# Leray formula

Cauchy–Fantappié formula

A formula for the integral representation of holomorphic functions $f ( z)$ of several complex variables $z = ( z _ {1} \dots z _ {n} )$, $n \geq 1$, which generalizes the Cauchy integral formula (see Cauchy integral).

Let $D$ be a finite domain in the complex space $\mathbf C ^ {n}$ with piecewise-smooth boundary $\partial D$ and let $\chi ( \zeta ; z ) : \partial D \rightarrow \mathbf C ^ {n}$ be a smooth vector-valued function of $\zeta \in \partial D$ with values in $\mathbf C ^ {n}$ such that the scalar product

$$\langle \zeta - z , \chi ( \zeta ; z ) \rangle = \sum _ {\nu = 1 } ^ { n } ( \zeta _ \nu - z _ \nu ) \chi _ \nu ( \zeta ; z ) \neq 0$$

everywhere on $\partial D$ for all $z \in D$. Then any function $f ( z)$ holomorphic in $D$ and continuous in the closed domain $\overline{D}\;$ can be represented in the form

$$\tag{* } f ( z) = \frac{( n- 1 )! }{( 2 \pi i ) ^ {n} } \int\limits _ {\partial D } \frac{f ( \zeta ) \delta ( \chi ( \zeta ; z )) \wedge d \zeta }{< \zeta - z , \chi ( \zeta ; z ) > ^ {n} } ,\ z \in D .$$

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 $\delta ( \chi ( \zeta ; z ))$ and $d \zeta$ are constituted according to the laws:

$$\delta ( \chi ( \zeta ; z )) = \sum _ {\nu = 1 } ^ { n- } 1 ( - 1 ) ^ { \nu - 1 } \chi _ \nu ( \zeta ; z ) d \chi _ {1} ( \zeta ; z ) \wedge \dots$$

$$\dots \wedge d \chi _ {\nu - 1 } ( \zeta ; z ) \wedge d \chi _ {\nu + 1 } ( \zeta ; z ) \wedge \dots \wedge d \chi _ {n} ( \zeta ; z )$$

and

$$d \zeta = d \zeta _ {1} \wedge \dots \wedge d \zeta _ {n} ,$$

where $\wedge$ is the sign of exterior multiplication (see Exterior product). By varying the form of the function $\chi$ 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 $z$ is outside $D$.

How to Cite This Entry:
Leray formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Leray_formula&oldid=47615
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article