Advertisement

Mathematics in Computer Science

, Volume 11, Issue 3–4, pp 341–350 | Cite as

Branch Structure and Implementation of Lambert W

  • David J. Jeffrey
Article
  • 81 Downloads

Abstract

We begin with a discussion of general design decisions made in implementing the Lambert W function in Maple . Many of these decisions are not system-specific and apply to any implementation of W; also they touch some of the fundamental issues in computer-algebra systems. A specific topic is the choice of a branch structure for W, and a new approach is presented that allows us to extend the definition of a function from the real line into the complex plane.

Keywords

Lambert W Branches 

Mathematics Subject Classification

Primary 99Z99 Secondary 00A00 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bernadin, L.: A Review of symbolic solvers. In: Wester, M. (ed.) Computer Algebra Systems: A Practical Guide. Wiley, Chichester (1999)Google Scholar
  2. 2.
    Bhamidi, S., Steele, J.M., Zaman, T.: Twitter event networks and the superstar model. Ann. Appl. Probab. 25(5), 2462–2502 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J.: Lambert’s \(W\) function in Maple. Maple Tech. Newsl. 9, 12–22 (1993)Google Scholar
  4. 4.
    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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    http://dlmf.nist.gov/4.13. Content as of April 2017
  6. 6.
    De Bruijn, N.G.: Asymptotic Methods in Analysis. North-Holland, New York (1961)Google Scholar
  7. 7.
    Flajolet, P., Knuth, D.E., Pittel, B.: The first cycles in an evolving graph. Dis. Math. 75, 167–215 (1989)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hayes, B.: Why W? Am. Sci. 93, 104–108 (2005)CrossRefGoogle Scholar
  9. 9.
    Jeffrey, D.J., Hare, D.E.G., Corless, R.M.: Unwinding the branches of the Lambert W function. Math. Sci. 21, 1–7 (1996)MathSciNetMATHGoogle Scholar
  10. 10.
    Jeffrey, D.J.: Indexed elementary functions in Maple. ACA 2015, Kalamata, Greece. http://math.unm.edu/~aca/ACA/2015/Education/Jeffrey.txt. Accessed 10 Apr 2017
  11. 11.
    Jordan, William B., Glasser, M.L.: Solution of problem 68–17. SIAM Rev. 12(1), 153–154 (1970)CrossRefGoogle Scholar
  12. 12.
    Euler, L.: De Formulis exponentialibus replicatis. Acta Acad. Sci. Imp. Petropolitinae 1, 1778, 38–60, also Opera Omnia: Ser. 1, 15, 268–297Google Scholar
  13. 13.
    Euler, L.: De serie Lambertina plurimisque eius insignibus proprietatibus. Acta Acad. Sci. Imp. Petropolitinae 1779, 1783, 29–51, also Leonhardi Euleri Opera Omnia, Ser. 1, Opera Mathematica, 6, 1921 (orig. date 1779), 350–369Google Scholar
  14. 14.
    Fritsch, F.N., Shafer, R.E., Crowley, W.P.: Algorithm 443: solution of the transcendental equation \(w e^w = x\). Commun. ACM 16, 123–124 (1973)CrossRefGoogle Scholar
  15. 15.
    Gonnet, G.H.: Expected length of the longest probe sequence in hash code searching. J. ACM 28(2), 289–304 (1981)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lambert, J.H.: Observationes variae in mathesin puram. Acta Helv. Phys. Math. Anat. Bot. Med. 3(5), 128–168 (1758)Google Scholar
  17. 17.
    Lambert, J.H.: Observations analytiques. Nouveaux mémoires de l’Académie Royale des Sciences et Belles-Lettres, Berlin, 1772, vol. 1, for 1770Google Scholar
  18. 18.
    Poisson, S.-D.: Suite du mémoire sur les intégrales définies et sur la sommation des séries. J. Ecole R. Polytech. 12, 404–509 (1823)Google Scholar
  19. 19.
    Pólya, G., Szegö, G.: Aufgaben und Lehrsätze aus der Analysis I, Springer (1925). English Translation: Problems and Theorems in Analysis, Springer (1972)Google Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsThe University of Western OntarioLondonCanada

Personalised recommendations