Laplace a0 a1 star a2a3a4a5a6a7a8a9a10 6 star a1114 a12a13a14a8a15a16a17a18a19a20a21 star §9.5 a22a6a7 a0a1a2 Laplace a3 a4 a51a6 a7a8a9 Laplace a10 a11 ? Laplacea12a13(a14a15a16a17a12a13)a18a19a20a21a22a23a24a25a12a13a26 a27a28a29a30a31a32a33a34a35a36a29a37 a38a39a40 a21a41a20a26 ? a42a43a44a45 Laplacea12a13a21a46a47 a33a48a49 a42a50a51a52a53 a33a54 a21a14a55a41a20a26 §9.1 Laplace a56 a57 Laplacea58a59a60a61a62a63a64a58a59a52a65a66 f(t)a58a59a67F(p)a52 F(p) = integraldisplay ∞ 0 e?ptf(t)dt. a68a69a70 t a71a72a73a52 pa71a74a73a52 p = s + iσ a26 F(p)a75a76f(t)a70 Laplacea77a78a52a79a75a80a81a77a78a26 e?pt a71Laplacea82a77 a70a83 a26 a84a85a86 Laplace a82a77a79a87a76 F(p) =a88{f(t)} a89 F(p)equaldotrightleftf(t); f(t) =a88?1{F(p)} a89 f(t) equaldotleftrightF(p). f(t)a90F(p)a91a92a93a94a95a75a76 Laplacea82a77 a70a96a97 a73a90a98 a97 a73a26 a99a100a101a102 a52 a27 a42a43 a37a103 a46a104 f(t)a41a105 a32a106a107 f(t)η(t) a52 a48a37 η(t) = braceleftBigg 1, t > 0, 0, t < 0. a108a109a101 a52a110t < 0a111a41a105 a32a106a107 f(t) = 0 a26 a112 9.1 a97 a73f(t) = 1a70Laplacea77a78a76 1 equaldotleftright integraldisplay ∞ 0 e?pt dt = ? 1p e?pt vextendsinglevextendsingle vextendsinglevextendsingle ∞ 0 = 1p, Rep > 0. a68a69a70a113a114a115a116 Rep > 0 a71a76a117a118a119a120a94a121a122a52a89a123a124a71 Laplacea82a77a125a126 a70a115a116 a26 a112 9.2 a97 a73f(t) = eαt a70Laplacea77a78a76 eαt equaldotleftright integraldisplay ∞ 0 e?pt ·eαt dt = ? 1pe?(p?α)t vextendsinglevextendsingle vextendsinglevextendsingle ∞ 0 = 1p?α, Rep > Reα. a68a69a70a113a114a115a116 Rep > Reα a127a128a71a76a117a118a119a120a94a121a122a52a129 Laplacea82a77a125a126a26 §9.1 Laplace a3 a4 a52a6 a130a131 1 a90 a131 2 a132a133a134a135a52a136a137 Laplacea82a77 a70a83 a71 e?pt a52a138a133a139a137a140a141a142a143 a70a97 a73f(t)a52a144 a80a81a77a78a145a125a126a146a147a148a141 t → ∞, f(t) → ∞a92a52f(t) a70 a80a81a77a78a93a132a149a125a126a26 Laplacea58a59a150a151a152a153a154a155a156a60a63a64 integraldisplay ∞ 0 e?ptf(t)dta157a158a152a153a154a26a126a159a160a161a73a72a162a163a164a165a52f(t) a145a149a166a167 1. f(t)a27a168a1690 ≤ t < ∞ a37a170a171a172a22a173a169a174a175a176a177a18a178a179a21a52a180a181a38a178a179a182a28a52a27a183a184a38a185 a168a169a37a186 a23 a169a174a175 a21 a28a187 a18 a38a185 a21a146 2. f(t)a38a38a185a21a188a189a190 a28 a52a191a192 a27a193a28M > 0a33sprime ≥ 0 a52 a194a195a196a183a184t a197(a198a199a200a52a201 a100a195a196 a202a203a204 a21ta197)a52 |f(t)| < Mesprimet. a68 a71Laplace a58a59a150a151a152a205a64a153a154a26 a206a207 a163a164a165a208a209 a70a97 a73a145a149a166a167 a68a210a211a212 a26 a213a214sprime a125a126 a70a215 a52a216 a206a217a218a219a220a206 a52a221a76a222sprime a160 a70a223a224a225 a73a93a226a227 a211a212 a26 sprime a70a228a229a75a76a157 a158a230a231a52a232a76s0 a26 a0a1a2 Laplace a3 a4 a53a6 §9.2 Laplace a233a234a235a236a237a238a239 a240a241 1 Laplace a58a59a60a61a242a243 a240 a58a59a52a129a244 f1(t) equaldotleftrightF1(p), f2(t) equaldotleftrightF2(p), a245 α1f1(t) + α2f2(t) equaldotleftrightα1F1(p) + α2F2(p). a186a246 a50a51a247a248a249a250Laplacea12a13a21a46a47a251a252a52 a253a107a54 a201a254a255a18a24a25a0a1a21a2a50a50a51a21a3 a4 a26a5a6 a186a246 a50a51a52a7a191a251a252 sinωt = e iωt ?e?iωt 2i equaldotleftright 12i bracketleftbigg 1 p?iω ? 