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Re: ForAll testing equality, and Limit evaluating wrong

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104573] Re: ForAll testing equality, and Limit evaluating wrong
  • From: dh <dh at metrohm.com>
  • Date: Wed, 4 Nov 2009 01:35:09 -0500 (EST)
  • References: <hcop0k$2d7$1@smc.vnet.net>


Hi Rui,

we show that the difference between xTransf2 and xTransf is zero for 

multiple of 1/24:



FullSimplify[xTransf2[n/24] - xTransf[n/24], Element[n, Integers]]



Note the second argument, specifying n to be integer.



Daniel



Rui 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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