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 Bergmann (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), New Series (in German), 1 (43) (6): 851–862, JFM  62.1220.04, Zbl  0016.17001.
• Chirka, E.M. (2001) [1994], "Bergman–Weil representation", Encyclopedia of Mathematics, EMS Press
• Shirinbekov, M. (2001) [1994], "Weil domain", Encyclopedia of Mathematics, EMS Press
• 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, S2CID  120807854, Zbl  0011.12301.