Wu Chong-shi a0a1a2a3a4 a5a6a7a8a9a10a11 (a1) Sturm-Liouville a12a13a14a15a16a17a18a19a20 §25.1 a21a22a23a24a25a26a27a28a29a30 a31a32 25.1 a33La34M a35a36a37a38a39a36a40a41a42a43a44a45 (a46a47) a48a49a50a51a52a53a54a40a41a42a43a44a45a55 a56a57a58 a40a41ua34va50a59a60 (v,Lu) = (Mv,u) a61 integraldisplay b a v?Ludx = integraldisplay b a (Mv)?udx, a62a63M a64La45a65a66a67a68 a69 25.1 a51L= ddx a50a53a64 integraldisplay b a v?dudxdx = v?u vextendsinglevextendsingle vextendsingle b a ? integraldisplay b a dv? dxudx. a70a71 a50a72ua34va73a74a75a76a77a78a79 y(a) = y(b) a80 a50 d dx a45a81a48a49a64? d dx a68 a36a37 25.1 a82a45a48a49M a34La64a83a35a81a48a49a50a84a35a85a86 M a64La45a81a48a49a50 a62 a52a53a55 a56 a40a41ua34va50a87a60 integraldisplay b a v?Mudx = bracketleftBiggintegraldisplay b a (Mu)?vdx bracketrightBigg? = bracketleftBiggintegraldisplay b a u?Lvdx bracketrightBigg? = integraldisplay b a (Lv)?udx, a70a71 a50La87a64M a45a81a48a49a68 a69 25.2 a33L= d 2 dx2 a50a88a89a90a91integraldisplay b a v?d 2u dx2dx = bracketleftBig v?uprime ? (v?)primeu bracketrightBigb a + integraldisplay b a parenleftBigd2v dx2 parenrightBig? udx. a70a71 a50a72a40a41ua34va73a74a75a39a92a93a92a94a95a76a77a78a79 α1y(a) +β1yprime(a) = 0, α2y(b) +β2yprime(b) = 0 (a96a82|α1|2 + |β1|2 negationslash= 0, |α2|2 + |β2|2 negationslash= 0)a97a98a99a78a79 y(a) = y(b), yprime(a) = yprime(b) a80 a50 d2 dx2 a45a81a48a49a100a64a101a102 a103 a68 Wu Chong-shi §25.1 a104 a105a106a107a108a109a110a111a112a113 a1142 a115 a31a32 25.2 a51a48a49La45a81a48a49a100a64a101a102 a103 a50a61a52a53a54a40a41a42a43a44a45a55 a56a57a58 a40a41ua34va50 a59a60 (v,Lu) = (Lv,u) a61 integraldisplay b a v?Ludx = integraldisplay b a (Lv)?udx, a62a63L a64a116a65a66a67a68 a69 25.3 a38a34a117 4a118a119a120a121a45a78a79a122a50a48a49 i ddx a100a64a102a81a48a49a68 integraldisplay b a v? parenleftbigg idudx parenrightbigg dx = ?i integraldisplay b a dv? dxudx = integraldisplay b a parenleftbigg idvdx parenrightbigg? udx. a48a49a45a102a81a123a50a124a64a34a39a36a45a40a41a42a43a125a126a38a39a127a45a68a128a129a50a130a131a124a64a132a133 ? a40a41a36a37a38a134a36a45a135a43a136a50 ? a40a41a137a60a75a138a45a139a140a123(a117a85a50a52a53a93a141a46a47a48a49a50a100a132a133a40a41a45a93a141a142a41a139a140a50 a143 a144 a47a145a139a140a146a85a86a64a147a77a135a43a50 a62 a132a133a40a41a148a149a150a151) a50 a84a152a50a153a154a136a124a64a155a53 Hilberta42a43a68a156a157a50a158a132a133 ? a40a41a74a75a39a36a45a76a77a78a79a50a61a124a64a159a155a38 Hilberta42a43a82a45a39a36a160a42a43a44a68 a161a162a163a164a165 a76a77a78a79a45a166a167a168a169a170a48a49a45a102a81a123a68 a39 a58 a48a49a50a120a52a53a171a39a95a40a41a64a102a81a45a50a172a52a53a173a39a95a40a41a50a100a150 a163a162 a64a102a81a45a68 a69 25.4 a33L= i ddx a50a174a175a76a77a78a79a176a177a178a39a179a45a180a181 y(b) = αy(a), αa35(a182)a129a41. a53a64 integraldisplay b a v?idudxdx =iv?u vextendsinglevextendsingle vextendsingle b a ? i integraldisplay b a dv? dxudx =i(αα? ? 1)u(a)v?