# Weierstrass Product Theorem

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## 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:

- $\displaystyle \sum_{n \mathop = 1}^\infty \size {\dfrac r {a_n} }^{1 + p_n}$

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

Let:

- $\displaystyle \map f z = \prod_{n \mathop = 1}^\infty \map {E_{p_n} } {\frac z {a_n} }$

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 $\displaystyle \prod_{n \mathop = 1}^\infty \map {E_{p_n} } {\frac z {a_n} }$ converges locally uniformly absolutely.

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

$\blacksquare$

## Also see

## Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.