δ a0 a1 star a2a3a4a5a6a7a8a9a10 4 star a11a12a13a14a15a16a17a18 Green a19a20a21a22a23a24 a25a26 2 a8a9a27a28 a0a1a2 δ a3 a4 1 a5a6a7 δ a8 a9 star a10a11a12a13a14a15 a16 a17a18a19 a20 δ a17a18a21 star δ a17a18a22a23a24a25a26a27P. A. M. Diraca28a29a30a31a15 a20 a32a33a34a24a25a26a35a36a37a38a39 a15a40a41 a21a42 a43a44 a41a45a46a47 a24a25a26a35 a15a12a48a49a50 a20 a51a52 a49a53a50a54a49a55a56a54 a57a58a59a60 a20 a24a25a61a62a63a64a21 star a32a18a26a65a20δ a17a18a43a44a66a67a68a69a17a18a12a70a31a71a72a73 a20 a52 a31a71a74a75a76a77a75a78a79 a20 a80a81 a40 a41a45a82a83a74a75a84a85 a20a86a87a88a89 a15a90a91a76 a24a25 a90a92 a22 a12a93a15 a21 star a72a41δ a17a18a20a43a44a94a95a96a97a25a36a98a15a18a26a24a25a99a100a20a101a102a103a104a15a105a106a21 star δ a17a18a22a12a107a16 a108a109a19 a15 a17a18 a20 a110a111 a16 a112a113a19 a15 a18a26a114a115a22a116a117a25 a83a15 a21a42 a15a118a119 a18a26 a25 a92 a20a120a121a122 a39a17 a75a123a15a124a125 a21 star a126a127a128a129 a24a25a26 a15a130a131a132a133 a20 a30a31 δ a17a18a15 a114a115 a20 a10a11 a42 a15a134a135a126a15a124a125 a122a136a137a138 a40 a41 a21 §10.1 δ a3 a4 2 §10.1 δ a139 a140 a67a94δ a17a18 a15 a24a25a141a142 a20 a29a143a92a49 a59 a54 a51a52 a49a55a56a15a55a56a75a144a145a146 a17a18 a15 a18a26a147a148a21 a94 a149a150a151a152 a20a153a120 a143a92a12a154a155a156 a21 a15710.1 a158a159a160a161a162a163a161a162a164a165 a166a16710.1 a20 a168a169a170a171a172a173a1740 < x < l a175a176a177a178a179a180a181a182a183a184a185 a20 a186 a182a187a188 1a189a190a191 a20 a169 a175a176 a192a170 a182a183 a20a193 a182a183a194a195a196a197a188 δl(x) = ? ??? ? ??? ?? 0, a198x < ?l2; 1 l , a198? l 2 < x < l 2; 0, a198x > l2. a199a200a201a202a203 a189 a169?l/2 < x < l/2 a177a204a205a181a196a197 f(x)a20a206a207a208a209a210a211a20a178integraldisplay ∞ ?∞ f(x)δl(x)dx = f(θl), ?1/2 ≤ θ ≤ 1/2. a212a213a174 a20a214 a184a215a216 a203 a210a217±∞ a21a218a219a < ?l/2, b > l/2 a20a220 a178integraldisplay b a f(x)δ(x)dx = f(θl), ?1/2 ≤ θ ≤ 1/2. a221 a188a222a215a223a224 a20 a198 l → 0a225 a20a220a226a227a228 a182a183a181a194a195a196a197 a20a229 a188 δ(x) = lim l→0 δl(x) = ? ??? ??? 0, a198x < 0; ∞, a198x = 0; 0, a198x > 0. a230a231 a20 a199a200a201a202a203 a189 a169 x = 0 a228 a204a205a181a196a197 f(x)a20a178integraldisplay ∞ ?∞ f(x)δ(x)dx = f(0). a212a213a174 a20a214 a184a215a216 a203 a210a217±∞ a21a218a219a < 0, b > 0 a20a220 a178integraldisplay b a f(x)δ(x)dx = f(0). a232a233 a20 a234a43a44a235a236a237a238 a15a55a56a75a144 a17a18a239a240 a94 a136a241a242a243a244a245 a17a18 a20 a246a247a248a65a249 a15a143a92 a21 a67a94 a42 a96 a15 a103a250 a155a156 a20 a95a96a251a252 a88a89a253 a70a15a90a91 a21 a0a1a2 δ a3 a4 3 star δ a196a197 a20 a254 a216 a217a255a0 a202a1a2 a181a196a197a3 a4a254a5 a178a6a7a196a197a8a9 a10 a187a11a176a181 a199a12a13a14 a20 a15a16a17 a20 a4 a6a7a181 a199a12a13a14 δ(x) = braceleftBigg 0, a198x negationslash= 0; ∞, a198x = 0 a169 a255a0 a202a1a2 a217 a5 a178 a202a1 a181 a21 star δ a196a197a18a19a181 a217 ( a201a202a20a21a22) a196a197a23a24a181a222a215 a21a4a25 a6a7a181 a16a196a197 a209 a19a218 a217 a169 a214 a184a26a27 a208 a28 a178 a202a1a21 integraldisplay ∞ ?∞ f(x)δ(x)dx = f(0), a29a30 a217 integraldisplay ∞ ?∞ δ(x)dx = 1. star a31a189 a214 a184 a12a32 a211a33 a188 integraldisplay ∞ ?∞ f(x)δ(x)dx = lim l→0 integraldisplay ∞ ?∞ f(x)δl(x)dx. a34a35 a27a181a36a195a37a38 a20a39a40δ a196a197a181a41a181 a20a42 a169a200a43a44a199 a196a197a23a24 a40a45 a22 a214 a184 a35 a27a54 a230a46a47 a222 a215a181a48a49 a21a50a200 a196a197a23a24 a217 a50a51 a178a52a53a54a181a204a205a55a56a181a196a197a57a58a181 a20 a25a59 a20 a169a35 a27 a208 a21a59a60δ a196 a197a198 a221(a201a202a20) a204a205 a21a22 a181a196a197a61 a211a20 a62a63a21a59 a210 a1 δ a196a197a181a64a197δprime(x)a3 a199a200a169x = 0 a228 a204a205 a254 a178a204a205a64a197a181 a201a202 a196a197 f(x) a20 a178 integraldisplay ∞ ?∞ f(x)δprime(x)dx = f(x)δ(x) vextendsinglevextendsingle vextendsingle ∞ ?∞ ? integraldisplay ∞ ?∞ fprime(x)δ(x)dx = ?fprime(0). a31a65 a20a220 a60δ a196a197a198 a221a66 a255 a181a204a205a196a197 a203a67 a40a45 a184a68 a214 a184 a21 star δ a196a197 a254 a216 a217 a6a7 a66 a255 a181a197 a209 a11a176a181 a199a12a13a14 a20 a69a254 a216a70 a66 a255 a196a197a71 a67a51 a178a72 a203 a54 a73 a210 a181a18 a74a75a21 star a76 a217 a51 a178 lim l→0 integraldisplay ∞ ?