(a) + integraldisplay b a parenleftbigg idvdx parenrightbigg? udx. a70a71a183 a60a76a77a78a79a82a45αa74a75αα? = 1a80a50a48a49 i ddx a184 a64a102a81a45a68 a31a32 25.3 a33La35a102a81a48a49a50 a62 a149a185 Ly(x) = λy(x) a63 a35a102a81a48a49a45a186a187a188a189a190a68 a191a192a193 a60a91a194a195a196a197a198a76a77a78a79a50a64a84a35a101a199a200a201a202a38a102a81a48a49La45a36a37a82a203a68 Wu Chong-shi a204a205a206a207a208 Sturm-Liouville a209a210a211 a108a109a110a111a112a113 a1143 a115 a102a81a48a49a45a186a187a188a189a190a137a60a122a212a213 a58a214 a132a45a215a186a123a216a217 ? a218a219 1 a102a81a48a49a45a186a187a188a220a221a222a38a68(a162a90) ? a218a219 2 a102a81a48a49a45a186a187a188a220a35a153a41a68 a223 a84a35 Ly = λy, a176a182a224a225 (Ly)? = λ?y?. a226 a53La64a102a81a48a49a50 a70a71 integraldisplay b a [y?Ly? (Ly)?y]dx = (λ?λ?) integraldisplay b a yy?dx = 0. a227 a84a35 integraldisplay b a yy?dxnegationslash= 0a50 a70a71 λ = λ?, a61a90a228a186a187a188λa35a153a41a68 square ? a218a219 3 a102a81a48a49a45a186a187a40a41a137a60a229a230a123a50a61a52a231 a162 a121a186a187a188a45a186a187a40a41a39a36a229a230a68 a223 a33λi a34λj a64 a162 a120a232a45 a57a58 a186a187a188a50a52a231a45a186a187a40a41a35 yi a34yj a50 Lyi = λiyi, Lyj = λjyj. a233a56a234 a186a187a188λi,λj a35a153a41a50a53a64integraldisplay b a [y?iLyj ? (Lyi)?yj]dx = (λj ?λi) integraldisplay b a y?iyjdx. a84a35λi negationslash= λj a50 a70a71 integraldisplay b a y?i (x)yj(x)dx = 0. a191a235 a100a90a91a203a186a187a40a41a45a229a230a123a68 square a226 a53a186a187a40a41a64a197a198a46a47a149a185a38a197a198a76a77a78a79a122a45a236a50 a70a71 a175a186a187a40a41a237 a71 a39 a58a238a239 a129a41a84 a160a240a221a64a186a187a40a41a68a130a131a100a150 a71a241 a72a242a243 a191a58 a129a41a84a160a50a244a228a52a53a55 a56 a39 a58 a186a187a188λi a50a73a60 integraldisplay b a y?i (x)yi(x)dx = 1. a191a235 a228 a234 a45a100a64a39 a58a245a246a247a248a249a250a251a252 a68 integraldisplay b a y?i (x)yj(x)dx = δij. Wu Chong-shi §25.1 a104 a105a106a107a108a109a110a111a112a113 a1144 a115 ? a218a219 4 a102a81a48a49a45a186a187a40a41 (a45a119a253) a254a177a39 a58 a118a255a40a41a0a50a61a55 a56 a39 a58 a38a135a43 [a,b] a82a60 a139a140a93a141a142a41a92a157a74a75a34a102a81a48a49La120a121a45a76a77a78a79a45a40a41f(x)a50 a1 a150a2a186a187a40a41{yn(x)} a3a4 a35 a161 a52a174a157a39a5a6a7a45a8a41 f(x) = ∞summationdisplay n=1 cnyn(x), (#) a96a82 cn = integraldisplay b a f(x)y?n(x)dx integraldisplay b a yn(x)y?n(x)dx . a9a10 a64a50a85a86a186a187a40a41a0a64a11a39a12a45a50 a62 a136a181a82a45a47a13a35 1a50 a3a4 a45a180a181a178a14a15a16a68(a162a90) a121 a235 a50a229a230a11a39a45a186a187a40a41a0a45a118a255a123a87a158a150 a71a17a18 a177 ∞summationdisplay n=1 yn(x)y?n(xprime) = δ(x?xprime). ? a226a136a19a45a123a2163a344a150 a71a20a234 a50 a183 a132a175a186a187a40a41 a241 a72a11a39a12a50 a62 a186a187a40a41a45a119a253a100a254a177a203a39 a58 a118a255a45a229a230a11a39a40a41a21a68a84a152a50a136a39a22a82a60a23a118a255a45a229a230a11a39a40a41a21a45a169a170 a1 a150 a241a24 a68 ? a191a192a25a80a26a27a28a39a29a150 a163 a123a50a61a52a231a53a39 a58 a186a187a188a150 a163 a60 a162a30 a39 a58( a31a123a147a23a45)a186a187a40a41a50 a84a174a150 a163 a156 a162a32 a152a229a230a68 a191 a29a33a180a175a38 25.3a22a169a170a68a172a61a244a85a152a50a124a158a150 a71a34a24 Schmidt a45 a229a230a12a35a36 (a37a38 18.1a22)a244a39a229a230a12a50a84a174a240a221a150 