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.
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 requiring in the above definition that f be a polynomial.)
The domain is called holomorphically convex if for every compact in, 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 note that being holomorphically convex is the same as being a domain of holomorphy (The Cartan–Thullen theorem). These concepts are more important in the case n > 1 of several complex variables.