Hadamard Factorization Theorem

Theorem
Let $f: \C \to \C$ be an entire function of finite order $\omega$.

Let $0$ be a zero of $f$ of multiplicity $m \ge 0$.

Let $\sequence {a_n}$ be the sequence of non-zero zeroes of $f$, repeated according to multiplicity.

Then:
 * $f$ has finite rank $p \le \omega$

and:
 * there exists a polynomial $g$ of degree at most $\omega$ such that:
 * $\ds \map f z = z^m e^{\map g z} \prod_{n \mathop = 1}^\infty E_p \paren {\frac z {a_n} }$

where $E_p$ denotes the $p$th Weierstrass elementary factor.

Proof
By Convergence Exponent is Less Than Order, $f$ has finite exponent of convergence $\tau \le \omega$.

By Relation Between Rank and Exponent of Convergence, $f$ has finite rank $p \leq \omega$.

Also see

 * Weierstrass Factorization Theorem
 * Order is Maximum of Exponent of Convergence and Degree
 * Definition:Hadamard's Canonical Factorization