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 **C***n* 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 2*n* − 1, and the intersections of *k* faces have codimension at least *k*.