Page 233 - Numerical Analysis Using MATLAB and Excel

Chapter 6 Fourier, Taylor, and Maclaurin Series

a = 4A
-------
n
nπ
,
,,
and for n = 3 7 11 15 , and so on, it becomes
– 4A
a = ----------
n
nπ
Then, the trigonometric Fourier series for the square waveform with even symmetry is

( n1 )
–
4A ⎛ 1 1 ⎞ 4A ----------------
2 1
-
-
ft() = ------- cos ωt --– cos 3ωt + -- cos 5ωt – … = ------- ∑ – ( 1 ) -- cos nωt (6.32)
-
π ⎝ 3 5 ⎠ π n
n = odd
Alternate Solution:
Since the waveform of Example 6.3 is the same as of Example 6.1, but shifted to the right by π 2⁄
radians, we can use the result of Example 6.1, i.e.,

4A 1 1
⎛
-
-
ft() = ------- sin ωt + -- sin 3ωt + -- sin 5ωt + … ⎞ (6.33)
π ⎝ 3 5 ⎠
⁄
⁄
and substitute ωt with ωt + π 2 , that is, we let ωt = ωτ + π 2 . With this substitution, relation
(6.33) becomes
4A ⎛ π⎞ 1 ⎛ π⎞ 1 ⎛ π⎞
-
-
f τ() = ------- sin ωτ + --- + -- sin 3 ωτ + --- + --sin 5 ωτ + --- + …
π ⎝ 2 ⎠ 3 ⎝ 2 ⎠ 5 ⎝ 2 ⎠
(6.34)
4A ⎛ π⎞ 1 ⎛ 3π⎞ 1 ⎛ 5π⎞
-
-
= ------- sin ωτ + --- + -- sin 3ωτ + ------ + -- sin 5ωτ + ------ + …
π ⎝ 2 ⎠ 3 ⎝ 2 ⎠ 5 ⎝ 2 ⎠
)
⁄
)
⁄
and using the identities sin ( x + π 2 = cos x , sin ( x + 3π 2 = – cos x , and so on, we rewrite
(6.34) as
4A 1 1
-
-
f τ() = ------- cos ωτ – -- cos 3ωτ + -- cos 5ωτ … (6.35)
–
π 3 5
and this is the same as (6.27).
Therefore, if we compute the trigonometric Fourier series with reference to one ordinate, and
afterwards we want to recompute the series with reference to a different ordinate, we can use the
above procedure to save time.

6−16 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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