Chapter 71
The complex or
exponential form
of a Fourier series
71.1 Introduction
The form used for the Fourier series in Chapters 66 to
70 consisted of cosine and sine terms. However, there is
another form that is commonly used—one that directly
gives the amplitude terms in the frequency spectrum and
relates to phasor notation. This form involves the use of
complex numbers (see Chapters 20 and 21). It is called
the exponential or complex form of a Fourier series.
71.2 Exponential or complex
notation
It was shown on page 226, equations (4) and (5) that:
e
j θ
= cos θ + j sin θ
(1)
and e
− j θ
= cos θ − j sin θ
(2)
Adding equations (1) and (2) gives:
e
j θ
+ e
− j θ
= 2 cosθ
from which, cos θ =
e j θ + e − j θ
2
(3)
Similarly, equation (1) – equation (2) gives:
e
j θ
− e
− j θ
= 2 j sin θ
from which,
sin θ =
e j θ − e − j θ
2 j
(4)
Thus, from page 630, the Fourier series f (x) over
any range L,
f (x) = a 0 +
∞
n=1
a n cos
2πnx
L
+ b n sin
2πnx
L
may be written as:
f (x) = a 0 +
∞
n=1
a n
e
j
2πnx
L + e
− j
2πnx
L
2
+ b n
e
j
2πnx
L − e
− j
2πnx
L
2 j
Multiplying top and bottom of the b n term by − j (and
remembering that j 2 =−1) gives:
f (x) = a 0 +
∞
n=1
a n
e
j
2πnx
L + e
− j
2πnx
L
2
− j b n
e
j 2πnx
L − e
− j 2πnx
L
2