Series expansion for the Fourier transform of a rational function in three dimensions

In Rashba-Dresselhaus spin-orbit coupled systems, the calculation of Green’s function requires the knowledge of the inverse Fourier transform of rational function $P(p)\mathrm{/}Q(p)$, where $P(p)$ takes the values $1$ and $p^2$, and where $$Q(p)=(p^2-\zeta {)}^2-{\alpha }^2(p_1^2+p_2^2)-{\beta }^2$$ with suitable parameters $\alpha$, $\beta \ge 0$, $\zeta \in \mathbb{C}$. While a two-dimensional problem, with $p=(p_1,p_2)$, has been recently solved [J. Br"{u}ning et al, J. Phys. A: Math. Theor. 40 (2007)], its three-dimensional analogue, with $p=(p_1,p_2,p_3)$, remains open. In this paper, a hypergeometric series expansion for the triple integral is provided. Convergence of the series dependent on the parameters is studied in detail.