The Chemistry Maths Book, Second Edition

15.4 Fourier series 427

Figure 15.4

Figure 15.5

Form 1 = 1 n 1 > 10 ,

0 Exercise 7

Periodicity

The trigonometric functions (15.31) are periodic functions of xwith period 2 π

and, therefore, every linear combination of them is also periodic. This means that the

functionf(x)can be extended to values outside the base interval−π 1 ≤ 1 x 1 ≤ 1 πby means

of the periodicity relation

f(x 1 + 12 π) 1 = 1 f(x) (15.40)

For example, the function

(15.41)

shown in Figure 15.4 can be extended by means of (15.40) to give the periodic

function shown in Figure 15.5.

Periodic functions occur in the mathematical modelling of physical systems that

exhibit periodic phenomena. For example, the function shown in Figure 15.5, with

f(x)as current and xas time, might represent the result of passing an alternating

electric current through a rectifier that allows current to flow in one direction only (a

half-wave rectifier). Despite the discontinuities (of the gradient) exhibited by the

function, it is one of an important class of functions that can be represented as Fourier

series. An example of such a function is discussed in Example 15.6.

In general, a function that satisfies the periodicity relation (15.40) can be expanded

as a Fourier series if it is single-valued and piecewise continuous; that is, continuous

except for a finite number of finite discontinuities and only a finite number of

maxima and minima in any finite interval (that is, every reasonably well-behaved

function).

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for

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• 1

−π +π

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