= Abel regularization of an oscillatory integral
{c}
{title2=$\lim_{a\downarrow0}\int_0^\infty e^{-ak}f(k)\,dk$}
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 $a\downarrow0$. This does not remove an unrelated additive divergence of the potential at low wavenumber.
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