# Laplace Transform Definition

Cuthbert Nyack

The Laplace Transform F(s) of f(t) is defined as
In this definition f(t) is assumed to be zero
for t < 0. The Laplace variable s (p also used) is a
complex variable which can take on all possible vluues.
The Laplace Transform is well suited for describing systems
with initial values and
transients. F(s) is a complex
function of a complex variable s. In the Fourier Transform,
the transformed variable w has a
familiar meaning as a frequency spectrum. The Laplace variable s
can be viewed as a generalisation of w
with the 2 being identical if the real part of s is restricted to
a value of zero (ie the value of F(s) along the imaginary
s axis is equivalent to the frequency spectrum of f(t).
The fact that the general variable s can take on
any complex value may seem intimidating, however as will be seen
in many cases of interest only the values of s at the poles of
F(s) are important in finding the inverse Laplace Transform.
Not all f(t) have Laplace Transforms. The above integral does
not converge for functions that increase faster
than exponential and these functions do not have a Laplace
Transform. For many functions f(t) the transform only converges
for some range of s eg for f(t) = e^{-at} the integral
converges only for s > - a.

The above definition is sometimes referred to as the one sided
transform. It is also possible to define a 2 sided transform

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Copyright 1996 © Cuthbert A. Nyack.