Integral representations and estimates for the remainder term for the summation of the double hypergeometric Appell series are constructed. The resulting formulas have applications in developing algorithms for computing the Appell functions and in C using analytic continuation formulas. The results have applications to problems in mathematical physics and computational function theory, including the construction of conformal mappings of complicated polygons based on the Christoffel-Schwarz integral.
S.I. Bezrodnykh (Wed,) studied this question.