Page 458 - Advanced engineering mathematics

438 CHAPTER 13 Fourier Series

1

0.8

0.6

0.4

0.2

0
–1 –0.5 0 0.5 1
x
FIGURE 13.11 Tenth partial sum of the
series of Example 13.8.

1
2
−12(nπx) sin(nπx) − 24nπx cos(nπx) + 24sin(nπx)
+ 2
(nπ) 5
0
2
2
n π − 6 n
=8 (−1) .
4
n π 4
The Fourier series is
1 8(−1) n
∞
2
2
+ (n π − 6) cos(nπx).
5 n π 4
4
n=1
4
By Theorem 13.1, this series converges to x for −1 ≤ x ≤ 1. Figure 13.11 shows the
twentieth partial sum of this series, compared with the function.
13.2.2 The Gibbs Phenomenon
A.A. Michelson was a Prussian-born physicist who teamed with E.W. Morley of Case-Western
Reserve University to show that the postulated “ether,” a fluid which supposedly permeated all of
space, had no effect on the speed of light. Michelson also built a mechanical device for construct-
ing a function from its Fourier coefficients. In one test, he used eighty coefficients for the series
of f (x) = x on [−π,π] and noticed unexpected jumps in the graph near the endpoints. At first,
he thought this was a problem with his machine. It was subsequently found that this behavior is
characteristic of the Fourier series of a function at a point of discontinuity. In the early twentieth
century, the Yale mathematician Josiah Willard Gibbs finally explained this behavior.
To illustrate the Gibbs phenomenon, expand f in a Fourier series on [−π,π], where
⎧
⎪−π/4 for −π ≤ x < 0
⎨
f (x) = 0 for x = 0
⎪
π/4 for 0 < x ≤ π.
⎩
This function has a jump discontinuity at 0, but its Fourier series on [−π,π] converges at 0 to
1 1 π π

( f (0+) + f (0−)) = − = 0 = f (0).
2 2 4 4

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October 14, 2010 14:57 THM/NEIL Page-438 27410_13_ch13_p425-464