= Cauchy principal value for an endpoint Jacobi weight
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{title2=$\operatorname{PV}\int_{-1}^1\frac{(1-t)^{\alpha-1}}{(1+t)^\alpha(t-x)}dt=\pi\cot(\pi\alpha)\frac{(1-x)^{\alpha-1}}{(1+x)^\alpha}$}
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.
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