Wu Chong-shi a0a1a2a1a3 a4 a5 a6 ( a7) a8Helmholtz a9a10a11a12a13a14a15a16a17a18a19a20a21a22a23a24a25a26a27a28a17a9a10 1 r d dr bracketleftbigg rdR(r)dr bracketrightbigg + bracketleftBig k2 ?λ? μr2 bracketrightBig R(r) = 0. a29a30k2 ?λ negationslash= 0 a22a31a19a32x = √k2 ?λr, y(x) = R(r)a22a33a9a10a19a34 (ν a35)Bessela9a10 1 x d dx bracketleftbigg xdy(x)dx bracketrightbigg + bracketleftbigg 1? ν 2 x2 bracketrightbigg y(x) = 0, a36a37μ = ν2 a38 a39a40a41a392 a42 a37a43 a24a44a45a46Bessela9a10a11x = 0a47a48a49a33a50 a38 a16a51a52a53a54a55a56a57a16 a43 a24a25a26a48a58 a30 a38 §22.1 Bessel a59a60a61 Neumann a59a60 Bessela62a63a64a65a66a67a68a69 x = 0a70x = ∞a71x = 0a72a73a74a67a68a22 x = ∞a72a75a73a74a67a68 a38 a76 a73a74a67a68 x = 0a77a22a78a79ρ = ±ν a38 a80ν negationslash= a81a82a83a22 Bessela62a63a84a65a66 (a85a86a87a88) a73a74a89a72 J±ν(x) = ∞summationdisplay k=0 (?)k k!Γ(k ±ν + 1) parenleftBigx 2 parenrightBig2k±ν . a90a91ν = a81a82na22a74Jn(x)a70J?n(x)a85a86a92a88a22 J?n(x) = (?)nJn(x), a93 a83a22 Bessela62a63a84a94a95a89a96a72 Jn(x)a22a94a97a89a74a98a99a100 Nn(x) = limν→n cosνpiJν(x)?J?ν(x)sinνpi = 2piJn(x)ln x2 ? 1pi n?1summationdisplay k=0 (n?k?1)! k! parenleftBigx 2 parenrightBig2k?n ?1pi ∞summationdisplay k=0 (?)k k!(k + n)! bracketleftbigψ(n + k + 1)+ψ(k + 1)bracketrightbigparenleftBigx 2 parenrightBig2k+n , a101a102a103a104 a22 a80 n = 0 a83a22a105a106a107a108a109a110a111a94a97a112a84a64a113a70 a38 Wu Chong-shi §22.1 Bessela114a115a116Neumanna114a115 a1172a118 a11922.1 a111a120a121a122a123 a124a125 a100a126a82a83a127a128a66 Jn(x)a84 a119a129 a38 a13022.1 Bessel a131a132 a127a128a66 Nn(x)a84 a119a129a133a119 22.2 a38 a13022.2 Neumann a131a132 a134 a82a108a109a110a72 Bessela135a82a84a136a137a108a109a110a22a138a139a98a140a141a121 Bessela135a82a84a95a142a143a144a86a145a22a146 a90a147 a141a88a148 (a133a149a95a150)a38a151a152 Bessela135a82a84 a134 a82a108a109a110a22a153a98a140a154a155a156a142a157a158a84a159a160a22a146 a90a161a162a163a164 a165a166a164a163a164a167 Bessela163a164a168a169a162a168a162a170 a38 Wu Chong-shi a171a172a173a172a174 a175 a114 a115 (a176) a1173a118 a177 22.1 a154a155a159a160 integraldisplay ∞ 0 e?axJ0(bx)dx, Rea > 0. a178 a179a180Bessel a135a82a84 a134 a82a108a181a22 a101a182 a112a159a160 a38 integraldisplay ∞ 0 e?axJ0(bx)dx = integraldisplay ∞ 0 e?ax ∞summationdisplay k=0 (?)k (k!)2 parenleftbiggbx 2 parenrightbigg2k dx = ∞summationdisplay k=0 (?)k (k!)2 parenleftbiggb 2 parenrightbigg2k integraldisplay ∞ 0 