Mathematical Fundamentals
127
Euler formula
complex sinusoid
De Moivre formula
0
5
10
-1
-0.5
0
0.5
1
Sine of an angle
angle [rad]
sin
0
5
10
15
-1
-0.5
0
0.5
1
Cosine of an angle
angle [rad]
cos
0
5
10
-6
-4
-2
0
2
4
6
Tangent of an angle
angle [rad]
tan
Figure 8: Trigonometric functions
A fundamental identity, that links trigonometry with exponential functions,
is the Euler formula
e
i
= cos + i sin ,
(23)
which expresses a complex number laying on the unit circumference as an ex-
ponential with imaginary exponent
. When is left free to take any real value,
the exponential (23) generates the so-called complex sinusoid.
Any complex number c having magnitude and argument can be repre-
sented in compact form as
c = e
i
,
(24)
and to it we can apply the usual rules of power functions. For instance, we can
compute the m-th power of c as
c
m
=
m
e
im
=
m
(cos m + i sin m) ,
(25)
thus showing that it is obtained by taking the m-th power of the magnitude and
multiplying by m the argument. The (25) is called De Moivre formula.
The order-m root of a number c is that number b such that b
m
= c. In
general, a complex number admits m order-m distinct complex roots
Moivre formula establishes that
the order-m roots of 1 are evenly distributed
9
The actual meaning of the exponential comes from the series expansion (20)
10
For instance, 1 admits two square roots (1 and -1) and four order-4 roots (1, -1, i, -i).
11
The reader is invited to justify this statement by an example. The simplest non-trivial
example is obtained by considering the cubic roots of 1.