Analytic continuation of the Appell function $F_1$ and integration of the associated system of equations in the logarithmic case

The Appell function $F_1$ (i.e., a generalized hypergeometric function of two complex variables) and a corresponding system of partial differential equations are considered in the logarithmic case when the parameters of $F_1$ are related in a special way. Formulas for the analytic continuation of $F_1$ beyond the unit bicircle are constructed in which $F_1$ is determined by a double hypergeometric series. For the indicated system of equations, a collection of canonical solutions are presented that are two-dimensional analogues of Kummer solutions well known in the theory of the classical Gauss hypergeometric equation. In the logarithmic case, the canonical solutions are written as generalized hypergeometric series of new form. The continuation formulas are derived using representations of $F_1$ in the form of Barnes contour integrals. The resulting formulas make it possible to efficiently calculate the Appell function in the entire range of its variables. The results of this work find a number of applications, including the problem of parameters of the Schwarz–Christoffel integral.