Wu Chong-shi a0a1a2a3a4 Green a5a6 (a1) §29.1 a7a8a9a10 Green a11a12a13a14a15a16a17 a18a19a20a21a22a23a24a25 Green a26a27a28a29a30a31a32a33a34a35a36a37a38 a25a39a40a41a42a43 Green a26a27a44a45a46a47a48 a25a49a50a51a52 Green a26a27 a25a53a54a41a55 1. Green a56a57a58a59a60a61a62a63a64a65 a66a67a68a69a70a71a72a73a25a74a75a76a77a78Poisson a79a80a81 a39a82a83a23a24a25Green a26a27 a55a84a85a39a86a25a87a88a89 a51a90a91 a32a44a92a93V a94 a25 a45 a72a95 a32 a96a69 a35a44 a82a97a98a85a99a100a39a22a25a101a100 (a98) a72a95a87a102 a32 a84a103a104a82a97a98 a105a50a106a107a98a55a108a82a97a109a110a111 a32 a112a113a114a115a39a116a87a72a95a117a118a119a117a120 a32 a104a114a82a97a98a25a72a121a122a110a123a106( a124 a50 0)a55a125a126a32a127 a22Green a26a27 a25a22a128a23a24a112a89a51a106a129 ( a44V a130 a106a129)a110a131a105a132a97 a92a93a94 a25 Poisson a79 a80 ?2G(r;rprime) = ? 1ε 0 bracketleftbigδ(r ?rprime) + σ(Σ)bracketrightbig, a133 a94σ(Σ)a134 a82a97a98Σ a85a25a101a100a98a72a95a135a136a55a123a137a110 a32( a22a138 a44V a130 a25)Green a26a27G(r;rprime)a33 a137a139 a134a140a141 a116a87a72a95a72a121a25a142a143a43a144a107 a45 a72a95 δ(r ?rprime) a25a72a121 G 0(r;rprime) a145 a82a97a98a85a25a101a100a72a95 σ(Σ) a25a72a121g(r;rprime) a32 G(r;rprime) = G0(r;rprime)g(r;rprime). ?2G0(r;rprime) = ? 1ε 0 δ(r?rprime), G0(r;rprime) = 14piε 0 1 |r?rprime| a146a51 a32G0(r;r prime) a44r = rprime a45a134 a66a147a148a25a55 a145 ?2g(r;rprime) = ? 1ε 0 σ(Σ). a125a50a101a100a72a95σ(Σ) a149 a87a102 a44a150 a98 Σa85 a32 a146 a51 a32g(r;r prime)a52a133a39a151a152a153 a27a44a150 a98 Σ a30a154 (a155a156a134a32a44 V a130)a134a157a157 a147a148a25a55 a158 a140a141 a116a87a159a160a161a76 a32a33 a115 G(r;rprime) = 14piε 0 1 |r?rprime| + g(r;r prime). a53a162 a81a163a164 a82a97a165a166 a32a167 a115a168a169a25a170a171a55 a149 a66a172 g(r;rprime)a25a173a174a175a176a177a113a114a115a146a66a168a55 a53a162a133a178 a164a179 a25a21a22a23a24 a32a180a181 Helmholtza79a80 a25Green a26a27a32 ?2 ?G(r;rprime) + k2 ?G(r;rprime) = ? 1ε 0 δ(r ?rprime), r,rprime ∈ V, ?G(r;rprime)vextendsinglevextendsingle Σ = 0. a167 a89a182a183 a38a184 a25 Green a26a27 a173a115 a145 Poisson a79a80 a25 Green a26a27 a168a169a25a147a148a41a42a55a185a20 r = rprime a45a154a32 ?G(r;rprime) a44V a130a134a157a157 a147a148a25a55a186 ?g(r;rprime) = ?G(r;rprime)?G(r;rprime), G(r;rprime)a134 a123a137 Poisson a79a80 a25 Green a26a27 a55a187 ?G(r;rprime) a145G(r;rprime)a146a188a189a25a22a128a23a24a32 Wu Chong-shi §29.1 a190a191a192a193 Greena194a195a196a197a198a199a200 a2012a202 a89a51a153a203 ?2?g(r;rprime) + k2?g(r;rprime) = k2G(r;rprime), r,rprime ∈ V, ?g(r;rprime)vextendsinglevextendsingleΣ = 0. a187a162 a140a204a79a80a205a206 a25 G(r;rprime) a44 r = rprime a45a134 a51 1/|r?rprime|a25a207a177a208a209a25 a32 a146a51 a32 ?g(r;rprime) a44 a139 a45 a39a22a147a148( a210a211?2?g(r;rprime) a113a203a212δ a26a27) a32a140a33a213 a183 ?G(r;rprime) a145 G(r;rprime) a39a169a32 a44r = r prime a45a214a134 a51 1/|r?rprime|a25a207a177a208a209a25a55a215a216a85 a32 a84a217a39a86a25 a36a37 a89a218 a32a44r = r prime a45a47a48a32 a39a22a115 ?G(r;rprime) ～ 1 4piε0 cos(k|r?rprime|) |r?rprime| . ? a163a219a92a93a94Greena26a27a44a45a46a157a25a49a50a32a145a39a219a92a93a94Greena26a27a66a168a55 ? a39a219a92a93a94a25 Greena26a27a134a157a157 a147a148a25 a32 a103 a38 a25a39a151a153 a27 a66a147a148a55 ? a140a134a220a221a222 a128a25 a32 a125a50 a223 a45a46a224 a25a41a42a225a66a123a168 a32 a39 a219a92a93a94 a25 a45a46 a216a226a85 a134a163a219a92a93a94 a25 a98 a46 a55 ? a66a227a228a229a32a230a219a92a93a94 a25 Green a26a27a167 a137a139a175a212a203a66a168a25a49a50a55 a53a162 a230a219a92a93a94 a25 Poisson a79a80a81 a39a82a83a23a24 a32a38 a25 Green a26a27G(x,y;xprime,yprime)a32a134 a22a128a23a24 bracketleftBig ?2 ?x2 + ?2 ?y2 bracketrightBig G(x,y;xprime,yprime) = ? 