Investigation Riemann–Hilbert boundary value problem with infinite index on circle

We consider the Riemann–Hilbert boundary value problem of analytic function theory with infinite index and the boundary condition on the circumference. The boundary condition coefficients are Holder’s continuous everywhere except one particular point where the coefficients have discontinuity of second kind (power order with the index is less than one). In this formulation the problem with infinite index is considered for the first time. As the result of the research, we obtained the formulas of the general solution of the homogeneous problem, investigated the existence and uniqueness of solutions, described the set of solutions in the case of non-uniqueness. In the study of solutions we applied the theory of entire functions and the geometrical theory of functions of complex variables.