|
Definition of Laplace Transforms
Let f(t) a function of the real variable t, such that t>= 0. The laplace transform F(s) of f is given by
F(s) = L(f(t) = ò 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.
|