Differential equations are the group of equations that contain derivatives. There are several different types of differential equations, but here we will be considering only the simplest types. As its name suggests, an ordinary differential equation (ODE) contains only ordinary derivatives (no partial derivatives) and describes the relationship between these derivatives of the dependent variable, usually called y, with respect to the independent variable, usually called x. The solution to such an ODE is therefore a function of x and is written y(x). For an ODE to have a closed-form solution, it must be possible to express y(x) in terms of the standard elementary functions such as x 2, exp x, In x, sin x, etc. The solutions of some differential equations cannot, however, be written in closed form, but only as an infinite series that carry no special names.

Ordinary differential equations may be separated conveniently into different categories according to their general characteristics. The primary grouping adopted here is by the order of the equation. The order of an ODE is simply the order of the highest derivative it contains. Thus, equations containing dy/dx, but no higher derivatives, are called first order, those containing d 2 y/dx 2 are called second order and so on. In this chapter we consider first-order equations and some of the more straightforward equations of second order.

Fourier transforms are eternal. They have not changed their nature since the last edition ten years ago: but the intervening time has allowed the author to correct errors in the text and to expand it slightly to cover some other interesting applications. The van Cittert–Zernike theorem makes a belated appearance, for example, and there are hints of some aspects of radio aerial design as interesting applications.

I also take the opportunity to thank many people who have offered criticism, often anonymously and therefore frankly, which has (usually) been acted upon and which, I hope, has improved the appeal both of the writing and of the contents.

History

Fourier transformation is formally an analytic process which uses integral calculus. In experimental physics and engineering, however, the integrand may be a set of experimental data, and the integration is necessarily done artificially. Since a separate integration is needed to give each point of the transformed function, the process would become exceedingly tedious if it were to be attempted manually, and many ingenious devices have been invented for performing Fourier transforms mechanically, electrically, acoustically and optically. These are all now part of history since the arrival of the digital computer and more particularly since the discovery – or invention – of the ‘fast Fourier transform’ algorithm or FFT as it is generally called. Using this algorithm, the data are put (‘read’) into a file (or ‘array’, depending on the computer jargon in use), the transform is carried out, and the array then contains the points of the transformed function. It can be achieved by a software program, or by a purpose-built integrated circuit. It can be done very quickly so that vibration sensitive instruments with Fourier transformers attached can be used for tuning pianos and motor engines, for aircraft and submarine detection and so on. It must not be forgotten that the ear is Nature's own Fourier transformer, and, as used by an expert piano-tuner, for example, is probably the equal of any electronic simulator in the 20–20 000-Hz range.

Fraunhofer diffraction

The application of Fourier theory to Fraunhofer diffraction problems, and to interference phenomena generally, was hardly recognized before the late 1950s. Consequently, only textbooks written since then mention the technique. Diffraction theory, of which interference is only a special case, derives from Huygens' principle: that every point on a wavefront which has come from a source can be regarded as a secondary source; and that all the wave fronts from all these secondary sources combine and interfere to form a new wavefront.

Some precision can be added by using calculus. In Fig. 3.1, suppose that at O there is a source of ‘strength’ q, defined by the fact that at A, a distance r from O, there is a ‘field’, E, of strength E = q/r. Huygens' principle is now as follows:

In elementary Fraunhofer diffraction theory we simplify. We assume the following.

• That only two dimensions need be considered. All apertures bounding the transparent part of the surface S are rectangular and of length unity perpendicular to the plane of the diagram.

• […]

As indicated at the start of the previous chapter, the differential calculus and its complement, the integral calculus, together form the most widely used tool for the analysis of physical systems. The link that connects the two is that they both deal with the effects of vanishingly small changes in related quantities; one seeks to obtain the ratio of two such changes, the other uses such a ratio to calculate the variation in one of the quantities resulting from a change in the other.

Any change in the value of any one property (or variable) of a physical system almost always results in the values of some or all of its other properties being altered; in general, the size of each consequential change depends upon the current values of all of the variables. As a result, during a finite change in any one of the values, that of x say, those associated with all of the other variables are continuously changing, making computation of the final situation difficult, if not impossible. The solution to this difficulty is provided by the integral calculus, which allows only vanishingly small changes, and, after any such change in one variable, brings all the other values ‘up to date’ (by infinitesimal amounts) before allowing any further change.

All scientists will know the importance of experiment and observation and, equally, be aware that the results of some experiments depend to a degree on chance. For example, in an experiment to measure the heights of a random sample of people, we would not be in the least surprised if all the heights were found to be different; but, if the experiment were repeated often enough, we would expect to find some sort of regularity in the results. Statistical methods are concerned with the analysis of real experimental data of this sort.

In this final chapter we discuss the subject of probability, which is the theoretical basis for most statistical methods. Our development of probability will be with an eye to its eventual applications in statistics, with little emphasis on the axioms and theorems approach favoured by most pure mathematicians.

We first discuss the terminology required, with particular reference to the convenient graphical representation of experimental results as Venn diagrams. The concepts of random variables and distributions of random variables are then introduced. It is here that the connection with statistics is made; we assert that the results of many experiments are random variables and that those results have some sort of regularity, represented by a distribution. Finally, the defining equations for some important distributions, together with some useful quantities that characterise them, are introduced and discussed.

This edition follows much advice and constructive criticism which the author has received from all quarters of the globe, in consequence of which various typos and misprints have been corrected and some ambiguous statements and anfractuosities have been replaced by more clear and direct derivations. Chapter 7 has been largely rewritten to demonstrate the way in which Fourier transforms are used in CAT scanning, an application of more than usual ingenuity and importance: but overall this edition represents a renewed effort to rescue Fourier transforms from the clutches of the pure mathematicians and present them as a working tool to the horny-handed toilers who strive in the fields of electronic engineering and experimental physics.

Since Mathematical Methods for Physics and Engineering by Riley, Hobson and Bence (Cambridge: Cambridge University Press, 1998), hereafter denoted by MMPE, was first published, the range of material it covers has increased with each subsequent edition (2002 and 2006). Most of the additions have been in the form of introductory material covering polynomial equations, partial fractions, binomial expansions, coordinate geometry and a variety of basic methods of proof, though the third edition of MMPE also extended the range, but not the general level, of the areas to which the methods developed in the book could be applied. Recent feedback suggests that still further adjustments would be beneficial. In so far as content is concerned, the inclusion of some additional introductory material such as powers, logarithms, the sinusoidal and exponential functions, inequalities and the handling of physical dimensions, would make the starting level of the book better match that of some of its readers.

To incorporate these changes, and others aimed at increasing the user-friendliness of the text, into the current third edition of MMPE would inevitably produce a text that would be too ponderous for many students, to say nothing of the problems the physical production and transportation of such a large volume would entail.