Wu Chong-shi a0a1a2a3a4 a5 a6 a7 (a1) §23.1 Bessel a8a9a10a11a12a13a14a15 a16a17a18a19a20a21a22a23a24a25a26a27a28a29 Bessel a30a31a32a33a34a35 a23a24a36 a37a38a39a40a41 a32a42a43a44a45a32 a40a46a47a48 a49a50a27a51a52a53a54a55a56a57a58a59a60a61a58a62a63a64a65a66a67a68a69a53a54a70a71a72a73a74a75a76a77a78a79a80a81a27a82a83 a84a62a85a55a86a87a88a89a90a91a92a93a94a62a95a96a97a98a99a100a101a102a103a101a104a105a106a36a71a72a83a84a62a86a107a75a108a109a27a110a84a77 a76a68a63a64a65a66a67a102a111a112a80a81a113a62a82a75a114a115a93a94a116a117a62a118a108a109a36a85a119a86a120a121a71a72a62a53a54a122a73a74a75a76 a77a78a79a80a81a27a82a123a85a55a124a125a77a95a96a126a94a55a105a62a127a61a36 a128a129a130a131a132a133a134a27a132a133a135a136a137a138a17 a42a43a44a45a32a139a140 a36a141a142a27a143a144a145 a30a31a146a147a148a149a150a151a152 ?2u ?t2 ?c 2 bracketleftbigg1 r ? ?r parenleftbigg r?u?r parenrightbigg + 1r2 ? 2u ?φ2 bracketrightbigg = 0, (23.1a) uvextendsinglevextendsingler=0a46a148, uvextendsinglevextendsingler=a = 0, (23.1b) uvextendsinglevextendsingleφ=0 = uvextendsinglevextendsingleφ=2pi, ?u?φ vextendsinglevextendsingle vextendsinglevextendsingle φ=0 = ?u?φ vextendsinglevextendsingle vextendsinglevextendsingle φ=2pi . (23.1c) a16a17a153a37 a32a151a152 a17 a147a148a149a150 (23.1b) a146 (23.1c) a32a154a155a156 a27a157a158a159a160a161a162 ω a35 a27a163a164 a30a31 (23.1a)a46a165a166a167 u(r,φ,t) = v(r,φ)eiωt. (23.2) a168a169a167a170a171a25 a30a31 (23.1)a172a147a148a149a150 (23.1b)a146(23.1c)a27a173a174k = ω/ca27a151 a160a175a164a157 1 r ? ?r parenleftbigg r?v?r parenrightbigg + 1r2 ? 2v ?φ2 + k 2v = 0, vvextendsinglevextendsingler=0 a46a148 vvextendsinglevextendsingler=a = 0, vvextendsinglevextendsingleφ=0 = vvextendsinglevextendsingleφ=2pi, ?v?φ vextendsinglevextendsingle vextendsinglevextendsingle φ=0 = ?v?φ vextendsinglevextendsingle vextendsinglevextendsingle φ=2pi . a176a174 v(r,φ) = R(r)Φ(φ) a27a145a177a178a179a27 a151 a164a157a180a20 a33a34a35 a23a24 Φprimeprime(φ) + m2Φ(φ) = 0, (23.3a) Φ(0) = Φ(2pi), Φprime(0) = Φprime(2pi) (23.3b) a146 1 r d dr bracketleftbigg rdR(r)dr bracketrightbigg + parenleftbigg k2 ? m 2 r2 parenrightbigg R(r) = 0, (23.4a) R(0)a46a148, R(a) = 0. (23.4b) Wu Chong-shi §23.1 Bessela181a182a183a184a185a186a187a188 a1892a190 a33a34a35 a23a24 (23.3) a191a192a193a194a195 a157a196a27a197a198a199a200 a32a33a34a35 m2, m = 0,1,2,3,···, a33a34a201a202a203 Φm(φ) = braceleftbiggcosmφ, sinmφ. a204a175a27a17 a33a34a35 a23a24 (23.4) a139 a27a205 a202m2 a152a191a206a32 a27a207k2 a152a33a34a35 a27a208a37a36 a160a175a209a210 k2 integraldisplay a 0 R(r)R?(r)rdr = m2 integraldisplay a 0 R(r)R?(r)drr + integraldisplay a 0 dR(r) dr dR?(r) dr rdr, a204a175a27a19a41a46 a33a34a35 k2 > 0a36a211a196a212a178a213 x = kr a27y(x) = R(r)a27a151 a160a175a168a144a145 a30a31 (23.4a)a214a203 Bessela30a31 a27a18a207a37a164a200 a32 a211a167 R(r) = CJm(kr) + DNm(kr). (23.5) a215a216a157 a147a148a149a150 (23.4b)a32 a153a37a27 R(0)a46 a148 a27a217D = 0 a218a219a220 a199a153a37R(a) = 0a27 a151 a164a157 Jm(ka) = 0. (23.6) a168m a221Bessela201a202Jm(x)a32a222ia20a223a166a136(a220a224 a157a225a226a227) a228 a212μ(m) i a27i = 1,2,3,···a27a229 a33a34a35 a23 a24(23.4) a32 a167 a152 a33 a34 a35 k2mi = parenleftBigg μ(m)i a parenrightBigg2 , i = 1,2,3,···, (23.7a) a33a34a201a202 Rmi(r) = Jm(kmir). (23.7b) a199 a152a151 a37a164a230 a42a43a44a45a32 a40a46a231a232 a32a233 a47a48 ωmi = μ (m) i a c, (23.8) a234 a139μ(m)i a152ma221Bessela201a202Jm(x)a32a222ia20a223a166a136a36 a17a235a236a37a167a196 a31a139 a27a237a238a235a239a157a230a46a240J ν(x) a166a136 a32a241 a29a70a242ν > ?1 a243a203a244a202a245 a27J ν(x) a46 a246a247 a193 a20a166a136a27a200a248a249a250a251 a152 a237 a202 a27a197a252a253a145a254a17a237a255a235a36 Wu Chong-shi a0a1a2a3a4 a5 a6 a7 (a1) a1893a190 Zeros of the functions Jν(z) & Nν(z) 1. Real zeros When ν is real, the functions Jν(z) & Nν(z) each have an infinite number of zeros, all of which are simple with the possible exception of z = 0. For non-negative ν the sth positive zeros of these functions are denoted by jν,s and nν,s respectively. s j0,s j1,s n0,s n1,s 1 2.40483 3.83171 0.89358 2.19714 2 5.52008 7.01559 3.95768 5.42968 3 8.65373 10.17347 7.06805 8.59601 4 11.79153 13.32369 10.22235 11.74915 5 14.93092 16.47063 13.36110 14.89744 6 18.07106 19.61586 16.50092 18.04340 7 21.21164 22.76008 19.64131 21.18807 8 24.35247 25.90367 22.78203 24.33194 9 27.49348 29.04683 25.92296 27.47529 10 30.63461 32.18968 29.06403 30.61829 2. McMahon’s expansions for large zeros jν,s,nν,s ～ β ? μ?18β ? 4(μ?1)(7μ?31)3(8β)3 ? 32(μ?1)(83μ 2 ?982μ+ 3779) 15(8β)5 ?64(μ?1)(6949μ 3 ?153855μ2 + 1585743μ?6277237) 105(8β)7 ?······, s greatermuch ν, μ = 4ν 2, β = ? ??? ??? parenleftbigg s+ ν2 ? 14 parenrightbigg pi, for jν,s parenleftbigg s+ ν2 ? 34 parenrightbigg pi, for nν,s 3. Complex zeros of Jν(z) When ν ≥ ?1 the zeros of Jν(z) are all real. If ν < ?1 and ν is not an integer the number of complex zeros of Jν(z) is twice the integer part of (?ν); if the integer part of (?