∞ f(x)δl(x)dx = f(0) a55a56a181a196a197a23a24 δl(x)a20a15a217a51a178 limn→∞ integraldisplay ∞ ?∞ f(x)δn(x)dx = f(0) a55a56a181a196a197a23a24 δn(x)a20a4a77a181a222a215a78 a217 δ a196a197 a21 §10.1 δ a3 a4 4 n√ pie ?n2x2 n pi 1 1 + n2x2 sinnx pix a15710.2 δ a79a80a163a81a82a83a84a85a86 star a178 a13δ a196a197a181a87 a75 a20 a69a12 a198 a34 a214 a184 a202a1a2a88 a211a33 a21a89a166 xδ(x) = 0, δ(?x) = δ(x), δprime(?x) = ?δprime(x), δ(ax) = 1|a|δ(x), g(x)δ(x) = g(0)δ(x) a220 a184a30 a12a32 a211a33 a188 integraldisplay ∞ ?∞ f(x)xδ(x)dx = 0, integraldisplay ∞ ?∞ f(x)δ(?x)dx = integraldisplay ∞ ?∞ f(x)δ(x)dx, integraldisplay ∞ ?∞ f(x)δprime(?x)dx = ? integraldisplay ∞ ?∞ f(x)δprime(x)dx, integraldisplay ∞ ?∞ f(x)δ(ax)dx = integraldisplay ∞ ?∞ f(x) bracketleftbigg 1 |a|δ(x) bracketrightbigg dx, integraldisplay ∞ ?∞ f(x)g(x)δ(x)dx = integraldisplay ∞ ?∞ f(x)bracketleftbigg(0)δ(x)bracketrightbigdx. star δ a196a197a90 a21a59 a18a19a58a91a87a196a197a181 a22a92a21a50a200 integraldisplay x ?∞ δ(x)dx = η(x), a93a94 a20 δ(x) = dη(x)dx . a0a1a2 δ a3 a4 5 star a69a21a59a199δ a196a197 a221 Laplacea10a95 a20 δ(t?t0) equaldotleftright integraldisplay ∞ 0 δ(t?t0)e?ptdt = e?pt0, t0 > 0. star δ a196a197a69a21a59a18a19a58a91a87a196a197a181 Fouriera214a184a21a93a188integraldisplay ∞ ?∞ δ(x)e?ikxdx = 1, a25a59 a20a206a207 Fourier a10a95 a181a96a97a98 a75 a20 a178 δ(x) = 12pi integraldisplay ∞ ?∞ eikxdk. a99a100a101a102a100 δ a103a104 star a169a105a106a174(x0, y0)a228a61a178a203a189a190a191a228a182a183a20a71a107a20a4a181a194a195a184a185a196a197 a220a217 δ(x?x0)δ(y?y0) a21 star a169a108a109a110a176 (x0, y0, z0)a61a178 a203 a189a190a191 a228 a182a183 a20 a4 a181a194a195a184a185a196a197 a220a217 δ(x?x0)δ(y ?y0)δ(z ? z0)a21 star a34a108a109a110a176a37a38 a20 a25a111a203a109 a228 a182a183 a12a32 a217 a108a109a110 a176a177a181 a106 a182a183a112 a113 a109 a228 a182a183 a220a217 a108a109a110 a176a177 a181 a173 a182a183 a21 a114 10.1 a115a116 ?21r = ?4piδ(r), a117 a208 ?2 ≡ ? 2 ?x2 + ?2 ?y2 + ?2 ?z2 a118 a188Laplacea27a119 a20r = radicalbigx2 + y2 + z2, δ(r) = δ(x)δ(y)δ(z) a21 a120 a121 a70a122 a106a123 a7a181 a20 a76 a217a124a125 δ a196a197a181a87 a75 a78 a12a32a34 a214 a184 a202a1a2a88 a211a33a20a42 a12a32a88 a115a116 integraldisplayintegraldisplayintegraldisplay V ?21rdxdydz = braceleftBigg 0, a198r = 0 negationslash∈ V ; ?4pi, a198r = 0 ∈ V. a198r negationslash= 0a225 a20 a172a126a22a92a21 a226 ? ?x 1radicalbig x2 + y2 + z2 = ? x (x2 + y2 + z2)3/2 , ?2 ?x2 1radicalbig x2 + y2 + z2 = 3x2 ?parenleftbigx2 + y2 + z2parenrightbig (x2 + y2 + z2)5/2 . a127 a211a20 ?2 ?y2 1radicalbig x2 + y2 + z2 = 3y2 ?parenleftbigx2 + y2 + z2parenrightbig (x2 + y2 + z2)5/2 , ?2 ?z2 1radicalbig x2 + y2 + z2 = 3z2 ?parenleftbigx2 + y2 + z2parenrightbig (x2 + y2 + z2)5/2 . §10.1 δ a3 a4 6 a108a75a128a129 a20a42a226 ?21r = 0, r negationslash= 0. a31 a67 a220 a115 a226 a3a198 a214 a184a130 a214 V a177a216a131a132a133 a228 r = 0 a225 a20a214 a184a134a188 0a21 a198 a214 a184a130 a214V a177a131a132a133 a228r = 0 a225 a20 a50a200 a196a1971/ra169r = 0a228a216a21a64a20a174a106a181a135a136a216a58a137 a21 a31 a225a216a138a139V a220 a47 a188a140a189 (a108a109)a110a176a21a141a142a226a227integraldisplayintegraldisplayintegraldisplay ?21rdxdydz = lima→0 integraldisplayintegraldisplayintegraldisplay ?2 1√r2 + a2dxdydz = ? lima→0 integraldisplayintegraldisplayintegraldisplay 3a2 (r2 + a2)5/2 r2drsinθdθdφ = ?12pi lima→0 integraldisplay ∞ 0 a2 (r2 + a2)5/2 r2dr, a143r = atanθ a20a42 a21 a115a116 a174a106 a181 a214 a184a8 a a170a13 a20 a231 integraldisplayintegraldisplayintegraldisplay ?21rdxdydz = ?12pi integraldisplay pi/2 0 tan2θparenleftbig 1 + tan2θparenrightbig3/2 dθ = ?12pi integraldisplay pi/2 0 sin2θcosθdθ = ?12pi· 13sin3θ vextendsinglevextendsingle vextendsinglevextendsingle pi/2 0 = ?4pi. square a0a1a2 δ a3 a4 7 §10.2 a144a145 δ a139a140a146a147a148a149a150 a151a152δ a196a197a181 a0 a152 a214 a184a18 a74a75 δ(x) = 12pi integraldisplay ∞ ?∞ eikxdk, a15 δ(x) = 12pi integraldisplay ∞ ?