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Publisher:

Dover Publications

Publication Date:

1981

Number of Pages:

224

Format:

Paperback

Price:

12.95

ISBN:

978-0486642215

Category:

Textbook

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

Allen Stenger

03/24/2011

This book tells the intriguing and entertaining story of asymptotic analysis through a series of well-chosen examples. Some of the examples are contrived to illustrate a method, but most are interesting in themselves. Several problems are given more than one solution; these include Stirling’s formula and the problem of the iterated sine (that is, the asymptotic behavior of the sequence defined by *x _{n+1}* = sin

The first two-thirds of the book deals with direct asymptotics, where we have an explicit but not easy-to-use expression for the function of interest and we seek a convenient approximate form; Stirling’s formula for the gamma function is a typical example. The rest of the book deals with problems where we lack the explicit form but have other information such as a functional equation or average asymptotic information; these problems include Tauberian theorems, differential equations, and recursions.

It is a concise book, and despite covering a wide variety of topics still omits several important ones: there is almost nothing on analytic number theory, a subject that is the source of many delicate and important asymptotic problems; asymptotic series occur in several spots but are never used for numerical work; and the asymptotics of differential equations get one chapter but the subject is so broad that it can only be sketched here.

The book is clearly written and is always easy to follow, even when going through intricate reasoning. The book doesn’t use the terminology “bootstrapping,” but its description of this technique is especially good (in bootstrapping we we start with a very crude estimate and use this to develop several progressively more precise estimates).

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

- Preface
- CHAPTER 1. INTRODUCTION
- 1.1. What is asymptotics?
- 1.2. The O-symbol
- 1.3. The o-symbol
- 1.4. Asymptotic equivalence
- 1.5. Asymptotic series
- 1.6. Elementary operations on asymptotic series
- 1.7. Asymptotics and Numerical Analysis
- 1.8. Exercises

- CHAPTER 2. IMPLICIT FUNCTIONS
- 2.1. Introduction
- 2.2. The Lagrange inversion formula
- 2.3. Applications
- 2.4. A more difficult case
- 2.5. Iteration methods
- 2.6. Roots of equations
- 2.7. Asymptotic iteration
- 2.8. Exercises

- CHAPTER 3. SUMMATION
- 3.1. Introduction
- 3.2. Case
*a* - 3.3. Case
*b* - 3.4. Case
*c* - 3.5. Case
*d* - 3.6. The Euler-Maclaurin sum formula
- 3.7. Example
- 3.8. A remark
- 3.9. Another example
- 3.10. The Stirling formula for the Γ-function in the complex plane
- 3.11. Alternating sums
- 3.12. Application of the Poisson sum formula
- 3.13. Summation by parts
- 3.14. Exercises

- CHAPTER 4. THE LAPLACE METHOD FOR INTEGRALS
- 4.1. Introduction
- 4.2. A general case
- 4.3. Maximum at the boundary
- 4.4. Asymptotic expansions
- 4.5. Asymptotic behaviour of the Γ-function
- 4.6. Multiple integrals
- 4.7. An application
- 4.8. Exercises

- CHAPTER 5. THE SADDLE POINT METHOD
- 5.1. The method
- 5.2. Geometrical interpretation
- 5.3. Peakless landscapes
- 5.4. Steepest descent
- 5.5. Steepest descent at end-point
- 5.6. The second stage
- 5.7. A general simple case
- 5.8. Path of constant altitude
- 5.9. Closed path
- 5.10. Range of a saddle point
- 5.11 Examples
- 5.12. Small perturbations
- 5.13. Exercises

- CHAPTER 6. APPLICATIONS OF THE SADDLE POINT METHOD
- 6.1. The number of class-partitions of a finite set
- 6.2. Asymptotic behaviour of
*d*_{n} - 6.3. Alternative method
- 6.4. The sum
*S(s, n)* - 6.5. Asymptotic behaviour of
*P* - 6.6. Asymptotic behaviour of
*Q* - 6.7. Conclusions about
*S(s, n)* - 6.8. A modified Gamma Function
- 6.9. The entire function
*G*_{0}(s) - 6.10. Conclusions about
*G(s)* - 6.11. Exercises

- CHAPTER 7. INDIRECT ASYMPTOTICS
- 7.1. Direct and indirect asymptotics
- 7.2. Tauberian theorems
- 7.3. Differentiation of an asymptotic formula
- 7.4. A similar problem
- 7.5. Karamata’s method
- 7.6. Exercises

- CHAPTER 8. ITERATED FUNCTIONS
- 8.1. Introduction
- 8.2. Iterates of a function
- 8.3. Rapid convergence
- 8.4. Slow convergence
- 8.5. Preparation
- 8.6. Iteration of the sine function
- 8.7. An alternative method
- 8.8. Final discussion about the iterated sine
- 8.9. An inequality concerning infinite series
- 8.10. The iteration problem
- 8.11. Exercises

- CHAPTER 9. DIFFERENTIAL EQUATIONS
- 9.1. Introduction
- 9.2. A Riccati equation
- 9.3. An unstable case
- 9.4. Application to a linear second-order equation
- 9.5. Oscillatory cases
- 9.6. More general oscillatory cases
- 9.7. Exercises

- INDEX

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