(Ebook) Asymptotic methods in analysis MCap 1st edition by de Bruijn 0486642216 9780486642215

(Ebook) Asymptotic methods in analysis MCap 1st edition by de Bruijn 0486642216 9780486642215

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ISBN-10 ‏ : ‎0486642216 

ISBN-13 ‏ : ‎ 9780486642215

Author: N. G. de Bruijn 

"A reader looking for interesting problems tackled often by highly original methods, for precise results fully proved, and for procedures fully motivated, will be delighted." — Mathematical Reviews.
Asymptotics is not new. Its importance in many areas of pure and applied mathematics has been recognized since the days of Laplace. Asymptotic estimates of series, integrals, and other expressions are commonly needed in physics, engineering, and other fields. Unfortunately, for many years there was a dearth of literature dealing with this difficult but important topic. Then, in 1958, Professor N. G. de Bruijn published this pioneering study. Widely considered the first text on the subject — and the first comprehensive coverage of this broad field — the book embodied an original and highly effective approach to teaching asymptotics. Rather than trying to formulate a general theory (which, in the author's words, "leads to stating more and more about less and less") de Bruijn teaches asymptotic methods through a rigorous process of explaining worked examples in detail.

Asymptotic methods in analysis MCap 1st Table of contents:

Ch. 1. Introduction
1.1. What is asymptotics?
1.2. The O-symbo
1.3. The o-symbo
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 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
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 Γ-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 dn
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 G0(s)
6.10. Conclusions about G(s)
6.11. Exercises
Ch. 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
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

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