Table of Laplace Transforms
f t L 1 F s F s L f t f t L 1 F s F s L f t
1 1
1. 1 2.
eat
s s a
n! p 1
p
3. tn, n 1,2,3, 4.
t , p > -1
sn 1
sp 1
1 1 3 5 2n 1
2
5. 6.
t tn , n 1,2,3,
3 1
2 2
2s 2n sn
a s
sin at cos at
7. 8.
s2 a2 s2 a2
2as
s2 a2
t sin at 2 t cos at
9. 10. 2
s2 a2
s2 a2
2a3 2as2
sin at at cos at sin at at cos at
11. 2 12. 2
s2 a2 s2 a2
s s2 a2 s s2 3a2
cos at at sin at cos at at sin at
13. 14.
2 2
s2 a2 s2 a2
s sin b a cos b s cos b asin b
sin at b cos at b
15. 16.
s2 a2 s2 a2
a s
sinh at cosh at
17. 18.
s2 a2 s2 a2
b s a
eat sin bt eat cos bt
19. 2 20. 2
s a b2 s a b2
b s a
eat sinh bt eat cosh bt
21. 2 22. 2
s a b2 s a b2
n!
1 s
f ct F
23. tneat , n 1, 2,3, n 1 24.
s a
c c
uc t u t c e cs t c
25. 26.
e cs
Heaviside Function Dirac Delta Function
s
uc t f t c e csF s uc t g t e csL g t c
27. 28.
n
ect f t F s c tn f t , n 1, 2,3,
29. 30. 1 F n s
t
1 F s
F u du f v dv
31. f t 32.
s 0
t
s
T
t
e st f t dt
f t g d F s G s f t T f t 0
33. 34.
0
1 e sT
f t sF s f 0 f t s2F s sf 0 f 0
35. 36.
n n 2 n 1
37. f t snF s sn 1 f 0 sn 2 f 0 sf 0 f 0
Table Notes
1. This list is not a complete listing of Laplace transforms and only contains some of
the more commonly used Laplace transforms and formulas.
2. Recall the definition of hyperbolic functions.
t t
et eet e
cosh t sinh t
22
3. Be careful when using normal trig function vs. hyperbolic functions. The only
difference in the formulas is the + a2 for the normal trig functions becomes a
- a2 for the hyperbolic functions!
4. Formula #4 uses the Gamma function which is defined as
x
t e xt 1 dx
0
If n is a positive integer then,
n 1 n!
The Gamma function is an extension of the normal factorial function. Here are a
couple of quick facts for the Gamma function
p 1 p p
p n
p p 1 p 2 p n 1
p
1
2
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