Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17(a3) a181a19 a20a21a22 a23a24a25a26a27a28a29a30a31a32a33a34a35a36a37 (a23) §9.1 a38a39a40a41a42a43a44a45a46a47a48 a49a50a51a52a53a54a55a56a53a57 a58a59a55a56a53a60a61a62a63a64a65a66a55a56a53a57a67a68a60a61a65a66a55a52a53a69a70a54a56a53a71a72a60a61a65a66a55a73a53a57 a74a75a76a58a59a77a78a56a53a79a80a81a82a66a55a83a84a71a85a86a67a87a88a89a90a91a92a93a94a95a96a97a98 a99a1009.1 a101a102z0a65a58a59 d2w dz2 + p(z) dw dz + q(z)w = 0 a55a56a53a71a78a76p(z) a103q(z)a104 a66a105a55a106a107a108a800 < |z ?z 0| < R a81a71a58a59a55a109a95a110a54a111a112a66a65 w1(z) =(z ?z0)ρ1 ∞summationdisplay k=?∞ ck(z ?z0)k, w2(z) =gw1(z)ln(z ?z0) + (z ?z0)ρ2 ∞summationdisplay k=?∞ dk(z ?z0)k, a113a114ρ 1, ρ2a103ga104 a65a115a116a57 star a101a102ρ1a69ρ2a65a117a116a71a118g = 0a71a78z0a53a75a58a59a55a66a55a52a53a69a70a54a56a53a57 star a101a102ρ1a69ρ2a67a65a117a116a71a69g negationslash= 0a71a78a58a59a55a66a75a119a120a121a116a71z0a53a75a113a73a53a57 a122a123a124a125a126a127a128a129a130a131a132a133a134a135a136a137a71a138a139a140a141a142a143a144a145a146a147a148a131a149a150a151a145a71a152a153a154 a155a156a157a145a146a131a158a159a160a161a133a57a162a163a164a165a131a166a146a132a167a71 a168a169a170a171 a71a172a173a154a174a175a176a177a178a179a180a181 a178a179a71a182a183a184a185a149a150a151a145a186a187a188a154a189a190a57 a101a102a191 a116a66a114a49a192a192a193a95a194a195a196a71a197a63a198a60a199a200a117a201a202a55ρa120a71a203a204 a191 a116a66a114a205a192a194a195a196a71 w1(z) = (z ?z0)ρ1 ∞summationdisplay k=0 ck(z ?z0)k, w2(z) = gw1(z)ln(z ?z0) + (z ?z0)ρ2 ∞summationdisplay k=0 dk(z ?z0)k. a206a65a71a207a208a209a210a211a212a112a213a214a60a199a82a204a213a116a55a215a216a217a218a219a57a220a221a71a72a222a223a224a96a225ρa120a57 a164a226a227a133a131a132a228a163 a40a41a48 a57a229g negationslash= 0a165a71w 2(z) a131a227a133a180 w 1(z)a230a231(a232 a173a233a146a179)a71 a162a234a235a236a237a156a132a57a229g = 0a165a71w 2(z) a131a160a161a133a167 a230a232 a233a146a179a71a238a176a132a131a227a133a239 a231 a57 a58a59a56a53a79a80a81a109a95a110a54a111a112a66 a104 a65a77a78a66a55a240a241a222a224a242a243a71a244a245a246a55a96a97(a67a88)a98 Wu Chong-shi §9.1 a10a11a247a248a249a250a251a252a253a12a16 a182a19 a99a1009.2 a58a59 d2w dz2 + p(z) dw dz + q(z)w = 0, a76a254a55a56a53z 0 a55a79a800 < |z ?z 0| < R a192a109a95a77a78a66 w1(z) =(z ?z0)ρ1 ∞summationdisplay k=0 ck(z ?z0)k, c0 negationslash= 0, (9.1) w2(z) =gw1(z)ln(z ?z0) + (z ?z0)ρ2 ∞summationdisplay k=0 dk(z ?z0)k, ga69d0 negationslash= 0 (9.2) a55a240a224a242a243a65z 0 a53a65 p(z)a55a67a255a0a94a1a55a52a53a71 a2 (z ?z0)p(z)a76z0a53a66a105a3 q(z)a55a67a255a0a4a1a55a52a53a71 a2 (z ?z0)2q(z)a76z0a53a66a105a57 a2z 0 a75a58a59a55a77a78a56a53a57 ρ1a103ρ2 a5 a75a77a78a66a55a6a7a57 a8 9.1 a9 a221a71z = 0 a103z = 1a104 a65a255a10a11a58a59 z(1?z)d 2w dz2 + [γ ?(1 + α + β)z] dw dz ?αβw = 0 a55a77a78a56a53a3 x = ±1a64 a104 a65Legendrea58a59 parenleftbig1?x2parenrightbig d2y dx2 ?2x dy dx + l(l + 1)y = 0 a55a77a78a56a53a57 a75a12a13a14a111a15a16a53a65a17a75a77a78a56a53a71a62a18a224a74a19a20 z = 1/ta71 a101a102t = 0a65a19a20a21a55a58a59a55a77 a78a56a53a71a2t = 0a53a65a19a20a21a55a58a59a55a56a53a71a118 t bracketleftbigg2 t ? 1 t2p parenleftbigg1 t parenrightbiggbracketrightbigg = 2? 