Is There a Closed Form for Airy Disk Eigenfunctions?

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SUMMARY

The discussion centers on the eigenfunctions of the Airy disk function, specifically the Bessel function \(\frac{J_1(x)}{x}\). Participants conclude that there is no known closed form for these eigenfunctions, as existing literature primarily discusses approximations, such as the first zero and angular calculations. The consensus is that while approximations exist, definitive closed forms have not been established in the academic publications reviewed.

PREREQUISITES
  • Understanding of Bessel functions, specifically \(\frac{J_1(x)}{x}\)
  • Familiarity with eigenfunction concepts in mathematical physics
  • Knowledge of approximation techniques in mathematical analysis
  • Basic grasp of Airy disk theory in optics
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Mathematicians, physicists, and optical engineers interested in the theoretical aspects of Bessel functions and their applications in wave phenomena.

marcusl
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Does anyone know if the eigenfunctions of the Airy disk function (or Bessel function) \frac{J_1(x)}{x} has a closed form?
 
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From what I have seen on the internet, the answer is probably no. There are many publications and neither mentioned one, only approximations like the first zero and calculation of angles.
 

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