Rapidly Convergent Integrals and Function Evaluation

  • Heba al Kafri
  • David J. JeffreyEmail author
  • Robert M. Corless
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10693)


We analyse integrals representing the Lambert W function, paying attention to computations using various rules. Rates of convergence are investigated, with the way in which they vary over the domain of the function being a focus. The first integral evaluates with errors independent of the function variable over a significant range. The second integral converges faster, but the rate varies with the function variable.


  1. 1.
    Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert W function. Adv. Comput. Math. 5, 329–359 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Iacono R., Boyd, J.P.: New approximations to the principal real-valued branch of the Lambert W-function. Adv. Comput. Math., 1–34 (2017)Google Scholar
  3. 3.
    Trefethen, L.N., Weideman, J.A.C.: The exponentially convergent trapezoidal rule. SIAM Rev. 56(3), 385–458 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kalugin, G.A., Jeffrey, D.J., Corless, R.M., Borwein, P.B.: Stieltjes and other integral representations for functions of Lambert W. Integr. Transforms Spec. Functions 23(8), 581–593 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Poisson, S.-D.: Suite du mémoire sur les intégrales définies et sur la sommation des séries. J. de l’École Royale Polytechnique 12, 404–509 (1823)Google Scholar
  6. 6.
    Weideman, J.A.C.: Numerical integration of periodic functions: a few examples. Am. Math. Mon. 109, 21–36 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Heba al Kafri
    • 1
  • David J. Jeffrey
    • 1
    Email author
  • Robert M. Corless
    • 1
  1. 1.Department of Applied MathematicsThe University of Western OntarioLondonCanada

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