Fourier transform of Bessel function

  • Thread starter bobred
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  • #1
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Homework Statement


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].

Homework 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]

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?
 

Answers and Replies

  • #2
954
117
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.
 
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  • #3
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Hi
I went over my notes a few times and got it.
Thanks
 
  • #4
173
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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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