ν) is odd two of these zeros lie on the imaginary axis. 4. Complex zeros of Nν(z) When ν is real the pattern of the complex zeros of Nν(z) depends on the non-integer part of ν. Attention is confined here to the case ν = n, a positive integer or zero. Wu Chong-shi §23.1 Bessela181a182a183a184a185a186a187a188 a1894a190 Zeros of Nn(z) The figure 23.1 shows the approximate distribution of the complex zeros of Nn(z) in the region |argz|≤pi. The figure is symmetrical about the real axis. The two curves on the left extend to infinity, having the asymptotes Imz = ±12 ln3 = ±0.54931...... There are an infinite number of zeros near each of these curves. The two curves extending from z = ?n to z = n and bounding an eye-shaped domain intersect the imaginary axis at the points ±i(na+ b), where Figure 23.1 Zeros of Nn(z) a = radicalBig t20 ?1 = 0.66274...... b = 12 radicalBig 1?t?20 ln2 = 0.19146...... and t0 = 1.19968...... is the positive root of cotht = t. There are n zeros near each of these curves. Complex zeros of N0(z) Complex zeros of N1(z) Real part Imaginary part Real part Imaginary part ?2.40302 0.53988 ?0.50274 0.78624 ?5.51988 0.54718 ?3.83353 0.56236 ?8.65367 0.54841 ?7.01590 0.55339 Wu Chong-shi a0a1a2a3a4 a5 a6 a7 (a1) a1895a190 a203 a230a17a145a177a178a179a8 a139a32 a198a239a27a9a10a153a28a29a235a130a164a157 a32a33a34a201a202a32 a223a11a12a19a240a134a36 a156 a130a27a13a14 a19a15a16 a203a17a18a32a19 a8a27a160a175 a18a245 a164a157 a33a34a201a202a32 a223a11a12a19a240a134a36 a20a21a27a22a23 a33a34a201a202 Rmi(r) = Jm(kmir) a204a24a25 a32 a144a145 a30a31a146a147a148a149a150 a27 1 r d dr bracketleftbigg rdJm(kmir)dr bracketrightbigg + parenleftbigg k2mi ? m 2 r2 parenrightbigg Jm(kmir) = 0, (23.9a) Jm(0)a46a148, Jm(kmia) = 0. (23.9b) a18a245 a27a176a22a23 a201a202 R(r) = Jm(kr)a204a24a25a32 a144a145 a30a31a146a147a148a149a150 a27 1 r d dr bracketleftbigg rdJm(kr)dr bracketrightbigg + parenleftbigg k2 ? m 2 r2 parenrightbigg Jm(kr) = 0, (23.10a) Jm(0)a46a148. (23.10b) a220 a199a234 a139a32 ka203a26a27 a237 a202 a27a204a175a19a28a29a30a27 a17a31 a46 J m(ka) = 0 a36 a176a239rJ m(kr)a146rJm(kmir) a145a32a33 a30a31(23.9a)a146 (23.10a)a27 Jm(kr) ddr bracketleftbigg rdJm(kmir)dr bracketrightbigg + parenleftbigg k2mi ? m 2 r2 parenrightbigg rJm(kmir)Jm(kr) = 0, Jm(kmir) ddr bracketleftbigg rdJm(kr)dr bracketrightbigg + parenleftbigg k2 ? m 2 r2 parenrightbigg rJm(kmir)Jm(kr) = 0, a34a35a27a173a17a36a37 [0, a]a235a38a145a27 a151 a164a157 parenleftbigk2 mi ?k 2parenrightbig integraldisplay a 0 Jm(kmir)Jm(kr)rdr = r bracketleftbigg Jm(kmir)dJm(kr)dr ?Jm(kr)dJm(kmir)dr bracketrightbiggvextendsinglevextendsingle vextendsinglevextendsingle r=a r=0 . a171a25 a147a148a149a150 (23.9b)a146(23.10b) a27a160a175a168a235a130 a32a241a39a214a203 parenleftbigk2 mi ?k 2parenrightbig integraldisplay a 0 Jm(kmir)Jm(kr)rdr = ?kmiaJm(ka)Jprimem(kmia). (23.11) a40a248a197a180a20a41a42a43 a43a44a45a46 a36 a222 a19a15a43 a43a152 k = kmj negationslash= kmi a36a141 a245a151 a46 J m(kmja) = 0 a27a47a169 (23.11)a170a32a48a49a203 0a36a50a220a199kmj negationslash= kmi a27a204a175integraldisplay a 0 Jm(kmir)Jm(kmjr)rdr = 0, kmi negationslash= kmj, (23.12) a51a197a198a199 a17a18a33a34a35a32a33a34a201a202 a17a36a37 [0, a]a235a175a52a53 ra223a11a36 a54a19a15a43 a43a152k = kmi a27a141a245(23.11)a170a32a180a49a55a2030a36a40a248a160a175a21a168(23.11)a170a32a180a49a18a56a175 k2mi ?k2 a27a10a57a128a131a154 k → kmi a27a141a142a151 a164a157 integraldisplay a 0 J2m(kmir)rdr = ? lim k→kmi kmia k2mi ?k2Jm(ka)J prime m(kmia) = a2 2 [J prime m (kmia)] 2 . (23.13) a141a223 a152a33a34a201a202 Jm(kmir)a32a58a30 a36 a59 a39 a168 a33a34a35 a23a24 (23.9) a139r = aa49a32a60a194a147a148a149a150 (23.9b)a61a203a222a62a63a243a222a64a63a147a148a149a150 a27 a65a160a175 a63a66 a253a28a29a36 a67a237a235a27a160a175a68a141 a64 a15a43 a43a69 a19a22a70 1 r d dr bracketleftbigg rdR(r)dr bracketrightbigg + parenleftbigg k2 ? m 2 r2 parenrightbigg R(r) = 0, (23.14a) Wu Chong-shi §23.1 Bessela181a182a183a184a185a186a187a188 a1896a190 R(0)a46a148, αRprime(a) + βR(a) = 0. (23.14b) a59 a39α negationslash= 0, β = 0a27a229a152a222 a19 a63a147a148a149a150a218 a59 a39 α = 0, β negationslash= 0a27a151a152a222a62a63a147a148a149a150a218 a59 a39 αa146β a55a17a2030a27a229a203a222a64a63a147a148a149a150 a36 a240a199Bessel a201a202a71a32a72a73a74 a27a141a75a76a77a23 a241 a29a70a59 a39a201a202f(r)a17a36a37[0, a]a235a78a79a27a80a76a46a46 a154 a20a131a225 a146 a131 a224 a27a229a160a81 a33a34a201a202 Jm(kir)a82a83 a27 f(r) = ∞summationdisplay i=1 biJm(kir), (23.15) a234 a139Jm(kir)a152a33a34a35 a23a24 (23.14) a32 a167a27a207 a82a83 a134 a202a203 bi = integraldisplay a 0 f(r)Jm(kir)rdr integraldisplay a 0 J2m(kir)rdr . (23.16) a141a142a164a157 a32a84a202 a17a36a37 [δ, a?δ] (δ > 0)a235 a152 a19a85a86a87 a32 a36a209a210 a195 a205a215a88a89[15], a22217.33a90 a36a91a88 a139 a65a92a46a93a94a95 a32a82a83 a41a96a36 Wu Chong-shi a0a1a2a3a4 a5 a6 a7 (a1) a1897a190 a97 23.1 a42a98 a22 a32a99a100 a36a101a46a19a20a246a247a102 a32a42a98 a22a27a103a104 a203 a a36a105a9a10a253a40a248a198a91a106a239a98 a132a133a134a27 z a255a51 a203a42a98 a22 a32 a255a36a59 a39a98 a22 a32a107 a130a108a109a110a111 a203 0a27a112a108a203u0f(r)a27a113a37a98 a22a114a108a109 a32 a145a254 a146 a178 a214 a36 a115 a116a10a108a109u a117φ, z a246a240a27a51u = u(r,t)a36a200a220 a41a167a23a24 ?u ?t ? κ r ? ?r parenleftbigg r?u?r parenrightbigg = 0, (23.17a) uvextendsinglevextendsingler=0a46a148, u vextendsinglevextendsingle r=a = 0, (23.17b) uvextendsinglevextendsinglet=0 = u0f(r) (23.17c) a118a41a36a119a120a121a130 a32 a19a28a28a29a27a122a123a22a23a169a41a167a23a24 a32 a19a28a167 u(r,t) = ∞summationdisplay i=1 ci J0 parenleftBig μira parenrightBig exp bracketleftbigg ?