Weierstrass factorization theorem: Difference between revisions

Revision as of 14:28, 14 March 2025

Revision as of 14:33, 14 March 2025

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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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<math display="inline">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>