Definition:Order of Entire Function

Definition
An entire function $f: \C \to \C$ is said to have order at most $\alpha$ if for all $\beta > \alpha$:


 * $f \left({s}\right) = \mathcal O \left({\exp\left({\left|{s}\right|^\beta}\right)}\right)$

where $\mathcal O$ is big-O notation.

$f$ is said to have order equal to $\alpha$ if $f$ has order at most $\alpha$, and $f$ does not have order at most $\gamma$ for any $\gamma < \alpha$.