Weierstrass Product Theorem

Theorem
Let $\left\langle{a_k}\right\rangle$ be a sequence of non-zero complex numbers such that:
 * $|a_n| \to \infty$ as $n \to \infty$

Let $\left\langle{p_n}\right\rangle$ be a sequence of non-negative integers for which the series:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \left({\dfrac r {|a_n|} }\right)^{1 + p_n}$

converges for every $r \in \R_{> 0}$.

The function:


 * $\displaystyle f(z) = \prod_{n \mathop = 1}^\infty E_{p_n} \left({\frac z {a_n} }\right)$

is entire with zeros only at points $a_n$ where $E_{p_n}$ are Weierstrass elementary factors defined as:


 * $E_n (z) = \begin{cases} (1 - z) & : n = 0 \\ (1 - z) \exp \left({\dfrac {z^1} 1 + \dfrac {z^2} 2 + \cdots + \dfrac {z^n} n}\right) & : n \ne 0 \end{cases}$