Weierstrass factorization theorem

The Weierstrass product theorem for states that exist at a given zeros distribution in a holomorphic function with exactly those zeros. The function can be explicitly constructed as a so-called Weierstrass product. The rate was found in 1876 by ​​Karl Weierstrass.

Motivation

For a finite number of zeros can immediately write down a polynomial that solves the problem, for example. In the case of ( countably ) infinitely many zeros, the product generally will not converge. Starting from the identity led Weierstrass therefore " convergence- generating " factors, by breaking off the series expansion and factors defined. has only one zero at, but, in contrast to lie on every compact subset of the unit circle as close to, if is chosen large enough. This also allows the convergence of an infinite product can be achieved.

Weierstrass product

There was a positive divisor in the field and a so- chosen consequence that; That is, the result passes through the zero point, with the exception of the support points of all the necessary multiplicity. It is called the divisor associated to the sequence. A product called Weierstrass product for the divisor, if the following holds:

• Holomorphic in
• Has exactly one zero, in and by the multiplicity
• The product converges normally on each compact subset of.

Product set in

For each positive divisor in exist Weierstrass products of the form. It should be the divisor associated to the sequence.

Inferences in

• There is a meromorphic function with the predetermined thereby zeros and poles for each divisor. Each divisor is a divisor.
• For every meromorphic function of two holomorphic functions without common zeros such that. In particular, the body of meromorphic functions is the quotient field of the ring of holomorphic functions is integrity.
• In the ring of holomorphic functions every non- empty subset has a greatest common divisor, although the ring is not factorial.

Generalization for arbitrary areas

There is a range and a positive divisor on with carrier and it denotes the set of all accumulation points of in. Then there exist the divisor Weierstrass products. They converge in general, so on a larger area than.

Generalization of Stein manifolds

A first generalization of the product applicable to other complex manifolds 1895 succeeded Pierre Cousin, who proved the theorem for cylinders in areas. For this reason the question of whether an appropriate meromorphic function can be constructed at a predetermined divisor, also called cousin problem.

Jean- Pierre Serre 1953 sparked the Cousin problem once and showed: In a Stein manifold, a divisor is exactly the divisor of a meromorphic function if its cohomology vanishes in Chernsche. In particular, in a Stein manifold with each divisor a principal divisor. This is the immediate consequence from the fact that in Stein manifolds following sequence is exact, where the sheaf of divisors may refer to: