Ytukyg

I need help finding the Fourier transform of the function

$$
rho(vec{r}) = alpha delta_{vec{r},0} left(lambdalambda' J_1
(beta |vec{r}|)Y_1(beta |vec{r}|) - pi^2 J_0
(beta |vec{r}|)Y_0(beta |vec{r}|) right),
$$

where $alpha,beta in mathcal{R}$ and $lambda,lambda'=pm1$. The $J_i(x)$ are the $i$-th Bessel functions of the first, the $Y_i(x)$ are the $i$-th Bessel functions of the second kind. $delta_{vec{r},0}$ denotes the Kronecker delta.

The vector $vec{r} = (x,y)^T$ is discrete because I'm working on a discrete set of points. I'm a physicist, so please don't hesitate to ask for more specific information.

I tried something like

$$
rho(vec{k}) = sum_vec{r} e^{-ivec{k}cdotvec{r}}rho(vec{r}),
$$

which should in fact not be too difficult because the only vector that contributes to the sum is $vec{r} = vec{0}$ due to the Kronecker delta in $rho(vec{r})$. The problem is that $rho(vec{r})$ diverges at this point. I need some mathematical advice here. Is there anybody who can show me how to compute the Fourier transform of $rho(vec{r})$?

I'm almost certain the Fourier Transform doesn't exist.

I'll rewrite your radially symmetric (density?) function, $rho(vec{r})$, to be continuous, as opposed to discrete, in 2D polar coordinates:

$$d(r) = {}^2delta(r) alpha left[lambda lambda' J_1(beta r) Y_1(beta r) - pi^2 J_0(beta r) Y_0(beta r)right]$$

Where ${}^2delta(r)$ is the 2D Dirac Delta function at the origin of the polar coordinate system. (For the impulse at the origin, $theta$ doesn't matter, so it doesn't appear as an argument.) In this analysis, ${}^2delta(r)$ represents the following limiting sequence of functions (that are asymmetrical about 0):

$${}^2delta(r) = lim_{epsilon rightarrow 0^+} dfrac{1}{piepsilon^2} quad 0 < r < epsilon$$

so that

$$int_0^{2pi} int_0^{infty} {}^2delta(r) space r space mathrm{d}r space mathrm{d}theta = 1$$

To separate the $r$ and $theta$ portions of ${}^2delta(r)$, so we can use a 1D Dirac Delta function, I'll use the following limiting sequence of functions (that are asymmetric about 0) for the 1D Dirac Delta function:

$$delta_a(r) = lim_{epsilon rightarrow 0^+} dfrac{1}{epsilon} quad 0 < r < epsilon$$

and then note
$$begin{align}\
int_0^{infty} delta_a(r) space mathrm{d}r &= 1\
&= int_0^{2pi} int_0^{infty} {}^2delta(r) space r space mathrm{d}r space mathrm{d}theta\
&= int_0^{infty} {}^2delta(r) space 2pi r space mathrm{d}r\
end{align}$$

so we have

$${}^2delta(r) = dfrac{delta_a(r)}{2pi r}$$

$d(r)$ can now be written without any dependence on $theta$ as

$$d(r) = dfrac{delta_a(r)}{2pi r} alpha left[lambda lambda' J_1(beta r) Y_1(beta r) - pi^2 J_0(beta r) Y_0(beta r)right]$$

The 2D Fourier Transform of a radially symmetric function can be expressed in terms of the Hankel Transform of order 0:

$$F(rho) = mathcal{F}left{f(r)right} = 2pi mathcal{H_0}left{f(r)right} = 2pi int_0^{infty} f(r) J_0(rho r) space r space mathrm{d}r$$

When attempting to compute the Fourier Transform

$$begin{align}mathcal{F}left{d(r)right} = D(rho) &= 2pi int_0^{infty} dfrac{delta_a(r)}{2pi r} alpha left[lambda lambda' J_1(beta r) Y_1(beta r) - pi^2 J_0(beta r) Y_0(beta r)right] J_0(rho r) space r space mathrm{d}r\
\
&= alpha int_0^{infty} delta_a(r) left[lambda lambda' J_1(beta r) Y_1(beta r) - pi^2 J_0(beta r) Y_0(beta r)right] J_0(rho r) mathrm{d}r\
\
&= alphalambda lambda' int_0^{infty} delta_a(r) J_1(beta r) Y_1(beta r) J_0(rho r) mathrm{d}r - alpha pi^2 int_0^{infty} delta_a(r)J_0(beta r) Y_0(beta r) J_0(rho r) mathrm{d}r\
\
&= alphalambda lambda' lim_{r rightarrow 0+} J_1(beta r) Y_1(beta r) J_0(rho r) - alpha pi^2 lim_{r rightarrow 0+} J_0(beta r) Y_0(beta r) J_0(rho r) \
\
&= alphalambda lambda'left(-dfrac1{pi}right) - alpha pi^2 (-infty) \
end{align}$$

it diverges.

Note that if the problematic term had converged, the Fourier Transform would simply be an infinite disk of constant height above the $rhophi$ plane. That should be no surprise for the transform of a Dirac Delta function.

The problematic term would converge, if there were a factor of $1/ln(r)$:

$$lim_{r rightarrow 0+} dfrac{J_0(beta r) Y_0(beta r)}{ln(r)} J_0(rho r) = dfrac{2}{pi}$$

I don't know the physical situation you are analyzing, and whether inclusion of such a term could ever be justified.