I have been reading some material about the relationship between Kepler's Equation for Elliptical Motion and Bessel Functions.(adsbygoogle = window.adsbygoogle || []).push({});

Kepler's Equation for Elliptical Motion is the following:

E = M +esin(E)

It can also be stated in terms of Bessel Functions of the first kind of integer order n:

[tex]E = M + \sum_{1}^{\infty }(\frac{2}{n})J_{n}(ne)sin(nM)[/tex]

where

[tex]J_{n}(x) = \frac{1}{\pi}\int_{0}^{\pi }cos(x sin(\theta) - n\theta )d\theta[/tex]

(the integral form)

My question is:

Can this relationship be stated the other way around (i.e. - Bessel Functions in terms of Kepler's Equation)?

So far, I have not come across one, but I am thinking it might be useful, if any insights gleaned through Kepler's Equation could then be applied to Bessel's Function.

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# Bessel Function FROM Kepler Equation?

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