Some Applications of Bochner-Martinelli Integral Representation

Abstract

In the space of ℂⁿ there is a well-known integral representation of Bochner-Martinelli [l][2]:

Theorem 1.1. Let D be a bounded domain in the space ℂⁿ of complex variables z₁,…,z_n whose boundary ∂D is a 2n-l dimensional smooth orientable manifold. If f(z) is a function holomorphic in D and continuous on ∂D (denoted by f(z) ∈ A(D)), then

$$ f(z) = \int\limits_{\partial D} {F(\zeta )K(\zeta ,Z)} ,\,\,\,\,Z\, \in \,D, $$

(1.1)

where

$$ K(\zeta ,Z) = \frac{{(n - 1)!}} {{(2\pi i)^n }}\frac{1} {{\left| {\left. {\zeta - z} \right|} \right.^{2n} }}\sum\limits_{k = 1}^n {(\overline \zeta _k - \overline z _k )\overline d \zeta _1 \wedge d\zeta _1 \wedge \cdots \wedge [\overline d \zeta _k ] \wedge \,d\zeta _k \wedge \cdots \wedge \overline d \zeta _n \wedge d\zeta _{n'} } \,(n > 1) $$

(1.2)

$$ \left| {\zeta - z} \right| = \sqrt {\sum\limits_{k = 1}^n {\left| {\zeta _k - z_k } \right|^2 } } . $$

(1.3)