Page 454 - Advanced engineering mathematics

434 CHAPTER 13 Fourier Series

x x
–3 – 2 – 1 0 1 2 3 –3 – 2 – 1 0 1 2 3

0 0
–2 –2

–4 –4

–6 –6

–8 –8

–10 –10
–12 –12

FIGURE 13.4 Fifth partial sum of the FIGURE 13.5 Tenth partial sum in Example
Fourier series in Example 13.4. 13.4.

x
–3 – 2 – 1 0 1 2 3
0

–2

–4

–6

–8
–10

–12

FIGURE 13.6 Twentieth partial sum in
Example 13.4.

and
−1
n
1 (e − e )(−1) (nπ)
x
b n = e sin(nπx)dx =− .
2
1 + n π 2
−1
x
The Fourier series of e on [−1,1] is
∞
1 (−1) n
−1
−1
(e − e ) + (e − e ) (cos(nπx) − nπ sin(nπx)).
2 1 + n π 2
2
n=1
x
Because e is continuous with a continuous derivative for all x, this series converges to

e x for −1 < x < 1
1 (e + e ) for x = 1 and for x =−1.
−1
2
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October 14, 2010 14:57 THM/NEIL Page-434 27410_13_ch13_p425-464