Definition:Order of Entire Function/Definition 2

Definition
Let $f: \C \to \C$ be an entire function. Let $f$ be not identically zero.

The order $\alpha \in \closedint 0 {+\infty}$ of $f$ is the infimum of the $\beta \ge 0$ for which:
 * $\ds \map \ln {\max_{\size z \mathop \le R} \size {\map f z} } = \map \OO {R^\beta}$

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

The order of $0$ is $0$.

Also see

 * Equivalence of Definitions of Order of Entire Function