Suppose we want to expand asymptotically a Fourier transform
$$ g(k) = \int_{-\infty}^{\infty} f(x) e^{i k x} $$
for large $k$. The function $f(x)$ is regular on the real axis but its analytic continuation to the complex plane may contain cuts, essential singularities, etc.
I know that if the integration bounds were finite ($a$ and $b$ instead of $-\infty$ and $\infty$), I could use integration by parts to obtain the large frequency asymptotics.
I also know that the methods of steepest descent and stationary phase method do not apply because the large number $k$ in the exponent multiplies a function of x that has no saddle point (it's linear).
So what would you do?