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Non-principal Branches of Lambert W. A Tale of 2 Circles

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Computer Algebra in Scientific Computing (CASC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14139))

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Abstract

The Lambert W function is a multivalued function whose principal branch has been studied in detail. Non-principal branches, however, have been much less studied. Here, asymptotic series expansions for the non-principal branches are obtained, and their properties, including accuracy and convergence are studied. The expansions are investigated by mapping circles around singular points in the domain of the function into the range of the function using the new expansions. Different expansions apply for large circles around the origin and for small circles. Although the expansions are derived as asymptotic expansions, some surprising convergence properties are observed.

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Notes

  1. 1.

    Citations of [1] as of July 2023: Google scholar 7283; Scopus 4588.

  2. 2.

    Note the plural. We regard each branch of \(W_k\) as a separate function with its own domain and range [14].

  3. 3.

    This whimsical Shakespearian reference emphasises the mathematical point that previous investigations have concentrated on the large circle and neglected the equally important small circle.

  4. 4.

    Indeed, some authors define an asymptotic series as one that does not converge [18].

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Authors and Affiliations

  1. Department of Mathematics, University of Western Ontario, London, ON, Canada

    Jacob Imre & David J. Jeffrey

Authors
  1. Jacob Imre
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  2. David J. Jeffrey
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Correspondence to David J. Jeffrey .

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Editors and Affiliations

  1. University of Lille, CRIStAL, Villeneuve d'Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Wilfrid Laurier University, Waterloo, ON, Canada

    Ilias Kotsireas

  4. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  5. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Imre, J., Jeffrey, D.J. (2023). Non-principal Branches of Lambert W. A Tale of 2 Circles. In: Boulier, F., England, M., Kotsireas, I., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2023. Lecture Notes in Computer Science, vol 14139. Springer, Cham. https://doi.org/10.1007/978-3-031-41724-5_11

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  • DOI: https://doi.org/10.1007/978-3-031-41724-5_11

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  • Online ISBN: 978-3-031-41724-5

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