Re: Getting simple answers from Reduce, ComplexExpand and FullSimplify

*To*: mathgroup at smc.vnet.net*Subject*: [mg57855] Re: Getting simple answers from Reduce, ComplexExpand and FullSimplify*From*: Pratik Desai <pdesai1 at umbc.edu>*Date*: Fri, 10 Jun 2005 02:29:18 -0400 (EDT)*References*: <200506090955.FAA29632@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Hugh Goyder wrote: >Below the expression ex has roots, rts, where n in an integer and the >constant a is in the interval 0 < a <1. We may check this using FullSimplify >as follows. > > > >ex=2 Cos[z]+I a Sin[z]; > >rts=z ->Pi(2n-1)/2 + I Log[(2+a)/(2-a)]/2; > >FullSimplify[0 == ex/.rts,Element[n,Integers]] > > > >My problem is to try and deduce the roots using computer algebra rather than >by hand. (I also have other expressions I would like to work on.) My attempt >at using Reduce is partially successful but leads to unfamiliar, and >difficult to interpret, ArcTanh functions > >Reduce[{ex == 0,0<a<1},z] > >An attempt with FullSimplify and ComplexExpand to crack the ArcTanh function >is again partially successful in giving the imaginary part I require. >However, I am stuck with Arg functions with complex arguments. The Arg >Functions are just equal to -Pi/2 but I cannot crack them further. Any >suggestions for simplifying the output from Reduce so that I get the simple >form I guessed at the start? Alternativly, are there more methods for >simplifing the complex output? > > > >FullSimplify[ComplexExpand[(2*I)*ArcTanh[a/2+(I/2)*Sqrt[4-a^2]]],{0<a<1}] > >FullSimplify[ComplexExpand[Arg[2-a-I Sqrt[4-a^2]]-Arg[ > >2+a+I Sqrt[4-a^2]]],{0<a<1}] > > > >Thanks > >Hugh Goyder > > > > Try using TrigToExp, some times CAS like humans needs a little help with hyperbolic trig expressions Clear[ex, z, a, sol1] TagSet[a, Conjugate[a], a] TagSet[a, Re[a], a] TagSet[a, Im[a], 0] ex[z_] := 2 Cos[z] + I* a* Sin[z] // TrigToExp Reduce[ex[z] == 0, z] C[1] â?? Integers && 2 + a != 0 && -2 + a != 0 && z == (-I/2)*((2*I)*Pi*C[1] + Log[(-2 + a)/(2 + a)]) You can use simplify or full simplify to get what you want at the end Best regards Pratik -- Pratik Desai Graduate Student UMBC Department of Mechanical Engineering Phone: 410 455 8134

**References**:**Getting simple answers from Reduce, ComplexExpand and FullSimplify***From:*"Hugh Goyder" <h.g.d.goyder@cranfield.ac.uk>