Fourier transform of $\exp(-\cosh(x))$

Question

I am trying to compute the Fourier transform $$ \hat{g}(\xi) = \int_{\mathbb{R}} e^{-\cosh(x)}e^{-2i\pi \xi x} dx $$ of the function $g(x)=\exp(-\cosh(x))$.

Are there any standard substitution tricks for such type of integrals?

Answer 1

The modified Bessel function of the second kind, $K_\alpha$ has an integral representation given by

$$K_\alpha(x)=\int_0^\infty e^{-x\cosh(t)}\cosh(\alpha t)\,dt$$

for $\text{Re}(x)>0$.

Now, let $\hat g(\xi)$, $\xi \in \mathbb{R}$, be given by

$$\begin{align} \hat g(\xi)&=\int_{-\infty}^\infty e^{-\cosh(x)}e^{-i2\pi \xi x}\,dx\\\\ &=2\int_0^\infty e^{-\cosh(t)} \cos(2\pi |\xi| t)\,dt\\\\ &=2\int_0^\infty e^{-\cosh(t)}\cosh(i2\pi |\xi| t)\,dt\\\\ &=2K_{i2\pi |\xi|}(1) \end{align}$$

And we are done!