a0 a1 star a2a3a4a5a6a7a8a9a10 4 star §6.5 a11a12a8a13a14a15 a16a17a18 a6a7 a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17 a181 a19 a20a21a22 a23a24a25a26a27a28a29a30a31a32a33a34a35a36a37 §6.1 a38a39a40a41a42a43a44a45a46a47a42a48a49a50a48 a51a52a53a54a55a56a57a58a59a60a61a62a63a64a65a66 d2w dz2 + p(z) dw dz + q(z)w = 0, (6.1) p(z)a67q(z)a68a69a70a71a72a73a74a75 ? a70a71a72a76a77a78a79a80a70a71a72a73a74a81a82a72a75 ? a83a84a77a85a70a71a76a72a76a86a87a77a78a79a80a70a71a73a74a72a76a86a87a81a82a72a75 a88a89a90a91a92a91a93a94a95a96a97a98 a85 a99a100a101a91a102a103a104a105a106a107a108z 0 a101a109a110a111a112a113a101a114a115a89a90 a75 a96a97a116a90p(z), q(z) a117z0a108a101a91a118a119a120a121a107a122a89a90a91a117z0a108a101a91a118a119a85 a123a124a125 a85 a120a121a107 a122a89a90a91a101a126a127 a85a128a129a85 a103Taylora89a90a130a103Laurenta89a90 a75 ? a131a132p(z), q(z)a133z0 a134 a76a86a85a135z0 a134 a68a69a70a71a72a136 a134 a75 ? a131a132p(z), q(z) a137a138a139a140a141a142a133z0 a134a143 a76a86a85a135z0 a134 a68a69a70a71a72a144 a134 a75 a1456.1 a146a147a148a70a71(Hypergeometric equation) z(1?z)d 2w dz2 + bracketleftbigγ ?(1 + α + β)zbracketrightbigdw dz ?αβw = 0 a72a73a74a77 p(z) = γ ?(1 + α+ β)zz(1?z) a67 q(z) = ? αβz(1?z). a133a140a149a150a151a85p(z)a67q(z)a140a152a142a144 a134a153 z = 0a67z = 1a75a154a155a85a156a157z = 0a67z = 1a77a146a147a148a70a71 a72a144 a134a158 a85a140a149a150a151a72a159a160 a134a161 a77a70a71a72a136 a134 a75 a1456.2 Legendre a70a71 parenleftbig1?x2parenrightbig d2y dx2 ?2x dy dx + l(l + 1)y = 0, a133a140a149a150a151a72a144 a134 a69x = ±1a75 a162a163a164a165a166 a150 a134 z = ∞a77 a143 a77a70a71(6.1)a72a144 a134 a85a135a167a168a169a170 a171a172 a72 a171a173z = 1/t a75 dw dz = ?t 2dw dt , d2w dz2 = t 4d2w dt2 + 2t 3dw dt . a174a175 a85a70a71(6.1) a171 a69 d2w dt2 + bracketleftbigg2 t ? 1 t2p parenleftbigg1 t parenrightbiggbracketrightbigg dw dt + 1 t4q parenleftbigg1 t parenrightbigg w = 0. (6.2) a131a132t = 0a77a70a71(6.2)a72a136 a134 (a144 a134 )a85a135a68 a165a166 a150 a134 z = ∞a77a70a71 (6.1)a72a136 a134 (a144 a134 )a75 §6.1 a3a4a5a6a7a8a9a10a11a12a7a176a177a178a176 a182a19 t = 0 (a179z = ∞)a69a70a71a136 a134 a72a180a181a77 p parenleftbigg1 t parenrightbigg = 2t+ a2t2 + a3t3 +···, q parenleftbigg1 t parenrightbigg = b4t4 + b5t5 +···, a179 p(z) = 2z + a2z2 + a3z3 +···, q(z) = b4z4 + b5z5 +···. a165a166 a150 a134 a77a146a147a148a70a71a67Legendrea70a71a72a144 a134 a75 a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17 a183 a19 §6.2 a60a61a57a182a183a184a185a62a186 a187a188 a85 a143a189a190a191a192a193a194a195a196 a72a82a197a75 a198a1996.1 a131a132p(z)a67q(z)a133a200|z ?z0| < Ra201a202a203a76a86a85a135a133 a175 a200a201a136a204a205a70a71a206a203a207a208 d2w dz2 + p(z) dw dz + q(z)w = 0, w(z0) = c0, wprime(z0) = c1 (c0, c1a69a209a210a136a74) a140a211a141a72a141a142a76w(z)a85a212a213w(z)a133a214a142a200a201a202a203a76a86a75 a215a216 a214a142a82a197a85a217a155a218w(z)a133z0 a134 a72a219a220|z ?z0| < Ra201a221a222a69Taylora223a74 w(z) = ∞summationdisplay k=0 ck(z ?z0)k. a224a225 a85a214a226(z ?z0) 0a227(z ?z0)1 a72a73a74c0a227c1 a228a229 a67a206a203a180a181a141a230a75 a231 a214a142a232a233a72a223a74a76a234a235a204a205a70a71a85a236a237a73a74a85a238a217a155a239a240a73a74ck a75a82a197a241 a191 a85a73a74 ck(k = 2,3,···)a242a217a243c0, c1 a244a245 a75 a1456.3 a239Legendrea70a71 parenleftbig1?x2parenrightbig d2y dx2 ?2x dy dx + l(l + 1)y = 0 a133x = 0 a134 a219a220a201a72a76a85a159a137la77a141a142a246a74a75 a186 x = 0 a77a70a71a72a136 a134 a85 a174a175 a85a217a247a76 y = ∞summationdisplay k=0 ckxk. a234a235a70a71a85a238a140 parenleftbig1?x2parenrightbig ∞summationdisplay k=0 ckk(k ?1)xk?2 ?2x ∞summationdisplay k=0 ckkxk?1 + l(l + 1) ∞summationdisplay k=0 ckxk = 0, a248 a197a249a212a85a250a251 ∞summationdisplay k=0 braceleftBig (k + 2)(k + 1)ck+2 ?bracketleftbigk(k + 1)?l(l + 1)bracketrightbigck bracerightBig xk = 0. a215a216Taylor a221a222a72a211a141a87a85a217a250 (k + 2)(k + 1)ck+2 ?[k(k + 1)?l(l + 1)]ck = 0, a179 ck+2 = k(k + 1)?l(l + 1)(k + 2)(k + 1) ck = (k ?l)(k + l + 1)(k + 2)(k + 1) ck. §6.2 a10a11a7a176a252a253a254a12a16 a184a19 a214a255a238a250a251a157a73a74a0a1a72a2a3a4a5a75a6a7a8a243a9a10a11a73a85a238a217a155a239a250a73a74 c2n = (2n?l?2)(2n + l?1)2n(2n?1) c2n?2 = (2n?l?2)(2n?l?4)(2n + l?1)(2n + l?3)2n(2n?1)(2n?2)(2n?3) c2n?4 = ··· = c0(2n)!(2n?l?2)(2n?l?4)···(?l) ×(2n + l?1)(2n + l?3)···(l + 1), c2n+1 = (2n?l?1)(2n + l)(2n + 1)(2n) c2n?1 = (2n?l?1)(2n?l?3)(2n + l)(2n + l?2)(2n + 1)(2n)(2n?1)(2n?2) c2n?3 = ··· = c1(2n + 1)!