Abel integral equation
= Abel integral equation
{c}
{title2=$g(x)=\int_x^b f(y)(y-x)^{-1/2}\,dy$}
This square-root-kernel <integral equation> is inverted by integrating $g(x)/\sqrt{x-z}$ over $z<x<b$ and interchanging integrations. The inner integral over $z<x<y$ equals $\pi$, so $\int_z^bg(x)/\sqrt{x-z}\,dx=\pi\int_z^bf(y)\,dy$. Differentiating recovers $f(z)=-\pi^{-1}d[\int_z^bg(x)/\sqrt{x-z}\,dx]/dz$. The changes of variables $x=p^2$, $y=\eta^2$ put the angular-ray equation into this form.