# Weierstrass Factorization Theorem

## Theorem

Let $f$ be an entire function.

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

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

### First Form

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

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

$\ds \map f z = z^m e^{\map g z} \prod_{n \mathop = 1}^\infty \map {E_{p_n} } {\frac z {a_n} }$

where:

$E_{p_n}$ are Weierstrass's elementary factors
the product converges locally uniformly absolutely on $\C$.

### Second Form

There exists a sequence $\sequence {p_n}$ of non-negative integers and an entire function $g$ such that:

$\ds \map f z = z^m e^{\map g z} \prod_{n \mathop = 1}^\infty \map {E_{p_n} } {\frac z {a_n} }$

where:

$E_{p_n}$ are Weierstrass's elementary factors
the product converges locally uniformly absolutely on $\C$.

## Proof

From Weierstrass Product Theorem, the function:

$\ds \map h z = z^m \prod_{n \mathop = 1}^\infty \map {E_{p_n} } {\frac z {a_n} }$

defines an entire function that has the same zeroes 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.