The fractional derivative of of order
(if it exists) can be defined in terms of the fractional
integral
as
(1)
|
where
is an integer
,
where
is the ceiling function. The semiderivative
corresponds to
.
The fractional derivative of the function is given by
(2)
| |||
(3)
| |||
(4)
| |||
(5)
| |||
(6)
|
for .
The fractional derivative of the constant function
is then given by
(7)
| |||
(8)
|
The fractional derivate of the Et-function is given by
(9)
|
for .
It is always true that, for ,
(10)
|
but not always true that
(11)
|
Fractional derivatives are implemented in the Wolfram Language as FractionalD.
A fractional integral can also be similarly defined. The study of fractional derivatives and integrals is called fractional calculus.