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Re: New Analytical Functions - Mathematica Verified

Daniel Lichtblau wrote:
> Mohamed Al-Dabbagh wrote:
> > Abstract: An attempt was made to analytically describe some basic
> > functions which are provided with the libraries of mathematical
> > functions of programming languages. This study will rely on Mathematica
> > in calculating all numerical results. This study provides global
> > methods to calculate some discrete functions, which were previously
> > calculated by using low-level language techniques to chop digits. Using
> > these definitions we can simulate digital signals without the need for
> > IF statements and conditions.
> >
> > A comparison were made between the formulas provided by this study and
> > some functions provided by Mathematica such as FractionalPart function
> > that converges very slowly or does NOT converge at all to correct
> > results. The formulas derived herein were applied to complex numbers
> > and some interesting results were obtained, such as the integer part of
> > a complex number equals the Floor of its real part, and will be without
> > imaginary part, i.e. the integer part of a complex number is a real
> > number!
> >
> >
> >
> > Mohamed Al-Dabbagh
> I'll point out some things with regard to your reformulation of a
> sawtooth in terms of elementary functions. It is not hard to define the
> derivative of FractionalPart as 1, and this is a.e. correct, that is,
> correct except at jumps, where the derivative, strictly speaking, is not
> defined. So a question to ask is what do you want the derivative to do
> at such points? Your gPaper, shown below, will give an Indeterminate at
> x = 12 (which one might regard as a good thing). So if that is the
> behavior you want, well and good.
> f[x_] := x^5
> Fr[x_] := 1/2 - ArcTan[Cot[Pi*x]]/Pi
> gPaper[x_] := Fr[f[x]]
> If we simply do
> Derivative[1][FractionalPart][x_] := 1 + DiracDelta[x-IntegerPart[x]]
> Derivative[n_Integer /; n>1][FractionalPart][x_] := 0

Will the above two definitions change the behaviour of FractionalPart

> gMath[x_] := FractionalPart[f[x]]
> we'll get an exact result for gMath''[61/5] that agrees with yours
> numerically but is in vastly simpler form. We'll also get a rational
> result for gMath''[12] that might or might not be what you'd want.

Is that really simpler??!! Moreover, you rather used an exact rational
form and you didn't use the approximate 12.2`400 (12.2 correct to 400
digits) to see what my experiment will be? Again, you should use my
definition for FractionalPart function to get precise results.

Now why you used [x-IntegerPart[x]]? Why didn't you simply use
FractionalPart? I will do experiments again, and will post results.. I
hope moderator wont take long to publish them.

Mohamed Al-Dabbagh

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