428 Chapter 15Orthogonal expansions. Fourier analysis
EXAMPLE 15.6The Fourier series of the function
(15.42)
The graph of the (extended) function, Figure 15.6, shows that the function is
discontinuous when xis a multiple of π, and the function can therefore be expanded
as the Fourier series (15.37),
everywhere other than at these points of discontinuity. It is a property of Fourier
series that the value of the series at a point of discontinuity is the mean value of the
function at the point. In the present case, this mean value is 1 2 2.
The coefficienta
0. By equation (15.38) withn 1 = 1 0,
The functionf(x)is equal to zero in the first integral on the right, and is equal to
unity in the second. Therefore,
(15.43)
The coefficientsa
n, n 1 > 10. By equation (15.38),
(15.44)
All the Fourier coefficientsa
nare zero except fora
01 = 11.
a f x nxdx nxdx
n===
−++111
0
0πππ
ππππZZ()cos cos
sinnnx
n
= 0
adx
001
== 1
+π
πZ
a fxdx fxdx fxdx
000111
==+
−+−+πππ
ππππZZZ() () ()
fx
a
anxbnx
nnn()=+( cos +sin )
=∑
012
∞fx
x
x
()=
,<<
,−<<
10
00
for
for
π
π
- 1
− 3 π − 2 π −π +π 2 π 3 π
0
x
f(x)
............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...............................................................................................................................................................................................
................................................................................................................................................................................................ ..................................................................................................................................................................................................................................................................................................................................................................................................
Figure 15.6