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My book "Asymptotic Expansions for Ordinary Differential Equations" published in 1965 is out of print. In the almost 20 years since then, the subject has grown so much in breadth and in depth that an account of the present state of knowledge of all the topics discussed there could not be fitted into one volume without resorting to an excessively terse style of writing. Instead of undertaking such a task, I have concentrated, in this exposi­ tion, on the aspects of the asymptotic theory with which I have been particularly concerned during those 20 years, which is the nature and structure of turning points. As in Chapter VIII of my previous book, only linear analytic differential equations are considered, but the inclusion of important new ideas and results, as well as the development of the neces­ sary background material have made this an exposition of book length. The formal theory of linear analytic differential equations without a parameter near singularities with respect to the independent variable has, in recent years, been greatly deepened by bringing to it methods of modern algebra and topology. It is very probable that many of these ideas could also be applied to the problems concerning singularities with respect to a parameter, and I hope that this will be done in the near future. It is less likely, however, that the analytic, as opposed to the formal, aspects of turning point theory will greatly benefit from such an algebraization.



Chapter I. Historical Introduction

Turning point theory is a branch of the asymptotic theory of ordinary differential equations that depend in a singular manner on a parameter. “Turning points” are certain exceptional points in that theory. Their precise definition in a general framework is, in itself, a nontrivial matter which will be discussed later on. While turning points are exceptional, their analysis is essential to a full understanding of the asymptotic nature of the solutions of such differential equations. This is a general situation in Mathematics. In an analogous way, analytic functions can only be understood through a study of their singularities; the solutions of an ordinary differential equation depend decisively on the location and type of their critical points, etc. Also, the mathematical formulation of many problems of Physics and Engineering involves turning point problems, and this has been the principal motivation for most of the early investigations.
Wolfgang Wasow

Chapter II. Formal Solutions

The principal objects of study in this book are linear homogeneous ordinary differential equations that involve one scalar parameter. By introducing the higher derivatives of the dependent variable or variables as additional unknown functions, any such equation, or system of equations, can be rewritten as an equivalent first-order system:
$$\begin{array}{*{20}{c}} {\frac{{d{y_j}}}{{dx}} = \sum\limits_{k = 1}^n {{a_{jk}}(x, \in } ){y_k},}&{j = 1,2, \ldots ,n.} \end{array}$$
The parameter has been designated by the letter ∈, because it will be small in most situations to be encountered. If the n X n matrix A (x, ∈) with entries a jk (x, ∈) and the column vector y with components y j , j = 1, 2,...,n, are introduced, equation (2.1-1) takes on the concise and convenient form
$$y\prime = A(x, \in )y.$$
The prime indicates differentiation with respect to x.
Wolfgang Wasow

Chapter III. Solutions Away From Turning Points

The formal, usually divergent series in powers of ∈, or of some root of E, which appear in the theorems of Chapter II, are related to true solutions of the differential equation in the sense that they are asymptotic representations of analytic functions which are such true solutions when substituted for those series. This fact will be established in the present section.
Wolfgang Wasow

Chapter IV. Asymptotic Transformations of Differential Equations

A decisive part of the method of the first chapter was the transformation of a formal differential equation
$${ \in ^h}y\prime = \left( {\sum\limits_{r = 0}^\infty {{A_r}(x){ \in ^r}} } \right)y$$
into another one of the same type,
$${ \in ^h}z\prime = \left( {\sum\limits_{r = 0}^\infty {{B_r}(x){ \in ^r}} } \right)z,$$
by a formal transformation
$$\begin{array}{*{20}{c}} {y = \left( {\sum\limits_{r = 0}^\infty {{P_r}(x){ \in ^r}} } \right)z,}&{{P_0}(x) invertible.} \end{array}$$
P0(x) invertible.
Wolfgang Wasow

Chapter V. Uniform Transformations at Turning Points: Formal Theory

The formal theory of Chapter II essentially dealt with global simplifying transformations of the given differential equation in regions from which all potential turning points were removed. In this chapter domains containing turning points will be considered, and the first question is: How far can the differential equation be simplified by formal transformations with well understood asymptotic properties in such regions. The essence of Langer’s method belongs here. His reduction of certain second-order equations with turning points led him to differential equations so simple as to be solvable by classical special functions ([36], [31], [38], [39] and other papers).
Wolfgang Wasow

