Investigation of infinitely rapidly oscillating distributions

Abstract We rigorously investigate the rapidly oscillating contributions in sinc-function representation of Dirac delta function and Fourier transform Coulomb potential. Beginning with a derivation standard integral Heaviside step function, we examine that contains sinc function. By contour integration, prove satisfies properties although it is divergent at nonzero points. This good pedagogical example demonstrating difference between distribution. In most textbooks, contribution potential into momentum space has been ignored by regulating oscillatory divergence screened Wentzel. performing inverse rigorously, demonstrate well-defined distribution indeed zero, even if an ill-defined Proofs are extended to exhibit Riemann–Lebesgue lemma can hold for which not absolutely integrable.