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From: Ilario980 on 2 Jul 2010 03:02
Bonjour Alain,
thanks for your reply.
Before professor Renfro post i didn't know that already exists the fractional integral operator; i have found the T operator:

- it gives correct result for fractional integration and for fractional derivation (the fractional derivative operator D is defined in terms of fractional integral)

- my sketch of proof is over real numbers but I think that more interesting results are over complex numbers, where T operator gives the expected result for of f(x)=x^k (fractional derivative does not in my simulations )


my proof is not complete (and may be, not so legible) but I believe that T operator should be condidered in new researchs.

Ilario M.
From: Ilario980 on 2 Jul 2010 03:23
In my last post I said that over complex numbers the fractional derivative doesn't give correct result, but I made a mistake in my simulations. I apologize.

Ilario M.
From: Dave L. Renfro on 4 Jul 2010 09:51
Ilario M. wrote:

> Thanks for the post professor Renfro.

For the record, I'm not a professor (I no longer teach),
but I appreciate the thought.

I found by accident two papers that will probably be
of interest to you:

Rufus Isaacs, "Iterates of fractional order", Canadian
Journal of Mathematics 2 #4 (1950), 409-416.
http://books.google.com/books?id=-fQU06O9d0wC&pg=PA409

G. W. Ewing and W. R. Utz, "Continuous solutions of the
functional equation f^n(x) = f(x)", Canadian Journal of
Mathematics 5 #1 (1953), 101-103.
http://books.google.com/books?id=9_w73mr2fyEC&pg=PA101

Both papers are freely available on the internet, at least
where I'm at (in the U.S.), at the URL's given above.

If you can't see these papers where you're at, let me know
and, when I have some free time, I'll type some of the
introductory remarks to indicate what the papers are about.

Dave L. Renfro
From: Ilario980 on 4 Jul 2010 07:47
Thank you very much Dave, I can see both papers.

Ilario M.
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