# Laplace Table

**Topics:**Dirac delta function, Laplace transform, Leonhard Euler

**Pages:**5 (594 words)

**Published:**March 1, 2013

F ( s ) = L { f ( t )} 1 s n! s n +1

Table of Laplace Transforms

f ( t ) = L -1 {F ( s )}

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

2. 4. 6. 8.

2

e at t p , p > -1 t

n- 1 2

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 2

s 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 Function

uc ( t ) g ( t )

t

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

( -1)

T 0

n

F ( n) ( s )

ò

F ( u ) du

ò f ( v ) dv

0

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

e - st f ( t ) dt

f ¢ (t ) f ( n) ( t )

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 )

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. et + e - t et - e - t cosh ( t ) = sinh ( t ) = 2 2 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 G ( t ) = ò e - x x t -1 dx 0 ¥

If n is a positive integer then,

G ( 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 G ( p + 1) = pG ( p ) p ( p + 1)( p + 2 )L ( p + n - 1) = æ1ö Gç ÷ = p è2ø G ( p + n) G ( p)

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