Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17( a18) a191a20 a21a22a23 a24a25a26a27a28a29a30a31a32a33a34a35a36a37a38 ( a39) §8.1 a40a41a42a43a44a45a46a47a48a49a44a50a51a52a50 a53a54a55a56a57a58a59a60a61a62a63a64a65a66a67a68 d2w dz2 + p(z) dw dz + q(z)w = 0, (8.1) p(z)a69q(z)a70a71a72a73a74a75a76a77 ? a72a73a74a78a79a80a81a82a72a73a74a75a76a83a84a74a77 ? a85a86a79a87a72a73a78a74a78a88a89a79a80a81a82a72a73a75a76a74a78a88a89a83a84a74a77 a90a91a92a93a94a93a95a96a97a98a99a100 a87 a101a102a103a93a104a105a106a107a108a109a110z 0 a103a111a112a113a114a115a103a116a117a91a92 a77 a98a99a118a92p(z), q(z) a119z0a110a103a93a120a121a122a123a109a124a91a92a93a119z0a110a103a93a120a121a87 a125a126a127 a87 a122a123a109 a124a91a92a93a103a128a129 a87a130a131a87 a105Taylora91a92a132a105Laurenta91a92 a77 ? a133a134p(z), q(z)a135z0 a136 a78a88a87a137z0 a136 a70a71a72a73a74a138 a136 a77 ? a133a134p(z), q(z) a139a140a141a142a143a144a135z0 a136a145 a78a88a87a137z0 a136 a70a71a72a73a74a146 a136 a77 a1478.1 a148a149a150a72a73(Hypergeometric equation) z(1?z)d 2w dz2 + bracketleftbigγ ?(1 + α + β)zbracketrightbigdw dz ?αβw = 0 a74a75a76a79 p(z) = γ ?(1 + α + β)zz(1?z) a69 q(z) = ? αβz(1?z). a135a142a151a152a153a87p(z)a69q(z)a142a154a144a146 a136a155 z = 0a69z = 1a77a156a157a87a158a159z = 0a69z = 1a79a148a149a150a72a73 a74a146 a136a160 a87a142a151a152a153a74a161a162 a136a163 a79a72a73a74a138 a136 a77 a1478.2 Legendre a72a73 parenleftbig1?x2parenrightbig d2y dx2 ?2x dy dx + l(l + 1)y = 0, a135a142a151a152a153a74a146 a136 a71x = ±1a77 Wu Chong-shi §8.1 a3a4a5a6a7a8a9a10a11a12a7a164a165a166a164 a192a20 a167a168a169a170a171 a152 a136 z = ∞a79 a145 a79a72a73(8.1)a74a146 a136 a87a137a172a173a174a175 a176a177 a74 a176a178z = 1/t a77 dw dz = ?t 2dw dt , d2w dz2 = t 4d2w dt2 + 2t 3dw dt . a179a180 a87a72a73(8.1)a176a71 d2w dt2 + bracketleftbigg2 t ? 1 t2p parenleftbigg1 t parenrightbiggbracketrightbigg dw dt + 1 t4q parenleftbigg1 t parenrightbigg w = 0. (8.2) a133a134t = 0a79a72a73(8.2)a74a138 a136 (a146 a136 )a87a137a70 a170a171 a152 a136 z = ∞a79a72a73 (8.1)a74a138 a136 (a146 a136 )a77 t = 0 (a181z = ∞)a71a72a73a138 a136 a74a182a183a79 p parenleftbigg1 t parenrightbigg = 2t + a2t2 + a3t3 +···, q parenleftbigg1 t parenrightbigg = b4t4 + b5t5 +···, a181 p(z) = 2z + a2z2 + a3z3 +···, q(z) = b4z4 + b5z5 +···. a170a171 a152 a136 a79a148a149a150a72a73a69Legendrea72a73a74a146 a136 a77 Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17( a18) a193a20 §8.2 a62a63a59a184a185a186a187a64a188 a189a190 a87 a145a191a192a193a194a195a196a197a198 a74a84a199a77 a200a2018.1 a133a134p(z)a69q(z)a135a202|z ?z0| < R a203a204a205a78a88a87a137a135 a180 a202a203a138a206a207a72a73a208a205a209a210 d2w dz2 + p(z) dw dz + q(z)w = 0, w(z0) = c0, wprime(z0) = c1 (c0, c1a71a211a212a138a76) a142a213a143a74a143a144a78w(z)a87a214a215w(z)a135a216a144a202a203a204a205a78a88a77 a217a218 a216a144a84a199a87a219a157a220w(z)a135z0 a136 a74a221a222|z ?z0| < R a203a223a224a71Taylora225a76 w(z) = ∞summationdisplay k=0 ck(z ?z0)k. a226a227 a87a216a228(z ?z0)0a229(z ?z0)1a74a75a76c0a229c1 a230a231 a69a208a205a182a183a143a232a77 a233 a216a144a234a235a74a225a76a78a236a237a206a207a72a73a87a238a239a75a76a87a240a219a157a241a242a75a76ck a77a84a199a243 a193 a87a75a76 ck(k = 2,3,···)a244a219a245c0, c1 a246a247 a77 a1478.3 a241Legendrea72a73 parenleftbig1?x2parenrightbig d2y dx2 ?2x dy dx + l(l + 1)y = 0 a135x = 0 a136 a221a222a203a74a78a87a161a139la79a143a144a248a76a77 a188 x = 0 a79a72a73a74a138 a136 a87 a179a180 a87a219a249a78 y = ∞summationdisplay k=0 ckxk. a236a237a72a73a87a240a142 parenleftbig1?x2parenrightbig ∞summationdisplay k=0 ckk(k ?1)xk?2 ?2x ∞summationdisplay k=0 ckkxk?1 + l(l + 1) ∞summationdisplay k=0 ckxk = 0, a250 a199a251a214a87a252a253 ∞summationdisplay k=0 braceleftBig (k + 2)(k + 1)ck+2 ?bracketleftbigk(k + 1)?l(l + 1)bracketrightbigck bracerightBig xk = 0. a217a218Taylor a223a224a74a213a143a89a87a219a252 (k + 2)(k + 1)ck+2 ?[k(k + 1)?l(l + 1)]ck = 0, a181 ck+2 = k(k + 1)?l(l + 1)(k + 2)(k + 1) ck = (k ?l)(k + l + 1)(k + 2)(k + 1) ck. a216a254a240a252a253a159a75a76a255a0a74a1a2a3a4a77a5a6a7a245a8a9a10a75a87a240a219a157a241a252a75a76 Wu Chong-shi §8.2 a10a11a7a164a11a12a13a12a16 a194a20 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). a7a245Γa14a76a74a89a15 Γ(z + 1) = zΓ(z), Γ(z + n + 1) = (z + n)(z + n?1)···(z + 1)zΓ(z), a219a157 a233c 2na69c2n+1 a16a17 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. a156a157a87Legendrea72a73a74a78a240a79 y(x) = c0y1(x) + c1y2(x), a161a139 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, 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. Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17( a18) a195a20 a230 a133a84a199a156a243a87a211a212a18a84a143a19 c0a69c1a87a240a143a84a219a157a241a242a72a73a74a143a144a85a78a77a85a86a79a87 ? a133a134a20c0 = 1, c1 = 0a87a240a252a253a85a78y1(x)a21 ? a133a134a20c0 = 0, c1 = 1a87a240a252a253a85a78y2(x)a77 a22a23 a87 a24a25a26a27a93y 1(x)a28y2(x) a105a29a121a116a30a103 a77 a31 a24a25a26a29a121a116a30a27a93a32a33 a87 a122a34a35a36 a37a32a98a99a103a38a93 a77 ? a133a134a220a78a235a139a74a138a76c0a69c1 a39a17 a79a211a212a40 a191 a138a76a87 a41 a198 a252a253a74a240a79a72a73a74a42a78a77 a3a43 a188a64a44a45a56a64a46a47 a77 a41 a198 a241a252a74a85a78a139a87y1(x)a48a49a142xa74a50a51a52a87y2(x)a48a49a142xa74a146 a51a52a87a181y1(x)a79xa74a50a14a76a87y2(x)a79xa74a146a14a76a77 a53 a241a78a74a54a73a55 a39 a87a216a79a82a56a8a9a10a75a139a48 a242a57a75a76ck+2a69cka87a58 a229ck+1 a170 a10a87 a179a180c 2na80a81a82c0a83a84a87c2n+1a80a81a82c1a83a84a77 a53a217a59a41 a55a243a87a72a73a74a78a74a60a70a89(a216a228a61a74a79a146a50a89)a87a62 a227a63a64 a79a72a73a74a60a70a89a74a5a65a77 a42a54a216a144a66a67a87a219a157 a39 a242a135a138 a136 a221a222a203a241a225a76a78a74a143a68a69a70a77a216a240a79 a155 ? a233(a72a73a138 a136 a221a222a203a74)a78a223a224a71Taylora225a76a87a236a237a206a207a72a73a21 ? a238a239a75a76a87a252a253a75a76a255a0a74a8a9a10a75a21 ? a5a6a7a245a8a9a10a75a87a241a242a75a76cka74a71a72 a246a73 