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DEFINITION AND NOTATION SOLUTION OF
INITIAL VALUE PROBLEMS SHIFTING AND
CHAPTER 3 THE HEAVISIDE FUNCTION CONVOLUTION
IMPULSES AND THE DELTA FUNCTION
The Laplace
Transform
3.1 Definition and Notation
The Laplace transform is an important tool for solving certain kinds of initial value problems,
particularly those involving discontinuous forcing functions, as occur frequently in areas such
as electrical engineering. It is also used to solve boundary value problems involving partial
differential equations to analyze wave and diffusion phenomena.
We will see that the Laplace transform converts some initial value problems to algebra
problems, leading us to attempt the following procedure:
Initial value problem =⇒ algebra problem
=⇒ solution of the algebra problem
=⇒ solution of the initial value problem.
This is often an effective strategy, because some algebra problems are easier to solve than initial
value problems. We begin in this section with the definition and elementary properties of the
transform.
The Laplace transform of a function f is a function L[ f ] defined by
∞
L[ f ](s) = e −st f (t)dt.
0
The integration is with respect to t and defines a function of the new variable s for all s
such that this integral converges.
Because the symbol L[ f ](s) may be awkward to write in computations, we will make the
following convention. We will use lowercase letters for a function we put into the transform and
the corresponding uppercase letters for the transformed function. In this way,
L[ f ]= F, L[g]= G, and L[h]= H
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