analytic function

The Solution of the Homogeneous Boundary Value Problem of Riemann with Infinite Index of Logarithmic Order on the Beam by a New Method

Authors: Salimov R. B., Khasanova E. N.

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. 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.

Asymptotic Values of Analytic Functions Connected with a Prime End of a Domain

Authors: Ganenkova E. G.

In 1954 M. Heins proved that for any analytic set A, containing the infinity, there exists an entire function with asymptotic set A. In the article we prove the following analog of Heins's theorem: for a multi-connected planar domain D with an isolated boundary fragment, an analytic set A, ∞∈A, and a prime end of D with impression p there exists an analytic in D function f such that A is the set of asymptotic values of f connected with p.