Series expansion for the Fourier transform of a rational function in three dimensions
Abstract
In Rashba-Dresselhaus spin-orbit coupled systems, the calculation of Green's function requires the knowledge of the inverse Fourier transform of rational function $P(p)/Q(p)$, 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 $\alpha$, $\beta\geq0$, $\zeta\in\mathbb{C}$. While a two-dimensional problem, with $p=(p_{1},p_{2})$, has been recently solved [J. Brüning 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2014
- DOI:
- 10.48550/arXiv.1410.5199
- arXiv:
- arXiv:1410.5199
- Bibcode:
- 2014arXiv1410.5199J
- Keywords:
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- Mathematical Physics;
- Condensed Matter - Quantum Gases;
- 33C65;
- 33C70;
- 33C90
- E-Print:
- Accepted for publication in Rep. Math. Phys