Weierstrass Product Theorem

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

Let $\sequence {p_n}$ be a sequence of non-negative integers for which the series:
 * $\ds \sum_{n \mathop = 1}^\infty \size {\dfrac r {a_n} }^{1 + p_n}$

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

Let:
 * $\ds \map f z = \prod_{n \mathop = 1}^\infty \map {E_{p_n} } {\frac z {a_n} }$

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

Then $f$ is entire and its zeroes 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 $\ds \prod_{n \mathop = 1}^\infty \map {E_{p_n} } {\frac z {a_n} }$ converges locally uniformly absolutely.

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

Also see

 * Weierstrass Factorization Theorem