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Cauchy integral formula in several variables
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(Theorem)
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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 \begin{equation*} f(z_1,\ldots,z_n) = \int_{\partial D_1} \cdots \int_{\partial D_n} \frac{f(\zeta_1,\ldots,\zeta_n)} {(\zeta_1 - z_1) \ldots (\zeta_n - z_n)} d\zeta_1 \ldots d\zeta_n .
\end{equation*}
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 $\partial D_1 \times \ldots \times \partial D_n$ .
- 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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"Cauchy integral formula in several variables" is owned by jirka.
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Cross-references: distinguished boundary, boundary, theorem, variable, closure, continuous, function, polydisc
There is 1 reference to this entry.
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 2015 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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Pending Errata and Addenda
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