We obtain the asymptotic expansion for large integer $n$ of a generalised sine-integral \[\int_0^\infty\left(\frac{\sin\,x}{x}\right)^{n}dx\] by utilising the saddle-point method. This expansion is shown to agree with recent results of J. Schlage-Puchta in {\it Commun. Korean Math. Soc.} {\bf 35} (2020) 1193--1202 who used a different approach.
An asymptotic estimate is obtained for another related sine-integral also involving a large power $n$. Numerical results are given to illustrate the accuracy of this approximation. We also revisit the asymptotics of Ball's integral involving the Bessel function $J_ν(x)$, which reduces to the above integral when $ν=1/2$.
The modular properties and the integral representations of the multiple elliptic gamma functions
We show the modular properties of the multiple 'elliptic' gamma functions , which are an extension of those of the theta function and the elliptic gamma function . The modular property of the theta function is known as Jacobi's transformation, and that of the elliptic gamma function was provided by Felder and Varchenko. In this paper we deal with the multiple sine functions , since the modular properties of the multiple elliptic gamma functions result from the equivalence between two ways to represent the multiple sine functions as infinite product . We also derive integral representations of …
