Definition:Order of Entire Function/Definition 3

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

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

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

The order of a constant function is $0$.