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fractional calculus can contribute to an understanding of complex webs in the physical,
social and life sciences.
Another way to introduce fractional operators without the torture of rigorous math-
ematics is through Cauchy's formula for n -fold integration of a function f
(
t
)
over a
,
fixed time interval ( a
t ):
t
τ 1 τ 1
a
τ 2 τ 2
a
τ n 1
t
1
n
1 f
(τ).
(5.28)
The n -fold integral on the left is denoted by the integral or anti-derivative operator
d
d
d
τ 3 ···
d
τ n f
n ) =
d
τ(
t
τ)
(
n
1
) !
a
a
a
t
1
(
a D n
n
1 f
[
f
(
t
) ]≡
d
τ(
t
τ)
(τ),
(5.29)
t
n
)
a
where for convenience we have again introduced the gamma function and extended the
notation to include the lower limit of the integral as an index on the lower left side of the
operator. The fractional integral analogue to this equation is given, just as in the case of
the derivative, by the same operator expression with a non-integer index,
t
1
(α)
a D α
τ) α 1 f
[
f
(
t
) ]≡
d
τ(
t
(τ),
(5.30)
t
a
with the restriction a
t . Note that the gamma function remains well defined for
non-integer and non-positive values of its argument by analytic continuation into the
complex domain. The corresponding fractional derivative is defined by
D t a D α n
) ]
a D t [
f
(
t
) ]≡
[
f
(
t
(5.31)
t
with the restrictions on the fractional index
α
n
<
0 and
α
n
+
1
>
0. Conse-
quently, for 0
1. Equation ( 5.30 ) is the Riemann-Liouville (RL)
formula for the fractional operator; it is the integral operator when interpreted as ( 5.30 )
and it is the differential operator interpreted as ( 5.31 ) when
<α<
1wehave n
=
α>
0
.
Fractal function evolution
Recall the fractal function introduced by Weierstrass and generalized by Mandelbrot
and others. Let us rewrite the generalized Weierstrass function (GWF) here,
1
a n [
b n t
W
(
t
) =
1
cos
(
) ] ,
(5.32)
n
=−∞
and recall further that the parameters a and b are related by
a μ =
b
(5.33)
and that in terms of the fractal dimension D
=
2
μ
the GWF can be written as
1
b ( 2 D ) n [
b n t
W
(
t
) =
1
cos
(
) ] .
(5.34)
n
=−∞
 
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