e?axx2kdx = ∞summationdisplay k=0 (?)k (k!)2 parenleftbiggb 2 parenrightbigg2k (2k)! a2k+1 = 1a ∞summationdisplay k=0 1 k! parenleftbigg ?12 parenrightbiggparenleftbigg ?32 parenrightbiggparenleftbigg ?52 parenrightbigg ··· parenleftbigg ?2k?12 parenrightbiggparenleftbiggb a parenrightbigg2k = 1a bracketleftBigg 1 + parenleftbiggb a parenrightbigg2bracketrightBigg?1/2 = 1√a2 + b2. a93a183a184a185 a84a186a68a72 a134 a82a187a70a22a187a70a83a153a188a188a189a64a95 a104 a84a113a190a191a192 a38 a146 a90a76a193a194 a187a70a83a195a189a187|b/a| < 1a38a196a195a137a197a198a199a22 a200a201a202a203 a22 a204a205 a84a159a160 a76 Rea > 0 a84a206a207a208a209a210a111a95a211a212a213a22 a214 a198 a76 Rea > 0 a84a206a207a209a210a215a89a216a71a198a159a160a121a84a217 a91a218a76a219 a95a209a210a215a89a216 a38a220a221 a89a216a222a223a84 a204a224 a22a195a98a140a106a107 a93 a66a113a190a191a192 a38 Wu Chong-shi §22.2 Bessela114a115a225a226a227a228a229 a1174a118 §22.2 Bessel a59a60a230a231a232a233a234 Bessela135a82 J±ν(x)a84a136a137 a147 a141a88a148a72 d dx [x νJν(x)] = xνJν?1(x), d dx bracketleftbigx?νJ ν(x) bracketrightbig = ?x?νJ ν+1(x). a235 a236a202a203 d dx [x νJν(x)] = xνJν?1(x). a100a139a22a237a238a239 Bessela135a82a84 a134 a82a108a109a110 Jν(x) = ∞summationdisplay k=0 (?)k k!Γ(k + ν + 1) parenleftBigx 2 parenrightBig2k+ν a121a240 a38 a138a241 a134 a82 a76a242a243a194 a212a213a22a244a140a98a140 a182 a112a245a246 a38 d dx [x νJν(x)] = d dx ∞summationdisplay k=0 (?)k k!Γ(k + ν + 1) x2k+2ν 22k+ν = ∞summationdisplay k=0 (?)k k!Γ(k + ν) x2k+2ν?1 22k+ν?1 = x νJν?1(x). a219a247 a22 d dx bracketleftbigx?νJ ν(x) bracketrightbig = d dx ∞summationdisplay k=0 (?)k k!Γ(k + ν + 1) x2k 22k+ν = ∞summationdisplay k=0 (?)k+1 k!Γ(k + ν + 2) x2k+1 22k+ν+1 = ?x ?νJν+1(x). (square) a76a93 a65a66 a147 a141a88a148a111a248a106 Jν(x)a249Jprimeν(x)a22a250a98a140a251a252a65a66a253a84 a147 a141a88a148a69 Jν?1(x)?Jν+1(x) = 2Jprimeν(x), Jν?1(x) + Jν+1(x) = 2νx Jν(x). a239 a93 a142 a147 a141a88a148a98a140a254a121a22a206a207a81a82a255a84 Bessela135a82a22a0a98a140 a152 J0(x) a70 J1(x)a108a181a121 a205 a38 a1a2 a72a22 a76 d dx bracketleftbigx?νJ ν(x) bracketrightbig = ?x?νJ ν+1(x) a111a3ν = 0a22a153a4a251a252 Jprime0(x) = ?J1(x). a220a221Nν(x) a84 a104a5 Nν(x) = cosνpiJν(x)?J?ν(x)sinνpi a6J ν(x)a84 a147 a141a88a148a22a98a140a7a121 Nν(x)a84 a147 a141a88a148a22a143 a129 a110a70 Jν(x)a8 a242 a92 a219 a38 d dx [x νNν(x)] = xνNν?1(x), Wu Chong-shi a171a172a173a172a174 a175 a114 a115 (a176) a1175a118 d dx bracketleftbigx?