a71 a228 a234 a39 a58 a118a255a45a229a230a11a39a40a41a21a68 ? a40a153a136a50a136a19a45 a3a4 a78a79a158a150 a71a41a42 a35a217a52a53a55 a56 a38[a,b] a82a148a149a150a151a45a40a41a50(#)a181a38a148 a1 a6a7 lim N→∞ integraldisplay b a vextendsinglevextendsingle vextendsinglef(x) ? Nsummationdisplay n=1 cnyn(x) vextendsinglevextendsingle vextendsingle 2 dx = 0 a45 a56 a37a122a240a221a177a43a68 a44a45a46 a168a50a136a19a23a53a102a81a48a49a186a187a188a45a222a38a123a34a186a187a40a41a45a118a255a123a45a169a170a50a186a168a158 a231a72a135a47a47a48a45 (a135a43a147a77a97a49a147a77a146a97a64a38a60a77a135a43a136a46a47a149a185a60a47a50) a34 a238 a47a48a45 (a135a43a60a77a50a157a46a47a149a185a38a135a43a136a147a47a50)a186a187a188a189a190 a191a57 a29a33a180a68a172 a226 a53a156 a193 a60a134a196a60 a23a45a90a91a50 a70a71 a87a100a51a52a135a47 a191a57 a95a186a187a188a189a190a68a174a157a50a35a203a53a54a45a149a55a50a38a60a23a45 a17 a54a82a73 a34a24 a203a60a77a135a43a45a180a181a68 Wu Chong-shi a204a205a206a207a208 Sturm-Liouville a209a210a211 a108a109a110a111a112a113 a1145 a115 §25.2 Sturm–Liouville a56a57a58a25a26a27a28a29a30 a38a59a19a213a60a82a50a130a131a169a170a61a213 a58 a129a46a47a149a185a45a186a187a188a189a190a68a62a63a45a46a47a149a185a60 Xprimeprime +λX = 0; d dx bracketleftbiggparenleftbig 1 ?x2parenrightbig dydx bracketrightbigg + bracketleftBig λ? m 2 1 ?x2 bracketrightBig y = 0; 1 r d dr parenleftbigg rdRdr parenrightbigg + bracketleftBig λ? m 2 r2 bracketrightBig R = 0. a101a131a150 a71 a11a64a35a122a19a45a39a179a180a181 d dx bracketleftbigg p(x)dydx bracketrightbigg + [λρ(x) ?q(x)]y = 0. (#) a191 a29a95a65a45a149a185 a63 a35 Sturm–Liouvillea65(a15 a63 S–L a65)a149a185a68 ? a162a66a67S–La65a149a185a82a45a40a41p(x), q(x)a34ρ(x)a155a68a35a73a64a153a40a41a50a174a157a73a74a75a220a132a45a139a140a123 a132a133a68 ? ρ(x)a50 a63 a35a69 a214 a40a41a68 ? a72a69 a214 a40a41ρ(x) =a129a41 a80 a50a150 a71 a176a35 1a68 ? a162a59a35a129a41a45a69 a214 a40a41a50a150 a71 a168a70a53a229a230a71a19a72a73a126a45a244 a24 (a191a80 a150 a71a74 Laplace a48a49a45a137a253 a17a75 a181a82a76a77 a234 a69 a214 a40a41a45a78a79a146 a74a80 a186a136 a46 a50a101a81a82a203a72a73a83a84a16a85a64a54a86a87a45a40a41a68a150 a71a63 a39a35a168a70a53a42a43a45a213a88a89a54a45 a162a1a90 a123) a50a87a150 a163 a168a70a53a189a190 a70 a62a63a45a91a92a123a216a45 a162a1 a90 a123(a117a85a50 a93 a84a47a94a45 a162a1a90) a68a84a152a50a100a130a131 a70 a23a95a45a91a92a189a190a174a96a50 a162a66a97 a33 ρ(x) ≥ 0a50 a174a157a50a231a72 a162 a59a35 0a68 a35a203a38a195a45a98a99a50a158a150 a71a100a101 a48a49 L≡ ? ddx bracketleftbigg p(x) ddx bracketrightbigg +q(x) (dividemultiply) a45a102a103a68 a191a235 a50 S–La65a149a185a100a150 a71a104 a195a177 Ly(x) = λρ(x)y(x). (##) S–La65a149a185a105a14a136 a241 a72a45a76a77a78a79a50a100a254a177 S–La65a149a185a45a186a187a188a189a190a68 λa63a35a186a187a188a68a52a53 a171a39 a58 a186a187a188λa50a74a75 S–La149a185a63a120a231a45a76a77a78a79a45 a238a239 a236a100a64a186a187a40a41a68 a74 a46a47a149a185a168 a20 a50 a226 a53ρ(x)a45a196a106a50 S–La65a149a185 (#)a97(##) a91a107 a162 a121a53a149a185 Lprimeu(x) = λu(x). (maltesecross) Wu Chong-shi §25.2 Sturm–Liouvillea209a210a211 a108a109a110a111a112a113 a1146 a115 a172a64a50a128a61a86a87a86a108 u(x) = radicalbig ρ(x)y(x), a100a150 a71 a175a149a185 (#)a12a35 (maltesecross)a50a96a82 Lprime = ? ddx bracketleftbigg φ(x) ddx bracketrightbigg +ψ(x), φ(x) = p(x)ρ(x), ψ(x) = ? 