Weierstrass factorization theorem: Difference between revisions

Revision as of 16:39, 30 October 2020

Revision as of 20:29, 14 January 2021

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The theorem, which is named for [[Karl Weierstrass]], is closely related to a second result that every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence.

A generalization of the theorem extends it to [[meromorphic function]]s and allows one to consider a given meromorphic function as a product of three factors: terms depending on the function's [[zeros and poles]], and an associated non-zero [[holomorphic function]].{{cncitation needed|date=April 2019}}

==Motivation==

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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>,

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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while the [[gamma function]] <math>\Gamma</math> has factorization

<math display=block>\frac{1}{\Gamma (z)}=e^{\gamma z}z\prod_{n=1}^{\infty }\left ( 1+\frac{z}{n} \right )e^{-z/n},</math>

<math>\gamma</math> is the [[Euler–Mascheroni constant]].{{cncitation needed|date=April 2019}} The cosine identity can be seen as special case of

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

for <math>s=\tfrac{1}{2}</math>.{{cncitation needed|date=April 2019}}

===Hadamard factorization theorem===