Laplace Limit

The value $e=0.6627434193\ldots$ (Sloane's A033259) for which Laplace's formula for solving Kepler's Equation begins diverging. The constant is defined as the value $e$ at which the function

$$f(x)={x\mathop{\rm exp}\nolimits (\sqrt{1+x^2}\,)\over 1+\sqrt{1+x^2}}$$

equals $f(\lambda)=1$. The Continued Fraction of $e$ is given by [0, 1, 1, 1, 27, 1, 1, 1, 8, 2, 154, ...] (Sloane's A033260). The positions of the first occurrences of $n$ in the Continued Fraction of $e$ are 2, 10, 35, 13, 15, 32, 101, 9, ... (Sloane's A033261). The incrementally largest terms in the Continued Fraction are 1, 27, 154, 1601, 2135, ... (Sloane's A033262), which occur at positions 2, 5, 11, 19, 1801, ... (Sloane's A033263).

See also Eccentric Anomaly, Kepler's Equation

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/lpc/lpc.html

Plouffe, S. ``Laplace Limit Constant.'' http://www.lacim.uqam.ca/piDATA/laplace.txt.

Sloane, N. J. A. Sequences A033259, A033260, A033261, A033262, and A033263 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.