128
D. Rocchesso: Sound Processing
regular functions
derivative
along the unit circumference, starting from 1 itself, and they are separated by
a constant angle 2/m.
At this point, we propose some problems for the reader:
Prove the following identities, which are corollaries of the Euler identity
cos =
e
i
+ e
-i
2
,
(26)
sin =
e
i
- e
-i
2i
.
(27)
Prove the "most beautiful formula in mathematics" [59]
e
i
+ 1 = 0 .
(28)
Prove, by means of the De Moivre formula, the following identities:
cos 2 = cos
2
- sin
2
,
(29)
sin 2 = 2 sin cos .
(30)
Prove, by the representation of unit-magnitude complex numbers e
i
, that
the following identities are true:
cos ( + ) = cos cos - sin sin ,
(31)
sin ( + ) = cos sin + sin cos .
(32)
A.7
Derivatives and Integrals
A.7.1
Derivatives of Functions
Given the function y = f (x) (for the moment, we only consider functions of
one variable), it might be interesting to find the places where local maxima and
minima are located. It is natural, in such a search, to focus on the slope of the
line that is tangent to the function curve, in such a way that local maxima and
minima are found where the slope of the tangent is zero (i.e., the tangent is
horizontal). This operation is possible for all regular functions, which are func-
tions without discontinuities and without sharp corners. Given this assumption
of regularity, the shape of the curve can be defined at any point, thus becom-
ing itself a function of the same independent variable. This function is called
derivative and is indicated with
y =
dy
dx
.
(33)
The notation (33) recalls how the local shape of a curve can be computed:
the tangent line is drawn, two distinct points are taken on this line, the ratio
between the differences of coordinates y and x of the points is formed. As we have
already seen in appendix A.6, this operation corresponds to the computation of
the trigonometric tangent, whose argument is the angle formed by the tangent
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