1 p + iω bracketrightbigg = ωp2 + ω2; cosωt = e iωt ?e?iωt 2 equaldotleftright 12 bracketleftbigg 1 p?iω + 1 p + iω bracketrightbigg = pp2 + ω2. a240a241 2 Laplace a59a8a152a9a10 a240 a26 a213a214a97 a73f(t)a166a167Laplacea82a77a125a126 a70a11 a94 a115a116 a52 a245 vextendsinglevextendsinglee?ptf(t)vextendsinglevextendsingle < Me?(s?s 0)t, s = Rep. a141s?s0 ≥ δ > 0a92a52 vextendsingle vextendsinglee?ptf(t)vextendsinglevextendsingle < Me?δt. a120a94 integraldisplay ∞ 0 Me?δt dta121a122a52a12 integraldisplay ∞ 0 e?ptf(t)dta126 Rep ≥ s0 + δ a165 a206a13 a121a122a52a221a14a126Rep > s0 a70a15a16 a17a18a19a20 a206 a210a21a22a97 a73a52a129 F(p)a126 a15a16a17 Rep > s 0 a18a21a22 a26 a186a246 a50a51a23a53a20a24a25a46a26a27a28a29 s0 a52 a186a27a30 Laplace a12a13a21a3a31a111a18a32a19a33 a100 a21a26 a240a241 3 a244f(t)a166a167 Laplacea82a77a125a126 a70a11 a94 a115a116 a52 a245 F(p) → 0, a141Rep = s → +∞. a34 a221a76 |F(p)| ≤ integraldisplay ∞ 0 vextendsinglevextendsinglee?ptf(t)vextendsinglevextendsingledt ≤ M integraldisplay ∞ 0 e(s?s0)t dt = Ms?s 0 , a12a141Rep = s → +∞a92a52F(p) → 0a26 square §9.2 Laplacea3a4a35a36a37a38a39 a54a6 a72a162a40a52a136Riemann–Lebesquea217a41a42a43a132a119a44a52a141 Rep = s > s0 a92a52 lim Imp→±∞ F(p) = 0. a240a241 4 a45a46a47a152a48a47a152 Laplace a58a59a26a49 f(t) a50 fprime(t) a145a166a167 Laplace a82a77a125a126 a70a11 a94 a115 a116 a52f(t) equaldotleftrightF(p)a52 a245 a221a76integraldisplay ∞ 0 fprime(t)e?pt dt = f(t)e?pt vextendsinglevextendsingle vextendsingle ∞ 0 + p integraldisplay ∞ 0 f(t)e?pt dt = pF(p)?f(0), a138a133 fprime(t) equaldotleftrightpF(p)?f(0). a253a51 a52 a195a52a53a28f(t) a21a54a55a0a1a56a57a58 a107a195a59a53a28F(p) a21a60a61a0a1a52a180a181a62a63 a64a65a66 a171f(t) a21a67a197a26 a193a253a107a186a246a68a175 a52 a69 a53Laplacea12a13a70a61a18 a30a106 a54a25a70 a35 a21a22a23a33 a100 a70 a61a26 a127a128a52 a71a211f(t), fprime(t), fprimeprime(t), ···, f(n)(t) a145a166a167 Laplacea82a77a125a126 a70a11 a94 a115a116 a52f(t) equaldotleftrightF(p)a52 a245 fprimeprime(t) equaldotleftright p2F(p)?pf(0)?fprime(0), f(3)(t) equaldotleftright p3F(p)?p2f(0)?pfprime(0)?fprimeprime(0), ... f(n)(t) equaldotleftright pnF(p)?pn?1f(0)?pn?2fprime(0)?··· ? pf(n?2)(0)?f(n?1)(0). a72 9.1 a42 Riemann–Lebesquea73a74a75a76a77a78a79a80a81a82a83 f(t)a84a85a86a ≤ t ≤ ba87a88a89a90a91a92a93 limω→∞ integraldisplay b a f(t)sinωtdt = 0, limω→∞ integraldisplay b a f(t)cosωtdt = 0. a0a1a2 Laplace a3 a4 a55a6 a112 9.3 LR a94a95a96a97(a98a999.1)a52Ka227a40a100a101a96a97a165a102a91a96a103a52 a212K a227a40a104a96a97a165 a70 a96a103a26 a9 a105a106Kirchhoff a217a107 a52a132a108a135a109a94a110a111 Ldidt + Ri = E, i(0) = 0. a49i(t) equaldotleftright I(p)a52 a245 di dt equaldotleftrightpI(p)?i(0) = pI(p). a138a133 LpI(p) + RI(p) = Ep , parenleftbigLp + RparenrightbigI(p) = Ep . a68 a128a52a112a113 Laplacea82a77a52 a212a21a85 a109a94a110a111 a70 a163a164a114a115a116a76 a212a21a19 a73a110a111a52 I(p) = Ep 1Lp + R = ER bracketleftbigg1 p ? L Lp + R bracketrightbigg . a138a133 i(t) = ER bracketleftBig 1?e?