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

*To*: mathgroup at smc.vnet.net*Subject*: [mg50636] 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 01:49:37 -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>*Reply-to*: adams at wolfram.com*Sender*: owner-wri-mathgroup at wolfram.com

This is because what looks simpler (and is simpler from the mathematical point of view) is not necessarily simpler when measured by Simplify's built-in complexity function. Simplify's measure of complextity is based on the number of leaves in the FullForm of the expression (and also on the size of integers involved). In[1]:= Im[Sqrt[-1 + eta^2]]//LeafCount Out[1]= 10 In[2]:= Sqrt[1 - eta^2]//LeafCount Out[2]= 11 The second form is computed by Simplify, but is rejected because it has a higher complexity than the first form. When the input is a pair, the complexity gain from transformation Im[Sqrt[-1 + eta^2]] -> Sqrt[1 - eta^2] is offset by complexity loss from transformation Re[Sqrt[-1 + eta^2]] -> 0. Both transformations are made in the same simplification step, namely the "evaluation with assumptions" (a.k.a. Refine). In[3]:= Refine[{Re[Sqrt[-1+eta^2]],Im[Sqrt[-1+eta^2]]},-1<eta<1] 2 Out[3]= {0, Sqrt[1 - eta ]} In[4]:= Refine[Re[Sqrt[-1+eta^2]],-1<eta<1] Out[4]= 0 In[5]:= Refine[Im[Sqrt[-1+eta^2]],-1<eta<1] 2 Out[5]= Sqrt[1 - eta ] Refine is designed to find a "standard" form of the input expression, rather than the "simplest" one. It uses a much smaller set of transformation rules than Simplify, and its transformations are so chosen that they do not reverse each other. Hence Refine does not need to use any complexity criteria, it simply uses all its transformation rules that apply under the given assumptions. I would suggest using Refine in addition to Simplify. Another possibility is to change the ComplexityFunction used by Simplify. In[8]:= cf[e_] := LeafCount[e]+10 Count[e, _Im, {0, Infinity}] In[9]:= Simplify[Im[Sqrt[-1 + eta^2]], -1<eta<1, ComplexityFunction->cf] 2 Out[9]= Sqrt[1 - eta ] My experience with changing the ComplexityFunction suggests that for each complexity function there are examples where an "obvious" simplification does not have a lower complexity as measured by the given complexity function. However, for a specific application, using a custom complexity function may be a good idea. For completeness let me give a Mathematica implementation of the "Automatic" complexity function used by Simplify. SimplifyCount[p_] := If[Head[p]===Symbol, 1, If[IntegerQ[p], If[p==0, 1, Floor[N[Log[2, Abs[p]]/Log[2, 10]]]+If[p>0, 1, 2]], If[Head[p]===Rational, SimplifyCount[Numerator[p]]+SimplifyCount[Denominator[p]]+1, If[Head[p]===Complex, SimplifyCount[Re[p]]+SimplifyCount[Im[p]]+1, If[NumberQ[p], 2, SimplifyCount[Head[p]]+If[Length[p]==0, 0, Plus@@(SimplifyCount/@(List@@p))]]]]]] In[11]:= SimplifyCount/@{Im[Sqrt[-1 + eta^2]], Sqrt[1 - eta^2]} Out[11]= {11, 12} Best Regards, Adam Strzebonski Wolfram Research Andrzej Kozlowski wrote: > *This message was transferred with a trial version of CommuniGate(tm) Pro* > This is indeed most peculiar and looks like a bug. However as a > workaround I suggest adding ComplexExpand as follows: > > > FullSimplify[ComplexExpand[Im[Sqrt[-1 + eta^2]]], > -1 < eta < 1] > > > Sqrt[1 - eta^2] > > This also works in version 4.2. > > Andrzej > > On 13 Sep 2004, at 21:56, Peter S Aptaker wrote: > >> Sadly it does not work in M4.2 which I tend to use "for varous reasons" >> >> >> Back to M5 for now: >> >> Simplify[{Re[Sqrt[-1+eta^2]],Im[Sqrt[-1+eta^2]]},-1<eta<1] is fine >> >> Unfortunately: >> >> >> Simplify[Im[Sqrt[-1 + eta^2]],-1<eta<1] >> >> and >> >> Simplify[{Im[Sqrt[-1+eta^2]],Im[Sqrt[-1+eta^2]]},-1<eta<1] >> >> both leave the Im[] >> >> Thanks >> Peter >> ----- Original Message ----- >> From: "Andrzej Kozlowski" <andrzej at akikoz.net> To: mathgroup at smc.vnet.net >> To: "peteraptaker" <psa at laplacian.co.uk> >> Cc: <mathgroup at smc.vnet.net> >> Sent: Monday, September 13, 2004 8:43 AM >> Subject: [mg50636] Re: [mg50617] Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + >> eta^2]]}, eta<1] >> >> > *This message was transferred with a trial version of >> CommuniGate(tm) Pro* >> > On 13 Sep 2004, at 15:19, peteraptaker wrote: >> > >> > > *This message was transferred with a trial version of CommuniGate(tm) >> > > Pro* >> > > Have I missed something - my apologies if this is answered in a FAQ >> > > I want to make the simple Re and Im parts simplify properly? >> > > >> > > test = >> > > {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]} >> > > >> > > FullSimplify[test, eta > 1] >> > > gives*{Sqrt[-1 + eta^2], 0} >> > > >> > > But >> > > FullSimplify[test, eta < 1] >> > > gives >> > > {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]} >> > > >> > > Needs["Algebra`ReIm`"] does not seem to help >> > > >> > > Real numbers demonstrate what should happen: >> > > test) /. {{eta -> 0.1}, {eta -> 2}} >> > > {{0., 0.99498743710662}, {Sqrt[3], 0}} >> > > >> > > >> > >> > There is nothing really strange here, Mathematica simply can't give a >> > single simple expression that would cover all the cases that arise. So >> > you have to split it yourself, for example: >> > >> > >> > FullSimplify[test, eta < -1] >> > >> > >> > {Sqrt[eta^2 - 1], 0} >> > >> > FullSimplify[test, eta == -1] >> > >> > {0, 0} >> > >> > >> > FullSimplify[test, -1 < eta < 1] >> > >> > {0, Sqrt[1 - eta^2]} >> > >> > >> > FullSimplify[test, eta == 1] >> > >> > >> > {0, 0} >> > >> > >> > FullSimplify[test, 1 <= eta] >> > >> > >> > {Sqrt[eta^2 - 1], 0} >> > >> > >> > or, you can combine everything into just two cases: >> > >> > FullSimplify[test, eta $B":(B Reals && Abs[eta] < 1] >> > >> > {Re[Sqrt[eta^2 - 1]], Im[Sqrt[eta^2 - 1]]} >> > >> > >> > FullSimplify[test, eta $B":(B Reals && Abs[eta] >= 1] >> > >> > {Sqrt[eta^2 - 1], 0} >> > >> > In fact you do not really need FullSimplify, simple Simplify will do >> > just as well. >> > >> > >> > Andrzej Kozlowski >> > Chiba, Japan >> > http://www.akikoz.net/~andrzej/ >> > http://www.mimuw.edu.pl/~akoz/ >> > >> > > >

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