Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]

Groups > comp.soft-sys.math.mathematica > #2375

Re: Complex arithmetic identity question

Show all headers | View raw

Am 13.05.2011 12:26, schrieb Ralph Dratman:
> 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

Hi Ralph,

you have to tell Solve, that a, b, c, d are real numbers:

Simplify[Solve[c + d*I == 1/(a + I*b) &&
   Element[a | b | c | d, Reals] && (a != 0 || b != 0),
 {c, d}], Element[a | b, Reals]]

or you calculate directly:

{Thread[{c, d} -> ComplexExpand[Through[{Re, Im}[1/(a + b I)]]]]}


Peter

Back to comp.soft-sys.math.mathematica | Previous | Next — Previous in thread | Find similar | Unroll thread

Thread

Complex arithmetic identity question Ralph Dratman <ralph.dratman@gmail.com> - 2011-05-13 10:26 +0000
  Re: Complex arithmetic identity question Bert RAM Aerts <bert.ram.aerts@gmail.com> - 2011-05-14 07:13 +0000
  Re: Complex arithmetic identity question Peter Pein <petsie@dordos.net> - 2011-05-14 07:14 +0000

csiph-web