SIN, COS, EXP AND LOG OF LIOUVILLE NUMBERS

Abstract

Abstract For any Liouville number $\alpha $ , all of the following are transcendental numbers: ${e}^\alpha $ , $\log _{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.