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Using this new definition of derivatives given by (
5.24
), we can solve the
1
/
2-derivative problem posed to Leibniz. Consider the definition (
5.24
)for
α
=−
β
=
1
/
2 so that
(
−
1
/
2
+
1
)
D
1
/
2
t
−
1
/
2
t
−
1
/
2
−
1
/
2
[
]=
t
(
−
1
/
2
+
1
−
1
/
2
)
=
(
/
)
1
2
t
−
1
=
,
0
(5.25)
(
0
)
where the last equality arises from the gamma function
. Thus, a particular
function is effectively a constant with regard to a certain fractional derivative; that is,
the fractional derivative of a monomial with a matching index vanishes, just like the
ordinary derivative of a constant.
Consider a second example. This time let us examine the fractional derivative of a
constant. Take the index of the monomial
(
0
)
=∞
β
=
0sothatthe1
/
2-derivative of a constant
is given by (
5.24
)tobe
(
0
+
1
)
D
1
/
2
t
t
−
1
/
2
[
1
]=
(
0
+
1
−
1
/
2
)
1
√
π
=
t
.
(5.26)
/
Thus, we see that the 1
2-derivative of a constant does not vanish, but is a function of the
independent variable. This result implies the broader conclusion that what is considered
a constant in the ordinary calculus is not necessarily a constant in the fractional calculus.
Now let us consider the response of Leibniz to the original question and determine
the 1
/
2-derivative of the monomial
t
,
β
=
1:
(
1
+
1
)
D
1
/
2
t
t
1
−
1
/
2
[
t
]=
(
1
+
1
−
1
/
2
)
t
π
,
=
(5.27)
the result obtained by Leibniz. Mandelbrot quotes Leibniz' letter
1
from which we
extract the piece:
...
John Bernoulli seems to have told you of my having mentioned to him a marvelous analogy
which makes it possible to say in a way the successive differentials are in geometric progression.
One can ask what would be a differential having as its exponent a fraction. You see that the result
can be expressed by an infinite series. Although this seems removed from Geometry, which does
not yet know of such fractional exponents, it appears that one day these paradoxes will yield
useful consequences, since there is hardly a paradox without utility.
After 310 years of sporadic development, the fractional calculus is now becoming
sufficiently well developed and well known that topics and articles are being devoted
to its “utility” in the physical sciences [
9
,
26
,
31
]. Of most interest to us is what the
1
This letter was translated by B. Mandelbrot and is contained in the “Historical sketches” of his second topic
[
16
].
