On the asymptotics and zeros of a class of Fourier integrals

Abstract

We obtain the asymptotic expansion of the Fourier integrals
\[\int_0^\infty t^{\nu-1}\, \raisebox{-.8ex}{\mbox{$\stackrel{\textstyle\cos}{\sin}$}}\,(xt) \,\exp\,(-t^n/n)\,dt\]
for large complex values of $x$ and integer $n>2$ by means of the asymptotic theory of the
Wright function. Asymptotic approximations for both the real and complex zeros of these integrals
are considered.
These results are extended to $p$-dimensional Fourier integrals of a similar structure.