Presented by:
Robert Corless
Date:
Tuesday 12th November 2019 - 14:00 to 15:00
Venue:
INI Seminar Room 2
Abstract:
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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