Weierstrass Factorization Theorem
 | This page has been identified as a candidate for refactoring of basic complexity. Until this has been finished, please leave {{Refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code. |
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.
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 $5.14$