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 (possibly infinite) 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.
[citation needed] It is clear that any finite set
of points in the complex plane has an associated polynomial
whose zeroes are precisely at the points of that set.
The converse is a consequence of the fundamental theorem of algebra: any polynomial function
[1] 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
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
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.
, they sharply fall to some small positive value.
Also note that for |z| < 1, The elementary factors,[2] also referred to as primary factors,[3] are functions that combine the properties of zero slope and zero value (see graphic): For |z| < 1 and
The utility of the elementary factors
be a sequence of non-zero complex numbers such that
, then the function is entire with zeros only at points
of multiplicity m. Let ƒ be an entire function, and let
be the non-zero zeros of ƒ repeated according to multiplicity; suppose also that ƒ has a zero at z = 0 of order m ≥ 0.
[a] Then there exists an entire function g and a sequence of integers
such that The trigonometric functions sine and cosine have the factorizations
[citation needed] The cosine identity can be seen as special case of
[citation needed] A special case of the Weierstraß factorization theorem occurs for entire functions of finite order.
a polynomial (whose degree we shall call
is the smallest non-negative integer such that the series
This is called Hadamard's canonical representation.
is called the genus of the entire function
If the order is a positive integer, then there are two possibilities:
