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

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

where $A$ is a real positive constant.

- Does this integral have a closed form or asymptotic expansion? $\int_0^\infty \frac{\sin(\beta u)}{1+u^\alpha} du$
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- Dirac Delta and Exponential integral
- Is Plancherel's theorem true for tempered distribution?
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- Find the solution of the Dirichlet problem in the half-plane y>0.

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.

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