Laplace_table.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)
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)
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