1ε 0 δ(x?xprime)δ(y ?yprime), (x,y),(xprime,yprime) ∈ S, G(x,y;xprime,yprime)vextendsinglevextendsingleC = 0 a25a128 a32 a133 a94C a134a231 a98a232a233 S a25a82a97a55 a220a221a234 a114 a32 G(x,y;xprime,yprime) = ? 12piε 0 ln radicalbig (x?xprime)2 + (y ?yprime)2 + g(x,y;xprime,yprime), a133 a94a81 a39a235 a134 a144a107 a45 a72a95 a44 a132a97 a92a93a94 a25a72a121 ( a236 a89a51a143a85a39 a204a237a27a32a124a127 a162a72a121a238 a45 a25a239 a124) a32 a44 a223 a45a46a224(a216a226a85a134a163a219a92a93a94 a25a240 a46)δ(x?xprime)δ(y?yprime)a157a134 a53 a27 a208a209a25a241 a81a230 a235g(x,y;xprime,yprime) a134 a82a97a85a25a101a100a72a95a99a100a25a72a121 a32a44 S a130a157a157 a147a148a55 Wu Chong-shi a242a243a244a245a246 Green a194a195(a243) a2013a202 2. Green a56a57a63a247a248a249 a250a251a252a39a217a253a98a114a91a25a128a177 u(r) = integraldisplayintegraldisplayintegraldisplay V prime G(rprime;r)ρ(rprime)drprime ?ε0 integraldisplayintegraldisplay Σprime f(Σprime)?primeG(rprime;r)vextendsinglevextendsingleΣprime ·dΣprime. a140a204 a170a171 a44a254a222a255 a138a85a115a0a128 a30a157 a43 a44a205a206 a25a174a1a87 a94a32 G(rprime;r) a2 a175r a157 a25a144a107 a45 a72a95 a44rprime a157 a25a72a121 a32a38a3 a85 a44a4a5a45rprime a157 a25a72a95ρ(rprime)drprime a32 a225a53 a4a5a45 a1a87 a32a6a7 a203r a157 a25a72a121a8 a53 a140a204 a23a24a25a9a10 a35a11 a52a91 Green a26a27 a25a53a54a41a55a125a50 a32a181 a171a12a132a97 a92a93 a25 Green a26a27a13 a169 a32a14a15 a177 G(rprime;r) = G(r;rprime) (#) a105a19a25a16 a32a13a17a32 a85a177 a33a18a19 a131a105 u(r) = integraldisplayintegraldisplayintegraldisplay V prime G(r;rprime)ρ(rprime)drprime ?ε0 integraldisplayintegraldisplay Σprime f(Σprime)?primeG(r;rprime)vextendsinglevextendsingleΣprime ·dΣprime, a174a1a87a25 a254a222a255 a138 a33 a39a20 a230a21 a20a55 a81a230 a235a25a98a1a87a108a69 a33a134 a76a22a82a97a98a85a25a101a100a98 a72a95a25a23a24a55 a25a26(#) a27 a55 a145a81a28 a39a29 a94 a25a30a31a39a169 a32a32a33a34 G(r;rprimeprime)a32a38 a188a189a25a22a128a23a24a108a69 a33a134 ?2G(r;rprimeprime) = ? 1ε 0 δ(r?rprimeprime), r,rprimeprime ∈ V, G(r;rprimeprime)vextendsinglevextendsingleΣ = 0. a35 a141a204a79a80 a87 a156a3 a51 G(r;rprimeprime) a145 G(r;rprime)a32 a123a36 a32 a69 a31a44 a232a233 V a130 a1a87 a32a33 a114a91 integraldisplayintegraldisplayintegraldisplay V bracketleftbigG(r;rprimeprime)?2G(r;rprime)?G(r;rprime)?2G(r;rprimeprime)bracketrightbigdr = ? 1ε 0 integraldisplayintegraldisplayintegraldisplay V bracketleftbigG(r;rprimeprime)δ(r ?rprime)?G(r;rprime)δ(r ?rprimeprime)bracketrightbigdr = ? 1ε 0 bracketleftbigG(rprime;rprimeprime)?G(rprimeprime;rprime)bracketrightbig. a37a38Green a39 a177 a32 a35a85a177a40 a206 a25a174a1a87a41a50a98a1a87 a32a33 a115 G(rprime;rprimeprime)?G(rprimeprime;rprime) = ?ε0 integraldisplayintegraldisplay Σ bracketleftbigG(r;rprimeprime)?G(r;rprime)?G(r;rprime)?G(r;rprimeprime)bracketrightbig·dΣ. a2 a120a82a97a165a166 a32 a19a42a114a203 a205a206 a25a98a1a87a50 0a55 a140 a169 a33 a182a183a20 G(rprime;rprimeprime) = G(rprimeprime;rprime), a35rprimeprime a19 a131a50r a32a140a33a134(#)a177a55 a181 a171 a134a81a163a164 a82a97a165a166 a32 a85a98a25a170 a37 a68a69a43a44a55 a53a162a133a178 a164a179 a25a21a22a23a24 a32a38a184 a25 Green a26a27a134a210 a68a69a115a53a54 a14a15 (#)a32a34a35 a173a174 a36a37 a55 Wu Chong-shi §29.2 a45a46a47a48a49a50 Helmholtza51a52a196Greena194a195 a2014a202 §29.2 a53a54a55a56a57a58 Helmholtz a59a60a13 Green a11a12 a234a163a219 a132a97 a92a93a94Helmholtza79a80 a25Green a26a27a32 a42 a44a163a219 a132a97 a92a93a94a234 a128 a79a80 ?2G(r;rprime) + k2G(r;rprime) = ? 