Abel regularization of an oscillatory integral (${\lim }_{a\downarrow 0}{\int }_0^{\infty }{e}^{-ak}f(k)dk$)

An exponential damping factor makes many otherwise conditionally convergent or divergent oscillatory integrals absolutely convergent. Removing the damping after integration defines an Abel limit when it exists. For the Bessel function $J_1$, ${\int }_0^{\infty }{e}^{-ak}{J}_1(kR)dk=R^{-1}(1-a/\sqrt{a^2+R^2})$, giving the Mestel disk radial field as $a\downarrow 0$. This does not remove an unrelated additive divergence of the potential at low wavenumber.