# Weierstrass factorization theorem

There are several different statements of this theorem, but in essence this theorem will allow us to prescribe zeros and their orders of a holomorphic function^{}. It also allows us to factor any holomorphic function into
a product of zeros and a non-zero holomorphic function. We will need to know
here how an infinite product converges. It can then be shown
that if ${\prod}_{k=1}^{\mathrm{\infty}}{f}_{k}(z)$ converges uniformly and absolutely (http://planetmath.org/AbsoluteConvergenceOfInfiniteProduct)
on compact subsets, then it converges to a holomorphic function given that all the ${f}_{k}(z)$ are holomorphic. This is what we will by the infinite product in what follows.

Note that once we can prescribe zeros of a function then we can also prescribe the poles as well and get a meromorphic function just by dividing two holomorphic functions $f/h$ where $f$ will contribute zeros, and $h$ will make poles at the points where $h(z)=0$. So let’s start with the existence statement.

###### Theorem (Weierstrass Product).

Let $G\mathrm{\subset}\mathrm{C}$ be a domain, let $\mathrm{\{}{a}_{k}\mathrm{\}}$ be a sequence of
points in $G$ with no accumulation points^{} in $G$, and let $\mathrm{\{}{n}_{k}\mathrm{\}}$ be any
sequence of non-zero integers (positive or negative).
Then there exists a function
$f$ meromorphic in $G$ whose poles and zeros are exactly at the points ${a}_{k}$
and the order of the pole or zero at ${a}_{k}$ is ${n}_{k}$
(a positive order stands for zero, negative stands for pole).

Next let’s look at a more specific statement with more . For one let’s start looking at the whole complex plane and further let’s forget about poles for now to make the following formulas simpler.

###### Definition.

We call

${E}_{0}(z)$ | $:=1-z,$ | ||

${E}_{p}(z)$ | $:=(1-z){e}^{z+\frac{1}{2}{z}^{2}+\mathrm{\cdots}+\frac{1}{p}{z}^{p}}\mathit{\hspace{1em}\hspace{1em}}\text{for}p\ge 1,$ |

Now note that for some $a\in \u2102\backslash \{0\}$, ${E}_{p}(z/a)$ has a zero (zero of order 1) at $a$.

###### Theorem (Weierstrass Factorization).

Suppose $f$ be an entire function^{} and let $\mathrm{\{}{a}_{k}\mathrm{\}}$ be the zeros of $f$
such that ${a}_{k}\mathrm{\ne}\mathrm{0}$ (the non-zero zeros of $f$). Let $m$ be the order of
the zero of $f$ at $z\mathrm{=}\mathrm{0}$ ($m\mathrm{=}\mathrm{0}$ if $f$ does not have a zero at $z\mathrm{=}\mathrm{0}$). Then
there exists an entire function $g$ and a sequence of non-negative
integers $\mathrm{\{}{p}_{k}\mathrm{\}}$ such that

$$f(z)={z}^{m}{e}^{g(z)}\prod _{k=1}^{\mathrm{\infty}}{E}_{{p}_{k}}\left(\frac{z}{{a}_{k}}\right).$$ |

Note that we can always choose ${p}_{k}=k-1$ and the product above will converge as needed, but we may be able to choose better ${p}_{k}$ for specific functions.

###### Example.

As an example we can try to factorize the function $\mathrm{sin}\mathit{}\mathrm{(}\pi \mathit{}z\mathrm{)}$, which has zeros at all the integers. Applying the Weierstrass factorization theorem directly we get that

$$\mathrm{sin}(\pi z)=z{e}^{g(z)}\prod _{k=-\mathrm{\infty},k\ne 0}^{\mathrm{\infty}}\left(1-\frac{z}{k}\right){e}^{z/k},$$ |

where $g\mathit{}\mathrm{(}z\mathrm{)}$ is some holomorphic function. It turns out that ${e}^{g\mathit{}\mathrm{(}z\mathrm{)}}\mathrm{=}\pi $, and rearranging the product we get

$$\mathrm{sin}(\pi z)=z\pi \prod _{k=1}^{\mathrm{\infty}}\left(1-\frac{{z}^{2}}{{k}^{2}}\right).$$ |

This is an example where we could choose the ${p}_{k}\mathrm{=}\mathrm{1}$ for all $k$ and thus we could then get rid of the ugly parts of the infinite product. For calculations in this example see Conway [1].

## References

- 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
- 2 Theodore B. Gamelin. . Springer-Verlag, New York, New York, 2001.

Title | Weierstrass factorization theorem |
---|---|

Canonical name | WeierstrassFactorizationTheorem |

Date of creation | 2013-03-22 14:19:31 |

Last modified on | 2013-03-22 14:19:31 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 8 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 30C15 |

Synonym | Weierstrass product theorem |

Related topic | MittagLefflersTheorem |

Defines | elementary factor |