Fourier transform in elliptic coordinates: Case of axial symmetry

Abstract

We perform the Fourier transform in two dimensional elliptic coordinates. The case of axial symmetry allows to reduce significantly the final transformation. The obtained integral formulae generalize the Fourier-Bessel transform due to relation between elliptic and polar coordinates. We show also the alternating definition using Mathieu functions. The application of the formulae obtained is the transitions of operator representations in Quantum Mechanics.