In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the theory of functions of several complex variables

Several complex variables

The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...

, a domain of holomorphy is a set which is maximal in the sense that there exists a holomorphic function

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

 on this set which cannot be extended

Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...

 to a bigger set.

Formally, an open set

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

  in the n-dimensional complex space is called a domain of holomorphy if there do not exist non-empty open sets and where is connected

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

, and such that for every holomorphic function on there exists a holomorphic function on with on

In the case, every open set is a domain of holomorphy: we can define a holomorphic function with zeros accumulating everywhere on the boundary

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

 of the domain, which must then be a natural boundary for a domain of definition of its inverse. For this is no longer true, as it follows from Hartogs' lemma.

Equivalent conditions

For a domain the following conditions are equivalent:

1. is a domain of holomorphy
2. is holomorphically convex
   Holomorphically convex hull
   In mathematics, more precisely in complex analysis, the holomorphically convex hull of a given compact set in the n-dimensional complex space Cn is defined as follows....
   
   
3. is pseudoconvex
4. is Levi convex - for every sequence of analytic compact surfaces such that for some set we have ( cannot be "touched from inside" by a sequence of analytic surfaces)
5. has local Levi property - for every point there exist a neighbourhood of and holomorphic on such that cannot be extended to any neighbourhood of

Implications are standard results. The main difficulty lies in proving , i.e. constructing a global holomoprhic function which admits no extension from non-extendable functions defined only locally. This is called the Levi problem and was first solved by Kiyoshi Oka

Kiyoshi Oka

was a Japanese mathematician who did fundamental work in the theory of several complex variables. He was born in Osaka. He went to Kyoto Imperial University in 1919, turning to mathematics in 1923 and graduating in 1924....

, and then by Lars Hörmander

Lars Hörmander

Lars Valter Hörmander is a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". He was awarded the Fields Medal in 1962, and the Wolf Prize in 1988...

 using methods from functional analysis and partial differential equations (a consequence of -problem).

Properties

• if are domains of holomorphy, then their intersection is also a domain of holomorphy
• if is an ascending sequence of domains of holomorphy, then their union is also a domain of holomorphy (see Behnke-Stein theorem)
• product of domains of holomorphy is a domain of holomorphy
• the first Cousin problem
  Cousin problems
  In mathematics, the Cousin problems are two questions in several complex variables, concerning the existence of meromorphic functions that are specified in terms of local data. They were introduced in special cases by P. Cousin in 1895...
  
   is always solvable in a domain of holomorphy; this is also true, with additional topological assumptions, for the second Cousin problem
  Cousin problems
  In mathematics, the Cousin problems are two questions in several complex variables, concerning the existence of meromorphic functions that are specified in terms of local data. They were introduced in special cases by P. Cousin in 1895...
  
  

See also

• Behnke–Stein theorem
  Behnke–Stein theorem
  In mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence G_k of domains of holomorphy is again a domain of holomorphy....
  
  
• Levi pseudoconvex
• solution of the Levi problem

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