# A new method for obtaining approximate solutions of the hyperbolic Kepler's equation

```@inproceedings{Avendano2015ANM,
title={A new method for obtaining approximate solutions of the hyperbolic Kepler's equation},
author={Martin E. Avendano and Ver'onica Mart'in-Molina and Jorge Ortigas-Galindo},
year={2015}
}```
• Published 20 February 2015
• Mathematics
We provide an approximate zero S̃(g, L) for the hyperbolic Kepler’s equation S − g arcsinh(S)−L = 0 for g ∈ (0, 1) and L ∈ [0,∞). We prove, by using Smale’s α-theory, that Newton’s method starting at our approximate zero produces a sequence that converges to the actual solution S(g, L) at quadratic speed, i.e. if Sn is the value obtained after n iterations, then |Sn − S| ≤ 0.5 n−1|S̃ − S|. The approximate zero S̃(g, L) is a piecewisedefined function involving several linear expressions and one…
1 Citations

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