + D0t = C0 + D0 lnr; a133λm = m2, m negationslash= 0a233 a57a45a90a128 Rm(r) = Cmemt + Dme?mt = Cmrm + Dmr?m. a219a70a57a61a45a63a67 a26a27a167a168 a129a130 a34a167a168 a140a141a142a143 (a35a36a142a143) a37 a223a147 a30 a90 u0(r,φ) = C0 + D0 lnr, um1(r,φ) = parenleftbigCm1rm + Dm1r?mparenrightbigsinmφ, um2(r,φ) = parenleftbigCm2rm + Dm2r?mparenrightbigcosmφ. a148a149a150a159a57a61a63a64a125a90a126a127 a37 a59a151a90 u(r,φ) = C0 + D0 lnr + ∞summationdisplay m=1 parenleftbigC m1rm + Dm1r?m parenrightbigsinmφ + ∞summationdisplay m=1 parenleftbigC m2rm + Dm2r?m parenrightbigcosmφ. a28a29a64a164a141a142a143 uvextendsinglevextendsingler=0a164a141, a91a128lnr a34r?m a70r = 0a251a152a40a51 a141 a37 a57 a25 a76a242a170 a37a39 a53 a152 a122a153a128 0a57 D0 = 0, Dm1 = 0, Dm2 = 0. a62a162a163a4a154 a37 a140a141a142a143a57a61a63a64 u(r,φ) vextendsinglevextendsingle vextendsingle r=a = C0 + ∞summationdisplay m=1 am(Cm1 sinmφ + Cm2 cosmφ) = f(φ). Wu Chong-shi §18.1 a106a107a108a109 a868a87 a67 a30 a37 a126a127a155 a40a156a17 a125a46a148a149 a39 a53 C 0, Cm1 a34Cm2 a124a171a172a6a75a76a39Fourier a157a158a37a36a159a160 a45a46 a39 a53C 0, Cm1 a34Cm2 a57a139 a155a156 a135a136a137a138a55 a37 a33a144a161a55a57a74 a40a162a156 a191a192a52a53 a37a15a163a119 a125a148a149 a39 a53a124 a164a165a166a167a168a122a123 d2Φ dφ2 + λΦ = 0, Φ(0) = Φ(2pi), Φprime(0) = Φprime(2pi), a164a202a169a170a166a167a168a119a166a167a171a172a173a174a175a119a124 star a191a192a52a531(a123a153a146a191a192a193λ0 = 0)a34 a191a192a52a53sinmφa239cosmφ (a123 a153 a146a191a192a193λ m = m2, m negationslash= 0) a40a15a163a37a229 integraldisplay 2pi 0 sinmφdφ = 0, integraldisplay 2pi 0 cosmφdφ = 0. star a123a153 a146a191a192a193λ m = m2 a37 a191a192a52a53 sinmφ, cosmφ a34 a123 a153 a146a191a192a193λ n = n2, n negationslash= ma37 a191a192a52 a53sinnφ, cosnφ a40 a175a175 a15a163a37a229 integraldisplay 2pi 0 sinnφsinmφdφ = 0, integraldisplay 2pi 0 sinnφcosmφdφ = 0, integraldisplay 2pi 0 cosnφcosmφdφ = 0. a164a202a165a170a176a177a166a167a168 λ m = m2 a119a178a177a166a167a171a172 sinmφ a179cosmφa180 a173a174a175a119 a229integraldisplay 2pi 0 sinmφcosmφdφ = 0. a91a42a57 a162a156 a191a192a52a53 a37a15a163a119 a76a181 integraldisplay 2pi 0 sin2mφdφ = pi, integraldisplay 2pi 0 cos2mφdφ = pi, a61a75a45a63 C0 = 12pi integraldisplay 2pi 0 f(φ)dφ, Cm1 = 1ampi integraldisplay 2pi 0 f(φ)sinmφdφ, Cm2 = 1ampi integraldisplay 2pi 0 