Asymptotic Expansion of Laplace-Fourier-Type Integrals

Abstract

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

13 pages, to be submitted to SciPost Physics