# Weil domain

From Encyclopedia of Mathematics

A special case of an analytic polyhedron. A bounded domain in -dimensional space is said to be a Weil domain if there exist functions , , holomorphic in a fixed neighbourhood of the closure , such that

1) ;

2) the faces of the Weil domain , i.e. the sets

have dimension ;

3) the edges of the Weil domain , i.e. the intersections of any () different faces, have dimension .

The totality of all -dimensional edges of a Weil domain is called the skeleton of the domain. The Bergman–Weil representation applies to Weil domains. These domains are named for A. Weil [1], who obtained the first important results for these domains.

#### References

[1] | A. Weil, "L'intégrale de Cauchy et les fonctions de plusieurs variables" Math. Ann. , 111 (1935) pp. 178–182 |

[2] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) |

[3] | V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian) |

#### Comments

#### References

[a1] | B.A. Fuks, "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1963) (Translated from Russian) |

[a2] | G.M. [G.M. Khenkin] Henkin, J. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1984) |

**How to Cite This Entry:**

Weil domain. M. Shirinbekov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Weil_domain&oldid=15119

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098