# Leray formula

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Cauchy–Fantappié formula

A formula for the integral representation of holomorphic functions of several complex variables , , which generalizes the Cauchy integral formula (see Cauchy integral).

Let be a finite domain in the complex space with piecewise-smooth boundary and let be a smooth vector-valued function of with values in such that the scalar product everywhere on for all . Then any function holomorphic in and continuous in the closed domain can be represented in the form (*)

Formula (*) generalizes Cauchy's classical integral formula for analytic functions of one complex variable and is called the Leray formula. J. Leray, who obtained this formula (see ), called it the Cauchy–Fantappié formula. In this formula the differential forms and are constituted according to the laws:  and where is the sign of exterior multiplication (see Exterior product). By varying the form of the function it is possible to obtain various integral representations from formula (*). One should bear in mind that, generally speaking, the Leray integral in (*) is not identically zero when is outside .

How to Cite This Entry:
Leray formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Leray_formula&oldid=11615
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article