1ε 0 δ(r ?rprime), r,rprime ∈ V. a14 a162a132a61a62 a157 a25a82a97a165a166 a32a31 a98 a32a36a37 a55 a140a204a79a80a134 a39 a204a63a64a65a79a80a32 a125a126 a32 a89a51a66a67 a234 a128 a63a64a65a79a80 a25a68a69a30a31 a32 ? a250a234 a203 a79a80 a25a39 a204a155 a128 a32 a103a35 a79a80a64a65 a41a241 ? a35G(r;rprime)a66a123a137a64a65 a23a24a25a70a71 a26a27a72a73 a55 a140a141a74 a30a31 a32a155a156a134a81a230a74 a30a31 a32a75a211 a85a76a115a77 a17a78 a227 a32a140a79 a66a80a173a174a25a81a82a55 ? a140 a112 a134 a39 a204a155a83 a25 a63a64a65a79a80 a43 a149a44 r = rprime a45a32a63a64a65 a235a84a66a50 0a55 ? a103a85a32 a187a162 a140a134a44 a132a97 a92a93a32 a89a51a86a108a110a87a88a89a68a90 a32 a51a91a87a208a92 Laplace a93a94 a25a66a95a41 a32 a104a23a24a114a91a91a87a25a96a41a55 a97a250a80a89a68 a231a98a32 ξ = x?xprime, η = y ?yprime, ζ = z ?zprime, a42a35 a45 a72a95a146 a44a45a124 a50a99a89a68 a15 a25 a75a45 a55a186 G(r;rprime) = g(ξ,η,ζ) a32 a162 a134a32g(ξ,η,ζ)a188a189a79a80 ?2ξ,η,ζg(ξ,η,ζ) + k2g(ξ,η,ζ) = ? 1ε 0 δ(ξ)δ(η)δ(ζ), a133 a94 ?2ξ,η,ζ ≡ ? 2 ?ξ2 + ?2 ?η2 + ?2 ?ζ2 a134 a51a100a101a89a68 ξ,η,ζ a50a22a95a102a25 Laplace a93a94 a55 a220a221 a90a203 a32 a95a103 a31 a25 a79a80a134a104a105 a66a95a25 a32 g(ξ,η,ζ) a149a134R = radicalbigξ2 + η2 + ζ2 a25a26a27a32g(ξ,η,ζ) = f(R). a125a126a32a181 a171a35a100a101a89a68 a15 (ξ,η,ζ)a105 a103a50a106a89a68 a15a32a211a79a80 a35a95a50 R negationslash= 0 a45a157 a25 a64a65a79a80 1 R2 d dR bracketleftBig R2df(R)dR bracketrightBig + k2f(R) = 0 (a75 a125 a134a44a44 R = 0 a45a149a107a44 a144a108a153 a27) a51a52 R = 0 a45a157 a25a82a97a165a166 ( a44 R = 0 a45a157 a115a39a144a107 a45 a72 a95)a55a126 a79a80a134 a238a151a106 Bessel a79a80a32a38 a25a109a128 a134 f(R) = A(k)e ikR R + B(k) e?ikR R . a37a38R = 0 a145 a132a61a62 a157 a25a82a97a165a166a22a203 a237a27 A(k)a145B(k)a55 a110a111a112a113a114a115B(k) a251a116a91Helmholtz a79a80 a25a216a226a117a118 a32a119a181a213a32a38a134 a187a120a121 a79a80a122 a172a87a123 a95a102(a87a123a124a111 a93 a116a87) a114a91a25a55a80a50a39 a204a180a125a32a126a127a35a234 a114a91a25a128 a44 a132a61a62 a157 a50a208a209a120a55 a124 a111 a93 a125 a125 a50e?iωt a32a211 a128a177 a94 a25 a81 a39a235a50a208a209a120 a32a81a230 a235a50a113a128a120a55a146a51 a32 a137a139a115 B(k) = 0a55 Wu Chong-shi a242a243a244a245a246 Green a194a195(a243) a2015a202 a129a217a25 a237a27 A(k)a33 a137a139a187R = 0 a157 a25a82a97a165a166 a127 a22 a32 a42a187R = 0 a157a45a46 a25a130a136 a127 a22a55 R = 0a131a63a132a133 a113a114a115A(k) a140 a111a225a66 a18 a100a109a35a128a177 a2 a120R = 0 a157 a25a82a97a165a166 a32a75 a125 a134f(R) a119 g(ξ,η,ζ) a44 R = 0 a157 a25a153 a27 a225a66 a107a44 a55a134a39 a79 a98 a32a135a184a136a122a137 a22 a32a138a134a11 a52 δ a26a27 a25a106a177 a214 a137 a139a84a1a87 a255 a138a217a124 a222 a128a55a162 a134a32a139 a22a69a110 a32 a137a108a35 a79a80a44 R = 0a47a48 a25a140a174a1 a130 a1a87 a32integraldisplayintegraldisplayintegraldisplay ?2ξ,η,ζf(R)dξdηdζ + k2 integraldisplayintegraldisplayintegraldisplay f(R)dξdηdζ = ? 1ε 0 . (maltesecross) a40 a206a81 a39a235a25a174a1a87a137a108a41a50a98a1a87 integraldisplayintegraldisplayintegraldisplay ?2ξ,η,ζf(R)dξdηdζ = integraldisplayintegraldisplay bracketleftBig ?