νN ν(x) bracketrightbig = ?x?νN ν+1(x). a9a10a147 a141a88a148 d dx [x νCν(x)] = xνCν?1(x), d dx bracketleftbigx?νC ν(x) bracketrightbig = ?x?νC ν+1(x) a84a135a82{Cν(x)}a11a12a100a13a135a82 a38 a98a140 a202a203 a69a13a135a82a95 a104 a72 Bessela62a63a84a89 a38 Bessela135a82a72a94a95a157a13a135a82a22 Neumanna135a82a72a94a97a157a13a135a82 a38 Bessela135a82 a147 a141a88a148a84 a151a152a14 a95a22a72a154a155a15Bessela135a82a84a159a160 a38a16 a189 a152 a241 a161a162a163a164a165a17a163a164 a167Bessela163a164a168a169a162 a84a18 a129 a38 a177 22.2 a154a155a159a160 integraldisplay 1 0 parenleftbig1?x2parenrightbigJ 0(μx)xdxa22a143a111J0(μ)=0a38 a178 a19 a152 a147 a141a88a148 d dx [x νJν(x)] = xνJν?1(x). a160a20a159a160a22a64 integraldisplay 1 0 parenleftbig1?x2parenrightbigJ 0(μx)xdx = integraldisplay 1 0 parenleftbig1?x2parenrightbig 1 μ d dx [xJ1(μx)]dx = parenleftbig1?x2parenrightbig 1μ [xJ1(μx)] vextendsinglevextendsingle vextendsingle 1 0 + 2μ integraldisplay 1 0 x2J1(μx)dx = 2μ2x2J2(μx) vextendsinglevextendsingle vextendsingle 1 0 = 2μ2J2(μ). a21 a3 a147 a141a88a148 Jν?1(x) + Jν+1(x) = 2νx Jν(x) a111ν = 1a22 J0(x) + J2(x) = 2xJ1(x), a101a22a23 a252 J0(μ) = 0a22a195a64 J2(μ) = 2μJ1(μ). a179a180a24 a251 integraldisplay 1 0 parenleftbig1?x2parenrightbigJ 0(μx)xdx = 4 μ3J1(μ). Wu Chong-shi §22.3 Bessela114a115a225a25a26a27a28 a1176a118 §22.3 Bessel a59a60a230a29a30a31a32 Bessela135a82a84a33a34a35a36a64a65 a183 a136a137a157a158 a38 a95 a183a37 a152 a241 x→0a22 Jν(x) = 1Γ(ν + 1) parenleftBigx 2 parenrightBigν + Oparenleftbigxν+2parenrightbig. a93 a98a140a237a238a138Bessela135a82a84 a134 a82a108a109a110a251a252 a38a38 a95 a183 a33a34a35a36 a37 a152 a241 x → ∞a22 Jν(x) ～ radicalbigg 2 pix cos parenleftBig x? νpi2 ? pi4 parenrightBig , |argx| < pi. a93 a66a39a110a84a141a7a40a41a189 a152 a252a206a207a255Bessela135a82a84a159a160a108a181a22a153a189 a152 a252a95 a183a1a42 a84a43a44(a45a68 a185 a22a249 a12a46a47 a149a48a185) a38 a49a50 a84a141a7a98 a133a51a22a52a53[3] a84a947a54 a38 a76a51a22a52a53[1] a111 a218 a120a121a122a81a82a255Bessel a135a82a33a34a35a36a84 a202a203 a38 a80x → 0, Reν > 0 a83a22Nν(x)a84a33a34a55a100a138 J?ν(x)a56 a104 a22 Nν(x) ～?Γ(ν)pi parenleftBigx 2 parenrightBig?ν . a198a57a241 N0(x)a22a98a138 Nn(x) = 2piJn(x)ln x2 ? 1pi n?1summationdisplay k=0 (n?k?1)! k! parenleftBigx 2 parenrightBig2k?n ? 1pi ∞summationdisplay k=0 (?)k k!