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 continuousfunctions 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.