On certain transcendental numbers
Making use of the classic Lindemann-Weierstraß theorem and the Gelfond-Schneider theorem, the author derives a lot of interesting new facts such as Theorem 1. Suppose \(\alpha\) and \(\beta\) are algebraic numbers. Then \(e^{\alpha} \cos \beta\) and \(e^{\alpha} \sin\beta\) are algebraically independent if and only if \(\alpha\) and \(\beta\) are linearly independent. Theorem 2 and 3 mention many results regarding transcendental numbers. For example, if \(\alpha\) and \(\beta\) are algebraic numbers, then \(e^{\alpha} \cos \beta\) for \((\alpha,\beta)\neq (0,0)\), \(e^{\alpha} \sin \beta\) and \(e^{\alpha} \tan \beta\) for \(\beta\neq 0\) are transcendental. Under the additional condition that \(\alpha\) i is irrational the author proves a.o. that \(\cos(\alpha \log\beta)\), \(\sin(\alpha \log\beta)\), and \(\tan(\alpha \log\beta)\), for \(\beta\neq 0\), are transcendental.
scientific article; zbMATH DE number 3922768