1tp parenleftbigg1 t parenrightbigg a103 t2 · 1t4q parenleftbigg1 t parenrightbigg = 1t2q parenleftbigg1 t parenrightbigg a76t = 0a53a66a105a71a22a2 z = ∞a53a65a19a20a23a58a59a55a56a53a71a118zp(z) a103z2q(z)a76z = ∞a53a66a105a71a78 a5 z = ∞a53a65a19a20a23a55a58a59a55a77a78a56a53a57a24a199a71a111a15a16a53z = ∞a64a104 a65a255a10a11a58a59 a103Legendre a58a59 a55a77a78a56a53a57 Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17(a3) a183a19 a25 a38a39a40a41a42a43a44a45a46a26a48a27a28 a98 ? a29 a77a78a66w 1(z) a69w 2(z)a30a31 a58a59 ? a32 a0a33a34a213a116a71a82a225a35a36 a103 a211a212a112a213 ? a37a38 a82a225a213a116a55a215a216a217a218a219 a39a40a55a82a66a0a59a71a198a65a41 a29 w1(z) a107a219a55a66 a30a31 a58a59a71 star a101a102 a61a42a62a63a82a204a109a95a110a54a111a112a66a71a220a221a43a44a45a46a47a48a71a205a192a222a224a85 a29 w2(z)a107a219a55 a66 a30a31 a58a59a57 star a101a102 a197a63a49a61a82a204a94a95a66( a49a101ρ1 = ρ2a63)a71a50a51a71a214a72a222a223a85a29 w2(z)a107a219a55a66(a197 a63a55ga94a96a67a750) a30a31 a58a59a82a66a57 a52a84a115a53a241a58a59a55a215a216a97a51a71a54a206a94a95a4a1a110a54a115a53a241a58a59 d2w dz2 + p(z) dw dz + q(z)w = 0, a101a102a55a56 a82a225a12a94a95a66w 1(z) a71a50a51a71a198a60a199 a32 a0a57a241 w2(z) = Aw1(z) integraldisplay z braceleftbigg 1 [w1(z)]2 exp bracketleftbigg ? integraldisplay z p(ζ)dζ bracketrightbiggbracerightbigg dz a58a82a225a59a4a66a57a197a65a60a75a197a109a95a66 a104a61a62 a58a59 d2w1 dz2 + p(z) dw1 dz + q(z)w1 =0, d2w2 dz2 + p(z) dw2 dz + q(z)w2 =0. a210w 2(z)a103w1(z) a241a63a64a197a109a95a58a59a71a85a201a65a71a45a60a204a66 w1d 2w2 dz2 ?w2 d2w1 dz2 + p(z) parenleftbigg w1dw2dz ?w2dw1dz parenrightbigg = 0, a2 d dz parenleftbigg w1dw2dz ?w2dw1dz parenrightbigg + p(z) parenleftbigg w1dw2dz ?w2dw1dz parenrightbigg = 0. a57a241a71a60a204 w1dw2dz ?w2dw1dz = Aexp bracketleftbigg ? integraldisplay z p(ζ)dζ bracketrightbigg . a109a67a68a199w2 1 a71a69a60a199a204a66 d dz parenleftbiggw 2 w1 parenrightbigg = Aw2 1 exp bracketleftbigg ? integraldisplay z p(ζ)dζ bracketrightbigg . (9.3) a85a57a241a94a86a71a214a204a66a70a246a55a71 a102 a57 Wu Chong-shi §9.1 a10a11a247a248a249a250a251a252a253a12a16 a184a19 a8 9.2 a82Legendrea58a59 parenleftbig1?x2parenrightbig d2y dx2 ?2x dy dx + l(l + 1)y = 0 a76x = 1a79a80a81a55a192a72a66a57 a48 a60x = 1a65Legendrea58a59a55a77a78a56a53a71a73a202a74 y(x) = (x?1)ρ ∞summationdisplay n=0 cn(x?1)n. a30a31 a58a59a71a214a192 ∞summationdisplay n=0 cnbracketleftbig(n + ρ)(n + ρ + 1)?l(l + 1)bracketrightbig(x?1)n+1 + 2 ∞summationdisplay n=0 cn(n + ρ)2(x?1)n = 0. a75a76a60a199a204a66a35a36a58a59 ρ(ρ?1) + ρ = 0 a103 a211a212a112a213 cn = ?n(n?1)?l(l + 1)2n2 cn?1. a35a36a58a59a55a66a65 ρ1 = ρ2 = 0. a164 a170a77Legendre a136a137a123x = 1 a78a79a80a81 a131a82 a168 a132a83a84a129a85a123a86 a80|x?1| < 2 a81 a132a87a131a71 a229a141a123x = 1 a78 a173a88a3a234a82a89a132a90 a168a91 a232 a173a233a146a179a71a92x = 1(a180x = ?1)a163a93 a78 a71a162a234 a123x = 1(a180x = ?1) a78a94a95 a57a96a97a235a156a82 a168 a132a57 a75a211a212a112a213a71a60a199a82a225Legendrea58a59a76x = 1a53a79a80a81a59a94a66a55a213a116a55 a32 a196a98a219 cn = (l + n)(l + 1?n)2n2 cn?1 = (l + n)(l + 1?n)2n2 (l + n?1)(l + 2?n)2(n?1)2 cn?2 = ······ = (l + n)(l + 1?n)2n2 (l + n?1)(l + 