Singular integral equation ($a(t)f(t)+b(t)\mathrm{PV}{\int }_{\Gamma }K(t,\zeta )f(\zeta )d\zeta =c(t)$)

An integral equation with a kernel singular on the integration contour needs a prescribed limiting interpretation such as a Cauchy principal value. For the Cauchy kernel, the Sokhotski–Plemelj theorem converts a linear scalar equation into a Riemann-Hilbert problem for its two analytic boundary values. Normalization, endpoint regularity and the index of that jump determine homogeneous freedom and solvability.