ξ,η,ζf(R) bracketrightBig ·dΣ, a125a50 a140 a169 a33 a89a51a9a141a142 a44 R = 0a45 a25 a234 a153a23a24a55 a124a140a204 a140a174a1a50a51 R = 0 a45 a50a106a143 a32 ρa50a144a145a25 a106a174 a32a211 integraldisplayintegraldisplayintegraldisplay ?2ξ,η,ζf(R)dξdηdζ = integraldisplayintegraldisplay bracketleftBig ?ξ,η,ζf(R) bracketrightBig ·dΣ = integraldisplayintegraldisplay df(R) dR R 2 sinθdθdφ vextendsinglevextendsingle vextendsingle R=ρ = ?4piA(k)(1 ?ikρ)eikρ. a81a230 a235a25a174a1a87a89a51a100a109 a93 a203 a32integraldisplayintegraldisplayintegraldisplay f(R)dξdηdζ = 4piA(k) integraldisplay ρ 0 eikRRdR = 4piA(k)k2 bracketleftBig (eikρ ?1)?ikρeikρ bracketrightBig . a35 a140a146 a170a171 a2 a9a91 (maltesecross)a177 a32a33 a115 ?4piA(k) = ? 1ε 0 , a146a51 a32A(k) = 1/4piε0, a122 ka132 a14 a55 a140 a169 a32 a147 a31a33a234 a203a20 a163a219 a132a97 a92a93 Helmholtza79a80 a25Green a26a27 g(ξ,η,ζ) = f(R) = 14piε 0 eikR R , a119 G(r;rprime) = 14piε 0 eik|r?rprime| |r?rprime| . a108k = 0a111 a32a140a204 a170a171 a33 a9a91 Poisson a79a80 a25Green a26a27 a55 a147 a31a32a34a35a213 a183 a32a140a204 a170a171 a134a44 a132a61a62 a157 a50a208a209a120 a32 a225a85 a124 a111 a93 a125 a125 a50 e?iωt a25a165a166a217a114a91 a25a55a89a51 a127a148a32a181 a171 a35a234 a132a61a62 a157 a50a113a128a120 a32 a225a85a68 a124 a111 a93 a125 a125 a50 e?iωt a32a211Greena26a27 a137a139 a134 G(r;rprime) = 14piε 0 e?ik|r?rprime| |r?rprime| . a181 a171 a134 a133a178a207a177a25a132a61a62a165a166 a32 a108a69 a236 a113a114a91a133a178a207a177a25a128a55 Wu Chong-shi §29.3 a149a150Poissona51a52 a242 a197a151a152a192a193a196 Greena194a195 a2016a202 §29.3 a153a154 Poisson a59a60a155a14a156a157a9a10a13 Green a11a12 a70a86a25a158a25 a134 a109a172a53a162a159 a130Poissona79a80a81 a39a82a83a23a24Green a26a27 a25 a36a37a32a32 a81a82a39 a146a234Greena26a27 a25 a237 a70 a79 a31a55 a159 a130Poissona79a80a81 a39a82a83a23a24 Green a26a27 a25a22a138 a134 ?22G(r;rprime) = ? 1ε 0 δ(r ?rprime), |r| < a, |rprime| < a, G(r;rprime)vextendsinglevextendsingler=a = 0, a133 a94 r2 = x2 + y2, ?22 = ? 2 ?x2 + ?2 ?y2. a250a81a82a39 a74 a68a69a25a30a31 a32 a42a251a116a91 a79a80a134 a39 a204a63a64a65a79a80a32 a146a51a35 Green a26a27 a66a123a137 a64a65 a23 a24a25a70a71 a26a27a72a73 a55 a160a70 a231 a98a161a89a68 a15a32 a89a68 a75a45a162a44 a159a143 a32 G(r;rprime) = R0(r) + ∞summationdisplay m=1 bracketleftbigR m1(r)cosmφ + Rm2(r)sinmφ bracketrightbig. a168a169 a32 a35δ a26a27a167 a66a139a163a70a71 a26a27a72a73a32 δ(r ?rprime) = δ(x?xprime)δ(y ?yprime) = 1rprimeδ(r ?rprime)δ(φ ?φprime) = 1rprimeδ(r ?rprime) braceleftbigg 1 2pi+ 1 pi ∞summationdisplay m=1 bracketleftbigcosmφcosmφprime + sinmφsinmφprimebracketrightbigbracerightbigg. a212 a44 a25a23a24 a33a134a181a164a234 a128 R 0(r), Rm1(r)a145Rm2(r) a55 ? a127 a22R 0(r) a25 a237a165 a87 a79a80 a22a128a23a24 a134 1 r d dr bracketleftbigg rdR0(r)dr bracketrightbigg = ? 12piε 0 1 rprimeδ(r ?r prime), R0(0)a115a97, R0(a) = 0. a108r negationslash= rprime a111 a32a79a80a134a64a65 a25 a32a44 a251a116a91a82a97a165a166 a31a32 a115a128 R0(r) = ? ? ? A0, r < rprime, B0 ln ra, r > rprime. a32 a37a38R 0(r) a44r = rprime a45 a25a147a148a41 a32 a42 R 0(r)a44r = rprime a45 a147a148 a32 a103 Rprime 0(r) a66a147a148( a38 a89a51a187 a79 a80a44r = rprime a45a141 a108a1a87a114a91) a32 dR0(r) dr vextendsinglevextendsingle vextendsinglevextendsingle rprime+0 rprime?0 = ? 12piε 0 1 rprime, Wu Chong-shi a242a243a244a245a246 Green a194a195(a243) a2017a202 a33 a89a51a22a203A 0 a145B0 a32 A0 = ? 