1radicalbigρ(x) ddx bracketleftBig p(x) ddx 1radicalbigρ(x) bracketrightBig + q(x)ρ(x). a149a185 (maltesecross)a72a221a87a158a64 S–La65a149a185a50 a183a162 a61a64a39a29 a9a109 a45S–La65a149a185a50a69 a214 a40a41a35 1 a45S–La65a149a185a68 a31a110 25.1 a52a53a55 a56 a40a41u1(x)a34u2(x)a50a59a60 u?1Lprimeu2 ?parenleftbigLprimeu1parenrightbig?u2 = ? ddx bracketleftBig φ(x) parenleftBig u?1du2dx ?u2du ?1 dx parenrightBigbracketrightBig , a96a82 Lprime = ddx bracketleftbigg φ(x) ddx bracketrightbigg ?ψ(x). a111a112a113 a66a67La68 L≡ ? ddx bracketleftbigg p(x) ddx bracketrightbigg +q(x) a84a35a38a86a108 u1(x) = radicalbigρ(x)y1(x), u2(x) = radicalbigρ(x)y2(x)a39a122a50a60 u?1Lprimeu2 ?parenleftbigLprimeu1parenrightbig?u2 = y?1Ly2 ? (Ly1)?y2. a70a71 a50a52a53a55 a56 a40a41y1(x)a34y2(x)a50 y?1Ly2 ? (Ly1)?y2 = ? ddx bracketleftBig p(x) parenleftBig y?1 dy2dx ?y2dy ?1 dx parenrightBigbracketrightBig . a31a110 25.2 a38a76a77a78a79 φ(x) parenleftBig u?1du2dx ?u2du ?1 dx parenrightBigvextendsinglevextendsinglevextendsingle vextendsingle b a = 0 a39a122a50a48a49Lprime a64a102a81a45a68 a175a36a92 1a45a114a170a34a36a92 2a115a116a127a168a50a43a61a228 a234 a217a38a76a77a78a79 p(x) parenleftBig y?1 dy2dx ?y2dy ?1 dx parenrightBigvextendsinglevextendsinglevextendsingle vextendsingle b a = 0 (circleasterisk) a39a122a50a48a49La87a64a102a81a45a68 Wu Chong-shi a204a205a206a207a208 Sturm-Liouville a209a210a211 a108a109a110a111a112a113 a1147 a115 a38a117a118a33a119a122a50a76a77a78a79 (circleasterisk)a163a138a177a43a120 ? a121a39a29a33a119a64a38a122a50x = aa34x = ba50 a1 a60 p(x) parenleftBig y?1 dy2dx ?y2dy ?1 dx parenrightBig = 0. (triangle) 1. a85a86y1 a34y2 a38 a57 a122a50 a1 a74a75a121a39a92a93a92a94a95a76a77a78a79a50 a62 (triangle) a181a177a43a68 a117a85a50a38x = aa50a50 αyi(a) ?βyprimei(a) = 0, i = 1,2, αa34βa1a35(a229)a153a41a50 a176a182a224a225a50a158a150 a71 a228a196 αy?i (a) ?βy?i prime(a) = 0, i = 1,2. a226 a53αa34β a162 a150 a163 a121 a80 a35 0a50a123a60vextendsingle vextendsinglevextendsingle vextendsingle y?1(a) y?prime1 (a) y2(a) yprime2(a) vextendsinglevextendsingle vextendsinglevextendsingle = y?1(a)yprime2(a) ?y2(a)y?prime1 (a) = 0. 2. a85a86p(x) a38a122a50(a117a85a50x = a)a124a35 0, a191a80x = aa50a64a149a185a45a47a50a68a97a36p(x), q(x) a34 ρ(x)a74a75a39a36a45a132a133a50a244a228x = aa50a64a149a185a45a229 a62 a47a50a50a174a157a121a39a236a60a77a50a121a93a236a147a77a68 a38a105a14a136a60a77a78a79a125 a28 a147a77a236a126a50a100a60 p(x) parenleftBig y?1 dy2dx ?y2dy ?1 dx parenrightBigvextendsinglevextendsinglevextendsingle vextendsingle x=a = 0. a117a85 p(a) = 0, pprime(a) negationslash= 0, ρ(x)a34(x?a)q(x)a1a38x = aa50a236a127 a97 p(a) = 0, pprime(a) = 0, pprimeprime(a) negationslash= 0, ρ(x)a34q(x)a1a38x = aa50a236a127, a191 a38a130a131a169a170a61a45a153a154a189a190a82a64 a163 a138a74a75a45a68 ? a173a39a29a33a119a64 p(x) parenleftBig y?1 dy2dx ?y2dy ?1 dx