View source for Minimum-variance unbiased estimator

In [[mathematics]], the '''Weierstrass factorization theorem''' in [[complex analysis]], named after [[Karl Weierstrass]], asserts that [[entire function]]s can be represented by a product involving their [[zero (complex analysis)|zeroes]]. In addition, every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence. 

A second form extended to [[meromorphic function]]s allows one to consider a given meromorphic function as a product of three factors: the function's poles, zeroes, and an associated non-zero holomorphic function.

==Motivation==

The consequences of the [[fundamental theorem of algebra]] are twofold.<ref name="knopp">{{citation|last=Knopp|first=K.|contribution=Weierstrass's Factor-Theorem|title=Theory of Functions, Part II|location=New York|publisher=Dover|pages=1–7|year=1996}}.</ref>
Firstly, any finite sequence <math>\{c_n\}</math> in the [[complex plane]] has an associated [[polynomial]] <math>p(z)</math> that has [[zeroes]] precisely at the points of that [[sequence]], <math>p(z) = \,\prod_n (z-c_n).</math>

Secondly, any polynomial function <math>p(z)</math> in the complex plane has a [[factorization]] 
<math>\,p(z)=a\prod_n(z-c_n),</math> 
where ''a'' is a non-zero constant and ''c''<sub>''n''</sub> are the zeroes of ''p''.

The two forms of the Weierstrass factorization theorem can be thought of as extensions of the above to entire functions. The necessity of extra machinery is demonstrated when one considers the product <math>\,\prod_n (z-c_n)</math> if the sequence <math>\{c_n\}</math> is not [[finite set|finite]]. It can never define an entire function, because the [[infinite product]] does not converge. Thus one cannot, in general, define an entire function from a sequence of prescribed zeroes or represent an entire function by its zeroes using the expressions yielded by the fundamental theorem of algebra.

A necessary condition for convergence of the infinite product in question is that each factor <math> (z-c_n) </math> must approach 1 as <math>n\to\infty</math>. So it stands to reason that one should seek a function that could be 0 at a prescribed point, yet remain near 1 when not at that point and furthermore introduce no more zeroes than those prescribed. Enter the genius of Weierstrass' ''elementary factors''. These factors serve the same purpose as the factors <math> (z-c_n) </math> above.

==The elementary factors==

These are also referred to as ''primary factors''.<ref name="boas">{{citation|last=Boas|first=R. P.|title=Entire Functions|publisher=Academic Press Inc.|location=New York|year=1954|isbn=0-8218-4505-5|oclc=6487790}}, chapter 2.</ref>

For <math>n \in \mathbb{N}</math>, define the ''elementary factors'':<ref name="rudin">{{citation|last=Rudin|first=W.|title=Real and Complex Analysis|edition=3rd|publisher=McGraw Hill|location=Boston|pages=301–304|year=1987|isbn=0-07-054234-1|oclc=13093736}}.</ref>

:<math>E_n(z) = \begin{cases} (1 -z) & \text{if }n=0, \\ (1-z)\exp \left( \frac{z^1}{1}+\frac{z^2}{2}+\cdots+\frac{z^n}{n} \right) & \text{otherwise}. \end{cases} </math>

Their utility lies in the following lemma:<ref name="rudin"/>

'''Lemma (15.8, Rudin)''' for |''z''|&nbsp;≤&nbsp;1, ''n''&nbsp;∈&nbsp;'''N'''<sub>o</sub>

:<math>\vert 1 - E_n(z) \vert \leq \vert z \vert^{n+1}.</math>

==The two forms of the theorem==
===Existence of entire function with specified zeroes===
Sometimes called the '''Weierstrass theorem'''.<ref name="mw-wst">{{MathWorld | urlname=WeierstrasssTheorem | title=Weierstrass's Theorem}}</ref>

