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

An additive Riemann-Hilbert problem seeks a holomorphic function on each side of an oriented contour $\Gamma$, with prescribed difference of boundary valuesThe plus side is on the left of the orientation. Conditions at infinity and at contour endpoints are part of the problem; the jump alone does not usually give uniqueness.

On a positively oriented smooth simple closed contour, for Hölder-continuous data $g$, the Cauchy transformhas the required jump by the Sokhotski–Plemelj theorem and vanishes at infinity. For the unit circle and $g=1$, the Cauchy integral formula gives $F=1$ inside and $F=0$ outside. This concrete solution checks both the sign of the jump and the normalization. Subtracting two solutions removes the jump, so analytic continuation and growth conditions can establish uniqueness.^[ref]