Numerical evaluation of Appell's F1 hypergeometric function

In this work we present a numerical scheme to compute the two-variable hypergeometric function F-1 (alpha, beta, beta, gamma; x, y) of Appell for complex parameters alpha, beta, beta ' and gamma, and real values of the variables x and y. We implement a set of analytic continuations that allow us to obtain the F-1 function outside the region of convergence of the series definition. These continuations can be written in terms of the Horn's G(2) function, Appell's F-2 function related, and the F-1 hypergeometric itself. The computation of the function inside the region of convergence is achieved by two complementary methods. The first one involves a single-index series expansion of the F-1 function, while the second one makes use of a numerical integration of a third order ordinary differential equation that represents the system of partial differential equations of the F-1 function. We briefly sketch the program and show some examples of the numerical computation. (C) 2001 Elsevier Science B.V. All rights reserved.