Fractional Derivative

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)

A fractional integral can also be similarly defined. The study of fractional derivatives and integrals is called fractional calculus.

Wolfram Web Resources