Relation Between Rank and Exponent of Convergence

Theorem

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

Let $k$ be its rank and $\tau$ be its exponent of convergence.

Then:

• $k=\tau=0$ if $f$ has finitely many zeroes.
• $k<\tau\leq k+1$ otherwise.

Proof

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