Definition:Order of Entire Function/Definition 1

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

The order $\alpha \in \closedint 0 {+\infty}$ of $f$ is the infimum of the $\beta \ge 0$ for which:
 * $\map f z = \map {\mathcal O} {\map \exp {\size z^\beta} }$

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

Also see

 * Equivalence of Definitions of Order of Entire Function