Asymptotic Expansion of Laplace-Fourier-Type Integrals

We study the asymptotic behaviour of integrals of the Laplace-Fourier type $P(k) = \int_\Omega$\mathrm{e}$^{-|k|^sf(x)}$\mathrm{e}$^{$\mathrm{i}$ kx}$\mathrm{d}$ x;, $with$k\in$\mathbb{R}$^d$in$d\ge1$dimensions,with$\Omega\subset$\mathbb{R}$^d$andsufficientlywell-behavedfunctions$f:\Omega\to$\mathbb{R}$$\mathrm{.}Ourmainresultis$ P(k) \sim \frac{$\mathrm{e}$^{-|k|^sf(0)}}{|k|^{sd/2}}\sqrt{\frac{(2\pi)^d}{\det A}} \exp\left(-\frac{k^\top A^{-1}k}{2|k|^s}\right) $for$|k|\to\infty$,where$A$istheHessianmatrixofthefunction$f$ at its critical point, assumed to be at $x_0=0$. In one dimension, the Hessian is replaced by the second derivative, $A=f^{\mathrm{\prime }\mathrm{\prime }}(0)$. We also show that the integration domain $\mathrm{\Omega }$ can be extended to ${\mathbb{R}}^d$ without changing the asymptotic behaviour.