Complex Power is of Exponential Order Epsilon

Theorem

Let:

$f: \hointr 0 \to \to \C: t \mapsto t^\phi$

be $t$ to the power of $\phi$, for $\phi \in \C$, defined on its principal branch.

Let $\map \Re \phi > -1$.

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

Proof

\(\ds \size {t^\phi}\)

\(=\)

\(\ds t^{\map \Re \phi}\)

Modulus of Positive Real Number to Complex Power is Positive Real Number to Power of Real Part

The result follows from Real Power is of Exponential Order Epsilon.

$\blacksquare$