A second form extended to meromorphic functions allows one to consider a given meromorphic function as a product of three factors: the function's poles, zeroes, and an associated non-zero holomorphic function.
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 whether the product
defines an entire function if the sequence, , is not finite. The answer is never, because the now-infinite product will 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: 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.
These are also referred to as primary factors.
For , define the elementary factors:
Their utility lies in the following lemma:
Lemma (15.8, Rudin) for |z| ≤ 1, n ∈ No
If is a sequence such that:
Then there exists an entire function that has (only) zeroes at every point of ; in particular, P is such a function:
If f is a function holomorphic in a region, , with zeroes at every point of then there exists an entire function g, and a sequence such that: