Definition:Rank of Entire Function

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

Let $\left\langle{a_n}\right\rangle$ be the sequence of nonzero zeroes of $f$, repeated according to multiplicity.

The rank of $f$ is the smallest integer $p\geq0$ for which the series $\displaystyle\sum_{n\mathop=1}^\infty |a_n|^{-p-1}$ 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