12piε 0 ln r prime a , B0 = ? 1 2piε0. a162 a134 R0(r) = ?? ?? ??? ? 12piε 0 ln r prime a , r < r prime, ? 12piε 0 ln ra, r > rprime. ? a127a22Rm1(r)a25a237a165 a87 a79a80 a22a128a23a24 a134bracketleftBig 1 r d dr parenleftbigg r ddr parenrightbigg ? m 2 r2 bracketrightBig Rm1(r) = ?δ(r ?r prime) piε0rprime cosmφ prime, Rm1(0)a115a97, Rm1(a) = 0. a108r negationslash= rprime a111 a32a79a80a134a64a65 a25 a32a44 a251a116a91a82a97a165a166 a31a32 a115a128 Rm1(r) = ?? ? ?? Am1 parenleftBigr a parenrightBigm , r < rprime, Bm1 bracketleftBigparenleftBigr a parenrightBigm ? parenleftBiga r parenrightBigmbracketrightBig , r > rprime. a37a38R m1(r)a44r = rprime a45 a25a147a148a41 a32 a42 R m1(r)a44r = rprime a45 a147a148 a32 a103Rprime m1(r) a66a147a148 a32 dRm1(r) dr vextendsinglevextendsingle vextendsinglevextendsingle rprime+0 rprime?0 = ? 1piε 0 1 rprime cosmφ prime, a22a203A m1 a145Bm1 a32 Am1 = ? 12piε 0 1 m bracketleftbiggparenleftBigrprime a parenrightBigm ? parenleftBiga rprime parenrightBigmbracketrightbigg cosmφprime, Bm1 = ? 12piε 0 1 m parenleftBigrprime a parenrightBigm cosmφprime. a162 a134 Rm1(r) = ? ?? ?? ? 12piε 0 1 m bracketleftBigparenleftBigrrprime a2 parenrightBigm ? parenleftBig r rprime parenrightBigmbracketrightBig cosmφprime, r < rprime, ? 12piε 0 1 m bracketleftBigparenleftBigrrprime a2 parenrightBigm ? parenleftBigrprime r parenrightBigmbracketrightBig cosmφprime, r > rprime. ? a127 a22R m2(r) a25 a237a165 a87 a79a80 a22a128a23a24 a134bracketleftBig 1 r d dr parenleftbigg r ddr parenrightbigg ? m 2 r2 bracketrightBig Rm2(r) = ?δ(r ?r prime) piε0rprime sinmφ prime, Rm2(0)a115a97, Rm2(a) = 0. a38a145Rm1(r)a188a189a25a237a165 a87 a79a80 a22a128a23a24a25a207a177a166a167a168a169a123a168 a32a149a134 a158 a63a64a65 a235 a94 a25 cosmφprime a103a105a20sinmφprime a32 a146a51 a32 Rm2(r) = ? ?? ?? ? 12piε 0 1 m bracketleftBigparenleftBigrrprime a2 parenrightBigm ? parenleftBig r rprime parenrightBigmbracketrightBig sinmφprime, r < rprime, ? 12piε 0 1 m bracketleftBigparenleftBigrrprime a2 parenrightBigm ? parenleftBigrprime r parenrightBigmbracketrightBig sinmφprime, r > rprime. Wu Chong-shi §29.3 a149a150Poissona51a52 a242 a197a151a152a192a193a196 Greena194a195 a2018a202 a140 a169 a32a33a234 a114a20a159 a130Poissona79a80a81 a39a82a83a23a24a25 Green a26a27a32 a108r < rprime a111 a32 G(r;rprime) = ? 12piε 0 braceleftbigg ln r prime a + ∞summationdisplay m=1 1 m bracketleftBigparenleftBigrrprime a2 parenrightBigm ? parenleftBig r rprime parenrightBigmbracketrightBig cosm(φ?φprime) bracerightbigg , a108r > rprime a111 a32 G(r;rprime) = ? 12piε 0 braceleftbigg ln ra + ∞summationdisplay m=1 1 m bracketleftBigparenleftBigrrprime a2 parenrightBigm ? parenleftBigrprime r parenrightBigmbracketrightBig cosm(φ?φprime) bracerightbigg , a85a98 a140a74a79 a31 a32 a35 Green a26a27 a66a123a137 a64a65 a23a24a25a70a71 a26a27a72a73a32 a39a40 a213 a76 a32 a114a91a25 a128a177a113 a134 a132a61a170 a27 a55a108a69 a32 a66a171a185 a44a172a146a155a83a173 a207a217a89a51a35a170 a27a234a145 a55 a180a181a32 a212 a44 a114a91 a25a128a177 a33a134a181 a126a55a66a172 a32a140a34a35a119a174a175a176 a170 a27a234a145 a25a177a178a55 a217a98 a32 a81a82a39 a74a79 a31 a32a38 a35a100a109 a7 a203a128a25a115a179a207a177a55 a180a181a218a182 a32 a39a183 a44 a109a110a159 a94a162 a85 a45 a72a95 a31a32a44 a159a184a85a96a69a203a212a101a100a72a95a55a159 a130a185a255 a39 a45 a25a72 a121 a32a33a134a45 a72a95a25a72a121 a145 