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# 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.

Fractional Calculus, Fractional Integral Equation, Riemann-Liouville Operator, Semi-Integral

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## 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.

## Referenced on Wolfram|Alpha

Fractional Integral

## Cite this as:

Weisstein, Eric W. "Fractional Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FractionalIntegral.html