I'm hoping to determine the value of the following integral: $$\int_{-\infty}^{\infty}\exp(\mathrm{i} n \cosh{x}) \, \mathrm{d}x$$

Here is a plot of the integrand as a function of $x$ with parameter $n$ varying from 0 to 10.

enter image description here

The integral appears to not converge. However, it is known that $$\int_{-\infty}^{\infty}\exp(\mathrm{i} n x) \, \mathrm{d}x = 2\pi \delta(n)$$

where $\delta(x)$ is the Dirac delta function. Is it possible for the first integral to be expressed similarly using Dirac delta notation?

  • 1
    $\begingroup$ This integral is related to the Hankel function for which an integral representation is $${H^{(1)}_{\nu}}\left(z\right)=\frac{e^{-\frac{1}{2}\nu\pi i}}{\pi i}\int_{-\infty}^{\infty}e^{iz\cosh t-\nu t}\mathrm{d}t$$ Here, $\nu=0$. $\endgroup$ – Paul Enta Dec 14 '18 at 17:09
  • $\begingroup$ @PaulEnta, then for real $z$ the integral doesn't exist in the usual sense, does it? Or is the function analytically continued somehow? $\endgroup$ – Yuriy S Dec 14 '18 at 17:10
  • $\begingroup$ @YuriyS The principal branch of the function has a branch cut along the negative reals. See here for example. $\endgroup$ – Paul Enta Dec 14 '18 at 17:15
  • 3
    $\begingroup$ The representation $$Y_{0}\left(x\right)=-\frac{2}{\pi}\int_{0}^{\infty}\cos\left(x\cosh t\right)\, \mathrm{d}t$$ from here may be also helpful. $\endgroup$ – Sangchul Lee Dec 14 '18 at 17:18

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