= Antiplane ligament Cauchy solution
{title2=$G(z)=\frac1{\pi\sqrt{a^2-z^2}}\left(\int_a^\infty-\int_{-\infty}^{-a}\right)\frac{\sqrt{t^2-a^2}\,p(t)}{t-z}\,dt$}
For two semi-infinite cracks separated by a ligament, choose $\chi(z)=\sqrt{a^2-z^2}>0$ on the ligament. Its upper crack values are $-i\sqrt{t^2-a^2}$ on the right and $+i\sqrt{t^2-a^2}$ on the left. Multiplication by $\chi$ converts the <Hilbert problem for an antiplane crack> to an additive jump; the <Sokhotski–Plemelj theorem> gives the displayed solution, assuming convergence. A homogeneous term $C/\chi$ transfers an extra resultant $\pi C$ through the ligament and has logarithmic displacement at infinity. The usual correction with no additional remote resultant excludes it. The far-field normalization is essential to uniqueness.
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