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Re: How to simplify to a result that is real

• To: mathgroup at smc.vnet.net
• Subject: [mg50810] Re: How to simplify to a result that is real
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Wed, 22 Sep 2004 00:11:19 -0400 (EDT)
• Organization: The University of Western Australia
• References: <cidt38$brv$1@smc.vnet.net> <200409180948.FAA00572@smc.vnet.net> <cion2b$r9a$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

In article <cion2b$r9a$1 at smc.vnet.net>,
 Richard Chen <richard at doubleprime.com> wrote:

> 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. 

I'm not sure if I follow you here, but ArcTan[u,v] is equal to 
ArcTan[v/u] only if u > 0. For example, try

   FullSimplify[ArcTan[u,v] == ArcTan[v/u], u > 0]

However, if u < 0 then

  ArcTan[u,v] - ArcTan[v/u] = Sign[v] Pi

(though I cannot see how to get FullSimplify to deduce this fact).

Cheers,
Paul

> 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
> > 
>

-- 
Paul Abbott                                   Phone: +61 8 6488 2734
School of Physics, M013                         Fax: +61 8 6488 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul

• References:
  • Re: How to simplify to a result that is real
    • From: p-valko@tamu.edu (Peter Valko)