I would like to approximate the following integral for large $t$ :

$I(t)=\int_0^{\pi}dx f(x)e^{iS(x)t}$

$S$ is real and $S'(0)=S'(\pi)=0$.

$f$ is real and $f(0)=f(\pi)=0$ and $f'(0)=f'(\pi)=0$.

Can one do this for this type of integral?

Thank you.


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