# Cauchy kernel

2010 Mathematics Subject Classification: Primary: 30-XX [MSN][ZBL]

The term refers usual to the function $\mathbb C^2 \setminus \Delta \ni (\zeta, z) \mapsto \frac{1}{\zeta-z}$, where $\Delta \subset \mathbb C^2$ is the diagonal $\{(z,\zeta): z=\zeta\}$. Such function is the kernel of the Cauchy integral, which gives a powerful identity for holomorphic functions of one complex variable. In the case of the unit circle one has the following relationship between the Cauchy kernel and the Hilbert kernel: if $\zeta = e^{i\tau}$ and $z = e^{it}$, with $\tau, t \in \mathbb S^1$, then $\frac{d\zeta}{\zeta-z} = \frac{1}{2} \left(\cot \frac{\tau-t}{2} + i\right)\, d\tau\, .$

Some authors use the term for the function $\frac{1}{2\pi i (\zeta-z)}\, .$