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Numerical evaluation of Mittag-Leffler functions

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Abstract

The Mittag-Leffler function is computed via a quadrature approximation of a contour integral representation. We compare results for parabolic and hyperbolic contours, and give special attention to evaluation on the real line. The main point of difference with respect to similar approaches from the literature is the way that poles in the integrand are handled. Rational approximation of the Mittag-Leffler function on the negative real axis is also discussed.

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References

  1. Filip, S.I., Nakatsukasa, Y., Trefethen, L.N., Beckermann, B.: Rational minimax approximation via adaptive barycentric representations. SIAM J. Sci. Comput. 40(4), A2427-2455 (2018). https://doi.org/10.1137/17M1132409

    Article  MathSciNet  MATH  Google Scholar 

  2. Garrappa, R.: Numerical evaluation of two and three parameter Mittag-Leffler functions. SIAM J. Numer. Anal. 53(3), 1350–1369 (2015). https://doi.org/10.1137/140971191

    Article  MathSciNet  MATH  Google Scholar 

  3. Garrappa, R., Poplizio, M.: Evaluation of generalized Mittag-Leffler functions on the real line. Adv. Comput. Math. 39, 205–225 (2013). https://doi.org/10.1007/s10444-012-9274-z

    Article  MathSciNet  MATH  Google Scholar 

  4. Gill, G., Straka, P.: MittagLeffleR: Using the Mittag-Leffler distributions in R (2017). https://strakaps.github.io/MittagLeffleR/

  5. Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions. Related Topics and Applications. Springer, New York (2014). https://doi.org/10.1007/978-3-662-43930-2

  6. Gorenflo, R., Loutchko, J., Luchko, Y.: Computation of the Mittag-Leffler function \(E_{\alpha ,\beta }(z)\) and its derivative. Frac. Calc. Appl. Anal. 5(4), 491–518 (2002). http://www.diogenes.bg/fcaa/volume6/fcaa61/aluchcor54.pdf

  7. Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag-Leffler functions and their applications. J. Appl. Math. (2011). https://doi.org/10.1155/2011/298628

    Article  MathSciNet  MATH  Google Scholar 

  8. Litvinov, G.L.: Error autocorrection in rational approximation and interval estimates. [a survey of results.]. Centr. Eur. J. Math. 1, 36–60 (2003). https://doi.org/10.2478/BF02475663

  9. López-Fernández, M., Palencia, C.: On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51(2), 289–303 (2004). https://doi.org/10.1016/j.apnum.2004.06.015

    Article  MathSciNet  MATH  Google Scholar 

  10. Luke, Y.L.: Computations of coefficients in the polynomials of Padé approximations by solving systems of linear equations. J. Comput. Appl. Math. 6(3), 213–218 (1980). https://doi.org/10.1016/0771-050X(80)90028-5

    Article  MathSciNet  MATH  Google Scholar 

  11. McLean, W.: MittagLefflerFunctions. https://github.com/billmclean/MittagLefflerFunctions.jl (2020)

  12. McNamee, J., Stenger, F., Whitney, E.L.: Whittaker’s cardinal function in retrospect. Math. Comput. (1971). https://doi.org/10.1090/S0025-5718-1971-0301428-0

    Article  MathSciNet  MATH  Google Scholar 

  13. Mittag-Leffler, G.M.: Une généralization de l’intégrale de Laplace-Abel. Comput. Rend. Acad. Sci. Paris 136, 537–539 (1903)

    MATH  Google Scholar 

  14. Nakatsukasa, Y., Séte, O., Trefethen, L.N.: The AAA algorithm for rational approximation. SIAM J. Sci. Comput. 40(3), A1494–A1522 (2018). https://doi.org/10.1137/16M1106122

    Article  MathSciNet  MATH  Google Scholar 

  15. Paris, R.B.: Exponential asymptotics of the Mittag-Leffler function. Proc. Roy. Soc. Lond. A 458, 3041–3052 (2002). https://doi.org/10.2307/3560097

    Article  MathSciNet  MATH  Google Scholar 

  16. Paris, R.B.: Asymptotics of the Mittag-Leffler function \({E}_a(z)\) on the negative real axis when \(a\rightarrow 1\). Arxiv preprint (2020). 2005.05737v1

  17. Podlubny, I.: Mittag-Leffler function. https://www.mathworks.com/matlabcentral/fileexchange/8738-mittag-leffler-function. MATLAB Central File Exchange

  18. Seybold, H., Hilfer, R.: Numerical algorithm for calculating the generalized Mittag-Leffler function. SIAM J. Numer. Anal. 47(1), 69–88 (2008). https://doi.org/10.1137/070700280

    Article  MathSciNet  MATH  Google Scholar 

  19. Weideman, J.A.C., Trefethen, L.N.: Parabolic and hyperbolic contours for computing the Bromwich integral. Math. Comp. 76(259), 1341–1356 (2007). https://doi.org/10.1090/S0025-5718-07-01945-X

    Article  MathSciNet  MATH  Google Scholar 

  20. Wong, R., Zhao, Y.Q.: Exponential asymptotics of the Mittag-Leffler function. Constr. Approx. 18, 355–385 (2002). https://doi.org/10.1007/s00365-001-0019-3

    Article  MathSciNet  MATH  Google Scholar 

  21. Zeng, C., Chen, Y.Q.: Global Padé approximations of the generalized Mittag-Leffler function and its inverse. Frac. Calc. Appl. Anal. 18, 1492–1506 (2015). https://doi.org/10.1515/fca-2015-0086

    Article  MATH  Google Scholar 

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  1. School of Mathematics and Statistics, University of New South Wales, Sydney, 2052, Australia

    William McLean

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McLean, W. Numerical evaluation of Mittag-Leffler functions. Calcolo 58, 7 (2021). https://doi.org/10.1007/s10092-021-00398-6

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  • DOI: https://doi.org/10.1007/s10092-021-00398-6

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