Wu Chong-shi a0a1a2a3a4 ( a5) a6 a7 a8 (a9) §24.1 a10a11a12a13 Bessel a14a12 a15a16a17a18a19a20a21a22a23a24 Bessel a25a26a27a28a29a26a30 a24 Bessel a25a26a31 a32a17a18 J 1/2(x)a27 J1/2(x) = ∞summationdisplay k=0 (?)k k!Γ(k + 3/2) parenleftBigx 2 parenrightBig2k+1/2 = radicalbigg 2 pix ∞summationdisplay k=0 (?)k (2k + 1)!x 2k+1 = radicalbigg 2 pix sinx. a33a34a35 J 1/2(x)a36a37a38a25a26a31a39a40a41a42a43a44 J?1/2(x) = radicalbigg 2 pix cosx. a45a46a47a35a48 J ν(x) a24a49a50a51 a43a52a53a54a55a56parenleftbigg 1 x d dx parenrightbigg xνJν(x) = xν?1Jν?1(x), parenleftbigg ?1x ddx parenrightbigg x?νJν(x) = x?(ν+1)Jν+1(x), a57a58a34a59a60 x?n+1/2J?n+1/2(x) = parenleftbigg1 x d dx parenrightbiggn x1/2J1/2(x) = parenleftbigg1 x d dx parenrightbiggnradicalbigg2 pi sinx, x?n?1/2Jn+1/2(x)= parenleftbigg ?1x ddx parenrightbiggn x?1/2J1/2(x)= parenleftbigg ?1x ddx parenrightbiggnradicalbigg2 pi sinx x . a61a62a35a63a64a20a50 a28a29a26a30 Bessela25a26a65a36a37a38a25a26 a35 a65a36a66a25a26a67a68a69a25a26 a24a70a71 a25a26a31 a72a73a35 J n+1/2(x)a74J?(n+1/2)(x)a36a75a76a77a52 a24a35 W[Jn+1/2(x), J?(n+1/2)(x)] = (?)n+1 2pix. a78N n+1/2(x)a74J?(n+1/2)(x)a75a76a79a52 a35 Nn+1/2(x) = cos(n + 1/2)pi ·Jn+1/2(x) ?J?(n+1/2)(x)sin(n + 1/2)pi = (?)n+1J?(n+1/2)(x). Wu Chong-shi §24.2 a80Bessela81a82 a832a84 §24.2 a85 Bessel a14a12 Helmholtza86a87?2u+k2u=0a88a89a90a91a53a92a93a94a95a96a97 a35a98a99a100a101a59a60a102a103 a93a86a87 1 r2 d dr parenleftBig r2dRdr parenrightBig + parenleftBig k2 ? λr2 parenrightBig R = 0. a88 a20a104a105a106 a92 λl = l(l + 1), l = 0,1,2,···a31 a15a16a57a17a18a107a50 a86a87 a24a108a109a110a111 a31 star k = 0a27 a49a50 a75a76a77a52 a109 a36 rl a67r?l?1 a31 (a112a11320a114) star k negationslash= 0a27 a58a115 a95a116x = kr a67y(x) = R(r)a35a117a86a87a95a118 1 x2 d dx parenleftBig x2 dydx parenrightBig + bracketleftBig 1? l(l + 1)x2 bracketrightBig y(x) = 0. a107a50 a86a87a119a118a89 Bessela86a87 a35a120a24a121a122 a67 Bessela86a87a123 a102 a79a124a31 star a89 Bessel a86a87a41a125 a49a50 a29a126 a35a20a50 a36 x = 0 a35a127a128a29a126 a35a20a50 a36 x = ∞a35a123 a127a128 a29a126 a35 a41a67 Bessela86a87a79a39a31 star a61a62a35a58a34a129a130a117a120a131a118 Bessela86a87a31 a132a133a60a107a50 a86a87a88 x = 0a126 a24a134 a91a86a87 ρ(ρ?1) + 2ρ?l(l + 1) = 0, a61a78a134 a91a118 ρ1 = l a67ρ2 = ?(l + 1) a35 a67 Bessela86a87 a24a134 a91 ρ = ±ν a135a39 a35a136a137a138a115 a95a116 y(x) = v(x)√x , a107 a40 a35a58a34a139a140a35 v(x)a24a103 a93a86a87a88x = 0a126 a24a134 a91 a57a141 a95a118 ρ = ± parenleftbigg l + 12 parenrightbigg , a67Bessela86a87 a24a22 a126a142a143 a20 a40a31 a107 a40 a35 v(x)a33a144a145a24a103 a93a86a87 a57 a36 1 x d dx parenleftbigg xdvdx parenrightbigg + bracketleftbigg 1? (l + 1/2) 2 x2 bracketrightbigg v = 0. a127 a36l + 1/2a30 a24 Bessel a86a87a31 a120a24a49a50 a75a76a77a52 a109a57 a36Jl+1/2(x) a67 Nl+1/2(x) a31a88 a62a146a147a47a35a57 a58a34a117 a89 Bessela86a87(17.84) a24 a75a76a77a52 a109a148 a118 jl(x) = radicalbigg pi 2xJl+1/2(x) = √pi 2 ∞summationdisplay n=0 (?)n n!