generalized Cauchy integral formula

Theorem.

Let $U\mathrm{\subset }\mathrm{C}$ be a domain with $C^{\mathrm{1}}$ boundary. Let $f\mathrm{:}U\mathrm{\to }\mathrm{C}$ be a $C^{\mathrm{1}}$ function that is $C^{\mathrm{1}}$ up to the boundary. Then for $z\mathrm{\in }U\mathrm{,}$

$$f(z)=\frac{1}{2\pi i}{\int }_{\partial U}\frac{f(w)}{w-z}𝑑w-\frac{1}{2\pi i}{\int }_U\frac{\frac{\partial f}{\partial \overline{z}}(w)}{w-z}𝑑\overline{w}\wedge dw.$$

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.

Title	generalized Cauchy integral formula
Canonical name	GeneralizedCauchyIntegralFormula
Date of creation	2013-03-22 17:46:41
Last modified on	2013-03-22 17:46:41
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	6
Author	jirka (4157)
Entry type	Theorem
Classification	msc 30E20
Synonym	generalized Cauchy formula