The solution of the homogeneous boundary value problem of Riemann with infinite index of logarithmic order on the beam by a new method

R. B. Salimov ; E. N. Khasanova

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 2, pp. 160-171

Résumé

In this paper we consider the homogeneous Riemann boundary value problem with infinite index of logarithmic order and boundary condition on the unlimited ray. This ray goes by the positive real axis and has a vertex at $(1,0)$. We solve the problem for analytic function with the cut along the ray. The value of the function at any point of the left bank equals the product of the coefficient and the value of the function at the corresponding point of the right bank of the cut. Let the modulus of the coefficient meet the Holder condition at each point of the ray. Let the argument of the coefficient meet the Holder condition at each point of the finite part of the ray. Let the argument of the coefficient come arbitrarily close to infinity with the logarithmic order. We obtain the formula of analytic on the upper half-plane function. The imaginary component of this function is infinitely large when the point of the ray tends to infinity. The order of the imaginary component of this function at infinity is equal to the order of the coefficient. Then we obtain the formula of analytic on the lower half-plane function. These two functions permit us to eliminate the infinite discontinuity of the argument of the coefficient of the boundary condition exactly as in the case of finite discontinuities of coefficient. The problem with the condition on the ray is reduced to the problem with the boundary condition on the real axis. We solve the latter problem by the Gakhov method. The solution depends on an arbitrary entire function with zero order. The modulus of the solution meet one and only one condition.