Γ(n + l + 3/2) parenleftBigx 2 parenrightBig2n+l nl(x) =(?)l+1j?l?1(x) = radicalbigg pi 2xNl+1/2(x) =(?)l+1 √pi 2 ∞summationdisplay n=0 (?)n n!Γ(n?l + 1/2) parenleftBigx 2 parenrightBig2n?l?1 , a93a149a119a118la30a89 Bessela25a26a67a89 Neumanna25a26a31 Wu Chong-shi a150a151a152a153a154 ( a155) a156 a81 a82 (a157) a833a84 a158a159a50 a89 Bessela25a26a67a89 Neumanna25a26(a130a121a112 a130 24.1)a24a160a161a122 a36a27 j0(x) = sinxx , n0(x) = ?cosxx , j1(x) = 1x2parenleftbigsinx?xcosxparenrightbig, n1(x) = ? 1x2parenleftbigcosx + xsinxparenrightbig, j2(x) = 1x3 bracketleftBigparenleftbig 3?x2parenrightbigsinx?3xcosx bracketrightBig ; n2(x) = ? 1x3 bracketleftBigparenleftbig 3?x2parenrightbigcosx + 3xsinx bracketrightBig . a16224.1 a163Bessela164a165jl(x)a166a163Neumanna164a165nl(x)a167a168a169a170a171a172a173a174a175a176a170 y = ±1/x a21 a124a177 a35 a41a178 a58a34a179a180 a89 Hankela25a26 h(1)l (x) = jl(x) + inl(x), h(2)l (x) = jl(x)?inl(x). a181 24.1 a117 a25a26eikr cosθ a182Legendrea183a184 a122a185a186 a31 a187 a188 eikr cosθ = ∞summationdisplay l=0 cl(kr)Pl(cosθ), a128a185a186 a53a26 cl(kr) = 2l+12 integraldisplay 1 ?1 eikrxPl(x)dx = 2l+12 ∞summationdisplay n=0 (ikr)n n! integraldisplay 1 ?1 xnPl(x)dx. a189a190 a113 19a114a113 4 a16a24a191a192a35a57 a125 cl(kr) = 2l + 12 ∞summationdisplay n=0 (ikr)l+2n (l + 2n)! integraldisplay 1 ?1 xl+2nPl(x)dx = 2l + 12 il ∞summationdisplay n=0 (?)n (l + 2n)!(kr) l+2n · (l + 2n)! 2l+2n n! √pi Γ(n + l + 3/2) = 2l + 12 il√pi ∞summationdisplay n=0 (?)n n!Γ(n + l + 3/2) parenleftbiggkr 2 parenrightbiggl+2n =(2l + 1)il jl(kr). Wu Chong-shi §24.2 a80Bessela81a82 a834a84 a33a34a35a193a194a57 a125 a185a186a122 eikr cosθ = ∞summationdisplay l=0 (2l + 1)il jl(kr)Pl(cosθ). a195a196 a61 a118eikr cosθ = eikz a36Helmholtza86a87 a24a109 parenleftbig?2 + k2parenrightbigeikr cosθ = 0, a136a137 a125 eikr cosθ = ∞summationdisplay l=0 Aljl(kr)Pl(cosθ). a197 a88 a24a110a111 a36a198a199 a179 a44a53a26 Al a200 Aljl(kr) = 2l + 12 integraldisplay 1 ?1 eikrxPl(x)dx = 2l + 12 bracketleftBigg 1 ikre ikrxPl(x) vextendsinglevextendsingle vextendsinglevextendsingle 1 ?1 ? 1ikr integraldisplay 1 ?1 eikrxPprimel(x)dx bracketrightBigg = 2l + 12 1ikr bracketleftbigeikr ?(?)le?ikrbracketrightbig+ O parenleftbigg 1 r2 parenrightbigg . a19a20 a86a201 a35 jl(kr) = 1kr cos parenleftbigg kr ? 12 parenleftbigg l + 12 parenrightbigg pi? pi4 parenrightbigg + O parenleftbigg 1 r2 parenrightbigg = 1kr cos parenleftbigg kr ? l + 12 pi parenrightbigg + O parenleftbigg 1 r2 parenrightbigg = 12kr bracketleftbig(?i)l+1eikr + il+1e?ikrbracketrightbig+ O parenleftbigg 1 r2 parenrightbigg = 12kr(?i)l+1 bracketleftbigeikr ?