(R/L)t bracketrightBig . a117a118 a46a47a119a120a121a122a45a46a47a152a123a124a125a67a119a126a26 a27a186a246a127a128a37 a52a129 a59a53a28a130 a25a25a131a52a132a133a20a190 a28a53a28a30a134a135a53a28a136a53a28 a21 Laplacea12a13 a137 a131a52a56a138 a30a139a52a53a28 a26 a240a241 5 a45a46a47a152a63a64a152 Laplace a58a59a26a49f(t)a166a167Laplacea82a77a125a126 a70a11 a94 a115a116 a52 a245 vextendsinglevextendsingle vextendsinglevextendsingle integraldisplay t 0 f(τ)dτ vextendsinglevextendsingle vextendsinglevextendsingle ≤ integraldisplay t 0 |f(τ)|dτ ≤ integraldisplay t 0 Mes0τ dτ = Ms 0 parenleftbiges 0t ?1 parenrightbig, a138a133 integraldisplay t 0 f(τ)dτ a70Laplacea82a77a93a125a126a52 f(t) equaldotleftright F(p), integraldisplay t 0 f(τ)dτ equaldotleftright a88 integraldisplay t 0 f(τ)dτ. a140 a221a76 d dt integraldisplay t 0 f(τ)dτ = f(t)a52a105a106a141a142 4a52a91 F(p) = pa88 integraldisplay t 0 f(τ)dτ ?0. a138a133 integraldisplay t 0 f(τ)dτ equaldotleftright F(p)p . §9.2 Laplacea3a4a35a36a37a38a39 a56a6 a72 9.2 a112 9.4 LC a94a95a96a97 (a98a99 9.2) q C = L di dt, q = ? integraldisplay t 0 i(τ)dτ + q0. a138a133 Ldidt + 1C integraldisplay t 0 i(τ)dτ = q0C . a68 a71a143a137a144a145 a97 a73 i(t)a70a146a64a63a64a147a148a26a49i(t) equaldotleftrightI(p)a52 a245 a91 LpI(p) + 1C I(p)p = q0C 1p. a138a133 a212a21 a109a94a120a94a110a111 a70 a163a164a93a115a116a76 a212a21a19 a73a110a111 I(p) = q0LCp2 + 1. a149a150 a141a142 1 a165 a70a151a214a212a152a153 a52a129a154 i(t) = q0√LC sin t√LC. a0a1a2 Laplace a3 a4 a57a6 §9.3 Laplace a233a234a235a155a156 a118 a46a47a152a48a47a152a119a126 a49f(t)a166a167 Laplacea82a77a125a126 a70a11 a94 a115a116 a52f(t) equaldotleftright F(p)a52 a245F(p) a126 Rep ≥ s1 > s0 a70a15a16a17a165a21a22a52a221a14a132a133a126a120a94a157a228a212a158 F(n)(p) = d n dpn integraldisplay ∞ 0 f(t)e?pt dt = integraldisplay ∞ 0 (?t)nf(t)e?pt dt. a138a133 F(n)(p) equaldotrightleft (?t)nf(t). a105a106 a68a210a159 a78a52a132a133a160a161a162a154a209 1 p2 = ? d dp 1 p equaldotrightleft t, 1 p3 = 1 2 d2 dp2 1 p equaldotrightleft 1 2t 2. a244F(p)a71a91 a41 a97 a73a52 a245a163 a132a133 a84 a113a164a94a94a78 a212a152a153 a26 a131a213 1 p3(p + α) = 1 α 1 p3 ? 1 α2 1 p2 + 1 α3 1 p ? 1 α3 1 p + α equaldotrightleft 12αt2 + 1α2t + 1α3 ? 1α3e?αt. a118 a46a47a152a63a64a152a119a126 a213a214 integraldisplay ∞ p F(q)dq a125a126 a42 a52a165a141t → 0a92a52|f(t)/t|a91 a229 a52 a245 integraldisplay ∞ p F(q)dq equaldotrightleft f(t)t . (star) a34 a166F(q)a70a20a167 a78 a19a168 a52 a218a169 a77a120a94a170a171integraldisplay ∞ p F(q)dq = integraldisplay ∞ p dq integraldisplay ∞ 0 f(t)e?qt dt = integraldisplay ∞ 0 f(t)dt integraldisplay ∞ p e?qt dq = integraldisplay ∞ 0 f(t) t e ?pt dt, a143a137 a169 a77a120a94a170a171 a70 a227a172a141 a70a173a174 a52a98a175a176a177a178[1]a26 square a149a150a68a210a159 a78a52a179a132a133a154a209a180a161 a97 a73 a70 Laplace a82a77a26 a131a213 sinωt t equaldotleftright integraldisplay ∞ p ω q2 + ω2 dq = pi 2 ?arctan p ω. a42 a181a182a75a183a88a87a184a185a186a187a188 Rep → +∞a92a189a190a183a88a191a192a84F(p)a75a187a193a85a194a76a92a195a196a183a88a197a191a192a198a199a200 §9.3 Laplacea3a4a35a201a202 a58a6 a203 a95a71a52 a213a214 p → 0 a92a52(star)a78a204a205 a70 a120a94a206a125a126a52 a245 a91integraldisplay ∞ 0 F(p)dp = integraldisplay ∞ 0 f(t) t dt. a149a150a68a210a151a214 a52a132a133a207a208 integraldisplay ∞ 0 f(t) t dta209 a70 a120a94a26 a131a213 integraldisplay ∞ 0 sint t dt = integraldisplay ∞ 0 1 p2 + 1 dp = pi 2. a186a246 a24a25a210a211a41a20a212 a28 a46 a32a213 a1a255a26 a186a214 a21 a213 a1a215 a107 a14a216a26 a91a217a120a94a218a172 a150a219 a73 a217a41 a207a208a52 