How Do You Solve Poisson's Equation Using Fourier Transforms?

[ultimateguy]

Homework Statement

Solve Poisson's equation, $$\bigtriangledown^2 \psi(\vec{r}) = \frac{- \rho (\vec{r})}{\epsilon_0}$$, by the following sequence of operations:

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

Homework Equations

Fourier transformation

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.

$$\bigtriangledown^2 \psi(\vec{r}) = \frac{- \rho (\vec{r})}{\epsilon_0}$$
$$\frac{-\vec{k}^2}{\sqrt{2\pi}} \int^\infty_{-\infty} \psi(\vec{r}) e^{i \vec{k} \cdot \vec{r}} d\vec{r} = \frac{-1}{\epsilon_0 \sqrt{2\pi}} \int^\infty_{-\infty} \rho (\vec{r}) e^{i \vec{k} \cdot \vec{r}} d\vec{r}$$
$$-\vec{k}^2 \Psi(\vec{k}) = \frac{-1}{\epsilon_0} P(\vec{k})$$
$$\Psi (\vec{k}) = \frac{1}{\epsilon_0 \vec{k}^2} P (\vec{k})$$

 

[mjsd]

seems ok...