= Cauchy reconstruction from a jump across a finite interval
{c}
{title2=$w(z)=\dfrac1{2\pi i}\int_0^1\dfrac{w_+(t)-w_-(t)}{t-z}\,dt$}
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 $\varepsilon$ contributes $O(\varepsilon^{1-\alpha})$, 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.
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