∞ coskxdk = 1pi integraldisplay ∞ 0 coskxdk, a69a21a59a35 a27 a210a214 a184 a21a2a106 a255 a48a153a189 a89a154 a37 a17 a116 a203a155 a181 a35 a27a156a157 a21 a114 10.2 a35 a27 a214 a184 integraldisplay ∞ ?∞ sinx x dx a21 a158 a159a160a161a162 a214 a184 F(λ) = integraldisplay ∞ ?∞ sinλx x dx, a163a164 a178 Fprime(λ) = integraldisplay ∞ ?∞ cosλxdx = 2piδ(λ). a25a59 F(λ) = 2piη(λ) + C, a117 a208C a188 a214 a184 a0 a197 a20a165a210 a21a166 a198 λ > 0a225 a20 F(λ) = 2pi+ C, F(?λ) = C. a159a160 a227F(λ)a217λ a181a167a196a197 a20 F(?λ) = ?F(λ), F(0) = 0, a42 a21 a210 a7C = ?pia21a93a94 F(λ) = ? ??? ??? pi, λ > 0; 0, λ = 0; ?pi, λ < 0. a29a30 a217a20 a198λ = 1a20a220a178 integraldisplay ∞ ?∞ sinx x dx = pi. a114 10.3 a35 a27 a214 a184I = integraldisplay ∞ ?∞ sin2x x2 + x + 1dx a21 a158 a39a40 a161a162 a214 a184 F(λ) = integraldisplay ∞ ?∞ eiλx x2 + x + 1dx, a4a168 a52 a22 a184a169a49 ?Fprimeprime(λ)?iFprime(λ) + F(λ) = 2piδ(λ). (star) §10.2 a170a171δ a3a4a172a173a174a175a176 8 a31 a217 a203 a189a29a177a181 a113 a20 a0 a22 a184a169a49a3 a117a178a179a180a181 a132a178 δ a196a197 a21 a31a182a29a177a55a18a183 a169a184 a169 a106 a3 ? a66λ negationslash= 0a58a20δ(λ) = 0a20a84a85 a22a185a186 a15 a20 ? a66λ = 0a58a20F(λ)a22a244a245a15 a20 limε→+0[F(0?ε)?F(0 + ε)] = 0, a187Fprime(λ) a188a189 a244a245a20integraldisplay 0+ε 0?ε bracketleftBig Fprimeprime(λ)+iFprime(λ)?F(λ) bracketrightBig dλ = ?2pi integraldisplay 0+ε 0?ε δ(λ)dλ = ?2pi. a23 a45 F(λ) a32 λ = 0 a49 a244a245a20a190 a66ε → +0 a58 a20 a65a191a192 a193a194a195a196 a76 a194a197a196 a15a77a75a198a199a45 0a20a45 a22 limε→+0Fprime(λ) vextendsinglevextendsingle vextendsingle 0+ε 0?ε = ?2pi. Fprime(λ)a32λ = 0a49a15a189 a244a245a200a20 a201Fprime(λ)a32λ = 0 a49a15 a192a202 a103a250a203 a32a187 a189a204 a60 a20 a205a206a207a208 a209a195a210 a74a75a84a85 (star)a15a211 a185a186a196 a94 δ a17a18a21 a183 a169a212 a227 a22 a184a169a49 (star)a181a213 a33 a174a21a93 a188a198 λ negationslash= 0a225δ(λ) = 0 a20 a25a59 F(λ) = ?? ? Aeλe?pii/6 + Beλe?5pii/6, λ > 0; Ceλe?pii/6 + Deλe?5pii/6, λ < 0. a159a160 a227F(λ) a178a214 a20A a215D a216a1880a112a217 a93 a188F(λ)a169λ = 0a228a204a205 a20 B = C; Fprime(λ)a169λ = 0a228a216a204a205a20 ?i?√3 2 B ? ?i +√3 2 C = ?2pi, a93a94 a213 a226 B = C = 2pi/√3. a25a59 F(λ) = ? ??? ??? 2pi√ 3e ?√3λ/2 e?iλ/2, λ > 0; 2pi√ 3e √3λ/2 e?iλ/2, λ < 0. a25a219 a213a181 a214 a184 a42 a188 I = ImF(2) = ?2pi√3e? √3 sin1. a0a1a2 δ a3 a4 9 Fprime(λ)a218λ = 0a219a220a221a222a223a224a225 a120 Fprime(λ) = ?? ?? ?? ? bracketleftBig 1 + i√3 bracketrightBig pie? √3λ/2 e?iλ/2, λ > 0; bracketleftBig 1? i√3 bracketrightBig pie √3λ/2 e?iλ/2, λ < 0, a226a227a228 a12a47a229 Fprime(λ) = bracketleftBig 1? i√3 ?2η(λ) bracketrightBig pie? √3|λ|/2 e?iλ/2. a230a231a22 a20 a32λ = 0 a49 a20F prime(λ) a15 a192a202 a103a250 a94 limε→+0Fprime(0±ε) = ?pii√3 ?pi. §10.3 a232a233a176a234a235a236a237a238a239a240 Greena3a4 10 §10.3 a241a242a150a243a244a245a246a247a248a249 Green a139a140 a32a65 a12a250 a35 a82a83a15a251a74a75a84a85 (star) a20a22a12a13 a230a252 a15a251a74a75a84a85 a20 a201 a211 a185a186a196 a94δ a17a18 a15a251a74a75a84a85 a21 a253 a107a84a85 a32a25 a92 a65 a76a254a41 a65a255a0a36a230a252 a15 a247 a120a200 a21 a29a143a92 a253 a107a84a85 a32a185a186 a137a1a2a3a4 a15a82a83 a99a100 a20 a44 a122 a83a15a134a135a126 a200 a53a5 a6a203 a20 a169a7a8 a181 a202a1a2 a20 a31a9 a0 a22 a184a169a49a181 a178a179a180a181 a217 a5 a178 a202a1 a181 a20a10 a77a218a12a32a60 a169a49a181 a178 a179a180a181 a211a33 a188 a203 a189a196a197a23a24a181a222a215 a20 a196a197a23a24 a208 a181a11 a203a181 a78 a51 a178a52a53a54a181a204a205a55(a93a94a20a169a47a222a215 a11a122 a20 a169a49 a217 a178 a202a1 a181) a20a10 a77a169 a213 a33 a22 a184a169a49 a46 a217 a47 a222a215 a21 a39a40δ a196a197a181a41a181 a220 a169a200a12 a226a10 a77a21 a59a172a126 a61 a211 a31a182a222a215a223a224a181 a22 a184a169a49a213 a33a13 a154 a20 a230 a216a216 a159a160a51 a130a181a196a197a23a24 a59 a125 a4 a181a222a215a48a49 a21 a6 a113 a20 a121a93 a188 δ a196a197a216 a217 a7a8a202a1a2 a181a196a197 a20 a31 a12 a226 a178a179a180a181 a188 δ a196a197a181 a22 a184a169a49a181 a33 a51 a178 a14 