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Re: Pi Formula

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92923] Re: Pi Formula
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Mon, 20 Oct 2008 07:32:09 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <gcn4ge$7ad$1@smc.vnet.net> <200810120833.EAA08815@smc.vnet.net> <200810181024.GAA15999@smc.vnet.net> <gdevc2$3mg$1@smc.vnet.net>

Artur wrote:

> Who know which another function as Simplify or FullSimplify to use to 
> following formula
> (Simplify do nothing but FullSimplify simplify too much).
> 1/8 (-2 I Sqrt[-7 - I] Log[1/5 ((1 - 2 I) + 2 Sqrt[-7 - I])] -
>    Log[(3 - 4 I)^Sqrt[7 + I]
>       5^((3 - I) (51 - 10 Sqrt[2])^(
>       1/4)) ((1 - 2 I) - 2 Sqrt[-7 - I])^((-1 - I) (51 - 10 Sqrt[2])^(
>       1/4)) (-5 I + (4 - 2 I) Sqrt[-7 - I])^(-2 Sqrt[
>       7 + I]) ((1 + 2 I) - 2 Sqrt[-7 + I])^(2 Sqrt[-7 + I])] -
>    I Sqrt[7 - I] Log[(-1 + Sqrt[-1 - I])^2] -
>    I Sqrt[7 - I] Log[(1 + Sqrt[-1 - I])^2] -
>    Sqrt[7 + I] Log[(-1 + Sqrt[-1 + I])^2] -
>    Sqrt[7 + I] Log[(1 + Sqrt[-1 + I])^2] + (1 - 2 I) Sqrt[1 - I]
>      Log[(1 + I) - Sqrt[1 - I]] - (2 - I) Sqrt[1 + I]
>      Log[(1 + I) - Sqrt[1 - I]] - (1 - 2 I) Sqrt[1 - I]
>      Log[(1 + I) + Sqrt[1 - I]] + (2 - I) Sqrt[1 + I]
>      Log[(1 + I) + Sqrt[1 - I]] +
>    I Sqrt[7 - I] Log[(-60 - 4 I) + 8 Sqrt[-1 - I] - 24 Sqrt[7 - I]] +
>    I Sqrt[7 - I] Log[(66 - 14 I) + 8 Sqrt[-1 - I] + 24 Sqrt[7 - I]] +
>    Sqrt[7 + I] Log[(-60 + 4 I) + 8 Sqrt[-1 + I] - 24 Sqrt[7 + I]] +
>    Sqrt[7 + I] Log[(66 + 14 I) + 8 Sqrt[-1 + I] + 24 Sqrt[7 + I]] +
>    I Log[((1 + 2 I) - 2 Sqrt[-7 + I])^(
>       2 Sqrt[7 - I]) ((-60 + 4 I) - 16 Sqrt[14 - 2 I])^-Sqrt[-7 -
>         I] ((-60 - 4 I) - 16 Sqrt[14 + 2 I])^-Sqrt[
>        7 - I] ((-(153/100) + (71 I)/100) + 2/25 Sqrt[287 - 359 I])^
>       Sqrt[-7 -
>        I] ((-(153/2500) - (71 I)/2500) + 2/625 Sqrt[287 + 359 I])^
>       Sqrt[7 - I]])

Arthur,

Calling the above expression "expr", we have

     In[5]:= FullSimplify[expr]

     Out[5]= Pi

If I have understood your correctly, you wish to have something more 
complicated than Pi, yet simpler than the original expr, which begs the 
  question: What do you expect? Perhaps *PowerExpand* is what you are 
looking for (though the leaf count is not that much different)?

     In[7]:= pw = PowerExpand[expr];

     In[8]:= LeafCount /@ {expr, pw}

     Out[8]= {590, 582}

Or you may want to tweak/build your own *ComplexityFunction*. See

http://reference.wolfram.com/mathematica/ref/ComplexityFunction.html


Regards,
-- Jean-Marc


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