Skip to main content
SpringerLink
Log in

Hyperanalytic Functions and Their Applications

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

A survey on the theory of hyperanalytic functions in the sense of Douglis is presented. Some applications of hyperanalytic functions are also given, in particular, to the description of solutions of elliptic systems in the plane. Special attention is devoted to some systems arising in the plane elasticity theory and linearized Stokes system of hydromechanics.

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.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. I. Vekua, Generalized Analytic Functions [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  2. A. Douglis, “Function theoretic approach to elliptic systems of equations in two variables,” Comm. Pure Appl. Math., 6, 259–289 (1953).

    MATH  MathSciNet  Google Scholar 

  3. J. Horvath, “A generalization of the Cauchy-Riemann equations,” Contrib. Differ. Equations, 1, 39–57 (1961).

    Google Scholar 

  4. B. Bojarski, “Theory of generalized analytic vectors,” Ann. Polon. Math., 17, No.3, 281–320 (1966).

    MathSciNet  Google Scholar 

  5. R. Gilbert and J. Buchanan, First Order Elliptic Systems: A Function Theoretic Approach, Academic Press, New York (1983).

    Google Scholar 

  6. R. Gilbert and G. Hile, “Generalized hypercomplex function theory,” Trans. Amer. Math. Soc., 195, 1–29 (1974).

    MathSciNet  Google Scholar 

  7. G. Hile, “Elliptic systems in the plane with order term and constant coefficients,” Comm. Pure Appl. Math., 3, 949–977 (1978).

    MATH  MathSciNet  Google Scholar 

  8. H. Begehr and R. Gilbert, “Pseudohyperanalytic functions,” Complex Variables Theory Appl., 9, 343–357 (1988).

    MathSciNet  Google Scholar 

  9. H. Begehr and R. Gilbert, Transformations, Transmutations, and Kernel Functions. I, II, Longman, Harlow (1992–1993).

    Google Scholar 

  10. I. Vekua, New Methods of Solving Elliptic Equations [in Russian], Gostekhizdat, Moscow (1948).

    Google Scholar 

  11. A. Bitsadze, Boundary-Value Problems for Second-Order Elliptic Equations [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  12. A. Soldatov, “Elliptic system of high order,” Differ. Uravn., 25, No.1, 136–142 (1989).

    MATH  MathSciNet  Google Scholar 

  13. R. Ieh, “Hyperholomorphic functions and higher order partial differential equations in the plane,” Pacif. J. Math., 142, No.2, 379–399 (1990).

    Google Scholar 

  14. N. Zhura, “On the general solution of Leray-Douglis-Nirenberg systems with constant coefficients in the plane,” Dokl. Ross. Akad. Nauk, 331, No.5, 546–549 (1993).

    MATH  Google Scholar 

  15. N. Zhura, “General boundary-value problem for elliptic systems in the sense of Douglis-Nirenberg in domain with piecewise-smooth boundary,” Dokl. Ross. Akad. Nauk, 330, No.2, 151–154 (1993).

    MATH  Google Scholar 

  16. F. Gantmakher, Theory of Matrices [Russian translation], Nauka, Moscow (1988).

    Google Scholar 

  17. A. Maltsev, Elements of Linear Algebra [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  18. M. Keldysh, Mathematics. Collected Papers [in Russian], Nauka, Moscow (1985).

    Google Scholar 

  19. A. Bitsadze, Elements of the Theory of Analytic Functions [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  20. M. Balk, “Polyanalytic functions and their applications,” In: Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Fundamental Directions [in Russian], 85, All-Russian Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1991).

    Google Scholar 

  21. G. Khimshiashvili, “Signature formulae for topological invariants,” Proc. A. Razmadze Math. Inst., 125, 1–121 (2001).

    MATH  MathSciNet  Google Scholar 

  22. G. Khimshiashvili, “Global geometric aspects of Riemann-Hilbert problems,” Mem. Differential Equations Math. Phys., 27, 1–114 (2002).

    MATH  MathSciNet  Google Scholar 

  23. A. Soldatov, “Function theoretic method in boundary-value problems in the plane. I: Smooth case,” Izv. Akad. Nauk SSSR, 55, No.5, 1070–1100 (1991).

    MATH  Google Scholar 

  24. A. Soldatov, “Function theoretic method in boundary-value problems in the plane. II: Piecewise smooth case,” Izv. Akad. Nauk SSSR, 56, No.3, 566–604 (1992).

