Riemann-Hilbert problem ($F_{+}-F_{-}=J$)

Find functions analytic on the two sides of a contour whose boundary values obey a prescribed jump, with suitable normalization at infinity and other distinguished points. An additive jump $J$ on a counterclockwise closed contour is solved by the Cauchy integral formula $F(k)=(2\pi i{)}^{-1}\int J(\zeta )/(\zeta -k)d\zeta$, subject to the normalization and compatibility conditions. The Sokhotski–Plemelj theorem verifies the jump.