a29a181a204a205a55a56 a21 a220 a113 a20 a0 a22 a184a169a49 a230a15 a20a10 a77 a139 a219 a38 a227a20 a4 a181 a33a217 a204a205a181 a20a16a217a33 a181 a203a20 a64a197 a217 a216a204a205a181 a21a121 a217 a50a200a203a20 a64a197a181a216a204a205 a20 a28a12 a226 a4a121 a54 a217 a178a179a180a181 δ a196a197a181 a0 a22 a184a169a49a181 a33 a21 a6a108 a20a0 a22 a184a169a49a181 a178a179a180a181 a188 δ a196a197 a20a17 a12 a226 a31a182 a178a179a180 a169a49a216 a217 a203a155 a223a224 a2 a181 a178a179a180 a169 a49a3a18a19 a169a12δ a196a197a181a20a187a188a21a181a189a30 a228 a192 a20 a169a49 a217 a179a180 a181a22a31 a12 a226 a31a182 a178a179a180 a0 a22 a184a169a49 a17a23 a141a142 a213 a33a24a25 a29a177a223a224 a2a62a63a21a59a26a126a27a28a29a30a31a32a33a34a35a36 a24 a37 10.4 a30a36a38a39a28a32a33a40a41a42a43 d2g dx2 = δ(x?t), x,t > 0, g(0) = 0, gprime(0) = 0. a44 a45a46a47a34a39a28a32a33a26a48a27a28a49a50a31a51 gprime(x) = η(x?t) + α(t). a52a53 η(x?t) = braceleftBigg 0, x < t; 1, x > t, a50a54a55a45a46a47a34a56a57a58a27a28a59a60a49a31a51 g(x) = (x?t)η(x?t) + α(t)x + β(t). a61a62a63a64a65g(x) a66x = ta67a68a69a70a71a72a73a74 a49a75a76a77a78 a72a79 a49 a66gprime(x) a80a81a82a83a84a85δ a86a87 a24 a88a89a90 δ a91 a92 11 a45a36a93a94a95a40a41a96a97a49a98a99a27a28a38a100 α(t) = 0, β(t) = 0, a101a102a103a104a30a31 g(x) = (x?t)η(x?t). a10510.4 a106a107g(x;t)a108a109a110a111 a25 a102a112a113a46a49a50a104a114a30a36 d2y dx2 = f(x), x > 0, y(0) = 0, yprime(0) = 0. a115a116a117a45a32a33a34a118a119a60a120 f(x)a28a36a121 f(x) = integraldisplay ∞ 0 f(t)δ(x?t)dt, x > 0, (a122a123 a25a124 a93a46) a50a104a114a31a51a36 y(x) = integraldisplay ∞ 0 g(x;t)f(t)dt = integraldisplay x 0 (x?t)f(t)dt. a125 a126 1. g(x)a127x = ta128a129a130a131a132a133a134 g(x)a135a136, gprime(x)a137a135a136 a49gprime(x)vextendsinglevextendsinglet+0 t?0 = 1. a66a138a139a140a72a141a142a143 a49a144a145 a66a146a147g(x)a72a148a149a150a151a152a153a154a80 a49a155a156a157a158a76 δ(x?t)a49a159 a66 x = ta67a81a82a160 g(x)a70a71a161gprime(x)a77a70a71 a49a61a162a61a163 a153a154a164a72a165a166a167a168a169 a24 2. a170a171a129 a44 a127x < ta172a173a174 0a24 a61a175a176a177a178 a160a139a140a179 a24 a75a76 x < t a180 a49 a153a154 a162a156a157 a72 a49a181a182 a160a183a164 g(x) = c1g1(x) + c2g2(x), a184a185a156a157a186a187a188a189a49a190a191a64 a153a154a192 c1g1(0) + c2g2(0) = 0, c1gprime1(0) + c2gprime2(0) = 0. §10.3 a193a194a195a196a197a198a199a200a201a202 Greena203a204 12 a205a206 a147g1(x)a161g2(x) a162a207 a179a208a146a72 a49 W[g1(x), g2(x)] ≡ vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle g1(x) g2(x) gprime1(x) gprime2(x) vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle negationslash= 0, a61a209a210a211 vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle g1(x) g2(x) gprime1(x) gprime2(x) vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle x=0 negationslash= 0, a181a182a49 a212a213 a72a153a154a192 a49a214a76c 1a161c2a72 a207 a179 a156a157a184a87 a153a154a192 a49 a81a82a215a160a216a164 c1 = 0, c2 = 0a49 a217 g(x) ≡ 0, x < t. a218a114a49a46a47a34a40a41a96a97a219a220a221a222a223 g(x)vextendsinglevextendsinglex<t = 0, gprime(x)vextendsinglevextendsinglex<t = 0. a115a224a225a226a227a228a229a39a28a32a33a34a230a231 a124 a93a49a31a51a34a56a232a233a234a235a236a237a121a238a34 a24 3. a239a240a129 a44a241a242a243a244a245a246 x a129a247a248 a49a249a250a251a252a253a254 a248 ta255 a61a0a78a1a178 a160a139a140a179 a49a181a182a49a182a2a190a3a4a61a163a5a6 a72a164a7 a76 g(x;t) a255 a37 10.5 a30a36a38a39a28a32a33a40a41a42a43 d2g(x;t) dx2 + k 2g(x;t) = δ(x?t), x,t > 0, g(0;t) = 0, dg(x;t)dx vextendsinglevextendsingle vextendsingle x=0 = 0. a44 a52a53a8x negationslash= t a9 a49a32a33a34a118a119a60a120a10 0a49a218a114 g(x;t) = bracketleftbigA(t)sinkx+ B(t)coskxbracketrightbigη(x?t). a61a62a11a12a63a64a65a212a213 a72 a176a177 a134 g(x,t)vextendsinglevextendsinglex<t = 0. g(x;t)a13 a230a227a14a15 4 a16a17a18 a34 a135a136a19a20 a49a103a21 x = t a22 g(x;t)a135a136, dg(x;t)dx a137a135a136 a49dg(x;t) dx vextendsinglevextendsingle vextendsinglevextendsingle t+0 t?0 = 1, a223a235 A(t)sinkt + B(t)coskt = 0, A(t)coskt?B(t)sinkt = 1k. a36a23a103a31 A(t) = 