Mellin Transform of Exponential

Theorem
Let $a$ be a complex constant and $e^t$ be the complex exponential.

Let $\mathcal M$ be the Mellin transform.

Then:
 * $\mathcal M \left\{ {e^{-at} }\right\} \left({s}\right) = a^{-s} \Gamma \left({s}\right)$

where $\Re \left({a}\right), \Re \left({s}\right) > 0$

Also see

 * Mellin Transform of Higher Order Exponential