Principal-value beta integral
= Principal-value beta integral
For $0<s<a$,
$$
\operatorname{PV}\int_0^\infty\frac{x^{s-1}}{1-x^a}dx=\frac\pi a\cot\frac{\pi s}{a}.
$$
= Principal-value beta integral
For $0<s<a$,
$$
\operatorname{PV}\int_0^\infty\frac{x^{s-1}}{1-x^a}dx=\frac\pi a\cot\frac{\pi s}{a}.
$$