## 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

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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