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Re: Simplifying Complex expression

  • To: mathgroup at smc.vnet.net
  • Subject: [mg43091] Re: Simplifying Complex expression
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Tue, 12 Aug 2003 04:43:05 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <bgv903$5i3$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <bgv903$5i3$1 at smc.vnet.net>,
 Helge Andersson <helge at envic.chalmers.se> wrote:

> I'm interested in one of the solution to the the following  equation:
> y=3 x^2-2 x^3
> in the x-intervall 0 to 1 and for y-values between 0 and 1
> With Solve I got three solutions where two of them are complex in general. 
> However, it is possible to plot each of them Using the Plot-command in the 
> desired domain which I interpretate as they are real-valued with my 
> specific conditions. My problem is to simplify the general complex solution 
> to one that is real in the domain 0 to 1. The solution that I specifically 
> is interested in is the following given in InputForm:
> 
> 1/2 - (1 - I*Sqrt[3])/(4*(1 + 2*Sqrt[-1 + y]*Sqrt[y] - 2*y)^(1/3)) -
>   ((1 + I*Sqrt[3])*(1 + 2*Sqrt[-1 + y]*Sqrt[y] - 2*y)^(1/3))/4
> 
> Could someone give me an idea to perform this (for example with Simplify 
> Command)

After computing the solutions,

  solns = x /. Solve[y == 3 x^2 - 2 x^3, x]

one can use ComplexExpand to obtain explicitly real forms,

  Simplify[ComplexExpand[Re[solns],TargetFunctions -> {Re, Im}], 0<y< 1]

One needs to verify that the imaginary part is identically zero.

  Simplify[ComplexExpand[Im[solns],TargetFunctions -> {Re, Im}], 0<y< 1]

Cheers,
Paul

-- 
Paul Abbott                                   Phone: +61 8 9380 2734
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