Weierstrass factorization theorem

In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that entire functions 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. The theorem is named after Karl Weierstrass.

A second form of the theorem extends to meromorphic functions and allows one to consider a given meromorphic function as a product of three factors: terms depending on the function's poles and zeroes, and an associated non-zero holomorphic function.

Motivation

The consequences of the fundamental theorem of algebra are twofold.^[1] Firstly, any finite sequence $\{c_{n}\}$ in the complex plane has an associated polynomial $p(z)$ that has zeroes precisely at the points of that sequence, $p(z)=\,\prod _{n}(z-c_{n}).$

Secondly, any polynomial function $p(z)$ in the complex plane has a factorization $\,p(z)=a\prod _{n}(z-c_{n}),$ 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 functions. The necessity of extra machinery is demonstrated when one considers the product $\,\prod _{n}(z-c_{n})$ if the sequence $\{c_{n}\}$ is not 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 for each z, the factors $(z-c_{n})$ must approach 1 as $n\to \infty$ . 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. Weierstrass' elementary factors have these properties and serve the same purpose as the factors $(z-c_{n})$ above.

The elementary factors

First 5 Weierstrass factors on the unit interval.

Plot of

E_{n}(x)

for n=0,...,4 and x in [-1,1].

These are also referred to as primary factors.^[2]

For $n\in \mathbb {N}$ , define the elementary factors:^[3]

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}}

In comparison, note that for $| z | < 1$ , it is true that

(1-z)=\exp(\ln(1-z))=\exp \left(-{\frac {z^{1}}{1}}-{\frac {z^{2}}{2}}-{\frac {z^{3}}{3}}-\cdots \right)

The utility of the elementary factors $E n (z)$ lies in the following lemma:^[3]

Lemma (15.8, Rudin) for $| z | \leq 1$ , $n\in \mathbb {N}$

\vert 1-E_{n}(z)\vert \leq \vert z\vert ^{{n+1}}.

The two forms of the theorem

Existence of entire function with specified zeroes

Sometimes called the Weierstrass theorem.^[4]

Let $\{a_{n}\}$ be a sequence of non-zero complex numbers such that $|a_{n}|\to \infty$ . If $\{p_{n}\}$ is any sequence of integers such that for all $r>0$ ,

\sum _{{n=1}}^{\infty }\left(r/|a_{n}|\right)^{{1+p_{n}}}<\infty ,

then the function

f(z)=\prod _{{n=1}}^{\infty }E_{{p_{n}}}(z/a_{n})

is entire with zeros only at points $a_{n}$ . If a number $z_{0}$ occurs in the sequence $\{a_{n}\}$ exactly $m$ times, then function $f$ has a zero at $z=z_{0}$ of multiplicity $m$ .

Note that the sequence $\{p_{n}\}$ in the statement of the theorem always exists. For example we could always take $p_{n}=n$ and have the convergence. Such a sequence is not unique: changing it at finite number of positions, or taking another sequence $p' n \geq p n$ , 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.^[3]
Note also that the case given by the fundamental theorem of algebra is incorporated here. If the sequence $\{a_{n}\}$ is finite then we can take $p_{n}=0$ and obtain: $\,f(z)=c\,{\displaystyle \prod }_{n}(z-a_{n})$ .

The Weierstrass factorization theorem

Sometimes called the Weierstrass product/factor theorem.^[5]

Let $ƒ$ be an entire function, and let $\{a_{n}\}$ be the non-zero zeros of $ƒ$ repeated according to multiplicity; suppose also that $ƒ$ has a zero at $z = 0$ of order $m \geq 0$ (a zero of order $m = 0$ at $z = 0$ means $ƒ (0) \neq 0$ ). Then there exists an entire function $g$ and a sequence of integers $\{p_{n}\}$ such that

f(z)=z^{m}e^{{g(z)}}\prod _{{n=1}}^{\infty }E_{{p_{n}}}\!\!\left({\frac {z}{a_{n}}}\right).

^[6]

Examples of factorization

$\sin \pi z=\pi z\prod _{n\neq 0}\left(1-{\frac {z}{n}}\right)e^{z/n}=\pi z\prod _{n=1}^{\infty }\left(1-\left({\frac {z}{n}}\right)^{2}\right)$
$\cos \pi z=\prod _{q\in \mathbb {Z} ,\,q\;{\text{odd}}}\left(1-{\frac {2z}{q}}\right)e^{2z/q}=\prod _{n=0}^{\infty }\left(1-\left({\frac {z}{n+{\tfrac {1}{2}}}}\right)^{2}\right)$

The cosine identity can be seen as special case of

${\frac {1}{\Gamma (s-z)\Gamma (s+z)}}={\frac {1}{\Gamma (s)^{2}}}\prod _{n=0}^{\infty }\left(1-\left({\frac {z}{n+s}}\right)^{2}\right)$

for $s={\tfrac {1}{2}}$ .

Hadamard factorization theorem

If $ƒ$ is an entire function of finite order $ρ$ then it admits a factorization

f(z)=z^{m}e^{g(z)}\displaystyle \prod _{n=1}^{\infty }E_{p}\!\!\left({\frac {z}{a_{n}}}\right)

where $g (z)$ is a polynomial of degree $q$ , $q \leq ρ$ and $p = [ρ]$ .^[6]

Notes

↑ Knopp, K. (1996), "Weierstrass's Factor-Theorem", Theory of Functions, Part II, New York: Dover, pp. 1–7 .
↑ Boas, R. P. (1954), Entire Functions, New York: Academic Press Inc., ISBN 0-8218-4505-5, OCLC 6487790 , chapter 2.
1 2 3 Rudin, W. (1987), Real and Complex Analysis (3rd ed.), Boston: McGraw Hill, pp. 301–304, ISBN 0-07-054234-1, OCLC 13093736 .
↑ Weisstein, Eric W. "Weierstrass's Theorem". MathWorld.
↑ Weisstein, Eric W. "Weierstrass Product Theorem". MathWorld.
1 2 Conway, J. B. (1995), Functions of One Complex Variable I, 2nd ed., springer.com: Springer, ISBN 0-387-90328-3

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Weierstrass theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Visualization of the Weierstrass factorization of the sine function due to Euler

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