# Tails of the Airy disk intensity formula?

The Airy disk intensity formula is given in the mathematical details section of the Wikipedia Airy disk article in terms of the Bessel function of the first kind.

I am interested in the asymptotic behavior of the tail of this function. Is it exponential or powerlaw? If powerlaw, what power? Can you give a Taylor series expansion? TIA.

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## migrated from physics.stackexchange.comDec 9 '12 at 23:52

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(Moderators: This is a very mathematical question. Feel free to migrate it to Math SE if you think it will get better answers there.) – Jim Graber Dec 9 '12 at 21:52

Answering one wiki entry with another: The asymptotic forms of the Bessel functions are described here. From the article, $$J_\alpha(x) \sim \sqrt{\frac{2}{\pi x}} \cos\left(x - \frac{\alpha\pi}{2} - \frac{\pi}{4}\right)$$ for large $x$, so $J_1(x)/x$ asymptotically approaches a sinusoid with power-law envelope proportional to $x^{-3/2}$. Thus the Airy pattern in intensity, $$I(\theta) = I_0 \left(\frac{2J_1(x)}{x}\right)^2, \quad x = ka\sin(\theta),$$ goes as the square of a sinusoid with an $x^{-3}$ envelope. Here $k$ is the wavenumber of the light, $a$ is the radius of the aperture, and $\theta$ is the angle away from the normal to the aperture.