  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
$\Omega\subset\mathbb{R}^d$ and sufficiently well-behaved functions
$f:\Omega\to\mathbb{R}$. Our main result is
  $ 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 $A$ 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, $A = f''(0)$. We also show that the
integration domain $\Omega$ can be extended to $\mathbb{R}^d$ without changing
the asymptotic behaviour.