f(φ)cosmφdφ. square Wu Chong-shi a92a93a94a95 a96a97a98a99a100 ( a101) a102a103a104 a105 a79a80a81 a869a87 a219a70a62a123a73a30a45a90a46a130 a157a37a182a183 a126a127a33a59 a183a34 a3a184a185a124 star a186 a176a57a194a187a58a166a167a168a122a123a188a57a164a202a165a176a177a166a167a168a189a178a177 ( a190a191a192a193 a119) a166a167a171a172a124 ? a123a153 a59a176a191a192a193a164 a145a194 a59a176 ( a118a119a51a38a37) a191a192a52a53a37 a219a195a57a196a128a197 a182 (a239a198a254)a124 ? a156 a235a123 a153 a59a176a191a192a193a164 na176a191a192a52a53a57a56a196a191a192a193a126a127 a40 na63a197a182a37 a57a239a240a241a197 a182a159 a128 na124 ? a123a146a131a199a180a181 a135a129a130 a37 a191a192a193a126a127a57a65a5a7a62 a40a131 a63a197 a182a37 a124 ? a70a131a199a180a181a135a129a130a37a191a192a193a126a127a157a57a156a235a140a141a142a143a40a59a200a131a200a201a202a57a56a123a153a59a176a191a192a193a57a7a62 a164a59a176a191a192a52a53a57a239a240a241a57a191a192a193a126a127a59a125 a40a166 a197 a182a37 a124a188 a133 a140a141a142a143 a40 a35a36a142a143 a233 a57a191a192a193 a126a127a203 a40 a197 a182a37 a124 star a186a204 a57a164a165a205a206a119a166a167a168a122a123a57a166a167a171a172a119a207a208a206a169a209a176a124 ? a123a153a32 a59a176a191a192a193 a37 a191a192a52a53a6 a145 a59a125 a15a163 a57 ? a139a40a59a125a75a76a45a46a210a133a37 a63a73a120a51a188a211a242a170 a15a163 a254a124 a110a76a80a137a212a57a110a126a127a128a100a140a213 λ m = m2, m = 1, 2, 3, ··· a81a76a77a214a92a138a143 eimφ a98 e?imφ, a215a216a217a218a141a128a219a220a76a77a78 ( a221a222λ0 = 0)a98a76a77a214a92a135a99a136a106 λm = m2, m = 0, ±1, ±2, ±3, ···, Φm(φ) = eimφ. a74a223a57a100a140a224a225a76a77a78a81a76a77a214a92a108a109a226a109a84a227a228a81 a229 integraldisplay 2pi 0 einφ(eimφ)?dφ = 0, n, m = 0,±1,±2,±3,···,a229n negationslash= m. a137 a229 a57a100a140a213a225a99a75a76a77a78 λ m = m2, m negationslash= 0 a81a230a75a76a77a214a92 e±imφ a124a84a227a228a81 a229 integraldisplay 2pi 0 eimφ(e?imφ)?dφ = 0. a231a232a113a233a81a76a77a214a92a84a234a214a92a57a233a235a236a81a227a228a237a238a239a240a241a128a242a239a81a99a75a76a77a214a92a138a234 a243a244a124 Wu Chong-shi §18.1 a106a107a108a109 a8610a87 ? a38 a146a125a90a126a127 a37a30 a90a57a242a170 a40 1, lnr, rm sinmφ, rm cosmφ, r?m sinmφa34r?m cosmφ. a145 a116 a10a245 a37a1a181 a135a129a130 a40 (a131a132)Laplacea129a130a124a70a50a137a52a53a147a135a157a57a169a170a246a70a247a117a57a90a248a52a53 