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Suppose I have a situation where I want to find an asymptotic expansion as $x \to \infty$ for an integral of the form: $$ \int_{a}^{b} f(t) e^{-\phi(t) x} \mathrm{d}t$$

Let us also suppose that $f(t)$ is bounded and $\phi(t)$ has a (or some) local minima on $[a,b]$. I understand that the general procedure is to expand $\phi(t)$ about its minima, find these contributions and neglect the rest.

My question is, how do we make the neglecting of the other parts of the integral rigorous? I've heard things being said like "the other parts are exponentially small and so never appear in the asymptotic expansion", but I can't justify this mathematically. Can anyone help?

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    $\begingroup$ I found the proofs in de Bruijn's Asymptotic Methods in Analysis and Miller's Applied Asymptotic Analysis in their chapters on the Laplace method to be rigorous and enlightening. The two have different flavors, so you might find yourself more comfortable with one than the other (or a combination of the two). $\endgroup$ – Antonio Vargas Mar 4 '14 at 15:05

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