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generalized Cauchy integral formula
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(Theorem)
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Theorem 1 Let $U \subset \mathbb{C}$ be a domain with $C^1$ boundary. Let $f \colon U \to \mathbb{C}$ be a $C^1$ function that is $C^1$ up to the boundary. Then for $z \in U,$ \begin{equation*} f(z) = \frac{1}{2\pi i} \int_{\partial U} \frac{f(w)}{w-z} dw - \frac{1}{2\pi i} \int_{U} \frac{\frac{\partial f}{\partial \bar{z}}(w)}{w-z} d\bar{w} \wedge dw . \end{equation*}
Note that $C^1$ up to the boundary means that the function and the derivative extend to be continuous functions on the closure of $U.$ The theorem follows from Stokes' theorem. When $f$ is holomorphic, then the second term
is zero and this is the classical Cauchy integral formula.
- 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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"generalized Cauchy integral formula" is owned by jirka.
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generalized Cauchy formula |
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Cross-references: Cauchy integral formula, term, holomorphic, Stokes theorem, theorem, closure, continuous functions, derivative, function, boundary, domain
There is 1 reference to this entry.
This is version 3 of generalized Cauchy integral formula, born on 2008-02-05, modified 2008-02-05.
Object id is 10236, canonical name is GeneralizedCauchyIntegralFormula.
Accessed 1845 times total.
Classification:
| AMS MSC: | 30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions) |
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Pending Errata and Addenda
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