Weierstrass factorization theorem

Define the notation $$E_n(z)=\left\{ \begin{array}{ll} 1-z &; n=0 \\ (1-z)e^{z+\frac{z^2}{2}+\frac{z^3}{3}+\ldots+\frac{z^n}{n}} &; \mathrm{otherwise} \end{array} \right.$$

Theorem: (Weierstrass factorization theorem) Let $f \colon \mathbb{C} \rightarrow \mathbb{C}$ be an entire function and let $\{a_n\}$ denote the set of zeros of $f$ repeated according to multiplicity. Suppose that $f$ has a zero at $z=0$ of order $m \geq 0$ (if $0$ is a zero of order $0$ means that $f(0) \neq 0$). Then there exists an entire function $g$ and a sequence of integers $\{p_n\}$ such that $$f(z) = z^m e^{g(z)} \displaystyle\prod_{k=1}^{\infty} E_{p_n} \left( \dfrac{z}{a_k} \right).$$

Proof: █

Examples of Weierstrass factorizations

Weierstrass factorization of sine
Weierstrass factorization of cosine
Weierstrass factorization of sinh
Weierstrass factorization of cosh
Gamma function Weierstrass product