Math 4564 Laplace Transform Table PDF [PDF]

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

1

3.

t n , n = 1, 2,3,K

5.

1 s n! s n +1

p

t

3 2

7.

sin ( at )

9.

t sin ( at )

11.

Table of Laplace Transforms

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

sin ( at ) - at cos ( at )

2s a 2 s + a2 2as

(s

2

+ a2 )

2a 3

2 2

2

cos ( at ) - at sin ( at )

15.

sin ( at + b )

17.

sinh ( at )

19.

e at sin ( bt )

21.

e at sinh ( bt )

23.

t ne at , n = 1, 2,3,K

25.

uc ( t ) = u ( t - c )

2 2

(s - a)

2

+ b2

b

(s - a)

2

-b

(s - a)

n +1

27.

uc ( t ) f ( t - c )

e - cs s - cs e F (s)

29.

ect f ( t )

F ( s - c)

31.

1 f (t ) t t

f ( t - t ) g (t ) dt

33.

ò

35.

f ¢ (t )

37.

f ( n) ( t )

0

2

n!

¥ s

4.

t p , p > -1

6.

t

8.

cos ( at )

10.

t cos ( at )

n - 12

, n = 1, 2,3,K

1 s-a G ( p + 1) s p +1 1 × 3 × 5L ( 2n - 1) p n+ 1

2n s 2 s 2 s + a2 s2 - a2

(s

+ a2 )

2

sin ( at ) + at cos ( at )

F ( u ) du

14.

cos ( at ) + at sin ( at )

16.

cos ( at + b )

18.

cosh ( at )

20.

e at cos ( bt )

22.

e at cosh ( bt )

24.

f ( ct )

26.

d (t - c ) Dirac Delta Function

28.

uc ( t ) g ( t )

30.

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

32.

ò f ( v ) dv

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

2

f (t + T ) = f (t )

sF ( s ) - f ( 0 )

36.

f ¢¢ ( t )

2 2

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

(s - a)

2

+ b2

s-a

(s - a)

2

- b2

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

( -1)

n

F ( n) ( s )

F (s) s

t

34.

2

2

0

F (s)G (s)

2

2as 2 2

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

ò

e at

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

2

2

Heaviside Function

2.

12.

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

13.

2

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

ò

T 0

e - st f ( t ) dt

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)