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In this paper, we describe the inhomogeneous Hilbert boundary-value problem of the theory of analytic functions with an infinite index and a boundary condition for a half-plane. The coefficients of the boundary condition are Holder-continuous everywhere except for a finite number of singular points at which the argument of the coefficient function has second-type discontinuities (of a power order with exponent

In the unit disk, we consider the Hilbert boundary value problem. Its coefficient is assumed to be Holder continuous everywhere on the unit circle excluding a finite set of points. At these points, the argument of the coefficient has non-removable discontinuity of logarithmic order. We obtain formulas for the general solution and fully describe the solvability of the problem in the class of functions, analytic and bounded in the unit disc. Our technique is based on the theory of entire functions of zero proximate order and the geometric theory of functions of a complex variable. We apply the obtained results to the study of the solvability of a boundary value problem for generalized analytic function.

We examine the existence of an univalent mapping of the upper half-plane with a fixed set of preimages of vertices to a polygonal domain.

We study a boundary-value problem for a generalized biaxisymmetric Helmholtz equation. Boundary conditions in this problem depend on equation parameters. By the method of separation of variables, using the Fourier-Bessel series expansion and the Hankel transform, we prove the unique solvability of the problem and establish explicit formulas for its solution.

We consider the so called Hilbert boundary value problem with infinite index in the unit disk. Its coefficient is assumed to be H¨older-continuous everywhere on the unit circle excluding a finite set of points. At these points its argument has power discontinuities of orders less than one. We obtain formulas for the general solution and describe completely the solvability picture in a special functional class. Our technique is based on the theory of entire functions and the geometric theory of functions.

We consider Riemann–Hilbert boundary value problem with infinite index in unit disk. Its coefficient is Holder-continuous everywhere on the unit circle excluding a finite set of points, where its argument has power discontinuities of order less one. The present article is the first research of this version of Hilbert boundary-value problem with infinite index. We obtain formulas for its general solution, investigate existence ad uniqueness of solutions, and describe the set of solutions in the case of non-uniqueness. Our technique is based on theory of entire functions and geometric theory of functions.

We consider a homogeneous Riemann–Hilbert boundary-value problem for upper halfplane in the situation where its coefficients have countable set of discontinuities of jump type and two-side curling at infinity of a logarithmic order. We obtain general solution and describe completely its solvability in a special class of functions for the case where the index of the problem has power singularity of a logarithmic order.

We consider the Hilbert boundary value problem for the upper half-plane in the situation where its coefficients have countably many discontinuities of jump type and two-side curling at infinity. We obtain the general solution and describe completely its solvability in a special class of functions for the case where the index of the problem has power singularity of order lesser than 1.

We consider the Hilbert boundary-value problem for the upper half-plane with a countable set of discontinuity points of coefficient of the boundary condition and with a two-side curling at infinity of a logarithmic order. We obtain formulas for the general solution to the problem.

We study a homogeneous Riemann—Hilbert boundary-value problem in the upper half of the complex plane with a countable set of coefficient discontinuities and two-side curling at infinity. We obtain a general solution in the case when the problem index has a power singularity of order ρ, 1/2 ≤ ρ < 1, and study the solvability conditions.

We consider the Hilbert boundary-value problem with a finite index for the case, when the coefficients in the boundary condition have two infinite sequences of discontinuity points of the first kind. We obtain a formula for the general solution and study the solvability issues.

In this paper we generalize the Schwarz—Christoffel formula for a conformal mapping of a half-plane onto a certain polygon with infinitely many vertices. We assume that the interior angles of the polygon (at unknown vertices) and the points of the real axis that are images of unknown vertices under the mentioned mapping are given.

We obtain a solution to the Hilbert boundary value problem in the theory of analytic functions on the half-plane in the case that the coefficients of the boundary condition have countably many discontinuity points of the first kind. We elaborate the two substantially different situations: the series consisting of the jumps of the argument of the coefficient function and the increments of its continuous part converges and this series diverges. Accordingly, Hilbert problems with finite and infinite indices result. We derive formulas for the general solution and investigate the pictures of solvability of these problems.

In this paper, we obtain a generalization of the method of regularizing multipliers for the solution of the Hilbert boundary-value problem with finite index in the theory of analytic functions to the case of an infinite power-behaved index. This method is used to obtain a general solution of the homogeneous Hilbert problem for the half-plane, a solution that depends on the existence and the number of entire functions possessing mirror symmetry with respect to the real axis and satisfying some additional constraints related to the singularity characteristic of the index. To solve of the inhomogeneous problem, we essentially use a specially constructed solution of the homogeneous problem whereby we reduce the boundary condition of the Hilbert problem to a Dirichlet problem.

The Riemann–Hilbert boundary value problem is one of the oldest boundary value problems of theory of analytic functions.Its complete solution (for the case of a finite index and continuous coefficients) was given by Hilbert in 1905.I n the present paper we study the inhomogeneous Riemann–Hilbert boundary value problem in the upper half of complex plane with strong singularities of boundary data.W e obtain general solution for the case where coefficients of the problem have a countable set of finite discontinuity points and two-side curling of order less than 1/2 at the infinity point.W e investigate also the solvability conditions.