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Need help with Fourier transformation

  1. Apr 2, 2007 #1
    1. The problem statement, all variables and given/known data
    Solve Poisson's equation, [tex]\bigtriangledown^2 \psi(\vec{r}) = \frac{- \rho (\vec{r})}{\epsilon_0}[/tex], by the following sequence of operations:

    a) Take the Fourier transform of both sides of this equation. Solve for the Fourier transform of [tex]\psi(\vec{r})[/tex].
    b) Carry out the inverse transform by using a three-dimensional analog of the convolution theorem.

    2. Relevant equations
    Fourier transformation

    3. The attempt at a solution

    I'm working on part a. I understand how to do a Fourier transform, but not so much in 3D. This is my work, I just want to make sure that I've done part a correctly so that I can move on to part b.

    [tex]\bigtriangledown^2 \psi(\vec{r}) = \frac{- \rho (\vec{r})}{\epsilon_0}[/tex]
    [tex]\frac{-\vec{k}^2}{\sqrt{2\pi}} \int^\infty_{-\infty} \psi(\vec{r}) e^{i \vec{k} \cdot \vec{r}} d\vec{r} = \frac{-1}{\epsilon_0 \sqrt{2\pi}} \int^\infty_{-\infty} \rho (\vec{r}) e^{i \vec{k} \cdot \vec{r}} d\vec{r}[/tex]
    [tex]-\vec{k}^2 \Psi(\vec{k}) = \frac{-1}{\epsilon_0} P(\vec{k})[/tex]
    [tex] \Psi (\vec{k}) = \frac{1}{\epsilon_0 \vec{k}^2} P (\vec{k})[/tex]
    Last edited: Apr 2, 2007
  2. jcsd
  3. Apr 2, 2007 #2


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    Homework Helper

    seems ok.......
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