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I'm trying to approximate $$\dfrac{1}{\pi}\int_0^{\pi}\cos(x\sin\theta-n\theta)d\theta$$

Where x goes to infinity I know to make it complex and then use the small angle approximation for $\sin\theta$ but I'm not sure how to proceed. Any help is greatly appreciated Thanks

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You can use the stationary phase approximation method to solve the integral.

Look at https://en.wikipedia.org/wiki/Stationary_phase_approximation.\

The essential idea is that because the frequency is so large, the integral will cancel out everywhere, except for points where the argument of the cosine are zero. Finding the first order approximation is pretty easy and is derived on the wiki and results in an accuracy of x^(-1/2). To get a better approximation is not terrible, but not very easy. Essentially you need to do a third and fifth order Taylor expansion and multiply the two expansion, and it gets ugly quickly.

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