1k coskt, B(t) = ?1k sinkt. a115a24a49a50a31a51 g(x;t) = 1k sink(x?t)η(x?t). a25a26a27 δ a203 a204 13 a10510.5 a106a107g(x;t)a108a109a110a111 a21a102a112a113a46a49a28a104a114a30a99 d2y dx2 + k 2y(x) = f(x), x > 0, y(0) = 0, yprime(0) = 0 a34a36 y(x) = 1k integraldisplay x 0 f(t)sink(x?t)dt. a38a39a28a32a33a40a41a42a43 Green a29 a100g(x;t)a34a112a30 a19a20 star a31a32g(x;t)a129a130a131a132 a38a39a28a32a33a40a41a42a43 d dx bracketleftbigg p(x)dg(x;t)dx bracketrightbigg + q(x)g(x;t) = δ(x?t), x,t > 0; g(0;t) = 0, dg(x;t)dx vextendsinglevextendsingle vextendsingle x=0 = 0 a34a36g(x;t)a14 d dx bracketleftbigg p(x)dg0(x;t)dx bracketrightbigg = δ(x?t), x,t > 0; g0(0;t) = 0, dg0(x;t)dx vextendsinglevextendsingle vextendsingle x=0 = 0 a34a36g 0(x;t) a59a98a230a227 a17a18 a34 a135a136a19a20a255 §10.3 a193a194a195a196a197a198a199a200a201a202 Greena203a204 14 star a127x < ta172g(x;t)a173a1740a255 a10a33a34a35a115a36a37a42a43a21a100a38a46a39a40a49a41a117a42a43 1. a86a87p(x)a66a44a45[0, ∞)a212a70a71 a49 2. pprime(x)a161q(x)a70a71a46a152a47a70a71 a49 3. p(x)a208a216a67a48 a49a50a51a40a46a34a52a53a41a117a49a54a42a43 4. a55a56 d dx bracketleftbigg p(x) ddx bracketrightbigg + q(x)a66x →?x (a57a180a45a58a59a60) a143a77a61a49a217 p(x) = p(?x), q(x) = q(?x). star g(x;t)a129a62a63a132 g(x;t) = g(?t;?x). a101g(x;t)a34a64a65 a19 a49a104a114a31a51g(x;t) a66 a10ta34 a29 a100a218a67a68a34a38a39a28a32a33a40a41a42a43 a255 a69a10g(?t;?x) a67a68 d d(?t) bracketleftbigg p(?t)dg(?t;?x)d(?t) bracketrightbigg + q(?t)g(?t;?x) = δ(x?t), t,x > 0, g(?t;?x)vextendsinglevextendsingle?t<?x = 0, dg(?t;?x)d(?t) vextendsinglevextendsingle vextendsingle ?t<?x = 0, a103 d dt bracketleftbigg p(t)dg(?t;?x)dt bracketrightbigg + q(t)g(?t;?x) = δ(x?t), t,x > 0, g(?t;?x)vextendsinglevextendsinglet>x = 0, dg(?t;?x)dt vextendsinglevextendsingle vextendsingle t>x = 0, a58a70a71a64a65a72a73 g(x;t) = g(?t;?x)a49a50a31a51 d dt bracketleftbigg p(t)dg(x;t)dt bracketrightbigg + q(t)g(x;t) = δ(x?t), t,x > 0; g(x;t)vextendsinglevextendsinglet>x = 0, dg(x;t)dt vextendsinglevextendsingle vextendsingle t>x = 0. star a74a75a76a77a174 δ a247a248a129a78a79a80a81a82a83a84a85a86a87a88 a49a89a243a90 a174a91a92a86a93a94a74a75a76a77a174a95a96a247a248 a250a97a98a99a100a101 a174a74a75a76a172a80a81a82a83a129 a44 a255 a15a102a49a101a38a39a28a32a33a40a41a42a43 d dx bracketleftbigg p(x)dg(x;t)dx bracketrightbigg + q(x)g(x;t) = δ(x?t), x,t > 0; g(0;t) = 0, dg(x;t)dx vextendsinglevextendsingle vextendsingle x=0 = 0 a25a26a27 δ a203 a204 15 a34a36g(x;t)a49a50a104a114a103a104a99a38a39a28a32a33a40a41a42a43 d dx bracketleftbigg p(x)dy(x)dx bracketrightbigg + q(x)y(x) = f(x), x > 0, y(0) = A, yprime(0) = B a34a36 a255 ? a77a177a155a156a157a158 f(x)a162a105a142a106 ? a1a77a177a160 a105a186a187 y(0), yprime(0) g(x;t)a65a10a17a107 a34a38a39a28a32a33a40a41a42a43 d dx bracketleftbigg p(x)dy(x)dx bracketrightbigg + q(x)y(x) = f(x), x > 0, y(0) = A, yprime(0) = B a34Green a29 a100 a255 Greena86a87a153a108 a162 a74a164 a155a156a157 a150a151a152a153a154a72a81a109a110a73a153a108 a49a1a162 a74a164 a155a156a157a111 a151a152a153a154 a72a110a73a153a108a255 a61 a81a112a215a113 a177 a150a151a152a153a154 a186a187a5a6 a72a141a142a255 a66 a87a114a115a116 a153a154a117a152a80a118a113 a177a111 a151a152a153a154a72 Greena86a87a164a108a255 §10.4 a193a194a195a196a197a119a199a200a201a202 Greena203a204 16 §10.4 a120a121a122a123a124a125a126a127a128a129 Green a130a131 a61 a81a132a113 a177a133 a81a109 a163a134 a72Greena86a87a49a217a150a151a152a153a154a135 a187a5a6 a72Greena86a87a49a136a162a155a156 a157a158a76δ a86a87 a72a150a151a152a153a154a66 a156a157 a135a137 a188a189 a143a72a164a255 a37 10.6 a30a36a38a39a28a32a33a138a41a42a43 d2g(x;t) dx2 = δ(x?t), a < x,t < b, g(a;t) = 0, g(b;t) = 0. a44 a115a37a42a43a14a15 4 a137a18 a23a139a21a223a98a36a96a97a49a140a21a117a30a34a235 a80a81a82a83a127a141a142a75a76a143a144 a99a100a145 a129 a44 a255a146a147a13 a234a104a114a148a149a99a32a33a34a35a36a49a234a150a49a50a138a151a96a97a98a99a27a28a38a100 a255 a37 10.4 a16a152a153a154 a99a155a38a39a28a32a33a34a35a36a49a103 g(x;t) = (x?t)η(x?t) + α(t)x + β(t). a94a95a138a151a96a97a49a227 aα(t) + β(t) = 