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For the finite crack use the branchcut on $[-a,a]$. For $-a<t<a$, its limiting values are ${\chi }^{+}(t)=i\sqrt{a^2-t^2}$ and ${\chi }^{-}(t)=-i\sqrt{a^2-t^2}$. Consequently $H=\chi G$ converts the sum condition of the Hilbert problem for an antiplane crack into an additive jump:By the Sokhotski–Plemelj theorem, the Cauchy integral with this jump isThus the finite antiplane crack Cauchy solution is $G=H/\chi$. For regular integrable loading, $H=O(z^{-1})$ and $G=O(z^{-2})$. A homogeneous term $C/\chi$ would instead produce a logarithmic potential. In the reflected real-loading class $C$ is real, and its $1/z$ term gives a nonzero displacement period around the crack. It is excluded for the induced, dislocation-free correction. A nonvanishing constant stress term would change the remote applied field and is likewise excluded.

For $x=x_1>a$, $\chi (x)=\sqrt{x^2-a^2}>0$, soThis is the additional stress, not the total stress; the applied field must still be added. The integral is real at these intact-axis points, so ${\sigma }_{13}(x,0)=0$. On the crack itself, the upper Cauchy principal value formula gives $H^{+}={\pi }^{-1}\mathrm{PV}\int \sqrt{a^2-t^2}p(t)/(x-t)dt-i\sqrt{a^2-x^2}p(x)$. Division by $i\sqrt{a^2-x^2}$ verifies $\mathrm{Re}{G}^{+}=-p$, checking the sign of the face traction cancellation.

For example, constant applied shear $p_0$ gives $H(z)=p_0(z-\chi (z))$ and $G(z)=p_0(z/\chi (z)-1)$. The total shear ahead of the crack is then $p_{0}x/\sqrt{x^2-a^2}$, providing a direct check of the normalization and the crack-tip singularity.