(?)le?ikrbracketrightbig+ O parenleftbigg 1 r2 parenrightbigg , a61a62a35 Al (?i) l+1 2 = 2l + 1 2i a202 Al = (2l + 1)il, a193a194a57a59a60a185a186a122 eikr cosθ = ∞summationdisplay l=0 (2l + 1)il jl(kr)Pl(cosθ). a41 a58a34a203a204a107a50a185a186a122a20a50a205a206a109a207 a27a208a209a210a211a212a209a210a213a214a31 a107 a36 a61 a118 a35a215a216a179a179 a79a217 a24 a97 a218a61a219 a118 e?iωt a35a220 r a67θ a118a89a90a91 a35a128a47a122a221a222 a36a223θ = 0( a202 a127 z a224)a86a223a225a226 a24a227 a201a228 a35 a228a26 a118k a35a78a229a222a230a20a184a231 a24 j l(kr) a128a232 a125a89a201a228 a24 a79a217 a61a219a35 jl(kr) ～ 1kr sin parenleftbigg kr ? lpi2 parenrightbigg . Wu Chong-shi a150a151a152a153a154 ( a155) a156 a81 a82 (a157) a835a84 a0a1a2a3a4 (a1) a233a234a235a236a237a238a239 (a5) a60a197 a88a118a240 a35a98a99a241a101a242a206a243a159a244a245a246a24a247a103 a93a86a87 a179a109a110a111a35a248a249a243a108a109a107a250a179a109a110a111a24 a20a244 a125a251a86a252 a35 a93a94a95a96a252a31 a107a244 a86a252 a35a253a73 a125 a20a179a24a254a190a255a0a35a1 a198 a35a2a108 a86a87a67 a179a109a255a0 a65a36 a75a76 a24a35a61a62a179a109a110a111a24a109a232 a125a3a4a76a31a88a113 15a114a231 a35a98a99a100a101a191a71a232a5a24a108a109a6 a87 a35 a93a7 a243a107 a244a109 a252a8a9 a179a109a110a111a24a2a108 a31 a22 a149a36 a35a100a101a134 a44 (a11215.1 a16)a107a244 a86a252a36a10a42a11a12a13a177 a137a190 a9 a108a109 a247a103 a93a86a87 a179a109a110a111a35 a88 a206a18a47a35a148a14 a9a92a15 a159a50a110a111 a27 1. a16a17a18a19a20a21a22a23a24a25a26 a35a27a28a29a30a35a31a32a33a34a35a36a35 a16a17a18a19a20a23a24a25a26a37 2. a24a26a19a20a38a26a21a22a23a24a39a40a41a42a43a23a44a16a17a45a46a47a48 a35a27a28a29a30a35a31a32a33a34a35a36a35 a16a17a45 a46a21a49a50a38a37 3. a16a17a45a46a21a22a23a24a51a25a52a53a54a31 a55a107a20 a114 a186a56a35a98a99a57a2a55a206a18a47a57a58a107a159a50a110a111a35a55a78 a118a93a94a95a96a252a59 a179a20a50a60a45a24a206a18a146a147 a31 a253a73a35a61a62a63a64a35a107a65a248a249a24 a41a66a36a67a93 a255a0 a31a88 a20a104a205a206a110a111 a231 a35a107a250a255a0 a36a42a11 a144a145a24 a31 §24.3 a68a69a70a71a72a14a12a70a71 1. a73a74a75a73a74a76a77 a188 a88a26a78 K a47a179a180a243 na79a80a81a76a77V a35a120a24a82a83 (a80a81) a190x, y, ···a160a84a31 a58a34a48 a68a79a85a96 a86a218 a231a85a96 a24a87a88a24a89a90 a43a91 a60 n a79a85a96 a86a218 a31a118 a62a35a32a179a180 n a79a85a96 a24 a73a74a31 a8a9 a45n a79a85a96 a86a218( a202 K a118 a45 a26a78)a35a88a92 a179a243a20a93a146{e i, i = 1, 2, ···, n}a94 a194a35a86a218 a231 a24 a63a64a20a50 a85a96xa65 a58a34a190a120 a88 a107a20a93a146a47a24a95a96 ( a90a91) x1, x2, ···, xn a160a84a35 x = x1e1 + x2e2 +···+ xnen = nsummationdisplay i=1 xiei. a8a9 a86a218 a231 a24 a85a96 xa67y a35a193a102a112 a24a97a98a179a180 a118 (x, y) = x1y1 + x2y2 +···+ xnyn = nsummationdisplay i=1 xiyi. a107 a36 a20a50a45 a26a31 a72a73 a125 (x, y) = (y, x) a67 (x, x) ≥ 0, a99a220a35a253a220a100a253x = 0 a97 a35a101 a125 (x, x) = 0a31a88 a62a146a147a47a35a57a58a34a179a180 a85a96 