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Laplace Transform Tables

This document outlines common Laplace transforms and properties of the Laplace transform. It lists 10 common transforms including those of 1, eat, tn, sin(bt), cos(bt), and the Heaviside function Hc(t). It then describes 6 main properties of the Laplace transform including linearity, translation, the transform of the derivative f'(t), second derivative f''(t), nth derivative f(n)(t), and taking the derivative of the transform.

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Ian Causseaux
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132 views2 pages

Laplace Transform Tables

This document outlines common Laplace transforms and properties of the Laplace transform. It lists 10 common transforms including those of 1, eat, tn, sin(bt), cos(bt), and the Heaviside function Hc(t). It then describes 6 main properties of the Laplace transform including linearity, translation, the transform of the derivative f'(t), second derivative f''(t), nth derivative f(n)(t), and taking the derivative of the transform.

Uploaded by

Ian Causseaux
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Common Laplace Transforms and Main Properties

Common Transforms.
(1)
1
L {1} (s) = , s > 0.
s
(2)
1
L eat (s) =

, s > a.
s−a
(3)
n!
L {tn } (s) = , n = 1, 2, ..., s > 0.
sn+1
(4)
b
L {sin(bt)} (s) = , s > 0.
s2 + b2
(5)
s
L {cos(bt)} (s) = , s > 0.
s2 + b2
(6)
n!
L eat tn (s) =

, n = 1, 2, ..., s > a.
(s − a)n+1

(7)
b
L eat sin(bt) (s) =

, s > a.
(s − a)2 + b2
(8)
s−a
L eat cos(bt) (s) =

, s > a.
(s − a)2 + b2
(9)
1
L {Hc (t)} (s) = e−cs , s > 0.
s
(10)
L {Hc (t)f (t − c)} (s) = e−cs L {f (t)} (s), s > 0.

For c > 0, the Heaviside function (or unit step function) Hc (t) in (9) and
(10) is the piecewise continuous function given by:

0, 0 ≤ t < c,
Hc (t) =
1, t ≥ c.

1
2

Some Properties of the Laplace Transform.


Suppose f and g are such that their corresponding Laplace transforms,

F (s) = L {f (t)} (s), G(s) = L {g(t)} (s)


exist, and let c, a ∈ R.
(1) (linearity)

L {f + g} (s) = F (s) + G(s), L {cf } (s) = cF (s).

(2) (translation)

L eat f (t) (s) = F (s − a).




(3) (transform of f 0 )

L f 0 (t) (s) = sF (s) − f (0).




(4) (transform of f 00 )

L f 00 (t) (s) = s2 F (s) − sf (0) − f 0 (0).




(5) (transform of f (n) , the nth−derivative of f )


n o
L f (n) (t) (s) = sn F (s) − sn−1 f (0) − sn−2 f 0 (0) − ... − f (n−1) (0).

(6) (derivative of the transform)


dn
L {tn f (t)} (s) = (−1)n F (s).
dsn

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