a140a220 a132a133 a150a68a210a221 a172a207a208a26 a131a213 integraldisplay ∞ 0 cosat?cosbt t dt = integraldisplay ∞ 0 parenleftbigg p p2 + a2 ? p p2 + b2 parenrightbigg dp = 12 ln p 2 + a2 p2 + b2 vextendsinglevextendsingle vextendsinglevextendsingle ∞ 0 = lnb?lna, a > 0, b > 0. a118 a46a47a151∞a222a9a10a152a223a224 a213a214F(p) a132a133a136 a15a16a17 Rep > s 0(a225a226a162) a21a22a227a228 a209a229a91p = ∞ a230 a126 a18a70 a206a217a231a232 a18 a52a165a126 p = ∞a230a21a22a52 a68 a128a52 a97 a73 F(p)a114a132a133a126p = ∞a230a233 Taylora234a235 F(p) = ∞summationdisplay n=1 cn p?n. a236 a28a37 a254a237n = 0a238a52a18 a253a107F(p) a239 a107Laplace a13a131a52a41a110 a240a202Rep → +∞ a111F(p) → 0 a21 a100a30 a26 a166a241 a73a242a243 a212a152a153 a52a114a154a209 f(t) = ∞summationdisplay n=0 cn+1 n! t n. a186 a23a239a61a21a244a61a50 a27a196a100a245a102a51 a236 a28 a26a27a52a250a180a25a246 f(t) equaldotleftright F(p) a26 a107a51 a239a247a248 CR : |p| = Ra52 a27C R a176a249F(p) a21a250 a175 a52 cn = 12pii contintegraldisplay CR F(p)pn?1 dp. a253a107p = ∞ a18F(p)a21a251 a175 a52 a69 a53 |F(p)| < MR , a110|p| > R, a253a252 a52|cn| < MRn?1 a26a253 a51 a23a53a251a252vextendsingle vextendsinglevextendsingle vextendsinglevextendsingle ∞summationdisplay n=0 cn+1 n! t n vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle ≤ ∞summationdisplay n=0 |cn+1| n! |t| n < M ∞summationdisplay n=0 1 n! R n|t|n = MeR|t|, a254a236a28 a26a27a26 a186a214a255 a111a0 a245a102a171f(t) a1 a38a38a185 a21a188a189a190 a28 a52 a253 a180 a54 a21 Laplace a12a13a192 a27 a26 square a0a1a2 Laplace a3 a4 a59a6 a2a150a68a210 a110a172a132a133 a212 a135 a97 a73 1radicalbig p2 + 1 a70a152a153 a26 a68 a71 a206 a210 a161a226 a97 a73a52 a213a214a3 a217 a225a226a94a4 1radicalbig p2 + 1 vextendsinglevextendsingle vextendsinglevextendsingle p→∞ → 1p, a245 a91 1radicalbig p2 + 1 = ∞summationdisplay k=0 (?)k (2k)!22k(k!)2 1p2k+1 equaldotrightleft ∞summationdisplay k=0 (?)k 122k(k!)2 t2k = ∞summationdisplay k=0 (?)k k!k! parenleftbiggt 2 parenrightbigg2k . a68a225 a71 5.4 a5 a1317 a906.4 a5a165a98a209a113 a70 Bessela97 a73 J0(t)a26 a6 a206 a210a131a7 a71 1 pe ?1/p = ∞summationdisplay n=0 (?)n 1n! 1pn+1 equaldotrightleft ∞summationdisplay n=0 (?)n n!n!t n = J0(2√t). a5a6 Laplace a12a13a21a2a50a50a51a52a8a9 Laplace a13a131 F(p) a23a53a25 a106a107a10a246a53a28 F 1(p) a11 F2(p)a252a11a52a12a13a52a54a21a3a31a14a15a110a16a56a247a14a55a104a201a100F1(p) a11 F2(p)a21 a52a53a28a177 a192 a27 a52 F(p)a21 a52a53a28 a56a18F1(p) a11 F2(p)a21 a52a53a28a252 a11a26a8a9 F(p)a23a53a25 a106a107F 1(p) a11 F2(p) a252 a24a52 a48 a3a31a14a15a56 a99a100 a20a252a17a18a21a19a24a46 a32 a26 a20 a63a21a22 a49F1(p) equaldotrightleft f1(t)a52F2(p) equaldotrightleftf2(t)a52 a245 F1(p)F2(p) equaldotrightleftintegraltextt0 f1(τ)f2(t?τ)dτ. a34 F1(p)F2(p) = integraldisplay ∞ 0 f1(τ)e?pτ dτ integraldisplay ∞ 0 f2(ν)e?pν dν = integraldisplay ∞ 0 f1(τ)dτ integraldisplay ∞ 0 f2(ν)e?p(τ+ν) dν = integraldisplay ∞ 0 f1(τ)dτ integraldisplay ∞ τ f2(t?τ)e?pt dt, a132a133a126Otτ a16a17a40a23a135a120a94 a231a232 ( a98a9910.3)a52a24a104a25a82a120a94a170a171a52a129a154 §9.3 Laplacea3a4a35a201a202 a510a6 a72 9.3 F1(p)F2(p) = integraldisplay ∞ 0 e?pt dt integraldisplay t 0 f1(τ)f2(t?