🤖 AI Summary
This work addresses the challenge of stable, high-precision evaluation of four classes of multivariate hypergeometric functions—Appell $F_1$, $F_3$, Lauricella $F_D^{(3)}$, and Saran $F_S^{(3)}$—over the entire complex parameter domain. Methodologically, it introduces a unified analytic continuation framework: first, systematically constructing cross-region continuation paths; second, integrating high-accuracy numerical ODE integration, Wronskian-based boundary matching, Padé approximants, and rigorous branch-cut handling, all implemented using both arbitrary-precision floating-point and interval arithmetic. The key contribution is overcoming classical convergence-domain limitations, achieving relative accuracy of $sim 10^{-15}$ across physically relevant parameter regions. The method has been successfully applied to obtain full-phase-space analytic expressions and numerical validation for multiple two-loop three- and four-point Feynman integrals in QED and QCD, thereby significantly enhancing both computational efficiency and reliability of higher-order Standard Model corrections.