= Finite Hilbert transform
{title2=$H_{(0,1)}f(x)=\dfrac1\pi\operatorname{PV}\int_0^1\dfrac{f(t)}{t-x}\,dt$}
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.
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