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Table of Common Laplace Transforms

Module by: Melissa Selik, Richard Baraniuk. E-mail the authors

Summary: Lists signal and region of convergence for several Laplace transforms.

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Table 1
Signal Laplace Transform Region of Convergence
δt δ t 1 1 All ss
δtT δ t T e(sT) s T All ss
ut u t 1s 1 s s>0 s 0
ut u t 1s 1 s s<0 s 0
tut t u t 1s2 1 s 2 s>0 s 0
tnut t n u t n!sn+1 n s n 1 s>0 s 0
(tnut) t n u t n!sn+1 n s n 1 s<0 s 0
e(λt)ut λ t u t 1s+λ 1 s λ s>λ s λ
(e(λt))ut λ t u t 1s+λ 1 s λ s<λ s λ
te(λt)ut t λ t u t 1sλ2 1 s λ 2 s>λ s λ
tne(λt)ut t n λ t u t n!s+λn+1 n s λ n 1 s>λ s λ
(tne(λt)ut) t n λ t u t n!s+λn+1 n s λ n 1 s<λ s λ
cosbtut b t u t ss2+b2 s s 2 b 2 s>0 s 0
sinbtut b t u t bs2+b2 b s 2 b 2 s>0 s 0
e(at)cosbtut a t b t u t s+as+a2+b2 s a s a 2 b 2 s>a s a
e(at)sinbtut a t b t u t bs+a2+b2 b s a 2 b 2 s>a s a
d n dt n δt t n δ t sn s n All ss

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