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Fourier transform of Bessel function

  • Thread starter bobred
  • Start date
173
0
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?
 
954
116
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.
 
173
0
Hi
I went over my notes a few times and got it.
Thanks
 
173
0
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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