Differential Equations
250
Section 4-10 : Table Of Laplace Transforms
f  t    1 F  s 
1.
1
2.
ea t
3.
t n , n  1, 2,3,
4.
t p , p > -1
5.
t
n  12
, n  1, 2,3,
t
7.
sin  at 
8.
cos  at 
9.
t sin  at 
11.
12.
1
s
1
sa
n!
s n 1
  p  1
s p 1

3
6.
10.
F  s     f  t 
t cos  at 
sin  at   at cos  at 
sin  at   at cos  at 
2s 2
1  3  5  2n  1 
n 1
2n s 2
a
2
s  a2
s
2
s  a2
2as
s
 a2 
2
s2  a2
s
 a2 
2
cos  at   at sin  at 
s
 a2 
2
cos  at   at sin  at 
15.
sin  at  b 
16.
cos  at  b 
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2
2as 2
s  a 
ss  a 
s  a 
s  s  3a 
s  a 
2 2
2
2
2 2
2
2
14.
2
2a 3
2
13.
2
2
2
2 2
s sin  b   a cos  b 
s2  a2
s cos  b   a sin  b 
s2  a2
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Differential Equations
251
a
s  a2
s
2
s  a2
b
17.
sinh  at 
18.
cosh  at 
19.
eat sin  bt 
20.
e at cos  bt 
21.
eat sinh  bt 
22.
eat cosh  bt 
23.
t ne at , n  1, 2,3,
s  a
24.
f  ct 
25.
uc  t   u  t  c 
1 s
F 
c c
26.
2
s  a
s  a
s  a
uc  t  g  t 
29.
ect f  t 
30.
t n f  t  , n  1, 2,3,
31.
1
f t 
t
32.

33.
 f  t    g   d
t
0
 b2
2
 b2
sa
s  a
2
 b2
n!
n 1
e  cs
s
e  cs
Dirac Delta Function
28.
2
b
 t  c 
uc  t  f  t  c 
 b2
sa
Heaviside Function
27.
2
e  cs F  s 
e  cs L  g  t  c 
F s  c
 1


s
n
F n  s 
F  u  du
F  s
s
f  v  dv
t
F  sG s
0

T
e  st f  t  dt
34.
f t  T   f t 
35.
f  t 
1  e  sT
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   sf  n  2  0   f  n 1  0 
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0
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Differential Equations
252
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.
cosh  t  
et  e  t
2
sinh  t  
et  e  t
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

  t    e  x x t 1 dx
0
If n is a positive integer then,
  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
  p  1  p  p 
p  p  1 p  2   p  n  1 
  p  n
  p
1
   
2
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