Asymptotic Methods in Analysis

N. G. de Bruijn

• Publisher: Dover Publications
• Year: 2014
• Print ISBN: 9780486642215
• eBook ISBN: 9780486150796
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

"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.
Most of the important asymptotic methods are covered here with unusual effectiveness and clarity: "Every step in the mathematical process is explained, its purpose and necessity made clear, with the result that the reader not only has no difficulty in following the rigorous proofs, but even turns to them with eager expectation." (Nuclear Physics).
Part of the attraction of this book is its pleasant, straightforward style of exposition, leavened with a touch of humor and occasionally even using the dramatic form of dialogue. The book begins with a general introduction (fundamental to the whole book) on O and o notation and asymptotic series in general. Subsequent chapters cover estimation of implicit functions and the roots of equations; various methods of estimating sums; extensive treatment of the saddle-point method with full details and intricate worked examples; a brief introduction to Tauberian theorems; a detailed chapter on iteration; and a short chapter on asymptotic behavior of solutions of differential equations. Most chapters progress from simple examples to difficult problems; and in some cases, two or more different treatments of the same problem are given to enable the reader to compare different methods. Several proofs of the Stirling theorem are included, for example, and the problem of the iterated sine is treated twice in Chapter 8. Exercises are given at the end of each chapter.
Since its first publication, Asymptotic Methods in Analysis has received widespread acclaim for its rigorous and original approach to teaching a difficult subject. This Dover edition, with corrections by the author, offers students, mathematicians, engineers, and physicists not only an inexpensive, comprehensive guide to asymptotic methods but also an unusually lucid and useful account of a significant mathematical discipline.

CHAPTER 1

INTRODUCTION

1.1. What is asymptotics?

This question is about as difficult to answer as the question “What is mathematics?” Nevertheless, we shall have to find some explanation for the word asymptotics.

It often happens that we want to evaluate a certain number, defined in a certain way, and that the evaluation involves a very large number of operations so that the direct method is almost prohibitive. In such cases we should be very happy to have an entirely different method for finding information about the number, giving at least some useful approximation to it. And usually this new method gives (as remarked by Laplace) the better results in proportion to its being more necessary: its accuracy improves when the number of operations involved in the definition increases. A situation like this is considered to belong to asymptotics.

This statement is very vague indeed. However, if we try to be more precise, a definition of the word asymptotics either excludes everything we are used to call asymptotics, or it includes almost the whole of mathematical analysis. It is hard to phrase the definition in such a way that Stirling’s formula (1.1.1) belongs to asymptotics, and that a formula like

does not. The obvious reason why the latter formula is not called an asymptotic formula is that it belongs to a part of analysis that already has got a name: the integral calculus. The safest and not the vaguest definition is the following one: Asymptotics is that part of analysis which considers problems of the type dealt with in this book.

A typical asymptotic result, and one of the oldest, is Stirling’s formula just mentioned:

For each n, the number n! can be evaluated without any theoretical difficulty, and the larger n is, the larger the number of necessary operations becomes. But Stirling’s formula gives a decent approximation

, and the larger n is, the smaller its relative error becomes.

Several proofs of (1.1.1) and of its generalizations will be given in this book (see secs. 3.7, 3.10, 4.5, 6.9).

We quote another famous asymptotic formula, much deeper than the previous one and beyond the scope of this book. If x is a positive number, we denote by π(x) the number of primes not exceeding x. Then the so-called Prime Number Theorem states that ¹)

The above formulas are limit formulas, and therefore they have, as they stand, litiile valu...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Preface
• Ch. 1. Introduction
• Ch. 2. Implicit Functions
• Ch. 3. Summation
• Ch. 4. The Laplace Method for Integrals
• Ch. 5. The Saddle Point Method
• Ch. 6. Applications of the Saddle Point Method
• Ch. 7. Indirect Asymptotics
• Ch. 8. Iterated Functions
• Ch. 9. Differential Equations
• INDEX