Bochner–Martinelli Formula

In mathematics, the Bochner–Martinelli formula is a generalization of the Cauchy integral formula to functions of several complex variables, introduced by and .

History

Bochner–Martinelli kernel

For , in $\C^n$ the Bochner–Martinelli kernel is a differential form in of bidegree defined by
:$\omega(\zeta,z) = \frac\frac \sum_(\overline\zeta_j-\overline z_j) \, d\overline\zeta_1 \land d\zeta_1 \land \cdots \land d\zeta_j \land \cdots \land d\overline\zeta_n \land d\zeta_n$

(where the term is omitted).

Suppose that is a continuously differentiable function on the closure of a domain in $\mathbb$^''n'' with piecewise smooth boundary . Then the Bochner–Martinelli formula states that if is in the domain then
:$\displaystyle f(z) = \int_f(\zeta)\omega(\zeta, z) - \int_D\overline\partial f(\zeta)\land\omega(\zeta,z).$

In particular if is holomorphic the second term vanishes, so
:$\displaystyle f(z) = \int_f(\zeta)\omega(\zeta, z).$

See also

*Bergman–Weil formula

Notes

References

*.
*.
*.
*
*.
*.
*.
*, (ebook).
*. The first paper where the now called Bochner-Martinelli formula is introduced and proved.
*. Available at th
SEALS Portal
. In this paper Martinelli gives a proof of Hartogs' extension theorem by using the Bochner-Martinelli formula.
*. The notes form a course, published by the Accademia Nazionale dei Lincei, held by Martinelli during his stay at the Accademia as "''Professore Linceo''".
*. In this article, Martinelli gives another form to the Martinelli–Bochner formula.

{{DEFAULTSORT:Bochner-Martinelli formula
Theorems in complex analysis
Several complex variables