Weierstrass factorization theorem: Difference between revisions

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where {{math|''a''}} is a non-zero constant and <math>\{c_n\}</math> is the set of zeroes of <math>p(z)</math>.<ref name="knopp">{{citation |last=Knopp |first=K. |title=Theory of Functions, Part II |pages=1–7 |year=1996 |contribution=Weierstrass's Factor-Theorem |___location=New York |publisher=Dover}}.</ref>

The two forms of the Weierstrass factorization theorem can be thought of as extensions of the above to entire functions. The necessity of additional terms in the product is demonstrated when one considers <math display="inline">\prod_n (z-c_n)</math> where 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. Instead, the theorem replaces these with other factors.

A necessary condition for convergence of the infinite product in question is that for each <math>z</math>, the factors replacing <math> (z-c_n) </math> must approach 1 as <math>n\to\infty</math>. So it stands to reason that one should seek a series of factor functions 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 <math> (z-c_n) </math> above.

==The elementaryElementary factors==

Consider the functions of the form <math display="inline">\exp\left(-\tfrac{z^{n+1}}{n+1}\right)</math> for <math>n \in \mathbb{N}</math>. At <math>z=0</math>, they evaluate to <math>1</math> and have a flat slope at order up to <math>n</math>. Right after <math>z=1</math>, they sharply fall to some small positive value. In contrast, consider the function <math>1-z</math> which has no flat slope but, at <math>z=1</math>, evaluates to exactly zero. Also note that for {{math|{{abs|''z''}} < 1}},

:<math>(1-z) = \exp(\ln(1-z)) = \exp \left( -\tfrac{z^1}{1} - \tfrac{z^2}{2} - \tfrac{z^3}{3} + \cdots \right).</math>

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[[File:First_5_Weierstrass_factors_on_the_unit_interval.svg|thumb|right|alt=First 5 Weierstrass factors on the unit interval.|Plot of <math>E_n(x)</math> for n = 0,...,4 and x in the interval [-1,1]''.]]

The ''elementary factors'',<ref name="rudin">{{citation|last=Rudin|first=W.|title=Real and Complex Analysis|edition=3rd|url=https://perso.telecom-paristech.fr/decreuse/_downloads/c22155fef582344beb326c1f44f437d2/rudin.pdf|publisher=McGraw Hill|___location=Boston|pages=301–304299–304|year=1987|isbn=0-07-054234-1|oclc=13093736}}</ref>

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>

are functions that combine the properties of zero slope and zero value (see graphic):

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For {{math|{{abs|''z''}} < 1}} and <math>n>0</math>, one may express it as

<math display="inline">\displaystyle E_n(z)=\exp\left(-\tfrac{z^{n+1}}{n+1}\sum_{k=0}^\infty\tfrac{z^k}{1+k/(n+1)}\right)</math> and one can read off how those properties are enforced.

The utility of the elementary factors <math display="inline">E_n(z)</math> lies in the following lemma:<ref name="rudin"/>

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'''Lemma (15.8, Rudin)''' for {{math|{{abs|''z''}} ≤ 1}}, <math>n \in \mathbb{N}</math>

:<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===

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: <math> \sum_{n=1}^\infty \left( r/|a_n|\right)^{1+p_n} < \infty,</math>

then the function

: <math>fE(z) = \prod_{n=1}^\infty E_{p_n}(z/a_n)</math>

is entire with zeros only at points <math>a_n</math>.<ref name="rudin"/> If a number <math>z_0</math> occurs in the sequence <math>\{a_n\}</math> exactly {{math|''m''}} times, then the function {{math|''fE''}} has a zero at <math>z=z_0</math> of multiplicity {{math|''m''}}.

* 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 {{math|''p''′_''n'' ≥ ''p''_''n''}}, will not break the convergence.

* The theorem generalizes to the following: [[sequences]] in [[open subsets]] (and hence [[Region (mathematics)|regions]]) of the [[Riemann sphere]] have associated functions that are [[Holomorphic function|holomorphic]] in those subsets and have zeroes at the points of the sequence.<ref name="rudin"/>

* Also 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===

Let {{math|''ƒ''}} be an entire function, and let <math>\{a_n\}</math> be the non-zero zeros of {{math|''ƒ''}} repeated according to multiplicity; suppose also that {{math|''ƒ''}} has a zero at {{math|1=''z'' = 0}} of order {{math|''m'' ≥ 0}}.{{efn|A zero of order {{math|1=''m'' = 0}} at {{math|1=''z'' = 0}} is taken to mean {{math|''&fnof;''(0) ≠ 0}} — that is, <math>f</math> does not have a zero at <math>0</math>.}}

Then there exists an entire function {{math|''g''}} and a sequence of integers <math>\{p_n\}</math> such that

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: <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>

* Also theThe 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>, <math>m=0</math> and <math>e^{g(z)}=c</math> to obtain: <math>\, f(z) = c\,{\displaystyle\prod}_n (z-a_n)</math>.

====Examples of factorization====

==== Examples of factorization= ===

The trigonometric functions [[sine]] and [[cosine]] have the factorizations

<math display=block>\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)</math>