Weierstrass factorization theorem


In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a product involving its zeroes. The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root.
The theorem, which is named for Karl Weierstrass, is closely related to a second result that every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence.
A generalization of the theorem extends it to meromorphic functions and allows one to consider a given meromorphic function as a product of three factors: terms depending on the function's zeros and poles, and an associated non-zero holomorphic function.

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 is a non-zero constant and are the zeroes of.
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 for each z, the factors 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.
Weierstrass' elementary factors have these properties and serve the same purpose as the factors above.

The elementary factors

Consider the functions of the form for. At, they evaluate to and have a flat slope at order up to. Right after, they sharply fall to some small positive value. In contrast, consider the function which has no flat slope but, at, evaluates to exactly zero. Also note that for,
File:First_5_Weierstrass_factors_on_the_unit_interval.svg|thumb|right|alt=First 5 Weierstrass factors on the unit interval.|Plot of for n = 0,...,4 and x in the interval .
The elementary factors
,
also referred to as primary factors''
,
are functions that combine the properties of zero slope and zero value :
For and, one may express it as
and one can read off how those properties are enforced.
The utility of the elementary factors lies in the following lemma:
Lemma for,

The two forms of the theorem

Existence of entire function with specified zeroes

Let be a sequence of non-zero complex numbers such that.
If is any sequence of integers such that for all,
then the function
is entire with zeros only at points. If a number occurs in the sequence exactly times, then function has a zero at of multiplicity.
Let be an entire function, and let be the non-zero zeros of repeated according to multiplicity; suppose also that has a zero at of order .
Then there exists an entire function and a sequence of integers such that

Examples of factorization

The trigonometric functions sine and cosine have the factorizations
while the gamma function has factorization
is the Euler–Mascheroni constant. The cosine identity can be seen as special case of
for.

Hadamard factorization theorem

If is an entire function of finite order and is the order of the zero of at =, then it admits a factorization
where is a polynomial of degree, and .