(2n?l?1)(2n?l?3)···(?l + 1) ×(2n + l)(2n + l?2)···(l + 2). a8a243Γa12a74a72a87a13 Γ(z + 1) = zΓ(z), Γ(z + n + 1) = (z + n)(z + n?1)···(z + 1)zΓ(z), a217a155 a231c 2na67c2n+1 a14a15 c2n = 2 2n (2n)! Γ parenleftbigg n? l2 parenrightbigg Γ parenleftbigg ?l2 parenrightbigg Γ parenleftbigg n + l + 12 parenrightbigg Γ parenleftbiggl + 1 2 parenrightbigg c0, c2n+1 = 2 2n (2n + 1)! Γ parenleftbigg n? l?12 parenrightbigg Γ parenleftbigg ?l?12 parenrightbigg Γ parenleftbigg n + 1 + l2 parenrightbigg Γ parenleftbigg 1+ l2 parenrightbigg c1. a154a155a85Legendrea70a71a72a76a238a77 y(x) = c0y1(x) + c1y2(x), a159a137 y1(x) = ∞summationdisplay n=0 22n (2n)! Γ parenleftbigg n? l2 parenrightbigg Γ parenleftbigg ?l2 parenrightbigg Γ parenleftbigg n + l + 12 parenrightbigg Γ parenleftbiggl + 1 2 parenrightbigg x2n, a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17 a185 a19 y2(x) = ∞summationdisplay n=0 22n (2n + 1)! Γ parenleftbigg n? l?12 parenrightbigg Γ parenleftbigg ?l?12 parenrightbigg Γ parenleftbigg n + 1 + l2 parenrightbigg Γ parenleftbigg 1 + l2 parenrightbigg x2n+1. a228 a131a82a197a154a241a85a209a210a16a82a141a17a206a180a181c0a67c1a85a238a141a82a217a155a239a240a70a71a72a141a142a83a76a75a83a84 a77a85 ? a131a132a18c0 = 1, c1 = 0a85a238a250a251a83a76y1(x)a19 ? a131a132a18c0 = 0, c1 = 1a85a238a250a251a83a76y2(x)a75 a224a225 a85a214a152a142a83a76y1(x)a67y2(x)a77a20a87 a165 a11a72a75 a21 a214a152a142a20a87 a165 a11a83a76a240a22a85a238a217a155a23 a24 a240a70a71a72a25a76a75 ? a131a132a218a26 a196 a76a233a137a72a136a74c0a67c1 a27a15 a77a209a210a28 a189 a136a74a85a26 a196 a250a251a72a238a77a70a71 a72a25a76a75 a4a29 a186a62a30a31a54a62a32a33 a75a26 a196 a239a250a72a83a76a137a85y1(x)a34a35a140xa72a36a37a38a85y2(x)a34a35a140xa72a144 a37a38a85a179y1(x)a77xa72a36a12a74a85y2(x)a77xa72a144a12a74a75 a21 a239a76a72a39a71a40 a27 a85a214a77a80a41a9a10a11a73a137a34 a240a42a73a74ck+2a67cka85a43 a227ck+1 a165 a11a85 a174a175c 2na78a79a80c0a81a82a85c2n+1a78a79a80c1a81a82a75 a21a215a44 a26 a40a241a85a70a71a72a76a72a45a68a87(a214a226a46a72a77a144a36a87)a85a47 a225a48a49 a77a70a71a72a45a68a87a72a6a50a75 a25a39a214a142a51a52a85a217a155 a27 a240a133a136 a134 a219a220a201a239a223a74a76a72a141a53a54a55a75a214a238a77 a153 ? a231(a70a71a136 a134 a219a220a201a72)a76a221a222a69Taylora223a74a85a234a235a204a205a70a71a19 ? a236a237a73a74a85a250a251a73a74a0a1a72a9a10a11a73a19 ? a6a7a8a243a9a10a11a73a85a239a240a73a74cka72a56a57 a244a58 a233(a243c0a67c1 a244a245 )a85 a21 a43a59a60a250a240a223a74a76a19 a80a41a9a10a11a73a141a82a77a20a87a72(a174a69a70a71a77a20a87a72)a85a154a155a59a60a72a223a74a76a141a82a217a155 a14a15 w(z) = c0w1(z)+ c1w2(z) a72a232a233a75 a61a162 a46a240a85a133a73a74a0a1a72a9a10a11a73a137a85a141a53a62a63a64a240a42 ck, ck+1, ck+2 a65 a142a66a219a72a73a74a85 a174a175 ck a62a63a64a67a68a41c0a67c1a85a59a60a239a250a72w1(z)a69w2(z)a238 a143 a62a34a35a140za72a36a37a38a69a144a37a38a75 a48 a243a136a204a205a70a71a72a38a223a74a76a70a85a217a155a250a251a70a71a133a141a82a71a220a201a72a76a233a75a72a73a74a217a155 a215a216a61a162 a85 a239a240a70a71a133 a143 a63a71a220a201a72a76a233a75a217a155 a190a191 a85a70a71a133 a143 a63a71a220a201a72a76a233a85 a75 a69a76a86a76a77a75 a174a175 a85a74a217 a21 a70a71a133a78a141a71a220a201a72a76a233a240a22a85a25a39a76a86a76a77a85a10a240a70a71a133a159a160a71a220a201a72a76a233a75 a1456.4 a79w1a77a70a71 d2w dz2 + p(z) dw dz + q(z)w = 0 (6.3) a72a76a85a133a71a220G1a201a76a86a75a80 tildewidew1a77w1a133a71a220G2a201a72a76a86a76a77a85a179 §6.2 a10a11a7a176a252a253a254a12a16 a186a19 w1 ≡ tildewidew1, z ∈ G1 intersectiontextG2, (6.4) a81 a190a191a153 tildewidew1 a82 a77a70a71(6.3)a72a76a75 a83 a79 d2 tildewidew1 dz2 + p(z) dtildewidew1 dz + q(z)tildewidew1 = g(z), g(z)a133G2 a201a76a86a75 a174 a69w1a77a70a71(6.3)a133a71a220G1 a201a72a76a85 a84 a133a85a71a220 G1 intersectiontextG2 a201a85 a82a86a87 a70a71 d2w1 dz2 + p(z) dw1 dz + q(z)w1 = 0. a43a133 a175 a85a71a220a201a85w1(z) ≡ tildewidew1(z)a85 a84 d2 tildewidew1 dz2 + p(z) dtildewidew1 dz + q(z)tildewidew1 = 0, z ∈ G1 intersectiontextG 2, a179g(z) ≡ 0, z ∈ G1 intersectiontextG2a75 a215a216 a76a86a12a74a72a211a141a87a85a88a179 a190 a250 g(z) ≡ 0, z ∈ G2, a89 a179 tildewidew1a133G2 a201 a86a87 a70a71 d2 tildewidew1 dz2 + p(z) dtildewidew1 dz + q(z)tildewidew1 = 0. square a145 6.5 a79 w1 a67 w2 a161 a77a70a71 (6.3) a72a152a142a20a87 a165 a11a76a85a213a242a133a71a220 G1 a201a76a86a75a80 tildewidew1a67 tildewidew2a205a84a77w1a67w2a133a71a220 G2 a201a72a76a86a76a77a85a179a133z ∈ G1 intersectiontextG2 a137 w1 ≡ tildewidew1, w2 ≡ tildewidew2. a81 a190a153 tildewidew1a67 tildewidew2 a82 a20a87 a165 a11a75 a83 a80a526.4a90a85 tildewidew1a67 tildewidew2 a82 a77a70a71(a133G2 a201)a72a76a75 a174 a69w1a67w2 a20a87 a165 a11a85 ?[w1,w2] ≡ vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle w1 w2 wprime1 wprime2 vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle negationslash= 0, z ∈ G1. a79 ?[tildewidew1, tildewidew2] ≡ vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle tildewidew1 tildewidew2 tildewidewprime1 tildewidewprime2 vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = g(z), g(z)a133G2 a201a76a86a75a80a41a133z ∈ G1 intersectiontextG2 a137a85 w1 ≡ tildewidew1, w2 ≡ tildewidew2, a84g(z) negationslash= 0, z ∈ G 1 intersectiontextG 2a75a82 a225a215a216 a76a86a12a74a72a211a141a87a85a238 a190 a250 g(z) negationslash= 0, z ∈ G2. a154a155a85 tildewidew1a67 tildewidew2(a133G2 a201) a82 a20a87 a165 a11a75square a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17 a187 a19 §6.3 a60a61a91a92a30a182a183a184a185a62a186 a34a93a94a95 a134 a87a72a144 a134 a75 a70a71a72a144 a134 a217a96a63a64a74a77a76a72a144 a134 a75 a143a97 a217a96a77a76a72a95 a134 a69 a44 a87a144 a134 a85 a98 a217a96a77a76a72a99 a134 a75 a169a69a133a70a71 a228 a135a144 a134 a219a220a201a239a76a72a67 a216 a85a100a37 