f(r)=\left[ \frac{1}{2}\delta(r-y_\Delta)\sqrt{1-(y_\Delta/r)^2} - \frac{1}{\pi} \frac{H(y_\Delta-r)}{\sqrt{y_\Delta^2-r^2}} \right] \Delta F-\frac{1}{\pi}\int_r^\infty\frac{d F}{dy}\frac{dy}{\sqrt{y^2-r^2}}.
where \delta is the Dirac delta function and H(x) the Heaviside step function.