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The Fractional Integral

April 2nd, 2009 |

Author: Merrin

Our results with repeated integration lead to the definition of the fractional integral. Taking this result

$\displaystyle D^{-n}f(x) =\cfrac{1}{(n-1)!} \int _{0}^{x}f(t)(x-t)^{n-1} dt$

And generalizing to fractional order is logical

$\displaystyle D^{-\nu } f(x)=\cfrac{1}{\Gamma (\nu )} \int _{0}^{x}\cfrac{f(t)}{(x-t)^{1-\nu } } dt$

The integral is improper such that nu greater than zero is required by p-integral convergence.

Also we must shore up our work with the integration limits. In general, the lower limit can be any number resulting in different flavors of fractional calculus. A notation which indicates the limits of fractional integration as well as the order is the following.

$\displaystyle _{a} D_{x}^{-\nu } f(x)=\cfrac{1}{\Gamma (\nu )} \int _{a}^{x}\cfrac{f(t)}{(x-t)^{1-\nu } } dt$

The fractional integral is called the Riemann-Liouville integral after developers of the theory. Taking these integrals is not going to be easy, but many have succeeded in numerous situations. Fractional integrals are only half the basics for fractional calculus. Now we turn our attention to fractional derivatives.

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