Cite this article as:

Salimov R. B. About New Approach to Solution of Riemann’s Boundary Value Problem with Condition on the Half-line in Case of Infinite Index. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2016, vol. 16, iss. 1, pp. 29-33. DOI: https://doi.org/10.18500/1816-9791-2016-16-1-29-33

# About New Approach to Solution of Riemann’s Boundary Value Problem with Condition on the Half-line in Case of Infinite Index

To solve a homogeneous Riemann boundary value problem with infinite index and condition on the half-line we propose a new approach based on the reduction of the considered problem to the corresponding task with the condition on the real axis and finite index. It is required to define a function Φ(z), analytic and bounded in the complex plane z, cut down on positive real semi-axis L^{+} , if the edge condition Φ^{+}(t) = G(t)Φ^{−}(t), t ∈ L^{+} is fulfilled, where Φ^{+}(t), Φ^{−} (t) are limit values of the function Φ(z), as z → tcorrespondingly on the left and on the right,G(t) is a given function, for which argument argG(t) = ν^{−} t^{ρ} +ν(t), t ∈ L^{+ }holds, here ν^{−} , ρ are given numbers, ν^{−} > 0, 1/2< ρ < 1, and ln|G(t)|, ν(t) are functions which satisfy the Holder condition. It is admitted that G(t) = 1 at t ∈ (−∞,0). The functions E^{+} (z) = e^{(α+iβ)zρ} , 0 ≤ argz ≤ π, E^{−} (z) = e^{(α−iβ)zρ} , −π ≤ argz ≤ 0 are used to avoid infinite gap of the argG(t), by the selection of real numbers α, β.

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