0, b?t + bα(t) + β(t) = 0. a36a23a31 α(t) = ?b?tb?a, β(t) = a(b?t)b?a . a156a150a50a30a99a33a38a39a28a32a33a138a41a42a43a34a36 g(x;t) = (x?t)η(x?t)? b?tb?a(x?a). a10510.6 a106a107g(x;t)a108a109a110a111 a14a154(a38a39a28a32a33a40a41a42a43) a34a36a59a24a49a115a37a42a43a34a36a28a230a227 a135a136a19a20 g(x;t)a21x = ta22a135a136, dg(x;t) dx a21x = t a22a137a135a136 a49dg(x;t) dx vextendsinglevextendsingle vextendsinglevextendsingle t+0 t?0 = 1. a69a10a49a157a49a56a158a49a115a159 a135a136a19a20 g(x;t)a21x = ta22a135a136 a14 dg(x;t) dx a21x = t a22a137a135a136 a25a26a27 δ a203 a204 17 a235a101a39a28a32a33a160a98a34 a255 a37 10.7 a30a36a38a39a28a32a33a138a41a42a43 d2g(x;t) dx2 + k 2g(x;t) = δ(x?t), a < x,t < b, g(a;t) = 0, g(b;t) = 0. a44 a70a71δ a29 a100a34 a19a20 δ(x?t) = 0, x negationslash= t a161a30a36a115a59a38a39a28a32a33a138a41a42a43 a255 ? a0a < x < t a180 a49 a151a152a153a154 a162a156a157 a72 a49 a66 x = a a67a72 a156a157 a135a137 a188a189 a72a162a163a143 a49a136 a72a164 a162 g(x;t) = A(t)sink(x?a), a < x < t. ? a0t < x < ba180 a49 a151a152a153a154 a1a162a156a157 a72 a49 a66x = ba67a72 a156a157 a135a137 a188a189 a72a162a163a143 a49a136 a72a164 a162 g(x;t) = B(t)sink(b?x), t < x < b. a70a71g(x;t)a21x = t a22 a34 a135a136a19a20 a49a104a114a31a51 B(t)sink(b?t)?A(t)sink(t?a) = 0, B(t)cosk(b?t) + A(t)cosk(t?a) = ?1k. a115a24a50a104a114a30a31 A(t) = ?1k sink(b?t)sink(b?a), B(t) = ?1k sink(t?a)sink(b?a). a156a150a49a50a31a51 Green a29 a100 g(x;t) = ? ??? ??? ?1k sink(b?t)sink(b?a) sink(x?a), a < x < t ?1k sink(t?a)sink(b?a) sink(b?x), t < x < b = ? 1k sink(b?t)sink(b?a) sink(x?a) + sink(x?t)k η(x?t). a101a102a54a104a164a99a49a8 k(b?a) = npi negationslash= 0 a9 a49a102a43a165a36 a255 §10.4 a193a194a195a196a197a119a199a200a201a202 Greena203a204 18 a10510.7 a106a107g(x;t)a108a109a110a111 a138a166a167a72a164a108 a52a53a155 a16a152a153 a30a31a34Green a29 a100a49a168a52a28a169a235a30a15a43 a16 a39a28a32a33a34a59a37a170 a36a49a69a102a49a140a21a171a172a149a99a39a28a32a33a34a35a36 g(x;t) = 1k sink(x?t)η(x?t) + α(t)sinkx+ β(t)coskx. a94a95a138a151a96a97a49a50a227 α(t)sinka + β(t)coska = 0, α(t)sinkb + β(t)coskb = ?1k sink(b?t). a36a23a103a31 α(t) = ?1k coskasink(b?a) sink(b?t), β(t) = 1k sinkasink(b?a) sink(b?t). a115a24a156a150a30a31a34 g(x;t)a8a234a14a46a47a34a56a57a219a220 a17a18a255 a173 a80a81a82a83a143 a98 a239a240 Greena247a248g(x;t)a129a174a175a132a133 d dx bracketleftBig p(x)dg(x;t)dx bracketrightBig + q(x)g(x;t) = δ(x?t), a < x,t < b, g(a;t) = 0, g(b;t) = 0 star Greena247a248g(x;t)a129a130a131a132a255 g(x;t)a14a101 d dx bracketleftBig p(x)dg0(x;t)dx bracketrightBig = δ(x?t), a < x,t < b, g0(a;t) = 0, g0(b;t) = 0 a218a98a176a34 Green a29 a100g 0(x;t) a230a227 a18 a24a34 a135a136a19a20a255 star Greena247a248g(x;t)a129a62a63a132a255 g(x;t) = g(t;x). a25a26a27 δ a203 a204 19 star a177a178a179a180g(x;t)a181a182a183 d dx bracketleftBig p(x)dy(x)dx bracketrightBig + q(x)y(x) = f(x), a < x,t < b, y(a) = A, y(b) = B a129 a44 a184 §10.5 a185a186a193a194a195a196a197a202 Greena203a204a196a187 20 §10.5 a188a189a120a121a122a123a124a129 Green a130a131a123a190 a148a191a232a38a39a28a32a33a40a41a42a43 d dx bracketleftbigg p(x)dy(x)dx bracketrightbigg + q(x)y(x) = f(x), x > 0, y(0) = A, yprime(0) = B. a42a43 a29 a100p(x)a21a192a193[0, ∞)a46 a135a136 a49pprime(x)a14q(x) a135a136a194 a28a195 a135a136 a49p(x)a165a196 a22a48 a29a197a49a49a50a51 a40a46a34a52a53a41a117a49a54a42a43 p(x) = p(?x), q(x) = q(?x)a49a103a198a199 d dx bracketleftbigg p(x) ddx bracketrightbigg + q(x) a21a200a201x →?x( a57a9 a193a202a203 a60)a204 a34a205 a137 a200 a255a17a107 a34Green a29 a100g(x;t)a67a68a114δ a29 a100a10a118a119a60a120a34a38 a39a28a32a33a40a41a42a43 d dx bracketleftbigg p(x)dg(x;t)dx bracketrightbigg + q(x)g(x;t) = δ(x?t), x,t > 0; g(0;t) = 0, dg(x;t)dx vextendsinglevextendsingle vextendsingle x=0 = 0. a76a65a3 a164y(x)a63g(x;t)a206a207a84a208 a49a209 a73a210a211g(x;t)a72a62a63a132 g(x;t) = g(?t;?x). (star) a212 a104a114a213a48a101Green a29 a100g(x;t)a34a230a231a214a215a93 g(x;t) = 1p(t) y1(t)y2(x)?y2(t)y1(x)y 1(t)yprime2(t)?y2(t)yprime1(t) η(x?t) a161a216a99 (star)a93 a255 a52a53a21 p(x) = p(?x) a14 