SIN, COS, EXP and LOG of Liouville numbers

TL;DR

The paper proves that applying exponential, logarithmic, trigonometric, and hyperbolic functions to Liouville numbers results in transcendental numbers, extending to U-numbers within Mahler's classes.

Contribution

It establishes the transcendence of various functions of Liouville and U-numbers, broadening understanding of their algebraic properties.

Findings

e^α, log_e α, sin α, cos α, tan α, sinh α, cosh α, tanh α, arcsin α are transcendental for Liouville α.

The transcendence results hold for all values of inverse functions where multiple values exist.

Results extend to U-numbers in Mahler's classes, not just Liouville numbers.

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.