Weierstrass Factorization Theorem

Theorem
Let $f$ be an entire function.

Let the sequence $\left\langle{a_n}\right\rangle$ be the zeros of ƒ where ${a_n}\ne0$ and repeated according to multiplicity.

Suppose also that $f$ has a zero at $z = 0$ of order $m ≥ 0$.

Then there exists an entire function $g$ and a sequence of integers $\left\langle{p_n}\right\rangle$ such that


 * $\displaystyle f(z)=z^me^{g(z)} \prod_{n \mathop = 1}^\infty E_{p_{n}} \left({\frac z a_{n}}\right)$

where $ E_{p_{n}}$ are Weierstrass's elementary factors defined by:


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