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Domain of holomorphy

In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain that is maximal in the sense that there exists a holomorphic function on this domain that cannot be extended to a bigger domain.

Formally, an open set in the -dimensional complex space is called a domain of holomorphy if there do not exist non-empty open sets and where is connected, and such that for every holomorphic function on , there exists a holomorphic function on with on .

Equivalently, for any such and , there exists a holomorphic on , such that cannot be analytically continued to .

In the case , every open set is a domain of holomorphy: we can define a holomorphic function that is not identically zero, but whose zeros accumulate everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its reciprocal. For this is no longer true, as it follows from Hartogs's extension theorem.

Equivalent conditions

For a domain the following conditions are equivalent:

  1. is a domain of holomorphy.
  2. is holomorphically convex.
  3. There exists a function holomorphic on that cannot be analytically continued beyond . That is, its domain of existence is .
  4. is pseudoconvex.
  5. 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).
  6. has local Levi property – for every point there exists a neighbourhood of and holomorphic on such that cannot be extended to any neighbourhood of

Implications are standard results (for , see Oka's lemma). The equivalence of 1, 2, 3 is the Cartan–Thullen theorem. The main difficulty lies in proving , i.e. constructing a global holomorphic function that admits no extension from non-extendable functions defined only locally. This is called the Levi problem (after E. E. Levi) and was first solved by Kiyoshi Oka, and then by Lars Hörmander 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).
  • If and are domains of holomorphy, then is a domain of holomorphy.
  • The first Cousin problem is always solvable in a domain of holomorphy; this is also true, with additional topological assumptions, for the second Cousin problem.

Examples

is trivially a domain of holomorphy.

In the case, every open set is a domain of holomorphy. A particular example is the open unit disk. Define the lacunary function .

it is holomorphic on the open unit disk by the Weierstrass M-test, and singular at all , which is dense on the unit circle, and therefore it cannot be analytically extended beyond the unit disk.

In the case, let where is open and is nonempty and compact. If is connected, then by the Hartogs's extension theorem, any function holomorphic on can be analytically continued to , which means is an open set that is not a domain of holomorphy. Thus, domain of holomorphy becomes a nontrivial concept in the case .

See also

References