Abstract
This chapter discusses the principal problem of the theory of boundary value problems of analytic functions namely the Riemann boundary value problem. It reviews the concept of the index of a function, which is of great value as an auxiliary tool. Various assertions are made in the chapter—(1) the index of a function that is continuous on a closed contour and does not vanish anywhere on it, is an integer or zero; the definition of the index immediately implies the statement, and (2) the index of a product of functions is equal to the sum of the indices of the factors. The index of a quotient is equal to the difference of the indices of the dividend and the divisor. The chapter also discusses the Riemann problem for a simply-connected domain, determination of sectionally analytic function in accordance with given jump, the canonical function of the homogeneous problem, the Riemann problem for the semi-plane, and Riemann boundary value problem with shift.
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