Getting simple answers from Reduce, ComplexExpand and FullSimplify

*To*: mathgroup at smc.vnet.net*Subject*: [mg57832] Getting simple answers from Reduce, ComplexExpand and FullSimplify*From*: "Hugh Goyder" <h.g.d.goyder at cranfield.ac.uk>*Date*: Thu, 9 Jun 2005 05:55:33 -0400 (EDT)*Organization*: University of Warwick, UK*Sender*: owner-wri-mathgroup at wolfram.com

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

**Follow-Ups**:**Re: Getting simple answers from Reduce, ComplexExpand and FullSimplify***From:*Pratik Desai <pdesai1@umbc.edu>

**Re: Getting simple answers from Reduce, ComplexExpand and FullSimplify***From:*Andrzej Kozlowski <andrzej@akikoz.net>