Fourier transform of Bessel functions

Question

I'm curious as to how the Fourier transform of the various types of Bessel functions would be calculated. The Wikipedia page on the Fourier transform gives the transform of $J_o(x)$ as being $\frac{2rect(\pi\zeta)}{\sqrt{1-4\pi^2\zeta^2}}$. I've searched the web some but I seem to be unable to find a derivation for that value. Since $J_o$ only "damps out" to zero at infinity I'm guessing the transform must be computed using some kind of generalized function representation of the Bessel function. Any references would be much appreciated.

Answer 1

Although this question was asked and answered quite a while ago, I thought that it might be useful to see an alternative development, one that uses classical analysis only and forgoes the use of Generalized Functions.

Here, we will find the Fourier Transform of $J_0(x)$ by first finding the Fourier Transform representation of $J_0(x)$ and subsequently invoking the Fourier Inversion Theorem.

We begin, as @Sasha began, with the integral representation

$$\begin{align} J_0(x)&=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ix\sin \phi}\,d\phi\\\\ &=\frac{1}{\pi}\int_{0}^{\pi}\cos (x\sin \phi)\,d\phi \tag 1 \end{align}$$

where we exploited the fact that real part of the integrand is an even function of $k$ while the imaginary part is odd. Making the substitutions

$$\phi= \begin{cases} \arcsin (k),&\text{for}\,\,0\le\phi\le\pi/2\\\\ \pi-\arcsin (k),&\text{for}\,\,\pi/2\le\phi\le\pi \end{cases}$$

into $(1)$ yields

$$\begin{align} J_0(x)&=\frac{2}{\pi}\int_{0}^{1}\frac{1}{\sqrt{1-k^2}}\cos (kx)\,dk \tag 2\\\\ &=\frac{1}{\pi}\int_{-1}^{1}\frac{1}{\sqrt{1-k^2}}\cos (kx)\,dk \tag 3\\\\ &=\frac{1}{\pi}\int_{-1}^{1}\frac{1}{\sqrt{1-k^2}}e^{ikx}\,dk \tag 4\\\\ &=\frac{1}{2\pi}\int_{-1}^{1}\frac{2}{\sqrt{1-k^2}}e^{ikx}\,dk \tag 5\\\\ &=\frac{1}{2\pi}\int_{-\infty}^{\infty}\left(\text{rect}\left(\frac{k}{2}\right)\frac{2}{\sqrt{1-k^2}}\right)e^{ikx}\,dk \tag 6\\\\ \end{align}$$

where $\text{rect}$ is the Rectangular Function. Finally, using the Fourier Inversion Theorem, we have

$$\bbox[5px,border:2px solid #C0A000]{\mathscr{F}\left(J_0(x)\right)(k)=\text{rect}\left(\frac{k}{2}\right)\frac{2}{\sqrt{1-k^2}}}$$

recovering the well-known result.

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NOTES:

In arriving at $(2)$, we wrote $\int_{0}^{\pi}\cos (x\sin(\phi))\,d\phi=\int_{0}^{\pi/2}\cos (x\sin(\phi))\,d\phi+\int_{\pi/2}^{\pi}\cos (x\sin(\phi))\,d\phi$, enforced the substitutions, and combined the resulting integrals.

In going from $(2)$ to $(3)$, we exploited the even property of the cosine.

In going from $(3)$ to $(4)$, we exploited the odd property of the sine.

In going from $(4)$ to $(5)$, we placed a factor of $1/2$ outside the integral and a factor of $2$ in the integrand.

In going from $(5)$ to $(6)$, we multiplied the integrand by the rectangle function and extended the limits.

Answer 2

Using integral representation: $$ J_0(x) = \frac{1}{2 \pi} \int_{-\pi}^\pi \mathrm{e}^{i x \sin \tau} \mathrm{d} \tau $$ Thus the Fourier transform: $$ \begin{eqnarray} \mathcal{F}_x(J_0(x))(\omega) &=& \int_{-\infty}^\infty J_0(x) \mathrm{e}^{i \omega x} \mathrm{d} x \\ &=& \frac{1}{2 \pi} \int_{-\pi}^\pi \mathrm{d} \tau \, \mathcal{F}_x(\mathrm{e}^{i x \sin \tau})(\omega) \\ &=& \frac{1}{2\pi} \int_{-\pi}^\pi \mathrm{d} \tau \, \left( 2 \pi \right) \delta\left( \omega + \sin(\tau) \right) \\ &=& \int_{-\pi}^\pi \mathbf{1}_{-1 \le \omega \le 1} \delta\left( \omega + \sin(\tau) \right) \,\, \mathrm{d} \tau \\ &=& \int_{-\pi}^\pi \mathbf{1}_{-1 \le \omega \le 1} \frac{1}{\vert \cos(\tau) \vert} \left( \delta\left( \arcsin \omega + \tau \right) + \delta\left( \arcsin \omega - \operatorname{sign}(\omega) \pi + \tau \right) \right)\,\, \mathrm{d} \tau \\ &=& \mathbf{1}_{-1 \le \omega \le 1} \frac{2}{\sqrt{1-\omega^2}} \end{eqnarray} $$