Chapter 67

Fourier series for a

non-periodic function

over range 2π

67.1 Expansion of non-periodic

functions

If a function f(x)is not periodic then it cannot be
expandedinaFourierseriesforallvalues ofx.However,
it is possible to determine a Fourier series to represent
the function over any range of width 2π.
Given a non-periodic function, a new function may
be constructed by taking the values of f(x)in the
given range and then repeating them outside of the
given range at intervals of 2π. Since this new func-
tion is, by construction, periodic with period 2π,
it may then be expanded in a Fourier series for
all values of x. For example, the function f(x)=x
is not a periodic function. However, if a Fourier
series for f(x)=x is required then the function
is constructed outside of this range so that it is
periodic with period 2πas shown by the broken lines in
Fig. 67.1.
For non-periodic functions, such as f(x)=x,thesum
of the Fourier series is equal tof(x)at all points in the
given range but it is not equal tof(x)at points outside
of the range.
For determining a Fourier series of a non-periodic func-
tion over a range 2π, exactly the same formulae for the
Fourier coefficients are used as in Section 66.3(i).

x

f(x)

f(x) 5 x

22  0 2 

2 

4 

Figure 67.1

67.2 Worked problemson Fourier

series of non-periodic functions

over a range of 2π

Problem 1. Determine the Fourier series to

represent the functionf(x)= 2 xin the range

−πto+π.

The functionf(x)= 2 xis not periodic. The function is

shown in the range−πtoπin Fig. 67.2 and is then

constructed outside of that range so that it is periodic of

period2π(seebrokenlines)withtheresultingsaw-tooth

waveform.

ForaFourierseries:

f(x)=a 0 +

∑∞

n= 1

(ancosnx+bnsinnx)