Calculating Fourier Transform on TI-89 for Non-Integrable Functions?

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Discussion Overview

The discussion revolves around calculating the Fourier transform of non-integrable functions, specifically the function 1/(1+t^2), and the challenges associated with this calculation using a TI-89 calculator. Participants explore various methods for addressing the integration difficulties encountered.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how to calculate the Fourier transform of 1/(1+t^2) due to integration issues with the TI-89.
  • Another suggests integration by parts as a potential method, while expressing concern about the complexity of the process.
  • A participant mentions that Mathematica provides a Fourier transform result of sqrt(pi/2)/e^abs(w), but they are unsure how to derive it manually.
  • There is a mention of a book providing a different answer, pi*e^abs(w), leading to uncertainty about which result is correct.
  • One participant proposes using contour integration as a method to evaluate the integral, referencing complex variable techniques.
  • A later reply confirms that contour integration is effective and provides a link to a related thread for further reference.
  • Another participant notes that the choice of contour in contour integration depends on the sign of omega, adding a layer of complexity to the discussion.

Areas of Agreement / Disagreement

Participants express differing opinions on the correct result of the Fourier transform, with some supporting the Mathematica output and others referencing the book's answer. The discussion remains unresolved regarding which method or answer is definitive.

Contextual Notes

There are unresolved issues regarding the assumptions made in the integration methods and the definitions of the Fourier transform being used, which may contribute to the differing results.

kolycholy
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how would i calculate Fourier transform of functions such as 1/(1+t^2)?
because if you try to integrate the product of the above function and e^(-jxt), you would realize it's nonintegrable or something
at least my ti-89 does not calculate it for me.
any other way?
 
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Integration by parts perhaps, although I can imagine that getting very messy very fast. Is it possible that you could integrate numerically? I think that would at least work in finding a Fourier series, but I'm not certain.
 
Mathematica can calculate the Fourier transform, according to that it's sqrt(pi/2)/e^abs(w)

however, I'm not completely sure how you would go about doing it manually. I would agree that integration by parts is the first step to try.
 
jbusc said:
Mathematica can calculate the Fourier transform, according to that it's sqrt(pi/2)/e^abs(w)

however, I'm not completely sure how you would go about doing it manually. I would agree that integration by parts is the first step to try.

the answer given in the book is pi*e^abs(w).
One of them is right. or maybe both of them are right?
 
You would need to solve this integral via contour integration (at least this is the first thing that comes to mind). Have a look on google for "functions of a complex variable" or "residue theorem" or simply "contour integral" or something like this. Anyway, contour integration allows you evaluate these types of integrals pretty quickly.

By the way the answers stated here differ by a factor of \pi^{1/2}. This is probably because one of you is using the forward Fourier transform as being defiend with a prefactor of 1/2\pi and the backward transform having a prefactor of 1, while the other one is using the prefactor [1/2\pi]^{1/2} for both the forward and backward transforms.
 
Hi George, thanks for the thread.
I should say that in the post on this thread, the contour you pick (semicircle in the upper or lower half plane) is determined by the sign of \omega.
 

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