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Re: Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]


Andrzej Kozlowski wrote:
> *This message was transferred with a trial version of CommuniGate(tm) Pro*
> On 14 Sep 2004, at 10:35, Andrzej Kozlowski wrote:
> 
>>  But there is still something that puzzles me, why
>>
>>
>> Simplify[Sqrt[1 - eta^2]]
>>
>> Sqrt[1 - eta^2]
>>
>> rather than Im[Sqrt[-1 + eta^2]]?
> 
> 
> 
> I think I can answer my own question:
> 
> f[Sqrt[x_]] := Im[Sqrt[-x]]
> 
> 
> Simplify[Sqrt[1 - eta^2], TransformationFunctions ->
>    {f, Automatic}]
> 
> 
> Im[Sqrt[eta^2 - 1]]
> 
> There are certain operations that are included among the transformation 
> rules that Simplify automatically uses but their "inverses" are not 
> included. That is one reason why Simplify will sometimes fail to find 
> the "simplest" expression even based on the current ComplexityFunction. 
> I think this also accounts why it is often useful to apply ComplexExpand 
> first in cases like this one.
> 
> 
> Simplify[ComplexExpand[Im[Sqrt[eta^2 - 1]]], -1 < eta < 1]
> 
> 
> Sqrt[1 - eta^2]
> 
> 
> Simplify[Im[Sqrt[eta^2 - 1]], -1 < eta < 1]
> 
> 
> Im[Sqrt[eta^2 - 1]]
> 
> Andrzej Kozlowski
> Chiba, Japan
> http://www.akikoz.net/~andrzej/
> http://www.mimuw.edu.pl/~akoz/
> 

Yes, this is exactly the reason. While Simplify selects its
answer based solely on the value of complexity function,
the transformations used by Simplify have been selected with
a "natural" notion of simplification in mind. This is why
the transformation

Im[Sqrt[a]] -> Sqrt[-a], for a<0

done by Refine as a composition of three transformations

Sqrt[a] -> I Sqrt[-a], for a<0
Im[I b] -> Re[b]
Re[c] -> c, for real c

is among transformations attempted by Simplify,
but the transformation

Sqrt[a] -> Im[Sqrt[-a]], for a>0

is not.


Best Regards,

Adam Strzebonski
Wolfram Research



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