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arXiv · 2202.11293

SIN, COS, EXP and LOG of Liouville numbers

Taboka Chalebgwa·Sidney A. Morris
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Abstract

For any Liouville number $\alpha$, all of the following are transcendental numbers: $\textrm{e}^\alpha$, $\log_\textrm{e}\alpha$, $\sin \alpha$, $\cos\alpha$, $\tan\alpha$, $\sinh\alpha$, $\cosh\alpha$, $\tanh\alpha$, $\arcsin\alpha$ and the inverse functions evaluated at $\alpha$ of the listed trigonometric and hyperbolic functions, noting that wherever multiple values are involved, every such value is transcendental. This remains true if "Liouville number" is replaced by "$U$-number", where $U$ is one of Mahler's classes of transcendental numbers.

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Taboka Chalebgwa, Sidney A. Morris. 2022-02-23. SIN, COS, EXP and LOG of Liouville numbers. https://arxiv.org/abs/2202.11293

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