Wu Chong-shi a0a1a2a3a4 Laplace a5a6 §26.1 Laplace a7a8 ? Laplacea9a10(a11a12a13a14a9a10)a15a16a17a18a19a20a21a22a9a10a23 a24a25a26a27a28a29a30a31a32a33a26a34 a35a36a37 a18a38a17a23 ? a39a40a41a42 Laplacea9a10a18a43a44 a30a45a46 a39a47a48a49a50 a30a51 a18a11a52a38a17a23 Laplacea53a54a55a56a57a58a59a53a54a49a60a61 f(t)a53a54a62F(p)a49 F(p) = integraldisplay ∞ 0 e?ptf(t)dt. a63a64a65 t a66a67a68a49 pa66a69a68a49 p = s + iσ a23 F(p) a70a71f(t) a65 Laplacea72a73a49a74a70a75a76a72a73a23 e?pt a66Laplacea77a72 a65a78 a23 a79a80a81 Laplace a77a72a74a82a71 F(p) =a83{f(t)} a84 F(p)equaldotrightleftf(t); f(t) =a83 ?1{F(p)} a84 f(t) equaldotleftrightF(p). f(t)a85F(p)a86a87a88a89a90a70a71 Laplacea77a72 a65a91a92 a68a85a93 a92 a68a23 a94a95a96a97 a49 a24 a39a40 a34a98 a43a99 f(t)a38a100 a29a101a102 f(t)η(t) a49 a45a34 η(t) = braceleftBigg 1, t > 0, 0, t < 0. a103a104a96 a49a105t < 0a106a38a100 a29a101a102 f(t) = 0 a23 a107 26.1 a92 a68f(t) = 1a65Laplacea72a73a71 1 equaldotleftright integraldisplay ∞ 0 e?pt dt = ? 1p e?pt vextendsinglevextendsingle vextendsinglevextendsingle ∞ 0 = 1p, Rep > 0. a63a64a65a108a109a110a111 Rep > 0 a66a71a112a113a114a115a89a116a117a49a84a118a119a66 Laplacea77a72a120a121 a65a110a111 a23 a107 26.2 a92 a68f(t) = e αt a65Laplace a72a73a71 eαt equaldotleftright integraldisplay ∞ 0 e?pt · eαt dt = ? 1pe?(p?α)t vextendsinglevextendsingle vextendsinglevextendsingle ∞ 0 = 1p?α, Rep > Reα. a63a64a65a108a109a110a111 Rep > Reα a122a123a66a71a112a113a114a115a89a116a117a49a124 Laplacea77a72a120a121a23 a125a12626.1 a85 a12626.2 a127a128a129a130a49 a131a132Laplace a77a72 a65a78 a66e?pt a49 a133 a128a134 a132a135a136a137a138a65a92 a68f(t)a49 a139 a75a76a72a73a140a120a121a141a142a143 a136 t → ∞, f(t) → ∞ a87a49f(t)a65a75a76a72a73a88a127a144a120a121a23 Wu Chong-shi §26.1 Laplacea145a146a147a148a149 a1502a151 Laplacea53a54a152a153a154a155a156a157a158a55a58a59 integraldisplay ∞ 0 e?ptf(t)dta159a160a154a155a156a23a121a161a162a163a68a67a164a165a166a167a49f(t) a140a144a168a169 1. f(t) a24a170a1710 ≤ t < ∞ a34a172a173a174a19a175a171a176a177a178a179a15a180a181a18a49a182a183a35a180a181a184a25a49a24a185a186a35a187 a170a171a34a188 a20 a171a176a177 a18 a25a189 a15 a35a187 a18a141 2. f(t)a35a35a187a18a190a191a192 a25 a49a193a194 a24a195a25M > 0a30sprime ≥ 0 a49 a196a197a198a185a186t a199(a200a201a202a49a203 a95a197a198 a204a205a206 a18ta199)a49 |f(t)| < Mesprimet. a63 a66Laplace a53a54a152a153a154a207a59a155a156a23 a208a209 a165a166a167a210a211 a65a92 a68a140a144a168a169 a63a212a213a214 a23 a215a216sprime a120a121 a65a217 a49a218 a208a219a220a221a222a208 