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.

Even a cursory look at the table of contents of this book will reveal that the material is arranged in an unusual way, starting immediately with the application of techniques (Part I) which are justified and explained in greater detail in later chapters (Parts II and III). Indeed, the book is not meant to be read sequentially and I have not attempted to force the material into a sequential exposition. Rather, I have organized it as a series of modules through which each reader can fashion an individual pathway according to background, needs and interests.

The widespread availability of computers and symbolic manipulation packages has made mathematics relevant to many more people than in the past. A serious problem, however, remains the high “cost of entry” of mathematics. While some might argue that in a perfect world people would spend the time required to learn all the mathematics they might need before applying it, this is a utopia certainly, and probably also a misguided ideal as any user of mathematics – having gone through many cycles of learning and applying – can testify.

Hopefully, the modular structure of the book will serve a variety of users addressing different needs ranging from a quick impression of a mathematical tool to a fairly deep understanding of its basis and “inner workings.” The many cross-references, detailed index and table of contents will render possible a non-systematic “navigation” through the material.