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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!

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

Can you offer guidance or do you also need help?

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