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SUMMARY:An Older Special Function meets a (Slightly) Newer One - Robert Co
 rless (University of Western Ontario)
DTSTART:20191112T140000Z
DTEND:20191112T150000Z
UID:TALK134734@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Euler invented the Gamma function in 1729\, and it remains one
  of the most-studied special functions\; see in particular Philip J. Davis
 &#39\;  <span>Chauvenet prize-winning article "Leonhard Euler&#39\;s Integ
 ral"\, 1959.&nbsp\; In 2016\, Jon Borwein and I started a survey of articl
 es on Gamma in the American Mathematical Monthly (including that beautiful
  paper by Davis)\; Jon died before our survey was finished\, but I finishe
 d it and it was published in 2018: "Gamma and Factorial in the Monthly". I
 n that survey\, we uncovered a surprising gap in the nearly three hundred 
 years of literature subsequent to Euler&#39\;s invention: almost nobody ha
 d studied the functional inverse of the Gamma function.&nbsp\; More\, we u
 ncovered Stirling&#39\;s original asymptotic series (the asymptotic series
  that "everyone knows" as Stirling&#39\;s is\, in fact\, due to de Moivre)
 \, and used it to find a remarkably accurate approximation to the principa
 l branch of the functional inverse of Gamma using what is now known as the
  Lambert W function.&nbsp\; This newer function was also invented by Euler
 \,&nbsp\; in 1783\, using a series due to Lambert in 1758\; since Euler di
 d not need yet another function or equation named after him\, we chose in 
 the mid 1990&#39\;s to name it after Lambert.&nbsp\; Facts about W may be 
 found at <a target="_blank" rel="nofollow" href="http://www.orcca.on.ca/La
 mbertW">http://www.orcca.on.ca/LambertW</a> and physical copies of that po
 ster are being couriered here\; not\, unfortunately in time for the talk\,
  but you&#39\;ll be able to get your very own copy (lucky you!) probably b
 y the end of the week.</span>  &nbsp\;  My talk will survey some of the Mo
 nthly articles on Gamma\, and introduce some of the facts about W that I f
 ind interesting.&nbsp\; 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.  &nbsp\;  The Monthly paper can be found at  <
 a target="_blank" rel="nofollow" href="https://www.tandfonline.com/doi/ful
 l/10.1080/00029890.2018.1420983">https://www.tandfonline.com/doi/full/10.1
 080/00029890.2018.1420983</a>  &nbsp\;  The Wikipedia articles on Gamma an
 d on Lambert W are substantive.  <br><br><br><br><br>
LOCATION:Seminar Room 2\, Newton Institute
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