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

Subject Matter

 * Analysis

Contents

 * Preface to the First Edition (October, 1957)
 * Preface to the Second Edition (February, 1961)


 * . 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


 * . 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


 * . 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


 * . 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


 * . 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


 * . 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


 * . 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


 * . 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


 * . 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



Source work progress
* : $1.1$ What is asymptotics? $(1.1.3)$, $(1.1.4)$