Bergman–Weil formula

In mathematics, the Bergman–Weil formula is an integral representation for holomorphic functions of several variables generalizing the Cauchy integral formula. It was introduced by Bergman (1936) and Weil (1935).

Weil domains

A Weil domain (Weil 1935) is an analytic polyhedron with a domain U in Cⁿ defined by inequalities f_j(z) < 1 for functions f_j that are holomorphic on some neighborhood of the closure of U, such that the faces of the Weil domain (where one of the functions is 1 and the others are less than 1) all have dimension 2n − 1, and the intersections of k faces have codimension at least k.

See also

• Andreotti–Norguet formula
• Bochner–Martinelli formula

References

• Bergmann, S. (1936), "Über eine Integraldarstellung von Funktionen zweier komplexer Veränderlichen", Recueil Mathématique (Matematicheskii Sbornik) N.S. (in German), 1 (43) (6): 851–862, JFM 62.1220.04, Zbl 0016.17001 .
• Chirka, E.M. (2001), "Bergman–Weil representation", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
• Shirinbekov, M. (2001), "Weil domain", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
• Weil, André (1935), "L'intégrale de Cauchy et les fonctions de plusieurs variables", Mathematische Annalen 111 (1): 178–182, doi:10.1007/BF01472212, ISSN 0025-5831, JFM 61.0371.03, MR 1512987, Zbl 0011.12301 .