标题： 解析延拓和Appell $F_1$， $F_3$，Lauricella $F_D^{(3)}$ 和Lauricella-Saran $F_S^{(3)}$的数值计算及其在费曼积分中的应用 显示英文标题

标题： 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

摘要： 我们对两个变量的超几何级数的研究进行了调查，即Appell$F_{1}$和$F_{3}$级数，并获得了其解析延拓的全面列表，足以覆盖整个实数$(x,y)$平面，除了它们的奇异点轨迹。 我们还推导了它们的三变量广义形式的解析延拓，即Lauricella$F_{D}^{(3)}$级数和Lauricella-Saran$F_{S}^{(3)}$级数，利用了$F_{1}$和$F_{3}$的解析延拓，这确保了整个实$(x,y,z)$空间都被覆盖，除了这些函数的奇点轨迹。 虽然这些研究是由于这些多变量超几何函数在费曼积分计算中频繁出现而产生的，但它们也可以在其他数学物理分支中出现时使用。 为了便于实际使用，我们提供了四个包：AppellF1$.$wl，AppellF3$.$wl，LauricellaFD$.$wl，以及LauricellaSaranFS$.$wl 在MATHEMATICA 中。 这些包适用于参数的通用和非通用值，在考虑它们在费曼积分计算中的实用性时。 我们明确展示了这些包在费曼积分计算背景下的各种物理应用，并使用其他包如 FIESTA 对结果进行比较。 在应用适当的数值计算约定后，我们发现从我们的包中得到的结果是一致的。 还提供了各种 Mathematica 笔记本，以展示不同的数值结果，并随本文一起提供。 显示英文摘要

摘要： 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 generalization, 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, 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.