From: hrubin@odds.stat.purdue.edu (Herman Rubin)
Subject: Re: Fractional-order differentiation
Date: 12 Feb 2001 21:18:08 -0500
Newsgroups: sci.math

In article <4gHHRMAte5h6EwND@borve.demon.co.uk>,
David Jones  <djones@REMOVE_THIS.borve.demon.co.uk> wrote:
>Hi. I was wondering whether any useful system has been worked out for
>fractional-order differentiation, and what it has been or might be used
>for. I mean a system where for example one can have not only first-order
>and second-order differentials, but also one-and-a-halfth order
>differentials etc. What with the use of fractional dimensionality in
>chaos, ISTM that fractional-order differentials might also have a use
>therein. I would be grateful for any info or advice on this.

There is considerable literature on this.  The place one
starts is with fractional integrals.  The identity

\int_a^y\int_a^x_n ... \int_a^x_2 f(x_1) dx_1 ... dx_n

	= \int_a^y (y-x)^{n-1} dx / \Gamma(n),

proved by induction, gives and extension to integrals
of arbitrary order for sufficiently well-behaved functions.
The derivative of the integral of order k is the integral
of order k-1 if k > 1.  One proceeds from there.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558