Hi! How can I calculate the Fourier transform of the potential generated by a point charge in the origin with the charge q?

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- February 28th 2011, 06:29 PMTriKriFourier transform of potential from point charge
Hi! How can I calculate the Fourier transform of the potential generated by a point charge in the origin with the charge q?

- February 28th 2011, 06:57 PMchisigma
How many dimensions?... one, two, three... or more?...

Kind regards

- February 28th 2011, 08:26 PMTriKri
Three dimensions! :)

- March 1st 2011, 06:07 AMchisigma
All right!... but for semplicity sake may be it is better to start with the two-dimensional case (Thinking)...

The potential function generated by a charge q in the origin is...

(1)

The two-dimension Fourier tranform of V(x,y) by definition is...

(2)

Now if we pass to polar coordinates the (2) becomes...

(3)

Now it is easy to verify that the integral...

(4)

... vanishes everiwhere with the only exception of the case where is so that the final result is...

(5)

Kind regards

- March 1st 2011, 12:47 PMTriKri
Thank you, but does the integral really vanish? If the exponent over e will be in the interval , so the real part of the integrand will be positive for all and then the integral can't vanish, can it?

- March 5th 2011, 08:20 AMTriKri
I'm still wondering what the Fourier transform is; the delta function can't be the answer since it's the Fourier transform of 1 (or at least of a real constant). The Fourier transform of 1/r in three dimensions has to be some kind of standard transform, hasn't it?