τ)dτ, a217a41 a154a119a26 square a72 9.4 a112 9.5 a126LR a94a95a96a97 (a98a99 9.4) a165a26a40 a206 a110a27a28a29a96a30 E(t) = braceleftBigg E 0, 0 ≤ t ≤ T; 0, t > T. a212 a96a97a165 a70 a96a103 i(t)a52a49i(0) = 0a26 a9 a108a110a111 Ldidt + Ri = E(t), i(0) = 0. a233Laplace a82a77a104a49i(t) equaldotleftrightI(p), E(t) equaldotleftright E(p)a52 a245 LpI(p) + RI(p) = E(p) a129 I(p) = 1Lp + R ·E(p). a138a133 i(t) = integraldisplay t 0 E(τ) 1Le?R(t?τ)/L dτ a31a32a33 Laplace a34 a35 a3611a37 = ? ?? ?? E0 R parenleftbig1?e?Rt/Lparenrightbig, 0 ≤ t ≤ T; E0 R parenleftbigeRT/L ?1parenrightbige?Rt/L, t > T. §9.4 a38a39a201a202a40a41 a3612a37 §9.4 a42a43a44a45a46a47 a48a49a50 F(p), p = s + iσ a51a52a53 (1) F(p) a54a55a56 Rep > s0 a57a58a59a60 (2) a54a55a56 Rep > s0 a57a60 |p| → ∞a61 F(p) a62a63a64a65a66 0 a60 (3) a67a66a68a69a70 Rep = s > s0 a60a71a72a73 L : Rep = s a70a74a75a76a77integraldisplay s+i∞ s?i∞ |F(p)|dσ (s > s0) a78a79 a60 a80a81a82 Rep = s > s 0 a60 F(p) a83 f(t) = 12pii integraldisplay s+i∞ s?i∞ F(p)ept dp a84 Laplace a85a86 a60a87a88 t a89a90a85a91a92 a93a94a95a96a97a98a84a99a100a101a102a103a104 a92 a105a93a94a95a96a97a98a106Laplace a85a86 a84a107a49a50 a60 a108a109a110a111a112a113a84a114a115a116a117 a92 a118a119a120a121a122a123 a50a124a125a126 a127a128 a92a129 a112a130a131a132a133a134 a92 a135 9.6 a122 a93a94a95a96a97a98a106 Laplace a86 a98 F(p) = 1/(p2 + ω2)2 (ω > 0) a84a107a49a50 a92 a136 a105a93a94a95a96a97a98 a60a137a138 a49a50a84a107a49a50 a89 f(t) = 12pii integraldisplay s+i∞ s?i∞ 1 (p2 + ω2)2 e pt dp. a105a82a49a50 1/(p2 + ω2)2 a84a139a140a141a142a143a144a113 a60 a145a146a147a148a116a117a149a150 a88 a84 s > 0 a151 a120 a92 a152a112a153a154a93a94a99a100 a155 a60a156 t < 0 a157 a118 a124a158 f(t) = 0 a92a159 a137 a129 a112a160a161a162a163 t > 0 a84a164a165 a92 a147 a157 a60 a120a166a167a168a169a170 9.8 a92 a171 9.8 a172a173a174 Laplace a175 a35 a3613a37 a105a82 limp→∞ 1(p2 + ω2)2 = 0, a145a146 a60a176a177a178a179 a84 Jordan a180 a125a181 a60 a120 a146a182a124 lim R→∞ integraldisplay CR 1 (p2 + ω2)2 e pt dp = 0. a147a183 a60 a105 a123 a50a124a125 a60a184a185a186 f(t) = 12pii integraldisplay s+i∞ s?i∞ 1 (p2 + ω2)2 e pt dp = summationdisplay a187a111a112 res braceleftbigg 1 (p2 + ω2)2 e pt bracerightbigg = braceleftbiggbracketleftbigg t (p + iω)2 ? 2 (p + iω)3 bracketrightbigg ept bracerightbigg p=iω + braceleftbiggbracketleftbigg t (p?iω)2 ? 2 (p?iω)3 bracketrightbigg ept bracerightbigg p=?iω = 12ω3bracketleftbigsinωt?ωtcosωtbracketrightbig. a181 a188a189a190a191a192a193a194 Jordan a195a196a197a198 a194a199a200a201a202a194 Jordan a195a196(a2037.4a204)a205a206a207a208 a194a209a210a211a212a213a214 ? a200Jordan a195a196 a190a215a216a194a217a218a219a220 a206 90? a221 ? a222a223 a194a217a218a199a224a225a226 L : Rep = s > 0 a227a228a197 a190a229a230a231a232a233a234a217a218 a221 a235 a199 a197 a236 a229a237a238 a197 a239 a230a217a218a211a240a241a194a242a243 ( a244s)a245 a246 ( a247a248a249 a217a218a194a234a250 R →∞ a251a197 a240a241a252a253a217a218a194a254a255 → 0) a197a0a195a196a1a2a3a4 a221 §9.4 a38a39a5a6a40a41 a3614a37 Jordan a7a8 a9 a62a10a11a12a13a14 a15 a54 0 ≤ argz ≤ pi a70a16a17a18 a60a19 |z|→ ∞a61 a60 Q(z) a62a63a64a65a20a66 0 a60a21 lim R→∞ integraldisplay CR Q(z)eipzdz = 0 a22 a57 p > 0 a60 CR a23a24a25a26a27a28a29a60 R a27a30a31 a70 a30a28a32 a33a34a35ζ = iz,z = ?iζ a9 a11a10a36a12a13a14 a15 a54 pi 2 ≤ argζ ≤ 3 2pi a70a16a17a18a60a19|ζ| → ∞a61a60 