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 this chapter, we shift our focus from behavior that is primarily social to behavior that is cognitive. Social cognition was discussed in Chapter 4, where the phenomena of attribution, conformity and self-construal were presented as social psychological manifestations of the cultural context. In Chapter 8, we shall return to a consideration of cultural aspects of cognition, where links between language and culture are explored. In this chapter we focus on the more traditional cognitive phenomena that deal with knowing and interpreting the world, using such notions as intelligence, abilities and styles. We begin with a brief overview of the historical legacy of thinking about how human populations are similar and different in their cognitive lives. In each of the subsequent sections we present four perspectives on relationships between cognition and culture, beginning with a set of conceptualizations that involve a unitary view of cognition (captured in the notion of general intelligence). Thereafter we present cognitive styles, which are general preferences to deal with the world in a particular way. The third perspective is one that focusses on the East–West contrasts in cognition, where there has been much recent research on differences in the cognitive life of western and East Asian populations. Finally, the fourth perspective is contextualized cognition, in which cognitions are seen as task-specific and embedded in sociocultural contexts.

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.

LIFE

Our evidence for St.'s life comes mainly from the Siluae, particularly 3.5, an epistle to his wife, and 5.3, the poem on his father's death. St. was a poet of two cities and two cultures. He was born around 50 ce in Naples, ‘practically a Greek city’ (quasi Graecam urbem, Tac. Ann. 15.33), where Hellenic culture was supported by Roman wealth and power. The city celebrated Greek-style games founded by Augustus; Greek and Latin, and probably Oscan, were spoken in the city, with Greek remaining the language of cultural prestige in many official contexts. St. was the son of an eminent grammaticus and Greek professional poet who, composing in a tradition of extemporaneous poetry, won prizes at the major Greek games and was honoured with a statue in the Athenian agora. St. was doubly privileged: born in a city with a rich cultural heritage, he was taught a demanding curriculum in Greek literature by his father in a period when such knowledge was crucial for social advancement.

At some point in St.'s youth father and son moved to Rome where St. senior had a successful career as grammaticus to the Roman elite and probably also to the imperial family. He enjoyed too the great honour of reciting his poem on the civil war of 68–9 in the temple of Jupiter Optimus Maximus on the Capitol (5.3.199–204). The father thus may have been closer to the imperial court than the son.

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.

At virtually the same time as the rise in cross-cultural studies of development, there has been a dramatic increase in interest in life span development, which covers not only the period from birth to maturity, but also continues through maturity to eventual demise (Baltes, Lindenberger and Staudinger, 2006). In this chapter, we examine cross-cultural variations in the developmental stages beyond the ones that were discussed in Chapter 2; these are childhood, adolescence and adulthood. After discussing cultural notions of childhood and adolescence, we will present evidence on how childhood experiences can explain cross-cultural variations in adulthood. In the section on adulthood, we will deal with mating, partnership and parenting across cultures. In the final section, we will discuss life span developmental and evolutionary approaches to late adulthood. The chapter concludes with reflections on the cross-cultural applicability of the developmental issues raised in the last two chapters.

Childhood and adolescence

As we have seen in the previous chapter, human development can be described in stages. There, we dealt with the first decade of life, comprising the two earliest stages, infancy and early childhood. While infancy is the period from birth to two years, childhood is mainly defined as the period after infancy and before sexual maturation.

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.

It is not unusual in the analysis of a physical system to encounter an equation in which an unknown but required function y(x), say, appears under an integral sign. Such an equation is called an integral equation, and in this chapter we discuss several methods for solving the more straightforward examples of such equations.

Before embarking on our discussion of methods for solving various integral equations, we begin with a warning that many of the integral equations met in practice cannot be solved by the elementary methods presented here but must instead be solved numerically, usually on a computer. Nevertheless, the regular occurrence of several simple types of integral equation that may be solved analytically is sufficient reason to explore these equations more fully.

We begin this chapter by discussing how a differential equation can be transformed into an integral equation and by considering the most common types of linear integral equation. After introducing the operator notation and considering the existence of solutions for various types of equation, we go on to discuss elementary methods of obtaining closed-form solutions of simple integral equations. We then consider the solution of integral equations in terms of infinite series and conclude by discussing the properties of integral equations with Hermitian kernels, i.e. those in which the integrands have particular symmetry properties.