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    The question: Use Parseval's Theorem to show that  \int_{-\mu}^{\mu} exp[\frac{-x^2}{2}] dx = \sqrt(\frac{2}{\pi} \int_{-\infty}^{\infty} exp[\frac{-s^2}{2}] \frac {sin\mu s}{s} ds

    My answer:

     \int_{-\mu}^{\mu} 1 dx = \frac{2}{\pi}\int_{-\infty}^{\infty} \frac{sin^2(\mu s)}{s^2}



\mu\pi = \int_{-\infty}^{\infty} \frac{sin^2(\mu s)}{s^2}

    I thought I had completed the question but obviously not. How do I get it into the required form?:confused:

    EDIT: forgot to mention the function.

    f(x) = 1 if  |x| \leq \mu and  f(x) = 0 if  |x| > \mu
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    What is your definition of Fourier transform, and what is your version of Parseval's theorem?
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    (Original post by DFranklin)
    What is your definition of Fourier transform, and what is your version of Parseval's theorem?
    Fourier transform:  \frac{1}{\sqrt(2\pi)} \int_{-\infty}^{\infty} f(x) e^i^s^x dx

    Parsevals' Identity:  \int_{-\infty}^{\infty} f(x)f(x) dx = \int_{-\infty}^{\infty} \tilde{f}(s)\tilde{f}(s) ds

    We were a given a different parseval's identity but the one given above was used for a worked example that is similar to this one. But in our worked one, we ended it where I did so I don't know how to continue it further.
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    Are you sure what you've written for Parseval's Identity is what you meant?
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    (Original post by DFranklin)
    Are you sure what you've written for Parseval's Identity is what you meant?
    The original one involved the complex conjugate of  g(x) and  \tilde{g}(s) on the other side. But we did an example in lectures and we used the one that I posted previously so I thought it'd be relevant to use that one for this question too as there was only a different variable involved and most of the question was directly similar. Have I missed the point somewhere?
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    The one with \int f(x) g(x)\,dx on the LHS is the useful one here.
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    Ok, I just realised I missed out important information when I typed this out the first time. I am revisiting the question now and I think I have the answer but I just want to check. I feel a bit iffy as it didn't really involve any calculations.

     f(x) = \frac{1}{\sqrt(2\pi)\epsilon} exp(\frac{-x^2}{2\epsilon^2})

    Fourier transform of f(x): \frac{1}{\sqrt(2\pi)} exp(\frac{-\epsilon^2s^2}{2})


     h(x) = 1,   if  |x| < \mu. . 0, otherwise.

    Fourier tranform of h(x) is :  \sqrt(\frac{2}{\pi}) \frac{sin \mu s}{s}

    I have literally just multiplied the f(x) and h(x) and then their corresponding transforms together and equated them with the appropriate limits and let my epsilon = 1 and I have shown what is asked to be shown. I just want to make sure if I did it the correct way. Can I just let epsilon = 1 like that?
 
 
 
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