Making Transcendence Transparent

An intuitive approach to classical transcendental number theory

Edward B. Burger, Robert Tubbs (Autoren)

Buch | Softcover

263 Seiten

2011 | Softcover reprint of hardcover 1st ed. 2004
Springer-Verlag New York Inc.
978-1-4419-1948-9 (ISBN)

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The Journey Ahead At the heart of transcendental number theory lies an intriguing paradox: While essen tially all numbers are transcendental, establishing the transcendence of a particular number is a monumental task. Thus transcendental numbers are an enigmatic species of number: We know they are all around us and yet it requires enormous effort to catch one. More often than not, they slip through our fingers and dissappear back into the dense jungle of numbers. Here we will venture to tame a few of these incredible creatures. In the pages ahead we offer an approach to transcendence that not only includes the intricate analysis but also the beautiful ideas behind the technical details. The phrase "classical transcendental number theory" in the title of this book refers to the most widely known results that were obtained in the nineteenth and early twentieth centuries. The reason for this focus is threefold. Firstly, this body of work requires only the mathematical techniques and tools familiar to advanced undergraduate mathematics students, and thus this area can be appreciated by a wide range of readers. Secondly, the ideas behind modem transcendence results are almost always an elaboration of the classical arguments we will explore here. And finally, and perhaps more importantly, this early work yields the transcendence of such admired and well-known numbers as e, rr, and even 2v'2.

Edward Burger is one of the authors of "The Heart of Mathematics," Winner of a 2001 Robert W. Hamilton Book Award. He will also be awarded the 2004 Chauvenet Prize, one of the most prestigious MAA prizes acknowledging an outstanding expository article. To read more about Ed Burger go to http://www.williams.edu/Mathematics/eburger/.

A prequel to transcendence.- Incredible numbers incredibly close to modest rational numbers.- The powerful power series for e.- Conjugation and symmetry as a means towards transcendence.- The analytic adventures of exp(z).- Debunking conspiracy theories for independent functions.- Class distinctions among complex numbers.- Extending our reach through periodic functions.- Transcending numbers and discovering a more formal e.- Selected highlights from complex analysis.

Zusatzinfo	50 Illustrations, black and white; IX, 263 p. 50 illus.
Verlagsort	New York, NY
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
ISBN-10	1-4419-1948-1 / 1441919481
ISBN-13	978-1-4419-1948-9 / 9781441919489
Zustand	Neuware