parenrightBigvextendsinglevextendsinglevextendsingle vextendsingle x=a = p(x) parenleftBig y?1 dy2dx ?y2dy ?1 dx parenrightBigvextendsinglevextendsinglevextendsingle vextendsingle x=b , a172 a162 a350a50 a191a80(triangle) a181a87a177a43a68a85a86 p(a) = p(b), q(a) = q(b), ρ(a) = ρ(b), a156a157 yi(a) = yi(b), yprimei(a) = yprimei(b), i = 1,2, a107a221a100a150 a71 a74a75 a191a58 a132a133a68 a191 a229a64a169a170a61a45a98a99a78a79a45a33a180a68 Wu Chong-shi §25.3 Sturm–Liouvillea209a210a211 a109a110a111a112a113a108a128a129a130a131 a1148 a115 §25.3 Sturm–Liouville a56a57a58a26a27a28a29a30a25a132a133a134a135 a52a231a39 a58 a186a187a188a60 a162a183 a39 a58 ( a31a123a147a23a45) a186a187a40a41a45a106a136a50 a63 a35a15a156a97a137a12a68 a226 a53S–La65a149a185a64a93a141a31a123a129a46a47a149a185a50 a70a71 a50a52a231a39 a58 a186a187a188a138a139 a183a163 a60 a57a58( a31 a123a147a23a45)a186a187a40a41a68 a38a117a118a78a79a122a50 S–La65a149a185a45a186a187a188a189a190a64a15a156a45a120a38a117a118a78a79a122a64 a238 a15a156a45a120 a31a110 25.3 a85a86S–La65a149a185a186a187a188a189a190a45a39 a58 a186a187a40a41a64a182a45a50a157a96a153a140a34a141a140a31a123a147a23a50 a62 a152a186a187a188a189a190a64a93 a214 a15a156a45a68 a223 a80a142 a36a92 a70 a33a50a186a187a40a41y(x)a64a182a45a50a96a153a140a34a141a140a47 a10 a35f(x)a34g(x)a50 y(x) = f(x) + ig(x). a62S–L a65a149a185a150 a71 a195a177 L(f + ig) = λρ(f + ig). a226 a53a48a49La64a153a48a49a50a69 a214 a40a41ρ(x)a64a153a40a41a50a157a186a187a188λa35a153a41a50a123a175a136a181a47 a10a143a144 a153a140a34a141 a140a50a100a228 a234 Lf = λρf, Lg = λρg. a191a46 a91f(x) a34g(x) a73a64a52a231a53a121a39 a58 a186a187a188λa45a186a187a40a41a50a101a131a45a31a123a147a23a123a38a36a92a45a199a145a78 a79a82a199a200a146a203a91a194a45a155a36a68 a158a220a147a90a91f(x) a34g(x) a87a74a75a148a186a187a188a189a190a45a76a77a78a79a68 a191a80a183 a132 a233a56a234 a76a77a78a79a87a64a31 a123a197a198a45a50a156a157a150 a163 a196a106a45a126a41a87a64a153a41a50a53a64a38a76a77a78a79a82a87a47 a10a143a144 a153a140a34a141a140a61a150a68 square a31a110 25.4 a33y1(x)a34y2(x)a73a64S–La65a149a185a186a187a188a189a190 Ly(x) = λρ(x)y(x). a45 a57a58 a153a45a31a123a147a23a45a186a187a40a41a50a156a157a38x = aa34x = ba50a73a16a149a74a75a76a77a78a79 p(x) parenleftBig y?1 dy2dx ?y2dy ?1 dx parenrightBigvextendsinglevextendsinglevextendsingle vextendsingle x=a = p(x) parenleftBig y?1 dy2dx ?y2dy ?1 dx parenrightBigvextendsinglevextendsinglevextendsingle vextendsingle x=b = 0, (#) a62y 1(x)a34y2(x) a162 a150 a163 a52a231a53a121a39 a58 a186a187a188λa68 a223 a24 a81a90a150a68a33y1(x)a34y2(x)a52a231a53a121a39 a58 a186a187a188λa50 Ly1 = λρy1, Ly2 = λρy2, a84a152 y1Ly2 ?y2Ly1 = 0, a233a56y 1(x)a34y2(x)a73a64a153a40a41a50y?1(x) = y1(x)a50y?2(x) = y2(x)a50 a70a71a80a142 a136a22a36a92 1a45a114a170a50a100a60 d dx bracketleftbigg p(x) parenleftbigg y1dy2dx ?y2dy1dx parenrightbiggbracketrightbigg = 0. a53a64 p(x) parenleftbigg y1dy2dx ?y2dy1dx parenrightbigg =a129a41C. Wu Chong-shi a204a205a206a207a208 Sturm-Liouville a209a210a211 a108a109a110a111a112a113 a1149 a115 a174 a80a142 a36a92a134a196a45a199a145a78a79 (#)a50a100a231a60 p(x) parenleftbigg y1dy2dx ?y2dy1dx parenrightbigg ≡ 