I have come across an integral that I would like to asymptotically evaluate (to leading order at least) which I have seen no mention of in standard textbooks.
I want to evaluate an integral of the form
$$ \int_{-1}^1 h(x) \frac{e^{kf(x)}}{e^{ikg(x)}-\lambda} d x $$
where $h$ is a smooth nicely behaved function, $f$ is a complex function, $g$ is a real function and $|\lambda|>1$. $f$ and $g$ depend on a parameter as well.
Ideas so far are to expand the integral using binomial theorem which then gives an infinite sum to evaluate, or try and write it in terms of $sin$, $cos$ and $sec$.
I would also like to think about the case where $|\lambda|<1$ and how this changes things.