a143a189a190a191a192a193a194a101 a141a142a82a197 a153 a198a1996.2 a131a132z0a77a70a71 d2w dz2 + p(z) dw dz + q(z)w = 0 a72a144 a134 a85a135a133p(z)a67q(z) a161 a76a86a72a102a232a71a2200 < |z ?z0| < Ra201a85a70a71a72a152a142a20a87 a165 a11a76a77 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, a159a137ρ1, ρ2a67g a161 a77a136a74a75 star a131a132ρ1 a69ρ2a77 a248 a74a85a213g = 0a85a135z0 a134 a69a70a71a72a76a72a95 a134 a69 a44 a87a144 a134 a75 star a131a132ρ1 a69ρ2 a143 a77 a248 a74a85a69 g negationslash= 0a85a135a70a71a72a76a69a103a203a12a74a85z0 a134 a69a159a99 a134 a75 a104 a117a129a105a106a107a108a109a110 a101a91a127a111a112a96a97 a85a113a114a115a116a117 a99a100a116a90a118a119a101a120a121a122a116 a85a123a124 a114 a92a125a126a116a90a101a127a128a129a130a127 a75a131a132a133 a98a101a89a90a91a134 a85 a105a135a125a136 a85 a137a138a114a115a139a140a141a142a143a144a145 a142a143 a85a146a147a148 a88a120a121a122a116a149a150a151a114a152a153 a75 a131a132a223a74a76a137a34a140a140a149a142a154a38a155a85a214a64a156a217a155a157 a248 a66 a48 a72ρa203a85a158a250a223a74a76a137a159a140a154a38a155a85 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. a41a77a85a6a7a8a243a9a10a11a73a238a217a155a239a250a73a74a72a56a57 a244a58 a233a75a47 a225 a85 a98 a167a168 a162 a82a240ρa203a75 a133a160 a126a127a101a91a161 a132 a91a92a186 a75a162g negationslash= 0 a98 a85w2(z) a101a126a127a144 w 1(z)a163a164(a165 a138a166a90a143) a85 a131a167a168 a95a169a125a91 a75a162g = 0a98a85w2(z)a101a129a130a127a134a163a165 a166a90a143 a85a170 a140a91a101a126a127a171 a164a75 a69a157a172a240a70a71a144 a134 a219a220a201a173a133 a228 a135a76a72a180a181a85 a143a174 a188 a18 w(z) = (z ?z0)ρ ∞summationdisplay k=0 ck(z ?z0)k, §6.3 a10a11a175a176a178a176a252a253a254a12a16 a188a19 a63a64a218p(z)a67q(z)a74a1330 < |z ?z0| < R a201a169Laurenta221a222 p(z) =(z ?z0)?m ∞summationdisplay k=0 ak(z ?z0)k, q(z) =(z ?z0)?n ∞summationdisplay k=0 bk(z ?z0)k. a80a41z = z0 a134 a77a70a71a72a95 a134 a87a144 a134 a85 a84m, n a167a69 a248 a74a85a213a138a139a140a141a142a69a154 a248 a74a75 a231w(z) a155a177p(z) a67q(z)a72a223a74 a244a58 a233a234a235a70a71a85a212a178a179 a174 a85 (z ?z0)ρ?2a85a238a250a251 ∞summationdisplay k=0 ck(k + ρ)(k + ρ?1)(z ?z0)k +(z ?z0)1?m ∞summationdisplay l=0 al(z ?z0)l ∞summationdisplay k=0 ck(k + ρ)(z ?z0)k +(z ?z0)2?n ∞summationdisplay l=0 bl(z ?z0)l ∞summationdisplay k=0 ck(z ?z0)k = 0. a180a181a182 a109 a125 a85a183a184a109 a127 a170a185a186a187 a142a101a116a90 a85a129a105a117 a125a126 ρ a188a189a190 a116a90 c k a101a127a128a129a130 a127 a85a106a107 a120a125a126a122a141 a182 a91w(z) a75 a191a169a103 a85a132 a122a192a125a99 a170 a140a141 a182 a91 a85 a193a194 a162a116 a103a195a196a125a126 a170 a140ρ a188a85a197ρa195a196a103a198a187a96a97a101a91a75 a72a73a199a200a141 a195 a26 a196 a250a251a72a201a233a75a133a201a233a202a203a204a140 a65 a155a85a205a73a72a59a206a37a38a155a205a84a69 c0ρ(ρ?1)(z ?z0)0, c0a0ρ(z ?z0)1?m, c0b0(z ?z0)2?n. a174a175 a85a69a157 a162 a96a239a250a152a142ρa203a85a214a142a201a233a202a203a72a59a206a37a38a141a82a770a37a38a85a179 1?m ≥ 0, 2?n ≥ 0. a173a207a208 a241a85z0 a48a49a77 p(z)a72 a143 a146a39a141a209a72a95 a134 a85 a179 (z ?z0)p(z)a133z0 a134 a76a86a19 q(z)a72 a143 a146a39a210a209a72a95 a134 a85 a179 (z ?z0)2q(z)a133z0 a134 a76a86a75 a214a211a144 a134 a68a69a70a71a72 a91a92a30a182 a85a212a135a85a68a69a213 a91a92a30a182 a75 a214a255 a27 a40a85a133a70a71a72 a228 a135a144 a134 a72a219a220a201a85a152a142a76a217a96 a161 a77 a228 a135a76a75a214a215a26 a196 a72a205a86 a98 a143 a78a79(a216a93a94a35a45a74a155a72 a228 a135a76a72a217a232)a85 a97 a77a85a214a142a218a94a219a77 a228a220 a72a75 a221 a195a196 a72a82a197 ( a143a190 )a75 a198a1996.3 a70a71 d2w dz2 + p(z) dw dz + q(z)w = 0, a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17 a189 a19 a133a205a72a144 a134 z0a72a219a2200 < |z ?z0| < Ra140a152a142 a228 a135a76 w1(z) =(z ?z0)ρ1 ∞summationdisplay k=0 ck(z ?z0)k, c0 negationslash= 0, (6.5) w2(z) =gw1(z)ln(z ?z0) (6.6) + (z ?z0)ρ2 ∞summationdisplay k=0 dk(z ?z0)k, ga69d0 negationslash= 0 (6.7) a72a222 a162 a180a181a77z0a69a70a71a72 a228 a135a144 a134 a75 ρ1a67ρ2a68a69 a228 a135a76a72a223 a63 a75 a224 a60a61a91a92a30a182a183a184a185a225a186a226a227 a153 ? a231 a228 a135a76w1(z)a69w2(z)a234a235a70a71 ? a25a39a236a237a73a74a85a239a240a46a228a67a9a10a11a73 ? a229a43a239a240a73a74a72a56a57 a244a58 a233 a51a230a72a239a76a39a71a85a156a77 a188a231w 1(z)a232a233a72a76a234a235a70a71a85 star a131a132a96a231a63a64a239a250a152a142a20a87 a165 a11a76a85a47 a225 a209a232a233a234a78 a15 a85a159a140a167 a162 a100 a231w 2(z)a232a233a72 a76a234a235a70a71a75 star a131a132a214a64a34a96a239a250a141a142a76(a52a131ρ1 = ρ2 a64)a85 a235a236 a85a238 a98 a167a168a100 a231w 2(z)a232a233a72a76(a214 a64a72ga141a82 a143 a690)a234a235a70a71a239a76a75 a215a216 a136a204a205a70a71a72a56a57a197a94a85a45a41a141a142a210a209a20a87a136a204a205a70a71 d2w dz2 + p(z) dw dz + q(z)w = 0, a131a132a237a238a239a240a157a141a142a76w1(z)a85 a235a236 a85a156a217a155a25a39a239a205 w2(z) = Aw1(z) integraldisplay z braceleftbigg 1 [w1(z)]2 exp bracketleftbigg ? integraldisplay z p(ζ)dζ bracketrightbiggbracerightbigg dz a40a239a240a240a210a76a75a214a77 a174 a69a214a152a142a76 a161a86a87 