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The Exponential Form of the Fourier Series

or

a = C + C – n (6.97)
n
n
Similarly,
1
C – C – n = -- a jb–( 2 - n n a – – jb ) n (6.98)
n
n
or
(
b = jC – C ) – n (6.99)
n
n
Symmetry in Exponential Series
1. For even functions, all coefficients C i are real

We recall from (6.89) and (6.90) that

1 ⎛ b n⎞ 1
-
C – n = -- a – ----- = -- a +( 2 - n jb ) n (6.100)
2 ⎝
⎠
n
j
and
1 ⎛ b n⎞ 1
C = -- a + ----- = --- a j– b( 2 n n ) (6.101)
-
2 ⎝
⎠
n
n
j
Since even functions have no sine terms, the b n coefficients in (6.100) and (6.101) are zero.
Therefore, both C – n and C n are real.

2. For odd functions, all coefficients C i are imaginary
Since odd functions have no cosine terms, the a n coefficients in (6.100) and (6.101) are zero.
Therefore, both C – n and C n are imaginary.

3. If there is half−wave symmetry, C = 0 for n = even
n
We recall from the trigonometric Fourier series that if there is half−wave symmetry, all even
harmonics are zero. Therefore, in (6.100) and (6.101) the coefficients a n and b n are both zero
for n = even , and thus, both C – n and C n are also zero for n = even .

4. If there is no symmetry, ft() is complex.

5. C – n = C ∗ always
n
This can be seen in (6.100) and (6.101)

Numerical Analysis Using MATLAB® and Excel®, Third Edition 6−31
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