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)