L
Fourier series
70
Fourier series for a non-periodic
function over range 2π
70.1 Expansion of non-periodic
functions
If a functionf(x) is not periodic then it cannot be
expanded in a Fourier series forallvalues ofx.How-
ever, 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 off(x)
in the given range and then repeating them out-
side of the given range at intervals of 2π. Since
this new function is, by construction, periodic with
period 2π, it may then be expanded in a Fourier
series for all values ofx. For example, the func-
tionf(x)=xis not a periodic function. However, if a
Fourier series forf(x)=xis required then the func-
tion is constructed outside of this range so that it is
periodic with period 2πas shown by the broken lines
in Fig. 70.1.
For non-periodic functions, such asf(x)=x, the
sum 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
function over a range 2π, exactly the same for-
mulae for the Fourier coefficients are used as in
Section 69.3(i).
70.2 Worked problems on 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. 70.2 and is
then constructed outside of that range so that it is
periodic of period 2π(see broken lines) with the
resulting saw-tooth waveform.
f(x)
f(x) = x
2 π
− 2 π 0 2 π 4 π x
Figure 70.1
f(x)
f(x) = 2 x
2 π
− 2 π
− 2 π −π^0 π 2 π 3 πx
Figure 70.2
For a Fourier series:
f(x)=a 0 +
∑∞
n= 1
(ancosnx+bnsinnx)
From Section 69.3(i),
a 0 =
1
2 π
∫π
−π
f(x)dx
=
1
2 π
∫π
−π
2 xdx=
2
2 π
[
x^2
2
]π
−π
= 0
an=
1
π
∫π
−π
f(x) cosnxdx=
1
π
∫π
−π
2 xcosnxdx
=
2
π
[
xsinnx
n
−
∫
sinnx
n
dx
]π
−π
by parts (see Chapter 43)