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2D Fourier of 3D (vector) function - laplace of it?

  1. Mar 23, 2012 #1
    Hi,
    I have a vector function (momentum [tex]\vec{p}(x,y,z)=\vec{p}(\vec{r},z)[/tex] for [tex]\vec{r}=(x,y)[/tex]) and need to transform it, also I use the fourier transform
    [tex]F(\vec{p})=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \vec{p}(\vec{r},z) exp^{i \vec{\chi}\cdot \vec{r}}d\vec{r}[/tex]
    and then
    when I have in the equation laplace operator,
    [tex]\Delta\vec{p}(x,y,z)=\Delta\vec{p}(\vec{r},z)[/tex]
    which I have transformed, what is the appropriate expression for it?
    [tex]\Delta\vec{p}(\vec{r},z)\rightarrow (FT)\rightarrow ???[/tex]
    is it this?
    [tex]\Delta\vec{p}(\vec{r},z)\rightarrow (FT) \rightarrow (\chi^2+\frac{\partial^2}{\partial z^2})\vec{\hat{p}}(\vec{\chi},z)[/tex]
    is this right?
    thank you for your help!
     
  2. jcsd
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