κ parenleftBigμi a parenrightBig2 t bracketrightbigg , (23.18) a234 a139μi a152J0(x)a32a222ia20a223a166a136a36a171a25a112a149a150 a27a46 u(r,t)vextendsinglevextendsinglet=0 = ∞summationdisplay i=1 ci J0 parenleftBig μi ra parenrightBig = u0f(r). a204a175 ci = 2u0a2J2 1(μi) integraldisplay a 0 f(r)J0 parenleftBig μi ra parenrightBig rdr. (23.19) a206a124 a230f(r) a32 a21a22 a43 a170a27 a151 a160a175a125a23a235a130 a32 a38a145a36a126a59a27a127 f(r) = 1? parenleftBigr a parenrightBig2 , (23.20) a128a46 ci = 2u0a2J2 1(μi) integraldisplay a 0 bracketleftbigg 1? parenleftBigr a parenrightBig2bracketrightbigg J0 parenleftBig μi ra parenrightBig rdr = 8u0μ3 iJ1(μi) . (23.21) a129a57a19a130a239a157a230a126 21.2 a139a32a131 a125 a241a39 a36 a168 a33 a24a164a157 a32a241a39 1?x2 = 8 ∞summationdisplay i=1 1 μ3iJ1(μi) J0(μix) a180 a49 a144a132a27a173a174 x = 1a27a92a160a175a133a23a19a20a46 a27a134a32a241a39 a70 ∞summationdisplay i=1 1 μ2i = 1 4. (23.22) Wu Chong-shi §23.1 Bessela181a182a183a184a185a186a187a188 a1898a190 a97 23.2 a42a135 a212a129a130a104a136a231a232 a32 a40a46a47a48a36a101 a42a135a32 a114a137a103a104a145a32 a203 aa146ba36a127a114a147a148(a114 a42) a40a41a27a137 a147a148 ( a137 a42) a9a220 a27a37 a42a135 a212a129a130a104a136a231a232 a32 a40a46a47a48a36 a115 a116a10a198a91a106a239a129a130a131a132a133a134a36a229a138a139 ( a140 a179)u = ue r a24a25 a32a141 a232 a30a31a203 ?2u ?t2 ?c 2?2u = 0. (23.23) a47 a203 ?2u≡?2(uer) = parenleftBig ?2u? ur2 parenrightBig er + 2r2 ?u?φeφ, a204a175 a30a31(23.23)a142a143 a199a143a144a145 a30a31a144 ?2u ?t2 ?c 2 bracketleftBig ?2u? ur2 bracketrightBig = 0, ?u?φ = 0. (23.23prime) a220 a169a160 a195 a27a104a136a138a139 a117 φa246a240a27u = u(r,t)a24a25a144a145a30a31a146a147a148a149a150 ?2u ?t2 ?c 2 bracketleftbigg1 r ? ?r parenleftbigg r?u?r parenrightbigg ? ur2 bracketrightbigg = 0, (23.24a) uvextendsinglevextendsingler=a = 0, ?u?r vextendsinglevextendsingle vextendsinglevextendsingle r=b = 0. (23.24b) a174u(r,t) = R(r)e?iωt, k = ω/ca27a171a25(23.24)a27a128a164a157 1 r d dr bracketleftbigg rdR(r)dr bracketrightbigg + bracketleftbigg k2 ? 