a101a100a72a95a25a72a121a25a142a143a55a253a186 a44a45 a72a95a146 a44a45a134 a53 a27 a208a209a25 a32 a103 a31 a186 a44 a159 a130a134a157a157 a147a148a25a55 a181 a171 a135a184a18a187a79a188 a110 a234 a203a101a100a72a95 a44 a159 a130 a146a99a100a25a72a121 a32 a108a69 a167a33a234 a203a20a189 a204 a159 a130Poissona79a80a81 a39a82a83a23a24a25 Green a26a27 a55 a212 a44a35 a81a82a25 a140a74a79 a31 (a54a50a72a12a31) a32 a133a190a70a191 a148a134 ? a192a132a133a193a63a194a195a196a197a198a199a200a201a202a63a59a196a197a203a204 a55 ? a103a205a16a213a32a33a206a207a208a209a210a63a59a196a197a63a211a212a201a202a208a213a214a65 a110 a133a215a216a217a63a218a200a59a196a197 (a199a200a219a220a221a63a59a196a197a32a222a199 a200a219a201a202a63 a223a223 a224a196a197) a63a211a212 a55 ? a140a204 a223a224 a224 a72a95a25a106a129a41 a32a33 a175a212 a44a38a145 a159 a130 a25a225a216a25 a45 a72a95a39a161 a32a44a209a210a18a7 a203 a145a75 a76a23a24a168a169a25a128a55 a22629.1 a227a228a229 ? a149a35 a159 a130 a25a72a95a87a102a66a95 a32a149a35a140a141a204a45 a72a95 a167a18 a99a100a203 a209a230 r = a a207a208 (a196a231a65 0) a25a232 a171 a32 a82a83a23a24a233 a63a234a58a235a199a249a32a33a18a236 a182 a140 a169a114a91a25a128 a145a75 a76a23a24a25a128 a44 a159 a130 a39a22a39a237a55 ? a89a51a183a44a110a228a238a91a32a140a204 a106a129a72a95 a181 a171 a107a44 a25a16 a32a38 a39a22a107a162 a209a239a32a210a211 a159 a130 a25a72a95a87a102 a33 a145a75 a76a25a23a24a66a168 a32a33 a66 a18a236 a182a106a129a41a55a119a186a103a39 a74a213 a31 a32 a187a162a101a100a72a95a25a72a121 a44 a159 a130a134a157 a157 a147a148a25 a32a44 a159 a130 a25 a185a164 a106a129 ( a45) a72a95a214 a66a89 a18 a99a100a168a169a25a232a171a55 ? a137a70a72a12a31a105a240a25a14a241a32a33a44 a162 a18a210a234 a203 a140a204 a106a129a72a95a25a72a102 a145a38 a25 a92a93 a107a88a55 a140a134a140a204 a106 a129a72a95 a134a210a107a44 a25a242 a94 a174a212a55 ? a37a38a53a54a41a25a251a116a32a236 a89a51 a34 a39a243a244a22 a32a181 a171 a140a204 a106a129a72a95 a107a44 a25a16 a32a38a236 a39a22a107a162a225a216a72 a95a146 a157 a25a144a145a25a245a246a240a85a55 Wu Chong-shi a242a243a244a245a246 Green a194a195(a243) a2019a202 a127a140a204 a106a129a72a95a25a107a88a50r 1 = (x1, y1)a32 a72a102a50e a32a38a145 a225a216 a45 a72a95a39a161 a32a44 a159 a130 a25a72a121 a33a134 G(r;rprime) = ? 12piε 0 bracketleftBig ln|r?rprime|+ eln|r?r1|+ C bracketrightBig , (maltesecross) a133 a94a237a27C a122a72a121a238 a45 a25a239a247a115 a14 a55a212 a44 a25a23a24 a33a134a35 a84 a35a234 a159a184 r = aa85a25a72a121a50 0 a32 ? 12piε 0 bracketleftBig ln|r?rprime|+ eln|r?r1|+ C bracketrightBig r=a = 0, a234 a203r 1, ea145C a55a248 a255a140a204a79a80 a137a139a53a159a184a85a25a39a249 a45a250 a105a19a55 a181 a171a160a70 a231 a98a161a89a68 a32 a42a186 x = rcosφ, xprime = rprime cosφprime, x1 = r1 cosφprime, y = rsinφ, yprime = rprime sinφprime, y1 = r1 sinφprime, a211a79a80 a41a50 lnbracketleftbiga2 + rprime2 ?2arprime cos(φ ?φprime)bracketrightbig+ eln bracketleftBig a2 + r21 ?2ar1 cos(φ?φprime) bracketrightBig + 2C = 0, a38 a137a139a53a39a249 φ a250 a105a19a55a251a70 a72a73 a177 lnbracketleftbig1 + t2 ?2tcosφbracketrightbig = lnbracketleftbig1?teiφbracketrightbig+ lnbracketleftbig1?te?iφbracketrightbig = ?2 ∞summationdisplay m=1 1 mt m cosmφ, |t| < 1, a33 a89a51 a34 a39a243a41a50 2lna + ln bracketleftBig 1 + parenleftBigrprime a parenrightBig2 ?2r prime a cos(φ ?φ prime) bracketrightBig + 2elnr1 + eln bracketleftBig 1 + parenleftbigg a r1 parenrightbigg2 ?2 ar 1 cos(φ?