Definition:Hadamard's Canonical Factorization

Definition
Let $f: \C \to \C$ be an entire function of finite rank $p\in\N$.

Let $0$ be a zero of $f$ of multiplicity $m\geq0$.

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

Its canonical representation is
 * $\displaystyle f \left({z}\right) = z^m e^{g(z)} \prod_{n \mathop = 1}^\infty E_p\left( \frac z{a_n} \right)$

where $g:\C\to\C$ is an entire function and $E_p$ denotes the $p$th Weierstrass elementary factor.

Also see

 * Weierstrass Factorization Theorem, why such a representation exists
 * Hadamard Factorization Theorem