0. a172a84a35p(x) negationslash≡ 0a50a123a60 y1dy2dx ?y2dy1dx ≡ 0, a61 Wbracketleftbigy1(x), y2(x)bracketrightbig ≡ vextendsinglevextendsingle vextendsinglevextendsingley1(x) y2(x) y?1(x) y?2(x) vextendsinglevextendsingle vextendsinglevextendsingle ≡ 0. a191a46 a91y1(x) a34y2(x) a31a123a120a23a50a151a199a145a78a79a152a153a68 a70a71 y 1(x) a34y2(x) a162 a150 a163 a52a231a53a121a39 a58 a186a187 a188a68 square a191a58 a36a92a100a154a155a130a131a50a38a39a92a93a92a94a95 (a197a198)a76a78a79a97(a34)a60a77a78a79a122a50S–La65a149a185 a186a187a188a189a190 a162 a150 a163 a64a15a156a45a68a100a186a38 a70 a169a170a61a45a213a29a95a65a45a76a77a78a79a174a96a50 a183 a60a38a98a99 a78a79a39a122a50a186a187a40a41a38a135a43a45a156a39 a58 a122a50a156 a162 a16a149a74a75(#)a50 a184 a60a150 a163a157a158 a15a156a106a136a68 Wu Chong-shi §25.4 a159Sturm–Liouvillea209a210a211 a108a109a110a111a112a113a160a161a162a163a164a165 a11410 a115 §25.4 a166 Sturm–Liouville a56a57a58a25a26a27a28a29a30a167a168a169a170a171a172 a240 a71a173 a45a174a175a176a189a190a35a117a68 a52a53 a57 a122a177a36 a173 a45a102 a226 a175a176a50a36a236a189a190a64 ?2u ?t2 ?a 2?2u ?x2 = 0, 0 <x<l,t> 0; uvextendsinglevextendsinglex=0 = 0, uvextendsinglevextendsinglex=l = 0, t> 0; uvextendsinglevextendsinglet=0 = φ(x), ?u?t vextendsinglevextendsingle vextendsingle t=0 = ψ(x), 0 <x<l. a80a14225.1 a22a3425.2a22a45a169a170a150a145a50a85a86a222a38a39 a58 S–L a149a185a45a186a187a188a189a190 LX = λρX, X(0) = 0, X(l) = 0, a178 a118a50 a226 a53a101a45a76a77a78a79a34a36a236a189a190a45a76a77a78a79a180a181a118a119a120a121a50a84a152a50a150 a71 a175a36a236a189a190a45a236u(x,t) a2a179a186a187a40a41a45a119a253 {Xn(x), n = 1, 2, 3, ···} (a35a149a55a127a37a50 a97 a33a186a187a40a41 a1 a199a11a39a12) a3a4a50 u(x,t) = ∞summationdisplay n=1 Tn(t)Xn(x). a191a192 a50a186a187a40a41a0a45a118a255a123a127a203a180a36a123a45a146 a24 a68a35a203a181a90 ∞summationtext n=1 Tn(t)Xn(x) a163a138a6a7 (a143a144a64a148 a1 a6a7) a234a236u(x,t)a50 a191a192 a45a133a34a220a147a182a63a183a184a186a187a40a41a68 a161a162 a150 a71 a147a92 a226 a185a186a187 a51a188 a58 a186a187a40a41a68 a189a62 a50 a190a191 a38a180a181a136a192a193a240 a163 a133 a234 a39 a58 a8a41a194a236a195a50a172a101 a161a162 a150 a163 a6a7 a234a196 a229a45a236u(x,t)a68 a175a236a181a197a198a149a185a50a60 ∞summationdisplay m=1 Tprimeprimem(t)Xm(x) ?a2 ∞summationdisplay m=1 Tm(t)Xprimeprimem(x) = 0. a24X? n(x)a237a136a181 a57 a122a50a221a126a38a135a43 [0, l]a136a151a47a50a100a228 a234 Tprimeprimen(t) ?a2 ∞summationdisplay m=1 (Xn, Xprimeprimem)Tm(t) = 0, m = 1,2,3,···. a199 a175a200a201a78a79a87a2 a191 a39a0a186a187a40a41 a3a4 a50a228 a234 Tn(0) = (Xn, φ), Tprimen(0) = (Xn, ψ). a85a86 a163 a138a133a196Tn(t)a50a197a202 a234 a236a181a82a50a72a221a100a133a196a203a36a236a189a190a45a236 u(x,t)a68 a191a192 a132a133a236a45a64a23a53a51a145a40a41{Tn(t),n = 1,2,3,···}a45a129a46a47a149a185a0a68a39a179 a46 a168a50 a191 a158a64 a143a144a203a204 a45a68 a205a206a207 a203a197a198a76a77a78a79a38a47 a165 a86a87a150a82a45a180a36a123a146 a24 a126a50 a238 a197a198a149a185a45a33a180a100a208a209a174a236a203a68 Wu Chong-shi a204a205a206a207a208 Sturm-Liouville a209a210a211 a108a109a110a111a112a113 