xa24a102a103bardblxbardbl bardblxbardbl = (x, x)1/2. a8a9 a70 n a79a85a96 a86a218a35 a198 a192a104a105a106a47a107a97a98a179a180a35a108a109a110 a44 a35a107 a97 a24 a85a96 a102a103a57a58 a42a135a36 a45 Wu Chong-shi §24.3 a111 a112a113a114a115 a81a82 a113a114 a836a84 a26a31a118 a243a105a116 a85a96 a102a103a104 a36 a45 a26 a35 a135a117a88 a105a116a102a103a179a180a24a158a118 a92 a35a48a97a98a179a180a119 a54a118 (x, y) = x?1y1 + x?2y2 +···+ x?nyn = nsummationdisplay i=1 x?i yi, a120 a231x?i a36xi a24a70a121a122a31 a72a73a35 a88 a70 a85a96 a86a218 a231 a35 (x, y) = (y, x)?. a107 a40 a24a97a98a89a90a72a73 a36a68a79a85a96 a24 a91 a98a24a123a124 a43a91a31a125a178a135a11a12a13a67a126a127 a35a22 a149a36a85a96 a24a97 a98a128a72a129a130 a9 a146a24 a92 a148 a31 a98a99a253a73a131a2a55a97a98a24a132a244a58 a42 a179a180 a231a126a127a44 a120a24a193a15a133a24a2a134 a31 a55a78 a135 a44 a20a50a136a206a131a24a97a98a179a180 (a34a194a57 a119a118 a97a98a136a206) a31 a137a138 24.1 (a179a180 a88 a45 a26a139 a70 a26a78 K a47a24) a85a96 a86a218 a231a85a96xa67y a24a97a98 (x, y)a36 a120a99a24 a91 a96a140a25a26 a35a144a145 a27 1. (x, y) = (y, x)? a37 2. (αx + βy, z) = α?(x, z) + β?(y, z)a35a141a142αa143β a21a46a144K a145a38a146a147a37 3. a148a149a150a151xa35(x, x) ≥ 0a37a152a153a154a152x = 0a155 a35(x, x) = 0 a31 a181 24.1 a215 x = ? ?? ?? ?? x1 x2 ... xn ? ?? ?? ?? a67 y = ? ?? ?? ?? y1 y2 ... yn ? ?? ?? ?? a36 a45 a26a78 a47a24 a15a85a96 a35P a118(a135a179a24)a8a69a156a157 a35 a8a69a158Pii a159 a118 a127a45 a26 a35a128a58a179a180 a85a96xa67y a24 a97a98 a118 (x, y) = parenleftBig x1, x2, ···, xn parenrightBig ? ?? ?? ?? P11 0 ··· 0 0 P22 ··· 0 ... ... 0 0 ··· Pnn ? ?? ?? ?? ? ?? ?? ?? y1 y2 ... yn ? ?? ?? ??. a181 24.2 a45 a95a96 ta24a33a125 a70 a53a26 a24 a183a184 a122a24a160a71a35 a88a183a184 a122 a4a252 a34a161 a183a184 a122 a67 a70 a26 a24a162 a252 a92a163a56 a20a50a70 a85a96 a86a218 a31a135a117a164 a188 0 ≤ t ≤ 1 a31 a215x(t) a67y(t)a36 a62 a85a96 a86a218 a231 a24a49a50 a85a96 ( a202 a183a184 a122)a35a128a120a99a24a97a98a58a34a179a180 a118 (x, y) = integraldisplay 1 0 x?(t)y(t)ρ(t)dt, a120 a231 a241a165 a25a26 ρ(x) ≥ 0a220negationslash≡ 0a31 Wu Chong-shi a166a167a168a169a170 ( a171) a172 a173 a174 (a175) a1767a177 a120a24a22a23a105a121 a36 ρ(x) ≡ 1a35 (x, y) = integraldisplay 1 0 x?(t)y(t)dt. a178a179a180a181a182a183a142 a38a1841a34a185a186a35a39a40a187a188 a35 a189a190 a21a191a38a192a193a38a194a147a195a196 a35 a23a197a194a147a143a198 a199a200 a38 a180a181a201 a21a191a46 a35a202a203 a1843a34a185a186a142a38 a189a204a205a206 a25a207a208a31 a88 a62a146a147a47a35a57a48 (x, x)1/2 = bardblxbardbl a119a118a85a96xa24a209( a202 a85a96x a24 a210a102a103a211) a31 a55a47 a201 a97a98a136a206 a231 a24 a113 1a67a1132a255a2a108a35a58a59 (x,αy) = α(x, y). a61a62 bardblαxbardbl = (αx, αx)1/2 = bracketleftBig αα?