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Table of Common Laplace Transforms

Module by: Melissa Selik, Richard Baraniuk

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

Signal Laplace Transform Region of Convergence
δt δ t 1 1 All ss
δt-T δ t T ⅇ-sT s T All ss
ut u t 1s 1 s ℜs>0 s 0
-u-t u t 1s 1 s ℜs<0 s 0
tut t u t 1s2 1 s 2 ℜs>0 s 0
tnut t n u t n!sn+1 n s n 1 ℜs>0 s 0
-tnu-t t n u t n!sn+1 n s n 1 ℜs<0 s 0
ⅇ-λtut λ t u t 1s+λ 1 s λ ℜs>-λ s λ
-ⅇ-λtu-t λ t u t 1s+λ 1 s λ ℜs<-λ s λ
tⅇ-λtut t λ t u t 1s-λ2 1 s λ 2 ℜs>-λ s λ
tnⅇ-λtut t n λ t u t n!s+λn+1 n s λ n 1 ℜs>-λ s λ
-tnⅇ-λtu-t t n λ t u t n!s+λn+1 n s λ n 1 ℜs<-λ s λ
cosbtut b t u t ss2+b2 s s 2 b 2 ℜs>0 s 0
sinbtut b t u t bs2+b2 b s 2 b 2 ℜs>0 s 0
ⅇ-atcosbtut a t b t u t s+as+a2+b2 s a s a 2 b 2 ℜs>-a s a
ⅇ-atsinbtut a t b t u t bs+a2+b2 b s a 2 b 2 ℜs>-a s a
dndtnδt t n δ t sn s n All ss

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