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Re: New Analytical Functions - Mathematica Verified


Daniel Lichtblau wrote:

> They will change the behavior of its derivatives. Actually I think I did
> not want to use the DiracDelta component in that particular definition
> (or else to use further derivatives thereof in the definition for higher
> derivatives of FractionalPart).
> ...........................
> ...........................
> Did you try it?
>

Dan,

Your usage of DiracDelta has made some remarkable correction for the
result, but on a very high cost of runtime! I have made some
experiments and wanted to publish them. You should remember that my
paper:

http://dabbagh2.fortunecity.com/disc

has proved that the derivative of the fractional part of any function
is the SAME as derivative of that function WITHOUT involving in the
calculation of fractional part!

When I used your improvement using Dirac Delta, a very substantial
improvement occurred on the numerical results. HOWEVER, this lead to
some very long delays. To the extent that calculating 5th derivative of
the FractionalPart(x^5) to 1000 places of decimal would take about ONE
HOUR!!!! Here are some results I arranged it in a page prepared for
you:

http://dabbagh2.fortunecity.com/lichtblau/

You will see how my formulas are A LOT faster. 


Mohamed Al-Dabbagh


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