Table of Laplace Transforms

Definition of Laplace Transforms

Let f(t) be a function of the real variable t, such that t ≥ 0. The laplace transform F(s) of f is given by the integral

F(s) = L(f(t) = 0 e -st f(t) dt


s is a complex variable. f(t) is called the original and F(s) is called the image function.

Table of Laplace Transforms.

f(t) F(s)
1 1 / s
t 1 / s 2
t n n! / s n+1 , (n = 1,2,3...)
t 1/2 Pi 1/2 / 2s 3/2
t -1/2 (Pi / s) 1/2
e -a t 1 / (s + a)
t e -a t 1 / (s + a) 2
sin a t a / (s 2 + a 2)
t sin a t 2 a s / (s 2 + a 2) 2
e -at sin b t b / (s + a) 2 + b 2
cos a t s / (s 2 + a 2)
t cos a t (s 2 - a 2) / (s 2 + a 2) 2
e -at cos b t (s + a) / (s + a) 2 + b 2
sinh a t a / (s 2 - a 2)
cosh a t s / (s 2 - a 2)
1 - cos a t a 2 / s (s 2 + a 2)
(2 / t)( t - cos a t) ln [ (s 2 + a 2) / s 2 ]
(2 / t)( t - cosh a t) ln [ (s 2 - a 2) / s 2 ]
(1 / t)( sin a t) arctan(a / s)



More references on integrals in calculus.






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