Boundary value problems of analytic function theory Problems of
finding an analytic function in a certain domain from a given
relation between the boundary values of its real and its
imaginary part. This problem was first posed by B. Riemann in
1857 . D. Hilbert  studied the boundary value problem formulated
as follows (the Riemann–Hilbert problem): To find the function
that is analytic in a simply-connected domain bounded by a
contour and that is continuous in , from the boundary condition
where and are given real continuous functions on . Hilbert
initially reduced this problem to a singular integral equation
in order to give an example of the application of such an
equation. The problem (1) may be reduced to a successive
solution of two Dirichlet problems. A complete study of the
problem by this method may be found in . The problem arrived at
by H. Poincaré  in developing the mathematical theory of tide
resembles problem (1). Poincaré's problem consists in
determining a harmonic function in a domain from th