generalized Cauchy integral formula
GeneralizedCauchyIntegralFormula
2008-02-05 14:07:36
2008-02-05 18:20:50
Theorem
Cauchy integral theorem
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\begin{thm}
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*}
\end{thm}
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.
\begin{thebibliography}{9}
\bibitem{Hormander:several}
Lars H\"ormander.
{\em \PMlinkescapetext{An Introduction to Complex Analysis in Several
Variables}},
North-Holland Publishing Company, New York, New York, 1973.
\bibitem{Krantz:several}
Steven~G.\@ Krantz.
{\em \PMlinkescapetext{Function Theory of Several Complex Variables}},
AMS Chelsea Publishing, Providence, Rhode Island, 1992.
\end{thebibliography}