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 $a(f)$, $b(f)$ such that


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