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I'd be interested in computing an asymptotic expansion when $h \rightarrow 0$, of an integral of the form $$ I_h = \int_{\mathbb{R}}{e^{\frac{i}{h}\varphi(x)}dx} $$ where $\varphi : \mathbb{R} \rightarrow \mathbb{C}$ extends to a holomorphic function on a neighbourhood of $\mathbb{R} \subset \mathbb{C}$ (I'm happy to assume that it extends to the whole of $\mathbb{C}$ if it simplifies matters).

I've heard of the existence of a "complex" stationary phase formula allowing to compute this expansion in terms of the critical points of $\varphi : \mathbb{C} \rightarrow \mathbb{C}$ but couldn't find a satisfactory reference for that. Any idea where I could find such a result?

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    $\begingroup$ A standard reference is F. W. J. Olver, Asymptotics and special functions (1974). $\endgroup$
    – Alexandre Eremenko
    Commented Oct 14 at 12:07

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See Asymptotic expansions of oscillatory integrals with complex phase (2010). The usual formulas carry through even if $\phi(x)$ is complex, provided that it has a non-negative imaginary part.

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