I have thoroughly revised all the ten chapters in the original edition, both making some changes due to comments from my readers and also checking for errors. I've also added a chapter on vectors which continues naturally from the present chapter on complex numbers.

I wrote the first version of this new chapter as an extension to the book's website (which is now at http://www.mathssurvivalguide.com) building up the pages there gradually. Their content was influenced by emails from visitors, often with particular problems with which they hoped for help. I've now extensively rewritten and rearranged this material. Writing in book form, it was possible to structure the content much more closely than on the Web so that it's easy to see the connections between the different areas and how results can be applied to later problems. The new chapter also has, of course, many practice exercises with complete solutions just as the earlier chapters have.

I'm once again very grateful to Rodie and Tony Sudbery and to David Olive for their helpful suggestions and comments. I must also thank all the people who emailed me, both with comments on the original ten chapters, and also with particular needs in using vectors which I've tried to fulfil here.

I hope that this two-way communication will continue. You can email me from the book's website if you would like to.

For systems that have some degree of symmetry, full exploitation of that symmetry is desirable. Significant physical results can sometimes be deduced simply by a study of the symmetry properties of the system under investigation. Consequently it becomes important, for such a system, to identify all those operations (rotations, reflections, inversions) that carry the system into a physically indistinguishable copy of itself.

The study of the properties of the complete set of such operations forms one application of group theory. Though this is the aspect of most interest to the physical scientist, group theory itself is a much larger subject and of great importance in its own right. Consequently we leave until the next chapter any direct applications of group theoretical results and concentrate on building up the general mathematical properties of groups.

Groups

As an example of symmetry properties, let us consider the sets of operations, such as rotations, reflections, and inversions, that transform physical objects, for example molecules, into physically indistinguishable copies of themselves, so that only the labelling of identical components of the system (the atoms) changes in the process. For differently shaped molecules there are different sets of operations, but in each case it is a well-defined set, and with a little practice all members of each set can be identified.

As simple examples, consider (a) the hydrogen molecule, and (b) the ammonia molecule illustrated in figure 24.1.

Throughout this book references have been made to results derived from the theory of complex variables. This theory thus becomes an integral part of the mathematics appropriate to physical applications. The difficulty with it, from the point of view of a book such as the present one, is that although it has many practical applications its underlying basis has a distinctly pure mathematics flavour.

Thus, to adopt a comprehensive rigorous approach would involve a large amount of groundwork in analysis, for example formulating precise definitions of continuity and differentiability, developing the theory of sets and making a detailed study of boundedness. Instead, we will be selective and pursue only those parts of the formal theory that are needed to establish the results used elsewhere in this book and some others of general utility.

In this spirit, the proofs that have been adopted for some of the standard results of complex variable theory have been chosen with an eye to simplicity rather than sophistication. This means that in some cases the imposed conditions are more stringent than would be strictly necessary if more sophisticated proofs were used; where this happens the less restrictive results are usually stated as well. The reader who is interested in a fuller treatment should consult one of the many excellent textbooks on this fascinating subject.

One further concession to ‘hand-waving’ has been made in the interests of keeping the treatment to a moderate length.

We have already discussed, in chapter 4, how complicated functions may be expressed as power series. However, this is not the only way in which a function may be represented as a series, and the subject of this chapter is the expression of functions as a sum of sine and cosine terms. Such a representation is called a Fourier series. Unlike Taylor series, a Fourier series can describe functions that are not everywhere continuous and/or differentiable. There are also other advantages in using trigonometric terms. They are easy to differentiate and integrate, their moduli are easily taken and each term contains only one characteristic frequency. This last point is important because, as we shall see later, Fourier series are often used to represent the response of a system to a periodic input, and this response often depends directly on the frequency content of the input. Fourier series are used in a wide variety of such physical situations, including the vibrations of a finite string, the scattering of light by a diffraction grating and the transmission of an input signal by an electronic circuit.

The Dirichlet conditions

We have already mentioned that Fourier series may be used to represent some functions for which a Taylor series expansion is not possible. The particular conditions that a function f(x) must fulfil in order that it may be expanded as a Fourier series are known as the Dirichlet conditions, and may be summarised by the following four points:

1. (i) the function must be periodic;

2. (ii) […]