Hadamard Factorization Theorem

Theorem
Let $f: \C \to \C$ be an entire function of order $1$.

Let $\rho_1, \rho_2, \ldots$ be an increasing enumeration of the zeros of $f$, counted with multiplicity.

Then there exist constants $a \left({f}\right)$, $b \left({f}\right)$ such that:


 * $\displaystyle f \left({z}\right) = \exp \left({a + b z}\right) \prod_{k \mathop = 1}^\infty \left({1 - \frac z {\rho_k} }\right) \exp \left({\frac z {\rho_k} }\right)$

Also see

 * Weierstrass Factorization Theorem