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ForAll testing equality, and Limit evaluating wrong

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104512] ForAll testing equality, and Limit evaluating wrong
  • From: Rui <rui.rojo at gmail.com>
  • Date: Tue, 3 Nov 2009 02:50:55 -0500 (EST)
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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