Fourier series and the dirchlet integral

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Discussion Overview

The discussion revolves around the use of Fourier series to evaluate the integral \(\int_{0}^{\infty} \frac{\sin(x)}{x}\ \mbox{d}x\) and seeks to identify suitable functions for Fourier transformation. The scope includes mathematical reasoning and exploration of different transformation techniques.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant proposes using a Fourier series to prove the integral's value and inquires about the appropriate function for Fourier transformation.
  • Another participant suggests trying to Fourier-transform \(\sin(x)\) and \(\sin(x)/x\) as potential approaches.
  • A later reply indicates that there are multiple methods to arrive at the answer, which do not solely rely on \(\sin(x)\) or \(\sin(x)/x\), expressing surprise at the lack of deeper exploration of the problem.
  • One participant acknowledges a misunderstanding regarding the use of Fourier transform versus Fourier series and agrees that transforming \(\sin(x)\) or \(\sin(x)/x\) does not seem straightforward.
  • There is a question about the alternative methods mentioned, with a suggestion that one might involve the Laplace transform of \(1/x\).

Areas of Agreement / Disagreement

Participants express differing views on the appropriate functions for Fourier transformation and the methods to evaluate the integral. There is no consensus on the best approach or the specific methods to be used.

Contextual Notes

Participants have not fully explored all assumptions or provided detailed mathematical steps for the proposed methods. The discussion reflects uncertainty regarding the effectiveness of different transformation techniques.

dirk_mec1
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Use a Fourier series to prove that: [tex]\int_{0}^{ \infty} \frac{\sin(x)}{x}\ \mbox{d}x = \frac{ \pi}{2}[/tex]


What function do I need to Fourier transform?
 
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There aren't many to choose from are they? I suggest you try Fourier-transforming sin(x) and sin(x)/x and see where it leads you.
 
Pere Callahan said:
There aren't many to choose from are they? I suggest you try Fourier-transforming sin(x) and sin(x)/x and see where it leads you.

Actually there are and I found that there are two ways of finding this answer both NOT with solely a sin(x) or sin(x)/x. I'm suprised that you didn't took the effort to look more closely to the question at hand.
 
dirk_mec1 said:
Actually there are and I found that there are two ways of finding this answer both NOT with solely a sin(x) or sin(x)/x. I'm suprised that you didn't took the effort to look more closely to the question at hand.

You're right. In particular I read Fouriertransform instead of Fourier Series. You're also right in saying that transforming sin(x) or sin(x)/x seems not to provide an easy solution to the problem.

Which two ways are you referring to? One is likely to Laplace-transform 1/x ..?
 

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