On certain transcendental numbers
type: Item
instance of: scholarly article
title: On certain transcendental numbers
zbMATH Open document ID: 0577.10032
author: M. L. Robinson
published in: Michigan Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11J85
zbMATH DE Number: 3922768
zbMATH Keywords: algebraic independence
zbMATH Keywords: transcendence
zbMATH Keywords: arithmetic nature of real
zbMATH Keywords: and imaginary parts
zbMATH Keywords: algebraic numbers
reviewed by: L. Kuipers
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029002967
OpenAlex ID: W2041980693
DOI: 10.1307/MMJ/1029002967
description: scientific article; zbMATH DE number 3922768