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. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Complex analysis is the branch of mathematics investigating functions of complex numbers. ... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ... 2-dimensional renderings (ie. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i represents the imaginary number, i2 = −1. ...

Let be a domain (an open and connected set), or alternatively for a more general definition, let G be an n dimensional complex analytic manifold. Further let stand for the set of holomorphic functions on G. For a compact set, we define the holomorphically convex hull of K as In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ...

(One obtaines the related concept of polynomially convex hull by requiring in the above definition that f be a polynomial, which is more than just asking for a holomorphic function.)

The domain G is called holomorphically convex if for every compact in G, is also compact in G. Sometimes this is just abbreviated as holomorph-convex. Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly...

When n = 1, any domain G is holomorphically convex since then for all compact Also note that for n=1 being holomorphically convex is the same as being a domain of holomorphy. These concepts are more important in the case n > 1 of several complex variables. The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ...

See also

• Stein manifold

In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a closed, complex submanifold of the vector space of n complex dimensions. ...

References

• 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.

This article incorporates material from Holomorphically convex on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...