- Weierstrass factorization theorem
In

mathematics , the**Weierstrass factorization theorem**incomplex analysis , named afterKarl Weierstrass , asserts thatentire function s can be represented by a product involving their 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 givenmeromorphic 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.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.] Firstly, any finite sequence,$\{c\_n\}$, in thecomplex plane has an associatedpolynomial that haszeroes precisely at the points of thatsequence ::$,prod\_n\; (z-c\_n).$

Secondly, any polynomial function in the complex plane, $p(z)$, has a

factorization :$,p(z)=aprod\_n(z-c\_n),$

where "a" is a non-zero constant and "c"

_{"n"}are the zeroes of "p".The two forms of the Weierstrass factorization theorem can be thought of as extensions of the above to

entire function s. The necessity of extra machinery is demonstrated when one considers whether the product:$,prod\_n\; (z-c\_n)$

defines an entire function if the sequence, $\{c\_n\}$, is not finite. The answer is never, because the now-

infinite product will 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: each factor, $(z-c\_n)$, must approach 1 as $n\; oinfty$. 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, $(z-c\_n)$, above.

**The elementary factors**These are also referred to as "primary factors".citation|last=Boas|first=R. P.|title=Entire Functions|publisher=Academic Press Inc.|location=New York|year=1954|isbn=0821845055|oclc=6487790, chapter 2.]

For $n\; in\; mathbb\{N\}$, define the "elementary factors":citation|last=Rudin|first=W.|title=Real and Complex Analysis|edition=3rd|publisher=McGraw Hill|location=Boston|pages=301–304|year=1987|isbn=0070542341|oclc=13093736.]

:$E\_n(z)\; =\; egin\{cases\}\; (1\; -z)\; mbox\{if\; \}n=0,\; \backslash \; (1-z)exp\; left(\; frac\{z^1\}\{1\}+frac\{z^2\}\{2\}+cdots+frac\{z^n\}\{n\}\; ight)\; mbox\{otherwise\}.\; end\{cases\}$

Their utility lies in the following lemma:

**Lemma (15.8, Rudin)**for |"z"| ≤ 1, "n" ∈**N**_{o}:$vert\; 1\; -\; E\_n(z)\; vert\; leq\; vert\; z\; vert^\{n+1\}.$

**The two forms of the theorem****equences define holomorphic functions**Sometimes called the

**Weierstrass theorem**MathWorld | urlname=WeierstrasssTheorem | title=Weierstrass's Theorem]If $lbrace\; z\_i\; brace\_i\; subset\; mathbb\{C\}-\{0\}$ is a sequence such that:

#$vert\; z\_i\; vert\; ightarrow\; infty$ as $i\; ightarrow\; infty$

#there is a sequence, $lbrace\; p\_i\; brace\_i\; subset\; mathbb\{N\}\_o$, such that for all "r" > 0, $sum\_\{i\}\; left(\; frac\{r\}\{vert\; z\_i\; vert\}\; ight)^\{1+p\_i\}\; <\; infty.$Then there exists an entire function that has (only) zeroes at every point of $lbrace\; z\_i\; brace$; in particular, "P" is such a function:: $P(z)=prod\_\{i=1\}^infty\; E\_\{p\_i\}left(frac\{z\}\{z\_i\}\; ight).$

*The theorem generalizes to:

sequences inopen subsets (and henceregions ) of theRiemann sphere have associated functions that areholomorphic in those subsets and have zeroes at the points of the sequence.

*Note also that the case given by the fundamental theorem of algebra is incorporated here. If the sequence, $\{\; z\_i\; \}$ is finite then setting $p\_i\; =\; 0$ suffices for convergence in condition 2, and we obtain: $,\; P(z)\; =\; prod\_n\; (z-z\_n)$.**Holomorphic functions can be factored**Sometimes called the Weierstrass Product/Factor/Factorization theorem.MathWorld | urlname=WeierstrassProductTheorem | title=Weierstrass Product Theorem] Sometimes called the Hadamard Factorization theorem; for example c.f. Boas.

If "f" is a function holomorphic in a region, $Omega$, with zeroes at every point of $lbrace\; z\_i\; brace\_i\; subset\; mathbb\{C\}-\{0\}$then there exists an entire function "g", and a sequence $lbrace\; p\_i\; brace\_i\; subset\; mathbb\{R\}\_o^+$ such that:

$f(z)=e^\{g(z)\}\; prod\_\{i=1\}^infty\; E\_\{p\_i\}left(frac\{z\}\{z\_i\}\; ight)$**References****See also***

Entire function

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