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Math Help - Fourier transformation

  1. #1
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    Fourier transformation

    Find the Fourier transformation of f(x)=(1-|x|)(H(x+1)-H(x-1)) where H(x) is the funcion of the Heaviside.
    I can't even guess where to start.
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  2. #2
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    Quote Originally Posted by totalnewbie View Post
    Find the Fourier transformation of f(x)=(1-|x|)(H(x+1)-H(x-1)) where H(x) is the funcion of the Heaviside.
    I can't even guess where to start.
    H(x+1) = 0 for x<-1 otherwise it is 1.
    H(x-1) = 0 for x< 1 otherwise it is 1.

    This tells us that,
    H(x+1) - H(x-1) = 0 - 0 = 0 for x<1
    H(x+1) - H(x-1) = 1 - 0 = 1 for -1<x<1
    H(x+1) - H(x-1) = 1 - 1 = 0 for 1<x

    Thus, the function H(x+1)-H(x-1) is non-zero on the interval -1<x<1

    So when you integrate on the interval (-oo,+oo) you can just do it on (-1,1)
    Because it is zero everywhere else and it does not matter.

    But furthermore, on the interval (-1,1) the Heaviside functions add up to 1.
    Which makes no difference when you multiply a number by them.

    Thus,
    1/sqrt(2*pi) * INT(-1,1) (1-|x|)*exp(-ist) dx
    Is the required Fourier Transform.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    H(x+1) = 0 for x<-1 otherwise it is 1.
    H(x-1) = 0 for x< 1 otherwise it is 1.

    This tells us that,
    H(x+1) - H(x-1) = 0 - 0 = 0 for x<1
    H(x+1) - H(x-1) = 1 - 0 = 1 for -1<x<1
    H(x+1) - H(x-1) = 1 - 1 = 0 for 1<x

    Thus, the function H(x+1)-H(x-1) is non-zero on the interval -1<x<1

    So when you integrate on the interval (-oo,+oo) you can just do it on (-1,1)
    Because it is zero everywhere else and it does not matter.

    But furthermore, on the interval (-1,1) the Heaviside functions add up to 1.
    Which makes no difference when you multiply a number by them.

    Thus,
    1/sqrt(2*pi) * INT(-1,1) (1-|x|)*exp(-ist) dx
    Is the required Fourier Transform.
    What do u mean by exp(-ist) ?
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  4. #4
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    Quote Originally Posted by totalnewbie View Post
    What do u mean by exp(-ist) ?
    It is a type for exp(-i x t) which is the kernel of the Fourier Transform (up to
    a randomly positioned constant, and arbitrary convention on what is the
    forward and what the backward transform) where the variable in the
    transformed domain is t.

    RonL
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