Definition:Order of Entire Function/Definition 1

Definition
Let $f: \C \to \C$ be an entire function.

The order $\alpha \in \left[{0 \,.\,.\, +\infty}\right]$ of $f$ is the infimum of the $\beta \ge 0$ for which:
 * $f \left({z}\right) = \mathcal O \left( \exp \left({\left\lvert{z}\right\rvert^\beta}\right) \right)$

or $\infty$ if no such $\beta$ exists, where $\mathcal O$ denotes big-O notation.

Also see

 * Equivalence of Definitions of Order of Entire Function