φprime) bracketrightBig + 2C = 2lna + 2elnr1 ?2 ∞summationdisplay m=1 1 m bracketleftBigparenleftBigrprime a parenrightBigm + e parenleftBig a r1 parenrightBigmbracketrightBig cosm(φ?φprime) + 2C = 0, a162 a134a32a33 a114a91 lna + elnr1 + C = 0 (#) a145 parenleftBig rprime a parenrightBigm + e parenleftBig a r1 parenrightBigm = 0, m = 1,2,3,···, a42 e = ? parenleftBigr1rprime a2 parenrightBigm a119 e = ? parenleftBigr1rprime a2 parenrightBig1 = ? parenleftBigr1rprime a2 parenrightBig2 = ? parenleftBigr1rprime a2 parenrightBig3 = ···, m = 1,2,3,···, a146a51 a32 e = ?1 a145 r1 = a 2 rprime a119 r 1 = parenleftBiga rprime parenrightBig2 rprime. a140 a169 a32a135a184 a25a44 a234 a203a20 a140a204 a106a129a72a95 a32a38 a107a162a225a216a72a95a146 a44 a144a145a25a245a246a240a85 a32 a225a85a188a189 rprimer1 = a2. a138a134 a188a189 a140a204a14a15 a25 a141a204a45a32a250 a54a50 a14 a162a159r = aa25a252a253a254a255a0a1a2a3a4a5a6a7a8 a9a10a11a12a13a14a10a11a15a16a17a18a19r = a a20a21a22a23 a17a7a24a25 a20 a10a26a27a8a7a28a29a27 a21 a0 a30ea12r 1 a20 a3a4a31a32(#) a33 a7a34a35a36a37a38 C = ?lna + lnr1 = ln arprime. Wu Chong-shi §29.3 a39a40Poissona41a42a43a44a45a46a47a48a49 Greena50a51 a5210a53 a54a30e, r 1 a12C a20 a3a4a31a55(maltesecross) a33 a7a56a57a58a37a38a19a59Poisson a60a61a62a63a64a65a66a67a20 Greena68a69 G(r;rprime) = ? 12piε 0 bracketleftBig ln|r?rprime|?ln vextendsinglevextendsingle vextendsingler? parenleftBiga rprime parenrightBig2 rprime vextendsinglevextendsingle vextendsingle+ ln arprime bracketrightBig , a70a71a72a28a73a74a75a76 a20a77a78a33 a7 G(r;rprime) = ? 14piε 0 braceleftBigg ln bracketleftBig r2 + rprime2 ?2rrprime cos(φ?φprime) bracketrightBig ?ln bracketleftBig r2 + parenleftBiga2 rprime parenrightBig2 ?2ra 2 rprime cos(φ?φ prime) bracketrightBig + 2ln arprime bracerightBigg . a30a17 a69a68a69a79a80a81 a7a58a35a36a82a83a7a84a85a86a87 a63a88a60a89 a38a90 a20 a3a4a0 Wu Chong-shi a43a91a92a93a94 Greena50a51(a91) a5211a53 a72a37a83a95a19a59Poisson a60a61a62a63a64a65a66a67a20 Greena68a69 a57a7a96a97a58a35a36a98a83 a63a99a20a100a101a66a67 ?22u(r) = ? 1ε 0 ρ(r), |r| < a, u(r)vextendsinglevextendsingler=a = f(φ) a20a101 a0a102a103a7a30 a60a61 a76 a20a104 a105a26a106a107a16 rprime a7 ?prime22u(rprime) = ? 1ε 0 ρ(rprime), |rprime| < a, u(rprime)vextendsinglevextendsinglerprime=a = f(φprime), a108 a63a60a109 a7a110a35a36a107a83 G(rprime;r) a111a112a113a114a115a20a100a101a66a67 a7 ?prime22G(rprime;r) = ? 1ε 0 δ(r ?rprime), |r| < a, |rprime| < a, G(rprime;r)vextendsinglevextendsinglerprime=a = 0, a54a116a117Green a68a69a20 a17a118a29(a24a35a36a82a16a8620.2 a119a20a120a121a122a123 a7a124a125a126a1 a109 a37a83 a20 G(r;rprime)a20a127a128 a77a78a33a129a130 a82a83) a7 G(r;rprime) = G(rprime;r), a131 a63a132 a106a107a16 ?prime22G(r;rprime) = ? 1ε 0 δ(r ?rprime), |r| < a, |rprime| < a, G(r;rprime)vextendsinglevextendsinglerprime=a = 0. a30a133a134 a60a61a135a136a137 a36 G(r;rprime)a12u(rprime)a7a27a138a7a54a72a19a59a139 a135 a7a58a38a90 integraldisplayintegraldisplay rprime<a ρ(rprime)G(r;rprime)drprime ?u(r) = ?ε0 integraldisplayintegraldisplay rprime<a bracketleftbigG(r;rprime)?prime 2 2u(rprime)?u(rprime)?prime 2 2G(r;rprime)bracketrightbigdrprime. a140a1 a109a20a109 a139 a135a141 a102a142a19a143 r = a a20a144 a139 a135 a7a145a146a31a32 a64a147a148a149 a7a58a150 u(r) = integraldisplayintegraldisplay rprime<a ρ(rprime)G(r;rprime)drprime + ε0 integraldisplay 2pi 0 bracketleftbigG(r;rprime)?primeu(rprime)?u(rprime)?primeG(r;rprime)bracketrightbig rprime=aadφ prime = integraldisplayintegraldisplay rprime<a ρ(rprime)G(r;rprime)drprime ?