# Weierstrass Factorization Theorem

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## 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 continued 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 continued 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**.

## Also see

- Hadamard Factorization Theorem, a refined version of this
- Weierstrass 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.

## Sources

- 2002: Robert E. Greene and Steven G. Krantz:
*Function Theory of One Complex Variable*: $8.2.4$ - 1973: John B. Conway:
*Functions of One Complex Variable*$\text {VII}$: Compact and Convergence in the Space of Analytic Functions: $\S5$: Weierstrass Factorization Theorem: Theorem $5.14$