SIN, COS, EXP AND LOG OF LIOUVILLE NUMBERS

Abstract

AbstractFor 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’, whereUis one of Mahler’s classes of transcendental numbers.