1r2 bracketrightbigg R(r) = 0, (23.25a) R(a) = 0, Rprime(b) = 0. (23.25b) a122a123a209a210a27 k = 0 a245a33a34a35 a23a24(23.25)a246a167a36a199 a152 a27a160a212a178a213 x = kr a146 y(x) = R(r) a27a207a168a30a31 (23.25a)a214a203Bessela30a31 a27 a220 a169a51a160a164a157 a30a31 (23.25a)a32 a211a167 R(r) = CJ1(kr) + DN1(kr). a171a25 a147a148a149a150 (23.25b) a27a51a164 CJ1(ka) + DN1(ka) = 0, CJprime1(kb) + DNprime1(kb) = 0. a141a160a175a145a70 a152 a240a199 C a146Da32a146a74 a171 a202a30a31a144 a27a46a165a166a167 a32a147 a145a148a153 a149a150a152vextendsingle vextendsinglevextendsingle vextendsinglevextendsingleJ1(ka) N1(ka)Jprime 1(kb) Nprime1(kb) vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle= 0. a141a142 a151 a37a164 a42a135 a212a129a130a104a136a231a232 a32 a40a46a47a48 ω i = kic a27a234 a139ki a152 J1(ka)Nprime1(kb)?N1(ka)Jprime1(kb) = 0 (23.26) a32a222ia20a223a119(a220a224 a157a225a226a227) a36a37a23C a146D a27a151 a160a22a23a34a198 a32 a40a46a231a232 a58 a170 ui(r,t) = [N1(kia)J1(kir)?J1(kia)N1(kir)]e?ikict. (23.27) Wu Chong-shi a0a1a2a3a4 a5 a6 a7 (a1) a1899a190 §23.2 Hankel a149a150 a121a130a13a14 a32 Jν(x)a146Nν(x)a251a160a175a239a30a151a22a98 a130 a141 a27a200a248 a32a152a153a82a83 a145a32 a152 Jν(x) ～ radicalbigg 2 pix cos parenleftBig x? νpi2 ? pi4 parenrightBig , Nν(x) ～ radicalbigg 2 pix sin parenleftBig x? νpi2 ? pi4 parenrightBig . a122a123a145a23a27a200a248a151a22 a32a98 a130 a141a139 a27a154a46a155a156 a141 a27 a219 a46 a31a157a141 a36a50a59 a39 a40a248a158a96 a32 a23a24 a139 a27a76a159 a172 a155 a156 a141a243a31a157a141 a27 a243 a153a37a210a160a36a145a155a156 a141a243a31a157a141 a27a141a180a20 a201a202a151a17a30 a128a230a36a141 a245 a116a10a198a242a212 a146a74 a144a161 H(1)ν (x) ≡Jν(x) + iNν(x) ～ radicalbigg 2 pix exp bracketleftBig i parenleftBig x? νpi2 ? pi4 parenrightBigbracketrightBig , H(2)ν (x) ≡Jν(x)?iNν(x) ～ radicalbigg 2 pix exp bracketleftBig ?i parenleftBig x? νpi2 ? pi4 parenrightBigbracketrightBig . a59 a39a162a161 a235a34a198 a32a245 a37a47a163 e?iωt a27a229H(1) ν (x) a171 a107 a155a156 a141 a27 H(2) ν (x) a171 a107a31a157a141 a36 H(1)ν (x)a146H(2)ν (x) a145a32a252a203a222 a19 a63a146a222a62a63 Hankela201a202 a36a200a248a251 a152 Bessela30a31a32 a167a27a251 a152a98 a201a202 a36 a69 a252 a203a222a64a63a98a201a202 a36 Wu Chong-shi §23.3 a164a165a166 Bessel a6a7 a18910a190 §23.3 a167a168a169 Bessel a149a150 a18a135a229a235a29a27a17Bessel a201a202a170a171Neumanna201a202a146Hankela201a202a32 a41a172 a139 a27a200a248 a32a173 a179 a33 a30 a151 a160 a175 a152a174a202 a36a50 a152 a27 a203 a230a237a239a235 a32a30 a128a27a197a199Bessel a201a202a32a173 a179 a203a175a176a202a32 a43 a43 a92 a152a35 a164a212a19a162a145 a177a28a29a27a173a178a19a130a41a172a180 a63a176a173 a179 a32 Bessela201a202 a36 a17a179a180 a10a18a143a144a145 a30a31a32 a41a167a23a24a23a155a27a30a181a25 a176a173 a179 a32Bessela201a202 a36a126a59a27a182a101a46 a42a98 a22 a114 a32Laplacea30a31 a41a167a23a24 1 r ? ?r parenleftbigg r?u?r parenrightbigg + 1r2 ? 