    MATH  MathSciNet  Google Scholar 

  25. N. Muskhelishvili, Singular Integral Equations [in Russian], Nauka, Moscow (1968).

    Google Scholar 

  26. A. Bitsadze, “On the unique solvability of Dirichlet problem for elliptic equations with partial derivatives,” Usp. Mat. Nauk, 3, 153–154 (1948).

    Google Scholar 

  27. M. Vishik, “On strongly elliptic systems of differential equations,” Mat. Sb., 29, 615–676 (1951).

    MATH  Google Scholar 

  28. A. Soldatov and S. Mitin, “On one class of strongly elliptic systems,” Differ. Uravn., 33, No.8, 1118–1122 (1997).

    MathSciNet  Google Scholar 

  29. Hua Loo Keng, Lin Wei, and Wu Ci-Quian, Second Order Systems of Partial Differential Equations in the Plane, Pitman, Boston-Melburn (1985).

    Google Scholar 

  30. G. Khimshiashvili, “Elliptic boundary-value problems for Cauchy-Riemann systems,” In: Proc. Symp. Anal. Numer. Meth. Quatrn. Cliff. Anal. (Seiffen, 1996), TU Bergakademie Freiberg, Freiberg (1996), pp. 77–83.

    Google Scholar 

  31. G. Khimshiashvili, Towards Homotopy Classification of Elliptic Boundary-Value Problems, Preprint 97-01, TU Bergakademie Freiberg, Freiberg (1996).

    Google Scholar 

  32. K. Chernykh, Introduction to Anisotropic Elasticity [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  33. V. Kupradze, Methods of Potential Theory in Elasticity Theory [in Russian], Fizmatgiz, Moscow (1963).

    Google Scholar 

  34. N. Muskhelishvili, Some Fundamental Problems of the Mathematical Theory of Elasticity [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  35. G. Lekhnitsky, Elasticity Theory for Anisotropic Bodies [in Russian], GITTL, Moscow (1950).

    Google Scholar 

  36. A. Soldatov, “Mixed problem of plane elasticity theory,” Differ. Uravn., 23, No.1, 161–168 (1987).

    MathSciNet  Google Scholar 

  37. S. Mitin, “On representation of solutions of anisotropic elasticity theory,” Differ. Uravn., 34, No.1, 94–100 (1998).

    MATH  MathSciNet  Google Scholar 

  38. H. Begehr and Wei Lin, “A mixed-contact problem in orthotropic elasticity,” In: Partial Differential Equations with Real Analysis, Longman Scientific and Technical (1992), pp. 219–239.

  39. R. Gilbert and Wei Lin, “Function theoretical solutions to problems of orthotropic elasticity,” J. Elasticity, 15, 143–154 (1985).

    Article  MathSciNet  Google Scholar 

  40. R. Gilbert and W. Wendland, “Analytic, generalized, hyperanalytic function theory and an application to elasticity,” Proc. Roy. Soc. Edinburgh, 73A, 317–371 (1975).

    MathSciNet  Google Scholar 

  41. A. Soldatov, “To elliptic theory for domains with piecewise smooth boundary on the plane,” In: Proc. Congr. ISAAC'97, 3–7 June 1997, University of Delaware, Newark, USA, Partial Diff. and Integr. Equations, Kluwer Acad. Publ. (1999), pp. 177–186.

  42. A. Douglis and L. Nirenberg, “Interior estimates for elliptic systems of partial differential equations,” Comm. Pure Appl. Math., 8, 503–538 (1955).

    MathSciNet  Google Scholar 

  43. I. Petrovsky, Systems of Partial Differential Equations. Collected Papers [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  44. N. Kochin, I. Kibel, and N. Rose, Theoretical Hydromechanics [in Russian], GIFMI, Moscow (1963).

    Google Scholar 

  45. N. Zhura, “Nonlocal boundary-value problem for a stationary linearized two-dimensional Stokes system,” Dokl. Akad. Nauk SSSR, 311, No.3, 524–527 (1990).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Novgorod State University, Russia

    A. Soldatov

Authors
  1. A. Soldatov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Soldatov, A. Hyperanalytic Functions and Their Applications. J Math Sci 132, 827–882 (2006). https://doi.org/10.1007/s10958-006-0023-z

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-006-0023-z

Keywords

Search

Navigation