2?n)2(n?1)2 ··· (l + 1)l2·12 c0 = 1(n!)2 Γ(l + n + 1)Γ(l ?n + 1) parenleftbigg1 2 parenrightbiggn c0. a99c 0 = 1 a71a214a82a225a12Legendrea58a59a55a59a94a66 Pl(x) = ∞summationdisplay n=0 1 (n!)2 Γ(l + n + 1) Γ(l ?n + 1) parenleftbiggx?1 2 parenrightbiggn , a5 a75la86a59a94a100 Legendrea121a116a57 a101a102 a224a101a102a82a59a4a66a71a78a202a74 y2(x) = gPl(x)ln(x?1) + ∞summationdisplay n=0 dn(x?1)n Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17(a3) a185a19 = g ∞summationdisplay n=0 1 (n!)2 Γ(l + n + 1) Γ(l ?n + 1) parenleftbiggx?1 2 parenrightbiggn ln(x?1) + ∞summationdisplay n=0 dn(x?1)n. a103a104a115a53a241a58a59 a191 a116a66a105a55a36a106a107a108a71a96a225a213a116g(a94a96a67a750) a103dn a2a60a57 a109a39a210a55a110a105a65a52a84a59a4a66a111a59a94a66a112a113a55a112a213a71a114a225 y2(x) = gPl(x) integraldisplay xbraceleftBigg 1 [Pl(ξ)]2 exp bracketleftBiggintegraldisplay ξ 2ζ 1?ζ2dζ bracketrightBiggbracerightBigg dξ = gPl(x) integraldisplay x 1 [Pl(ξ)]2 dξ 1?ξ2 = gPl(x) integraldisplay x dξ 1?ξ2 + gPl(x) integraldisplay xbraceleftbigg 1 [Pl(ξ)]2 ?1 bracerightbigg dξ 1?ξ2, a115a116a13a14a117a67a59a4a196a76|x?1| < 2 a81a66a105a71a60a76a60a199 a29 a59a4a66a74a75 y2(x) = g2Pl(x)ln x + 1x?1 + ∞summationdisplay n=0 dn(x?1)n. a99g = 1a71a118a96a225d n a71a119a21a214a60a199a82a225Legendrea58a59a55a59a4a66 Ql(x) = 12Pl(x) bracketleftbigg ln x + 1x?1 ?2γ ?2ψ(l + 1) bracketrightbigg + ∞summationdisplay n=0 1 (n!)2 Γ(l + n + 1) Γ(l ?n + 1) parenleftbigg 1 + 12 +···+ 1n parenrightbiggparenleftbiggx?1 2 parenrightbiggn , a5 a75la86a59a4a100Legendrea121a116a71a113a114γa65Eulera116a71ψ(z)a65Γa121a116a55a54a116a53a120a57a75a206a121a116P l(x)(a121 a122a66a123a124a246a21a71a254a65a199x = ?1 a103x = ∞a75a73a53a55a119a120a121a116) a103Ql(x)a55a119a120a54a55 a192a125a96a54a55a126 a96a71a203a210a63a127a224a128a63a129a130a57 Wu Chong-shi §9.1 a10a11a247a248a249a250a251a252a253a12a16 a186a19 a245a246a198a71a94a245a82a115a53a241a58a59 d2w dz2 + p(z) dw dz + q(z)w = 0 a76a77a78a56a53a79a80a81a55a66a55a94a131a107a108a71a62a63a132a133a98a76a134a51a135a136a245a71a58a59a55a59a4a66a67a137a54a116a196a3a76a134 a51a135a136a245a71a58a59a55a59a4a66a60a61a137a54a116a196a3a76a134a51a135a136a245a71a58a59a55a59a4a66a94a96a137a54a116a196a57 a138 a139 a140a126a96a58a59a76a77a78a56a53a141a55a109a95a35a36 Reρ 1 ≥ Reρ2 a71a78 a220ρ 1 ?ρ2 negationslash= a117a116a63a71 a59a4a66a94a96a67a137a54a116a196a3 a220ρ 1 = ρ2 a63a71 a59a4a66a94a96a137a54a116a196a3 a220ρ 1 ?ρ2 = a77a117a116a63a71 a59a4a66a60a61a137a54a116a196a57 a75a12a142a143a144a244a71a67a145a146a74 z = 0a53a65a254a55a77a78a56a53a57a206a65a71a76z = 0a53a55a79a80a81a71a60 a29 a58a59a55 a213a116a74Laurent a147a148 p(z) = ∞summationdisplay l=0 alzl?1, q(z) = ∞summationdisplay l=0 blzl?2. a74a66a75 w(z) = zρ ∞summationdisplay k=0 ckzk. a30a31 a58a59a71a214a192 ∞summationdisplay k=0 ck(k + ρ)(k + ρ?1)zk+ρ?2 + ∞summationdisplay l=0 alzl?1 ∞summationdisplay k=0 ck(k + ρ)zk+ρ?1 + ∞summationdisplay l=0 blzl?2 ∞summationdisplay k=0 ckzk+ρ = 0, a2 ∞summationdisplay k=0 ck(k + ρ)(k + ρ?1)zk+ρ?2 + ∞summationdisplay k=0 ksummationdisplay l=0 bracketleftbiga l(k + ρ?l) + bl bracketrightbigc k?lzk = 0. a33a34a149a219a109a67a119a150a86a195a71a2 z0a55a213a116a71a60a204 c0 [ρ(ρ?1) + a0ρ + b0] = 0. a75a206c 