a70a71 d2w1 dz2 + p(z) dw1 dz + q(z)w1 =0, d2w2 dz2 + p(z) dw2 dz + q(z)w2 =0. a243w2(z)a67w1(z)a205a84a241a214a152a142a70a71a85a100a66a242a85a233a217a250a251 w1d 2w2 dz2 ?w2 d2w1 dz2 + p(z) parenleftbigg w1dw2dz ?w2dw1dz parenrightbigg = 0, a179 d dz parenleftbigg w1dw2dz ?w2dw1dz parenrightbigg + p(z) parenleftbigg w1dw2dz ?w2dw1dz parenrightbigg = 0. §6.3 a10a11a175a176a178a176a252a253a254a12a16 a1810a19 a239a205a85a217a250 w1dw2dz ?w2dw1dz = Aexp bracketleftbigg ? integraldisplay z p(ζ)dζ bracketrightbigg . a152a203a156a155w21 a85a243a217a155a250a251 d dz parenleftbiggw 2 w1 parenrightbigg = Aw2 1 exp bracketleftbigg ? integraldisplay z p(ζ)dζ bracketrightbigg . (6.8) a100a239a205a141a37a85a238a250a251a26 a196 a72a218a132a75 a1456.4 a224a225 a85z = 0a67z = 1 a161 a77a146a147a148a70a71 z(1?z)d 2w dz2 + [γ ?(1 + α + β)z] dw dz ?αβw = 0 a72 a228 a135a144 a134 a19 x = ±1a74 a161 a77Legendrea70a71 parenleftbig1?x2parenrightbig d2y dx2 ?2x dy dx + l(l + 1)y = 0 a72 a228 a135a144 a134 a75 a69a157 a163a164a165a166 a150 a134 a77a212a69 a228 a135a144 a134 a85a63a255 a162 a169 a171a173z = 1/t a85a131a132t = 0a77 a171a173 a60a72a70a71a72 a228 a135a144 a134 a85a179t = 0 a134 a77 a171a173 a60a72a70a71a72a144 a134 a85a213 t bracketleftbigg2 t ? 1 t2p parenleftbigg1 t parenrightbiggbracketrightbigg = 2? 1tp parenleftbigg1 t parenrightbigg a67 t2 · 1t4q parenleftbigg1 t parenrightbigg = 1t2q parenleftbigg1 t parenrightbigg a133t = 0 a134 a76a86a85 a89 a179z = ∞ a134 a77 a171a173a244 a70a71a72a144 a134 a85a213zp(z)a67z2q(z)a133z = ∞ a134 a76a86a85a135a68 z = ∞ a134 a77 a171a173a244 a72a70a71a72 a228 a135a144 a134 a75a154a155a85 a165a166 a150 a134 z = ∞a74 a161 a77a146a147a148a70a71a67Legendrea70a71 a72 a228 a135a144 a134 a75 a1456.5 a239Legendrea70a71 parenleftbig1?x2parenrightbig d2y dx2 ?2x dy dx + l(l + 1)y = 0 a133x = 1a219a220a201a72a140a245a76a75 a186 a174x = 1 a77Legendrea70a71a72 a228 a135a144 a134 a85 a84a48 a79 y(x) = (x?1)ρ ∞summationdisplay n=0 cn(x?1)n. a234a235a70a71a85a238a140 ∞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. a80 a175 a217a155a250a251a46a228a70a71 ρ(ρ?1) + ρ = 0 a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17 a1811 a19 a67a9a10a11a73 cn = ?n(n?1)?l(l + 1)2n2 cn?1. a46a228a70a71a72a76a77 ρ1 = ρ2 = 0. a133 a125a246Legendrea96a97 a117x = 1a108a109a110a111a101a247a105a91a248a249a109 a103 a117a250 a110|x?1| < 2a111a91a118a101 a85 a162a116a117x = 1 a108a138a251 a19a167 a247a198a91 a182 a105a107 a165 a138a166a90a143 a85a189x = 1(a144x = ?1)a132a252 a108 a85a131a167 a117x = 1(a144x = ?1)a108a253a254a75a255a0a168 a125a247a105a91 a75 a80a9a10a11a73a85a217a155a239a240Legendrea70a71a133x = 1 a134 a219a220a201a240a141a76a72a73a74a72a25a155a1a233 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. a18c0 = 1a85a238a239a240a157Legendrea70a71a72a240a141a76 Pl(x) = ∞summationdisplay n=0 1 (n!)2 Γ(l + n + 1) Γ(l?n + 1) parenleftbiggx?1 2 parenrightbiggn , a68a69la37a240a141a2 Legendrea12a74a75 a131a132 a162a3a4 a239a240a210a76a85a135 a48 a79 y2(x) = gPl(x)ln(x?1)+ ∞summationdisplay n=0 dn(x?1)n = 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. a5a6 a136a204a205a70a71a223a74a76a70a72a228a7a54a55a85a82a240a73a74g(a141a82 a143 a690)a67dna179a217a75 a8 a51a243a72a9a70a77 a215a216 a240a210a76 a227 a240a141a76a0a1a72a11a73a85 a14 a240 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, §6.3 a10a11 a175a176a12a13a14a15a16a17a18 a1912 a20 a21a22a23a24a25a26a27a28a29a30|x?1| < 2 a31a32a33a34a35a36a37a38a39 a27a28 a32a40a41 y2(x) = g2Pl(x)ln x + 1x?1 + ∞summationdisplay n=0 dn(x?1)n. a42g = 1 a34a43a44a45dn a34a46a47a48a37a38a49a45 Legendrea50a51a52 a27a28 a32 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 , a53 a41la54 a27a28a55Legendre a56a57a34 a58a59γ a60Eulera57a34ψ(z)a60Γa56a57a52a61a57a62a63a64 a65a66 a56a57Pl(x)(a67 a68a69a70a71a72 a47a34a73a60a38 x = ?1 a74 x = ∞a41a75a76a52a77a78a56a57) a74 Ql(x) a52a77a78a79a80a81a82a44a79a52a83 a44a34a84a85a86a87a88a89a90a91a92a64 a93a72a94a95a96a93 a49a97a62a98a50a51 d2w dz2 + p(z) dw dz + q(z)w = 0 a30a99a100a101 a76a102a103a31a52a32a52 a96a104a105a106 a34a107a86a108a109a110 a30a111a112a113a114a93 a34a50a51a52 a27a28 a32a115a116a61a57 a29a117a30a111 a112a113a114a93 a34a50a51a52 a27a28 a32a37a118a116a61a57 a29a117a30a111a112a113a114a93 a34a50a51a52 a27a28 a32 a96 a44a116a61a57 a29 a64 a119 a120 a121 a83a44a50a51 a30a99a100a101 a76a122a52a123a124a125a126 Reρ1 ≥ Reρ2 a34 a100 a127ρ 1 ?ρ2 negationslash=a128a57a86a34 a27a28 a32 a96 a44a115a116a61a57 a29a117 a127ρ 1 = ρ2a86a34 a27a28 a32 a96 a44a116a61a57 a29a117 a127ρ 1 ?