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WuChong-shi Laplace §26.1Laplace •Laplace ( ) ! #$% & •’()*Laplace +, -.’/012 3 4& Laplace56789:;561=f(t)56F(p)1F(p)=Z∞0e−ptf(t)dt.?@AtBCD1pBED1p=s+iσ F(p)FGf(t)ALaplaceHI1JFKLHI e−ptBLaplaceMHAN OPQLaplaceMHJRGF(p)=S{f(t)}TF(p):f(t);f(t)=S−1{F(p)}Tf(t);F(p).f(t)UF(p)VWXYZFGLaplaceMHA[\DU]\D ^_‘a1 ’(b+cf(t)&d eff(t)η(t)1-η(t)=(1,t0,0,t0.gh‘1it0j&d eff(t)=0 k26.1\Df(t)=1ALaplaceHIG1;Z∞0e−ptdt=−1pe−pt ∞0=1p,Rep0.?@AlmnoRep0BGpqrsYtu1TvwBLaplaceMHxyAno k26.2\Df(t)=eαtALaplaceHIGeαt;Z∞0e−pt·eαtdt=−1pe−(p−α)t ∞0=1p−α,RepReα.?@AlmnoRepReαz{BGpqrsYtu1|LaplaceMHxy }~26.1U~26.2 1 LaplaceMHANBe−pt1 A\Df(t)1 KLHI xy t→∞,f(t)→∞W1f(t)AKLHIX xy WuChong-shi§26.1Laplace 2 Laplace56 7:;Z∞0e−ptf(t)dt y¡¢£DC⁄¥ƒ§1f(t) ¤'1.f(t) “«0≤t∞‹›fi fl« –†‡ · 1¶•#· ‚ 1 „”#»“«… « – ‰ #» 2.f(t)##» ¿ 1`´ ˆ M0 s0≥01˜¯˘„”t˙(¨ ˚1¸_¯˘ ˝˛ t˙)1|f(t)|Mes0t.?BLaplace56 ˇ; — ¥ƒ§ A\D ¤'? s0xyA 1 — —1 G s0¢AÆ ªDX s0A FG ŁØ1ŒGs0 WuChong-shiº Laplace 3 §26.2Laplace æ ı 26.1Laplace5678 łı561|øf1(t);F1(p),f2(t);F2(p),œα1f1(t)+α2f2(t);α1F1(p)+α2F2(p).…ß/0 Laplace +, 1 f3¸ //0 …ß/01 ` sinωt=eiωt−e−iωt2i;12i1p−iω−1p+iω=ωp2+ω2;cosωt=eiωt−e−iωt2;121p−iω+1p+iω=pp2+ω2.ı 26.2Laplace6 ı …ß/0 2 + s01… Laplace j _ ı 26.3øf(t)¤'LaplaceMHxyA Yno1œF(p)→0, Rep=s→+∞.ı 26.4 Laplace56 f(t)!f0(t) ¤'LaplaceMHxyA Yno1f(t);F(p)1œf0(t);pF(p)−f(0).¯# f(t) $% &’f¯(# F(p) )* 1¶•+,-./›f(t) 0˙ ˆ f…ß1–122Laplace 3* e$ 3 _3* z{14 f(t),f0(t),···,f(n)(t) ¤'LaplaceMHxyA Yno1f(t);F(p)1œf00(t);p2F(p)−pf(0)−f0(0),f(3)(t);p3F(p)−p2f(0)−pf0(0)−f00(0),...f(n)(t);pnF(p)−pn−1f(0)−pn−2f0(0)−···−pf(n−2)(0)−f(n−1)(0).WuChong-shi§26.2Laplace 5678 4 ı 26.5 :; Laplace56 f(t)¤'LaplaceMHxyA Yno1œZt0f(τ)dτ;F(p)p.WuChong-shiº Laplace 5 §26.3Laplace 9:; =f(t)¤'LaplaceMHxyA Yno1f(t);F(p)1œF(p)yRep≥s1s0A?@§AB1 C ysYD EF(n)(p)=dndpnZ∞0f(t)e−ptdt=Z∞0(−t)nf(t)e−ptdt. F(n)(p):(−t)nf(t).FG? HI1 IJKL 1p2=−ddp1p:t,1p3=12d2dp21p:12t2.øF(p)BVM\D1œN OOPYYI QR ~ 1p3(p+α)=1α1p3−1α21p2+1α31p−1α31p+α:12αt2+1α2t+1α3−1α3e−αt.; :; = Z∞pF(q)dqxyS1T t→0W1|f(t)/t|V 1œZ∞pF(q)dq:f(t)t.(F)UV? HI1W L X£\DALaplaceMH ~ sinωtt;Z∞pωq2+ω2dq=π2−arctanpω.YZB1 p→0W1(F)IZ[AsY\xy1œVZ∞0F(p)dp=Z∞0f(t)tdt.UV? ] 1 ^_Z∞0f(t)tdt‘AsY ~ Z∞0sinttdt=Z∞01p2+1dp=π2.…ß ab& c + d …e d ff g VhsYViD M^_ jEk1lm noKV? pq^_ ~ Z∞0cosat−cosbttdt=Z∞0pp2+a2−pp2+b2dpSrstuvwxyz{|Rep→+∞}~ uv F(p)t{ } uv WuChong-shi§26.3Laplace 6 =12lnp2+a2p2+b2 ∞0=lnb−lna,a0,b0. Laplace //01 Laplace F(p) 2 ef ß# F1(p) F2(p) 1 13 i 4c¸_F1(p) F2(p) # ‡´ 1F(p) # F1(p) F2(p) # F(p) 2 efF1(p) F2(p) 1- ^_ + : F1(p):f1(t)1F2(p):f2(t)1œF1(p)F2(p):Zt0f1(τ)f2(t−τ)dτ.¡¢QRHIø\DF(p),p=s+iσ¤'c(1)F(p) “£Reps0e⁄1(2) “£Reps01|p|→∞jF(p) ¥ƒ§˘01(3)¯˘2# Rep=ss01¤' L:Rep=s “« Zs+i∞s−i∞|F(p)|dσ(ss0) 1œ Rep=ss01F(p)Bf(t)=12πiZs+i∞s−i∞F(p)eptdpALaplaceMH1 §tGCM‹ ¡¢QRHI LaplaceMHA[\D1›!E?@fiAfl sY — UViD M–^_