# Bessel Function FROM Kepler Equation?

1. Apr 7, 2012

### DuncanM

I have been reading some material about the relationship between Kepler's Equation for Elliptical Motion and Bessel Functions.

Kepler's Equation for Elliptical Motion is the following:

E = M + e sin(E)​

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

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

where

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

(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.