ρ2 = a99 a128a57a86a34 a27a28 a32a37a118a116a61a57 a29 a64 a41a129a130a131a132a133a34a115a134a135a40 z = 0a76a60a73a52 a99a100a101 a76a64 a66 a60a34 a30 z = 0 a76a52a102a103a31a34a37a39a50a51a52 a136 a57a137Laurenta138a139 p(z) = ∞summationdisplay l=0 alzl?1, q(z) = ∞summationdisplay l=0 blzl?2. a40a32a41 w(z) = zρ ∞summationdisplay k=0 ckzk. a140a141 a50a51a34a48a81 ∞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, a142a143a144 a145a146a147a148a149a150a151a152 a11 a17a153a154a155a18a156 a1913 a20 a157 ∞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. a158a159a160a161 a123 a26 a46a162a54a163a34 a157 z0 a52 a136 a57a34a37a164 c0 [ρ(ρ?1)+ a0ρ+ b0] = 0. a65a66c 0 negationslash= 0a34a165a38 ρ(ρ?1)+ a0ρ+ b0 = 0. a166 a48a60a125a126a50a51a34a91a92 a58a59 a52 a0 a74b0 a41 a0 = limz→0zp(z), b0 = limz→0z2q(z). a167a168 a125a126a50a51a37a38a49a45a123a124a125a126a34 ρ1 a74ρ2 a64 a169a170 Reρ1 ≥ Reρ2 a64 a171a158a159zn a52 a136 a57a34a164 (n + ρ)(n + ρ?1)cn + nsummationdisplay l=0 bracketleftbiga l(n + ρ?l)+ bl bracketrightbigc n?l = 0, a157 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. a166a172a173 a164a45a129 a136 a57a174a175a52a176a177a178 a136 a64 a179a180a181a182a183a184a185a186 a34a187a188a189a190a191 a186a192 c n a193a194a195a196a197a198 a64a199a200a34a201 cn a193a196a197a198a202a203 a170 a204a205ρ a64 a182ρ = ρ 1 a206a207 a34a208a188a190a191a209 w1(z)a64a210 a182ρ = ρ 2 a206a207 a34a211a188a190a191a209 w2(z)a64a199 ρ1 ?ρ2 negationslash= a212 a192a213 a34a187a214a215a216 a217a218 a193 (a219a220a221a222a223 a185 a193 ) a224a209a64 a127ρ 1 = ρ2 a86a34a225a226 a166a172a227 a118a164 a69 a107 a96 a124a32a64a165a38a34 a166 a86 a27a28 a32 a96 a44a116a61a57 a29 a64 a127ρ 1 ?ρ2 = a99 a128a57ma86a34a61 a66a27a28 a32a52 a136 a57 c(2)m a34a81 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. §6.3 a152a228a229a230a12a13a14a15a16a17a18 a1914a20 a91a92m + ρ2 = ρ1 a34a165a38a81 0·c(2)m + msummationdisplay l=1 bracketleftBig al(ρ1 ?l)+ bl bracketrightBig c(2)m?l = 0. a35a36 a127 msummationdisplay l=1 bracketleftBig al(ρ1 ?l)+ bl bracketrightBig c(2)m?l negationslash= 0a86a34 c(2)m a231 a32 a117 a127 msummationdisplay l=1 bracketleftBig al(ρ1 ?l)+ bl bracketrightBig c(2)m?l = 0a86a34 c(2)m a232 a92a64 star a61 a66a27a96a233a113a234 a34a50a51a52 a27a28 a32a235 a96 a44a116a61a57 a29 a64 star a61 a66a27a28a233a113a234 a34a50a51a52 a27a28 a32 a96 a44a115a116a61a57 a29 a34 a127 a226a236a118a237a238a49a32a64 a227 a60 a166 a86a38a47a52a239 a29 a136 a57c(2)n (n > m)a240a107a86a241a242 a66 c 0(2)a74c (2) m a64 a27a28 a32w2(z)a173a81a123 a29 a34 a96a29a99a158a66 c(2) 0 a34 a96 a29a99a158a66c(2) m a64 a171a243a244 a98a33 a96a93 a34a48a240a245a246a34c(2)m+n a74c(2)m a174a175a52a178 a136a247c(1) n a74c (1) 0 a174a175a52 a178 a136a248a70a96a172 a34a35a36a34 a247c(2) m a249 a99a158 a52 a29a99a250 a48a60 a27a96 a32 (a46a77a37a118a251 a96 a124a97a57a252a57)a34a35a253 a115a134 a42c(2) m = 0a64 a142a143a144 a145a146a147a148a149a150a151a152a228a17a153a154a155a18a156 a1915 a20 §6.4 Bessel a254a255a0a1 Bessela50a51 d2w dz2 + 1 z dw dz + parenleftbigg 1? ν 2 z2 parenrightbigg w = 0 a60a97a133a52a97a62a98a50a51a174 a96 a34 a58a59ν a60a97a57a34Reν ≥ 0a64 a21a22a23a24 a34z = 0a60a50a51a52 a99a100a101 a76a34z = ∞ a60a50a51a52a2 a99a100a101 a76a64 a3a4a5a6Bessel a50a51 a30z = 0 a76a52a102a103|z| > 0 a31a52a32a64a40 w(z) = zρ ∞summationdisplay k=0 ckzk, c0 negationslash= 0, a140a141Bessel a50a51a34a164 ∞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, a82a7zρ?2 a34 a157 a164 ∞summationdisplay k=0 ck bracketleftbig(k + ρ)2 ?ν2bracketrightbigzk + ∞summationdisplay k=0 ckzk+2 = 0. a167a168a8 a57a138a139a52a9 a96 a79a34 a157 a37 a158a159a136 a57a64 a65 a46a162a54a163 z 0 a29 a52 a136 a57a34a10a35a41 c0 negationslash= 0a34a48a164 a69a11a12a13a14 a34 ρ2 ?ν2 = 0. a35a253a49a164 ρ1 = ν, ρ2 = ?ν. a35a41Reν ≥ 0a34a165a38Reρ1 ≥ Reρ2 a64 a65z1 a52 a136 a57a34a164 c1 bracketleftbig(ρ+ 1)2 ?ν2bracketrightbig = 0 a157 c1(2ρ+ 1) = 0. a35a36 c1 = 0, a127ρ negationslash= ?1/2; (6.9a) c1 a232 a92, a127ρ = ?1/2. (6.9b) a38a47a39a15 a69 a34 a157 a84 ρ = ?1/2a34a16a37a38 a42c 1 = 0a64 a65zn a52 a136 a57a34a164 cnbracketleftbig(ρ+ n)2 ?ν2bracketrightbig+ cn?2 = 0 a157 cnn(2ρ+ n)+ cn?2 = 0, a35a36a34a164 a69a17a18a19a20 cn = ? 1n(n + 2ρ)cn?2. a21a22a23 a85a176a177a178 a136 a34a48a37a38a49a164 c2n = ? 1n(n + ρ) 122c2n?2 §6.4 Bessela152a228a17a18 a1916a20 = (?)2 1n(n?1)(n + ρ)(n + ρ?1) 124c2n?4 = ··· = (?) n n! 1 (ρ + 1)n 1 22nc0, (6.10) 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. (6.11) a85ρ1 = ν a140a141a34 a157 a164 w1(z) = c0zν ∞summationdisplay k=0 (?)k k!(ν + 1)k parenleftBigz 2 parenrightBig2k . a42c 0 = 1 2νΓ(ν + 1) a34a48a81a32 Jν(z) = ∞summationdisplay k=0 (?)k k!Γ(k + ν + 1) parenleftBigz 2 parenrightBig2k+ν . (6.12) a85ρ2 = ?ν a140a141a34a81 w2(z) = c0z?ν ∞summationdisplay k=0 (?)k k!(?ν + 1)k parenleftBigz 2 parenrightBig2k , a127ν negationslash=(a99) a128a57a86a34a235a37 a42 c 0 = 2ν/Γ(?ν + 1)a34a24a164 J?ν(z) = ∞summationdisplay k=0 (?)k k!Γ(k ?ν + 1) parenleftBigz 2 parenrightBig2k?ν . (6.13) a246 a30a25a26a5a6a96a93 ρ = ?1/2 a52 a113a234 a64a27 a72a28a29a30a69 a34 a166 a86a16a226a37a38 a42 c 1 = 0 a64a35a41a31a32 c1 negationslash= 0a34 a100 c2n+1 = (?) n (3/2)n(1)n 1 22nc1, a91a92 a69(1) n = n!a34 a166a172a30w 2(z) a59a227 a115a33a60 a171a34a35a96a29 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). a157a30w 2(z) a59a227 a115a33a60 a171a36a35a37a27a96 a32a64 a38a21a22 a38a41a246 a30 a80 a29a248 a249 a129a49a32 Bessela50a51a52 a232a39 a34a35a41 a37a72 a52a40a49a45a129a123a124a125a126a34a43a10a61 a41a66a42a96 a124a125a126a34a235a43a49a45a129a44 a41 a52a32a64 a127ν negationslash= a128a57a86a52a40a31a36a34a35a41 a166 a86a49a45a52a123a124a32Jν(x)a74 a142a143a144 a145a146a147a148a149a150a151a152a228a17a153a154a155a18a156 a1917 a20 J?