 Finite Hilbert transform

The finite Hilbert transform is a singular integral operator whose integration is restricted to a finite interval. The displayed convention uses the denominator $t-x$; reversing that denominator reverses the sign. The Cauchy principal value deletes equal intervals on the two sides of the interior point $x$. Endpoint behavior matters: the integrable function $1/\sqrt{x(1-x)}$ belongs to the null space for this interval. This nontrivial null space is one reason that an inversion formula needs a normalization or a stated endpoint class.