Weil domain

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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.


[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)



[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098