1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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]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
    I went over my notes a few times and got it.
  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


    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted