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## Main Question or Discussion Point

Hi,

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

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