Page 46 - Essentials of applied mathematics for scientists and engineers

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36 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
3.2.4 Examples of Fourier Series
Example 3.1. Determine a Fourier sine series representation of f (x) = x on the interval
(0, 1). The series will take the form

∞

x = c j sin( jπx) (3.28)
j=0

since the sin( jπx) forms an orthogonal set on (0, 1), multiply both sides by sin(kπx)dx and
integrate over the interval on which the function is orthogonal.

1 1
∞

x sin(kπx)dx = c j sin( jπx)sin(kπx)dx (3.29)
k=0
x=0 x=0

Noting that all of the terms on the right-hand side of (2.20) are zero except the one for which
k = j,

1 1

2
x sin( jπx)dx = c j sin ( jπx)dx (3.30)
x=0 x=0

After integrating we find

(−1) j+1 c j
= (3.31)
jπ 2

Thus,

∞ j+1
(−1)

x = 2sin( jπx) (3.32)
jπ
j=0
This is an alternating sign series in which the coefficients always decrease as j increases, and
it therefore converges. The sine function is periodic and so the series must also be a periodic
function beyond the interval (0, 1). The series outside this interval forms the periodiccontinuation
of the series. Note that the sine is an odd function so that sin( jπx) =− sin(− jπx). Thus the
periodic continuation looks like Fig. 3.2. The series converges everywhere, but at x = 1itis
identically zero instead of one. It converges to 1 − ε arbitrarily close to x = 1.