Series Expansion for the Fourier Transform of a Rational Function in Three Dimensions

In Rashba–Dresselhaus spin-orbit coupled systems, the calculation of the Green function requires the knowledge of the inverse Fourier transform of rational function P(p)/Q(p), where P(p) takes the values 1 and p2, and where Q(p)=(p2-ζ)2-α2(p12+p22)-β2with suitable parameters α, β ≥ 0, ζ ∈ ℂ. While a two-dimensional problem, with p = (p1, p2), has been recently solved [J. Brüning et al., J. Phys. A: Math. Theor.40 (2007)], its three-dimensional analogue, with p = (p1, p2, p3), remains open. In this paper, a hypergeometric series expansion for the triple integral is provided. Convergence of the series depending on the parameters is studied in detail.