Re: New Analytical Functions - Mathematica Verified

*To*: mathgroup at smc.vnet.net*Subject*: [mg66853] Re: New Analytical Functions - Mathematica Verified*From*: "Mohamed Al-Dabbagh" <mohamed_al_dabbagh at hotmail.com>*Date*: Thu, 1 Jun 2006 06:55:41 -0400 (EDT)*References*: <200605280104.VAA23436@smc.vnet.net><e5jss1$e73$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

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! > > > > http://dabbagh2.fortunecity.com/disc/ > > > > 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 function? > 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

**Follow-Ups**:**Re: Re: New Analytical Functions - Mathematica Verified***From:*Daniel Lichtblau <danl@wolfram.com>