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MathGroup Archive 2007

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Re: Simplification

  • To: mathgroup at smc.vnet.net
  • Subject: [mg75439] Re: Simplification
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Tue, 1 May 2007 03:18:21 -0400 (EDT)
  • References: <f146na$m9e$1@smc.vnet.net>

> oo = Product[Cos[(2^j*Pi)/1023], {j, 0, 9}]/Product[Cos[(2^j*Pi)/1025],=
 {j, 0, 9}];

Having received a couple of responses mentioning that

N@oo
-1

I think I ought to restate my queries.

My basic question is how can someone SIMPLIFY oo to -1.
That is, I want a series of steps including functions like
(Full)Simplify,
RootReduce, TrigToExp, TrigReduce etc.

Note that in another CAS (during to the well known policy of this
forum I can't
mention its name) someone working as follows can take the desired
simplification.

 Product(cos(Pi*2^j/1023), j= 0..9)/ Product(cos(Pi*2^j/1025), j=
0=2E.9):
 p:=value(%);
 convert(p, sin);
 simplify(%);

              Pi       2 Pi      4 Pi      8 Pi      16 Pi
  p := - cos(----) cos(----) cos(----) cos(----) cos(-----)
             1023      1023      1023      1023      1023

            32 Pi      64 Pi      128 Pi      256 Pi      511 Pi
        cos(-----) cos(-----) cos(------) cos(------) cos(------)
            1023       1023        1023        1023        1023

           / /     Pi       2 Pi      4 Pi      8 Pi      16 Pi
          /  |cos(----) cos(----) cos(----) cos(----) cos(-----)
         /   \    1025      1025      1025      1025      1025

            32 Pi      64 Pi      128 Pi      256 Pi      512 Pi \
        cos(-----) cos(-----) cos(------) cos(------) cos(------)|
            1025       1025        1025        1025        1025  /


                       512 Pi      1022 Pi       Pi
                   sin(------) sin(-------) sin(----)
                        1023        1023        1025
                 - ----------------------------------
                        Pi       511 Pi      1024 Pi
                   sin(----) sin(------) sin(-------)
                       1023       1023        1025


                                  -1

However, no matter what have I tried I was not able to find a
simplification procedure
in Mathematica.

The expression

o1=-(Sin[(512*Pi)/1023]*Sin[(1022*Pi)/1023]*
              Sin[Pi/1025])/(Sin[Pi/1023]*Sin[(511*Pi)/
              1023]*Sin[(1024*Pi)/1025])

which appeared by application of the function convert of the other CAS

can be simplified to -1 in Mathematica

-(Sin[(512*Pi)/1023]*Sin[(1022*Pi)/1023]*
              Sin[Pi/1025])/(Sin[Pi/1023]*Sin[(511*Pi)/
              1023]*Sin[(1024*Pi)/1025]) // TrigToExp // ExpandAll //
Together
-1

So, the question may be put how someone can write oo as o1.
(although I am sure Andrzej Kozlowski will not be very fond of this
idea!)

My second query is how can someone show that oo==-1 (or oo-1==0) is
equal to true.
Even I consider this much more simpler task, again my (any!) knowledge
of Mathematica
give again pure results.

Thanks in advance for any response.
I really appreciate any help I got and will (I hope!) get

Dimitris

=CF/=C7 dimitris =DD=E3=F1=E1=F8=E5:
> This appeared in another forum.
>
> (Converting to Mathematica InputForm.)
>
> In[2]:=
> oo = Product[Cos[(2^j*Pi)/1023], {j, 0, 9}]/Product[Cos[(2^j*Pi)/
> 1025], {j, 0, 9}];
>
> The expression can be simplified to -1.
>
> Indeed, adopted by someone's reply, in another CAS, we simply have
>
> Product(cos(Pi*2^j/1023), j= 0..9)/ Product(cos(Pi*2^j/1025), j=
> 0..9):
>  p:=value(%):
>  convert(p, sin):
>  simplify(%);
>                                                    -1
>
> However, no matter what I tried I was not able to succeed in
> simplifying above expression
> to -1 with Mathematica, in reasonable time. Futhermore, even the much
> more simpler of
> showing oo==-1 didn't work.
>
> So I would really appreciate if someone pointing me out:
> 1) A way to show (in Mathematica!) that oo is simplified to -1
> 2) That the equality oo==-1 (or oo-1==0 alternatively) can be
> simplified
> to True.
>
> Any ideas?
>
> BTW, I found the function convert of the other CAS, very useful.
> Has anyone implementated a similar function in Mathematica?
> (I ain't aware of a Mathematica built-in function, similar to convert
> from the other CAS.)
>
> Dimitris



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