0
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Could someone explain me how to determine the following limit:

$\lim_{t\to+\infty}\int_{0}^{\pi} dx \cos\left(At\sin\left(\frac x2\right)\right)\cos^2\left(\frac{2n-1}2x\right)$

for $A\in \mathbb R^+$ and $n\in \mathbb N$? Is it eventually possible to reduce the integral to a Bessel function?

Thanks in advance.

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  • $\begingroup$ What does mean $\lfloor.\rfloor$? $\endgroup$ – Nosrati Sep 15 '18 at 13:47
  • $\begingroup$ If you mean the square brackets, these were meant to be the parentheses including the argument of the cosine. I just corrected it. $\endgroup$ – Graz Sep 15 '18 at 13:55
  • $\begingroup$ If all you need is just the limit, you should get your $0$ using integration by parts (preliminary $x \mapsto \pi - x$ might be of help). And yes, there is a link to Bessel functions (through pretty straightforward simplifications). $\endgroup$ – metamorphy Sep 15 '18 at 14:19
  • $\begingroup$ Thanks for the hint. Though is not obvious to me whether one can write the integral in terms of a closed expression involving Bessel functions (I guess the of first kind J_2). The difficulty arises from the multiplicative coefficients in the trigonometric functions. $\endgroup$ – Graz Sep 15 '18 at 15:47

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