Wikipedia / Domain of holomorphy
Wikipedia / 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 is called a domain of holomorphy is connected such that for every holomorphic function on on .
Equivalently, for any such and on 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 this is no longer true, as it follows from Hartogs's extension theorem.
Equivalent conditions
the following conditions are equivalent:
is a domain of holomorphy.
that cannot be analytically continued beyond . That is, its domain of existence is .
is pseudoconvex.
is Levi convex ( cannot be "touched from inside" by a sequence of analytic surfaces).
has local Levi property
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 \bar{\partial}-problem).
Properties
is also a domain of holomorphy.
is also a domain of holomorphy (see Behnke–Stein theorem).
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.
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.
is connected, then by the Hartogs's extension theorem can be analytically continued to is an open set that is not a domain of holomorphy. Thus, domain of holomorphy becomes a nontrivial concept in the case .