Re: ForAll testing equality, and Limit evaluating wrong

*To*: mathgroup at smc.vnet.net*Subject*: [mg104557] Re: ForAll testing equality, and Limit evaluating wrong*From*: ADL <alberto.dilullo at tiscali.it>*Date*: Wed, 4 Nov 2009 01:32:07 -0500 (EST)*References*: <hcop0k$2d7$1@smc.vnet.net>

You forgot and underscore in xTransf definition: ClearAll[xTransf, xTransf2]; xTransf2[f_] := 36*Sinc[6*Pi*f]^2; xTransf[f_] := Limit[(-Cos[12*r*Pi] + Cos[24*r*Pi] + 12*r*Pi*Sin[24*r*Pi])/(E^(24*I*r*Pi)*(2*r^2*Pi^2)), r -> f]; FullSimplify[xTransf[f/24] == xTransf2[f/24], Element[f, Integers]] ==> True (7.0 for Microsoft Windows (32-bit) (February 18, 2009)) ADL On Nov 3, 9:18 am, Rui <rui.r... at gmail.com> wrote: > I want to prove that xTransf2[f]==xTransf[f] for all f multiple of > 1/24. > xTransf2[f_]:=36 Sinc[6 Pi f]^2 and > xTransf[f]:=Limit[(E^(-24 I r \[Pi]) (-Cos[12 r \[Pi]] + Cos[24 r \ > [Pi]] + > 12 r \[Pi] Sin[24 r \[Pi]]))/(2 r^2 \[Pi]^2), r->f] > > If I do > ForAll[f \[Element] Integers, YTransf[f/24] == YTransf2[f/24]] > I don't get a result... I can't find a way. > In fact, I get > (144 E^(-I f \[Pi]) (-2 Cos[(f \[Pi])/2] + 2 Cos[f \[Pi]] + > f \[Pi] Sin[f \[Pi]]))/(f^2 \[Pi]^2) == 36 Sinc[(f \[Pi])/4]^= 2 > > They are clrealy equal, at least on the 48 points closest to 0, > because if I do > And @@ ((xTransf[1/24 #] == xTransf2[1/24 #]) & /@ Range[-24, 24]) > I get "True" > > Any pretty way to be certain? > > I've also realised that Mathematica has evaluated Limits with > variables, making the "Limit" disappear when for some values of the > variables I could get an indetermined result with the evaluated > version. For example, the Limit in xTransf > xTransf[f] > I get > (E^(-24 I f \[Pi]) (-Cos[12 f \[Pi]] + Cos[24 f \[Pi]] + > 12 f \[Pi] Sin[24 f \[Pi]]))/(2 f^2 \[Pi]^2) > without the Limit. > So, if I do > xTransf[f]/.f->0 > I get errors but if I do xTransf[0] I get 36 > ... > Hope you can help :)