File name: Table Of Laplace Transforms Pdf

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Table Of Laplace Transforms Pdf

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Table of Laplace transforms f(t) L(f(t)) or F(s) 1. 1 1 s 2. eat 1 s−a 3. tn n! sn+1 n≥0 integer 4. eattn n! (s−a)n+1 n≥0 integer 5. sinkt k s2 +k2 6. coskt s s2 +k2 7. eatsinkt k (s−a)2 +k2 8. . Table of Laplace Transforms f(t) L{f(t)} =F(s) 1. 1 1 s 2. t 1 s2 3. tn n! sn+1, n apositiveinteger 4. sinkt k s2 +k2 5. coskt s s2 +k2 6. sin2kt 2k2 s(s2 +4k2) 7. cos2kt s2 +2k2 s(s2 +4k2) 8. eat 1 . Table of Laplace Transforms f(t) L(f(t)) f(t) L(f(t)) 1 1 s t 1 s2 Derivatives t2 2 s3 y L(y) tn n! sn+1 y0 sL(y) y(o) eat 1 s a y00 s2L(y) sy(o) y0(0) tneat n! (s a)n+1 cos(!t) s s2 +!2 sin(!t)! s2 +!2 t . Table of Laplace Transforms f(t) L(f(t)) f(t) L(f(t)) 1 1 s t 1 s2 Derivatives t2 2 s3 y L(y) tn n! sn+1 y0 sL(y) y(o) eat 1 s a y00 s2L(y) sy(o) y0(0) tneat n! (s a)n+1 cos(!t) s s2 +!2 sin(!t)! s2 +!2 t-Shift cosh(at) s s2 a2 f(t) F(s) sinh(at) a s2 a2 u a(t)f(t a) e asF(s) eat cos(!t) s a (s a)2 +!2 eat sin(!t)! (s a)2 +!2 s-Shift (t a) e as. Table of Laplace Transforms f(t) L[f(t)] = F(s) 1 1 s (1) eatf(t) F(s a) (2) U(t a) e as s (3) f(t a)U(t a) e asF(s) (4) (t) 1 (5) (t stt 0) e 0 (6) tnf(t) (1)n dnF. We'll give two examples of the correct interpretation. First, suppose that f is the constant 1, and has no discontinuity at t = 0. In other words, f is the constant function with value 1. Then we have 0 f = 0, and f(0¡) = 1 (since there is no jump in f at t = 0). Now let's apply the derivative formula above. Table of Laplace Transforms f(x) F(s) = L[f(x)] c c s, s > 0 erx 1 s−r, s > r cos βx s s2 +. 2 Speci c Transforms f(t) = L 1fFg(t) F(s) = Lffg(s) 1 1 s t n! sn+1 p1 t p ˇ s p t p ˇ 2 1 s3=2 eat 1 s a sin(at) a s2+a2 cos(at) s s2+a2 eatsin(bt) b (s a)2+b2 eatcos(bt) s a (s a)2+b2 u(t ec) (for c 0) cs. Table of Laplace transforms f(t) L(f(t)) or F(s) 1. 1 1 s 2. eat 1 s−a 3. tn n! sn+1 n≥0 integer 4. eattn n! (s−a)n+1 n≥0 integer 5. sinkt k s2 +k2 6. coskt s s2 +k2 7. eatsinkt k (s−a)2 +k2 8. eatcoskt s−a (s−a)2 +k2 9. 1 √ t r π s u(t−a) e−as s a≥0 δ(t−a) e−as a≥0.