# Book:N.G. de Bruijn/Asymptotic Methods in Analysis/Third Edition

## N.G. de Bruijn: Asymptotic Methods in Analysis (3rd Edition)

Published $\text {1970}$, Dover Publications

ISBN 0-486-64221-6.

### Contents

Preface to the First Edition (October, 1957)
Preface to the Second Edition (February, 1961)
Ch. 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
Ch. 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
Ch. 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 $\Gamma$-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
Ch. 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 $\Gamma$-function
4.6. Multiple integrals
4.7. An application
4.8. Exercises
Ch. 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
Ch. 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 $\map S {s, n}$
6.5. Asymptotic behaviour of $P$
6.6. Asymptotic behaviour of $Q$
6.7. Conclusions about $\map S {s, n}$
6.8. A modified Gamma Function
6.9. The entire function $\map {G_0} s$
6.10. Conclusions about $\map G s$
6.11. Exercises
Ch. 7. INDIRECT ASYMPTOTICS
7.1. Direct and indirect asymptotes
7.2. Tauberian theorems
7.3. Differentiation of an asymptotic formula
7.4. A similar problem
7.5. Karamata's method
7.6. Exercises
Ch. 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
Ch. 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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