Cauchy reconstruction from a jump across a finite interval ($w(z)=\frac{1}{2\pi i}{\int }_0^1\frac{w_{+}(t)-w_{-}(t)}{t-z}dt$)

Suppose $w$ is holomorphic off $[0,1]$, tends to zero at infinity, has sufficiently regular boundary values in the interval interior, and has endpoint growth $O(|z-a{|}^{-\alpha })$ for some $\alpha <1$ at each endpoint $a$. If its jump is integrable, the displayed reconstruction follows by deforming a Cauchy integral formula contour onto the two sides of the interval. An endpoint circle of radius $\epsilon$ contributes $O({\epsilon }^{1-\alpha })$, which vanishes. The two oppositely oriented banks leave precisely the jump. Without decay at infinity, an entire remainder can survive; a nonzero constant is the simplest example.