Parseval's theorem help. Watch

Preeka
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#1
Report Thread starter 7 years ago
#1
 f(x) = \frac{1}{\epsilon} (1 - \frac{|x|}{\epsilon} , |x| < \epsilon and 0 otherwise.

Fourier transform is given by: \sqrt(\frac{2}{\pi}) (\frac{ 1 - cos(s\epsilon)}{\epsilon^2 s^2})

I have to use parseval's theorem to show that  \int_{-\infty}^{\infty} \frac{sin^4s}{s^4} ds = \frac{2\pi}{3}

I am given a hint that I need to use the fourier transform in two places and choose a suitable value for \epsilon. I'm unsure how to use it twice. I've just ended up muddling up what I'm doing and I'm not sure where I'm headed with that. So far, all I've got is that I would put epsilon as 1 at the end. Any help would be really appreciated for this. Thank you.
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mathemite
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Report 7 years ago
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Hi,

I've come into contact with a lot of these problems and I have found them quite challenging as well Can't say I exactly know what to do but you have to realise that the fourier transform of a sinc function i.e. (sin(x)/x) is a top hat function which makes the integral much simpler. But its actually changing the function into a top hat I find difficult but I've attached some helpful notes I think you will find hat you looking for on pages 24 and 25.
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