ε0 integraldisplay 2pi 0 f(φprime)?G(r;r prime) ?rprime vextendsinglevextendsingle vextendsingle rprime=a adφprime. a151a97a7a152a153 a20a62a63a154a77a155 a19a59a10a11 a135a156a20a157a158a159a62a160a154a161 a86a162 a104 a19a143a1 a20a163a164 a10a11a165 a164a20 a10a166a7 a163a164 a10a11 a20a135a156 a96a97a167a168 a100a20a64a147a148a149 (a19a143a1a10a166a65a20a135a156)a150a169a0a102a95a170a171a172a173a82a83a19a143a1a20 a10a11 a135a156 a7a35a36a30 a62a160a154 a76 a20a144 a139 a135 a54a106a107a16 integraldisplay 2pi 0 f(φprime)?G(r;r prime) ?rprime vextendsinglevextendsingle vextendsingle rprime=a adφprime = ? integraldisplay 2pi 0 f(φprime) lim ?r→0 1 ?r bracketleftBig G(r;rprime)vextendsinglevextendsinglerprime=a??r ?G(r;rprime)vextendsinglevextendsinglerprime=a bracketrightBig adφprime = ? integraldisplayintegraldisplay rprime<a f(φprime) lim ?r→0 G(r;rprime) ?r bracketleftBig δ(rprime ?a + ?r)?δ(rprime ?a) bracketrightBig rprimedrprimedφprime = ? integraldisplayintegraldisplay rprime<a f(φprime)G(r;rprime)δprime(rprime ?a)rprimedrprimedφprime, Wu Chong-shi §29.3 a39a40Poissona41a42a43a44a45a46a47a48a49 Greena50a51 a5212a53 a84a174a7a58a35a36a140a1 a109a20 a3a4a107a16 u(r) = integraldisplayintegraldisplay rprime<a bracketleftbigρ(rprime) + ε 0f(φprime)δprime(rprime ?a) bracketrightbigG(r;rprime)drprime. a84 a77 a6a7a19a143a1 a20a163a164 a10a11a175a176a58a86 ε 0f(φprime)δprime(rprime ?a) a0 a68a69δ prime(rprime ?a) a20 a83a177a7a5a6a72a19a143a1 a20a163 a164 a10a11a102a178a28a179a0 a84a180a110a35a36a181a182a83 a63 a134a183a184a185a186a0a187a188a150a108a189 a63 a134 a100a101a66a67 ?22u(r) = ? 1ε 0 bracketleftBig ρ(r) + ε0f(φ)δprime(r ?a) bracketrightBig , |r| < a, u(r)vextendsinglevextendsingler=a = 0, a151a97a7a24a124a30a190a150a191a174 a20a101 a0 ? a84a5a6a7a72a181a131δ a68a69a192a193 a98 a69a20 a87a194a195a7a196a197a198 a64a147a148a149a20a100a101a66a67 a7a124a35a36a106a107a102a197a198 a20a64a147a148a149 a7a199a200a201a72 a60a61 a76a202a27 a112 a173a203a204 a63a154a120a121a20 a196a197a198 a154 a7a72a205a206a59a207a208a208a102 0 a209 a199 a72 a64a147 a1a200a102 0 (a14a210 a69a65 a102a211a212) a20 a196a197a198 a154 a0a1a2 a135a213 a58a194a214a95a84 a88 a196a197a198 a154a20 a107 a89 a0 ? a96a97a7a196a197a198a64a147a148a149 a35a36a215 a141 a102 a60a61a20a120a121a123a33a20 a196a197a198 a154 a7a216a217a145a200a218a219a220a35a36a221a222a196 a197a198 a64a147a148a149 (a223 a107 a64a147a109 a1 a20a224a20a135a156) a167a60a61a196a197a198a154 (a205a206a59a207a20a224a20a135a156) a20 a205 a136 a0 a225a226a140a196a197a198 a64a147a148a149 a106a107a16 a60a61a20 a196a197a198 a154 a7a24 a223 a107 a20a227 a97a86a228a72a18 a64a147a109a20a224 a0 a36a1a229a230a95 Green a68a69a20 a133 a88a101a89 a0 a62a63a88a101a89 a86a231a27 a112 a197a198 a66a67a20a232a233a68a69a80a81 a0a84 a88a101a89a234 a117a235a236a237a7a238 a23 a86a38a90 a20a101 a239a239a86a211a212a240 a69 a0 a62a160a88a101a89 a86a10a241 a89 a7 a193 a76a242a243a188a86a140 a64a147 a1 a20a163a164 a10a11a117 a63 a134a8a9 a20a23 a10a11 (a118a102a241 a10a11)a31a244a0a84a199a150a72a245a246a196a247 a120a121a20a248a249a123a250(a251a252a253a123 a7a254a211 a147a255a0 a7a8a8)a195a1a125a14 a177a0a225a226a2a3a90a191a4a226a117 a248 a134 ( a209 a200a86 a63 a134) a241a10a11a162a8a5a173a31a244 a64a147 a1 a20a163a164 a10a11a7 a17 a255a0a20a248a249a123a250a227 a97a150a27a96a6a7 a20a8a9 a0 a111 a36a5a7a10a241 a89a20a10a23 a86a35a36a168a83a150 a8a123a33 a20a101 a7a238 a23 a86 a234 a117a235a236a150 a8 a0