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MathGroup Archive 2004

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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg50769] 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 21:39:27 -0400 (EDT)
  • References: <cidt38$brv$1@smc.vnet.net> <200409180948.FAA00572@smc.vnet.net> <200409190755.DAA17960@smc.vnet.net> <1C034D6C-0A39-11D9-A57C-000A95B4967A@akikoz.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Actually, I think your question is more interesting that I had at first 
realized. Having thought about this I now think that when you evaluate

ComplexExpand[exp[a,b,c,...],TargetFunctions->{Re,Im}], where a,b,c,... 
are parameters, the following thing happens. First
ComplexExpand[exp[a,b,c,...],{a,b,c,...},TargetFunctions->{Re,Im}] is 
evaluated in terms of Re[a], Im[a], Re[b], Im[b] ...and then all the 
Im[a],Im[b]  are set to 0 and all Re[a],Re[b] ... are set to a, b .... 
That is why they do not appear in the final answer. I came to this 
conclusion by considering the following puzzle:


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


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



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


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


The puzzle is: why are these answers different and how are they arrived 
at?

My answer is that they are evaluated as follows:


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

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


ComplexExpand[a/(b + c*I), {a, b, c},
    TargetFunctions -> {Im}] /. {Im[a] -> 0, Im[b] -> 0,
    Im[c] -> 0}


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

Note that the very long expression resulting from evaluating

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

does indeed contain only terms of the form Re[a] ...



and the expression resulting form evaluating:

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

only terms of the frm Im[a], Im[b] ....


Andrzej





On 19 Sep 2004, at 21:40, Andrzej Kozlowski wrote:

> 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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