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Introduction to Digital Filters
Introduction to Laplace Transform Analysis
Existence of the Laplace Transform
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Existence of the Laplace Transform
A function
has a Laplace transform whenever it is of
exponential order. That is, there must be a real number 
such that
has a
Laplace transform for all finite values of 
and 
. Let's
look at this case more closely.
The Laplace transform of a causal, growing exponential function
![$\displaystyle x(t) = \left\{\begin{array}{ll}
A e^{\alpha t}, & t\geq 0 \\ [5pt]
0, & t<0 \\
\end{array}\right.,
$](http://www.dsprelated.com/josimages/filters/img1672.png)
![\begin{eqnarray*}
X(s) &\isdef & \int_0^\infty x(t) e^{-st}dt
= \int_0^\infty A...
...alpha \\ [5pt]
\infty, & \sigma<\alpha \\
\end{array} \right.
\end{eqnarray*}](http://www.dsprelated.com/josimages/filters/img1673.png)
Thus, the Laplace transform of an exponential
is
, but this is defined only for
re
.
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Previous: Introduction to Laplace Transform AnalysisNext: Analytic Continuation
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.
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