0 negationslash= 0 a71a24a199 ρ(ρ?1) + a0ρ + b0 = 0. a197a214a65a35a36a58a59a71a129a130a113a114a55a 0a103b0 a75 a0 = limz→0zp(z), b0 = limz→0z2q(z). a52a84a35a36a58a59a60a199a82a225a109a95a35a36a71ρ 1a103ρ2 a57 a151 a91Reρ1 ≥ Reρ2 a57 Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17(a3) a187a19 a85a33a34zna55a213a116a71a204 (n + ρ)(n + ρ?1)cn + nsummationdisplay l=0 bracketleftbiga l(n + ρ?l) + bl bracketrightbigc n?l = 0, a2 bracketleftbig(n + ρ)(n + ρ?1) + a 0(n + ρ) + b0 bracketrightbigc n + nsummationdisplay l=1 bracketleftbiga l(n + ρ?l) + bl bracketrightbigc n?l = 0. a197a18a45a204a225a12a213a116a112a113a55a211a212a112a213a57 a182a183a184a185a149a150a151a145a71a152a153a92a143a144a145a146 c n a131a158a159a160a161a133a57a229a141a71a123 c n a131a160a161a133a167 a168a91 a232 a173ρa57a185ρ = ρ 1 a134a135a71a154a153a143a144a132w 1(z) a57a155a185ρ = ρ 2 a134a135a71a156a153a143a144a132w 2(z) a57a229 ρ1 ?ρ2 negationslash= a157 a146a165a71a152a156a157a158a136a137a131(a238a176a159a160a154a151a131) a161 a132a57 a220ρ 1 = ρ2 a63a71 a9 a221a197a18a49a61a204a66a62a94a95a66a57a24a199a71a197a63a59a4a66a94a96a137a54a116a196a57 a220ρ 1 ?ρ2 = a77a117a116ma63a71a54a206a59a4a66a55a213a116 c(2) m a71a192 bracketleftbig(m + ρ 2)(m + ρ2 ?1) + a0(m + ρ2) + b0 bracketrightbigc(2) m + msummationdisplay l=1 bracketleftbiga l(m + ρ2 ?l) + bl bracketrightbigc(2) m?l = 0. a129a130m + ρ 2 = ρ1 a71a24a199a192 0·c(2)m + msummationdisplay l=1 bracketleftBig al(ρ1 ?l) + bl bracketrightBig c(2)m?l = 0. a60a76 a220 msummationdisplay l=1 bracketleftBig al(ρ1 ?l) + bl bracketrightBig c(2)m?l negationslash= 0a63a71 c(2)m a111a66a3 a220 msummationdisplay l=1 bracketleftBig al(ρ1 ?l) + bl bracketrightBig c(2)m?l = 0a63a71 c(2)m a43a130a57 star a54a206a59a94a162a135a107a71a58a59a55a59a4a66a64a94a96a137a54a116a196a57 star a54a206a59a4a162a135a107a71a58a59a55a59a4a66a94a96a67a137a54a116a196a71a220a221a72a61a101a102a82a66a57a49a65a197a63a199a21a55a163a196 a213a116c(2) n (n > m)a164 a62a63a83a165a206c 0(2)a103c (2)m a57a59a4a66w 2(z) a45a192a109a196a71a94a196a77a33a206 c(2) 0 a71a94 a196a77a33a206c(2) m a57a85a166a167a241a105a94a245a71a214 a164a168a169 a71c(2) m+na103c (2) m a112a113a55a112a213a111c(1) n a103c (1) 0 a112a113a55 a112a213a47a123a94a18a71a60a76a71a111c(2) m a48a77a33a55a196a77a170a214a65a59a94a66(a119a119a60a61a171a94a95a115a116a172a116)a71a60 a38 a67a145a99c(2) m = 0 a57 Wu Chong-shi §9.2 Bessela10a11a12a16 a188a19 §9.2 Bessel a173a174a175a176 Bessela58a59 d2w dz2 + 1 z dw dz + parenleftbigg 1? ν 2 z2 parenrightbigg w = 0 a65a115a244a55a115a53a241a58a59a112a94a71a113a114νa65a115a116a71Reν ≥ 0a57a115a116a13a14a71z = 0a65a58a59a55a77a78a56a53a71z = ∞ a65a58a59a55a177a77a78a56a53a57 a70a178a50a51Bessela58a59a76z = 0a53a55a79a80|z| > 0 a81a55a66a57a74 w(z) = zρ ∞summationdisplay k=0 ckzk, c0 negationslash= 0, a30a31Bessela58a59a71a204 ∞summationdisplay k=0 ck(k+ρ)(k+ρ?1)zk+ρ?2+ ∞summationdisplay k=0 ck(k+ρ)zk+ρ?2+ ∞summationdisplay k=0 ckzk+ρ?ν2 ∞summationdisplay k=0 ckzk+ρ?2=0, a125a179zρ?2a71a2a204 ∞summationdisplay k=0 ck bracketleftbig(k + ρ)2 ?ν2bracketrightbigzk + ∞summationdisplay k=0 ckzk+2 = 0. a52a84 a191 a116 a147a148 a55a180a94a54a71a2a60a33a34a213a116a57 a75a119a150a86a195z0a196a55a213a116a71a118a60a75c 0 negationslash= 0 a71a214a204a66a6a7 a38a39 a71 ρ2 ?