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Re: Weird result in Mathematica 6

• To: mathgroup at smc.vnet.net
• Subject: [mg76430] Re: [mg76393] Weird result in Mathematica 6
• From: Carl Woll <carlw at wolfram.com>
• Date: Tue, 22 May 2007 02:47:07 -0400 (EDT)
• References: <200705211001.GAA10071@smc.vnet.net>

Sebastian Meznaric wrote:

>I was playing around with Mathematica 6 a bit and ran this command to
>solve for the inverse of the Moebius transformation
>
>FullSimplify[
> Reduce[(z - a)/(1 - a\[Conjugate] z) == w && a a\[Conjugate] < 1 &&
>   w w\[Conjugate] < 1, z]]
>
>This is what I got as a result:
>-1 < w < 1 && -1 < a < 1 && z == (a + w)/(1 + w Conjugate[a])
>
>Why is Mathematica assuming a and w are real? The Moebius
>transformation is invertible in the unit disc regardless of whether a
>and w are real or not. Any thoughts?
>  
>
 From the help:

Reduce[expr,vars] assumes by default that quantities appearing 
algebraically in inequalities are real, while all other quantities are 
complex.

You can try adding a domain specification:

In[144]:= FullSimplify[
 Reduce[(z - a)/(1 - Conjugate[a] z) == w && a Conjugate[a] < 1 &&
   w Conjugate[w] < 1, z, Complexes]]

Out[144]= -1 < Re[w] < 1 && -Sqrt[1 - Re[w]^2] < Im[w] < Sqrt[
  1 - Re[w]^2] && -1 < Re[a] < 1 && -Sqrt[1 - Re[a]^2] < Im[a] < Sqrt[
  1 - Re[a]^2] && z == (a + w)/(w Conjugate[a] + 1)

Carl Woll
Wolfram Research

• References:
  • Weird result in Mathematica 6
    • From: Sebastian Meznaric <meznaric@gmail.com>