a49a223a71a224sprime a162 a65a225a226a227 a68a88a228a229 a213a214 a23 sprime a65a230a231a70a71a159 a160a232a233a49a234a71s0 a23 Wu Chong-shi a235a236a237a238a239 Laplace a145a146 a1503a151 §26.2 Laplace a7a8a240a241a242a243a244 a245a246 26.1 Laplace a53a54a55a56a247a248 a245 a53a54a49a124a249 f1(t) equaldotleftrightF1(p), f2(t) equaldotleftrightF2(p), a250 α1f1(t) + α2f2(t) equaldotleftrightα1F1(p) + α2F2(p). a188a251 a47a48a252a253a254a255Laplacea9a10a18a43a44a0a1a49 a2a102a51 a203a3a4a15a21a22a5a6a18a7a47a47a48a18a8 a9 a23a10a11 a188a251 a47a48a49a12a193a0a1 sinωt = e iωt ? e?iωt 2i equaldotleftright 1 2i bracketleftbigg 1 p? iω ? 1 p + iω bracketrightbigg = ωp2 + ω2; cosωt = e iωt ? e?iωt 2 equaldotleftright 1 2 bracketleftbigg 1 p? iω + 1 p + iω bracketrightbigg = pp2 + ω2. a245a246 26.2 Laplace a54a13a154a14a15 a245 a23 a188a251 a47a48a16a50a17a17a18a43a19a20a21a22 s0 a49 a188a24a23 Laplace a9a10a18a8a24a106a15a25a16a26 a95 a18a23 a245a246 26.3 a249f(t)a168a169Laplacea77a72a120a121 a65a27 a89 a110a111 a49 a250 F(p) → 0, a136Rep = s → +∞. a245a246 26.4 a28a29a30a154a31a30a154 Laplace a53a54a23a32f(t) a33fprime(t) a140a168a169Laplacea77a72a120a121 a65a27 a89 a110a111 a49f(t) equaldotleftright F(p)a49 a250 fprime(t) equaldotleftrightpF(p) ?f(0). a197a34a35a25f(t) a18a36a37a5a6a38a39 a102a197a40a35a25 F(p) a18a41a42a5a6a49a182a183a43a44 a45a46a47a173f(t) a18 a48 a199a23 a195a2a102a188a251a49a177 a49a50a50Laplacea9a10a51a42a15 a23a101 a36a22a51 a32 a18a19a20a26 a95 a51a42a23 a122a123a49a52 a213 f(t), fprime(t), ···, f(n)(t) a140a168a169 Laplacea77a72a120a121 a65a27 a89 a110a111 a49 f(t) equaldotleftright F(p)a49 a250 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). Wu Chong-shi §26.2 Laplacea145a146a147a53a54a55a56 a1504a151 a245a246 26.5 a28a29a30a154a58a59a154 Laplace a53a54a23a32f(t)a168a169Laplacea77a72a120a121 a65a27 a89 a110a111 a49 a250 integraldisplay t 0 f(τ)dτ equaldotleftright F(p)p . Wu Chong-shi a235a236a237a238a239 Laplace a145a146 a1505a151 §26.3 Laplace a7a8a240a57a58 a59 a29a30a154a31a30a154a60a61 a32 f(t) a168a169 Laplace a77a72a120a121 a65a27 a89 a110a111 a49 f(t) equaldotleftright F(p) a49 a250 F(p) a121 Rep ≥ s1 > s0 a65a62a63a64a167a65a66a49a223a67a127a128a121a115a89a68a230a214a69 F(n)(p) = d n dpn integraldisplay ∞ 0 f(t)e?pt dt = integraldisplay ∞ 0 (?t)nf(t)e?pt dt. a133 a128 F(n)(p) equaldotrightleft (?t)nf(t). a70a71a63a212a72 a73a49a127a128a73a74a75a76a211 1 p2 = ? d dp 1 p equaldotrightleftt, 1 p3 = 1 2 d2 dp2 1 p equaldotrightleft 1 2t 2. a249F(p)a66a86a77 a92 a68a49 a250a78 a127a128 a79a79a80 a89a89a73 a214a81a82 a23 