Hi,(adsbygoogle = window.adsbygoogle || []).push({});

I have a question about the fourier transform of [itex]\frac{1}{|\mathbf{r_1} - \mathbf{r_2}|}[/itex] over a finite cube of unit volume. Where [itex]|\mathbf{r_1} - \mathbf{r_2}|[/itex] is [itex]\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}[/itex]

I know it looks like

[itex]\sum_\mathbf{k} f_k e^{-i\mathbf{k}\cdot (\mathbf{r_1}-\mathbf{r_2})}[/itex]

where f_k is the fourier coefficient

[itex]f_k = \frac{1}{V} \int_V \frac {e^{-i\mathbf{k} \cdot \mathbf{r} } } {|\mathbf{r}| } d\mathbf{r}[/itex]

over the volume {-1,1}{-1,1}{-1,1}

My question is, what happens when [itex]\frac{1}{|\mathbf{r_1} - \mathbf{r_2}|}[/itex] is not radially symmetric. Say [itex]|\mathbf{r_1} - \mathbf{r_2}|[/itex] is

[itex]\sqrt{(x_1-x_2)^2 + a(y_1-y_2)^2 + b(z_1-z_2)^2}[/itex]

would the expression then become

[itex]\sum_\mathbf{k} f_k e^{-i\mathbf{k}\cdot (x_1-x_2)}e^{-i\mathbf{k}\cdot a(y_1-y_2)}e^{-i\mathbf{k}\cdot b(z_1-z_2)}[/itex]

and would the coefficient f_k be affected? My guess is yes it would be over the interval {-1,1},{-a,a},{-b,b}

Is this correct?

Thanks

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Fourier transform of 1/|r|

Loading...

Similar Threads for Fourier transform |
---|

I Repetitive Fourier transform |

I Proving the Linear Transformation definition |

I Units of Fourier Transform |

A Help with Discrete Sine Transform |

**Physics Forums | Science Articles, Homework Help, Discussion**