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.