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


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/


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