I am trying to compute the two dimensional fourier transform ( defined as $\int f(r,\theta) \, e^{ -i \, \boldsymbol{k} \cdot \boldsymbol{x}} d\boldsymbol{x} $ ).
the function that i am trying to transform is rather easy:
$ f(r, \theta) = \frac{cos(2 \theta)}{r^2}$
where $\theta$ is the angle from the $x$ axis and $r$ the distance from the origin.
following wikipedia (https://en.wikipedia.org/wiki/Hankel_transform)
i was able to obtain as result:
$\hat{f} (\boldsymbol{k})=\int \frac{cos(2 \theta)}{r^2} \, e^{ -i \, \boldsymbol{k} \cdot \boldsymbol{x}} d\boldsymbol{x} = -\frac{\pi}{2} $
Here you can find the steps in case i made some mistake:
first i consider the wave vector $k_x$ aligned with the real $x$ axis, so that $\boldsymbol{k} \cdot \boldsymbol{x}= k \, r \cos{\theta}$, where $k =\sqrt{\boldsymbol{k} \cdot \boldsymbol{k}}$, and $r =\sqrt{\boldsymbol{x} \cdot \boldsymbol{x} }$.( I am not completely sure the above step is completely legitimate)
Then i rewrite the above integral in polar coordinates (note that the $r$ coming the area in polar coordinates simplifies with the $r$ in the denominator):
$ \int_0^{\infty} \int_0 ^{2 \pi} \frac{cos(2 \theta)}{r} \, e^{ -i \, k \, r \cos{\theta}} d\theta dr $
then i rewrite the cosine in its exponential form:
$ \frac{1}{2}\int_0^{\infty} \int_0 ^{2 \pi} \frac{1}{r} \, e^{ -i \, 2 \, \theta} e^{ -i \, k \, r \cos{\theta}} d\theta dr + \frac{1}{2}\int_0^{\infty} \int_0 ^{2 \pi} \frac{1}{r} \, e^{ i \, 2 \, \theta} e^{ -i \, k \, r \cos{\theta}} d\theta dr $
now integrating with mathematica:
$\int_0 ^{2 \pi} \frac{1}{r} \, e^{ \pm i \, 2 \, \theta} e^{ -i \, k \, r \cos{\theta}} d\theta= - \pi \, J_2(k \, r)$
with $J_2$ the Bessel function of the first kind and second order.
therefore we are left with:
$ -\frac{\pi}{2}\int_0^{\infty} \frac{1}{r} J_2(k \, r) dr - \frac{\pi}{2}\int_0^{\infty} \frac{1 }{r} \, J_2(k \, r) dr $
which again according to mathematica gives:
$ -\pi\int_0^{\infty} \frac{1}{r} J_2(k \, r) dr= -\frac{\pi}{2}$
that gives the final result $\hat{f}(\boldsymbol{k}) = -\frac{\pi}{2} $
However i don't understand how you invert the fourier transform to obtain again the function $\; f(r,\theta)$ anti transforming $ \; \hat{f} (\boldsymbol{k})$.
In particular i have no clue where the $\cos{(2\theta)}$ dependence comes from when I back transform the function $ \; \hat{f} (\boldsymbol{k})= -\frac{\pi}{2}$
Indeed, from my knowledge, the inverse fourier transform of a constant gives a dirac delta in real space. Any help?