(k + n)! bracketleftbigψ(n + k + 1)+ψ(k + 1)bracketrightbigparenleftBigx 2 parenrightBig2k+n , a237a238a251a252 N0(x) ～ 2pi ln x2. a244a140a22a58a59ν a72a60a100a81a82a22 Nν(x)a76x = 0a68a61a72a240a62a84 a38 a153a98a140 a202a203 a22 a80 x → ∞ a83a22Neumanna135a82a84a33a34a108a109a110a72 Nν(x) ～ radicalbigg 2 pix sin parenleftBig x? νpi2 ? pi4 parenrightBig , |argx| < pi. Wu Chong-shi a171a172a173a172a174 a175 a114 a115 (a176) a1177a118 §22.4 a63a60a64 Bessel a59a60a230a65a66a59a60a61a67a68a69a70 Bessela62a63 1 x d dx bracketleftbigg xdy(x)dx bracketrightbigg + bracketleftbigg 1? ν 2 x2 bracketrightbigg y(x) = 0 a111a84ν2 ≡ μa22a40a41a72a138a137a71a72a73a197 Φprimeprime + μΦ = 0, Φ(0) = Φ(2pi), Φprime(0) = Φprime(2pi) a56 a104 a84a22μ = m2, m = 0,1,2,···a38a214a139a22a137a150 a1a2a74a75 a81a82a255 Bessela135a82 a1 a64a84a86a145 a38 1. Jn(x) a168a76a77a163a164a78a79a80(a133a94 7a81a146 7.4) exp bracketleftbiggx 2 parenleftbigg t? 1t parenrightbiggbracketrightbigg = ∞summationdisplay n=?∞ Jn(x)tn, 0 < |t| < ∞. 2. Jn(x) a168a162a170a82a83 Jn(x) = 1pi integraldisplay pi 0 cos(xsinθ ?nθ)dθ. a235 a76Bessel a135a82a84a84a85a135a82a35a36a110a111a3t = eiθ a22a195a251a252 eixsinθ = ∞summationdisplay n=?∞ Jn(x)einθ. a93 a195a72a135a82 eixsinθ a84a86a87a88a35a36a110 (a89a82 a129 a110) a38a241a72a22a138a86a87a88a35a36a84a148a82a39a110a22a195a4 a202 a251 Jn(x), = 12pi integraldisplay pi ?pi eixsinθ parenleftbigeinθparenrightbig? dθ = 12pi integraldisplay pi ?pi [cos(xsinθ ?nθ) + isin(xsinθ ?nθ)]dθ. a76a90a91 a159a160a84a92a159a135a82a111a22a93a20a72a67a135a82a22a159a160a100 0a22a244a140a195 a202 a251 Jn(x)a84a159a160a108a181 a38 square Jn(x)a84a159a160a108a181a22 a218 a98a140 a152 a205 a154a155a15 Bessela135a82a84a159a160 a38 a146 a90 a57a241a146 22.1 a111a84a159a160a22a64integraldisplay ∞ 0 e?axJ0(bx)dx = integraldisplay ∞ 0 e?ax bracketleftbigg 1 2pi integraldisplay pi ?pi eibxsinθdθ bracketrightbigg dx = 12pi integraldisplay pi ?pi dθ integraldisplay ∞ 0 e?(a?ibsinθ)xdx = 12pi integraldisplay pi ?pi dθ a?ibsinθ. a152a94 a82 a104a224 a154a155 a93 a66a159a160a22integraldisplay ∞ 0 e?axJ0(bx)dx = 12pii contintegraldisplay |z|=1 2dz ?bz2 + 2az + b = 1?bz + a vextendsinglevextendsingle vextendsingle z=(a?√a2+b2)/b = 1√a2 + b2. a195a137a197a198a199a22 a93a183a184a185 a189a95 a179a180 Bessel a135a82a84 a134 a82a108a109a110a96 a200a201 a142a22 a214 a100a97 a76 a84a154a155a98a99 a203a100 a22 a58a101 a134 a82a187a70a96a102a64a43a44a86 a38 a93a183a184a185 a84 a38 a95a66a103a77a72a58a105a104a89a216a222a223 a38 Wu Chong-shi §22.4 a105a115a106Bessela114a115a225a107a108a114a115a116a109a110a111a112 a1178a118 a90a91a76 Bessel a135a82a84a84a85a135a82a35a36a110 exp bracketleftbiggx 2 parenleftbigg t? 1t parenrightbiggbracketrightbigg = ∞summationdisplay n=?