Computational Methods and Function Theory

, Volume 6, Issue 1, pp 165–182 | Cite as

Conjugate Harmonic Functions in Euclidean Space: a Spherical Approach

  • Fred Brackx
  • Hennie De Schepper


Given a harmonic function U in a domain Ω in Euclidean space, the problem of finding a harmonic conjugate V, generalizing the well-known case of the complex plane, was considered in [4] in the framework of Clifford analysis. By the nature of the given construction, which is genuinely cartesian, this approach lead to geometric constraints on the domain Ω. In this paper we consider the problem in a larger class of domains, by a spherical approach. Starting from a real-valued function u, and singling out the radial direction, we explicitly construct a harmonic function of the form w = e r v, with v ∈ span(e θ 1,…, e θ m−1), such that u+w is monogenic, i.e. a null solution of the Dirac operator. As an illustration, the construction is applied to important classes of homogeneous monogenic polynomials and functions. Finally, it is investigated to which extent the approach also applies to the complex plane case.


Clifford analysis 

2000 MSC

46F10 30G35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. Brackx, R. Delanghe, On harmonic potential fields and the structure of monogenic functions, Zeit. Anal. Anw. 22 (2003), 261–273.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    F. Brackx, B. De Knock, H. De Schepper and D. Eelbode, Some insights on the interplay between the Hilbert transform and conjugate harmonic functions, in: T. Tsimos et al. (eds), Proceedings of the ICNAAM 2005 Conference, Wiley Verlag — VCH, Weinheim, 2005, 915–917.Google Scholar
  3. 3.
    F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Pitman Publishers, Boston-London-Melbourne, 1982.MATHGoogle Scholar
  4. 4.
    F. Brackx, R. Delanghe and F. Sommen, On conjugate harmonic functions in Euclidean space, Math. Meth. Appl. Sci. 25 (2002), 1553–1562.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    F. Brackx, H. De Schepper and D. Eelbode, A new Hilbert transform on the unit sphere in ℝm, to appear in Complex Variables and Elliptic Equations.Google Scholar
  6. 6.
    F. Brackx and N. Van Acker, A conjugate Poisson kernel in Euclidean space, Simon Stevin 67 (1993), 3–14.MATHGoogle Scholar
  7. 7.
    D. Constales, A conjugate harmonic to the Poisson kernel in the unit ball of ℝn, Simon Stevin 62 (1988), 289–291.MathSciNetMATHGoogle Scholar
  8. 8.
    R. Delanghe, F. Sommen and V. Souček, Clifford Algebra and Spinor-Valued Functions — A Function Theory for the Dirac Operator, Kluwer Academic Publishers, Dordrecht, 1992.MATHCrossRefGoogle Scholar
  9. 9.
    J. Gilbert and M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambrigde University Press, Cambridge, 1991.MATHCrossRefGoogle Scholar
  10. 10.
    K. Gürlebeck and W. Sprößig, Quaternionic and Clifford Calculus for Physicists and Engineers, Mathematical Methods in Practice, Wiley, Chichester, 1997.MATHGoogle Scholar
  11. 11.
    V. V. Kravchenko and M. Shapiro, Integral Representations for Spatial Models of Mathematical Physics, Pitman Research Notes in Mathematics Series 351, Longman, Harlow, 1996.MATHGoogle Scholar
  12. 12.
    M. Mitrea, Clifford Wavelets, Singular Integrals and Hardy Spaces, Lecture Notes in Mathematics 1575, Springer-Verlag, Berlin, 1994.Google Scholar
  13. 13.
    C. A. Nolder, Conjugate harmonic functions and Clifford algebras, J. Math. Anal. Appl. 302 (1) (2005), 137–142.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    T. Qian, Th. Hempfling, A. McIntosh and F. Sommen (eds.), Advances in Analysis and Geometry: New Developments Using Clifford Algebras, Birkhäuser Verlag, Basel-Boston-Berlin, 2004.MATHGoogle Scholar
  15. 15.
    M. Riesz, Clifford Numbers and Spinors, Lecture Series vol.38, Institute for Physical Science and Technology, Maryland, 1958.Google Scholar
  16. 16.
    R. Rocha-Chávez, M. Shapiro and L. Tovar Sánchez, On the Hilbert operator for α-hyperholomorphic function theory in ℝ2, Complex Var. Theory Appl. 43 (1) (2000), 1–28.MATHCrossRefGoogle Scholar
  17. 17.
    J. Ryan, Basic Clifford analysis, Cubo Math. Educ. 2 (2000), 226–256.MATHGoogle Scholar
  18. 18.
    J. Ryan and D. Struppa (eds.), Dirac Operators in Analysis, Addison Wesley Longman Ltd., Harlow, 1998.MATHGoogle Scholar
  19. 19.
    M. Shapiro, On the conjugate harmonic functions of M. Riesz-E. Stein-G. Weiss, in: Topics in complex analysis, differential geometry and mathematical physics, St. Konstantin, 1996, 8–32, World Sci. Publishing, River Edge, NJ, 1997.Google Scholar
  20. 20.
    E. Stein and G. Weiss, On the theory of harmonic functions of several variables, Part I: The theory of H p spaces, Acta Math. 103 (1960), 25–62.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Heldermann  Verlag 2006

Authors and Affiliations

  1. 1.Clifford Research Group, Department of Mathematical Analysis, Faculty of EngineeringGhent UniversityGentBelgium

Personalised recommendations