Mathematical Fundamentals

125

eigenfunctions

factorial

radians

0

0.5

1

1.5

2

-4

-3

-2

-1

0

1

Figure 6: Logarithms expressed in the fundamental bases 2 (solid line) and 10

(dashed line)

the exponentials expressed in base e are eigenfunctions for the derivative op-

erator. In other words, differential linear operators do not alter the form of

these exponentials. Moreover, the exponential with base e admits an elegant

translation into an infinite series of addenda

e

x

= 1 +

x

1!

+

x

2

2!

+

x

3

3!

+ . . . ,

(20)

where n! is the factorial of n and is equal to the product of all integers ranging

from 1 to n. It can be proved that the infinite sum on the right-hand side of (20)

gives meaning to the exponential function even where its argument is complex.
A.6

Trigonometric Functions

Trigonometry describes the relations between angles and segments subtended

by these angles. The main trigonometric functions are easily visualized on the

complex plane, as in fig. 7, where the unit circle is explicitly represented.
**R**

**I**

P

Q

O

cos

sin

Figure 7: Trigonometric functions on the complex plane

An angle cuts on the unit circle an arc whose length is defined as the

measure in radians of the angle. Since the circumference has length 2, the 360

o