Advertisement

Journal of Mathematical Chemistry

, Volume 38, Issue 4, pp 629–635 | Cite as

Properties of the Mittag-Leffler Relaxation Function

  • Mário N. Berberan-Santos
Article

Abstract

The Mittag-Leffler relaxation function, E α (−x), with 0 ≤ α ≤ 1, which arises in the description of complex relaxation processes, is studied. A relation that gives the relaxation function in terms of two Mittag-Leffler functions with positive arguments is obtained, and from it a new form of the inverse Laplace transform of E α (−x) is derived and used to obtain a new integral representation of this function, its asymptotic behaviour and a new recurrence relation. It is also shown that the fastest initial decay of E α (−x) occurs for α =1/2, a result that displays the peculiar nature of the interpolation made by the Mittag-Leffler relaxation function between a pure exponential and a hyperbolic function.

Keywords

Mittag-Leffler function Laplace transform relaxation kinetics 

AMS (MOS) classification

33E12 Mittag-Leffler functions and generalizations 44A10 Laplace transform 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Mittag-Leffler, G.M. 1903C. R. Acad. Sci. Paris (Ser. II)13670Google Scholar
  2. 2.
    Mittag-Leffler, G.M. 1903C. R. Acad. Sci. Paris (Ser. II)137554Google Scholar
  3. 3.
    Érdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G. 1955Higher Transcendental FunctionsMcGraw-HillNew YorkVol. IIIGoogle Scholar
  4. 4.
    Miller, K.S. 1993Integral Transform Spec. Funct.141Google Scholar
  5. 5.
    Gorenflo, R., Loutchko, J., Luchko, Y. 2002Fract Calc. Appl. Anal.5491Google Scholar
  6. 6.
    Glöckle, W.G., Nonnenmacher, T.F. 1995Biophys J.6846PubMedGoogle Scholar
  7. 7.
    Hilfer, R., Anton, L. 1995Phys Rev E51R848CrossRefGoogle Scholar
  8. 8.
    Weron, K., Kotulski, M. 1996Physica A232180CrossRefGoogle Scholar
  9. 9.
    Weron, K., Klauzer, A. 2000Ferroelectrics23659Google Scholar
  10. 10.
    Metzler, R., Klafter, J. 2000Phys. Rep.3391CrossRefGoogle Scholar
  11. 11.
    Hilfer, R. 2002Chem. Phys.284399CrossRefGoogle Scholar
  12. 12.
    Saxena, R.K., Mathai, A.M., Haubold, H.J. 2002Astrophys. Space Sci.282281CrossRefGoogle Scholar
  13. 13.
    Metzler, R., Klafter, J. 2002J. Non-Cryst. Solids30581CrossRefGoogle Scholar
  14. 14.
    Metzler, R., Klafter, J. 2003Biophys. J.852776PubMedGoogle Scholar
  15. 15.
    Sjögren, L. 2003Physica A32281CrossRefGoogle Scholar
  16. 16.
    Crothers, D.S.F., Holland, D., Kalmykov, Y.P., Coffey, W.T. 2004J. Mol. Liquids11427CrossRefGoogle Scholar
  17. 17.
    Berberan-Santos, M.N. 2005J. Math. Chem.38165CrossRefGoogle Scholar
  18. 18.
    Berberan-Santos, M.N. 2005J. Math. Chem.38265CrossRefGoogle Scholar
  19. 19.
    Pollard, H. 1948Bull. Am. Math. Soc.541115Google Scholar
  20. 20.
    Feller, W. 1949Trans. Am Math. Soc.6798Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Centro de Química-Física MolecularInstituto Superior TécnicoLisboaPortugal

Personalised recommendations