# Fourier transform of Bessel function

1. Jan 2, 2015

### bobred

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

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

3. The attempt at a solution
I have found the Fourier transform of $f(x)$ using trig substitution I just cant see how to get the FT of $J_0(x)$.
Any hints as to where I should begin?

2. Jan 2, 2015

### Fightfish

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.

3. Jan 4, 2015

### bobred

Hi
I went over my notes a few times and got it.
Thanks

4. Jan 4, 2015

### bobred

Considering the second derivative of show the Fourier transform of is

I have done similar for using rules for derivatives of Fourier transforms but can't see where to start, where the numerator comes from.