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Let $M$ be a compact oriented Riemannian manifold without boundary.

Set $f(x)=a(x)+\sqrt{-1}b(x)$ be a complex-valued function on $M$, where $a(x),b(x)$ are real-valued function on $M$.

Then, how to compute the Oscillatory integrals below? $$I(t)=\int_{M}e^{-\sqrt{-1}tf(x)}dv=\int_{M}e^{tb(x)}e^{-\sqrt{-1}ta(x)}dv$$ where $dv$ be a smooth density on $M$.

Especially, I am interested in discussing the asymptotic behaviour of $I(t)$ when $t\rightarrow\infty$.

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    $\begingroup$ If you look at mathoverflow.net/questions/88640/…, you'll realize that the question you ask is far too general to have a good answer. :( $\endgroup$
    – fedja
    Commented May 5, 2013 at 14:22

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