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MathGroup Archive 2004

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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg50805] Re: Re: [mg50788] Re: [mg50735] Re: How to simplify to a result that is real
  • From: Richard Chen <richard at doubleprime.com>
  • Date: Wed, 22 Sep 2004 00:11:06 -0400 (EDT)
  • References: <20040921115025.QQQW18891.lakermmtao10.cox.net@smtp.east.cox.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Your procedure indeed works. However, the repeated use of
ComplexExpand with TrigToExp sandwitched in between is hardly
something people can come up with regularly. 

ArcTan[x,y] is indeed different from ArcTan[y/x] in general. 
But the problem is that under our assumptions of a>b>0,0<c<Pi they
are the same. It is the defect of the current version of mathematica
not being able to recognize this that is the cause of the problem.
Otherwise, the relatively simple command

FullSimplify[ComplexExpand[
Integrate[1/(a + b Cos[t]), {t, 0, c}, 
Assumptions -> {a > b > 0, 0 < c < Pi}], 
TargetFunctions -> {Re, Im}], {a > b > 0, 0 < c < Pi}]

will suffice to yield the final concise result.

Richard

On Tue, Sep 21, 2004 at 07:50:24AM -0400, Bob Hanlon wrote:
> 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: [mg50805] 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: [mg50805] [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
> > > > 
> > > 
> > > 
> > 
> 


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