Skip to main content
SpringerLink
Log in

On the modulus of continuity of solid derivatives of a Cauchy-type integral

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We establish sufficient conditions for the existence of solid derivatives of a continuous extension of a Cauchy-type integral onto the closure of a domain and obtain an estimate for the moduli of continuity of these derivatives. We prove that the Newton-Leibniz formula is true for certain classes of Jordan rectifiable curves.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. O. F. Gerus, “Estimate of the modulus of continuity of a Cauchy-type integral in a domain and on its boundary,” Ukr. Mat. Zh., 48, No. 10, 1321–1328 (1996).

    Article  MathSciNet  Google Scholar 

  2. O. F. Gerus, “On the modulus of continuity of a Cauchy-type integral in a closed domain,” in: Abstracts of the International Conference. “Functional Spaces, Theory of Approximations, and Nonlinear Analysis” (Moscow, April–May 1995) [in Russian], Steklov Mathematical Institute, Moscow (1995), pp. 94–95.

    Google Scholar 

  3. O. Lehto and K. I. Virtanen, Quasikonforme Abbildingen, Springer-Verlag, Berlin, 32, No. 995 (1965).

    Google Scholar 

  4. F. D. Gakhov, Boundary-Value Problems [in Russian], Fizmatgiz Moscow (1963).

  5. P. M. Tamrazov, Smoothness and Polynomial Approximations [in Russian], Naukova Dumka, Kiev (1975).

    MATH  Google Scholar 

  6. S. M. Nikol’skii, Course in Mathematical Analysis [in Russian], Vol. 1, Nauka, Moscow (1973).

    Google Scholar 

  7. T. S. Salimov, “A direct estimate for a singular Cauchy integral along a closed curve,” Nauch. Tr. MV SSO Azer.SSR., Ser. Fiz.Mat. Nauk, No. 5, 59–75 (1979).

  8. J. L. Walsh and W. E. Swell, “Sufficient conditions for various degrees of approximation by polynomials,” Duke Math. J., 6, No. 3, 658–705 (1940).

    Article  MathSciNet  Google Scholar 

  9. P. M. Tamrazov, “Contour and solid structural properties of holomorphic functions of a complex variable,” Usp. Mat. Nauk, 28, No. 1, 131–161 (1973).

    MATH  MathSciNet  Google Scholar 

  10. V. I. Belyi, “Conformal mappings and approximation of analytic functions in domains with quasiconformal boundary,” Mat. Sb., 102, No. 3, 331–361 (1977).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    O. F. Gerus

Authors
  1. O. F. Gerus
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 476–484, April, 1998.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gerus, O.F. On the modulus of continuity of solid derivatives of a Cauchy-type integral. Ukr Math J 50, 539–549 (1998). https://doi.org/10.1007/BF02487386

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02487386

Keywords

Search

Navigation