Weierstrass product set
The Weierstrass product theorem for states that at a given zeros distribution in a holomorphic function exists with exactly those zeros. The function can be explicitly constructed as a so-called Weierstrass product . The sentence was found by Karl Weierstrass in 1876 .
motivation
For a finite number of zeros one can immediately write a polynomial that solves the problem posed, for example . In the case of (countable) infinitely many zeros, the product will generally no longer converge. Based on the identity , Weierstrass therefore introduced "convergence-generating" factors by breaking off the series development and defining factors . has only one zero at , but in contrast to any compact subset of the unit circle, it can be as close as desired , provided that it is chosen large enough. In this way the convergence of an infinite product can also be achieved.
Weierstrass product
Let there be a positive divisor in the area and a sequence chosen so that . This means that the sequence passes through all points of the carrier with the necessary multiplicity with the exception of the zero point . It is called the sequence belonging to the divisor . A product is called a Weierstrass product for the divisor if:
- holomorphic in
- has exactly one zero, in and from the multiplicity
- The product converges normally on any compact subset of .
Product set in
Weierstrass products of the form exist for every positive divisor in . Let the sequence belonging to the divisor be .
Consequences in
- For each divisor there is a meromorphic function with the given zero and pole positions. Each divisor is a main divisor.
- For every meromorphic function there are two holomorphic functions without common zeros such that . In particular, the field of the meromorphic functions is the quotient field of the integrity ring of the holomorphic functions.
- In the ring of holomorphic functions, every non-empty subset has a greatest common factor , although the ring is not factorial.
Generalization for any area
Let there be a range and a positive divisor on with carrier and let it denote the set of all accumulation points of in . Then there are Weierstrass products for the divisor in . So they generally converge on a larger area than .
Generalization for Stein's manifolds
A first generalization of the product theorem for other complex manifolds was made by Pierre Cousin in 1895 , who proved the theorem for cylinder domains in . For this reason, the question of whether a suitable meromorphic function can be constructed for a given divisor is also referred to as the cousin problem .
Jean-Pierre Serre finally solved the cousin problem in 1953 and showed: In a Steinian manifold a divisor is the divisor of a meromorphic function if and only if its Chern cohomology class in vanishes. In particular, in a Steinian manifold, every divisor has a main divisor. This is the direct consequence of the fact that in Stein's manifolds the following sequence is exact , where the sheaf of divisors denotes:
literature
- Reinhold Remmert, Georg Schumacher: Function theory 2. Springer, Berlin 2007, ISBN 978-3-540-57052-3 .
- Hans Grauert, Reinhold Remmert: Theory of Stein Spaces. Springer, Berlin 2004, ISBN 3-540-00373-8 .