ν(x)a45a79 a231 a178a64a46a60a34 a96 a124a47a225a52a48a49a60a110 a127 ν = 0 a86a34 a37a72 a52a49a32a33a51 a227 a60a50a45a129a107 a96 a124a32 J0(x) = ∞summationdisplay k=0 (?)k k!k! parenleftBigx 2 parenrightBig2k . a166a51 a47a34 a166 a86a52 a27a28 a32 a41a52 a116a81a61a57 a29 a34 a157 y2(x) = gJ0(x)lnx+ ∞summationdisplay k=0 dkxk, g negationslash= 0. a115a53a31a36a34 a127 ν = n, n = 1,2,3,··· a86a34a27 a72 a16a226 a227 a60a49a45a129 a96 a124a32a34a54 a96 a15a55a34a52a40a56a57a81 a76a58a32a64 a166 a60a35a41a34a27 a72a59a60 a52a40a49a45a129a123a124a115a107a52a125a126a78a34a253a10a34a235a52a40a49a45a129a123a124 a234a161 a15a55 a43a115a44a107a52a32a64a46a60a34a48a49a43a2a31a36a64 star a61 a203 a34a199ν = n, n = 1,2,3,···a213a34a62 a192 a209 J?n(x) = ∞summationdisplay k=0 (?)k k!Γ(k?n + 1) parenleftBigx 2 parenrightBig2k?n a202a63 k = 0,1,···,n?1a64a65 a193 a186a192a66a67 0 a34a68a69a70 a67z = 0,?1,?2,···a71a69Γa72 a192 a193a203a73a74a75 a64a76a189 J?n(x) = ∞summationdisplay k=n (?)k k!Γ(k?n + 1) parenleftBigx 2 parenrightBig2k?n . a77k?n = l a34a187 a205 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), a78 a61 a203 a209Jn(x)a221a222a79 a185 a64 star a61a80a34a201a62 a192 a209a81 a202 a34 a82a83a84 a69a85a200a86a87 a170 a62 a192 a209 a193a88 a65 a186a192a89a670 a34 a90a91 a201a92a201a223a93 a202a94 a179 a216a68a220a87 a170 a110a201a95a215J?ν(x) (ν = 1,2,3,···)a213a34 a96c 0 = 2ν/Γ(1?ν)a34 a97a97 a169a170 a216c0 = 0a64 star a61a98a34a99a100a199a200a188a189a101a102a103a68a220 a89a104a105 a193 a169a170 a34a106a190 y2(x) a193 a62 a192 a202a63a107 k = 0,1,···,n?1 a108 a65 a193 a186a192a89a67 0 a34 a90 a68a109a110a95a111a112 k = na65a113a114 a186a192a66a115a67 a223a116a64a68a117a188a189a112 a183a184a185a186 c2k = ? 1k(k?ν) 122c2k?2 a118 a215a64a199ν = na213a34a119a200c2n a223a93a120a34a70a121a189a122a64a65 a186a192 a117a71a123a124a93a120a64 a38 a37 a52a98a33a125a47a34 a127 ν = n, n = 1,2,3,··· a86a34Bessela50a51 d2y dx2 + 1 x dy dx + parenleftbigg 1? n 2 x2 parenrightbigg y = 0 §6.4 Bessela152a228a17a18 a1918a20 a52 a27a28 a32a235 a96 a44a116a81a61a57 a29 a34 a157 y2(x) = gJn(x)lnx+ ∞summationdisplay k=0 dkxk?n, g negationslash= 0. a126a100a37 a39y2(x)a140a141Bessela50a51a34 a157 a37a44a45 a136 a57a64 a93a72a127a128 a49 Bessela50a51 a27a28 a32a52a129 a96a233 a50a130a64a41a36a34a131a132a133 Jν(z)a74 J?ν(z) a52Wronskia134a135 a161 a38a98a33a73 a60 a52a45a79a44a178a79a64a136a137 a69 Bessel a50a51a52 a136 a57p(z) = 1/z a34a138(6.8)a161a48a37a38a164 a69 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 . a41a129a44a45a139a98a97a57 Aa34 a227 a87a39Jν(z)a74J?ν(z)a52 a8 a57a32(6.12)a74(6.13)a140a141a34a140a45 W [Jν(z), J?ν(z)] ≡ Jν(z)Jprime?ν(z)?J?ν(z)Jprimeν(z) a59z?1 a29 a52 a136 a57 a157 a37a64 a166a227 a55a141a239 a8 a57 a59 a52 a27a96a29 a64a35a36a34 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. a166a172 a48a164 a69 W [Jν(z), J?ν(z)] = ? 2piz sinpiν. (6.14) a37a72 a52a132a133 a59 a85 a69 a129 Γa56a57a52a79a142 Γ(ν)Γ(1?ν) = pisinpiν. (6.14)a161a171a54a51a47a34a127ν = n, n = 0,1,2,···a86Jν(z)a74J?ν(z)a45a79a44a178a64a46a60a31a32a39Bessela50 a51a52 a27a28 a32 a42 a41 Jν(z)a74J?ν(z)a52a45a79a143a144a34 w2(z) = c1Jν(z)+ c2J?ν(z), a227 a88a145a146a147 a127 a52a143a144 a136 a57a34a84a164W [Jν(z), w2(z)]a61 a232a148 ν a149a115a410a34 a166a172 a52w2(z)a48 a96 a44(a61 a232a148 ν a149)a247 Jν(z)a45a79 a231 a178a64a41a36a34 a59a60 a48 a42a27a28 a32a41 w2(z) = cJν(z)?J?ν(z)sinpiν , a166a172a173 a81 W [Jν(z), w2(z)] = 2piz. a41a129a150a151 a166a172 a44a152a52 w2(z) a81a92a152 (sinnpi = 0 a34a98a153a41 0) a34a43a91a92 a69 J ?n(z) = (?)nJn(z)a34 a59a60 a173a41a127a154a96a105 a145 a42a136 a57 ca34a84a164w2(z) a59a52a98a155 a30 ν = n a86a235a410a34a156a31 a42c = cospiν a157 a37a64 a166a172 a164 a69Bessel a50a51a52 a27a28 a32 a173 a60 Nν(z) = cospiνJν(z)?J?ν(z)sinpiν , (6.15) a53 a41ν a157Neumanna56a57a64 a127ν = n, n = 0,1,2,··· a86a34(6.15)a161a41a115a44 a161 a34a37a158l’hospitala130 a100 a49a159 a160 a34 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 a142a143a144 a145a146a147a148a149a150a151a152a228a17a153a154a155a18a156 a1919 a20 = 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, (6.16) a58a59ψ(ζ) a60Γa56a57a52a61a57a62a63a34 ψ(ζ) ≡ dlnΓ(ζ)dζ = Γ prime(ζ) Γ(ζ). a43a10a82a44a34 a127 n = 0 a86 a41a127 a7a161(6.16)a161a25a26a27a28a29a52a81 a160 a74a64 ?§6.5 ν = n, n = 0,1,2,3, · · · a162a163Bessela152a228a17a142a145a18 a1920a20 ?§6.5 ν = n, n = 0,1,2,3,··· a164a165 Bessel a254a255a0a166a167a1 a127ν = n, n = 0,1,2,3,··· a86a34Bessela50a51 d2y dx2 + 1 x dy dx + parenleftbigg 1? n 2 x2 parenrightbigg y = 0 a52 a27a28 a32 a96 a44a116a81a61a57 a29 a34 a157 y2(x) = gJn(x)lnx+ ∞summationdisplay k=0 dkxk?n, g negationslash= 0. a127ν = 0 a86a34 y2(x) = gJ0(x)lnx+ ∞summationdisplay k=0 dkxk, g negationslash= 0. a49a62a63a34a164 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)· 1x2 + ∞summationdisplay k=0 dkk(k?1)xk?2. a140a141a168 a157Bessela50a51 d2y dx2 + 1 x dy dx + y = 0, a157 a164 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. a91a92J0(x)a235a60 a168 a157Bessela50a51a52a32a34 a37a161a27a96 a134 a59lnx a27a50a169a170a31a171 a29 a174a74 a41 a41 0a34a165a38 g ∞summationdisplay k=0 (?)k k!k! k 22k?2x 2k + ∞summationdisplay k=0 dkk2xk + ∞summationdisplay k=0 dkxk+2 = 0. a166a172 a80 a29a140a141 a129 J0(x)a52 a8 a57 a51a173a161 a64 a246 a30a158a159 a239 a29 a52 a136 a57a64a61 a66 x0 a29 a34a81g ·0 + d0 ·0 = 0a34a165a38 g a232 a92, d0 a232 a92. a171a158a159x1 a29 a52 a136 a57a34a164 d1 = 0. a65x2 a29 a52 a136 a57a34a37a38a49a164?g + 4d2 + d0 = 0a34a165a38 d2 = ?14d0 + 14g. a142a143a144 a145a146a147a148a149a150a151a152a228a17a153a154a155a18a156 a1921 a20 a31a36a237a238a34a48a37a38a98a90a164 a69a174 a54a163 a29 a74 a101 a54a163 a29 a52 a136 a57a64 a158a159 x2k a29 a52 a136 a57a34a164 g · (?) k k!k! 2k 22k?1 + d2k(2k) 2 + d2k?2 = 0, a66 a60a164 a69 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 . a158a159x2k+1 a29 a52 a136 a57a34a24a164 a69 (2k + 1)2d2k+1 + d2k?1 = 0, a65a66d 1 = 0a34a35a36a34 d2k+1 = 0. a46a47a34a48a49a45a129 ν = 0a86a52 a27a28 a32 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 . a175 a134a52a176a130a60 a42 g = 2pi, d0 = ?2pibracketleftbigln2 +ψ(1)bracketrightbig, a58a59 a52ψa56a57a48a60Γa56a57a52a61a57a62a63a64 a65Γ a56a57a52a79a142 Γ(z + 1) = zΓ(z)a34 a21a22 a151a47 ψ(z + n) =ψ(z)+ 1z + 1z + 1 +···+ 1z + n?1, a166a172 a164 a69 a52a32 (a177a41N0(x))a48a37a38a178 a249 N0(x) = 2piJ0(x)ln x2 ? 2pi ∞summationdisplay k=0 (?)k k!k!ψ(k + 1) parenleftBigx 2 parenrightBig2k . a171a5a6n = 1,2,3,··· a52 a113a234 a64a39y2(x)a140a141na157Bessela50a51a34 a157 a164 g bracketleftbiggd2J n(x) dx2 + 1 x dJn(x) dx + parenleftbigg 1? n 2 x2 parenrightbigg Jn(x) bracketrightbigg lnx ?§6.5 ν = n, n = 0,1,2,3, · · · a162a163Bessela152a228a17a142a145a18 a1922a20 + 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. a91a92Jn(x)a235a60na157Bessela50a51a52a32a34 a37a161a27a96 a134 a59lnx a27a50a169a170a31a171 a29 a174a74 a41 a41 0a34a165a38 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, a179a180 a178 a249 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. a246 a30 a55 a158a159a160a161 a123 a26 a239 a29 a52 a136 a57a64 a65x0 a29 a52 a136 a57a34a164 d0 ·0 = 0a34a165a38 d0 a232 a92. a65x1 a29 a52 a136 a57a34 d1(1?2n) = 0a34a165a38 d1 = 0. a65x2k+1 a29 a52 a136 a57a34a81 d2k+1(2k + 1)(2k?2n + 1)+ d2k?1 = 0, a165a38 d2k+1 = ? 1(2k + 1)(2k?2n + 1)d2k?1 = ··· = 0. a61 a66x2k a52 a136 a57a34a87a88a181a90 k < n, k = n, k > na182 a233a113a234 a64 star a127k < na86a34 d2k2k(2k?2n) + d2k?2 = 0. a165a38 d2k = 1k(n?k) 122d2k?2 = 1k(k ?1)(n?k)(n?k + 1) 124d2k?4 a142a143a144 a145a146a147a148a149a150a151a152a228a17a153a154a155a18a156 a1923 a20 = ··· = (n?k?1)!k!(n?1)! 122kd0. a89a90a60 d2n?2 = 1[(n?1)!]2 122(n?1)d0. star a127k = na86a34 1 2n?1(n?1)!g + d2n ·0 + d2n?2 = 0. a165a38 d2n a232 a92, g = ?2n?1(n?1)!d2n?2 = ? 12n?1(n?1)!d0. star a127k > na86a34 (?)k?n (k ?n)!k! 2k?n 22k?n?1g + d2k2k(2k?2n)+ d2k?2 = 0. a165a38 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 + 1k?n + 1k?n?1 +···+ 11 parenrightBig 1 22k?n+1g. a35a36 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 ?§6.5 ν = n, n = 0,1,2,3, · · · a162a163Bessela152a228a17a142a145a18 a1924a20 + 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 + 1k?n + 1k ?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 + 1k + 1k?1 +···+ 1 parenrightBigparenleftBigx 2 parenrightBig2k+n . a183 a97 a42 g = 2pi, d2n = ? 12npin!bracketleftbig2ln2 +ψ(n + 1)+ψ(1)bracketrightbig. a166a172a173 a81 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)bracketrightbig parenleftBigx 2 parenrightBig2k+n , n = 1,2,3,···. a158a159a96a93 a27 a72 a164 a69 a52N0(x)a52 a51a173a161 a34a37a38a15a45a34 a166a172 a52 Nn(x)a235a147a85 a66n = 0 a34 a227 a88a184a32a41 a166 a86 a185 a7a161 a27a28a29 a52a81 a160 a74a64 a142a143a144 a145a146a147a148a149a150a151a152a228a17a153a154a155a18a156 a1925 a20 ?§6.6 a254a255a186a187a188a189a190a191a192a0a1 a27 a72a5a6 a129 a30 a50a51 a99a100a101 a76a102a103a31a52a32a64 a59a60a193a194 a34a50a51 d2w dz2 + p(z) dw dz + q(z)w = 0 a30a99a100a101 a76a52a102a103a31a52a123a124a32a43a60 a99a100 a32a64a31a32a50a51a52 a101 a76 z = z0 a60a2 a99a100a101 a76a34 a100a30a52 a76a52a102 a1030 < |z ?z0| < R a31a46a77 a227 a118a81 a96 a124 a99a100 a32a64a39p(z)a74q(z)a300 < |z ?z0| < R a31a52Laurenta138a139 p(z) = (z ?z0)?m ∞summationdisplay k=0 ak(z ?z0)k, q(z) = (z ?z0)?n ∞summationdisplay k=0 bk(z ?z0)k a195a99a100 a32(6.5)a140a141a50a51a34 a100a127 a10a53 a127 m a74na196a197 m > 1,a43a10m ≥ n?1 a86a34a164 a69 a52a125a126a50a51a41 a96 a54a50a51a64 a166 a86a37a38a198a199 a99a100a101 a76a122a52a49a32 a105a106 a49a32a34a115 a171a200a201 a64 a31a32 a231a202a203 a76z = ∞a60a2 a99a100a101 a76a34 a126a100a37a41a127 a137a204a205 z = 1/ta34a226a47 a30t = 0 a76a52a102a103a31 a5 a6a157 a37a64a46a60a34a235a37a38a206a207a208 w(z) = zρ ∞summationdisplay k=0 ckz?k a107a86a39p(z)a74q(z)a30z = ∞a76a52a102a103a31a137 Laurenta138a139 p(z) = zm ∞summationdisplay k=0 akz?k, q(z) = zn ∞summationdisplay k=0 bkz?k. a140a141 a62a98a50a51a34 a100 a164 a69 ∞summationdisplay k=0 ck(k?ρ)(k?ρ?1)z?k+z1+m ∞summationdisplay l=0 alz?l ∞summationdisplay k=0 ck(k?