Page 47 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

ORTHOGONAL SETS OF FUNCTIONS 37

1

-1 1 2 3

FIGURE 3.2: The periodic continuation of the function x represented by the sine series

Example 3.2. Find a Fourier cosine for f (x) = x on the interval (0, 1). In this case
∞

x = c n cos(nπx) (3.33)
n=0
Multiply both sides by cos(mπx)dx and integrate over (0, 1).

1 1
∞

x cos(mπx)dx = c n cos(mπx)cos(nπx)dx (3.34)
n=0
x=0 x=0
and noting that cos(nπx) is an orthogonal set on (0, 1) all terms in (2.23) are zero except when
n = m. Evaluating the integrals,

2
c n [(−1) − 1]
= (3.35)
2 (nπ) 2
There is a problem when n = 0. Both the numerator and the denominator are zero there.
However we can evaluate c 0 by noting that according to Eq. (3.26)

1
1

xdx = c 0 = (3.36)
2
x=0
and the cosine series is therefore
n
∞
1 [(−1) − 1]
x = + 2 cos(nπx) (3.37)
2 (nπ) 2
n=1
The series converges to x everywhere. Since cos(nπx) = cos(−nπx)itisanevenfunction and
its periodic continuation is shown in Fig. 3.3. Note that the sine series is discontinuous at x = 1,
while the cosine series is continuous everywhere. (Which is the better representation?)