Page 463 - Advanced engineering mathematics

13.3 Sine and Cosine Series 443

2x
The cosine series for e on [0,1] is
∞
n
2
1 e (−1) − 1
2
(e − 1) + 4 cos(nπx).
2
2 4 + n π 2
n=1
This series converges to
⎧
⎪e 2x for 0 < x < 1
⎨
1 for x = 0
⎪
e for x = 1.
⎩ 2
2x
2x
This Fourier cosine series converges to e for 0≤ x ≤1. Figure 13.14 shows e and the fifth
partial sum of this cosine series.
13.3.2 Sine Series
We can also write an expansion of f on [0, L] that contains only sine terms. Now reflect the
graph of f on [0, L] through the origin to create an odd function h on [−L, L], with h(x)= f (x)
for 0 < x ≤ L (Figure 13.15).

7

6

5
4

3

2

1
0 0.2 0.4 0.6 0.8 1
x
FIGURE 13.14 Fifth partial sum of the
cosine series in Example 13.9.
y

x

FIGURE 13.15 Odd extension of a func-
tion defined on [0, L].

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