(x, x) bracketrightBig1/2 = |α|bardblxbardbl. a63 a199 a20a50 a123a212a85a96a213 a34a120a24a214a57 a56a118 a210a124 a217 a102a103a211a24 a85a96 a35 a139a119a118 a215a216a217 a24 a85a96 a35 parenleftBig x bardblxbardbl, x bardblxbardbl parenrightBig = 1. star a179a180a243a97a98a24a85a96 a86a218 a119a118 a97a98a86a218 a31 star a232a125 a97a98a24a45 a85a96 a86a218 a119a118a218 a159a65a219a86a218 (Euclidean space) a37 star a232a125 a97a98a24a70 a85a96 a86a218 a119a118a220 a86a218 (unitary space) a31 2. a221a222a223 a88a224a225 a243a97a98a179a180a194a35a57a58a34a226a227 a85a96a221a222 a24a89a90 a31 star a253a220a100a253 (x, y) = 0a97 a35a49 a85a96x, y a127a228 a31 star a212a85a96a67 a63 a199a85a96a65 a127a228 a31 a137a13824.2 a215 a8a9 a33 a125 a24i a67ja35(xi, xj) = δij a35a128a119a85a96 a93{x 1,x2,···}a36a221a222a215a216 a24 a31 a52a53a229a23a38a194a147a23a24a21a230a54a231a232a38 a35a202 a21a233a234a235a236a237a198a238a230a54a44a239a240a241a194a147 a35 α1x1 + α2x2 + α3x3 +··· = 0, a242 a23a24a25 αj = 0, j = 1,2,3,···. a243 a40na244a194a147a195a196 a142 a38a150a151a23a44 na197a52a53a229a23a194a147a245a39a40a246a240a247a195a196a38a248 a35a249 a234a221a222 a215a216a250(a192 a249 a221a222a251a252a250) a31 a253a254 a52a53a229a23a248 a35 a231 a190 a31a183 a190 a145a192a191a255a145 a35 a245a51a25a0a1a38a2 a185 a54a31 Wu Chong-shi §24.3 a111 a112a113a114a115 a81a82 a113a114 a838a84 3. a3a4a223 a137a138 24.3 a88a125a5a79a85a96 a86a218 a231 a35 a198 a192a20a93a127a228a6a20a24 a85a96(a119a118 a20a50 a221a222a215a216a80a81a7)a35 a99 a135a8a9a88 a19a20a50a10a11a24a127a228a6a20 a85a96 a160 a94a231 a35a128 a119 a138a127a228a6a20 a85a96 a160 a36a3a4 a24 a31 star a88a125a5a79 a24 a85a96 a86a218 a231 a35a20a50 a142a12a13a14 a228a6a15 a85a16 a160a17 a85a16a13a18a19a20a21a22 a86a23 a13a24a19a25a26a27 star a28a29a30a31 a17a32a33a33a34a35a36a37a38a39a40 a16a41 a23 a13a24a19 a32a42a43a36a44a45a46a47a48a15a49a50 a12a13a14a51a52 a15a40 a16( a15a49a53a54 a13a55a56a57a58a59a60 a40 a16 a49) a61a62a63a41 a23 a13a24a19 a32a43a64a65a66 a18 a40 a16a41 a23 a13 a15a49a67 a27 star a68 a15 a18a69a70a41 a23 V a17a32a71a15a49 a14a51a52 a15 a13 a40 a16 {xi,i = 1, 2, ···, k}, a44 a62a63a72 a36a73a50 a12 a32a36a15 a18a74a75a76a28a13a30a31a27 a75a77a13a62a78a79 a71a80a81a82 a18a83 1. a84a85a86a84x = 0a87 a32(x i, x) = 0, i = 1, 2, ···, k a27 2. a88a89a90a91a92x ∈ V a32a93a94x = ksummationdisplay i=1 (xi, x)xi a27 3. Bessela95a96a97a98a92a96a99a100a101 a32a102 a88a89a90a91a92 x ∈ V a32a93a94 bardblxbardbl2 = ksummationdisplay i=1 |(xi, x)|2. 