Antiplane ligament Cauchy solution ($G(z)=\frac{1}{\pi \sqrt{a^2-z^2}}\left({\int }_a^{\infty }-{\int }_{-\infty }^{-a}\right)\frac{\sqrt{t^2-a^2}p(t)}{t-z}dt$)

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.