a37a249 a147a239a250a147a59a125 a40 Laplacea129a130a37 a90a124a38 reiφ a65a190 a40 a50a137a53z = x + iy a57a61a75a76a65a46a57a73a30 a37 a10 a183a30 a90 a15a40 a90a248a52a53 z0, lnz, zm a34 z?m a37a249 a147a239a240a250a147a124a188a164a141a142a143 a15a40 a211a169a170a251a252a253a70 a149a246|z| < a a182a145a247a247 a90a248 a37 a52a53 lnz a34 z?m a124 a254a58a59a60a57a38a73a30a45a63 a37a39 a53a162a163a64a90a255 a157 a57a74a75a76a63a64 u(r,φ) = 12pi integraldisplay 2pi 0 f(φprime)dφprime + 1pi ∞summationdisplay m=1 parenleftBigr a parenrightBigm sinmφ integraldisplay 2pi 0 f(φprime)sinmφprimedφprime + 1pi ∞summationdisplay m=1 parenleftBigr a parenrightBigm cosmφ integraldisplay 2pi 0 f(φprime)cosmφprimedφprime = 12pi integraldisplay 2pi 0 f(φprime) bracketleftBig 1 + 2 ∞summationdisplay m=1 parenleftBigr a parenrightBigm cosm(φ?φprime) bracketrightBig dφprime. a144 a134 a57 a133r < aa233a0 a53a1a2a124a174a154a3a52a53a4a160a128a50a21a53a52a53a57 a162a156 a236a5 a0 a53 a37 a45 a34a6 a255a61a75a76a45a46 a0 a53 a37a34 a57a65a66a61a63a64 u(r,φ) = a 2 ?r2 2pi integraldisplay 2pi 0 f(φprime) r2 + a2 ?2arcos(φ?φprime)dφ prime. a10a176a234a235a196a128Poisson a7 a135 a6 a255a8a242a38Laplace a9a10a11a12a13a14a15a16a17a18a19a20a21a14a22a23a24a25a18a19f(φ) a14a7a26a27 a28a29a30a8a31 a22a32a33a34a14 Cauchya7a26a35a36 a8a37a38a39a40a41a42a43a44a45 ( a463.7a47)a8a48u(r,φ)a49 a50a51 a22a32a33a34a14 a29a52a53a54a52 a27 a42a55a56a57a58a59 a16a60a61a62a22a32a33a34a14 a29a52a53a54a52a63a64a65 Laplace a9a10 a14a22a66a67a14a68a69a27 Wu Chong-shi a70a71a72a73 a74a75a76a77a78 ( a79) a80a81a82 a83a84a85a86 a8711 a88 §18.2 Helmholtz a89a90a91a92a93a94a95a96a97a98a99a100a101 a102a103a104 a69a105 a8 Helmholtz a9a10a14a106a107a108a36 a51 1 r ? ?r parenleftbigg r?u?r parenrightbigg + 1r2 ? 2u ?θ2 + ?2u ?z2 + k 2u = 0. a109a110 a33a34 a51a111a43a112a113a114 a14a33a34a27 a115a116a117a118a119a120a121a122a123a123 a26a124a125a16 a43a112a113a114a8a126a127a59a128a129a130 a14a131 a43a112a113a114 a26a124a27 a132u(r,θ,z) = v(r,θ)Z(z)a8a133a134 a9a10 a8a135a136 Z bracketleftBig1 r ? ?r parenleftbigg r?v?r parenrightbigg + 1r2 ? 2v ?θ2 + k 2v bracketrightBig + vd 2Z dz2 = 0. a137a39 1 v bracketleftBig1 r ? ?r parenleftbigg r?v?r parenrightbigg + 1r2 ? 2v ?θ2 + k 2v bracketrightBig = ? 