a235(a245c0a69c1 a246a247 )a87 a53 a58a74a75a252a242a225a76a78a21 a82a56a8a9a10a75a143a84a79a76a89a74(a179a71a72a73a79a76a89a74)a87a156a157a74a75a74a225a76a78a143a84a219a157 a16a17 w(z) = c0w1(z) + c1w2(z) a74a234a235a77 a135a75a76a255a0a74a8a9a10a75a139a87a143a68a77a78a79a242a57ck, ck+1, ck+2 a80 a144a81a221a74a75a76a87 a179a180c k a77a78a79a82 a83 a56c0a69c1a87a74a75a241a252a74w1(z)a84w2(z)a240 a145 a77a48a49a142za74a50a51a52a84a146a51a52a77 Wu Chong-shi §8.2 a10a11a7a164a11a12a13a12a16 a196a20 a63 a245a138a206a207a72a73a74a52a225a76a78a85a87a219a157a252a253a72a73a135a143a84a86a222a203a74a78a235a77a87a88a89a219a157 a217a218a90a167 a87 a241a242a72a73a135 a145 a78a86a222a203a74a78a235a77a219a157 a192a193 a87a72a73a135 a145 a78a86a222a203a74a78a235a87 a91 a71a78a88a92a93a77 a179a180 a87a89a219 a53 a72a73a135a94a143a86a222a203a74a78a235a242a95a87a42a54a78a88a92a93a87a9a242a72a73a135a161a162a86a222a203a74a78a235a77 a1478.4 a96w1a79a72a73 d2w dz2 + p(z) dw dz + q(z)w = 0 (8.3) a74a78a87a135a86a222G1a203a78a88a77a97 tildewidew1a79w1a135a86a222G2a203a74a78a88a92a93a87a181 w1 ≡ tildewidew1, z ∈ G1 intersectiontextG2, (8.4) a98 a192a193a155 tildewidew1 a99 a79a72a73(8.3)a74a78a77 a100 a96 d2 tildewidew1 dz2 + p(z) dtildewidew1 dz + q(z)tildewidew1 = g(z), g(z)a135G2 a203a78a88a77a179a71w1a79a72a73(8.3)a135a86a222G1 a203a74a78a87a101a135a102a86a222 G1 intersectiontextG2 a203a87 a99a103a104 a72a73 d2w1 dz2 + p(z) dw1 dz + q(z)w1 = 0. a58a135 a180 a102a86a222a203a87w1(z) ≡ tildewidew1(z)a87 a101 d2 tildewidew1 dz2 + p(z) dtildewidew1 dz + q(z)tildewidew1 = 0, z ∈ G1 intersectiontextG 2, a181g(z) ≡ 0, z ∈ G1 intersectiontextG2a77 a217a218 a78a88a14a76a74a213a143a89a87a105a181 a192 a252 g(z) ≡ 0, z ∈ G2, a106 a181 tildewidew1a135G2 a203 a103a104 a72a73 d2 tildewidew1 dz2 + p(z) dtildewidew1 dz + q(z)tildewidew1 = 0. square a147 8.5 a96 w1 a69 w2 a163 a79a72a73 (8.3) a74a154a144a76a89 a170 a10a78a87a215a244a135a86a222 G1 a203a78a88a77a97 tildewidew1a69 tildewidew2a207a86a79w1a69w2a135a86a222 G2 a203a74a78a88a92a93a87a181a135z ∈ G1 intersectiontextG2 a139 w1 ≡ tildewidew1, w2 ≡ tildewidew2. a98 a192a155 tildewidew1a69 tildewidew2 a99 a76a89 a170 a10a77 a100 a82a67 8.4a107a87 tildewidew1a69 tildewidew2 a99 a79a72a73(a135G2 a203)a74a78a77 a179 a71w1a69w2 a76a89 a170 a10a87 ?[w1,w2] ≡ vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle w1 w2 wprime1 wprime2 vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle negationslash= 0, z ∈ G1. a96 ?[tildewidew1, tildewidew2] ≡ vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle tildewidew1 tildewidew2 tildewidewprime1 tildewidewprime2 vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = g(z), g(z)a135G2 a203a78a88a77a82a56a135z ∈ G1 intersectiontextG2 a139a87 w1 ≡ tildewidew1, w2 ≡ tildewidew2, a101g(z) negationslash= 0, z ∈ G 1 intersectiontextG 2a77a99 a227a217a218 a78a88a14a76a74a213a143a89a87a240 a192 a252 g(z) negationslash= 0, z ∈ G2. a156a157a87 tildewidew1a69 tildewidew2(a135G2 a203) a99 a76a89 a170 a10a77square