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Approximating the Functional Inverse of $\Gamma$

Banff International Research Station for Mathematical Innovation and Discovery

This work arose out of joint work with Jonathan M. Borwein (1951--2016), undertaken while he was the Distinguished Visiting Scholar in Residence at Western (April-August 2016). That work was a survey of articles in the American Mathematical Monthly on the~$\Gamma$ function, of which there were nearly fifty. The survey identified some gaps: visualization, computation, and perhaps most surprisingly \textsl{nothing} on the functional inverse of~$\Gamma$, which we denote~$y=\invG(x)$ if~$x=\Gamma(y)$. This is multivalued with infinitely many branches. In this talk, I show that one can invert Stirling's original approximation (which is more accurate than the popular approximation by that name) to give a surprisingly accurate asymptotic formula for~$\invG$.

Mathematics; Approximations and expansions; Commutative rings and algebras; Classical analysis

video/mp4

Author affiliation: University of Western Ontario

2017-06-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61974

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/