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Fourier transform of Bessel function

  1. Jan 2, 2015 #1
    1. The problem statement, all variables and given/known data
    Noting that [itex]J_0(k)[/itex] is an even function of [itex]k[/itex], use the result of part (a) to
    obtain the Fourier transform of the Bessel function [itex]J_0(x)[/itex].

    2. Relevant equations
    In (a) I am asked to show that the Fourier transform of
    [tex]f(x)=\dfrac{1}{\sqrt{1-x^{2}}}[/tex]
    is
    [tex]\tilde{f}(k)=\sqrt{\pi/2}J_0(-k)[/tex]
    where
    [tex]J_0(x)=\frac{1}{\pi}\int_{0}^{\pi} e^{i x \cos \theta}d \theta[/tex]

    3. The attempt at a solution
    I have found the Fourier transform of [itex]f(x)[/itex] using trig substitution I just cant see how to get the FT of [itex]J_0(x)[/itex].
    Any hints as to where I should begin?
     
  2. jcsd
  3. Jan 2, 2015 #2
    Have you heard of the Fourier inversion theorem?
    Make use of that, and the hint that question provided about the even nature of the Bessel function.
     
  4. Jan 4, 2015 #3
    Hi
    I went over my notes a few times and got it.
    Thanks
     
  5. Jan 4, 2015 #4
    Considering the second derivative of png.latex?J_0%28x%29.png show the Fourier transform of png.latex?J_2%28x%29.png is

    D_2%28x%29=%5Csqrt%7B%5Cfrac%7B2%7D%7B%5Cpi%7D%7D%5Cfrac%7B1-2k%5E2%7D%7B%5Csqrt%7B1-k%5E2%7D%7D.png

    I have done similar for png.latex?J_1%28x%29.png using rules for derivatives of Fourier transforms but can't see where to start, where the numerator png.latex?1-2k%5E2.png comes from.
     
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