A function

has a Laplace transform whenever it is of
exponential order. That is, there must be a
real number 
such that

As an example, every
exponential function

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
is given by
Thus, the Laplace transform of an exponential

is

, but this is defined only for
re

.
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