I would like to calculate "explicitly" the following integral, which is a Fourier transform: let $\alpha>0$ be a parameter, for $x\in \mathbb R$, we define $$ I(\alpha, x)=\int_\mathbb R \cos(xt) e^{-\alpha \cosh t} dt. $$ It is easy to see that $I(\alpha, \cdot)$ is a rapidly decreasing (even) smooth function, but I would like to have an explicit formula.
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7$\begingroup$ dlmf.nist.gov/10.32.E9 $\endgroup$– NemoCommented Apr 16, 2020 at 17:23
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$\begingroup$ @user82588 --- neat find --- Mathematica is unable to evaluate it in closed form. $\endgroup$– Carlo BeenakkerCommented Apr 16, 2020 at 17:28
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