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An Older Special Function meets a (Slightly) Newer One

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CAT - Complex analysis: techniques, applications and computations

Euler invented the Gamma function in 1729, and it remains one of the most-studied special functions; see in particular Philip J. Davis' Chauvenet prize-winning article “Leonhard Euler's Integral”, 1959.  In 2016, Jon Borwein and I started a survey of articles on Gamma in the American Mathematical Monthly (including that beautiful paper by Davis); Jon died before our survey was finished, but I finished it and it was published in 2018: “Gamma and Factorial in the Monthly”. In that survey, we uncovered a surprising gap in the nearly three hundred years of literature subsequent to Euler's invention: almost nobody had studied the functional inverse of the Gamma function.  More, we uncovered Stirling's original asymptotic series (the asymptotic series that “everyone knows” as Stirling's is, in fact, due to de Moivre), and used it to find a remarkably accurate approximation to the principal branch of the functional inverse of Gamma using what is now known as the Lambert W function.  This newer function was also invented by Euler,  in 1783, using a series due to Lambert in 1758; since Euler did not need yet another function or equation named after him, we chose in the mid 1990's to name it after Lambert.  Facts about W may be found at http://www.orcca.on.ca/LambertW and physical copies of that poster are being couriered here; not, unfortunately in time for the talk, but you'll be able to get your very own copy (lucky you!) probably by the end of the week.   My talk will survey some of the Monthly articles on Gamma, and introduce some of the facts about W that I find interesting.  I will have more material than will fit in an hour, so exactly which topics get covered will depend at least in part on the interests of the audience.   The Monthly paper can be found at https://www.tandfonline.com/doi/full/10.1080/00029890.2018.1420983   The Wikipedia articles on Gamma and on Lambert W are substantive.




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