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_new/filters/img1674.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_new/filters/img1675.png)
Thus, the Laplace transform of an exponential
is
, but this is defined only for
re
.
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