Definition:Order of Entire Function/Definition 2
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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-$\OO$ notation
The order of $0$ is $0$.