Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|cite|improve this question

2 Answers 2

up vote 8 down vote accepted

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

share|cite|improve this answer
Thanks, that helps! –  Bitrex Nov 2 '11 at 21:06
Hi, I am trying to do this problem, but instead I am trying $$ \int_{-\infty}^\infty J^3_0(x)e^{i\omega x} dx. $$ Any idea how I would go about this? THanks... –  Integrals Mar 12 '14 at 4:45

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

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

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.


In arriving at $(2)$, we wrote $\int_{0}^{\pi}\cos (x\phi)\,d\phi=\int_{0}^{\pi/2}\cos (x\phi)\,d\phi+\int_{\pi/2}^{\pi}\cos (x\phi)\,d\phi$, 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.

share|cite|improve this answer
Thank you! I receive your notification. I'm not able to use this site as much as I'd like these days unfortunately, but your answer is great. –  Bitrex Jul 17 at 1:24
You're welcome! My pleasure. It gives another way to see the transform. –  Dr. MV Jul 17 at 1:49

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.