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

• To: mathgroup at smc.vnet.net
• Subject: [mg50764] Re: [mg50750] Re: [mg50735] Re: How to simplify to a result that is real
• From: Andrzej Kozlowski <andrzej at akikoz.net>
• Date: Sun, 19 Sep 2004 21:39:20 -0400 (EDT)
• References: <cidt38\$brv\$1@smc.vnet.net> <200409180948.FAA00572@smc.vnet.net> <200409190755.DAA17960@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```I am not sure if I have understood you correctly, but normally,
ComplexExpand[expression, TargetFunctions -> {Re, Im}] will explicitly
include Re and Im  only if expression contains some symbols that are
specified to be complex, e.g:

ComplexExpand[(a + b*I)/(c + d*I), TargetFunctions ->
{Im, Re}]

(a*c)/(c^2 + d^2) + I*((b*c)/(c^2 + d^2) -
(a*d)/(c^2 + d^2)) + (b*d)/(c^2 + d^2)

but here explicit Re and Im are present:

ComplexExpand[(a + b*I)/(c + d*I), {a},
TargetFunctions -> {Im, Re}]

(Im[a]*d)/(c^2 + d^2) + (b*d)/(c^2 + d^2) +
(c*Re[a])/(c^2 + d^2) + I*((Im[a]*c)/(c^2 + d^2) +
(b*c)/(c^2 + d^2) - (d*Re[a])/(c^2 + d^2))

With other target functions this is not necessarily always the case,
e.g.

ComplexExpand[(a + b*I)/(c + d*I), TargetFunctions ->
{Abs}]

(a*c)/Abs[c + I*d]^2 + I*((b*c)/Abs[c + I*d]^2 -
(a*d)/Abs[c + I*d]^2) + (b*d)/Abs[c + I*d]^2

Andrzej Kozlowski
Chiba, Japan
http://www.akikoz.net/~andrzej/
http://www.mimuw.edu.pl/~akoz/

On 19 Sep 2004, at 16:55, DrBob wrote:

> *This message was transferred with a trial version of CommuniGate(tm)
> Pro*
> It's interesting that the output of ComplexExpand[ch, TargetFunctions
> -> {Re, Im}] in this case doesn't include Re or Im. What is
> ComplexExpand really doing, here?
>
> Bobby
>
> On Sat, 18 Sep 2004 05:48:55 -0400 (EDT), Peter Valko
> <p-valko at tamu.edu> 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
>>
>>
>>
>
>
>
> --
> DrBob at bigfoot.com
> www.eclecticdreams.net
>
>

```

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