# Weierstrass Factorization Theorem

## Theorem

Let $f$ be an entire function.

Let $0$ be a zero of $f$ of multiplicity $m\geq0$.

Let the sequence $\left\langle{a_n}\right\rangle$ consist of the nonzero zeroes of $f$, repeated according to multiplicity.

### First Form

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 $r \in \R_{> 0}$.

Then there exists an entire function $g$ such that:

$\displaystyle f \left({z}\right) = z^m e^{g \left({z}\right)} \prod_{n \mathop = 1}^\infty E_{p_n} \left({\frac z {a_n}}\right)$

where:

$E_{p_n}$ are Weierstrass's elementary factors

and the product converges locally uniformly absolutely on $\C$.

### Second Form

Then there exists a sequence $\left\langle{p_n}\right\rangle$ of non-negative integers and an entire function $g$ such that:

$\displaystyle f \left({z}\right) = z^m e^{g \left({z}\right)} \prod_{n \mathop = 1}^\infty E_{p_n} \left({\frac z {a_n}}\right)$

where:

$E_{p_n}$ are Weierstrass's elementary factors

and the product converges locally uniformly absolutely on $\C$.

## Proof

From Weierstrass Product Theorem, the function:

$\displaystyle h \left({z}\right) = z^m \prod_{n \mathop = 1}^\infty E_{p_n} \left({\frac z {a_n}}\right)$

defines an entire function that has the same zeros as $f$ counting multiplicity.

Thus $f / h$ is both an entire function and non-vanishing.

As $f / h$ is both holomorphic and nowhere zero there exists a holomorphic function $g$ such that:

$e^g = f / h$

Therefore:

$f = e^g h$

as desired.

$\blacksquare$

## Also known as

Some sources give this as the Weierstrass factor theorem or the Weierstrass infinite product theorem.

## Source of Name

This entry was named for Karl Weierstrass.

## Historical Note

The Weierstrass Factorization Theorem was proved by Karl Weierstrass during his work investigating the properties of entire functions.

This theorem finally provided a satisfactory context for Euler Formula for Sine Function.