Fourier transform of Bessel function

Question

Please help me find the the representation in $k$-space (Fourier transform) of the following function:
$$
f(k_t, k_z, m) = J_m(k_trho)exp(ik_zz)exp(imphi)/N
$$
The properties of $f$ given below might come in handy. The equations are from a paper by S.J. Van Enk and G. Nienhuis(2007) (https://doi.org/10.1080/09500349414550911) eqns.35 and 36 under the 4th subheading.
$$
(hat{P}^2_x + hat{P}^2_y)f = hbar^2k_t^2f, hat{P}_zf = hbar k_zf hat{L}_zf = hbar m f
$$

mike stone · Answer

The $k_z$ is irrelevent, so this is just a two-dimensional problem.
I suspect that this is a homework problem, so I'll just give a hint:
Start by showing that if  you have a function of the form $f(r,theta)=f(r)e^{-iltheta}$ then the integral definition of the Bessel function $J_l(kr)$ shows that the  two dimensional Fourier transform is
$$
tilde f({bf k})= 2pi i^l e^{-iltheta_k} int_0^infty J_l(kr)f(r)r dr.
$$
Here $k$, $theta_k$ are the polar coordinates of the vector ${bf k}=(k_x,k_y)$.
Now you can use the Hankel transform formula
$$
int_0^infty J_l(kr) J_l(k'r) rdr = frac 1 k delta(k-k'),
$$
(which is just a disguised version of the Fourier inversion theorem) to evaluate your FT.

Fourier transform of Bessel function

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