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A Riemann-Hilbert problem seeks holomorphic functions off an oriented contour, with prescribed relations between their limiting boundary values and with specified behavior at infinity, endpoints or other distinguished points. The plus side lies to the left when following the contour orientation; the minus side lies to the right.

A standard matrix form prescribes a jump matrix $J$ and asks forThe matrix multiplication order is part of the problem. Some versions also prescribe poles and their residues. The contour, jump, normalization and local singularity conditions together determine the analytic problem.^[ref]

For an additive Riemann-Hilbert problem, the condition is $F_{+}-F_{-}=g$. On a positively oriented smooth closed contour, the Cauchy integralhas that jump under the usual regularity assumptions, by the Sokhotski–Plemelj theorem. For the unit circle and $g=1$, it gives $F=1$ inside and $F=0$ outside. This checks the sign convention and the vanishing exterior normalization. A multiplicative matrix jump generally needs additional analysis beyond this scalar Cauchy formula.