# Inverse Fourier Transform of Bessel Functions

1. Jan 5, 2006

### jdstokes

I want to solve the partial differential equation
$\Delta f(r,z) = f(r,z) - e^{-(\alpha r^2 + \beta z^2)}$
where $\Delta$ is the laplacian operator and $\alpha, \beta > 0$
In full cylindrical symmetry, this becomes
$\frac{\partial_r f}{r} + \partial^2_rf + \partial^2_z f = f - e^{-(\alpha r^2 + \beta z^2)}$
Applying the fourier transform along the cylindrical symmetry axis one obtains the following ODE
$d^2_r\hat{f} + \frac{1}{r}d_r\hat{f} - (k_z^2 + 1) \hat{f} = \mathcal{F}\{e^{-(\alpha r^2 + \beta z^2)}\}$
where
$\mathcal{F} \equiv \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dz\, e^{i k_z z}$.
The solution to the homogeneous part, according to Mathematica is
$\hat{f} = C_1 J_0(ir\sqrt{k_z^2 + 1}) + C_2 Y_0(-ir\sqrt{k_z^2 + 1})$
for arbitrary constants C_1, C_2 and J_0 and Y_0 are Bessel functions of the first and second kind, respectively.
I tried to take the inverse fourier transform of $\hat{f}$ using Mathematica, but it won't give a result. Is anyone aware of, or know of a book containting this integral transform? If not, how about the limiting case as $k_z \rightarrow \infty$
$\hat{f} = C_1 J_0(irk_z) + C_2 Y_0(-irk_z)$
Thanks.
James

2. Jan 5, 2006

3. Jan 5, 2006