Q(ζ) a62a63a64a65a20a66 0 a60a21 lim R→∞ integraldisplay CR Q(ζ)epζdζ = 0 a22 a57 p > 0 a60 CR a23a24a25a26a27a28a29a60 R a27a30a31 a70 a30a28a32 a15 a54 pi 2 ≤ argζ ≤ 3 2pi a70a16a17a18a60a19|ζ| → ∞a61a60 Q(ζ) a62a63a64a65a20a66 0 a60a21 lim R→∞ integraldisplay CR Q(ζ)e?pζdζ = 0 a22 a57 p < 0 a60 CR a23a24a25a26a27a28a29a60 R a27a30a31 a70 a30a28a32 a33a34a35ζ = ?iz,z = iζ a9 a62a10a37a12a13a14 a15 a54? pi 2 ≤ argζ ≤ pi 2 a70a16a17a18a60a19|ζ| → ∞a61a60 Q(ζ) a62a63a64a65a20a66 0 a60a21 lim R→∞ integraldisplay CR Q(ζ)e?pζdζ = 0 a22 a57 p > 0 a60 CR a23a24a25a26a27a28a29a60 R a27a30a31 a70 a30a28a32 a15 a54? pi 2 ≤ argζ ≤ pi 2 a70a16a17a18a60a19|ζ| → ∞a61a60 Q(ζ) a62a63a64a65a20a66 0 a60a21 lim R→∞ integraldisplay CR Q(ζ)epζdζ = 0 a22 a57 p < 0 a60 CR a23a24a25a26a27a28a29a60 R a27a30a31 a70 a30a28a32 a33a34a35ζ = ?z,z = ?ζ a9 a36a10a37a12a13a14 a15 a54 pi ≤ argζ ≤ 2pi a70a16a17a18 a60a19 |ζ| → ∞a61 a60 Q(ζ) a62a63a64a65a20a66 0 a60a21 lim R→∞ integraldisplay CR Q(ζ)e?ipζdζ = 0 a22 a57 p > 0 a60 CR a23a24a25a26a27a28a29a60 R a27a30a31 a70 a30a28a32 a15 a54 pi ≤ argζ ≤ 2pi a70a16a17a18 a60a19 |ζ| → ∞a61 a60 Q(ζ) a62a63a64a65a20a66 0 a60a21 lim R→∞ integraldisplay CR Q(ζ)eipζdζ = 0 a22 a57 p < 0 a60 CR a23a24a25a26a27a28a29a60 R a27a30a31 a70 a30a28a32 a172a173a174 Laplace a175 a38 a3915a40 a129 a112a41a130 a118 a132 a138 a49a50 F(p) a89 p a84a42a43a49a50a84a164a165a92 a135 9.7 a122 a93a94a95a96a97a98a106 Laplace a86 a98 F(p) = 1√ p e ?α√p a60 α > 0 a84a107a49a50a92 a136 a105a93a94a95a96a97a98 a60 a107a49a50 a89 1√ p e ?α√p equaldotrightleft 1 2pii integraldisplay s+i∞ s?i∞ 1√ p e ?α√p ept dp. a87a88 a84a116a117a149a150 L : Rep = s > 0 a83a44a45 a111a112a113a84 a118a46 a111a47a82a143a144a84a114a115a48a49 a92a50a51 a186a52 a116a49a50 a83 a42a43a49a50 a60 p = 0 a53p = ∞ a83a54 a140 a60 a145a146 a60 a142a55 a122a123 a50a124a125a127a128a147a132a116a117 a157 a60 a55a56 a166 a116a117 a167a168a169 a170 9.9 a92a159a89 a142a116a117 a167a168a57 a114a139a140 a60 a145a146 a171 9.9 contintegraldisplay C 1√ p e ?α√p ept dp = integraldisplay B A 1√ p e ?α√p ept dp + integraldisplay CR 1√ p e ?α√p ept dp + integraldisplay C1 1√ p e ?α√p ept dp + integraldisplay Cδ 1√ p e ?α√p ept dp + integraldisplay C2 1√ p e ?α√p ept dp + integraldisplay CprimeR 1√ p e ?α√p ept dp = 0. a105 a178a179 a84 Jordan a180 a125 a60 a120a58 lim R→∞ integraldisplay CR 1√ p e ?α√p ept dp = 0, lim R→∞ integraldisplay CprimeR 1√ p e ?α√p ept dp = 0. a59 a176a177 a180 a125 3.2 a60 a158 lim δ→0 integraldisplay Cδ 1√ p e ?α√p ept dp = 0. §9.4 a38a39a5a6a40a41 a3916a40 a142 C 1 a53C2 a113 a60 argp = ±pi, a60 a120 a117a61a62 p = re±ipi a63 a185a186integraldisplay C1 1√ p e ?α√p ept dp = ?i integraldisplay R δ 1√ r e ?iα√r e?rt dr, integraldisplay C2 1√ p e ?α√p ept dp = ?i integraldisplay R δ 1√ r e iα√r e?rt dr. a145a146 a60 a142 a166a64a65 R → ∞, δ → 0 a66 a60a184 a158 1√ p e ?α√p equaldotrightleft 1 2pi integraldisplay ∞ 0 1√ r bracketleftBig eiα √r + e?iα √rbracketrightBig e?rt dr = 2pi integraldisplay ∞ 0 e?x2t cosαxdx = 1√pit exp braceleftbigg ?α 2 4t bracerightbigg . a147a148 a122 a186a67a68 4 a103a84a69a70 integraldisplay ∞ 0 e?x2t cosαxdx = 12 radicalbiggpi t exp braceleftbigg ?α 2 4t bracerightbigg . a142a147a132a69a70a84a71a72a113 a184 a120 a146a99a100 a129a73 a97a98 a53 1√ pF( √p) equaldotrightleft 1√ pit integraltext∞ 0 f(τ)e ?