The problem of analytic continuation is considered for the Lauricella function F(N) D , a generalized hypergeometric functions of N complex variables. For an arbitrary N a complete set of formulae is given for its analytic continuation outside the boundary of the unit polydisk, where it is defined originally by an N-variate hypergeometric series. Such formulae represent F(N) D in suitable subdomains of CN in terms of other generalized hypergeometric series, which solve the same system of partial differential equations as F(N) D . These hypergeometric series are the N-dimensional analogue of Kummer's solutions in the theory of Gauss's classical hypergeometric equation. The use of this function in the theory of the RiemannHilbert problem and its applications to the SchwarzChristoffel parameter problem and problems in plasma physics are also discussed. © 2018 Institute of Physics Publishing.All rights reserved.