Fourier transform of exp(cos)

Question

How do I calculate the Fourier transform ($t \rightarrow \omega$) of the following:

$\exp(A\cos(\omega_0 t))$

$A$ is a real constant, and $\omega_0$ is a real and positive constant. I know that this gives a Bessel function, but how can I show this?

Answer 1

The modified Bessel Function of second kind $K_z(a)$ can be expressed as a Fourier transform see here:

$$K_{z}(a)=\frac{1}{2}\int_{-\infty}^{\infty}\exp(-a\cosh t)\cosh(zt){\rm d}t=K_{-z}(a)$$