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The integral I want to solve is:

$$\int_{\mathbb{R}}dy\frac{\cos\left(Nk_1\arccos(f(y))\right)\exp(ik_2Ny)}{|\sqrt{1-(f(y)^2)}|}$$ when $f(y)=2+\frac{2\pi^2(E-\lambda)}{\ell_p^2}-\cos(y)$,

where $N, k_1, k_2, \ell_p, E, \lambda$ are constants. You could simply write $f(y)=a-\cos(y)$ where $a$ is a constant if that makes it easier. Also the term in the absolute value sign does not equal zero (so, $f(y)\neq\pm1$. I've tried looking into composite functions, but this doesn't seem to help because of the extra $\exp$ on the numerator. I have a feeling I would need to use complex analysis and solve via the method of resides because of the $\exp(ik_2Ny)$ term and the singularity in the denominator. I can't solve it numerically, because after I solve the integral There is a sum for the $k_1,k_2$ terms. Any suggestions?

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  • $\begingroup$ any chance that $N$ is big? $\endgroup$ – tired Aug 5 '16 at 11:59
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    $\begingroup$ well, it's used as a semiclassical parameter (from quantum mechanics) so in the end you could take the limit as N tends to infinity, but not before. I was thinking about maybe using the method of stationary phase? $\endgroup$ – Lewis Proctor Aug 5 '16 at 17:50
  • $\begingroup$ yes indeed, i see no chance for a complete closed form solution $\endgroup$ – tired Aug 5 '16 at 19:05

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