infinity at these values of . This means that the radius of convergence of the Maclaurin series is and the series will not converge for values of larger than this. The series can also be used for the hyperbolic case, in which case the radius of convergence is The series for when converges when .

While this solution is the simplest in a certain mathematical sense,, other solutions are preferable for most applications. Alternatively, Kepler's equation can be solved numerically.

The solution for was found by Karl Stumpff

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