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

**References**:**Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]***From:*psa@laplacian.co.uk (peteraptaker)