Find the leading behavior of $\int_0^{\pi/2} e^{i s \cos(x)} dx$ as $s \to \infty$

I wanted to use Laplace's method, but then I saw the $i$ and now I don't know what to do. And furthermore the methods I was seeing seemed to depend in $\frac{d}{dx} \cos (x) \neq 0$ when $x = 0$ (that is where cosine takes its maximum, though in this context "maximum" doesn't make sense since $i \cos(x)$ is a complex number and thus does not have a "maximum"), so even then I seem to be stuck.

  • $\begingroup$ Are you familiar with the method of stationary phase or the saddle point method? $\endgroup$ – Antonio Vargas Aug 8 '17 at 17:42
  • $\begingroup$ hint: the contours of stationary phase are lines paralell to the imaginary axis $\endgroup$ – tired Aug 8 '17 at 22:03

The modulus of $$f(s)\stackrel{\text{def}}{=}\int_{0}^{\pi/2}e^{is\cos x}\,dx = \int_{0}^{\pi/2}e^{is\sin x}\,dx = \int_{0}^{1}e^{isx}\frac{dx}{\sqrt{1-x^2}} =\frac{\pi}{2}\left[J_0(s)+i H_0(s)\right]$$ decays like $\frac{C}{\sqrt{s}}$ for $s\to +\infty$. You may study how to apply Laplace method in the last section of these notes. An equivalent approach is to devise a differential equation fulfilled by $f(s)$ and to derive such bound from such a differential equation.

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