ν2 = 0. a60 a38 a82a204 ρ1 = ν, ρ2 = ?ν. a60a75Reν ≥ 0a71a24a199Reρ 1 ≥ Reρ2 a57 a75z1a55a213a116a71a204 c1 bracketleftbig(ρ + 1)2 ?ν2bracketrightbig = 0 a2 c1(2ρ + 1) = 0. a60a76 c1 = 0, a220ρ negationslash= ?1/2; (9.4a) c1a43a130, a220ρ = ?1/2. (9.4b) a199a21 a29a181 a66a71a2a203ρ = ?1/2a71a182a60a199a99c 1 = 0 a57 a75zna55a213a116a71a204 cn bracketleftbig(ρ + n)2 ?ν2bracketrightbig+ cn?2 = 0 a2 cnn(2ρ + n) + cn?2 = 0, a60a76a71a204a66a183a184a185a186 cn = ? 1n(n + 2ρ)cn?2. Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17(a3) a189a19 a207a208a209a210a211a212a112a213a71a214a60a199a82a204 c2n = ? 1n(n + ρ) 122c2n?2 = (?)2 1n(n?1)(n + ρ)(n + ρ?1) 124c2n?4 = ··· = (?) n n! 1 (ρ + 1)n 1 22nc0, (9.5) c2n+1 = ? 1(n + 1/2)(n + ρ + 1/2) 122c2n?1 = (?) 2 (n + 1/2)(n?1/2)(n + ρ + 1/2)(n+ ρ?1/2) 1 24c2n?3 = ··· = (?)n 1(3/2) n 1 (ρ + 3/2)n 1 22nc1 = 0. (9.6) a210ρ 1 = ν a30a31 a71a2a204 w1(z) = c0zν ∞summationdisplay k=0 (?)k k!(ν + 1)k parenleftBigz 2 parenrightBig2k . a99c 0 = 1 2νΓ(ν + 1) a71a214a192a66 Jν(z) = ∞summationdisplay k=0 (?)k k!Γ(k + ν + 1) parenleftBigz 2 parenrightBig2k+ν . (9.7) a210ρ 2 = ?ν a30a31 a71a192 w2(z) = c0z?ν ∞summationdisplay k=0 (?)k k!(?ν + 1)k parenleftBigz 2 parenrightBig2k , a220ν negationslash=(a77)a117a116a63a71a64a60a99 c 0 = 2ν/Γ(?ν + 1) a71a69a204 J?ν(z) = ∞summationdisplay k=0 (?)k k!Γ(k ?ν + 1) parenleftBigz 2 parenrightBig2k?ν . (9.8) a187a240a50a51a94a245ρ = ?1/2a55a135a107a57a23a246a188 a56a189 a66a71a197a63a182a221a60a199a99 c 1 = 0 a57a60a75 a101a102c1 negationslash= 0a71 a78 c2n+1 = (?) n (3/2)n(1)n 1 22nc1, a129a130a66(1) n = n! a71a197a18a76w 2(z) a114a49a67a0a65a85a190a87a94a196 z?1/2 ∞summationdisplay n=0 c2n+1z2n+1 = c1 ∞summationdisplay n=0 (?)n n!Γ(n + 3/2) parenleftBigz 2 parenrightBig2n+1/2 · radicalbiggpi 2 = c1 radicalbiggpi 2J1/2(z). a2a76w 2(z) a114a49a67a0a65a85a191a87a70a59a94a66a57 Wu Chong-shi §9.2 Bessela10a11a12a16 a1810a19 a169 a76a65a17 a55a56 a47a48a12a82a66Bessela58a59a55a43a44a192 ? a70a246a55a193a82a225a12a109a95a35a36a71 ? a54a202a206a194a94a95a35a36a71a64a104 a82a225a12a201a202a55a66a57 star a220ν negationslash=a117a116a63a55a193a101 a76a71a60a75a197a63a82a225a55a109a95a66J ν(x)a103 J?ν(x) a110a54a111a112a57 star a220ν = 0a63a71a70a246a55a82a66a0a59a49a65a195a225a12a62a94a95a66 J0(x) = ∞summationdisplay k=0 (?)k k!k! parenleftBigx 2 parenrightBig2k . a197a217a89a71a197a63a55a59a4a66a202a196a137a192a54a116a196a71a2 y2(x) = gJ0(x)lnx+ ∞summationdisplay k=0 dkxk, g negationslash= 0. star a220ν = n, n = 1,2,3,···a63a71a23a246a182a221a49a65a82a225a12a94a95a66a57 ? a82a168a71a229ν = n, n = 1,2,3,···a165a71a166a146a132 J?n(x) = ∞summationdisplay k=0 (?)k k!Γ(k ?n + 1) parenleftBigx 2 parenrightBig2k?n a167a197k = 0,1,···,n?1 a198 a179a131a145a146a199a1630a71a164a85a162a163z = 0,?1,?2,···a172a85 Γa200 a146a131 a168a201a202 a78 a57a203a92 J?n(x) = ∞summationdisplay k=n (?)k k!Γ(k ?n + 1) parenleftBigx 2 parenrightBig2k?n . a204k ?n = la71a152a173 J?n(x) = ∞summationdisplay l=0 (?)n+l (n + l)!Γ(l + 1) parenleftBigx 2 parenrightBig2(n+l)?n = (?)n ∞summationdisplay l=0 (?)l l!