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Math Help - Fourier Integral

  1. #1
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    Nov 2011
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    Fourier Integral

    Hi! There was a first part to this question to find the fourier series repesentation, I can do that fine but then I cant seem to get around this integral. I am studying for exams tomorrow so I'm going through past papers and this question comes up a lot so if anyone could help it would be really helpful! Thanks

    The Fourier transforms of the function f(x) are given by
    ∞ ∞
    A(λ) = ∫ f(x) cosλx dx, B(λ) = ∫ f(x) sin λx dx
    -∞ -∞

    with the Fourier Integral representation of f(x) given by

    f(x) = 1/pi ∫ A(λ) cosλx dx + B(λ)sin λx dx
    0

    The non-periodic function f is defined by

    f(x) =1, −1 < x < 1
    0, otherwise.
    Obtain a Fourier integral representation for f(x).
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  2. #2
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    Joined
    Nov 2011
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    Re: Fourier Integral

    Think I finally got it.. Anyone able to tell me if it's right?



    A(λ) = ∫(-∞ to ∞) f(x) cos(λx) dx
    .......= ∫(-∞ to -1) f(x) cos(λx) dx + ∫(-1 to 1) f(x) cos(λx) dx + ∫(1 to ∞) f(x) cos(λx) dx
    .......= ∫(-∞ to -1) 0 * cos(λx) dx + ∫(-1 to 1) 1 * cos(λx) dx + ∫(1 to ∞) 0 * cos(λx) dx
    .......= ∫(x = -1 to 1) cos(λx) dx
    .......= (1/λ) sin(λx) {for x = -1 to 1}
    .......= 2 sin(λ)/λ.

    Similarly,
    B(λ) = ∫(-∞ to ∞) f(x) sin(λx) dx
    .......= ∫(x = -1 to 1) sin(λx) dx
    .......= (-1/λ) cos(λx) {for x = -1 to 1}
    .......= 0.

    Hence, f(x) = (1/π) ∫(x = 0 to ∞) [2 sin(λ)/λ] cos(λx) dx.
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