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Re: Complex arithmetic identity question

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  Re: Complex arithmetic identity question Bill Rowe <readnews@sbcglobal.net> - 2011-05-14 07:13 +0000

#2369 — Re: Complex arithmetic identity question

On 5/13/11 at 6:24 AM, ralph.dratman@gmail.com (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?

in one step? Yes. But simple, is in the eye of the beholder. A
one liner solution is:

In[11]:= Solve[
  Thread[List @@ (c + I d) == List @@ ComplexExpand[1/(a + I
b)]], {c,
   d}]

Out[11]= {{c -> a/(a^2 + b^2), d -> -(b/(a^2 + b^2))}}

ComplexExpand converts 1/(a + b I) to x + I y format

List@@ is short hand for Apply[List, ...] and simply replaces
head Plus with head List. That is it converts x +I y to {x, I y}

The Thread part sets up two equations, equating the real and
complex parts.

Finally, Solve simply solves for c and d.

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