Γ(n + l + 1) parenleftBigx 2 parenrightBig2l+n = (?)nJn(x), a180a82 a168 a132J n(x) a159a160a239a151a57 ? a82a89a71a123a166a146a132a155a167a71a205a171a206a85a207a141a208a209a91a166a146a132a131a210a179a145a146a230a1630a71a152a122a123a153a123a154a211a167a212 a182a158a164a176a209 a91 a98a123a213a157J ?ν(x) (ν = 1,2,3,···) a165a71a214c 0 = 2ν/Γ(1?ν) a71a215a215a151 a91 a158c 0 = 0 a57 ? a82a216a71a126a127a229a141a153a92a217a218a219a164a176a230a220a221 a131a151 a91 a71a222a143y 2(x) a131a166a146a167a197a130k = 0,1,···,n?1 a223a179a131a145a146 a230 a1630a71a152a164a224a225a213a226a227 k = na179a228a229a145a146a199a230a163a154a174a57a164a231a153a92a227a149a150a151a145 c2k = ? 1k(k ?ν) 122c2k?2 a232a157a57a229ν = na165a71a233a141c 2n a154a211a234a71a162a234a92a235 a198 a179a145a146a231a172a236a237a211a234a57 Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17(a3) a1811a19 a199a70a55a241a105a238a89a71a220ν = n, n = 1,2,3,···a63a71Bessela58a59 d2y dx2 + 1 x dy dx + parenleftbigg 1? n 2 x2 parenrightbigg y = 0 a55a59a4a66a64a94a96a137a192a54a116a196a71a2 y2(x) = gJn(x)lnx+ ∞summationdisplay k=0 dkxk?n, g negationslash= 0. a239a78a70 a29y2(x)a30a31Bessela58a59a71a2a60a96a225a213a116a57 a245a246a91a92a82Bessela58a59a59a4a66a55a93a94a162a58a105a57a75a76a71a41a240a241 J ν(z)a103 J?ν(z) a55Wronski a242a243 a219a199a241a105a254a244a55a110a54a201a112a54a57a245a246a66 Bessela58a59a55a213a116p(z) = 1/za71a247(9.3)a219a214a60a199a204a66 W [Jν(z), J?ν(z)] ≡ vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle Jν(z) J?ν(z) Jprimeν(z) Jprime?ν(z) vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = Aexp bracketleftbigg ? integraldisplay z dζ ζ bracketrightbigg = Az . a75a12a96a225a57a241a115a116Aa71a49a127 a29Jν(z)a103J?ν(z)a55a191 a116a66(9.7) a103(9.8)a30a31 a71a248a225 W [Jν(z), J?ν(z)] ≡ Jν(z)Jprime?ν(z)?J?ν(z)Jprimeν(z) a114z?1a196a55a213a116a2a60a57a197a49a58a249a163 a191 a116a114a55a59a94a196a57a60a76a71 A = 1Γ(1 + ν) 12ν 1Γ(1?ν) ?ν2?ν ? 1Γ(1?ν) 12ν 1Γ(1 + ν) ν2ν = ? 2νΓ(1 + ν)Γ(1?ν) = ? 2Γ(ν)Γ(1?ν) = ?2pisinpiν. a197a18a214a204a66 W [Jν(z), J?ν(z)] = ? 2piz sinpiν. (9.9) a70a246a55a240a241a114a210a66a12 Γa121a116a55a54a250 Γ(ν)Γ(1?ν) = pisinpiν. (9.9)a219a85a86a217a89a71a220ν = n, n = 0,1,2,···a63Jν(z)a103J?ν(z)a110a54a201a112a57a68a65a101a102a29Bessela58 a59a55a59a4a66a99a75 J ν(z)a103J?ν(z) a55a110a54a251a252a71 w2(z) = c1Jν(z) + c2J?ν(z), a49a224a253a254a255a220a55a251a252a213a116a71a203a204W [J ν(z), w2(z)] a54a43a11ν a0 a67a750a71a197a18a55w 2(z) a214a94a96(a54a43a11 ν a0)a111 Jν(z)a110a54a111a112a57a75a76a71a1a244a214a99a59a4a66a75 w2(z) = cJν(z)?J?ν(z)sinpiν , a197a18a45a192 W [Jν(z), w2(z)] = 2piz. a75a12a2a88a197a18a96a3a55w 2(z) a192a130a3(sinnpi = 0a71a241a4a750)a71a118a129a130a66 J ?n(z) = (?)nJn(z) a71a1a244 a45a202a220 a37 a94a107a253a99a213a116ca71a203a204w 2(z) a114a55a241a5a76ν = na63a64a750a71 a49a101 a99c = cospiν a2a60a57a197a18 Wu Chong-shi §9.2 Bessela6a7a8a9 a1012a11 a204a66Bessela58a59a55a59a4a66a45a65 Nν(z) = cospiνJν(z)?J?ν(z)sinpiν , (9.10) a5 a75ν a1Neumanna121a116a57a220ν = n, n = 0,1,2,···a63a71(9.10)a219a75a67a96a219a71a60a103l’Hospitala105a78a82a52 a193a71 Nn(z) = limν→nNν(z) = limν→n cosνpiJν(z)?J?ν(z)sinνpi = 1pi bracketleftbigg?J ν(z) ?ν ?(?) n?J?ν(z) ?ν bracketrightbigg ν=n = 2piJn(z)ln z2 ? 1pi n?1summationdisplay k=0 (n?k ?1)! k! parenleftBigz 2 parenrightBig2k?n ? 1pi ∞summationdisplay k=0 (?)k k!