Mathematical Methods for Physics and Engineering : A Comprehensive Guide

FOURIER SERIES

(a)

(b)

(c)

(d)

0

0

0

0

L

L

L

L

2 L

2 L

2 L

Figure 12.4 Possible periodic extensions of a function.

12.5 Non-periodic functions

We have already mentioned that a Fourier representation may sometimes be used

for non-periodic functions. If we wish to find the Fourier series of a non-periodic

function only within a fixed range then we maycontinuethe function outside the

range so as to make it periodic. The Fourier series of this periodic function would

then correctly represent the non-periodic function in the desired range. Since we

are often at liberty to extend the function in a number of ways, we can sometimes

make it odd or even and so reduce the calculation required. Figure 12.4(b) shows

the simplest extension to the function shown in figure 12.4(a). However, this

extension has no particular symmetry. Figures 12.4(c), (d) show extensions as odd

and even functions respectively with the benefit that only sine or cosine terms

appear in the resulting Fourier series. We note that these last two extensions give

a function of period 2L.

In view of the result of section 12.4, it must be added that the continuation

must not be discontinuous at the end-points of the interval of interest; if it is

the series will not converge to the required value there. This requirement that

the series converges appropriately may reduce the choice of continuations. This

is discussed further at the end of the following example.

Find the Fourier series off(x)=x^2 for 0 <x≤ 2.

We must first make the function periodic. We do this by extending the range of interest to
− 2 <x≤2insuchawaythatf(x)=f(−x) and then lettingf(x+4k)=f(x), wherekis
any integer. This is shown in figure 12.5. Now we have an even function of period 4. The
Fourier series will faithfully representf(x) in the range,− 2 <x≤2, although not outside
it. Firstly we note that since we have made the specified function even inxby extending