How do I compute the 3D Fourier transform of a point charge potential?

[Marthius]

I need to take the 3-Dimensional Fourier transform of a point charge potential. I have an integral of this form, but I am unsure as to how to approach this integral.

$$\int e^{-i\vec{x}\vec{k}} \frac{q}{|\vec{x}-\vec{x'}|}d^3x$$

A push in the right direction would be appreciated.

 

[Meir Achuz]

Introduce the variable $${\vec r}={\vec x}-{\vec x}'$$.

 

[mathman]

Change from Cartesian to spherical coordinates.

 

[Meir Achuz]

It was not in Cartesian coordinates.

 

[mathman]

It was not in Cartesian coordinates.

What do you mean by d³x?

 

[Meir Achuz]

d^3x usually just means the volume differential, not necessarily in cartesian coords.

 

[mathman]

In order to carry out the integration, the volume differential should be expressed explicitly in some coordinate system. I recommend using spherical.