 Cauchy principal value for an endpoint Jacobi weight

For $0<\alpha <1$ and $-1<x<1$, the endpoint powers are integrable and the interior pole is taken as a Cauchy principal value. The analytic function $w(z)=(z-1{)}^{\alpha -1}/(z+1{)}^{\alpha }$, positive for $z>1$ and decaying like $1/z$, has boundary values $-e^{\pm i\pi \alpha }(1-t{)}^{\alpha -1}/(1+t{)}^{\alpha }$. Its jump gives a Cauchy reconstruction, while the mean of its boundary values and the Sokhotski–Plemelj formula give the displayed identity. At $\alpha =1/2$ the principal value is zero.