= Additive Riemann-Hilbert problem
{title2=$F_+-F_-=g$}
An additive Riemann-Hilbert problem seeks a <holomorphic function> on each side of an oriented <contour> $\Gamma$, with prescribed difference of <boundary values>
$$
F_+(z)-F_-(z)=g(z).
$$
The 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 transform>
$$
F(z)=\frac1{2\pi i}\int_\Gamma\frac{g(\zeta)}{\zeta-z}\,d\zeta
$$
has 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.<Riemann-Hilbert Problems by Percy Deift>[Lecture 2, Cauchy operators and boundary values]{ref}
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