∞ Jn(x)tn, 0 < |t| < ∞. a111a3t = ieiθ a22a153a98a140a251a252 eixcosθ = ∞summationdisplay n=?∞ Jn(x)ineinθ = J0(x) + 2 ∞summationdisplay n=1 inJn(x)cosnθ. a1a2 a72a22 a90a91a21 a3x = kra22a241a72a195a64 eikrcosθ = J0(kr) + 2 ∞summationdisplay n=1 inJn(kr)cosnθ. a113a193 a110a111a84 r a70θ a224a89a100a13a114a79a148a111a84a114a79 a124a125 a22 a101a102a113 k a224 a89a100a115a82a22 a219 a83a99a92a116a84a83a117 a214a118 a100 e ?iωt a22a74 a193 a110a65 a91 a61a160 a2 a57 a151 a241a115a119a120a63a92a116 a214a118 a84a121a117a20a160a69a122 a91 a72a123a73 xa124a62a125a126a127a128 a84 a243a194 a115a22 a214 a100a129a84a130a92a116 a194 a72 krcosθ ?ωt =a41a82; a198 a90a91a131 a112a111a84 J0(kr)a70Jn(kr)a132a133a84a72a13 a194 a115 a38 a214 a139a22 a93 a66a35a36a110a84a207 a5 a195a72 a243a194 a115a134a13 a194 a115 a35a36 a38 a97 a76a205 a89a135a100a136a137Jν(kr)a132a133a84a72a13 a194 a115 a38 a138Bessela135a82a84a33a34a35a36 Jν(x) ～ radicalbigg 2 pix cos parenleftBig x? νpi2 ? pi4 parenrightBig , |argx| < pi. (maltesecross) a98a140a254a121a22 a80 ra10a138a139 a83a22 Jν(kr)a244a132a133a84a115a119a120a63a84a92a116a195a72 cos parenleftBig kr?νpi2 ?pi4 parenrightBig e?iωt = 12 braceleftbigg exp bracketleftBig i parenleftBig kr?νpi2 ?pi4?ωt parenrightBigbracketrightBig +exp bracketleftBig ?i parenleftBig kr?νpi2 ?pi4 +ωt parenrightBigbracketrightBigbracerightbigg , a130a92a116 a194 a72a13 a194 kr? νpi2 ? pi4 ?ωt =a41a82, a160 a2 a132a133a84a72a130a92a116 a194a140a141 a83a117a58a142a143 a139 a249a212a144a84a240a62a249a145a146a84a13 a194 a115 a38 a198 a102 a22a138a241 (maltesecross)a110a111a153 a15a64a147 √r a85a148a95a84a149a150 a214a118 a22a115a119a120a63a84a4a151a152a153a195a147ra85a148a95a22a98a72a138a241a154a13a84a155 a194 a159a147ra85 a73a95a22a244a140a156a116a83a117a215a40a120a157a66a154a13 a194 a151a120a84a0a4 a125 a58 a124 a38 a93 a195a72a158a22 (maltesecross)a110a132a133a84a153a72a95a66a58 a159a160 a84a13 a194 a115 a38 a219a247 a22 Nν(x) a218 a98a140 a152 a205 a132a161a13 a194 a115a22 a218 a72a240a62a84a13 a194 a115a70a145a146a84a13 a194 a115a84a162a163 a38 a128a164a165a166a167a168a169a170a171a172a173a174a175a171a176a177a178a179a180a181a182a183a172a173a174a175a184 eiωta185a186a187a188a189a190a191a192 a167a193 xa194 a166a167a164a165a171a180