Editing Weierstrass factorization theorem (section)

==Motivation==

It is clear that any finite set <math>\{c_n\}</math> of points in the [[complex plane]] has an associated [[polynomial]] <math display="inline">p(z) = \prod_n (z-c_n)</math> whose [[zeroes]] are precisely at the points of that set. The converse is a consequence of the [[fundamental theorem of algebra]]: any polynomial function <math>p(z)</math> in the complex plane has a [[factorization]] 
<math display="inline">p(z) = a\prod_n(z-c_n),</math> 
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 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.