a11411 a115 a85a86 a67 a36a236a189a190a82a45a149a185 a104 a35 ?2u ?t2 ?a 2?2u ?x2 = f(x,t), a178 a118a50a106a38 a20 a168a50a133a236a61a185a156 a193 a60a210a211a45a212a48a50 a162 a121a39a124 a183 a38a53a132a175a149a185a45 a238 a197a198a213 f(x,t)a87a2 a186a187a40a41 a3a4 a50a53a64a50a197a198a45a129a46a47a149a185a0a86a177a203 a238 a197a198a45a149a185a0 Tprimeprimen(t) ?a2 ∞summationdisplay m=1 (Xn,Xprimeprimem)Tm(t) = (Xn,f), m = 1,2,3,···. f(x,t)a146a35xa45a40a41a50a34{Xn(x)}a231a72a214a53a121a39a58a40a41a42a43a68 a121 a235 a50a87a100 a162a204 a92a236a50a85a86a36a236a189a190a45a76a77a78a79a64 a238 a197a198a45a50a100a215a216a220a147a175a76a77a78a79a197a198a12a68 ? a234a106a38a35a30a50a130a131a217a214a47a127a203a197a198a76a77a78a79a38a47a165a86a87a150a82a45a180a36a123a146a24a68 ? a52a53a186a187a40a41a50 a218 a203a132a133a101a74a75a34a36a236a189a190a120a121a45a76a77a78a79a219a50a52a53a101 a70 a74a75a45a46a47a149a185 a183 a64a132a133a220a147a64 S–La65a149a185a50a172a52a53a149a185a45a137a253a180a181a156 a193 a60a155a68a68 ? a186a187a40a41a74a75a45a46a47a149a185 a162 a121a50 {Xn(x), n = 1,2,3,···} a45a180a181 a162 a121a50a84a174a228 a234 a45a23a53Tn(t) a45a129a46a47a149a185a0a45a180a181a87 a162 a120a121a50a133a228a45 Tn(t)a87 a162 a120a121a68 ? a36a236a189a190a45a236a45a222a38a220a39a123a50a181a90a203a138a126a133a228a45a236a120a121a68 ? a221a132 a233a56 a50a156 a162 a64a55a88a129a46a47a149a185a0a73a64a88a89a133a236a45a68 ? a38a153a154a133a236a61a185a82a50a100a221a132a222a72 a185 a242a243a186a187a40a41a0{Xn(x), n = 1,2,3,···}a50a244a228Tn(t) a45a133 a236a189a190 a190 a150 a163a185 a15a16a68 ? a138a15a16a45a33a180a100a64a132a133 (Xn,Xprimeprimem) = 0, a72nnegationslash= m, a108a223a224 a46 a50 (Xn, Xprimeprimem) = ?λmδnm. a84a152a50Tn(t)a74a75a129a46a47a149a185 Tprimeprimen(t) +a2λnTn(t) = 0 a97 Tprimeprimen(t) +a2λnTn(t) = (Xn, f). a174 a162 a64a129a46a47a149a185a0a68 ? a225a226 a234 a186a187a40a41a0{Xn(x), n = 1,2,3,···}a64a229a230a11a39a45a50 (Xn, Xm) = δmn, a136a54a132a133a100a232a227a53 (Xn, Xprimeprimem) = ?λm(Xn, Xm) a61 (Xn, Xprimeprimem +λmXm) = 0 Wu Chong-shi §25.4 a159Sturm–Liouvillea209a210a211 a108a109a110a111a112a113a160a161a162a163a164a165 a11412 a115 a191a56a228 a217a186a187a40a41a231a72a74a75a129a46a47a149a185 Xprimeprimen(x) +λnXn(x) = 0, a191 a229a64a130a131 a24 a47 a165 a86a87a150a45a73a229a35a36a228 a234 a45a46a47a149a185a68 a70a71 a50a47 a165 a86a87a150a100a35a130a131a230a231a203a39 a58 a242a243a186a187a40a41a0a45a138a232a149a233a68 a186a187a40a41a45a118a255a123a64a38a92a170a136a181a90a203a39a36a150 a71 a175a36a236a189a190a45a236a2a54a186a187a40a41a0 a3a4(a191 a64a60a78a79a45a50a36a236a189a190a34a186a187a40a41a132a74a75a120a121a45a197a198a76a77a78a79) a50 a242 a24 a194a120a231a197a198a189a190a45a186a187a40a41a195 a62 a181a90a203a150 a71 a149a55 a185 a133a196 a3a4 a126a41 (a153a154a136a64a40a41)a50 a181a90a203 a191 a29a236a150a38a153 a24 a136a45a150a234a123a68 a38a235a198a92a236a203a47 a165 a86a87a150a45a236a237a153a216a126a50a133a236a238a46a47a149a185a36a236a189a190 a80 a100a239a228a203a178a211a45a102 a226 a68 a191a162a240 a253a106a38a244a228a130a131a52a53a241a29a95a65a45a36a236a189a190(a149a185a197a198a97 