Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Inverse Fourier Transform of Bessel Functions

  1. Jan 5, 2006 #1
    I want to solve the partial differential equation
    [itex]\Delta f(r,z) = f(r,z) - e^{-(\alpha r^2 + \beta z^2)}[/itex]
    where [itex]\Delta[/itex] is the laplacian operator and [itex]\alpha, \beta > 0[/itex]
    In full cylindrical symmetry, this becomes
    [itex]\frac{\partial_r f}{r} + \partial^2_rf + \partial^2_z f = f - e^{-(\alpha r^2 + \beta z^2)}[/itex]
    Applying the fourier transform along the cylindrical symmetry axis one obtains the following ODE
    [itex]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)}\}[/itex]
    [itex]\mathcal{F} \equiv \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dz\, e^{i k_z z}[/itex].
    The solution to the homogeneous part, according to Mathematica is
    [itex]\hat{f} = C_1 J_0(ir\sqrt{k_z^2 + 1}) + C_2 Y_0(-ir\sqrt{k_z^2 + 1})[/itex]
    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 [itex]\hat{f}[/itex] 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 [itex]k_z \rightarrow \infty[/itex]
    [itex]\hat{f} = C_1 J_0(irk_z) + C_2 Y_0(-irk_z)[/itex]
  2. jcsd
  3. Jan 5, 2006 #2

    Dr Transport

    User Avatar
    Science Advisor
    Gold Member

  4. Jan 5, 2006 #3
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook