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.

REFERENCES:

Kilbas, A. A.; Srivastava, H. M.; and Trujiilo, J. J. Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier, 2006.

Love, E. R. "Fractional Derivatives of Imaginary Order." J. London Math. Soc. 3, 241-259, 1971.

Miller, K. S. "Derivatives of Noninteger Order." Math. Mag. 68, 183-192, 1995.

Oldham, K. B. and Spanier, J. The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. New York: Academic Press, 1974.

Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, 1993.