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Fractional Calculus

The study of an extension of derivatives and integrals to noninteger orders. Fractional calculus is based on the definition of the fractional integral as

D^(-nu)f(t)=1/(Gamma(nu))int_0^t(t-xi)^(nu-1)f(xi)dxi,

where Gamma(nu) is the gamma function. From this equation, fractional derivatives can also be defined.

See also

Derivative, Fractional Derivative, Fractional Differential Equation, Fractional Integral, Fractional Integral Equation, Integral, Multiple Integral, Riemann-Liouville Operator

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References

Butzer, P. L. and Westphal, U. "An Introduction to Fractional Calculus." Ch. 1 in Applications of Fractional Calculus in Physics (Ed. R. Hilfer). Singapore: World Scientific, pp. 1-85, 2000.Kilbas, A. A.; Srivastava, H. M.; and Trujiilo, J. J. Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier, 2006.McBride, A. C. Fractional Calculus. New York: Halsted Press, 1986.Nishimoto, K. Fractional Calculus. New Haven, CT: University of New Haven Press, 1989.Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, 1993.Oldham, K. B. and Spanier, J. The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. New York: Academic Press, 1974.

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Fractional Calculus

Cite this as:

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

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