How to handle the large $r$ limit of this integral?

[praharmitra]

I am doing some calculation and am now stuck with an integral of the form

$$\lim_{r \to \infty} \int_{-1}^1 dt f(t) e^{i r (t-1)}$$

for some function $f(t)$. I don't know what the exact form of $f(t)$ is.

Is there any way to address this integral? Similar to the saddle-point method perhaps? The saddle-point method does not work here right? since the argument of the exponential does not have a minima.

How should I go about this?

Can we say that this integral is dominated by a certain value of $t$, say at $t=1$? Why or why not?

 

[arildno]

This is a classic case where we can what we call the "method of stationary phase"
You can check out the basic ideas on Wikipedia:
https://en.wikipedia.org/wiki/Stationary_phase_approximation

Ooops, it's been too long since I did this sort of thing, it won't work, of course...