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 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