Laplace Table

f (t ) = L 1. 3. 5. 7. 9. 11. 1

-1

{F ( s )}

Table of Laplace Transforms F ( s ) = L { f ( t )} f ( t ) = L -1 { F ( s )} 1 s n! s n +1 2. 4. 6. 8.
2

F ( s ) = L { f ( t )} 1 s-a G ( p +1) s p +1 1 × 3 × 5L ( 2n - 1) p 2n s 2 s 2 s + a2 s2 - a2
2 n+ 1

e at t p , p > -1 t
n-1 2

t n , n = 1, 2,3,K t sin ( at ) t sin ( at ) sin ( at ) - at cos ( at ) cos ( at ) - at sin ( at )sin ( at + b ) sinh ( at ) e at sin ( bt ) e at sinh ( bt ) t ne at , n = 1, 2, 3, K uc ( t ) = u ( t - c )
Heaviside Function

p
2s a 2 s + a2 2as
2
3 2

, n = 1, 2,3,K

cos ( at ) t cos (at ) sin ( at ) + at cos ( at ) cos ( at ) + at sin ( at ) cos ( at + b ) cosh ( at ) e at cos ( bt ) e at cosh ( bt ) f ( ct )

(s

+ a2 )

10. 12.

(s

+ a2 )

2

13. 15. 17. 19. 21.23. 25. 27. 29. 31. 33. 35. 37.

(s + a ) s (s - a ) (s + a )
2 2 2 2 2 2 2 2

2a 3

14. 16. 18. 20. 22. 24. 26. 28. 30. 32. 34. 36.

(s + a ) s ( s + 3a ) (s + a )
2 2 2 2 2 2 2 2

2as 2s sin ( b ) + a cos ( b ) s2 + a2 a 2 s - a2 b

s cos ( b ) - a sin ( b ) s2 + a2 s 2 s - a2 s-a

(s - a)

2

+ b2 -b
n +1 2

(s - a)

2

+ b2 - b2

b

s-a

(s - a)

2

(s -a)

2

n!

(s - a)

1 æsö Fç ÷ c ècø e - cs e - cs L { g ( t + c )}

uc ( t ) f ( t - c ) ect f ( t ) 1 f (t ) t

e - cs s - cs e F (s) F (s - c)
¥ s

d (t - c )
Dirac Delta Functionuc ( t ) g ( t )
t 0

t n f ( t ) , n = 1, 2,3,K

ò

F ( u ) du

ò

f ( v ) dv

( -1)n F ( n) ( s ) F (s)
s

ò

t 0

f ( t - t ) g (t ) dt

F (s)G ( s) sF ( s ) - f ( 0 )

f( t + T ) = f (t ) f ¢¢ ( t )

ò

T 0

e- st f ( t ) dt

f ¢(t ) f ( n) ( t )
© 2004 Paul Dawkins

1 - e - sT s 2 F ( s ) - sf ( 0 ) - f ¢ ( 0 )

s n F ( s ) - s n-1 f ( 0 ) - s n- 2 f ¢( 0 ) L - sf ( n -2) ( 0 ) - f ( n-1) ( 0 ) 1

Table Notes
1. This list is not inclusive and only contains some of the more commonly used Laplace transforms and formulas. 2. Recall the...