# Fourier transform of Bessel function

#### 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?

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#### 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.

• bobred

#### bobred

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

#### 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.

"Fourier transform of Bessel function"

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