Strange Fourier transform into something else how?

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Homework Help Overview

The discussion revolves around a Fourier transform problem involving a triple integral and its evaluation. Participants are exploring the transition from a complex integral expression to a simpler form, specifically relating to the Fourier transform of a function involving an exponential term and a denominator with a squared term.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants discuss various methods for evaluating the integral, including contour integration and the residue theorem. There are attempts to change the integral into spherical coordinates and simplify the integrand. Questions arise about the correctness of transformations and the implications of rotating vectors in the context of Fourier transforms.

Discussion Status

The discussion is ongoing, with participants providing suggestions for simplification and integration techniques. Some have expressed uncertainty about specific steps, while others have offered guidance on contour integration and the geometry of the integration path. There is a recognition of the need to clarify the treatment of singularities in the complex plane.

Contextual Notes

Participants are working under the constraints of homework rules, focusing on understanding the mathematical principles involved without arriving at a final solution. There are indications of confusion regarding the application of contour integration and the handling of angular integrations.

  • #31
Well I used the residues method and arrived at e^(-kr)/2 pi r so I seem to be a factor of 2 out..

I'm pretty sure it's not a human error so either I have the right answer now or I missed something?
 
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  • #32
MadMax said:
Well I used the residues method and arrived at e^(ikr)/2 pi r so I seem to be a factor of 2 out..

I'm pretty sure it's not a human error so either I have the right answer now or I missed something?

Not human? Hmm? This is a pretty well known result, so I doubt the math is wrong. You've skipped over a lot of steps here and there's any number of places to drop a 2. For one thing, did you put in a factor of 1/2 when you changed the integration range from 0->infinity to -infinity->infinity? BTW also change the i in your answer to (-1).
 
  • #33
ohh! it'll be the limits then. yes I guess the factor of two comes in because the new limits on rho mean that we no longer need to integrate from 0 to 2 pi in phi but only from 0 to pi.

Thank you very very much Dick! You rock. :D
 
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  • #34
Noooo. It's because the original problem only called for integrating over positive q and in using the contour we integrate over all q. Phi is still 0 to 2*pi. Anyway, good luck with the next residue problem!
 
  • #35
ahh ok, yes. :)

thanks again.
 

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