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
- Equivalence of Definitions of Order of Entire Function
- Definition:Genus of Entire Function
- Definition:Rank of Entire Function
- Definition:Type of Entire Function
- Definition:Exponent of Convergence
- Hadamard Factorization Theorem
- Results about the order of an entire function can be found here.