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function that is continuous everywhere but which has no derivatives. Such functions are
discussed subsequently.
Following the approach taken previously, we assume a trial solution of the form
t
μ
,
Z
(
t
)
=
A
(
t
)
(2.34)
and insert it into (
2.33
). As before we determine the separate equalities for the scaling
parameters and obtain the power-law index as the ratio of the scaling parameters
ln
a
ln
b
,
μ
=
(2.35)
the same ratio as we found in the discussion of the distribution of mass within the large
sphere. The unknown function
A
may be written in general as the Fourier series
(
2.14
) and is a function periodic in the logarithm of time with a period ln
b
.
In the literature
Z
(
t
)
is called a homogeneous function [
5
]. The scaling index given by
(
2.35
) is related to the fractal dimension obtained from the geometric argument given
for the distribution of mass within a sphere (
2.31
) and elsewhere. The homogeneous
solution to the scaling relation can be written as
(
t
)
∞
A
k
t
H
k
Z
(
t
)
=
,
(2.36)
k
=−∞
where the complex scaling index is given by
k
ln
b
.
H
k
=−
μ
+
i
2
π
(2.37)
The complex exponent in (
2.36
) indicates a possible periodic component of the homo-
geneous function superposed on the more traditional power-law scaling. There exist
phenomena that have a slow modulation (
k
1); recall that we saw the suggestion of
this in the scaling of the bronchial tubes and we discuss the fractal nature of such time
series more fully in subsequent chapters. For the moment it is sufficient to see that we
can have a process that is fractal in time just as we have a process that is fractal in space.
Functions of the form (
2.36
) can be continuous everywhere but are not necessarily
differentiable anywhere. Certain of these continuous non-differentiable functions have
been shown to be fractal. In his investigation of turbulence in 1926 Richardson observed
that the velocity field of the atmospheric wind is so erratic that it probably cannot be
described by an analytic function. In his paper [
63
], Richardson asked “does the wind
possess a velocity?,” addressing the non-differentiability of the wind field. He suggested
a Weierstrass function as a candidate to represent the wind field. Scientists have since
come to realize that Richardson's intuition was superior to nearly a century of analysis
regarding the nature of turbulence.
A generalization of Weierstrass' original function has the form
=
∞
1
a
n
[
b
n
t
(
)
=
−
(
)
]
,
W
t
1
cos
(2.38)
n
=−∞
where
a
and
b
are real parameters and this form is chosen to simplify some of the
algebra. Using the properties of trigonometric functions it is easy to see that the