Let <math>\{a_n\}</math> be a sequence of non-zero complex numbers such that <math>|a_n|\to\infty</math>.
If <math>\{p_n\}</math> is any sequence of integers such that for all <math>r>0</math>,
: <math> \sum_{n=1}^\infty \left( r/|a_n|\right)^{1+p_n} < \infty,</math>
then the function
: <math>f(z) = \prod_{n=1}^\infty E_{p_n}(z/a_n)</math>
is entire with zeros only at points <math>a_n</math>. If number <math>z_0</math> occurs in sequence <math>\{a_n\}</math> exactly ''m'' times, then function ''f'' has a zero at <math>z=z_0</math> of multiplicity ''m''.

* Note that the sequence <math>\{p_n\}</math> in the statement of the theorem always exists. For example we could always take <math>p_n=n</math> and have the convergence. Such a sequence is not unique: changing it at finite number of positions, or taking another sequence ''p'<sub>n</sub> ≥ p<sub>n</sub>,'' will not break the convergence.
* The theorem generalizes to the following: [[sequences]] in [[open subsets]] (and hence [[regions]]) of the [[Riemann sphere]] have associated functions that are [[holomorphic]] in those subsets and have zeroes at the points of the sequence.<ref name="rudin"/>
* Note also that the case given by the fundamental theorem of algebra is incorporated here. If the sequence <math>\{a_n\}</math> is finite then we can take <math>p_n = 0</math> and obtain: <math>\, f(z) = c\,{\displaystyle\prod}_n (z-a_n)</math>.

===The Weierstrass factorization theorem===
Sometimes called the Weierstrass product/factor theorem.<ref name="mw-wpt">{{MathWorld | urlname=WeierstrassProductTheorem | title=Weierstrass Product Theorem}}</ref> 

Let ''ƒ'' be an entire function, and let <math>\{a_n\}</math> be the non-zero zeros of ''ƒ'' repeated according to multiplicity; suppose also that ''ƒ'' has a zero at ''z''&nbsp;=&nbsp;0 of order ''m''&nbsp;≥&nbsp;0 (a zero of order ''m''&nbsp;=&nbsp;0 at ''z''&nbsp;=&nbsp;0 means ''&fnof;''(0)&nbsp;≠&nbsp;0).
Then there exists an entire function ''g'' and a sequence of integers <math>\{p_n\}</math> such that 

: <math>f(z)=z^m e^{g(z)} \prod_{n=1}^\infty E_{p_n}\!\!\left(\frac{z}{a_n}\right).</math><ref name="conway">{{citation|last=Conway|first=J. B.|title=Functions of One Complex Variable I, 2nd ed.|publisher=Springer|location=springer.com|year=1995|isbn=0-387-90328-3}}</ref>

====Examples of factorization====

*  <math>\sin \pi z = \pi z \prod_{n\neq 0} \left(1-\frac{z}{n}\right)e^{z/n} = \pi z\prod_{n=1}^\infty \left(1-\frac{z^2}{n^2}\right)</math>
*  <math>\cos \pi z = \prod_{q \in \mathbb{Z}, \, q \; \text{odd} } \left(1-\frac{2z}{q}\right)e^{2z/q} = \prod_{n=0}^\infty \left( 1 - \frac{4z^2}{(2n+1)^2} \right) </math>

===Hadamard factorization theorem===
If ''ƒ'' is an entire function of finite [[Entire function|order]] ''ρ'' then it admits a factorization
: <math>f(z) = z^m e^{g(z)} \displaystyle\prod_{n=1}^\infty E_p(z/a_n)</math>
where ''g(z)'' is a polynomial of degree ''q'', ''q&nbsp;≤&nbsp;ρ'' and ''p=[ρ]'' .<ref name="conway" />

==See also==
*[[Mittag-Leffler's theorem]]

==Notes==
{{reflist}}

==External links==
* {{springer|title=Weierstrass theorem|id=p/w097510}}

[[Category:Theorems in complex analysis]]