1Z d 2Z dz2 . a138 a36a14a139a140 a51ra63θ a14a33a34 a8a141z a142a68a143 a144 a140 a51z a14a33a34 a8a141 r a145θa146a142a68a27 a137a39a147a148a149a150a138a151a152 a141r, θ a142a68a153 a141z a142a68a14a154a34a27a155 a42a43 a154a34a156a25 λa8a157a136a62 1 r ? ?r parenleftbigg r?v?r parenrightbigg + 1r2 ? 2v ?θ2 + parenleftbigk2 ?λparenrightbigv = 0, d2Z dz2 + λZ = 0. a59a132v(r,θ) = R(r)Θ(θ) a8 a153 a136 a62 Θ(θ) bracketleftbigg1 r d dr parenleftbigg rdRdr parenrightbigg +parenleftbigk2 ?λparenrightbigR bracketrightbigg + R(r)r2 d 2Θ dθ2 = 0. a131a140a158 a39r2/R(r)Θ(θ) a8a159a160a161a8 a153a162 a136 a62 r2 R(r) bracketleftbigg1 r d dr parenleftbigg rdRdr parenrightbigg +parenleftbigk2 ?λparenrightbigR bracketrightbigg = ? 1Θ(θ) d 2Θ dθ2 . a59 a60a61a62 a8a138 a36a14a139a140 a56a51r a14a33a34 a8a141θ a142a68a143 a144 a140 a56a51θ a14a33a34 a8a141r a142a68a27 a137a39a147a148a149a150a138 a151a152a141r a142a68a153 a141θ a142a68a14a154a34 a8 a156a25 μa27 a151a51 a153 a136 a62 1 r d dr parenleftbigg rdRdr parenrightbigg + parenleftBig k2 ?λ? μr2 parenrightBig R = 0, d2Θ dθ2 + μΘ = 0. a42a163a8a157a164a165a166 Helmholtz a9a10a14a26a124 a113a114 a27 Wu Chong-shi §18.3 Helmholtza167a168a169a170 a84a85a86a171a172a74a75a76a77 a8712 a88 §18.3 Helmholtz a89a90a91a173a93a94a95a96a97a98a99a100a101 a174a103a104 a69a105 a8 Helmholtz a9a10a14a106a107a108a36 a51 1 r2 ? ?r parenleftbigg r2?u?r parenrightbigg + 1r2 sinθ ??θ parenleftbigg sinθ?u?θ parenrightbigg + 1r2sin2θ ? 2u ?φ2 + k 2u = 0. a132u(r,θ,φ) = R(r)S(θ,φ) a8a133a134 a9a10 a8a135a136 S(θ,φ) bracketleftbigg 1 r2 d dr parenleftBig r2dR(r)dr parenrightBig + k2R(r) bracketrightbigg + R(r)r2 bracketleftbigg 1 sinθ ? ?θ parenleftBig sinθ?S(θ,φ)?θ parenrightBig + 1sin2θ ? 2S(θ,φ) ?φ2 bracketrightbigg = 0. a131a140a158 a39r2/R(r)S(θ,φ)a8a159a160a161a8a157a38a39a136 a62 r2 R(r) bracketleftbigg 1 r2 d dr parenleftBig r2dR(r)dr parenrightBig + k2R(r) bracketrightbigg = ? 1S(θ,φ) bracketleftbigg 1 sinθ ? ?θ parenleftBig sinθ?S(θ,φ)?θ parenrightBig + 1sin2θ ? 2S(θ,φ) ?φ2 bracketrightbigg . a138 a36a14a139a140 a56a51 r a14a33a34 a8a141 θ, φ a142a68a143 a144 a140 a56a51 θ, φ a14a33a34 a8a141 r a142a68a27 a137a39a147a148a149a150a138a151a152 a141r a142a68a153 a141θ, φ a142a68a14a154a34a27a155 a42a43 a154a34a156a25 λa8a157a136a62 1 r2 d dr parenleftBig r2dR(r)dr parenrightBig + parenleftbigg k2 ? λr2 parenrightbigg R(r) = 0, 1 sinθ ? ?θ parenleftBig sinθ?S(θ,φ)?θ parenrightBig + 1sin2θ ? 2S(θ,φ) ?