Exponential is of Exponential Order Real Part of Index

Theorem
Let $f\left({t}\right) = e^{\psi t}$ be the complex exponential function, where $t \in \R, \psi \in \C$.

Let $a = \operatorname{Re}\left({\psi}\right)$.

Then $e^{\psi t}$ is of exponential order $a$.

Proof
The result follows from the definition of exponential order with $M = 1$, $K = 2$, and $a = a$.