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MathGroup Archive 2003

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Re: This should evaluate to zero

  • To: mathgroup at smc.vnet.net
  • Subject: [mg39947] Re: [mg39926] This should evaluate to zero
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 13 Mar 2003 03:00:24 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Just out of curiosity I tried to see if I can force Simplify to return 
0 with a suitable choice of ComplextityFunction and 
TransformationFunctions. Here is one way that works:

h[Log[x_] + a_ Log[y_]] := Log[FullSimplify[x*y^a]]


Simplify[ArcSinh[1] - Log[3363 + 2378*Sqrt[
     2]]/10,ComplexityFunction->(
             Count[#,ArcSinh,Infinity,Heads->True]&),
TransformationFunctions->Composition[h,TrigToExp]]


0

Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/


On Wednesday, March 12, 2003, at 05:59  pm, Andrzej Kozlowski wrote:

>
> On Wednesday, March 12, 2003, at 04:28  pm, jf alcover wrote:
>
>> Bonjour,
>> Methinks that the following expression should evaluate to zero,
>> but it does not, even with FullSimplify :
>>     ArcSinh[1]-Log[3363+2378*Sqrt[2]]/10
>> Could anyone explain ?
>>
>>
>>
>
> Presumably, the reason is that the built in transformation rules that 
> FullSimplify tries are simply insufficient to reach that answer. 
> However:
>
>
> Log[Exp[TrigToExp[ArcSinh[1]-Log[3363+2378*Sqrt[2]]/10]]//FullSimplify]
>
> 0
>
> We can get a different perspective on the difficulties by trying a 
> somewhat different approach. We first define a new transformation 
> function:
>
> h[Log[x_] + a_ Log[y_]] := Log[x*y^a]
>
> Of course this only makes sense for suitable values of x,y and a but 
> since the rule holds in our case we won't bother about this point. Now 
> consider the following:
>
>
> v=Simplify[TrigToExp[ArcSinh[1] - Log[3363 + 2378*Sqrt[2]]/10],
>   TransformationFunctions -> {Automatic, h}]
>
>
> Log[(1 + Sqrt[2])/(3363 + 2378*Sqrt[2])^(1/10)]
>
> The expression inside the Log is in fact 1. Mathematica can show this 
> but if we now simply apply FullSimplify we will get back to formula we 
> started with:
>
>
> FullSimplify[v]
>
>
> ArcSinh[1] - (1/10)*Log[3363 + 2378*Sqrt[2]]
>
>
> However, if we apply FullSimplify only to the expression inside the 
> Log we get the right answer:
>
>
> Map[FullSimplify,v]
>
> 0
>
> I think this kind of problems are unavoidable.  FullSimplify can only 
> perform a limited number of transformations on any given expression, 
> and there will always be cases when the needed transformation, or 
> right order of performing transformations or the correct way of 
> grouping an expression into subexpressions isn't tried without human 
> guidance.
>
> Andrzej Kozlowski
> Yokohama, Japan
> http://www.mimuw.edu.pl/~akoz/
> http://platon.c.u-tokyo.ac.jp/andrzej/
>
>



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