φ2 + λS(θ,φ) = 0. a59a132S(θ,φ) = Θ(θ)Φ(φ) a8a175a48a136 a62 Φ bracketleftbigg 1 sinθ d dθ parenleftBig sinθdΘ(θ)dθ parenrightBig + λΘ bracketrightbigg + Θsin2θ d 2Φ dφ2 = 0. a59a128a138 a36a131a140a158 a39 sin2θ/ΘΦa8a160a161a8 a153 a38a39a136 a62 sin2θ Θ bracketleftbigg 1 sinθ d dθ parenleftBig sinθdΘ(θ)dθ parenrightBig + λΘ bracketrightbigg = ?1Φ d 2Φ dφ2 . a42a163a8a138 a36a14a139a140 a56a51θ a14a33a34 a8a141φ a142a68a143 a144 a140 a56a51φ a14a33a34 a8a141θ a142a68a27 a137a39a147a148a149a150a138a151 a152a141θ a142a68a153 a141φ a142a68a14a154a34 a8 a156a25μa8a151a51a153 a164a165a166a128θa52 a26 a63φa52 a26a14a26a124 a8a136 a62a14a131 a43 a154 a176 a26a9a10 a51 1 sinθ d dθ parenleftBig sinθdΘ(θ)dθ parenrightBig + parenleftbigg λ? μsin2θ parenrightbigg Θ = 0, d2Φ dφ2 + μΦ = 0. a42a163 a153a177a178 a164a165a166 Helmholtz a9a10a11 a174a103a104 a69a105a14a26a124 a113a114 a27 a42a55a179a180a181a182 a16a183a154a46a184a185a186a108 a8a135u = u(r,θ) a141φ a142a68a14a186a108a27 a42a157a51a187a8a188a43a189 a22a20a21 a11a190a191a192a193a194a195a196a197a198 a57a113 a27a11 a42 a183a186a108a199 a8 Helmholtz a9a10a14a108a36 a157a200a201 a25 1 r2 ? ?r parenleftbigg r2?u?r parenrightbigg + 1r2 sinθ ??θ parenleftbigg sinθ?u?θ parenrightbigg + k2u = 0. a132u(r,θ) = R(r)Θ(θ) a8a133a134 a9a10 a8a135a136 Θ(θ) bracketleftbigg 1 r2 d dr parenleftBig r2dR(r)dr parenrightBig + k2R(r) bracketrightbigg + R(r)r2 1sinθ ??θ parenleftBig sinθ?Θ(θ)?θ parenrightBig = 0. Wu Chong-shi a70a71a72a73 a74a75a76a77a78 ( a79) a80a81a82 a83a84a85a86 a8713 a88 a131a140a158 a39r2/R(r)Θ(θ) a8a159a160a161a8a157a38a39a136 a62 r2 R(r) bracketleftbigg 1 r2 d dr parenleftBig r2dR(r)dr parenrightBig + k2R(r) bracketrightbigg = ? 1Θ(θ) 1sinθ ddθ parenleftBig sinθdΘ(θ)dθ parenrightBig . a138 a36a14a139a140 a56a51r a14a33a34 a8a141θ a142a68a143 a144 a140 a56a51θ a14a33a34 a8a141r a142a68a27 a137a39a147a148a149a150a138a151a152a141r a142 a68a153 a141θ a142a68a14a154a34 a8 a156a202λa8a42a163a157a164a165a166a26a124 a113a114 a14a195a203a27 a136 a62a14a131 a43 a154 a176 a26a9a10 a8a204a205a206 a10 a63a207a208 a14 a164a209a210a211a212a213 a16 a43a141a206a214 a197 (a135a215a216)θ a217a68a14a154 a176 a26 a206 a10 a51 1 sinθ d dθ parenleftBig sinθdΘ(θ)dθ parenrightBig + λΘ(θ) = 0, a218 a25Legendrea206a10 a212a147a51a219a220 Legendrea206 a10 1 sinθ d dθ parenleftBig sinθdΘ(θ)dθ parenrightBig + parenleftbigg λ? μsin2θ parenrightbigg Θ = 0 a14a184a185a186a108 (μ = 0)a27