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Re: Complex arithmetic identity question

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  Re: Complex arithmetic identity question peter lindsay <pl0@me.com> - 2011-05-14 07:10 +0000

#2384 — Re: Complex arithmetic identity question

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On 13 May 2011, at 11:24, Ralph Dratman wrote:

> Hi. I am new to the mailing list, trying to learn how to persuade
> Mathematica to perform helpful algebraic manipulations.
> 
> Here is a very very simple example. I think this should be easy, but so far
> I have not been able to figure it out.
> 
> Suppose  c + I d = 1/(a +I b), where a, b, c, d are reals. Solve for c and d
> in terms of a and b.
> 
> For a human, the solution is easy. Multiply top and bottom by the complex
> conjugate, then set real part equal to real part, imaginary part equal to
> imaginary part.
> 
> The result is  c -> a/(a^2+b^2),  d -> -b/(a^2+b^2).  But is there a simple
> way to get Mathematica to give me that answer in one step?
> 
> If I help by solving for c by hand and plugging that in, Mathematica knows
> how to finish the job:
> 
>   In[22]:= Solve[1/(a + I b) == a/(a^2 + b^2) + I d, d]
> 
> Out[22]= {{d -> -(b/(a^2 + b^2))}}
> 
> 
> But I was hoping Mathematica would be able to go all the way from the
> equation to the solution in one step, with the use of Solve or something
> similar.
> 
> 
> Is this possible, and if so, how? If not, why not?
> 
> 
> Thank you.
> 
> Ralph Dratman

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