= Finite antiplane crack Cauchy solution
{title2=$G(z)=\frac1{\pi\sqrt{z^2-a^2}}\int_{-a}^a\frac{\sqrt{a^2-t^2}\,p(t)}{z-t}\,dt$}
Choose $\chi(z)=\sqrt{z^2-a^2}\sim z$ at infinity, cut on the finite crack. Its upper crack value is $i\sqrt{a^2-t^2}$, so $H=\chi G$ has jump $-2i\sqrt{a^2-t^2}p(t)$. The <Sokhotski–Plemelj theorem> gives the displayed <Cauchy integral> solution. It decays as $z^{-2}$ for integrable weighted traction. A possible homogeneous term $C/\chi$ would give a logarithmic displacement period; it is removed for a dislocation-free finite crack with no extra remote loading.
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