Fourier transform of Airy function

Question

I am looking for the Fourier transform of the Airy function $\left(\dfrac{2J_1(x)}{x}\right)^2$ where $J_1$ is the Bessel function of the first kind of order one.

Thank you.

Answer 1

I'm not sure which of the Airy functions this is related to, but Wolfram Alpha and Mathematica give the result

$$\mathcal{F}_x\left[\left(\frac{2 J_1(x)}{x}\right)^2\right](\omega)=\sqrt{\frac{2}{\pi}} \omega\, (\theta(-\omega-2)-\theta(\omega-2)+2 \theta(\omega)-1)\, G_{2,2}^{2,0}\left(\frac{\omega^2}{4}| \begin{array}{c} 1,2 \\ -\frac{1}{2},\frac{1}{2} \\ \end{array} \right)\tag{1}$$

in terms of the MeijerG function (see WolframAlpha evaluation) which expands in terms of Elliptic E and Elliptic K functions as

$$\mathcal{F}_x\left[\left(\frac{2 J_1(x)}{x}\right)^2\right](\omega)=\left\{\begin{array}{cc} \frac{4 \sqrt{2} \left(\left(\omega^2+4\right)\, E\left(1-\frac{\omega^2}{4}\right)-2 \omega^2\, K\left(1-\frac{\omega^2}{4}\right)\right)}{3 \pi^{3/2}} & -2<\omega<2 \\ 0 & \text{Otherwise} \\ \end{array}\right.\tag{2}$$

where the Fourier transform is defined as

$$\mathcal{F}_x[f(x)](\omega)=\frac{1}{\sqrt{2 \pi}}\int\limits_{-\infty}^\infty f(x)\, e^{i \omega x}\,dx\,.\tag{3}$$

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The figure below illustrates a plot of the right-side of formula (2) above.