Canonical factorization of a rational scalar Riemann-Hilbert jump ($G=X_{+}/X_{-}$)

A rational jump with no zeros or poles on the contour can be split into factors analytic and nonvanishing on the two sides. The exterior factor may carry a power of $z$ recording the winding number. For a circle enclosing $0,\pm 1$, the jump $G=z/(z^2-1)$ has the canonical factors $X_{+}=1$, $X_{-}=z-z^{-1}$. The exterior growth of $X_{-}$ imposes compatibility conditions when a Cauchy transform is required to vanish at infinity.