a126a215 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. a59 a29a30a154a58a59a154a60a61 a215a216 integraldisplay ∞ p F(q)dq a120a121a83a49a84 a136t → 0 a87a49|f(t)/t|a86 a231 a49 a250 integraldisplay ∞ p F(q)dq equaldotrightleft f(t)t . (star) a85a86a63a212a72 a73a49a87a127a128a76a211a88a163 a92 a68 a65 Laplace a77a72a23 a126a215 sinωt t equaldotleftright integraldisplay ∞ p ω q2 + ω2 dq = pi 2 ? arctan p ω. a89 a90a66a49 a215a216 p → 0 a87a49(star) a73a90a91 a65 a115a89a92a120a121a49 a250 a86integraldisplay ∞ 0 F(p)dp = integraldisplay ∞ 0 f(t) t dt. a85a86a63a212a93a216 a49a127a128a94a95 integraldisplay ∞ 0 f(t) t dta96 a65 a115a89a23 a126a215 integraldisplay ∞ 0 sint t dt = integraldisplay ∞ 0 1 p2 + 1 dp = pi 2. a188a251 a21a22a97a98a38a17a99 a25 a43 a29a100 a6a4a23 a188a101 a18 a100 a6a102 a102 a11a103a23 a86a104a115a89 a86a105 a68 a219 a77a94a95a224a106a69a107a49a108a109a127a128a110a111a75 a86a63a212a112a113 a94a95a23 a126a215 integraldisplay ∞ 0 cosat? cosbt t dt = integraldisplay ∞ 0 parenleftbigg p p2 + a2 ? p p2 + b2 parenrightbigg dp a83 a114a115a116a117a118a119a120a121a122a123a124 Rep→+∞a125a126a127 a117a118a128a129a130 F(p) a116a123a131a132a133a134 a125a135a136 a117a118a137a128a129a138a139a140 Wu Chong-shi §26.3 Laplacea145a146a147a141a142 a1506a151 = 12 ln p 2 + a2 p2 + b2 vextendsinglevextendsingle vextendsinglevextendsingle ∞ 0 = lnb? lna, a > 0, b > 0. a10a11 Laplace a9a10a18a7a47a47a48a49a143a144 Laplace a10a145 F(p) a16a50a22 a101a102a146a251a35a25 F 1(p) a147 F2(p)a148a147a49 a149a150 a49 a51 a18a8a24a151a152a105a153a154a252a11a52a99a203 a95F 1(p)a147F2(p)a18 a34a35a25a179 a194 a24 a49 F(p) a18 a34a35a25 a154a15F1(p) a147 F2(p) a18 a34a35a25 a148a147a23a143a144F(p) a16a50a22 a101a102F 1(p) a147F2(p) a148a21a49 a45 a8a24a151a152a154 a94a95 a17a1a155a156a18a157a21a43 a29 a23 a158 a58a159a160 a32F1(p) equaldotrightleftf1(t)a49F2(p) equaldotrightleft f2(t)a49 a250 F1(p)F2(p) equaldotrightleft integraldisplay t 0 f1(τ)f2(t?τ)dτ. a161a162a81a82a72 a73 a249 a92 a68F(p), p = s + iσ a168a169a99 (1) F(p)a24a170a163 Rep > s0 a34a101a164a49 (2)a24a170a163 Rep > s0 a34a49|p| → ∞a106F(p)a19a165a166a167 a198 0 a49 (3)a197a198a50a35a18 Rep = s > s0 a49a168a169a7 L : Rep = sa18a170a171a21a22integraldisplay s+i∞ s?i∞ |F(p)|dσ (s > s0) a19a20a49 a250 a134 a132 Rep = s > s 0 a49F(p)a66 f(t) = 12pii integraldisplay s+i∞ s?i∞ F(p)ept dp a65Laplace a77a72a49 a139 a167ta71a67a77a172a23 a131a161a162a81a82a72 a73 a214Laplace a77a72 a65a91a92 a68a49 a173 a33a69 a63a64a174a65a175a176 a115a89a23 a208a209 a127 a85a86a105 a68 a219 a77a177 a94a95a23