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 . ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$

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. ${\displaystyle a_{1},\dots a_{n}\;\in \mathbb {C} }$ ${\displaystyle \left(1-{\frac {z}{a_{1}}}\right)\cdots \left(1-{\frac {z}{a_{n}}}\right)}$ ${\displaystyle 1-z=\exp(\log(1-z))=\exp \left(-\sum _{k=1}^{\infty }{\frac {z^{k}}{k}}\right),\quad z\in \mathbb {C} ,|z|<1,}$ ${\displaystyle E_{n}(z):=(1-z)\exp \left(\sum _{k=1}^{n}{\frac {z^{k}}{k}}\right)}$ ${\displaystyle E_{n}}$ ${\displaystyle 1}$ ${\displaystyle 1-z}$ ${\displaystyle 1}$ ${\displaystyle n}$

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: ${\displaystyle D}$ ${\displaystyle \Omega \subseteq \mathbb {C} }$ ${\displaystyle a_{k}}$ ${\displaystyle D=D(0)\cdot 0+\sum _{k}a_{k}}$ ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle z^{D(0)}\prod _{k\geq 1}f_{k}(z)}$ ${\displaystyle D}$

• ${\displaystyle f_{k}}$ holomorphic in ${\displaystyle \Omega }$
• ${\displaystyle f_{k}}$has exactly one zero, in and from the multiplicity ${\displaystyle a_{k}}$ ${\displaystyle 1}$
• The product converges normally on any compact subset of . ${\displaystyle \textstyle \prod _{k}f_{k}}$ ${\displaystyle \Omega }$

Product set in ${\displaystyle \mathbb {C} }$

Weierstrass products of the form exist for every positive divisor in . Let the sequence belonging to the divisor be . ${\displaystyle D}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle z^{D(0)}\prod _{k\geq 1}E_{k-1}\left({\frac {z}{a_{k}}}\right)}$ ${\displaystyle a_{k}}$ ${\displaystyle D}$

Consequences in ${\displaystyle \mathbb {C} }$

• 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. ${\displaystyle h}$ ${\displaystyle f,g}$ ${\displaystyle h=f/g}$
• 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 . ${\displaystyle \Omega \subseteq \mathbb {C} }$ ${\displaystyle D}$ ${\displaystyle \Omega }$ ${\displaystyle T}$ ${\displaystyle T':={\overline {T}}\!\setminus \!T}$ ${\displaystyle T}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle D}$ ${\displaystyle \mathbb {C} \!\setminus \!T'}$ ${\displaystyle \Omega }$

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 . ${\displaystyle \mathbb {C} ^{n}}$

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: ${\displaystyle X}$ ${\displaystyle H^{2}(X,\mathbb {Z} )}$ ${\displaystyle H^{2}(X,\mathbb {Z} )=0}$ ${\displaystyle {\mathcal {D}}}$

${\displaystyle 0\to {\mathcal {O}}^{*}(X)\to {\mathcal {M}}^{*}(X)\to {\mathcal {D}}(X)\rightarrow H^{2}(X,\mathbb {Z} )\to 0}$

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 .