Cauchy-type integral

Summary

A Cauchy-type integral is a function $\Phi(z) = \frac{1}{2\pi i} \int_\Gamma \frac{\varphi(\tau)}{\tau - z} \, d\tau$ built from a curve $\Gamma$ and a function $\varphi$ on it. It has the form of Cauchy's integral formula, but $\varphi$ is arbitrary and need not be the boundary value of a holomorphic function. It is a holomorphic function off $\Gamma$, and in general it jumps across $\Gamma$ by exactly $\varphi$. This makes it the basic tool for boundary value problems of Riemann–Hilbert type and for singular integral equations.