Chapter VI. Uniform Transformations at Turning Points: Analytic Theory

The results of Chapter V are of limited interest by themselves. Only if the formal series constructed there are asymptotic representations of asymptotic transformations in sets of the x-plane that contain a turning point has really useful information been attained.
Wolfgang Wasow

Chapter VII. Extensions of the Regions of Validity of the Asymptotic Solutions

Except for the formal theory of Chapter II and Theorem 3.3-1, all results proved so far have been local. They are valid in unspecified, possibly small, neighborhoods of a point. Several techniques exist for the study of the asymptotic properties of the solutions in larger domains. Most of those methods have to be adapted to specific examples, because general theories are either too difficult or so cumbersome as to yield little insight. Here is a short list of the various approaches available.
Wolfgang Wasow

Chapter VIII. Connection Problems

The results of Chapter VII extend the regions in which the solutions of the differential equations are asymptotically known, but they do not solve the connection problems, central or lateral, as defined in Section 7.1. Only for relatively simple equations such as those analyzed in Chapter VI are the theorems of Chapter VII of decisive usefulness, by leading to a reduction of the given equation to well known special equations in full neighborhoods of the turning point. In general, less elegant methods such as the stretchings and matchings to be investigated in this chapter have to be turned to. The existing theory is still quite incomplete and rather involved. Most of these investigations are best studied in the original papers such as Sibuya [77], [78]; Iwano and Sibuya [29]; Iwano [26], [28]; Olver [62], [63], [65]; Nakano [49]; and Nishimoto [53]—[58]. In this account only differential equations of the form
$${ \in ^h}\frac{{dy}}{{dx}} = \left( {\begin{array}{*{20}{c}} 0&1 \\ {\phi (x, \in )}&0 \end{array}} \right)y$$
with a turning point at x = 0 will be discussed. The known results for systems of higher order are less satisfying and more complicated.
Wolfgang Wasow

Chapter IX. Fedoryuk’s Global Theory of Second-Order Equations

In Section 2.3 a general method for finding formal solutions was illustrated by applying it to the simple equation
$$\in y\prime = \left( {\begin{array}{*{20}{c}} 0&1 \\ {a(x)}&0 \end{array}} \right)y,$$
which is equivalent to the scalar equation ∈2una(x)u = 0. The result was the Liouville—Green approximation (2.3-24), which is the matrix form of
$$u(x, \in ) = \left[ {{{(a(x))}^{ - 1/4}} + \cdots } \right]\exp \left\{ { \pm \frac{1}{ \in }\int {{{(a(x))}^{1/2}}dx} } \right\}.$$
Wolfgang Wasow

Chapter X. Doubly Asymptotic Expansions

In the preceding chapter the concept of “doubly asymptotic expansions” was introduced through an important special case. Whenever in the applications of the asymptotic theory one has to characterize functions of x and ∈ by their behavior near x = ∞ in some unbounded region of the x-plane, the knowledge of doubly asymptotic expansions is extremely helpful. Unfortunately, all known results of some generality in this direction require that the coefficients of the differential equation to be solved be polynomials in x. The matter has been explored in a number of papers by Leung, listed in the bibliography. The presentation here is based on Leung’s work.
Wolfgang Wasow

Chapter XI. A Singularly Perturbed Turning Point Problem

Differential equation problems that depend on small parameter ∈ in such a way that the order of the equation is lower for ∈ = 0 than for ∈≠ 0 but remains positive are now commonly called “singular perturbation problems.” The condition that the order remain positive for ∈ = 0 is not a very distinguishing property of the differential equation as such. The equation
$$\begin{array}{*{20}{c}} { \in u\prime \prime - 2xu\prime + ku = 0,}&{k a constant,} \end{array}$$
k a constant, for instance, which will be examined closely in the next section, becomes
$${ \in ^2}v\prime \prime - \left( {{x^2} - \in (1 + k)} \right)v = 0$$
under the simple change of variables
$$u = {e^{{x^2}/{2_ \in }}}v.$$
Wolfgang Wasow

Chapter XII. Appendix: Some Linear Algebra for Holomorphic Matrices

The theory of linear ordinary differential equations involves a considerable amount of linear algebra. If, as in this book, the differential equations are analytic, it is essential to know which of the common algebraic operations can be carried out in the ring of functions of one complex variable that are holomorphic in some given region. In this section, some useful general theorems, less simple than their analogs for constant matrices, will be proved.
Wolfgang Wasow


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