Weierstrass factorization theorem: Difference between revisions

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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" |pages=299/>

'''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>