Weierstrass Product Theorem

Theorem
Suppose $\left\langle{a_k}\right\rangle$ is 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:


 * $\displaystyle \sum_{n = 1}^\infty \left({\frac r {|a_{n}|}}\right)^{1+p_{n}}$

converges for every real number, 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 Weierstrauss elementary factors defined by:


 * $E_{n}(z)=\begin{cases}(1-z)&{\text{if }}n=0,\\(1-z)\exp \left({\frac {z^{1}}{1}}+{\frac  {z^{2}}{2}}+\cdots +{\frac  {z^{n}}{n}}\right)&{\text{otherwise}}.\end{cases}$