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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 :)



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