﻿ About New Approach to Solution of Riemann’s Boundary Value Problem with Condition on the Half-line in Case of Infinite Index | Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics

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

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Russian
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517.54

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

Abstract:

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 tρ +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.

References

1. Salimov R. B. , Karabasheva E. N. The new approach to solving the Riemann boundary value problem with infinite index. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 2, pp. 155–165 (in Russian).

2. Gakhov F. D. Boundary value problems. Moscow, Nauka, 1977, 640 p. (in Russian).

3. Markushevich A. I. The theory of analytic functions, in 2 vol. Vol. 2. Moscow, Nauka, 1968, 624 p. (in Russian).

4. Govorov N. V. Riemann’s boundary problem with infinite index. Moscow, Nauka, 1986, 239 p. (in Russian).

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