 Solution

Let $C(z)=(2\pi i{)}^{-1}{\int }_L\varphi (\tau )/(\tau -z)d\tau$, with interior and exterior boundary values $C_{+},C_{-}$. The Sokhotski–Plemelj formula gives $\varphi =C_{+}-C_{-}$ and $C_{+}+C_{-}=(\pi i{)}^{-1}\mathrm{PV}{\int }_L\varphi (\tau )/(\tau -t)d\tau$. Write $K=(\pi i{)}^{-1}{\int }_L(\tau +1/\tau )\varphi (\tau )d\tau$ for the nonlocal scalar moment. Substitution into the integral equation yieldsThus the associated scalar Riemann-Hilbert problem isThe functions are analytic in their respective interior/exterior domains, and $K$ must ultimately equal the moment of the recovered jump; it is not an independently specified free constant.