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Using Parseval's Theorem to evaluate an integral -- Help please

  1. Mar 30, 2015 #1
    1. The problem statement, all variables and given/known data
    By applying Parseval's (Plancherel's) theorem to the function 7a9819bffa30e110228e3910e13a9c1.png are given by:

    f(x) = -1 for -2 ≤ x < 0
    1 for 0 ≥ x < 2
    0 otherwise

    determine the value of the following integral.

    ∫ dk sin^4(k)/(k^2) (Integral between ±infinity)

    2. Relevant equations

    Parseval's Theorem, i.e the integral of the modulus squared of a function is equal to the integral of the modulus squared of its fourier transform.

    Fourier Transform formula.

    3. The attempt at a solution

    I've tried multiple times to try and arrive at the correct answer of pi/2 but I just can't do it. Is the fourier transform:

    ∫-dx e^(ikx) + ∫ dx e^(ikx)? (First integral is between -2 and 0, second between 0 and 2). Because I can't get the correct answer doing that.

    And for the equation of the function, is it just sgn(x), with the integral between -2 and 2? How else do I write f(x) given those boundaries? If I use sgn(x) between -2 and 2, I get 4 for the left hand side. But I still can't derive that integral.

    If someone would just start me off with the correct fourier transform equation, i'll go away and crunch the integration myself.

    Thank you.
     
    Last edited by a moderator: Apr 19, 2017
  2. jcsd
  3. Mar 31, 2015 #2

    Svein

    User Avatar
    Science Advisor

    Yes, but then you need to adjust the exponential factor, since eikx is periodic with period 2π, not 4. Try eikπx/2.
     
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