2u ?φ2 + ?2u ?z2 = 0, (23.28a) uvextendsinglevextendsingleφ=0 = uvextendsinglevextendsingleφ=2pi, ?u?φ vextendsinglevextendsingle vextendsingle φ=0 = ?u?φ vextendsinglevextendsingle vextendsingle φ=2pi , (23.28b) uvextendsinglevextendsinglez=0 = 0, uvextendsinglevextendsinglez=h = 0, (23.28c) uvextendsinglevextendsingler=0a46a148, uvextendsinglevextendsingler=a = f(φ,z). (23.28d) a81a183a145a177a178a179a8 a32 a133a184 a19 a8a27a174 u(r,φ,z) = R(r)Φ(φ)Z(z), a171a25 a30a31(23.28a)a175a172a147a148a149a150 (23.28b)a146(23.28c)a27a145a177a178a179a27a151a31 a164a157 a33a34a35 a23a24 Φprimeprime(φ) + μΦ(φ) = 0, (23.29a) Φ(0) = Φ(2pi), Φprime(0) = Φprime(2pi) (23.29b) a146 Zprimeprime(z) + λZ(z) = 0, (23.30a) Z(0) = 0, Z(h) = 0 (23.30b) a175 a172a185 a144a145 a30a31 1 r d dr parenleftbigg rdRdr parenrightbigg + parenleftBig ?λ? μr2 parenrightBig R = 0. (23.31) a220a33a34a35 a23a24 (23.29)a27a160a175a164a157 a33 a34 a35 μm = m 2, m = 0,1,2,3,···, (23.32a) a33a34a201a202 Φm(φ) = Am cosmφ + Bm sinmφ, (23.32b) a234 a139Am a146Bm a152a26a27a185a202 a36a176 a220a33a34a35 a23a24 (23.30)a27 a219 a160a37a164 a33 a34 a35 λn = parenleftBignpi h parenrightBig2 , n = 1,2,3,···, (23.33a) a33a34a201a202 Zn(z) = sin npih z. (23.33b) a141a142a27 a185 a144a145 a30a31 (23.31)a151 a178a70 1 r d dr parenleftbigg rdRdr parenrightbigg + bracketleftbigg ? parenleftBignpi h parenrightBig2 ? m 2 r2 bracketrightbigg R = 0. (23.31prime) Wu Chong-shi a0a1a2a3a4 a5 a6 a7 (a1) a18911a190 a212a178a213x = (npi/h)r a146y(x) = R(r)a27a151 a160a175a168a169 a30a31a214a203 1 x d dx parenleftbigg xdydx parenrightbigg + parenleftBig ?1? m 2 x2 parenrightBig y = 0. (23.34) a141a20 a30a31 a252 a203a176a173 a179Bessel a30a31 a27a47 a203 a176a212a178a213t = ix a151 a160a175a168a200 a214a203 Bessela30a31 a36a199 a152 a27 a30a31 (23.31prime)a32 a211a167 a151a152 R(r) = CJm parenleftbigginpi h r parenrightbigg + DNm parenleftbigginpi h r parenrightbigg . (23.35) a141a75 a151 a23a16a230 a173 a179 a203a175a176a202a32 Bessela201a202a146Neumanna201a202 a36 a19a28a29a30a27a242 Bessel a201a202a32a173 a179 a203a175a176a202 xeipi/2 (xa203 a237 a202)a245 a27 a201a202a35 a65 a152a174a202 a36 Jν(xeipi/2) = ∞summationdisplay k=0 (?)k k!Γ(k + ν + 1) parenleftBigx 2e ipi/2 parenrightBig2k+ν = eiνpi/2 ∞summationdisplay k=0 1 k!Γ(k + ν + 1) parenleftBigx 2 parenrightBig2k+ν . a141a142a27a128 a17a179 a41a172 a222 a19 a63a176a173 a179 Bessel a201a202 Iν(x) ≡ e?iνpi/2Jν(xeipi/2) = ∞summationdisplay k=0 1 k!