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Re: Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]
*To*: mathgroup at smc.vnet.net
*Subject*: [mg50644] Re: [mg50617] Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]
*From*: Adam Strzebonski <adams at wolfram.com>
*Date*: Wed, 15 Sep 2004 01:50:07 -0400 (EDT)
*References*: <200409130619.CAA14342@smc.vnet.net> <8C2E6168-0558-11D9-A0AA-000A95B4967A@akikoz.net> <00bc01c49991$4d2d5260$4f604ed5@lap5100> <656B2636-0588-11D9-A0AA-000A95B4967A@akikoz.net> <4145BB27.9070409@wolfram.com> <50B003A8-05EE-11D9-A0AA-000A95B4967A@akikoz.net> <CCE795EA-0601-11D9-B7F2-000A95B4967A@akikoz.net>
*Reply-to*: adams at wolfram.com
*Sender*: owner-wri-mathgroup at wolfram.com
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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