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[parent] Cauchy integral formula in several variables (Theorem)

Let $ D = D_1 \times \ldots \times D_n \subset {\mathbb{C}}^n$ be a polydisc.

Theorem 1   Let $ f$ be a function continuous in $ \bar{D}$ (the closure of $ D$). Then $ f$ is holomorphic in $ D$ if and only if for all $ z = (z_1,\ldots,z_n) \in D$ we have
$\displaystyle f(z_1,\ldots,z_n) = \int_{\partial D_1} \cdots \int_{\partial D_n... ...s,\zeta_n)} {(\zeta_1 - z_1) \ldots (\zeta_n - z_n)} d\zeta_1 \ldots d\zeta_n .$    

As in the case of one variable this theorem can be in fact used as a definition of holomorphicity. Note that when $ n > 1$ then we are no longer integrating over the entire boundary of the polydisc but over the distinguished boundary, that is over $ \partial D_1 \times \ldots \times \partial D_n$.

Bibliography

1
Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.
2
Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.



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Cross-references: distinguished boundary, boundary, variable, closure, continuous, function, polydisc
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This is version 3 of Cauchy integral formula in several variables, born on 2005-11-03, modified 2005-11-07.
Object id is 7466, canonical name is CauchyIntegralFormulaInSeveralVariables.
Accessed 1610 times total.

Classification:
AMS MSC: 32A10 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Holomorphic functions)
 32A07 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Special domains )
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