Editing Weierstrass factorization theorem (section)

==Elementary 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>

[[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=299–304|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):

:<math>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} </math>

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

'''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>

===Existence of entire function with specified zeroes===
Let <math>\{a_n\}</math> be a sequence of non-zero [[complex number]]s such that <math>|a_n|\to\infty</math>.
If <math>\{p_n\}</math> is any sequence of nonnegative integers such that for all <math>r>0</math>,
: <math> \sum_{n=1}^\infty \left( r/|a_n|\right)^{1+p_n} < \infty,</math>
then the function
: <math>E(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|''E''}} 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''′<sub>''n''</sub> ≥ ''p''<sub>''n''</sub>}}, 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"/>