I am searching for a solution to the following integral:
$$ \int_0^{2\pi} e^{-i m \alpha} e^{i x \cos{\alpha}+i y \sin{\alpha}} d\alpha $$
where $m$ is an integer. I am hoping or expecting that the above integral might be part of an identity equalling something involving Bessel functions, like:
$$ 2 \pi J_m\left(\sqrt{x^2+y^2}\right), $$
but have not been able to find this identity explicitly written anywhere. The above might be the correct answer but I'd like some proof or confirmation.
Close matches:
I know the following two identities:
$$ \int_0^{2\pi} e^{-i m \alpha} e^{i y \sin{\alpha}} d\alpha = 2 \pi J_m(y)$$
$$ \int_0^{2\pi} e^{i x \cos{\alpha}+i y \sin{\alpha}} d\alpha = 2 \pi J_0\left(\sqrt{x^2+y^2}\right), $$
which are particular cases of the more general integral that I am looking for. I am hoping that with the help of these two results I should be able to derive a solution for the desired integral above. If I understood how the second one can be derived from the first one for $m=0$ then I could try to do it for general $m$. Or if someone might directly point me to a proof or source of such an identity for $J_m$. I would be extremely grateful, thanks!
Research context:
For some context about my question, if it helps: we are trying to solve a research problem in my group involving solutions to the wave equation in the context of "translationally invariant" Bessel beams. Special beams of light called Bessel beams do not change their shape when they propagate, they are called invariant, they do not diffract, and this is precisely because the Bessel function can be described as a superposition of plane waves ($e^{i \bf{k} \cdot \bf{r}}$, known solutions to the wave equation) which are all propagating at the same angle with respect to the z-axis, i.e.
$$ \int{ e^{i \bf{k} \cdot \bf{r}} d\bf{k}}, $$
where $\mathbf{r} = (x,y,z)$ is the position vector, and $\mathbf{k} = (k_x,k_y,k_z) = (k_t \cos \alpha, k_t \sin \alpha, k_z)$ are wave-vectors of plane waves whose value of $k_z$ is identical, and have the same transverse wave-vector $k_t = (k_x^2+k_y^2)^{1/2}$, i.e. plane waves propagating at the same angle with respect to the $z$ axis, and this results in:
$$ \int_0^{2\pi} e^{i x k_t \cos{\alpha}+i y k_t \sin{\alpha} + i k_z z} d\alpha = 2 \pi J_0\left(k_t \sqrt{x^2+y^2}\right) e^{i k_z z} = F(x,y) e^{i k_z z}.$$
Thanks to the fact that $k_z$ is equal for all the plane-wave components, the $e^{i k_z z}$ term can be taken outside of the integral, and therefore the beam has a shape $F(x,y)$ in the transverse plane which does not change with $z$ (only its phase changes). This forms a Bessel beam. But nothing requires the amplitude of all plane waves to be the same, we could in general have a combination of plane waves with arbitrary amplitudes $f(\alpha)$:
$$ \int_0^{2\pi} f(\alpha) e^{i x k_t \cos{\alpha}+i y k_t \sin{\alpha} + i k_z z} d\alpha$$
and since $f(\alpha)$ must be $2\pi$ periodic, it can be expanded into an infinite Fourier series of $e^{i m \alpha}$ with integer $m$, giving rise to my original question/integral. With this piece of the puzzle one can then write the mathematical expression for an entire family of translationally invariant beams that combine plane waves with arbitrary amplitudes but same angle with respect to $z$.