**Student Mathematical Library**

Volume: 71;
2014;
183 pp;
Softcover

MSC: Primary 05;
Secondary 68; 11; 60

Print ISBN: 978-1-4704-0904-3

Product Code: STML/71

List Price: $39.00

Individual Price: $31.20

**Electronic ISBN: 978-1-4704-1661-4
Product Code: STML/71.E**

List Price: $39.00

Individual Price: $31.20

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#### Supplemental Materials

# Asymptopia

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*Joel Spencer*

with Laura Florescu, New York University, NY

This beautiful book is about how to estimate large quantities—and why. Building on nothing more than first-year calculus, it goes all the way into deep asymptotical methods and shows how these can be used to solve problems in number theory, combinatorics, probability, and geometry. The author is a master of exposition: starting from such a simple fact as the infinity of primes, he leads the reader through small steps, each carefully motivated, to many theorems that were cutting-edge when discovered, and teaches the general methods to be learned from these results.

—László Lovász, Eötvös-Loránd University

This is a lovely little travel guide to a country you might not even have heard about - full of wonders, mysteries, small and large discoveries ... and in Joel Spencer you have the perfect travel guide!

—Günter M. Ziegler, Freie Universität
Berlin, coauthor of "Proofs from THE BOOK"

Asymptotics in one form or another are part of
the landscape for every mathematician. The objective of this book is
to present the ideas of how to approach asymptotic problems that arise
in discrete mathematics, analysis of algorithms, and number theory. A
broad range of topics is covered, including distribution of prime
integers, Erdős Magic, random graphs, Ramsey numbers, and
asymptotic geometry.

The author is a disciple of Paul Erdős, who taught him about
Asymptopia. Primes less than \(n\), graphs with \(v\)
vertices, random walks of \(t\) steps—Erdős was
fascinated by the limiting behavior as the variables approached, but
never reached, infinity. Asymptotics is very much an art. The various
functions \(n\ln n\), \(n^2\), \(\frac{\ln n}{n}\),
\(\sqrt{\ln n}\), \(\frac{1}{n\ln n}\) all have distinct
personalities. Erdős knew these functions as personal
friends. It is the author's hope that these insights may be passed on,
that the reader may similarly feel which function has the right
temperament for a given task. This book is aimed at strong
undergraduates, though it is also suitable for particularly good high
school students or for graduates wanting to learn some basic
techniques.

Asymptopia is a beautiful world. Enjoy!

#### Readership

Undergraduate and graduate students interested in asymptotic techniques.

#### Reviews & Endorsements

The style and the beauty make this book an excellent reading. Keep it on your coffee table or/and bed table and open it often. Asymptopia is a fascinating place.

-- Péter Hajnal, ACTA Sci. Math.

#### Table of Contents

# Table of Contents

## Asymptopia

- Cover Cover11 free
- Title page iii4 free
- Table of contents v6 free
- Preface ix10 free
- A Reader's Guide xiii14 free
- 0. An Infinity of Primes 116 free
- 1. Stirling's Formula 520
- 2. Big Oh, Little Oh, and All That 2742
- 3. Integration in Asymptopia 3752
- 4. From Integrals to Sums 4762
- 5. Asymptotics of Binomial Coefficients 5772
- 6. Unicyclic Graphs 7186
- 7. Ramsey Numbers 93108
- 8. Large Deviations 103118
- 9. Primes 115130
- 10. Asymptotic Geometry 125140
- 11. Algorithms 137152
- 12. Potpourri 151166
- 13. Really Big Numbers! 173188
- Bibliography 179194
- Index 181196 free
- Back Cover Back Cover1202