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Re: Re: How to simplify to a result that is real
*To*: mathgroup at smc.vnet.net
*Subject*: [mg50844] Re: [mg50821] Re: How to simplify to a result that is real
*From*: DrBob <drbob at bigfoot.com>
*Date*: Thu, 23 Sep 2004 05:27:31 -0400 (EDT)
*References*: <20040921115025.QQQW18891.lakermmtao10.cox.net@smtp.east.cox.net> <ciqv1p$ihc$1@smc.vnet.net> <200409220851.EAA23147@smc.vnet.net>
*Reply-to*: drbob at bigfoot.com
*Sender*: owner-wri-mathgroup at wolfram.com
The ArcTan[x,y] form is not a "problem". It's just as simple as ArcTan[y/x] (once common factors have been removed), and it doesn't suffer from so many branch-cut and pole issues. A better question is how to make sure this is the form we reach, rather than the other one.
Bobby
On Wed, 22 Sep 2004 04:51:59 -0400 (EDT), Paul Abbott <paul at physics.uwa.edu.au> wrote:
> In article <ciqv1p$ihc$1 at smc.vnet.net>,
> Richard Chen <richard at doubleprime.com> wrote:
>
>> 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.
>
> How about
>
> SetOptions[Integrate, GenerateConditions -> False];
> Simplify[Integrate[1/(a + b Cos[t]), {t, 0, c}], a > b > 0]
>
> FullSimplify yields a slightly simpler result in terms of ArcCot instead
> of ArcTan.
>
> Cheers,
> Paul
>
--
DrBob at bigfoot.com
www.eclecticdreams.net
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