Mathematical Fundamentals
125
eigenfunctions
factorial
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
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