Higher Engineering Mathematics

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)