Gergô Nemes

Error bounds for the asymptotic expansion of the Gamma function

The asymptotic expansion of the complex Gamma function is an extension of the so-called Stirling
formula. The first proof of the expansion for positive variables dates back to Laplace. Since the
mid-20th century, this expansion has become a standard textbook example to illustrate various
techniques for deriving asymptotic series. It was not, however, until the end of the 20th century
that numerically computable error bounds were found for this important expansion, despite the fact
that for the related log-Gamma expansion, error bounds were known since the mid-19th century.

Ever
since I encountered these new error bounds, I felt that there must be an even simpler and nicer bound
for the remainder. During the talk I will provide you with a heuristic argument that gives hope for
the existence of a better error bound, and I will give an outline of the rigorous proof as well.