4. Parsevala103a104a100a101 a32a102 a88a89a90a91a92 x,y ∈ V a32a93a94 (y, x) = ksummationdisplay i=1 (y, xi)(xi, x). a72a105a106 a36 a14a51a52 a15a40 a16 a49a50 a12a13a107a108a20 a44a109a110a32a111a43a112a36a50a113a114a115 a13a27 4. a116a117a118a119 a116a117a118a119 a36a15a120a53a54 a13 a40 a16a41 a23 a83a118a119a121a122a123a124a116a117 a32a125a126a127a128a129a32a36a130a131 a68 a15a130a132a23( a133 a126a130a134a135a32a136 a133a137 a132a23 a ≤ x ≤ b) a138a13a139a140a141a19f(x)a32a34a142a70a108 integraldisplay b a vextendsinglevextendsinglef(x)vextendsinglevextendsingle2dx a143a68(a144a116a117f(x)a145 a146a147a148a149) a27 star a130a131a150a151 f1 a152 f2 a13a153a79f1 + f2 a154 a36a155 a141a19a25a153 a32 (f1 + f2)(x) = f1(x) + f2(x), star a150a151f a152 a139a19αa13a19a156αf a36 (αf)(x) = αf(x), a66a157 a13a158a159a160a70a141a19a13a161a162 a32a163a164 a153a79 a152 a19a156 a36a165 a137a13 a32a111a166 a13 a126a167a168a15 a18 a40 a16a41 a23 a27 a53 a78 a36a32a111 a133 vextendsingle vextendsinglef1(x) + f2(x)vextendsinglevextendsingle2 +vextendsinglevextendsinglef1(x)?f2(x)vextendsinglevextendsingle2 = 2bracketleftbig|f1(x)|2 +|f2(x)|2bracketrightbig, Wu Chong-shi a169a170a171a172a173 ( a174) a175 a176 a177 (a178) a1799a180 a181a182a32a155 a18a158a159a160a70a141a19a183 a152a184 a36 a158a159a160a70a13 a32 vextendsinglevextendsinglef 1(x) + f2(x) vextendsinglevextendsingle2 ≤ 2bracketleftbig|f 1(x)|2 +|f2(x)|2 bracketrightbig. 5. a116a117a121a185 a148 a186a187 24.4 a136f 1(x)a152 f2(x)a36a141a19a41 a23a17 a13 a155 a18a141a19 a32 a72a105a13a69a70 a36 (f1, f2) = integraldisplay b a f?1(x)f2(x)dx. a188 a89 vextendsingle vextendsinglef1(x)vextendsinglevextendsingle2 +vextendsinglevextendsinglef2(x)vextendsinglevextendsingle2 ?2vextendsinglevextendsinglef1(x)vextendsinglevextendsingle·vextendsinglevextendsinglef2(x)vextendsinglevextendsingle = bracketleftbig|f1(x)|?|f2(x)|bracketrightbig2 ≥ 0, a189a190 vextendsinglevextendsinglef? 1(x)f2(x) vextendsinglevextendsingle = vextendsinglevextendsinglef 1(x) vextendsinglevextendsingle·vextendsinglevextendsinglef 2(x) vextendsinglevextendsingle ≤ 1 2 bracketleftbig|f 1(x)|2 +|f2(x)|2 bracketrightbig, a191a192a193a194 integraldisplay b a vextendsinglevextendsinglef? 1(x)f2(x) vextendsinglevextendsingledx a195a196a27a197 a189a198 vextendsinglevextendsingle vextendsingle integraldisplay b a f?1(x)f2(x)dx vextendsinglevextendsingle vextendsingle ≤ integraldisplay b a vextendsinglevextendsinglef? 1(x)f2(x) vextendsinglevextendsingledx, a191a192a32a199a200f 1(x)a201f2(x)a202a103a203 a193a32a204a205a206a207 a92a208 a193a209a210a211 a195a196a27 a68 a166a67a212 a138 a32 a160 a182a130a131 a141a19 f(x)a13 a144 a213a214a149 bardblfbardbl = (f, f)1/2, a215 a133a141a19f(x)a13a216a117a27 star a30a31 a36 a83a217a218a219a121a185 a148a186a187a220a32a221a222 (f, f) = 0a32f(x) a223a224a225a226a217a227a228a229a119a230a231a231a232 0a27a233 a28 a36a32f(x) a160 a182 a68 a71a234 a18a235a138 a35 a133 0a32a236a66a237a35a133 0a238a141a19a140 a34a35a239a240a241 a70a108a140 a32a181a182 a184 a160 a182a71 (f, f) = 0 a27 star a242 a126a128a129a32a243a244(f, f) = 0a32a245f(x) a160 a182 a68a246 a214 a133a247a238a235a161a138a248a74a247a140a27 a181a182a249a250a129 (f, f) = 0 a251a252a253f(x) a254a255a231a231a232 0a27 star a243a244a0a77a1 a131 a238a247a141a19a238a2a3 a32a4a5a6 