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3D Fourier Transform 
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#1
Mar512, 07:00 AM

P: 29

The problem statement, all variables and given/known data
Calculate the Fourier transform of f(x) = (1 + x^{2})^{1}, x[itex]\in[/itex]ℝ^{3} The attempt at a solution As far as I can tell, the integral we are supposed to set up is: Mod note: Fixed your equation. You don't want to mix equationwriting methods. Just stick to LaTeX. $$\int \frac{e^{2\pi i (\vec{k}\cdot\vec{x})}}{1+\vec{x}^2}dV = \int \frac{e^{2\pi i r(k_1\sin\theta\cos\phi + k_2 \sin\theta\sin\phi + k_3\cos\theta)}}{1+r^2} r^2\sin\theta\,d\theta\,d\phi\,dr$$but I have no idea how to perform this integral. Any ideas appreciated! (Also, sorry about the fractions  I have no idea why they aren't working because I have no tex experience). 


#2
Mar512, 09:10 AM

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The first step is to use that your f(x) doesn't depend on the direction of the vector x. So the fourier transform won't depend on the direction of k. So you can choose a k that points along the zaxis. That simplifies thing a lot.



#3
Mar512, 08:24 PM

P: 29

*facepalm* Of course, that at least makes the angular part of the integral simple. After the angular integral I ended up with:
$$\frac{2}{k}\int \frac{r\sin2\pi kr}{1+r^2}\,dr$$ I don't think this is integrable, but that makes sense based on the way the question was posed.I think it ought to be square integrable though. I tried to compute this using residues  I was hoping to get something analogous to the 1D Fourier transform $$f(x) = \frac{1}{1 + x^2}, \hat{f}(k) = e^{2\pi xk}$$ but from the look of the residue I have so far it doesn't seem like it will be so aesthetically pleasing. At any rate, thanks for your help, and thanks vela for editing my equation. I've never used LaTex so I was guessing. 


#4
Mar512, 08:33 PM

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3D Fourier Transform



#5
Mar512, 09:52 PM

P: 29

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