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The Riemann–Hilbert Boundary Value Problem with a Countable Set of Coefficient Discontinuities and Two-side Curling at Infinity of the Order Less Than 1/2

  • R. B. Salimov
  • P. L. Shabalin
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)

Abstract

The Riemann–Hilbert boundary value problem is one of the oldest boundary value problems of theory of analytic functions.Its complete solution (for the case of a finite index and continuous coefficients) was given by Hilbert in 1905.I n the present paper we study the inhomogeneous Riemann–Hilbert boundary value problem in the upper half of complex plane with strong singularities of boundary data.W e obtain general solution for the case where coefficients of the problem have a countable set of finite discontinuity points and two-side curling of order less than 1/2 at the infinity point.W e investigate also the solvability conditions.

Keywords

Riemann–Hilbert boundary value problem infinite index curlings entire functions 

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References

  1. 1.
    V. Volterra, Sopra alkune condizioni caratteristische per functioni di variabile com-plessa. Ann. Mat. (2), T. 11, (1883).Google Scholar
  2. 2.
    I.E. Sandrigaylo, On the Hilbert boundary value problem with infinite index for half-plane. Izvestiya akademii nayk Belorusskoy SSSR, Seriya Fis.-Mat. Nauk, No. 6, (1974), 16-23.Google Scholar
  3. 3.
    P.Yu. Alekna, The Hilbert boundary value problem with infinite index of logarithmic order for half-plane. Litovskiy Matematicheskiy Sbornik. No. 1, (1977), 5-12.MathSciNetGoogle Scholar
  4. 4.
    N.I. Musheleshvili, Singular integral equations. Nauka, Moscow, 1968.Google Scholar
  5. 5.
    N.V. Govorov, On the Riemann boundary value problem with infinite index. Dokl. AN USSR 154, No. 6, (1964), 1247-1249.MathSciNetGoogle Scholar
  6. 6.
    N.V. Govorov, Inhomogeneous Riemann boundary value problem with infinite index. Dokl. AN USSR 159, No. 5, (1964), 961-964.MathSciNetGoogle Scholar
  7. 7.
    N.V. Govorov, The Riemann boundary value problem with infinite index. Nauka, Moscow, 1986.zbMATHGoogle Scholar
  8. 8.
    R.B. Salimov and P.L. Shabalin, Method of regularizing multiplier for solving of the uniform Hilbert problem with infinite index. Izvestiya vuzov, Matematika. No. 4, (2001), 76-79.MathSciNetGoogle Scholar
  9. 9.
    F.D. Gahov, Boundary value problems. Nauka, Moscow, 1977.Google Scholar
  10. 10.
    R.B. Salimov and P.L. Shabalin, On solving of the Hilbert problem with infinite index. Mathematical notes. 73 (5), (2003), 724-734.MathSciNetCrossRefGoogle Scholar
  11. 11.
    A.I. Markushevich, Theory of analytic functions. V. 2. Nauka, Moscow, 2008.Google Scholar
  12. 12.
    R.B. Salimov and P.L. Shabalin, The Hilbert problem: the case of infinitely many discontinuity points of coefficients. Siberian Mathematical Journal. Vol. 49, No. 4, (2008), 898-915.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    B. Ya. Levin, Distribution of the roots of entire functions Gostehizdat, Moscow, 1956.Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  • R. B. Salimov
    • 1
  • P. L. Shabalin
    • 1
  1. 1.Kazan State University of Architecture and EngineeringKazanRussian Federation

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