q(x) = q(?x) a34a96a97 a204 a49a32a33 (star) a34a36a37a217 a19 a165a72a36a59a98a230a227a218a219 a19 a49a59a37a36a59a98a235a219 a29 a100a49a220a59a37a59a98a235a218 a29 a100a49a15a102 y1(?x) = y1(x), y2(?x) = ?y2(x). a221a222a220a223a59a159a32a224a49a225 a137 a228a229a36a34a230a231a214a215a93a49a29a116a235a213a48a226a39a28a32a33a114a229a40a41a96a97a99a227a161 a35a228 a255 (star)a93a21x < t (a69a102?t < ?x)a9a229 a234a121a238 (a69a10a36a230a231a10 0) a255 a218a114 a204 a47a116a41a35a228a221a93a21 x > t a9 a28a121a238 a255 a152a232g(x;t) a235a38a39a28a32a33a40a41a42a43 d dx bracketleftbigg p(x)dg(x;t)dx bracketrightbigg + q(x)g(x;t) = δ(x?t), x,t > 0; g(0;t) = 0, dg(x;t)dx vextendsinglevextendsingle vextendsingle x=0 = 0. a25a26a27 δ a203 a204 21 a34a36 a255 a58a233a95 g(?x;?tprime)a49a225a235a38a39a28a32a33 d d(?x) bracketleftbigg p(?x)dg(?x;?t prime) d(?x) bracketrightbigg + q(?x)g(?x;?tprime) = δ(?x + tprime), x,tprime > 0 a103 d dx bracketleftbigg p(x)dg(?x;?t prime) dx bracketrightbigg + q(x)g(?x;?tprime) = δ(x?tprime), x,tprime > 0 a34a36 a255 a115a224 a137a234 a42a43 tprime > t > 0 a255 a71g(?x;?tprime)a14g(x;t)a28a235a236a114a36a32a33a49 a17a237 a49a50a31a51 g(?x;?tprime) ddx bracketleftbigg p(x)dg(x;t)dx bracketrightbigg ?g(x;t) ddx bracketleftbigg p(x)dg(?x;?t prime) dx bracketrightbigg = ddxp(x) bracketleftbigg g(?x;?tprime)dg(x;t)dx ?g(x;t)dg(?x;?t prime) dx bracketrightbigg = g(?x;?tprime)δ(x?t)?g(x;t)δ(x?tprime). a21a192a193 [0,∞)a46a238a28a49a52a53a51 g(x,t) vextendsinglevextendsingle vextendsingle x<t = 0, dg(x,t)dx vextendsinglevextendsingle vextendsingle x<t = 0, a14 g(?x,?tprime) vextendsinglevextendsingle vextendsingle ?x<?tprime = 0, dg(?x,?t prime) dx vextendsinglevextendsingle vextendsingle ?x<?tprime = 0 a103 g(?x,?tprime) vextendsinglevextendsingle vextendsingle x>tprime = 0, dg(?x,?t prime) dx vextendsinglevextendsingle vextendsingle x>tprime = 0, a50a227 g(?t;?tprime)?g(tprime;t) = braceleftbigg p(x) bracketleftbigg g(?x;?tprime)dg(x;t)dx ?g(x;t)dg(?x;?t prime) dx bracketrightbiggbracerightbiggx=∞ x=0 = 0. a45tprime a239 a149a121xa49x > ta49a50a219a121a33 (star)a93a34a220a240a35a228 a255 square a101g(x;t)a34a64a65 a19 a49a104a114a31a51g(x;t) a66 a10ta34 a29 a100a218a67a68a34a38a39a28a32a33a40a41a42a43 a255 a69a10g(?t;?x) a67a68 d d(?t) bracketleftbigg p(?t)dg(?t;?x)d(?t) bracketrightbigg + q(?t)g(?t;?x) = δ(x?t), t,x > 0, g(?t;?x)vextendsinglevextendsingle?t<?x = 0, dg(?t;?x)d(?t) vextendsinglevextendsingle vextendsingle ?t<?x = 0, a103 d dt bracketleftbigg p(t)dg(?t;?x)dt bracketrightbigg + q(t)g(?t;?x) = δ(x?t), t,x > 0, §10.5 a185a186a193a194a195a196a197a202 Greena203a204a196a187 22 g(?t;?x)vextendsinglevextendsinglet>x = 0, dg(?t;?x)dt vextendsinglevextendsingle vextendsingle t>x = 0, a58a70a71a64a65a72a73 (star)a49a50a31a51 d dt bracketleftbigg p(t)dg(x;t)dt bracketrightbigg + q(t)g(x;t) = δ(x?t), t,x > 0; g(x;t)vextendsinglevextendsinglet>x = 0, dg(x;t)dt vextendsinglevextendsingle vextendsingle t>x = 0. a241a173 a80a81a82a83 a97a98 a239a240 d dx bracketleftbigg p(x)dy(x)dx bracketrightbigg + q(x)y(x) = f(x), y(0) = A, yprime(0) = B a129 a44 a180a242a243a129 Greena247a248g(x;t)a244a245 a148a45a32a33 a16 a34a233a200a246 x a239 a149a121ta49 d dt bracketleftbigg p(t)dy(t)dt bracketrightbigg + q(t)y(t) = f(t), t > 0, y(0) = A, yprime(0) = B. a18a9 a149a99g(x;t) a66 a10ta34 a29 a100a218a67a68a34a32a33a14a40a41a96a97a49 d dt bracketleftbigg p(t)dg(x;t)dt bracketrightbigg + q(t)g(x;t) = δ(x?t), t,x > 0; g(x;t)vextendsinglevextendsinglet>x = 0, dg(x;t)dt vextendsinglevextendsingle vextendsingle t>x = 0. a45a36a32a33a28a235a236a114 y(t)a14g(x;t)a49a58 a17a237 a49 y(t) ddt bracketleftbigg p(t)dg(x;t)dt bracketrightbigg ?g(x;t) ddt bracketleftbigg p(t)dy(t)dt bracketrightbigg = ddtp(t) bracketleftbigg y(t)dg(x;t)dt ?g(x;t)dy(t)dt bracketrightbigg = y(t)δ(x?t)?g(x;t)f(t). a238a28a59a60a49a50a31a51 y(x) = integraldisplay ∞ 0 g(x;t)f(t)dt + braceleftbigg p(t) bracketleftbigg y(t)dg(x;t)dt ?g(x;t)dy(t)dt bracketrightbiggbracerightbigg∞ 0 = integraldisplay x 0 g(x;t)f(t)dt?p(0) bracketleftbigg Adg(x;t)dt ?Bg(x;t) bracketrightbigg t=0 . a247a212a248a249 