Estimation of the Remainder Term of the Appell Hypergeometric Series F2

Integral representations and estimates for the remainder terms of the Appell double hypergeometric series $${{F}_{2}}$$ are constructed. The found formulas can be used to develop algorithms for computing the Appell functions $${{F}_{1}}$$ and $${{F}_{3}}$$ in $${{\mathbb{C}}^{2}}$$ by applying analytic continuation formulas. The results have applications to problems in mathematical physics and computational function theory, including the construction of conformal mappings of complex polygons based on the Schwarz–Christoffel integral.