In mathematics, the Bergman–Weil formula is an integral representation for
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s of several variables generalizing the
Cauchy integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
. It was introduced by and .
Weil domains
A Weil domain is an
analytic polyhedron In mathematics, especially several complex variables, an analytic polyhedron is a subset of the complex space of the form
:P = \
where is a bounded connected open subset of , f_j are holomorphic on and is assumed to be relatively compact in ...
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
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals the ...
at least ''k''.
See also
*
Andreotti–Norguet formula The Andreotti–Norguet formula, first introduced by , is a higher–dimensional analogue of Cauchy integral formula for expressing the derivatives of a holomorphic function. Precisely, this formula express the value of the partial derivative of any ...
*
Bochner–Martinelli formula
References
*.
*
*
*.
{{DEFAULTSORT:Bergman-Weil formula
Theorems in complex analysis
Several complex variables