Transcendental Numbers

ISBN-13: 9781493908318 / Angielski / Miękka / 2014 / 217 str.

M. Ram Murty; Purusottam Rath

Opis

This book provides an introduction to the topic of transcendental numbers for upper-level undergraduate and graduate students. The text is constructed to support a full course on the subject, including descriptions of both relevant theorems and their applications. While the first part of the book focuses on introducing key concepts, the second part presents more complex material, including applications of Baker's theorem, Schanuel's conjecture, and Schneider's theorem. These later chapters may be of interest to researchers interested in examining the relationship between transcendence and "L"-functions. Readers of this text should possess basic knowledge of complex analysis and elementary algebraic number theory.

Recenzje

From the book reviews:

"Transcendental Number Theory ... though terse, has not had a significant competitor for nearly four decades, but the present volume by Murty (Queen's Univ., Canada) and Rath (Chennai Mathematical Institute, India) surpasses it in certain ways. ... Summing Up: Highly recommended. Upper-division undergraduates and above." (D. V. Feldman, Choice, Vol. 52 (5), January, 2015)

"This is an excellent book which can be used for a one- or two-semester upper undergraduate course or first or second year graduate course in transcendental numbers. ... There are 28 chapters in 205 pages resulting in an average of 7 pages per chapter. Yet each of these chapters covers a major technique, a major historical development or a major advanced topic." (Russell Jay Hendel, MAA Reviews, September, 2014)

"The book under review provides an introduction to the fascinating topic of transcendental numbers for senior undergraduate and first-year graduate students. ... Each chapter ends with a number of related exercises helping the beginning student develop her/his practical skills. ... The utmost lucid and detailed presentation of the material will be very helpful to beginners in the field, who are led from the historical origins of the subject up to the forefront of current research." (Werner Kleinert, zbMATH, Vol. 1297, 2014)

Spis treści

1. Liouville’s theorem.- 2. Hermite’s Theorem.- 3. Lindemann’s theorem.- 4. The Lindemann-Weierstrass theorem.- 5. The maximum modulus principle.- 6. Siegel’s lemma.- 7. The six exponentials theorem.- 8. Estimates for derivatives.- 9. The Schneider-Lang theorem.- 10. Elliptic functions.- 11. Transcendental values of elliptic functions.- 12. Periods and quasiperiods.- 13. Transcendental values of some elliptic integrals.- 14. The modular invariant.- 15. Transcendental values of the j-function.- 16. More elliptic integrals.- 17. Transcendental values of Eisenstein series.- 18. Elliptic integrals and hypergeometric series.- 19. Baker’s theorem.- 20. Some applications of Baker’s theorem.- 21. Schanuel’s conjecture.- 22. Transcendental values of some Dirichlet series.- 23. Proof of the Baker-Birch-Wirsing theorem.- 24. Transcendence of some infinite series.- 25. Linear independence of values of Dirichlet L-functions.- 26. Transcendence of values of modular forms.- 27. Transcendence of values of class group L-functions.- 28. Periods, multiple zeta functions and (3).      

O autorze

M. Ram Murty is a professor of mathematics at Queen's University. Purusottam Rath is a professor of mathematics at the Chennai Mathematical Institute.