Fractional Integral

Denote the th derivative  and the -fold integral . Then

(1)

Now, if the equation

(2)

for the multiple integral is true for , then

			(3)
			(4)

Interchanging the order of integration gives

(5)

But (3) is true for , so it is also true for all  by induction. The fractional integral of  of order  can then be defined by

(6)

where  is the gamma function.

More generally, the Riemann-Liouville operator of fractional integration is defined as

(7)

for  with  (Oldham and Spanier 1974, Miller and Ross 1993, Srivastava and Saxena 2001, Saxena 2002).

The fractional integral of order 1/2 is called a semi-integral.

Few functions have a fractional integral expressible in terms of elementary functions. Exceptions include

			(8)
			(9)
			(10)
			(11)

where  is a lower incomplete gamma function and  is the Et-function. From (10), the fractional integral of the constant function  is given by

			(12)
			(13)

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

REFERENCES:

Miller, K. S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley, 1993.

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.

Saxena, R. K.; Mathai, A. M.; and Haubold, H. J. "On Fractional Kinetic Equations." 23 Jun 2002. http://arxiv.org/abs/math.CA/0206240.

Srivastava, H. M. and Saxena, R. K. "Operators of Fractional Integration and Their Applications." Appl. Math. and Comput. 118, 1-52, 2001.