a238 a197a198a50a76a77a78a79a197a198a97 a238 a197a198)a45 a133a236a60a203a39 a58a242 a39a45a178a235a198a45a243a244a50a87a158 a17 a106a38a245 a42 a203a52a53a171a246a36a236a189a190a45a133a236a247a248a68 a117a85a50a52a53a16a85a249a44a45a250a36a189a190a50 ?2u = f, x2 +y2 +z2 < 1; uvextendsinglevextendsinglex2+y2+z2=1 = 0,? a34a24 a249a72a73a126a133a236a50a2a179a61a125a45a251a150a50a231a72a175 u(r,θ,φ) a2 a194a120a231a197a198a189a190a45a186a187a40a41a195 Yml (θ,φ) a3a4 a50 u(r,θ,φ) = ∞summationdisplay l=0 lsummationdisplay m=?l Rlm(r)Yml (θ,φ), a221a126a50 a80a142 a149a185 1 r2 d dr bracketleftbigg r2dRlmdr bracketrightbigg ? l(l+ 1)r2 Rlm(r) = integraldisplayintegraldisplay Ym?l (θ,φ)f(r,θ,φ)sinθdθdφ (a96a82a45a151a47a182a63a252a58 4pia43a253a253)a254a255a0a1a2 Rlm(0)a3a0, Rlm(1) = 0 a4a5R lm(r)a6 a7a8a9a10a11a12a13a14a15a16a17a18a19a20a21a22a11a23a24a25a15a16a26a17a27a28a29a30a31a4a32a33a34a35a23a24 a25a36a11a37a38a39a40 a6 a41a42a43a44a11a14a45a26a46a47a28a48a10a18a49a50a51a52a22a26a53a54a55a56a57a58a59a60a32a61a62a11a24a25 a255a0a1a2 a26a63a64a26 a65a66a67a68u(r,θ,φ)a41a7a18a49a50a51a52a22a69a70 a6 a37a38a71a72a26a66a67a73a4a32a50a51a74a61a62 ??2w = λw, x2 +y2 +z2 < 1; wvextendsinglevextendsinglex2+y2+z2=1 = 0, Wu Chong-shi a75a76a77a78a79 Sturm-Liouville a80a81a82a83a84a85a86a87a88 a8913a90 a91a10a50a51a74 λnl = k2nl, n = 1,2,3,···, l = 0,1,2,··· a254 a50a51a52a22 wnlm(r,θ,φ) = jl(knlr)Yml (θ,φ), a92a93k nl a17l a94a95Bessela52a22jl(x)a11a96na19a97a98a99a6 a100a101a26a7a8a11a50a51a74a102m = 0,±1,···,±l a103a104 a26a105a106a107a71a26a59a50a51a74a61a62a17a108m a109 a110a11a26 a109 a110a111a112 2l+ 1 a6 a113a114a26a68u(r,θ,φ)a41w nlm(r,θ,φ) a69a70a26 u(r,θ,φ) = ∞summationdisplay n=1 ∞summationdisplay l=0 lsummationdisplay m=?l cnlm jl(knlr)Yml (θ,φ), a115a116a117a13a14a15a16a26a65a91a10 ?k2nlcnlm integraldisplay 1 0 j2l (knlr)r2dr = integraldisplay 1 0 jl(knlr)r2dr integraldisplayintegraldisplay Ym?l (θ,φ)f(r,θ,φ)sinθdθdφ, a118a119 a95 Bessela52a22a11a60a120a121a3a104a122 a47a26a66a67a4a91 integraldisplay 1 0 j2l (knlr)r2dr = pi2k nl integraldisplay 1 0 J2l+1/2(knlr)rdr = pi4k nl bracketleftbigJprime l+1/2(knl) bracketrightbig2 = 12bracketleftbigjprimel(knl)bracketrightbig2, a123a67 cnlm = ? 2 k2nl bracketleftbigjprimel(knl)bracketrightbig2 integraldisplay 1 0 jl(knlr)r2dr integraldisplayintegraldisplay Ym?l (θ,φ)f(r,θ,φ)sinθdθdφ. ? a124a4a32a50a51a52a22a125a26a126a127a128a10a129a130a131a255a0a1a2a26a132a133a124a134a105a10a95a135a136a21a125a5a137a11a138a139a1a2a254 a3a0a1a2a6 a7a140a26a65a141a142a129a32u(r,θ,φ)a56a57a58a7a143 a255a0a1a2a6 ? a7a144a32a145a11a146a99a17a147a129a54a48a10a148a149a11a50a51a52a22a150a26a118a50a151a152a54a153a154a4a32a12a13a14a15a16a6 ? a7a17a67a155a156a129a18a157a158a22a69a70a112a115a159a11a6 ? a160 a114a11 a122 a47a97a161a162a35a68R lm(r) a56a41 a95Bessela52a22jl(knlr)a69a70a6 ? a7a144a163a145a56a3a164a165a166a167a53a149a128a35a151a168a125a169a11a170a60a61a62a26a110a171a172a54a4a161a173a11a50a51a74a61a62a3a32a26 a174a175a172a54a4 0a151a17a50a51a74 a6 a67a176a11a163a145a26a162a113a66a67a15a177a178a179a180a10a92a181a182a183a39a184a11a185a186a187a188a26a189a190a186a191a44a176a11a18a60a187a188 a6