 Suppose that formula_1 is a closed simple contour in the complex plane dividing the plane into two parts denoted by formula_2 (the inside) and formula_3 (the outside), determined by the index of the contour with respect to a point.The classical problem, considered in Riemann's PhD dissertation (see ), was that of finding a function

analytic inside formula_2 such that the boundary values of "M"+ along formula_1 satisfy the equation

for all formula_8, where "a", "b", and "c" are given real-valued functions .By the Riemann mapping theorem, it suffices to consider the case when formula_1 is the unit circle .