Analytical three-dimensional Fourier transform

In summary, the person is struggling with a specific task in Byron & Fuller's mathematical methods for classical and quantum physics and is unable to find the necessary information in the book or through Google. They mention problem 5.9 and have attempted to solve it using spherical coordinates, but got stuck. They also mention trying to use Gauss or Green's theorem, but have not been successful. They are requesting help and guidance on how to solve this task.
  • #1
johnkay
10
0
Hello, I am trying to do some self-studying in Byron & Fuller mathematical methods for classical and quantum physics. I have slightly ran aground on this one task of finding 3d Fourier transforms and I can't find the info in the book itself to free me. Google has neither been very fruitfull.

Homework Statement



It is problem 5.9 in the book and can be seen here (obviously I only need to really 'solve' a.)

http://dl.dropbox.com/u/2507151/screenshot.png"

The Attempt at a Solution



I tried going into spherical coordinates and ended up with a big mess and I was not quite sure what to do there anyway.

Then I wondered if I perhaps could make the problem into some almost one-liner through Gauss or Green's theorem or the like. But I have not been quite successful there either.

any pointers on how this should be done?
 
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  • #2
Your first approach sounds right. Show us what you did.
 

1. What is an analytical three-dimensional Fourier transform?

An analytical three-dimensional Fourier transform is a mathematical tool used in signal processing and image analysis to transform a three-dimensional signal or image from its original domain (usually time or space) to a representation in the frequency domain. This allows for analysis and manipulation of the signal or image in terms of its frequency components.

2. How is an analytical three-dimensional Fourier transform different from a regular Fourier transform?

An analytical three-dimensional Fourier transform is a generalization of the regular Fourier transform, which is typically used for one-dimensional signals or images. The main difference is that the analytical three-dimensional Fourier transform operates on a three-dimensional signal or image, taking into account variations in all three dimensions, rather than just one. This allows for a more comprehensive analysis of the signal or image.

3. What is the purpose of using an analytical three-dimensional Fourier transform?

The purpose of using an analytical three-dimensional Fourier transform is to analyze and manipulate three-dimensional signals or images in the frequency domain. This can provide insights into the underlying structure of the signal or image, and can also be used for tasks such as filtering, noise reduction, and compression.

4. What are some applications of analytical three-dimensional Fourier transform?

Analytical three-dimensional Fourier transforms are used in a variety of applications, including medical imaging, radar and sonar signal processing, seismic data analysis, and computer vision. They are also used in the development of 3D graphics and virtual reality technologies.

5. Are there any limitations to using an analytical three-dimensional Fourier transform?

Like any mathematical tool, there are limitations to using an analytical three-dimensional Fourier transform. One limitation is that it assumes the signal or image is stationary, meaning it does not change over time. It also requires the signal or image to be sampled at regular intervals, and may not be suitable for non-periodic signals or images with sharp edges or discontinuities.

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