Γ(k + ν + 1) parenleftBigx 2 parenrightBig2k+ν . (23.36) a41a32 a152 a197a199 a244a202a221a32a222 a19 a63a176a173 a179 Bessel a201a202 a27a186a187a253a46 In(x) = i?nJn(ix). (23.37) a47a169a27a242x a146ν a55a203 a237 a202a245 a27 I ν(x)a32a201a202a35 a65 a152 a237 a202 a36 a18 a142a27 a220 a199 I ν(x)a146I?ν(x) a251 a152a176a173 a179 Bessel a30a31(23.34)a32 a167a27a207a80a27a215a216a157 I?n(x) = In(x), (23.38) a160a175a41a172 a222a62a63a176a173 a179 Bessel a201a202a203 Kν(x) = pi2sinνpi bracketleftBig I?ν(x)?Iν(x) bracketrightBig . (23.39) a141a142a27a242ν a203a244a202na245 a27K n(x)a180 a10a46 a27 a172a27a80 a117 In(x)a146a74 a246a240a36 Kn(x) = limν→nKν(x) = 12 n?1summationdisplay k=0 (?)k (n?k?1)!k! parenleftBigx 2 parenrightBig2k?n + (?)n+1 ∞summationdisplay k=0 1 k!(n+k)! bracketleftbigg ln x2 ? 12ψ(n+k+1)?12ψ(k+1) bracketrightbiggparenleftBigx 2 parenrightBig2k+n . (23.40) a141a75 a180a188 a41a27a242 n = 0 a245 a27a198a189a190 a48a49a222 a19a191 a32 a46 a154a146 a36 a19223.2 a139 a77a23a230a121a193a20 I n(x)a146Kn(x)a32 a192 a43 a36 a220Iν(x) a146 Kν(x) a32 a41a172a27a122a123a22a23a200a248a17 x → 0 a245a32a152a153a194a203 a36a41a32 a152 a27a59 a39 ν ≥ 0 a27a229 Iν(x)a152 a46 a148a32 a27a207 K ν(x)a152 a246 a148a32 a36a242 x →∞ a245 a27a200a248 a32a152a153a194a203a219a152 Iν(x) ～ radicalbigg 1 2pixe x, (23.41) Kν(x) ～ radicalbiggpi 2xe ?x. (23.42) Wu Chong-shi §23.3 a164a165a166 Bessel a6a7 a18912a190 a17a237a239 a139 a27 a185a185 a119a120a141a162 a152a153a194a203a195 a106a23a204a196a153 a32 a167a36a126a59a27a17a235a130 a32 a41a167a23a24 (23.28) a139 a27 a220 a199 a46 a148a149a150u vextendsinglevextendsingle r=0 a46 a148a32a154a155 a27a17a167a170(23.35) a139a151 a19a41a46D = 0a36a199 a152 a27a17 a147a148a149a150(23.28b) (23.28c) a146 a46 a148a149a150a32a154a155a156 a27 a30a31 (23.28a)a32 a41a167 a151a152 umn(r,φ,z) = (Amn cosmφ + Bmn sinmφ) Im parenleftBignpi h r parenrightBig sin npih z. (23.43) a168a249a250a41a167a197a198a199a30a27a164a157a19a28a167a27a176a200a239 a147a148a149a150 (23.28d)a51a160a41a23a197a198a134a202 a36 a20123.2 a202a203a204Bessela205a206In(x)a207Kn(x) Iν(x)a146Kν(x)a32 a234a208 a74a209(a126a59a27a210a211a240a134)a27a242a10a251a160a175a220Jν(x)a146Nν(x)a32 a34a198 a74a209 a133a23a27 a169a158a18a16a36 a175a235a41a172 a32a176a173 a179 Bessel a201a202 a27 a175a212a152 a17a213a214 x a203 a237 a202a32a149a150a156 a181a178 a32 a36a50 a152 a27a141a15 a154a155a149 a150 a173 a17a152 a148a153 a32 a36 a72 a249a160a175a68 I ν(x) a32 a41a172 (23.36) a215a147 a157a216a46a217 a146a32a174 a129a130|argx| < pia235a36a34a198 a253a27K ν(x)a32 a41a172(23.39)a65 a151a215a147 a157a230 a18 a19a36a218 a139 a36