(n + k)! bracketleftbigψ(n + k + 1) +ψ(k + 1)bracketrightbigparenleftBigz 2 parenrightBig2k+n , |argz| < pi, (9.11) a113a114ψ(ζ)a65Γa121a116a55a54a116a53a120a71 ψ(ζ) ≡ dlnΓ(ζ)dζ = Γ prime(ζ) Γ(ζ). a118a118a125a96a71a220n = 0a63a202a220a179a12(9.11)a219a117a67a59a4a196a55a192a193 a103 a57 Wu Chong-shi a13a14a15 a16a17a18a19a20a21a22 a6a7a8a23a24a25a9a26 ( a16) a1013a11 ?§9.3 ν = n, n = 0,1,2,3,··· a27a28 Bessel a29a30a31a32a33a34 a35ν = n, n = 0,1,2,3,··· a36a37Bessela38a39 d2y dx2 + 1 x dy dx + parenleftbigg 1? n 2 x2 parenrightbigg y = 0 a40a41a42a43a44a45a46a47a48a49a50 a37a51 y2(x) = gJn(x)lnx+ ∞summationdisplay k=0 dkxk?n, g negationslash= 0. a35ν = 0 a36a37 y2(x) = gJ0(x)lnx+ ∞summationdisplay k=0 dkxk, g negationslash= 0. a52a53a54 a37a55 dy2(x) dx = g dJ0(x) dx lnx + gJ0(x)· 1 x + ∞summationdisplay k=0 dkkxk?1, d2y2(x) dx2 = g d2J0(x) dx2 lnx + 2g dJ0(x) dx · 1 x ?gJ0(x)· 1 x2 + ∞summationdisplay k=0 dkk(k ?1)xk?2. a56a57a58a59Bessel a38a39 d2y dx2 + 1 x dy dx + y = 0, a51a55 g bracketleftbiggd2J 0(x) dx2 + 1 x dJ0(x) dx + J0(x) bracketrightbigg lnx + g ∞summationdisplay k=0 (?)kk k!k! parenleftBigx 2 parenrightBig2k?2 + ∞summationdisplay k=0 dkk(k ?1)xk?2 + ∞summationdisplay k=0 dkkxk?2 + ∞summationdisplay k=0 dkxk = 0. a60a61J 0(x)a62a63 a58a59Bessel a38a39 a40a43 a37a64a65 a41a44a66a67lnx a68a38a69a70a71a72 a50a73a74a75a76 0 a37a77a78 g ∞summationdisplay k=0 (?)k k!k! k 22k?2x 2k + ∞summationdisplay k=0 dkk2xk + ∞summationdisplay k=0 dkxk+2 = 0. a79a80a81a82a56a57a83J 0(x) a40a84a49a85a86 a65a87 a88a89a90a91a92a50a40a93a49 a87 a48a94x0 a50 a37 a47g ·0 + d 0 ·0 = 0a37a77a78 ga95 a61, d 0a95 a61. a96a90a91x1 a50a40a93a49 a37a55 d1 = 0. a97x2 a50a40a93a49 a37a98a78 a52 a55?g + 4d2 + d0 = 0a37a77a78 d2 = ?14d0 + 14g. Wu Chong-shi ?§9.3 ν = n, n = 0,1,2,3, · · · a99a100Bessela101a102a103a104 a16a105 a10614 a107 a108a109a110a111 a37a112a98a78a113a114a55a115a116a117a118 a50a74a119 a117a118 a50a40a93a49 a87 a90a91 x2k a50a40a93a49 a37a55 g · (?) k k!k! 2k 22k?1 + d2k(2k) 2 + d2k?2 = 0, a94 a63a55a115 d2k = ? 1(2k)2d2k?2 ? (?) k k!k! 1 22k 1 kg = ? 1(2k)2 bracketleftbigg ? 1(2k ?2)2d2k?4 ? (?) k?1 (k ?1)!(k ?1)! 1 22k?2 1 k ?1g bracketrightbigg ? (?) k k!k! 1 22k 1 kg = (?) 2 k2(k ?1)2 1 24d2k?4 ? (?)k k!k! g 22k bracketleftbigg1 k + 1 k ?1 bracketrightbigg = ··· = (?) k k!k! 1 22kd0 ? (?)k k!k! g 22k bracketleftbigg1 k + 1 k ?1 +···+ 1 bracketrightbigg . a90a91x2k+1 a50a40a93a49 a37a120a55a115 (2k + 1)2d2k+1 + d2k?1 = 0, a97a94d 1 = 0a37a121 a109 a37 d2k+1 = 0. a122a123 a37a112 a52a124a83ν = 0 a36 a40a41a42a43 y2(x) = gJ0(x)lnx + d0 ∞summationdisplay k=0 (?)k k!k! parenleftBigx 2 parenrightBig2k ?g ∞summationdisplay k=1 (?)k k!k! parenleftbigg1 k + 1 k ?1 +···+ 1 parenrightbiggparenleftBigx 2 parenrightBig2k . a125a66a40a126a127 a63a128 g = 2pi, d0 = ?2pibracketleftbigln2 +ψ(1)bracketrightbig, a129a67a40ψ a130 a49 a112a63Γa130 a49a40a48a49a53a54 a87 a97Γ a130 a49a40a131a132 Γ(z + 1) = zΓ(z) a37a133a134a135a136 ψ(z + n) =ψ(z) + 1z + 1z + 1 +···+ 1z + n?1, a79a137 a55a115 a40a43( a138 a76N 0(x))a112a98a78a139a140 N0(x) = 2piJ0(x)ln x2 ? 2pi ∞summationdisplay k=0 (?)k k!k!