 Abel regularization of an oscillatory integral

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.