Definition:Order of Entire Function

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Definition

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

Let $\ln$ denote the natural logarithm.


Definition 1

The order $\alpha \in \closedint 0 {+\infty}$ of $f$ is the infimum of the $\beta \ge 0$ for which:

$\map f z = \map \OO {\map \exp {\size z^\beta} }$

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


Definition 2

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 $0$.


Definition 3

Let $f$ be non-constant.


The order $\alpha \in \closedint 0 {+\infty}$ of $f$ is the limit superior:

$\ds \limsup_{R \mathop \to \infty} \frac {\ds \ln \ln \max_{\cmod z \mathop \le R} \cmod f} {\ln R}$

The order of a constant function is $0$.


Finite Order

$f$ has finite order if and only if its order is a real number.


Also see

  • Results about the order of an entire function can be found here.