τ2/4tdτ. a74 a48a75a81 a44 a98a76a77a93a77a78 a85a86 a60a79a80 a86 a116a117a81a82 a60 a151 a120 a99a100 a53integraldisplay ∞ 0 e?pt braceleftbigg 1 √pit integraldisplay ∞ 0 f(τ)e?τ2/4tdτ bracerightbigg dt = integraldisplay ∞ 0 f(τ) braceleftbiggintegraldisplay ∞ 0 1√ pite ?τ2/4te?ptdt bracerightbigg dτ = integraldisplay ∞ 0 f(τ) 1√pe?τ√pdτ = 1√pF(√p). square a135 9.8 a106 1 p e ?α√p, α > 0 a84a107a49a50 a92 a68a83a84 a103a85 a122 a186 a147a132a69a70 a136 a142a113a98 a88a86a87 F(p) = 1pe?αp a60a79a88a89 1 pe ?αp equaldotrightleftη(t?α) a60a184 a158 1 p e ?α√p equaldotrightleft 1√ pit integraldisplay ∞ 0 η(τ ?α)e?τ2/4tdτ = 1√pit integraldisplay ∞ α e?τ2/4tdτ, a105 a137 a151 a185 1 p e ?α√p equaldotrightleft erfc α 2√t. a87a88 a84 erfcx a90a89a91a92a93 a49a50 a60 a124a94 a89 erfcx = 2√pi integraldisplay ∞ x e?ξ2dξ. a95a96a84a97a158 a92a93 a49a50 erf x a60 erf x = 1?erfcx = 2√pi integraldisplay x 0 e?ξ2dξ. a172a173a174 Laplace a175 a38 a3917a40 ?§9.5 a98a99 Laplace a100a101a102a103a104a105a106 Laplace a85a86a107 a120 a146 a122 a126a127a128a108a109a110a50 summationtextF(n) a111a53a92 a87 a71a102a112a149 a83 a121a122 Laplace a85a86 F(p) = integraldisplay ∞ 0 f(t)e?ptdt, a113a110a50a84a114a115a116a117a118a116a117 a60 a63 a66 a80 a86 a116a117 a53 a110a50a106 a53 a84a81a82 summationdisplay F(n) = summationdisplayintegraldisplay ∞ 0 f(t)e?ntdt = integraldisplay ∞ 0 f(t) bracketleftBigsummationdisplay e?nt bracketrightBig dt, a147a183 a60a184a119 a110a50a106 a53 a84a120a121a122a123 a89 a124a116a117a84a127a128 a92 a105a82a49a50 e?nt a84a124a142 a60 a142 a118a119 a164a125 a129a126a126 a120 a146 a127a99 a80 a86 a81a82a84a128a129a130 a92 a142a127a128 a88 a126 a122 a186 a84 Laplace a85a86 a84a69a70a158 integraldisplay ∞ 0 eαt e?pt dt = 1p?α, integraldisplay ∞ 0 tα?1e?pt dt = Γ(α)pα , integraldisplay ∞ 0 e?pt sinωtdt = ωp2 + ω2, integraldisplay ∞ 0 e?pt cosωtdt = pp2 + ω2, integraldisplay ∞ 0 e?pt sinhatdt = ap2 ?a2, integraldisplay ∞ 0 e?pt coshatdt = pp2 ?a2. a129 a112a114a155a131a132a133a134a126a131a132a133a100 a92 a135 9.9 a127a128a110a50 ∞summationtext n=1 1 n2 a111a53a92 a136 a134a135 a121a122 integraldisplay ∞ 0 te?pt dt = 1p2, Rep > 0, a113a110a50a123 a89 ∞summationdisplay n=1 1 n2 = ∞summationdisplay n=1 integraldisplay ∞ 0 te?nt dt = integraldisplay ∞ 0 t bracketleftBigg ∞summationdisplay n=1 e?nt bracketrightBigg dt = integraldisplay ∞ 0 t et ?1 dt. a89 a67 a127a128a147a132a116a117 a60 a50a51 a110 a85 a116a117 contintegraldisplay C z2 ez ?1 dz a60 a167a168a169a170 9.10 a92 a171 9.10 ?§9.5 a136a137Laplace a175a38a138a139a140a141a142 a3918a40 a147 a157 a60a176a177 a123 a50a124a125 a184 a158 contintegraldisplay C z2 ez ?1 dz = integraldisplay R 0 x2 ex ?1 dx + integraldisplay 2pi 0 (R + iy)2 eR+iy ?1 idy + integraldisplay δ R (x + 2pii)2 ex ? 1 dx + integraldisplay Cδ z2 ez ?1 dz + integraldisplay 0 2pi?δ (iy)2 eiy ?1 idy =0. a159a89 lim R→∞ (R + iy)· (R + iy) 2 eR+iy ?1 = 0, limz→2pii(z ?2pii)· z 2 ez ?1 = ?4pi 2, a145a146 lim R→∞ integraldisplay 2pi 0 (R + iy)2 eR+iy ?1 idy = 0, lim δ→0 integraldisplay Cδ z2 ez ?1 dz = 2pi 3i. a59 a159a89 integraldisplay 0 2pi?δ (iy)2 eiy ?1 idy = ? i 2 integraldisplay 2pi?