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

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Discussion Overview

The discussion revolves around computing the 3D Fourier transform of a point charge potential, specifically addressing the integral form and the appropriate coordinate systems for evaluation.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant presents the integral for the 3D Fourier transform of a point charge potential and expresses uncertainty about how to approach it.
  • Another participant suggests introducing a new variable, defining it as the difference between the position vectors.
  • Changing from Cartesian to spherical coordinates is proposed as a method to simplify the integration process.
  • Some participants clarify that the original integral was not in Cartesian coordinates, questioning the use of the volume differential notation.
  • There is a discussion about the meaning of d^3x, with one participant asserting that it typically refers to the volume differential without implying a specific coordinate system.
  • A recommendation is made to express the volume differential explicitly in a coordinate system, with a preference for spherical coordinates for the integration.

Areas of Agreement / Disagreement

Participants express differing views on the coordinate system used and the interpretation of the volume differential, indicating that there is no consensus on these aspects of the problem.

Contextual Notes

The discussion highlights the need for clarity regarding the coordinate system and the expression of the volume differential, which remain unresolved.

Marthius
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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.
 
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Introduce the variable {\vec r}={\vec x}-{\vec x}'.
 
Change from Cartesian to spherical coordinates.
 
It was not in Cartesian coordinates.
 
Meir Achuz said:
It was not in Cartesian coordinates.

What do you mean by d3x?
 
d^3x usually just means the volume differential, not necessarily in cartesian coords.
 
In order to carry out the integration, the volume differential should be expressed explicitly in some coordinate system. I recommend using spherical.
 

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