For reasons that are explained in the preface to Essential Mathematical Methods for the Physical Sciences the text of the third edition of Mathematical Methods for Physics and Engineering (MMPE) (Cambridge: Cambridge University Press, 2006) by Riley, Hobson and Bence, after a number of additions and omissions, has been republished as two slightly overlapping texts. Essential Mathematical Methods for the Physical Sciences (EMMPS) contains most of the more advanced material, and specifically develops mathematical methods that can be applied throughout the physical sciences; an augmented version of the more introductory material, principally concerned with mathematical tools rather than methods, is available as Foundation Mathematics for the Physical Sciences. The full text of MMPE, including all of the more specialized and advanced topics, is still available under its original title.

As in the third edition of MMPE, the penultimate subsection of each chapter of EMMPS consists of a significant number of problems, nearly all of which are based on topics drawn from several sections of that chapter. Also as in the third edition, hints and outline answers are given in the final subsection, but only to the odd-numbered problems, leaving all even-numbered problems free to be set as unaided homework.

This book is the solutions manual for the problems in EMMPS. For the 230 plus oddnumbered problems it contains, complete solutions are available, to both students and their teachers, in the form of this manual; these are in addition to the hints and outline answers given in the main text.

In Chapter 14, we developed the basic theory of the functions of a complex variable, z = x + iy, studied their analyticity (differentiability) properties and derived a number of results concerned with values of contour integrals in the complex plane. In this current chapter we will show how some of those results and properties can be exploited to tackle problems arising directly from physical situations or from apparently unrelated parts of mathematics.

In the former category will be the use of the differential properties of the real and imaginary parts of a function of a complex variable to solve problems involving Laplace's equation in two dimensions, whilst an example of the latter might be the summation of certain types of infinite series. Other applications, such as the Bromwich inversion formula for Laplace transforms, appear as mathematical problems that have their origins in physical applications; the Bromwich inversion enables us to extract the spatial or temporal response of a system to an initial input from the representation of that response in “frequency space” – or, more correctly, imaginary frequency space.

Some other topics that could have been considered, had space permitted, are the location of the (complex) zeros of a polynomial, the approximate evaluation of certain types of contour integrals using the methods of steepest descent and stationary phase, and the so-called “phase-integral” solutions to some differential equations. However, for these and many more, the brief outlines given in the final section of this chapter will have to suffice.

The reader will be familiar with how, through Taylor series (see Section A.6 of Appendix A), 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.

Since Mathematical Methods for Physics and Engineering (Cambridge: Cambridge University Press, 1998) by Riley, Hobson and Bence, 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 to increase 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.

In the previous two chapters, we introduced the most important second-order linear ODEs in physics and engineering, listing their regular and irregular singular points in Table 7.1 and their Sturm–Liouville forms in Table 8.1. These equations occur with such frequency that solutions to them, which obey particular commonly occurring boundary conditions, have been extensively studied and given special names.

In this chapter, we discuss these so-called “special functions” and their properties. Inevitably, for each set of functions in turn, the discussion has to cover the differential equation they satisfy, their polynomial or power series form with some particular examples, their orthogonality and normalization properties, and their recurrence relations. In addition, as first introduced in this chapter, most sets possess a Rodrigues' formula and a generating function.

Although each of these aspects needs to be treated in sufficient detail for the enquiring reader to be satisfied about the validity of the results stated, their serial presentation, for one set of functions after another, tends to become rather overwhelming. Consequently it is suggested that once the reader has become familiar with the general nature of each of the aspects, by studying, say, Sections 9.1 to 9.3 on Legendre functions, associated Legendre functions and spherical harmonics, he or she may treat other sets of functions more lightly, turning in the first instance to the summary beginning on p. 377, and only referring to the detailed derivations, proofs and worked examples in Sections 9.4 to 9.9 when specific needs arise.

In this chapter, we turn to the study of statistics, which is concerned with the analysis of experimental data. In a book of this nature we cannot hope to do justice to such a large subject; indeed, many would argue that statistics belongs to the realm of experimental science rather than in a mathematics textbook. Nevertheless, physical scientists and engineers are regularly called upon to perform a statistical analysis of their data and to present their results in a statistical context. This justifies the inclusion of the subject in a book such as this, but we will concentrate on those aspects of direct relevance to the presentation of experimental data.

Experiments, samples and populations

We may regard the product of any experiment as a set of N measurements of some quantity x or of some set of quantities x, y, …, z. This set of measurements constitutes the data. Each measurement (or data item) consists accordingly of a single number xi or a set of numbers (xi, yi, …, zi), where i = 1, …, N. For the moment, we will assume that each data item is a single number, although our discussion can be extended to the more general case.

As a result of inaccuracies in the measurement process, or because of intrinsic variability in the quantity x being measured, one would expect the N measured values xi, x2, …, xN to be different each time the experiment is performed. We may therefore consider the xi as a set of N random variables.

In this chapter and the next, the solution of differential equations of types typically encountered in the physical sciences and engineering is extended to situations involving more than one independent variable. A partial differential equation (PDE) is an equation relating an unknown function (the dependent variable) of two or more variables to its partial derivatives with respect to those variables. The most commonly occurring independent variables are those describing position and time, and so we will couch our discussion and examples in notation appropriate to them.

As in the rest of this book, we will focus our attention on the equations that arise most often in physical situations. We will restrict our discussion, therefore, to linear PDEs, i.e. those of first degree in the dependent variable. Furthermore, we will discuss primarily second-order equations. The solution of first-order PDEs will necessarily be involved in treating these, and some of the methods discussed can be extended without difficulty to third- and higher-order equations. We shall also see that many ideas developed for ODEs can be carried over directly into the study of PDEs.

Initially, in the current chapter, we will concentrate on general solutions of PDEs in terms of arbitrary functions of particular combinations of the independent variables, and on the solutions that may be derived from them in the presence of boundary conditions. We also discuss the existence and uniqueness of the solutions to PDEs under given boundary conditions.

In the previous chapter the solution of both homogeneous and non-homogeneous linear ODEs of order ≥ 2 was discussed. In particular we developed methods for solving some equations in which the coefficients were not constant but functions of the independent variable x. In each case we were able to write the solutions to such equations in terms of elementary functions, or as integrals. In general, however, the solutions of equations with variable coefficients cannot be written in this way, and we must consider alternative approaches.

In this chapter we discuss a method for obtaining solutions to linear ODEs in the form of convergent series. Such series can be evaluated numerically, and those occurring most commonly are named and tabulated. There is in fact no distinct borderline between this and the previous chapter, since solutions in terms of elementary functions may equally well be written as convergent series (i.e. the relevant Taylor series). Indeed, it is partly because some series occur so frequently that they are given special names such as sin x, cos x or exp x.

Since, in this chapter, we shall be concerned principally with second-order linear ODEs we begin with a discussion of this type of equation, and obtain some general results that will prove useful when we come to discuss series solutions.