Riemann–Hilbert problem

For the original problem of Hilbert concerning the existence of linear differential equations having a given monodromy group, see Hilbert's twenty-first problem.

In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Specifically, a Riemann–Hilbert problem is a boundary value problem for a holomorphic function on the complement of an oriented contour in the complex plane, with prescribed jump discontinuities across the contour. Classical scalar forms go back to Riemann and Hilbert, and modern matrix-valued Riemann–Hilbert problems play a central role in integrable systems, orthogonal polynomials, random matrix theory, inverse monodromy, and asymptotic analysis. Several existence theorems for Riemann–Hilbert problems have been produced by Mark Krein, Israel Gohberg and others.

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