This chapter collects in a simplified form some ideas and techniques extensively used in Part I of the book. This material will be revisited in greater detail in later chapters, but the brief summary given here may be helpful to readers who do not have the time or the inclination to tackle the more extensive treatments.

§2.1 continues the considerations of the final section of the previous chapter and further explains the fundamental idea underlying the method of eigenfunction expansions. While this method may be seen as an extension of the elementary “separation of variables” procedure (cf. §3.10), the geometric view advocated here provides a powerful aid to intuition and should greatly facilitate an understanding of “what is really going on” in most of the applications of Part I; the basis of the method is given in some detail in Chapters 19 and 21 of Part III.

§2.2 is a reminder of a useful method to solve linear non-homogeneous ordinary differential equations. Here the solution to some equations that frequently arise in the applications of Part I is derived. A more general way in which this technique may be understood is through its connection with Green's functions. This powerful idea is explained in very simple terms in §2.4 and, in greater detail, in Chapters 15 and 16.

Green's functions make use of the notion of the so-called “δ-function,” the principal properties of which are summarized in §2.3. A proper theory for this and other generalized functions is presented in Chapter 20.

This book is not meant to be read sequentially. The material is organized according to a modular structure with abundant cross-referencing and indexing to permit a variety of pathways through it.

Each chapter in Part I, Applications, is devoted to a particular technique: Fourier series, Fourier transform, etc. The chapters open with a section summarizing very briefly the basic relations and proceeds directly to show on a variety of examples how they are applied and how they “work.”

A fairly detailed exposition of the essential background of the various techniques is given in the chapters of Part II, Essential Tools. Other chapters here describe general concepts (e.g., Green's functions and analytic functions) that occur repeatedly elsewhere. The last chapter on matrices and finite-dimensional linear spaces is included mostly to introduce Part III, Some Advanced Tools. Here the general theory of linear spaces, generalized functions and linear operators provides a unified foundation to the various techniques of Parts I and II.

The book starts with some general remarks and introductory material in Part 0. Here the first chapter summarizes the basic equations of classical field theory to establish a connection between specific physical problems and the many examples of Part I in which, by and large, no explicit reference to physics is made. The last section of this chapter provides a very elementary introduction to the basic idea of eigenfunction expansion.