Definition:Hadamard's Canonical Factorization

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

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

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

The canonical representation of $f$ is:
 * $\ds \map f z = z^m e^{\map g z} \prod_{n \mathop = 1}^\infty E_p \paren {\frac z {a_n} }$

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

If $f$ has finitely many zeroes, the product is understood to be finite.

Also see

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