Fractional Integral

Denote the nth derivative D^n and the n-fold integral D^(-n). Then


Now, if the equation


for the multiple integral is true for n, then


Interchanging the order of integration gives


But (3) is true for n=1, so it is also true for all n by induction. The fractional integral of f(t) of order nu>0 can then be defined by


where Gamma(nu) is the gamma function.

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


for nu>0 with _aD_t^0f(t)=f(t) (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

D^(-nu)t^lambda=(Gamma(lambda+1))/(Gamma(lambda+nu+1))t^(lambda+nu)  for lambda>-1,nu>0

where gamma(a,x) is a lower incomplete gamma function and E_t(nu,a) is the Et-function. From (10), the fractional integral of the constant function f(t)=c is given by


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

See also

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

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

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Fractional Integral

Cite this as:

Weisstein, Eric W. "Fractional Integral." From MathWorld--A Wolfram Web Resource.

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