The Riemann–Hilbert problem and the Jacobi-type formulas for the Lauricella hypergeometric function

The research presents differential relations for the Lauricella function, which are a generalization of the well-known Jacobi identity for the Gauss hypergeometric function. Using Jacobi-type formulas for the Lauricella function, a new representation of the solution of the Riemann–Hilbert problem in the upper half-plane is obtained for the case of piecewise constant data in the form of the Schwarz–Christoffel type integral. An application is given to modeling the effect of magnetic reconnection and to solving the problem of parameters of the Schwarz–Christoffel integral.