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I am reading a paper (arXiv:1404.6407, by Galkin, Golyshev and Iritani) where the authors need to use the statement that for $z\to 0+$ (and in fact in a sector) the integral $$ \int_{1+\rm i \mathbb R} \Gamma(s)^N z^{Ns}\,ds $$ is asymptotically $e^{-\frac Nz}$ times something of polynomial growth. Here $N$ is a positive integer. There is a precise asymptotic formula which, as far as I can trace it, goes back to Barnes (1906) or perhaps earlier.

I would like to learn some modern self-contained arguments and/or methods for proving this and/or similar statements. Unfortunately, all the references I found so far just end up quoting Barnes (or Meijer, who in turn quotes Barnes).

What would be a good graduate level text that deals with these problems?

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  • $\begingroup$ Look at Braaksma in Compositio 1963. eudml.org/doc/88877 This is what Luke cites for instance. The sketch in Section 2.2, with some bits from Section 3 involving $\Gamma$-expansions, is already probably enough for the ideas. Much of the 100 pages is chasing down uniformity IIRC. Historically, I think that Stokes might have actually have obtained the asymptotic term, and Barnes the error (or an expansion), but I must admit parsing Barnes was hard when I tried $\endgroup$
    – post.as.a.guest
    Commented Oct 27, 2016 at 1:30
  • $\begingroup$ Thank you, I will take a look at it. I saw that reference in Luke's book. Was under the impression that the idea was to approximate $\Gamma(s)^N$ by $\Gamma(Ns-i)$, will look in more detail. $\endgroup$
    – Lev Borisov
    Commented Oct 27, 2016 at 1:33
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    $\begingroup$ Have you already looked in the book "Asymptotic Expansions of Integrals" of Bleistein and Handelsman? Its reprint is from 1975, so maybe not that modern, but I have a vague memory that it treated some Barnes integrals. BTW, have you also looked in Paris and Kaminski's "Asymptotics and Mellin-Barnes Integrals" (2001)? $\endgroup$
    – M.G.
    Commented Oct 27, 2016 at 8:02
  • $\begingroup$ Great, thank you! I have not looked at these. $\endgroup$
    – Lev Borisov
    Commented Oct 27, 2016 at 11:08

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