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. Several existence theorems for Riemann–Hilbert problems have been produced by Krein, Gohberg and others (see the book by Clancey and Gohberg (1981)).
Suppose that Σ is a closed simple contour in the complex plane dividing the plane into two parts denoted by Σ+ (the inside) and Σ− (the outside), determined by the index of the contour with respect to a point. The classical problem, considered in Riemann's PhD dissertation (see Pandey (1996)), was that of finding a function
analytic inside Σ+ such that the boundary values of M+ along Σ satisfy the equation
for all z ∈ Σ, where a, b, and c are given real-valued functions (Bitsadze 2001).
By the Riemann mapping theorem, it suffices to consider the case when Σ is the unit circle (Pandey 1996, §2.2). In this case, one may seek M+(z) along with its Schwarz reflection:
On the unit circle Σ, one has , and so
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