Integrable singularity

A function has an integrable singularity at a point if it becomes unbounded there but its absolute value has a finite improper integral over a sufficiently small surrounding interval, one-sided at an endpoint. The endpoint behavior of the arcsine distribution's probability density function is an example.

An endpoint integrable singularity with $f(x)\sim C/\sqrt{x-x_0}$ as $x\downarrow x_0$, for finite nonzero $C$. It is an integrable singularity because ${\int }_{x_0}^{x_0+ϵ}(x-x_0{)}^{-1/2}dx=2\sqrt{ϵ}$. One common origin is a stationary change of variables: if $y-y_0=b{z}^2+O(z^4)$ with $b>0$ and a smooth nonzero probability density for $z$ at zero, the two inverse branches give a probability density for $y$ proportional to $(y-y_0{)}^{-1/2}$.