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 is defined as follows.

Let be a domain (an open and connected set), or alternatively for a more general definition, let be an dimensional complex analytic manifold. Further let stand for the set of holomorphic functions on For a compact set , the holomorphically convex hull of is

One obtains a narrower concept of polynomially convex hull by taking instead to be the set of complex-valued polynomial functions on G. The polynomially convex hull contains the holomorphically convex hull.

The domain is called holomorphically convex if for every compact subset is also compact in . Sometimes this is just abbreviated as holomorph-convex.

When , any domain is holomorphically convex since then is the union of with the relatively compact components of . Also, being holomorphically convex is the same as being a domain of holomorphy (The Cartan–Thullen theorem). These concepts are more important in the case of several complex variables (n > 1).

See also


  • Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.
  • Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

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