Motivation
The consequences of the fundamental theorem of algebra are twofold. Firstly, any finite sequence in the complex plane has an associated polynomial that has zeroes precisely at the points of that sequence,
Secondly, any polynomial function in the complex plane has a factorization where a is a non-zero constant and cn are the zeroes of p.
The two forms of the Weierstrass factorization theorem can be thought of as extensions of the above to entire functions. The necessity of extra machinery is demonstrated when one considers the product if the sequence is not 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.
A necessary condition for convergence of the infinite product in question is that each factor must approach 1 as . So it stands to reason that one should seek a function 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. Enter the genius of Weierstrass' elementary factors. These factors serve the same purpose as the factors above.
Read more about this topic: Weierstrass Factorization Theorem
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