PHYSICS AND MATHEMATICS

Solution of homogeneous Riemann boundary-value problem with a condition on a real axis and an infinite index of logarithmic order with the new method

R. B. Salimov, T. Yu. Gorskaya

Kazan State University of Architecture and Engineering

Abstract: We consider the homogeneous Riemann boundary-value problem with the boundary condition on the real axis for a function analytic in the complex plane except for points of the real axis. In the boundary condition, the limit value of the desired analytic function at any point on the real axis when approaching from above is represented as the product of the value of a given function called the coefficient, and the limit value of the function at the specified point at the bottom approaching. We assume that the coefficient modulus satisfies the Hцlder condition everywhere on the real axis, including the infinitely distant point, and the coefficient argument satisfies the Hцlder condition on any finite part of the axis and increases indefinitely as the degree of logarithm coordinates of the axis point with unlimited distance from the origin. The authors derived the formula that determines an analytic function in the upper half-plane the imaginary part of which as the coordinate of the axis point tends to positive infinity is infinitely large of the same order as the argument of the coefficient of the boundary condition. Next, the corresponding function is constructed in the lower half-plane, then analytical functions are introduced the imaginary parts of which turn into the infinity of the same order as the argument of the coefficient of the boundary condition when the points of the negative real axis are removed to infinity. The use of these functions allows us to eliminate the infinite gap of the argument of the coefficient of the boundary condition in the same way as it is done in the case of finite discontinuities of this coefficient. Based on techniques similar to those used by F.D. Gakhov, the problem is reduced to a problem with a boundary condition on the real axis and a finite index. Gakhov’s method is used to solve the last problem. The solution found depends on an arbitrary entire function of order zero, the modulus of which is subject to additional conditions, while in the case of a finite index the solution of the problem depends on an arbitrary polynomial of degree not higher than the index of the problem.

Keywords: Riemann's boundary value problem, analytic function, infinite index, logarithmic order.

DOI: 10.23670/IRJ.2019.85.7.001