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【正文】 ics, National University of Kaohsiung Unit step function Chapter 5 Laplace Transforms . Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Dirac’s Delta function Chapter 5 Laplace Transforms ))](()([1)( katuatukatf k ??????以?xún)蓚€(gè)單位階梯函數(shù)來(lái)表示 163。[eatcos(wt)] 22 w)as(as????Rule 2: Let a be a positive constant. Let f(t) be given, with f(t) = 0 if t 0. Define g(t) by g(t) = f(ta), then 163。[f(t)] = F(s) 163。[f(t)] sdxe2dtte 0sx02/1st 2 ???? ?? ? ?? ?????? ? ? ??0 2/1t dtte)21(Chapter 5 Laplace Transforms . Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 5 Laplace Transforms . Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Laplace Transform Rule 1: if then 163。[f(t)] ?? ? ?? ?? ??0sx02/1st dxe2dtte 2{163。1[F(s)] Chapter 5 Laplace Transforms . Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Laplace Transform t1f(t) = 163。1[G(s)] 4. 163。1[F(s)+G(s)] = 163。[af(t)] = a163。(coswt)+ i163。(eiwt) = 163。[f(t)]+ 163。[sin(wt)] 22 wsw??Chapter 5 Laplace Transforms 利用分部積分 . Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Laplace Transform Theorem : 1. 163。[f(t)] }wss)]was i n (w)wac o s (s[wse{limdt)wtc o s (elimdt)wtc o s (e2222saaa0sta0st??????????????? ???If s 0 )s(Fwss22 ???163。39。(tn) 1ns!n??)t(f)t(y)t(Qy)t(P)t(y 39。1[F(s)] = f(t) The Laplace transform of f(t) = t is 163。[f(t)] )s(Fdt)t(felimdt)t(fe a0sta0st ??? ?? ???? ?163。. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 5 Laplace Transforms Laplace Transform 1. 拉普拉斯轉(zhuǎn)換乃 算子演算法 (operational calculus),它將微積分演算變成代數(shù)演算 .(為特殊的傅立葉轉(zhuǎn)換 ) 2. 拉普拉斯轉(zhuǎn)換在工程上用於機(jī)械以及電力的驅(qū)動(dòng)力問(wèn)題 ,特別是當(dāng)驅(qū)動(dòng)力為不連續(xù) ,脈衝或是正弦 ,餘弦及更複雜的周期性函數(shù) . 3. 拉普拉斯轉(zhuǎn)換可直接解問(wèn)題 ,求解初值問(wèn)題時(shí)無(wú)需先求通解 ,且解非齊次微分方程時(shí)亦無(wú)需先求對(duì)應(yīng)之齊次方程式之解 . 4. 偏微分方程式也能以拉普拉斯轉(zhuǎn)換處理 . . Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 5 Laplace Transforms Laplace Transform The Laplace transform 163。[f(t)] of a function f(t) is defined to be 163。[f(t)] = F(s) 163。[f(t)] )]sa1(ses1[limt d telimt d te2sa2aa0sta0st ????? ??????? ? ??If s 0 )s(Fs12 ??163。39。 ???. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Laplace Transform The Laplace transform of f(t) = cos(wt) is 163。[cos(wt)] 22 wss??同理可證 163。[f(t)+g(t)] = 163。[g(t)] Whenever all three Laplace transf
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