APPENDIX

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. e at
s s-a
n! G ( p + 1)
3. t n , n = 1, 2,3,K 4. t p , p > -1
s n+1 s p +1
p 1 × 3 × 5L ( 2n - 1) p
, n = 1, 2,3,K
n - 12
5. t 3 6. t n+ 1
2s 2
2n s 2
a s
7. sin ( at ) 8. cos ( at )
s + a2
2
s + a2
2

2as s2 - a2
9. t sin ( at ) 10. t cos ( at )
(s + a2 ) (s + a2 )
2 2 2
2

2a 3 2as 2
11. sin ( at ) - at cos ( at ) 12. sin ( at ) + at cos ( at )
(s + a )
2 2 2
(s + a )
2 2 2

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

13. cos ( at ) - at sin ( at ) 14. cos ( at ) + at sin ( at )

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

s sin ( b ) + a cos ( b ) s cos ( b ) - a sin ( b )

15. sin ( at + b ) 16. cos ( at + b )
s2 + a2 s2 + a2
a s
17. sinh ( at ) 18. cosh ( at )
s - a2
2
s - a2
2

b s-a
19. e at sin ( bt ) 20. e at cos ( bt )
(s - a) (s - a)
2 2
+ b2 + b2
b s-a
21. e at sinh ( bt ) 22. e at cosh ( bt )
(s - a) (s - a)
2 2
-b 2
- b2
n! 1 æsö
23. t n e at , n = 1, 2, 3, K 24. f ( ct ) Fç ÷
(s - a)
n +1
c ècø
uc ( t ) = u ( t - c ) e - cs d (t - c )
25. 26. e - cs
Heaviside Function s Dirac Delta Function
27. uc ( t ) f ( t - c ) e F (s)
- cs
28. uc ( t ) g ( t ) e - cs L { g ( t + c )}
29. ect f ( t ) F (s - c) 30. t n f ( t ) , n = 1, 2,3,K ( -1)n F ( n) ( s )
1 ¥ t F (s)
31. f (t ) ò F ( u ) du 32. ò f ( v ) dv
t s 0
s
T
t
ò e- st f ( t ) dt
33. ò f ( t - t ) g (t ) dt F (s)G ( s) 34. f ( t + T ) = f (t ) 0
0
1 - e - sT
35. f ¢(t ) sF ( s ) - f ( 0 ) 36. f ¢¢ ( t ) s 2 F ( s ) - sf ( 0 ) - f ¢ ( 0 )

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

© 2004 Paul Dawkins 1