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

1. Mar 23, 2012

### mogul

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