Analytic continuations and numerical evaluation of the Appell F_1, F_3, Lauricella F_D^(3) and Lauricella-Saran F_S^(3) and their application to Feynman integrals
Description
We present our investigation of the study of two variable hypergeometric series, namely Appell F_1 and F_3 series, and obtain a comprehensive list of its analytic continuations enough to cover the whole real (x, y) plane, except on their singular loci. We also derive analytic continuations of their 3-variable generalisation, the Lauricella F_D^(3) series and the Lauricella-Saran F_S^(3) series, leveraging the analytic continuations of F_1 and F_3, which ensures that the whole real (x, y, z) space is covered, except on the singular loci of these functions. While these studies are motivated by the frequent occurrence of these multivariable hypergeometric functions in Feynman integral evaluation, they can also be used whenever they appear in other branches of mathematical physics. To facilitate their practical use, for analytical and numerical purposes, we provide four packages: AppellF1.wl, AppellF3.wl, LauricellaFD.wl, and LauricellaSaranFS.wl in Mathematica. These packages are applicable for generic as well as non-generic values of parameters, keeping in mind their utilities in the evaluation of the Feynman integrals. We explicitly present various physical applications of these packages in the context of Feynman integral evaluation and compare the results using other packages such as FIESTA. Upon applying the appropriate conventions for numerical evaluation, we find that the results obtained from our packages are consistent. Various Mathematica notebooks demonstrating different numerical results are also provided along with this paper.