ψ(k + 1) parenleftBigx 2 parenrightBig2k . a96a141a142n = 1,2,3,···a40a143a144 a87a145y2(x)a56a57na59Bessela38a39a37a51a55 g bracketleftbiggd2J n(x) dx2 + 1 x dJn(x) dx + parenleftbigg 1? n 2 x2 parenrightbigg Jn(x) bracketrightbigg lnx + g2 ∞summationdisplay k=0 (?)k(2k + n) k!(k + n)! parenleftBigx 2 parenrightBig2k+n?2 + ∞summationdisplay k=0 dk(k ?n)(k ?n?1)xk?n?2 + ∞summationdisplay k=0 dk(k ?n)xk?n?2 + parenleftbigg 1? n 2 x2 parenrightbigg ∞summationdisplay k=0 dkxk?n = 0. a60a61J n(x)a62a63n a59Bessel a38a39 a40a43 a37a64a65 a41a44a66a67lnx a68a38a69a70a71a72 a50a73a74a75a76 0 a37a77a78 g ∞summationdisplay k=0 (?)k k!(k + n)! 2k + n 22k+n?1x 2k+n?2 + ∞summationdisplay k=0 dk[(k ?n)2 ?n2]xk?n?2 + ∞summationdisplay k=0 dkxk?n = 0, Wu Chong-shi a104a146a147 a16a17a18a19a20a21a22 a101a102a103a23a24a25 a105 a26 (a16) a10615a107 a148a149 a139a140 g ∞summationdisplay k=0 (?)k k!(k + n)! 2k + n 22k+n?1x 2k+2n + ∞summationdisplay k=0 dkk(k ?2n)xk + ∞summationdisplay k=0 dkxk+2 = 0. a88a89a150a90a91a151 a65a152a153 a92a50a40a93a49 a87 a97x0 a50a40a93a49 a37a55d0 ·0 = 0a37a77a78 d0a95 a61. a97x1 a50a40a93a49 a37d1(1 ?2n) = 0a37a77a78 d1 = 0. a97x2k+1 a50a40a93a49 a37 a47 d2k+1(2k + 1)(2k ?2n + 1) + d2k?1 = 0, a77a78 d2k+1 = ? 1(2k + 1)(2k?2n + 1)d2k?1 = ··· = 0. a48a94x2k a40a93a49 a37a154a155a156a114k < n, k = n, k > na157a158 a143a144 a87 star a35k < na36a37 d2k2k(2k?2n) + d2k?2 = 0. a77a78 d2k = 1k(n?k) 122d2k?2 = 1k(k ?1)(n?k)(n?k + 1) 124d2k?4 = ··· = (n?k ?1)!k!(n?1)! 122kd0. a159 a114a63 d2n?2 = 1[(n?1)!]2 122(n?1)d0. star a35k = na36a37 1 2n?1(n?1)!g + d2n ·0 + d2n?2 = 0. a77a78 d2na95 a61, g = ?2n?1(n?1)!d2n?2 = ? 12n?1(n?1)!d0. star a35k > na36a37 (?)k?n (k ?n)!k! 2k ?n 22k?n?1g + d2k2k(2k ?2n) + d2k?2 = 0. Wu Chong-shi ?§9.3 ν = n, n = 0,1,2,3, · · · a99a100Bessela101a102a103a104 a16a105 a10616 a107 a77a78 d2k = ? 1k(k ?n) 122d2k?2 ? (?) k?n k!(k ?n)! 2k ?n 22k?n?1 1 4k(k ?n)g = ? 1k(k ?n) 122d2k?2 ? (?) k?n k! (k ?n)! parenleftbigg1 k + 1 k ?n parenrightbigg 1 22k?n+1g = (?) 2 k(k ?1)(k ?n)(k ?n?1) 1 24d2k?4 ? (?)k?n k!(k ?n)! parenleftbigg1 k + 1 k ?1 + 1 k ?n + 1 k ?n?1 parenrightbigg 1 22k?n+1g = ··· = (?) k?nn! k!(k ?n)! 1 22k?2nd2n ? (?) k?n k!(k ?n)! parenleftBig1 k + 1 k ?1 +···+ 1 n + 1 + 1 k ?n + 1 k ?n?1 +···+ 1 1 parenrightBig 1 22k?n+1g. a121 a109 y2(x) =gJn(x)lnx + ∞summationdisplay k=0 d2kx2k?n =gJn(x)lnx? g2 n?1summationdisplay k=0 (n?k ?1)! k! parenleftBigx 2 parenrightBig2k?n + d2n ∞summationdisplay k=n (?)k?n2nn! k!(k ?n)! parenleftBigx 2 parenrightBig2k?n ? g2 ∞summationdisplay k=n+1 (?)k?n k!(k ?n)! parenleftBig1 k + 1 k ?1 +···+ 1 n + 1 + 1 k ?n + 1 k ?n?1 +···+ 1 parenrightBigparenleftBigx 2 parenrightBig2k?n =gJn(x)lnx? g2 n?1summationdisplay k=0 (n?k ?1)! k! parenleftBigx 2 parenrightBig2k?n + d2n2nn! ∞summationdisplay k=0 (?)k k!(k + n)! parenleftBigx 2 parenrightBig2k+n ? g2 ∞summationdisplay k=1 (?)k k!(k + n)! parenleftBig 1 k + n + 1 k + n?1 +···+ 1 n + 1 + 1 k + 1 k ?1 +···+ 1 parenrightBigparenleftBigx 2 parenrightBig2k+n . a160a161 a128 g = 2pi, d2n = ? 12npin!bracketleftbig2ln2 +ψ(n + 1) +ψ(1)bracketrightbig. a79a137a162a47 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 , a129a67n = 1,2,3,··· a87 a90a91a44a163 a68a164a55a115 a40 N 0(x) a40a85a86 a65a37a98a78a165 a124 a37 a79a80a40 N n(x) a62a166a167 a94 n = 0a37a168a155a169 a43a76a79 a36a170a171a172 a41a42a50a40a47a173a74 a87