I am trying to solve the integral $$\int_ {Cylinder}e^{-i\vec{k}\vec{r}}dV=\int_0^Rrdr\int_0^{2\pi}d\phi\int_0^Le^{-ik_zz}e^{-i(k_xx+k_yy)}dz$$ I tried to rewrite it using polar coordinates and solved it using Mathematica: $$\int_0^Rrdr\int_0^{2\pi}d\phi\int_0^Le^{-ik_zz}e^{-ik_rr\cos\phi}dz$$ where I used that $k_xx+k_yy = |k_r||r|\cos(\phi)$.
The result, however, is not correct and gives a regularized hypergeometric function and not the expected Bessel function.
Where did I go wrong?
Edit: I have also found various resources citing a so-called Hankel transform, but I have never heard of it before.
This is what I did, put $x=r\cos\phi$ and $y=r\sin\phi$:
\begin{equation} \int_0^L e^{(-ik_zz)}e^{(-ik_xrcos\phi)}e^{(ik_yrsin\phi)}dz=\frac{1}{k_z}\bigg(-(i e^{(-i (k_z L + k_x r \cos\phi) - k_y r \sin\phi))} )(-1 + e^{(i k_z L)})\bigg) \end{equation}
This is then integrated on dr:
\begin{equation} \int_0^R \frac{1}{k_z}\bigg(-(i e^{(-i (k_z L + k_x r \cos\phi) - k_y r \sin\phi))} )(-1 + e^{(i k_z L)})\bigg)rdr \end{equation}
and it gives:
\begin{equation} \frac{(e^{-i (k_z L+b R \cos\phi -k_y R +\sin\phi)} (-1+e^{i k_z L}) (k_x R \cos\phi+i (-1+e^{i R (k_x \cos\phi -k_y \sin\phi)}+i k_y R \sin\phi)))}{(k_z (k_x \cos\phi -k_y \sin\phi^2))} \end{equation}
Then you do the $d\phi$ integration. But Mathematica does not seem to solve that.