# Calculate 3D Fourier Transform for f(x) = (1 + |x|2)-1 on ℝ3

• Ansatz7
In summary, the conversation is about calculating the Fourier transform of the function f(x) = (1 + |x|2)-1 using an integral. The first step is to choose a k vector that points along the z-axis to simplify the integral. After the angular integral, the resulting equation is not integrable. However, it is still square integrable and can be computed using residues with a similar form to the 1D Fourier transform. It is also mentioned that careful attention must be paid to signs and factors.
Ansatz7
Homework Statement
Calculate the Fourier transform of f(x) = (1 + |x|2)-1, x$\in$ℝ3

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 equation-writing 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).

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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 z-axis. That simplifies thing a lot.

*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

$$f(x) = \frac{1}{1 + x^2}, \hat{f}(k) = e^{-2\pi x|k|}$$

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.

Ansatz7 said:
*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

$$f(x) = \frac{1}{1 + x^2}, \hat{f}(k) = e^{-2\pi x|k|}$$

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.

I'm not checking the details here, so I hope you are keeping track of all of the signs and factors. But that looks integrable to me. There are poles at i and -i. You'll have to split the sin up into exponentials so you can decide which half-plane to close the contours in, but it looks routine to me.

Deleted

## 1. What is a 3D Fourier Transform?

A 3D Fourier Transform is a mathematical operation that converts a three-dimensional function into the frequency domain. It decomposes a signal into its constituent frequencies and represents it as a sum of sinusoidal functions. It is commonly used in scientific fields such as image processing, physics, and engineering.

## 2. How does a 3D Fourier Transform work?

The 3D Fourier Transform works by applying a complex mathematical algorithm to a three-dimensional function. It involves breaking down the function into a series of sinusoidal functions with different frequencies, amplitudes, and phases. These sinusoidal functions are then combined to reconstruct the original function in the frequency domain.

## 3. What are the applications of 3D Fourier Transform?

3D Fourier Transform has a wide range of applications in various fields such as medical imaging, geophysical exploration, radar and sonar imaging, and crystallography. It is also used in digital signal processing, pattern recognition, and data compression.

## 4. What are the limitations of 3D Fourier Transform?

One of the main limitations of 3D Fourier Transform is that it assumes the function being transformed is stationary and infinite. In reality, most signals are time-dependent and have a finite duration. Additionally, 3D Fourier Transform is sensitive to noise and can produce artifacts in the reconstructed image.

## 5. Are there any alternative methods to 3D Fourier Transform?

Yes, there are alternative methods to 3D Fourier Transform such as the Wavelet Transform and the Short-Time Fourier Transform. These methods offer different advantages and are used in specific applications. For example, the Wavelet Transform is better suited for analyzing non-stationary signals, while the Short-Time Fourier Transform is commonly used in audio and speech processing.

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