Advertisement

Boundary value formula for the Cauchy integral on elliptic curve

  • Mukhiddin I. Muminov
  • A. H. M. Murid
Article

Abstract

In this paper we consider a Cauchy integral on elliptic curve \(\Gamma \) parameterized by equation \(\eta (t)=a \cos t+ib \sin t, a,b>0\). We drive a formula for the boundary values of the Cauchy integral when integral function is Hölder continuous on \(\Gamma \). Hence we extend Hilbert transform to elliptic curves.

Keywords

Cauchy integral Boundary value Hilbert transform 

Mathematics Subject Classification

Primary 30E25 30C75 Secondary 45E05 

Notes

Acknowledgements

We thank unknown referee for a careful reading of the first manuscript and useful comments.

References

  1. 1.
    Gakhov, F.D.: Boundary Value Problems. Pergamon, Oxford (1966)CrossRefGoogle Scholar
  2. 2.
    Pandey, J.N.: The Hilbert Transform of Schwartz Distributions and Applications. Wiley, New York (1996)zbMATHGoogle Scholar
  3. 3.
    Hahn, S.L.: Hilbert Transforms in Signal Processing. Artech House, Boston (1996)zbMATHGoogle Scholar
  4. 4.
    King, F.W.: Hilbert Transforms, vol. 1. Cambridge University Press, Cambridge (2009)zbMATHGoogle Scholar
  5. 5.
    Maz’ya, V.G., Nikol’skii, S.M.: Analysis IV. Encyclopedia of Mathematical Sciences, vol. 27. Springer, Berlin (1991)Google Scholar
  6. 6.
    Begehr, H., Mohammed, A.: The Schwarz problem for analytic functions in torus related domains. Appl. Anal. 85, 1079–1101 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kufner, A., Kadlec, J.: Fourier Series, ILIFFE Books, Academia. Prague, London (1971)zbMATHGoogle Scholar
  8. 8.
    Henrici, P.: Applied and Computational Complex Analysis, vol. 3. Wiley, New York (1986)zbMATHGoogle Scholar
  9. 9.
    Wegmann, R.: Methods for numerical conformalmappings. In: Kuhnau, R. (ed.) Handbook of Complex Analysis: Geometric Function Theory, vol. 2, p. 351477. Elsevier B.V, Amsterdam (2005)Google Scholar
  10. 10.
    Dwight, H.B.: Tables of Integrals and Other Mathematical Data. The MacMillan Company, New York (1961)zbMATHGoogle Scholar
  11. 11.
    Wen, G.-C.: Conformal Mappings and Boundary Value Problems. AMS, Providence (1992)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, Faculty of ScienceUniversiti Teknologi MalaysiaJohor BahruMalaysia
  2. 2.UTM Centre for Industrial and Applied Mathematics (UTM-CIAM), Institute for Scientific and Industrial Research (ISIR)Universiti Teknologi MalaysiaJohor BahruMalaysia

Personalised recommendations