# generalized Cauchy integral formula

###### Theorem.

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 $z\in U,$

 $f(z)=\frac{1}{2\pi i}\int_{\partial U}\frac{f(w)}{w-z}dw-\frac{1}{2\pi i}\int_% {U}\frac{\frac{\partial f}{\partial\bar{z}}(w)}{w-z}d\bar{w}\wedge dw.$

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.

## References

• 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
• 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title generalized Cauchy integral formula GeneralizedCauchyIntegralFormula 2013-03-22 17:46:41 2013-03-22 17:46:41 jirka (4157) jirka (4157) 6 jirka (4157) Theorem msc 30E20 generalized Cauchy formula