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I got a simple cubic equation as input, but Mathematica does not give all solutions, why is this the case here?

f[α_] := Sqrt[2] α (Δ + 2g^2/Ω(α Conjugate[α]))

Solve[f[α] == 0, α]
{{α -> 0}}

There should also be the solution where $\alpha \in$ Real, so that $|\alpha|^2 = -\frac{\Delta\Omega}{2g^2}$ under the condition $\Delta\Omega < 0$.

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marked as duplicate by m_goldberg, rcollyer, belisarius, Yves Klett, rm -rf Sep 17 '13 at 18:46

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

2  
Please include your code in InputForm, not $\LaTeX$. –  rm -rf Sep 16 '13 at 14:05
3  
If you replace (\[Alpha] Conjugate[\[Alpha]]) with Abs[\[Alpha]]^2 then Solve will do better. It really does not know how to work with Conjugate (it's hardly cognisant of Abs, but in this case "hardly" > "not at all"). –  Daniel Lichtblau Sep 16 '13 at 14:12
2  
So what is your problem? Since it does not return all solutions, have you tried various options (utilities) of Solve? This review should clarify the issue: What is the difference between Reduce and Solve?. But at first you should try Solve[ f[α] == 0, α, Reals], it yields your solution. –  Artes Sep 16 '13 at 15:37

1 Answer 1

I think you should instruct Mathematica (as well as us) what the variables, such as delta, g and omega, are assumed to be: imaginary or real, and if real, positive or negative or both? I will assume in the following that they are real, positive.

\[Alpha] = a + I*b;

then the expression for the left-hand parts of equations:

 exprRe = Simplify[
  Re[Sqrt[2] \[Alpha] (\[CapitalDelta] + 
      2 g^2/\[CapitalOmega] (\[Alpha] Conjugate[\[Alpha]]))], \
{Element[a, Reals], 
   Element[b, Reals], \[CapitalDelta] > 0, \[CapitalOmega] > 0, g > 0}]


exprIm = Simplify[
  Im[Sqrt[2] \[Alpha] (\[CapitalDelta] + 
      2 g^2/\[CapitalOmega] (\[Alpha] Conjugate[\[Alpha]]))], \
{Element[a, Reals], 
   Element[b, Reals], \[CapitalDelta] > 0, \[CapitalOmega] > 0, 
   g > 0}]`

The outcome is:

(*    (Sqrt[2] a (2 a^2 g^2 + 
   2 b^2 g^2 + \[CapitalDelta] \[CapitalOmega]))/\[CapitalOmega]  *)

(* (Sqrt[2] b (2 a^2 g^2 + 
   2 b^2 g^2 + \[CapitalDelta] \[CapitalOmega]))/\[CapitalOmega]  *)

Let us now solve them:

Solve[{exprRe == 0, exprIm == 0}, {a, b}]

Here is the result:

{{b -> -(Sqrt[-2 a^2 g^2 - \[CapitalDelta] \[CapitalOmega]]/(
    Sqrt[2] g))}, {b -> 
   Sqrt[-2 a^2 g^2 - \[CapitalDelta] \[CapitalOmega]]/(
   Sqrt[2] g)}, {a -> 0, b -> 0}}
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