a55a250a154a118a251a252a84 a49a144a145 g(x;t) a253a254a72 a77a162a156a157a186a187a188a189a49 a165 a2 a167 a77a255 a251a0a1 t = ∞a46t = xa180a72 a158 a255 a206 a147y(t) a66t = ∞a46t = xa180a72 a87a187a162a2a3 a72 a49a4a5a190a77 a251a6 a25a26a27 δ a203 a204 23 a51a71 a152a232 a246 (a11a3a86a87 f(x) a7 a186a187y(0) a161yprime(0)) a182a8 g(x;t)a206a207a84y(x)a72a9a72a255 a140a21a191a232a38a39a28a32a33a138a41a42a43 d dx bracketleftBig p(x)dy(x)dx bracketrightBig + q(x)y(x) = f(x), a < x < b, y(a) = A, y(b) = B. a17a107 a34 Green a29 a100g(x;t) a10 a67a68 d dx bracketleftbigg p(x)dg(x;t)dx bracketrightbigg + q(x)g(x;t) = δ(x?t), a < x,t < b, g(a;t) = 0, g(b;t) = 0. a42a43 a13 a234a235a171a71 g(x;t)a103a104a99y(x) a255a13 a234a148a117a35a228Green a29 a100g(x;t)a34a64a65 a19 g(x;t) = g(t;x). a212 a13 a234a235a58a233a95a59a37 Green a29 a100g(x;tprime)a49a225a235 d dx bracketleftbigg p(x)dg(x;t prime) dx bracketrightbigg + q(x)g(x;tprime) = δ(x?tprime), a < x,tprime < b, g(a;tprime) = 0, g(b;tprime) = 0 a34a36 a255 a71g(x;tprime)a14g(x;t)a28a235a236a114a36a32a33a49 a17a237 a49a58a21a192a193 [a, b]a46a238a28a49a103a31 g(t;tprime)?g(tprime;t) = braceleftbigg p(x) bracketleftbigg g(x;tprime)dg(x;t)dx ?g(x;t)dg(x;t prime) dx bracketrightbiggbracerightbiggb a . a94a95a138a151a96a97a49a238a103a31a51a46a93a11a32a10 0a49a218a114a49 g(t;tprime) = g(tprime;t). a45tprime a201a121xa49a115a169a12a50a235 (star)a93 a255 a35a13 a255 square a21a102a112a113a46a49a149a99 g(t;x)a67a68a34a38a39a28a32a33a138a41a42a43 d dt bracketleftbigg p(t)dg(t;x)dt bracketrightbigg + q(t)g(t;x) = δ(x?t), a < t,x < b, g(t;x)vextendsinglevextendsinglet=a = 0, g(t;x)vextendsinglevextendsinglet=b = 0. a70a71g(x;t)a34a64a65 a19 (star)a49a45g(t;x)a201a121g(x;t)a49a50a171a31a51 g(x;t)a66 a10 ta34 a29 a100a218a67a68a34a38a39a28 a32a33a138a41a42a43 d dt bracketleftbigg p(t)dg(x;t)dt bracketrightbigg + q(t)g(x;t) = δ(x?t), a < t,x < b, g(x;t)vextendsinglevextendsinglet=a = 0, g(x;t)vextendsinglevextendsinglet=b = 0. §10.5 a185a186a193a194a195a196a197a202 Greena203a204a196a187 24 a10a33a45a38a39a28a32a33a138a41a42a43a34a36a71 a17a107 a34 Green a29 a100g(x;t) d dx bracketleftBig p(x)dy(x)dx bracketrightBig + q(x)y(x) = f(x), a < x < b, y(a) = A, y(b) = B. a214a14a99a161a49a28a148a117a45a32a33 a16 a34a233a200a246 a239 a149a121 ta49 d dt bracketleftbigg p(t)dy(t)dt bracketrightbigg + q(t)y(t) = f(t), a < t < b, y(a) = A, y(b) = B. a58a149a99g(x;t) a66 a10ta34 a29 a100a218a67a68a34a38a39a28a32a33a138a41a42a43a49a103 d dt bracketleftbigg p(t)dg(x;t)dt bracketrightbigg + q(t)g(x;t) = δ(x?t), a < t,x < b, g(x;t)vextendsinglevextendsinglet=a = 0, g(x;t)vextendsinglevextendsinglet=b = 0. a71g(x;t)a14f(t)a28a235a236a114a36a32a33a49 a17a237 a49a58a21a192a193 [a, b]a46a64ta238a28a49a15a40a49a103a31 y(x) = integraldisplay b a g(x;t)f(t)dt + braceleftbigg p(t) bracketleftbigg y(t)dg(x;t)dt ?g(x;t)dy(t)dt bracketrightbiggbracerightbiggb a = integraldisplay b a g(x;t)f(t)dt + Bp(b) dg(x;t)dt vextendsinglevextendsingle vextendsinglevextendsingle t=b ?Ap(a) dg(x;t)dt vextendsinglevextendsingle vextendsinglevextendsingle t=a . a21a156a150a59a16a49a70a71a33a138a151a96a97 a255 a37 10.8 a71Green a29 a100a32a224a30a36a138a41a42a43 d2y(x) dx2 ?k 2y(x) = f(x), ?∞ < x < ∞, k > 0, y(x)vextendsinglevextendsinglex→±∞ → 0. a44 a10a102a49a148a117a30a99a102a42a43a34 Green a29 a100a49a225a235 d2g(x;t) dx2 ?k 2g(x;t) = δ(x?t), ?∞ < x < ∞, k > 0, g(x;t)vextendsinglevextendsinglex→±∞ → 0 a34a36 a255a137a17 a30a99 g(x;t) = ? 12ke?k|x?t|. a25a26a27 δ a203 a204 25 a10510.8 a106a107g(x;t)a108a109a110a111 a58a18a19a227a151a192a193a34a20a224a49a104a114a30a31 y(x)a14g(x;t)a23a193a34a72a73a49a103 y(x) = integraldisplay ∞ ?∞ g(x;t)f(t)dt + bracketleftbigg y(t)dg(x;t)dt ?g(x;t)dy(t)dt bracketrightbigg∞ ?∞ . a94a95g(x;t)a34a214a215a93a114a229y(x)a14g(x;t)a67a68a34a138a151a96a97a49a50a31a51 y(x) = ? 12k integraldisplay ∞ ?∞ e?k|x?t|f(t)dt. a167a132a215a113 a177a65 a81a109 a163a134 a72a135a137 a188a189a49a217a86a87 a66a21a67a72 a87a187a11a3(a156a157 a46 a155a156a157) a255 a1 a118 a255a182a162a22a23a163a134 a72a135a137 a188a189 ( a24 a144a49a86a87 a66a21a67a72a25 a87a187a11a3) a255