Complex Power is of Exponential Order Epsilon

Theorem
Let:


 * $f: \left[{0\,.\,.\,\to}\right) \to \C: t \mapsto t^\phi$

be $t$ to the power of $\phi$, for $\phi \in \C$

Let the choice of branch of $f$ be arbitrary, such that $\arg \phi$ is unambiguous.

Then $f$ is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude.