 Solution

With $K=1$, the additive jump is $g(t)=1/[2(t^2-1)]$, analytic inside the contour. Its Cauchy integral has $D_{+}(z)=g(z)$ and $D_{-}(z)=0$. Therefore $C_{+}=(z^2-1)g=1/2$, $C_{-}=0$, andThe moment is indeed $(\pi i{)}^{-1}{\int }_L(\tau +1/\tau )(1/2)d\tau =1$, as required. The principal-value Cauchy sum is $1/2$, and the original left side becomes $[(t^2+t-1)-(t^2-t-1)]/2-1=t-1$. Thus the recovered function satisfies both the equation and the self-consistency condition, and the homogeneous negative-index argument proves uniqueness.