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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, Riemann Hilbert problems are a class of problems that arise, inter alia, 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 is a closed simple contour in the complex plane dividing the plane into two parts denoted by + (the inside) and (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 M_+(z) = u(z) + i v(z)! analytic inside + such that the boundary values of M+ along satisfy the equation a(z)u(z) - b(z)v(z) = c(z) ! for all z , where a, b, and c are given real-valued functions (Bitsadze 2001).