Definition:Rank of Entire Function

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