Book:N.G. de Bruijn/Asymptotic Methods in Analysis/Third Edition
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N.G. de Bruijn: Asymptotic Methods in Analysis (3rd Edition)
Published $\text {1970}$, Dover Publications
- ISBN 0-486-64221-6
Subject Matter
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
Further Editions
- 1958: N.G. de Bruijn: Asymptotic Methods in Analysis
- 1961: N.G. de Bruijn: Asymptotic Methods in Analysis (2nd ed.)
Source work progress
- 1970: N.G. de Bruijn: Asymptotic Methods in Analysis (3rd ed.) ... (previous) ... (next): $1.1$ What is asymptotics? $(1.1.3)$, $(1.1.4)$