Fourier series and the dirchlet integral

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SUMMARY

The discussion centers on using Fourier series to prove the integral \(\int_{0}^{\infty} \frac{\sin(x)}{x} \, \text{d}x = \frac{\pi}{2}\). Participants suggest exploring the Fourier transforms of \(\sin(x)\) and \(\frac{\sin(x)}{x}\), but also note that these functions alone do not yield a straightforward solution. Alternative methods mentioned include the Laplace transform of \(1/x\), indicating multiple approaches to solving the integral.

PREREQUISITES
  • Understanding of Fourier series and transforms
  • Familiarity with Laplace transforms
  • Knowledge of integral calculus
  • Basic concepts of real analysis
NEXT STEPS
  • Research the properties of Fourier series and their applications
  • Study the Laplace transform and its relation to improper integrals
  • Explore advanced techniques in integral calculus
  • Investigate the convergence of Fourier series for various functions
USEFUL FOR

Mathematicians, physics students, and anyone interested in advanced calculus and integral transforms will benefit from this discussion.

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


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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