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 $\{z_n\}$ denote the set of zeros of $f$ repeated according to multiplicity. Suppose that $f$ has a zero of order $m \geq 0$ (if $m=0$ it means $f(0)\neq 0$). 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

Theorem

The following formula holds: $$\sin(z) = z \displaystyle\prod_{k=1}^{\infty} \left( 1 - \dfrac{z^2}{k^2\pi^2} \right),$$ where $\sin$ denotes the sine function and $\pi$ denotes pi.

Theorem

The Weierstrass factorization of $\cos(x)$ is $$\cos(x) = \displaystyle\prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right).$$