Transcendental and Algebraic Numbers

by Gelfond, A. O.; Boron, Leo F.

• ISBN13: 9780486495262
• ISBN10: 0486495264
• Format: Paperback
• Copyright: 2015-03-18
• Publisher: Dover Publications

Summary

Primarily an advanced study of the modern theory of transcendental numbers, this text focuses on the theory's fundamental methods and explores its connections with other problems in number theory. Topics include the Thue-Siegel theorem; the Hermite-Lindemann theorem on the transcendency of the exponential function; the transcendency of the Bessel functions; more.

Author Biography

Alexander Osipovich Gelfond (1906–68) was a prominent Soviet mathematician.

Table of Contents

Foreword	vii
The approximation of algebraic irrationalities	1	(40)
Introduction	1	(10)
Auxiliary lemmas	11	(10)
Fundamental theorems	21	(14)
Applications of the fundamental theorems	35	(6)
Transcendence of values of analytic functions whose Taylor series have algebraic coefficients	41	(56)
Introduction. The Hermite and Lindemann theorems	41	(9)
Further development of the ideas of Hermite and Lindemann	50	(10)
Auxiliary propositions and definitions	60	(18)
General theorem on the algebraic independence of values of an E-function and consequences of it	78	(15)
Problems concerning transcendence and algebraic independence, over the rational field, of numbers defined by infinite series or which are roots of algebraic or transcendental equations	93	(4)
Arithmetic properties of the set of values of an analytic function whose argument assumes values in an algebraic field; transcendence problems	97	(81)
Integrity of analytic functions	97	(5)
The Euler-Hilbert problem	102	(16)
Problems concerning measure of transcendence and further development of methods	118	(14)
Formulation of fundamental theorems and auxiliary propositions	132	(23)
Proof of the fundamental theorems	155	(23)
Literature	178	(11)
Index	189