Sokhotski–Plemelj theorem

The Sokhotski–Plemelj theorem (Polish spelling is Sochocki) is a theorem in complex analysis, which helps in evaluating certain integrals. The real-line version of it (see below) is often used in physics, although rarely referred to by name. The theorem is named after Julian Sochocki, who proved it in 1868, and Josip Plemelj, who rediscovered it as a main ingredient of his solution of the Riemann-Hilbert problem in 1908.

Let C be a smooth closed simple curve in the plane, and ${\displaystyle \varphi }$ an analytic function on C. Then the Cauchy-type integral

defines two analytic functions of z, ${\displaystyle \phi _{i}}$ inside C and ${\displaystyle \phi _{e}}$ outside. The Sokhotski–Plemelj formulas relate the limiting boundary values of these two analytic functions at a point z on C and the Cauchy principal value ${\displaystyle {\mathcal {P}}}$ of the integral:

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