Definition:Rank of Entire Function
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Definition
Let $f: \C \to \C$ be an entire function.
Let $\sequence {a_n}$ be the sequence of non-zero zeroes of $f$, repeated according to multiplicity.
The rank of $f$ is:
- the smallest positive integer $p \ge 0$ for which the series $\ds \sum_{n \mathop = 1}^\infty \size {a_n}^{-p - 1}$ converges
or:
- $\infty$ if there is no such integer.
If $f$ has finitely many zeroes, its rank is $0$.
Also see
- Definition:Exponent of Convergence
- Definition:Order of Entire Function
- Definition:Genus of Entire Function
- Hadamard Factorization Theorem
- Relation Between Rank and Exponent of Convergence
Sources
- 1973: John B. Conway: Functions of One Complex Variable: $\text {XI}$: Entire Functions: $\S 2$: The genus and order of an entire function: Definition $2.1$