Re: Re: Re: Re: How to simplify to a result that is real

*To*: mathgroup at smc.vnet.net*Subject*: [mg50802] Re: Re: [mg50788] Re: [mg50735] Re: How to simplify to a result that is real*From*: Bob Hanlon <hanlonr at cox.net>*Date*: Wed, 22 Sep 2004 00:11:01 -0400 (EDT)*Reply-to*: hanlonr at cox.net*Sender*: owner-wri-mathgroup at wolfram.com

I overfocused on getting rid of the second form of ArcTan. In the last step use FullSimplify rather than just Simplify. Bob Hanlon > > From: Bob Hanlon <hanlonr at cox.net> To: mathgroup at smc.vnet.net > Date: 2004/09/21 Tue AM 07:23:53 EDT > To: Richard Chen <richard at doubleprime.com>, <mathgroup at smc.vnet.net> > Subject: [mg50802] Re: [mg50788] Re: [mg50735] Re: How to simplify to a result that is real > > ArcTan[x, y] and ArcTan[y/x] are in general not the same. > > Plot3D[ArcTan[y/x],{x,-2,2},{y,-2,2}]; > > Plot3D[ArcTan[x,y],{x,-2,2},{y,-2,2}]; > > ch=Integrate[1/(a+b Cos[t]),{t,0,c},Assumptions->{a>b>0,0<c<Pi}]; > > ch = Simplify[ComplexExpand[ch, > TargetFunctions->{Re,Im}],{a>b>0,0<c<Pi}]; > > ch = TrigToExp[ch]; > > ch = Simplify[ComplexExpand[ch, > TargetFunctions->{Re,Im}],{a>b>0,0<c<Pi}] > > (ArcTan[Sqrt[(a - b)/(a + b)]*Tan[c/2]] + > ArcTan[((a - b)*Tan[c/2])/ > Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] > > > Bob Hanlon > > > > > From: Richard Chen <richard at doubleprime.com> To: mathgroup at smc.vnet.net > > Date: 2004/09/21 Tue AM 03:49:05 EDT > > To: mathgroup at smc.vnet.net > > Subject: [mg50802] [mg50788] Re: [mg50735] Re: How to simplify to a result that is > real > > > > I have seen quite a few responses here. I'll just Thank all of you > > with one response. > > > > I still feel that at this stage, simplifying expressions > > in mathematica is still a kind of art and requires intimate knowledge > > of how esoteric mathematica options work. For example, > > I just tried this technique on a problem which is essentially > > the same: > > > > ch = Integrate[1/(a + b Cos[t]), {t, 0, c}, > > Assumptions -> {a > b > 0, 0 < c < Pi}] > > > > FullSimplify[ComplexExpand[ch, TargetFunctions -> {Re, Im}], { > > a > b > 0, 0 < c < Pi}] > > > > This time, mathematica comes back with an expression involving > > ArcTan[u,v] which is just ArcTan[v/u], by direct inspection. > > So the 2 terms involving different ArcTan are actually the same. > > But I cannot easily make Mathematica to recognize that they > > are the same. Even if I use the rule ArcTan[u_,v_]->ArcTan[v/u] > > it still does not think the 2 expressions are the same. It is easier > > to simply copy and paste an anwser than manipulate mathematica > > to get a simpler result. > > > > Perhaps future versions of mathematica will be smarter and does not > > require as much intervention from the user. > > > > Thanks > > > > Richard > > > > On Sat, Sep 18, 2004 at 05:48:55AM -0400, Peter Valko wrote: > > > Richard Chen <richard at doubleprime.com> wrote in message news: > <cidt38$brv$1 at smc.vnet.net>... > > > > The command: > > > > > > > > Integrate[1/(1 + e Cos[t]), {t, 0, a}, > > > > Assumptions -> {-1 < e < 1, 0 < a < Pi}] > > > > > > > > leads to a complex valued result. I could not make > > > > mathematica to render the result in a form that is > > > > purely real. ComplexExpand, Refine all do not seem to work. > > > > > > > > Does anyone know how to make mathematica to simplify this > > > > result into a real form? > > > > > > > > Thanks for any info. > > > > > > > > Richard > > > > > > > > > > > > Richard, > > > > > > I think this will work: > > > > > > > > > ch = Integrate[1/(1 + e Cos[t]), {t, 0, a}, Assumptions -> {-1 < e < > > > 1, 0 < a < Pi}] > > > > > > FullSimplify[ComplexExpand[ch, TargetFunctions -> {Re, Im}], {-1 < e < > > > 1, 0 < a < Pi}] > > > > > > > > > The result is > > > > > > (-2*ArcTan[((-1 + e)*Tan[a/2])/Sqrt[1 - e^2]])/Sqrt[1 - e^2] > > > > > > > > > Peter > > > > > > > >