Definition of Laplace Transform
Section 7.1
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Transformations in General
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The Laplace Transform Definition
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Examples
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Criteria for Laplace Transform Existence
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Transforming a Piecewise Defined Function
Transformations
A common theme in mathematics is to transform one sort of
object into another sort of object with which one can more easily
work, or from which one can determine specific information. We
have already seen such transformations in this class.
D f   f '
The differential operator transforms a function into a derivative
in a very specific, and useful, way. One might ask if there is an
integration transformation which would also prove useful. The
answer to that is yes, the Laplace Transform is a type of
integration transformation.
Definitions and Examples
Definition 7.1
Let f be a function defined for t greater than or equal to zero.
Then, the integral
∞
L { f t }=∫0 e
−st
f t  dt
is called the Laplace Transform of f, provided that it converges.
Examples
Use the definition above to find the Laplace transform of each
function.
L {4 }
L {cos 3 t }
L {e }
L { f t  gt }
kt
Basic Laplace Transforms
Some basic functions for which we will wish to know the Laplace
transforms are listed below.
Theorem 7.1
a)
1
L {1 }=
s
c)
1
L {e }=
s−a
e)
g)
at
s
L {cos kt }= 2 2
s k
k
L {sinh kt }= 2 2
s −k
b)
d)
f)
n!
L {t }= n1
s
n
k
L {sin kt }= 2 2
s k
s
L {cosh kt }= 2 2
s −k
Existence of Laplace Transforms
Definition 7.2
A function f is said to be of exponential order c if there exists
constants c, M > 0, and T > 0 such that | f(t) | <= Mect for all t > T.
Definition
A function f is piecewise continuous on an interval of for any
subinterval, there are at most finitely man points at which f has
discontinuities, and those discontinuities are finite.
Existence of Laplace Transforms
Continued...
Theorem 7.2
If f is piecewise continuous on [0, ∞ ) and of exponential order c,
then L { f t } exists for s > c.
Example
Find the Laplace Transform of the following function.
{
2 t1 0≤t3
f t =
0
3t