ρ)z?k+z2+n ∞summationdisplay l=0 blz?l ∞summationdisplay k=0 ckz?k=0. a165a38a34a50a51 a227 a81 a96 a124 a99a100 a32a52a209a210a60 m ≥ 0,a43a10m ≥ n + 1, (6.17) a157a30z = ∞ a76p(z)a52a157a211 a66q(z) a52a157a64a91a92a34a50a51 a227 a81 a96 a124 a99a100 a32a52a209a210a34a61 a66 a81 a160 a203 a122a74 a231a202 a203 a76a34 a30a234a161a37 a81a165a115a107a64 a246 a30a5a6 a129 a96a233a113a234 a34 a157 a50a51a2 a99a100a101 a76a102a103a31a115a212 a30a99a100 a32a34 a179 a50a51a81 a96 a124 a99a100 a32a34a46 a59 a60 a88a49 a166 a124 a99a100 a32a174a213a52a129 a96 a32a64 a166 a86a34 z = z0 a41a52a60a32a52 a3 a79 a101 a76 (a127a226a236a37a118a60a75a76) a64a41a129 a214 a178a32 a30z = z 0 a76a52 a166a233a101a215 a79a34a115a134a135a40 w(z) = eQ(z)v(z), (6.18) a84a164z = z0 a60eQ(z) a52 a3 a79 a101 a76a34a253 v(z)a37a38a178 a249 a99a100 a32a52 a234a161 v(z) = (z ?z0)ρ ∞summationdisplay k=0 ck(z ?z0)k, c0 negationslash= 0. a166a233a234a161 a52a32 a53 a41a97a83a32a64a216 a166a233 a32 a140a141 a50a51a34 a100 a164 v(z)a52a50a51 d2v dz2 + p ?(z)dv dz + q ?(z)v = 0. ?§6.6 a152a228a217a229a230a218a219a220a221a17a18 a1926 a20 a58a59 p?(z) = p(z)+ 2Qprime(z), q?(z) = q(z) + p(z)Qprime(z)+ Qprimeprime(z)+ [Qprime(z)]2 . (6.19) a171a167a168 a50a51a81 a99a100 a32a52a88a49a34a48a81a37a118a222 a127a223 a145a146 Q(z)a34a43 a154 a253a49a45a32 w(z)a64 a93a72a171 a55 a5a6 Bessel a50a51 d2w dz2 + 1 z dw dz + parenleftbigg 1? ν 2 z2 parenrightbigg w = 0. a21a22 a15a45a34 a30 a2 a99a100a101 a76z = ∞a122a34m = ?1, n = 0a34a115a196a197a50a51a81 a99a100 a32a52a88a49(6.17)a34a35a253 a30 z = ∞a52a102a103a31a34a50a51a46a77 a227 a118a81a97a83a32a64a35a36a137a204a205 (6.18)a34 a100 a50a51(6.19) a59a52 a136 a57a41 p?(z) = 1z + 2Qprime(z), (6.20) q?(z) = 1? ν 2 z2 + Qprime(z) z + Q primeprime(z) + [Qprime(z)]2 . (6.21) a41a129a150a151z = ∞a60eQ(z) a52 a3 a79 a101 a76a34a37 a42Q(z) a41za52a77 a29a161 a34 a166 a235a107a86a196a197a129p?(z)a52a157m ≥ 0 a52a88a49 a117a171a154a96a105 a34a41a129a84a164 p?(z)a52a157a211 a66q?(z) a52a157a34 a100a224a185a42 Q(z) = λz a43a10 1 + λ2 = 0, a157λ = ±i a64 a166a172 a34v(z)a165a196a197a52a50a51a48a60 d2v dz2 + parenleftbigg1 z + 2λ parenrightbigg dv dz + parenleftbiggλ z ? ν2 z2 parenrightbigg v = 0. a40 v(z) = zρ ∞summationdisplay l=0 clz?l, a100 a81 ∞summationdisplay l=0 bracketleftbig(ρ?l)2 ?ν2bracketrightbigc lz?l + λ ∞summationdisplay l=0 bracketleftbig(2ρ+ 1)?2lbracketrightbigc lz?l+1 = 0. (6.22) a158a159z1 a29 a52 a136 a57a34 a157 a164 2ρ+ 1 = 0 a157 ρ = ?1/2. a165a38a34a31a32 a227a42a69 a32a52a225 a29 a34 a100a30 a231a202a203 a76 z = ∞a226a227a34Bessela50a51a52a32 a173 a37a178 a249 w(z) = c0e±iz radicalbigg 1 z [1 +···]. a41a129a49a45a128a124a32a34a37a39 ρ = ?1/2a140a141(6.22)a161a34 a171a158a159z?k+1 a29 a52 a136 a57a34a48a164 a69 a176a177a178 a136 ck = ?4ν 2 ?(2k?1)2 22k 1 2λck?1. a21a22a23 a85a176a177a178 a136 a34a48a37a38a164 a69a136 a57 ck a52a228a229 a51a173a161 a110 ck = (ν, k)(?2λ)kc0, a58a59(ν, 0) = 1 a34 (ν, k) = bracketleftbig4ν2?(2k?1)2bracketrightbigbracketleftbig4ν2?(2k?3)2bracketrightbig···bracketleftbig4ν2?32bracketrightbigbracketleftbig4ν2?12bracketrightbig 22kk! . a142a143a144 a145a146a147a148a149a150a151a152a228a17a153a154a155a18a156 a1927 a20 a183 a97a208 c0 = radicalbigg2 piexp bracketleftBig ?λ parenleftBigνpi 2 + pi 4 parenrightBigbracketrightBig , a66 a60a48a37a38a49a164 Bessela50a51 a30 a231a202a203 a76 z = ∞a102a103a31a52a32a64 a46a60a34 a21a22a23a24 a34 a166 a124 a8 a57a32a52a230a231a232a233 R = lim k→∞ vextendsinglevextendsingle vextendsinglevextendsingleck?1 ck vextendsinglevextendsingle vextendsinglevextendsingle = 0. a166 a125a47a34 a30a96a104a113a114a93 a34 a166a172 a164 a69 a52 a8 a57a32a60a245a234a52a34 a235 a2 ν a60a232 a101 a57a34 a157 ν = n +1/2 a34 a166 a86a138 z?n?1 a29a132a34a239a29a52a136a57a149a41 0a34 a8 a57a236 a24 a41a77 a29a161 a34 w(z) = radicalbigg 2 piz exp bracketleftbigg λ parenleftbigg z ? n + 12 pi parenrightbiggbracketrightbigg nsummationdisplay k=0 (n + 1/2, k) (?2λz)k . a39λ = ±ia140a141a34a48a164 a69 Bessel a50a51a52a123a124a32 w1(z) = (?i)n+1 radicalbigg 2 pize iz nsummationdisplay k=0 (?)k(n + 1/2, k) (2iz)k , (6.23) w2(z) = in+1 radicalbigg 2 pize ?iz nsummationdisplay k=0 (n + 1/2, k) (2iz)k . (6.24) a31a32νa115a60a232 a101 a57a34a164 a69 a52 a8 a57a49a237 a37 a60Bessela50a51a52a32(a48a49 a37 a34a60Hankela56a57)a127|z|→∞ a86a52a238a227a138a139a34 H(1)ν (z) ～ radicalbigg 2 piz exp bracketleftBig i parenleftBig z ? νpi2 ? pi4 parenrightBigbracketrightBig ∞summationdisplay k=0 (?)k(ν, k) (2iz)k H(2)ν (z) ～ radicalbigg 2 piz exp bracketleftBig ?i parenleftBig z ? νpi2 ? pi4 parenrightBigbracketrightBig ∞summationdisplay k=0 (ν, k) (2iz)k (?2pi< argz <pi). a216a73 a60a239a240 a137a45a79a143a144a34a236a37a38a164 a69 Bessel a56a57a74Neumanna56a57 a30|z|→∞, ?pi< argz <pi a86a52 a238a227a138a139a34 Jν(z) = H (1) ν (z)+ H (2) ν (z) 2 ～ radicalbigg 2 piz bracketleftbigg cos parenleftBig z?νpi2 ?pi4 parenrightBig ∞summationdisplay k=0 (?)k(ν,2k) (2z)2k ?sin parenleftBig z?νpi2 ?pi4 parenrightBig ∞summationdisplay k=0 (?)k(ν,2k+1) (2z)2k+1 bracketrightbigg , Nν(z) = H (1) ν (z)?H (2) ν (z) 2i ～ radicalbigg 2 piz bracketleftbigg sin parenleftBig z?νpi2 ?pi4 parenrightBig ∞summationdisplay k=0 (?)k(ν,2k) (2z)2k +cos parenleftBig z?νpi2 ?pi4 parenrightBig ∞summationdisplay k=0 (?)k(ν,2k+1) (2z)2k+1 bracketrightbigg .