Fourier Transform of $\exp{(A\sin(x))}$

Question

I would like to know how to calculate the Fourier transform of

$$e^{A\sin(x)}$$

where $A$ is a real positive constant.

Answer 1

The Fourier series of $e^{A\sin x}$ is given by:

$$ e^{A\sin x}= I_0(A) + 2\sum_{n\geq 0}(-1)^n I_{2n+1}(A)\sin((2n+1) x)+2\sum_{n\geq 1}(-1)^n I_{2n}(A)\cos(2nx) $$ where $I_n$ is a modified Bessel function of the first kind.
Now you may recover the Fourier transform from the Fourier series.