  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.