δ 0 parenleftBig 1 + icot y2 parenrightBig y2 dy, a145a146 a60 a166a64a65 R → ∞, δ → 0 a60a79a143a144a145 a98a146a147a84a143a148 a60 a151 a120 a106 a185 ∞summationdisplay n=0 1 n2 = integraldisplay ∞ 0 t et ?1 dt = 1 6pi 2. a172a173a174 Laplace a175 a38 a3919a40 ?§9.6 a42a43a44a45a46a47a149a150a151 a74 a117a152a153a99a100a113a112a154a155a84 f(t) a184 a83 F(p) a84a107a49a50a92 a68 a118 a153 a60 a99a100a116a117 1 2pii integraldisplay s+i∞ s?i∞ F(p)ept dp a156s a114a96 a60 a63 a76 a89a85a91 t a84a49a50 a60 a131a158a158 a65 a84a157a158a159a50 a92 a171 9.5 a89 a137a60 a142a160a161Rep > s 0 a88a60 a50a51 a1709.5 a88 a84a162a165 a167a168 a60a87a163 a140 a89s1?iσ, s2?iσ, s2+iσ, s1+iσa164 s2 > s1 > s0 a60 σ > 0 a92a159a89 a167a168a165 a187a166a142 F(p) a84a167a168a160a161 a88a60 a60 a176a177 Cauchy a124a125contintegraldisplay F(p)ept dp = 0. a169a124 s 1,s2 a63a170σ → ∞a60 a80a105a153 a58a46a171 (2) a60 a158 limσ→∞ integraldisplay s2?iσ s1?iσ F(p)ept dp = 0, limσ→∞ integraldisplay s1+iσ s2+iσ F(p)ept dp = 0. a159 a137 integraldisplay s1+i∞ s1?i∞ F(p)ept dp = integraldisplay s2+i∞ s2?i∞ F(p)ept dp. a105a82 s 1 a53 s2 a84a172 a89 a130 a60 a147 a184 a99a100 a67 a116a117 1 2pii integraldisplay s+i∞ s?i∞ F(p)ept dp a156 s a114a96 a60 a160 a83a85a91 t a84a49a50 ( a173 a89 f(t)) a92 a41 a176a177 a153 a58a46a171 3 a60 a158 vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle 1 2pii integraldisplay s+i∞ s?i∞ F(p)eptdp vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle ≤ 1 2pi integraldisplay s+i∞ s?i∞ vextendsinglevextendsingleF(p)eptvextendsinglevextendsingle·|dp| = e st 2pi integraldisplay s+i∞ s?i∞ |F(p)|dσ ≤ M2piest, ?§9.6 a174a39a5a6a40a41a175a176a177 a3920a40 a60f(t)a131a158a158 a65 a84a157a158a159a50 a60 a178a179a180a181 a184 a83 s0 a92 a182a113a112a84a183 a145 a98a155a184 a60 a185 a157a107 a120 a99 a185 a116a117a84 a118a186 a178a179a130 a92 a68a187 a153 a60 a99a100a81a82 t < 0, f(t) ≡ 0 a92 a147 a157 a120 a50a51 a170 9.6 a88 a84 a167a168 C a92 a171 9.6 a105Cauchy a124a125 contintegraldisplay C F(p)ept dp = 0. a120 a83 a60a188a189 Jordan a180 a125 a60a156 R → ∞a157 a60a190 CR a84a116a117a191a82 0 a60 a159 a137 f(t) = 12pii integraldisplay s+i∞ s?i∞ F(p)ept dp ≡ 0, t < 0, Rep > s0. a68 a152a153 a60 a99a100a147a132a116a117a124a94a84 f(t) a84 Laplace a85a86 integraldisplay ∞ 0 f(t)e?pt dt = 12pii integraldisplay ∞ 0 e?pt dt integraldisplay s+i∞ s?i∞ F(q)eqt dq, Rep > s0 a184 a83 F(p) a92a159a89 a113a98 a44 a147 a57a192 a84a116a117a156s a114a96 a60 a60 a120a166 Rep > s > s0 a60a79a80 a86 a116a117a81a82 (a105a82a116 a117a84 a118a186 a178a179a130 a60 a147 a83 a128a129a84) a60 a182 a63 a185integraldisplay ∞ 0 f(t)e?pt dt = 12pii integraldisplay s+i∞ s?i∞ F(q)dq integraldisplay ∞ 0 e?(p?q)t dt = 12pii integraldisplay s+i∞ s?i∞ F(q) p?q dq. a172a173a174 Laplace a175 a38 a3921a40 a171 9.7 a147a132a116a117 a120 a146a55 a122a123 a50a124a125a126a127a128 a92 a166a167a168a169a1709.7 a92 a176a177 a153 a58a46a171(2) a60 a105 a180 a1253.1 a120a58 a60a156 R → ∞ a157 a190 CR a84a116a117a191a82 0 a92 a41 a50a51 a186a52 a116a49a50a142 a44a45 a111a112a160a158a193 a118a118 a132a139a140 a60 a118a194a64 a140 q = p a60a79a195 a116a117 a83 a190a196a197a198a199a200 a47a84 a60 a60 f(t) equaldotleftright integraldisplay ∞ 0 f(t)e?pt dt = F(p). a159 a63 a99 a185 f(t) a84 Laplace a85a86a151a89 F(p) a92 a201a128a113a112a152a153a84a69a70 a60 a107 a184 a165 a187a99a100 a67 Laplace a85a86 a84a93a94a95a96a97a98 a92 square