We provide an approximate zero S~(g,L)\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{S}(g,L)$$\end{document} for the hyperbolic Kepler’s equation S-garcsinh(S)-L=0\documentclass[12pt]{minimal}
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\begin{document}$$S-g\,{{\mathrm{arcsinh}}}(S)-L=0$$\end{document} for g∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$g\in (0,1)$$\end{document} and L∈[0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$L\in [0,\infty )$$\end{document}. We prove, by using Smale’s α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}-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\documentclass[12pt]{minimal}
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\begin{document}$$S_n$$\end{document} is the value obtained after n iterations, then |Sn-S|≤0.52n-1|S~-S|\documentclass[12pt]{minimal}
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\begin{document}$$|S_n-S|\le 0.5^{2^n-1}|\widetilde{S}-S|$$\end{document}. The approximate zero S~(g,L)\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{S}(g,L)$$\end{document} is a piecewise-defined function involving several linear expressions and one with cubic and square roots. In bounded regions of (0,1)×[0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$(0,1) \times [0,\infty )$$\end{document} that exclude a small neighborhood of g=1,L=0\documentclass[12pt]{minimal}
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\begin{document}$$g=1, L=0$$\end{document}, we also provide a method to construct simpler starters involving only constants.
