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Title: Table of Laplace Transforms
Description: Here is a list of Laplace transforms for a differential equations class. This table gives many of the commonly used Laplace transforms and formulas.
Description: Here is a list of Laplace transforms for a differential equations class. This table gives many of the commonly used Laplace transforms and formulas.
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f ( t ) = L -1 {F ( s )}
1
...
t n , n = 1, 2,3,K
5
...
sin ( at )
9
...
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
...
sinh ( at )
19
...
e at sinh ( bt )
23
...
uc ( t ) = u ( t - c )
2 2
(s - a)
2
+ b2
b
(s - a)
2
-b
(s - a)
n +1
27
...
ect f ( t )
F ( s - c)
31
...
ò
35
...
f ( n) ( t )
0
2
n!
¥
s
4
...
t
8
...
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 + b )
18
...
e at cos ( bt )
22
...
f ( ct )
26
...
uc ( t ) g ( t )
30
...
ò f ( v ) dv
(s + a )
s ( s + 3a )
(s + a )
2 2
2
f (t + T ) = f (t )
sF ( s ) - f ( 0 )
36
...
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
...
(s + a )
s(s - a )
(s + a )
2
13
...
This list is not a complete listing of Laplace transforms and only contains some of
the more commonly used Laplace transforms and formulas
...
Recall the definition of hyperbolic functions
...
Be careful when using “normal” trig function vs
...
The only
difference in the formulas is the “+ a2” for the “normal” trig functions becomes a
“- a2” for the hyperbolic functions!
4
...
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)
Title: Table of Laplace Transforms
Description: Here is a list of Laplace transforms for a differential equations class. This table gives many of the commonly used Laplace transforms and formulas.
Description: Here is a list of Laplace transforms for a differential equations class. This table gives many of the commonly used Laplace transforms and formulas.