a254a255a231a231a232 0a121a116a117a7a232a8a116a117 a32a9a10a32a66a11a130a131 a238 a69a70 a112 a154a12 a162a69a70a13a14a15a238a16 3a109a44a17a27 a18a19 a208 a193 a92 a211a20a21 a203 a192a22a210a23a24a25a198 (f1, f2) = integraldisplay b a f?1(x)f2(x)ρ(x)dx, a26 a98ρ(x) ≥ 0a85negationslash≡ 0a27a27a28 a32a94a29a30 a97a31a32 a200a33a34a35 a92a36a37a27a38a39a40 a32a29 a89 a18a19 a202a103a203 a193 a92 a200a41a209a35a42 a36a37 a198a200a41a193a194 integraldisplay b a vextendsinglevextendsinglef(x)vextendsinglevextendsingle2ρ(x)dx a195a196a27 Wu Chong-shi §24.3 a43 a44a45a46a47 a176a177 a45a46 a17910a180 6. a116a117a121a48a49a50a51a52 a53 a141a54f(x) a152 g(x)a55a56 (f, g) ≡ integraldisplay b a f?(x)g(x)dx = 0, a245a215 a72a105 a36 ( a68 a132a23 [a, b] a138)a48a49a238a27 a53 a141a54 f(x) a152 a72a57 a58 a238a69a70 (f, f) ≡ integraldisplay b a f?(x)f(x)dx = 1, a59a60 bardblfbardbl = 1, a245a215f(x)a36 a50a51a61a238a27 a43a53a163a164 a141a54a161a162 {fi}a32a62a71 (fi, fj) ≡ integraldisplay b a f?i (x)fj(x)dx = δij, a245a215a166 a141a54a161a162 a36 a48a49a50a51a238a27 a63 24.3 a141a54a161a162 braceleftbigeinx/√2pi, n = 0,±1,±2,···bracerightbig a68 a132a23 [?pi, pi] a138 a36a64 a51a52a65a238a27 7. a48a49a50a51a116a117a66a121a67a68a52a69a70 a243a244a163a164 ( a141a54a41 a23 a15a238) a71a72a141a54 f(x) a32a73a160 a182a74a75a168a64 a51a52a65a141a54a161{fi, i = 1,2,···}a238 a57a58 a49 a162 f(x) = ∞summationdisplay i=1 cifi(x), (maltesecross) a245a215a64 a51a52a65a141a54a161{fi, i = 1,2,···}a36a50a76a238a27 a64 a51a52a65a141a54a161a238 a50a76 a58a2a3 a73a36 a152 a71a72a141a54 a36a73 a160 a182a77a78 a141a54a161a79a80a25a81a82a238a27 star a16a65 a32 a65a83 a129 a61 a32a66a84 a141a54a161a85 a78a252a71 a59a86a87 a84 a141a54 a32a73a245 (maltesecross) a88 a35 a160 a250a163 a71a72f(x)a89 a168a65 a27 a66 a65a233a28a90a91a92a105 a32 a141a54a41 a23a36 a59a86a24a238 a40a93 a41 a23 a27 star a16a94 a32(maltesecross) a88a85 a78a163a132a23[a, b] a69a238a95a65a235xa106 a168a65a32a96a97a129a32a163a164a132a23[a, b] a69a238a95a65a235xa32 a98 a54 ∞summationtext i=1 cifi(x)a106a85 a78a99a100a164 f(x) a27 a66a101a99a100 a58 a215 a133a102a235 a99a100 a27 star a133a103 a152 a1 a131 a247a141a54a238a2a3a25a104a85 a32a112 a160 a182a105(maltesecross) a88a14a106a133a107a108 a155a109 a25a110a65 a84 a1 a131 a238a247a141a54 a32 a111a112a113a129a32a105a98 a54 ∞summationtext i=1 cifi(x)a14a106a133a158a89a99a100a164 f(x)a32a60 limn→∞ integraldisplay b a vextendsinglevextendsingle vextendsinglef(x)? nsummationdisplay i=1 cifi(x) vextendsinglevextendsingle vextendsingle 2 dx = 0. (#) star a16a114 a32a115 a141a54a161{fi, i = 1,2,···}a238 a64 a51a52a65a58 a32 a160 a17a116 ci = integraldisplay b a f?i (x)f(x)dx = (fi, f). (circleasterisk) Wu Chong-shi a169a170a171a172a173 ( a174) a175 a176 a177 (a178) a17911a180 star a16a117 a32 integraldisplay b a vextendsinglevextendsingle vextendsinglef(x)? nsummationdisplay