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The paper considers the issue of constructing basises in the solution space of the system of partial differential equations, which is satisfied by the Lauricella hypergeometric function, depending on  N  complex variablesand having complex parameters,  b ,  c . For an arbitrary number  N  of variables, we have obtained explicit representations for such basis functions in the vicinity of pointsandin terms of the Horn type hypergeometric series in  N  variables. For some of these functions we have obtained formulas of analytic continuation. The found continuation formulas are important for calculating the solution of the Riemann – Hilbert problem with piecewise constant coefficients and studying its geometrical meaning. Besides, these formulas are effective for solving the parameters problem for the Schwarz – Christoffel integral and calculating conformal mapping of complex-shaped polygons.

Constructing basises in solution space of the system of equations for the Lauricella Function F D (N)