\begin{definition}[Definition:Hadamard's Canonical Factorization]
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.
{{NamedforDef|Jacques Salomon Hadamard|cat = Hadamard}}
\end{definition}