Weierstrass Product Theorem

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

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

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

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

where $E_{p_n}$ are Weierstrass elementary factors.

Then $f$ is entire and its zeros are the points $a_n$, counted with multiplicity.

Proof
By:
 * Locally Uniformly Absolutely Convergent Product is Locally Uniformly Convergent
 * Infinite Product of Analytic Functions is Analytic
 * Zeroes of Infinite Product of Analytic Functions

it suffices to show that the product $\displaystyle \prod_{n \mathop = 1}^\infty E_{p_n} \left({\frac z {a_n} }\right)$ converges locally uniformly absolutely.

By Bounds for Weierstrass Elementary Factors and Weierstrass M-Test, this is the case.

Also see

 * Weierstrass Factorization Theorem