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Fractional Calculus

Denote the $n$th Derivative $D^n$ and the $n$-fold Integral $D^{-n}$. Then

D^{-1} f(t)=\int_0^t f(\xi)\,d\xi.
\end{displaymath} (1)

Now, if
D^{-n} f(t)={1\over(n-1)!} \int_0^t (t-\xi)^{n-1}f(\xi)\,d\xi
\end{displaymath} (2)

is true for $n$, then
$\displaystyle D^{-(n+1)}f(t)$ $\textstyle =$ $\displaystyle D^{-1}\left[{{1\over(n-1)!} \int_0^t (t-\xi)^{n-1}f(\xi)\,d\xi}\right]$  
  $\textstyle =$ $\displaystyle \int_0^t \left[{{1\over(n-1)!}\int_0^x (x-\xi)^{n-1}f(\xi)\,d\xi}\right]\,dx.$  

Interchanging the order of integration gives
D^{-(n+1)}f(t)={1\over n!}\int_0^t (t-\xi)^nf(\xi)\,d\xi.
\end{displaymath} (4)

But (2) is true for $n=1$, so it is also true for all $n$ by Induction. The fractional integral of $f(t)$ can then be defined by
D^{-\nu}f(t)={1\over\Gamma(\nu)}\int_0^t (t-\xi)^{\nu-1}f(\xi)\,d\xi,
\end{displaymath} (5)

where $\Gamma(\nu)$ is the Gamma Function.

The fractional integral can only be given in terms of elementary functions for a small number of functions. For example,

$\displaystyle D^{-\nu} t^{-\lambda}$ $\textstyle =$ $\displaystyle {\Gamma(\lambda+1)\over\Gamma(\lambda+\nu+1)} t^{\lambda+\nu}
\quad{\rm for\ } \lambda>-1, \nu>0$ (6)
$\displaystyle D^{-\nu} e^{at}$ $\textstyle =$ $\displaystyle {1\over\Gamma(\nu)} e^{at} \int_0^t x^{\nu-1}e^{-ax}\,dx\equiv E_t(\nu,a),$ (7)

where $E_t(\nu,a)$ is the Et-Function. The fractional derivative of $f$ (if it exists) can be defined by
D^\mu f(t)=D^m[D^{-(m-\mu)}f(t)].
\end{displaymath} (8)

An example is
$\displaystyle D^\mu t^\lambda$ $\textstyle =$ $\displaystyle {\Gamma(\lambda+1)\over\Gamma(\lambda+m-\mu+1)}$  
  $\textstyle =$ $\displaystyle {\Gamma(\lambda+1)\over\Gamma(\lambda-\mu+1)} t^{\lambda-\mu}\quad {\rm for\ } \lambda>-1, \mu>0$  
$\displaystyle D^\rho E_t(\nu,a)$ $\textstyle =$ $\displaystyle E_t(\nu-\rho,a) \qquad{\rm for\ } \nu>0, \rho\not=0.$ (10)

It is always true that, for $\mu,\nu>0$,
D^{-\mu} D^{-\nu} f(t) = D^{-(\mu+\nu)},
\end{displaymath} (11)

but not always true that
D^\mu D^\nu=D^{\mu+\nu}.
\end{displaymath} (12)

See also Derivative, Integral


Love, E. R. ``Fractional Derivatives of Imaginary Order.'' J. London Math. Soc. 3, 241-259, 1971.

McBride, A. C. Fractional Calculus. New York: Halsted Press, 1986.

Miller, K. S. ``Derivatives of Noninteger Order.'' Math. Mag. 68, 183-192, 1995.

Nishimoto, K. Fractional Calculus. New Haven, CT: University of New Haven Press, 1989.

Spanier, J. and Oldham, K. B. The Fractional Calculus. New York: Academic Press, 1974.

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© 1996-9 Eric W. Weisstein