i=1 cifi(x) vextendsinglevextendsingle vextendsingle 2 dx = (f, f)? nsummationdisplay i=1 c?i(fi, f)? nsummationdisplay i=1 ci(f, fi) + nsummationdisplay i=1 vextendsinglevextendsinglec i vextendsinglevextendsingle2 = (f, f)? nsummationdisplay i=1 vextendsinglevextendsinglec i vextendsinglevextendsingle2, a111a166a32a249a44 a141a54a161{fi, i = 1,2,···} a36a50a76 a238 a32a9a10a32a118a119 (#) a88 a32 a154 a71 (f, f) = ∞summationdisplay i=1 vextendsinglevextendsinglec n vextendsinglevextendsingle2 = ∞summationdisplay i=1 vextendsinglevextendsingle(f i, f) vextendsinglevextendsingle2. a66 a154 a36 a116a117a66{fi, i = 1,2,···}a121a67a68a52a120a121 a32 a59 a215 Parseval a146a122 a27 a105 a141a54a161{fi, i = 1,2,···}a238 a109a110a123a124a32a125a136 a72 a36a64 a51a52a65a238 a32a236a35 a65 a130a50a76a32 a184a126a127a128 a77 a66 a84 a141a54a161a238a57a58 a49 a162 ∞summationtext i=1 aifi(x)a61a129a130f(x)a27a76a68a238a30a31 a36 a83 a243a131a132a133a49 a162a82a54 ai (a134na59a60)a32 a160 a182a116a135 a55a136a129a130 a32a137a138 a110 vextenddoublevextenddouble vextenddoublef(x)? nsummationdisplay i=1 aifi(x) vextenddoublevextenddouble vextenddouble 2 ≡ integraldisplay b a vextendsinglevextendsingle vextendsinglef(x)? nsummationdisplay i=1 aifi(x) vextendsinglevextendsingle vextendsingle 2 dx a248a139a140a141 a142a143 Parseval a159a144a238a145a146 a32 a160 a182a17a116 integraldisplay b a vextendsinglevextendsingle vextendsinglef(x) ? nsummationdisplay i=1 aifi(x) vextendsinglevextendsingle vextendsingle 2 dx = (f, f)? nsummationdisplay i=1 a?i(fi, f)? nsummationdisplay i=1 ai(f, fi) + nsummationdisplay i=1 vextendsinglevextendsinglea i vextendsinglevextendsingle2 = (f, f)? nsummationdisplay i=1 a?ici ? nsummationdisplay i=1 aic?i + nsummationdisplay i=1 a?iai = (f, f) + nsummationdisplay i=1 vextendsinglevextendsinglea i ?ci vextendsinglevextendsingle2 ? nsummationdisplay i=1 c?ici, a111a166a32a147a i = ci ≡ (fi, f)a148 a32a138 a110a65 a130 a248a139a140a140 a32 (f, f)? nsummationdisplay i=1 vextendsinglevextendsinglec i vextendsinglevextendsingle2 ≥ 0, a43a142a32a149a253a150 a54 na238a151a153 a32a138 a110a152a61a152a140a27 a236 a59a153 a243a131a32a73a71 (f, f) ≥ ∞summationdisplay i=1 vextendsinglevextendsinglec i vextendsinglevextendsingle2. a66a64a154a36 a116a117a118a119a155a121 Bessel a224a156a157a27 a114a158a163 a85 a164 a141a54a161 a36a50a76 a238a159a160a27 a116a117a118a119a121a67a68a52a69a70a27a161a162 a188a163a164 a208a92 a18a19a165 a100a92 Cauchy a166a167a92a168a169a170a171a172a196 a42 a163a164 a208 a32a173a174a42 a163a164a198a175a176 a92a27 a202a103a203 a193a18a19a177 a100a92 a163a164 a40 a175a176 a92a27 a178a179a32a180a175a176 a92a208 a193a163a164a174a198Hilbert a163a164 a27a27a181a182a183 a32 a196a184a185a186a98 a94a25a187 a92 a35a188 a27 a189a190 a92a191a192 a32a193a194a195a196 a40a196 Hilbert a163a164a92a197a198a208 a22a199 a92a27