Solving for conjugate of a long complex function

Question

I have the following function of ω

f[ω_] := (2 Sqrt[Γ] (4*g2^2 + (κ1 - 2*I*ω) (κ2 - 2*I*ω)))/(4*g2^2 (Γ - 
2*I*ω) + (4*
 g1^2 + (Γ - 2*I*ω) (κ1 - 
2*I*ω)) (κ2 - 2*I*ω))

And I wish to obtain the poles for the denominator of the function:

wroots1 = x /. Solve[(Denominator[f[ω]] /. {ω -> x}) == 0, x] 

The result is of the following:

{-(1/6) I (Γ + κ1 + κ2) + (I (-16 (\
Γ + κ1 + κ2)^2 + 
   48 (4 g1^2 + 
      4 g2^2 + Γ κ1 + Γ...}

Basically a really long ugly solution. My goal is to obtain the conjugate of wroots1, however, when I do it straightforwardly:

wroots2 = 
Simplify[Conjugate[wroots1], 
Assumptions -> {Γ ∈ 
  Reals, κ1 ∈ Reals, κ2 ∈ Reals, 
 g1 ∈ Reals, 
 g2 ∈ Reals, ω ∈ Reals, 
 4 (-16 (Γ + κ1 + κ2)^2 + 
      48 (4 g1^2 + 
         4 g2^2 + Γ κ1 + (Γ + \
κ1) κ2))^3 + 
   4096 (-36 g1^2 (Γ + κ1 - 
         2 κ2) + (2 Γ - κ1 - \
κ2) (36 g2^2 + (Γ + κ1 - 
            2 κ2) (Γ - 
            2 κ1 + κ2)))^2 > 0}];

I am returned with:

{1/48 I (8 (Γ + κ1 + κ2) + 
8 2^(1/3) (-12 g1^2 - 
   12 g2^2 + (Γ + κ1 + κ2)^2 - 
   3 (κ1 κ2 + Γ (κ1 + \
κ2))) Conjugate[
  1/(-36 g1^2 Γ + 72 g2^2 Γ + 
     2 Γ^3 - 36 g1^2 κ1 - 
     36 g2^2 κ1 - 3 Γ^2 κ1 - 
     3 Γ κ1^2 + 2 κ1^3 + 
     72 g1^2 κ2 - 36 g2^2 κ2 - 
     3 Γ^2 κ2 + 
     12 Γ κ1 κ2 - 
     3 κ1^2 κ2 - 3 Γ κ2^2 - 
     3 κ1 κ2^2 + 2 κ2^3...}

Clearly, Conjugate refuses to take the conjugate of said function. I tried doing

wroots2 = wroots1 /. {I -> -I}

But that doesn't work as well. I'm at lost at what to do here and I could use any help I can get. Thank you very much in advance.

Answer 1

If you Conjugate the equation before Solve, you'll get the complex-conjugated solutions (complex analysis is nice!):

wroots1 = w /. Solve[Denominator[f[w]] == 0, w];
wroots1C = w /. Solve[ComplexExpand[Conjugate[Denominator[f[w]]]] == 0, w];

This way you don't have to complex-conjugate the solutions. The downside is that the order of the solutions in wroots1 and wroots1C may not be the same.

Answer 2

You can use ComplexExpand, but the cube root term (what I define as cRoot below) interferes with its logic. The following could be a work-around (suppressing all output):

wroots2 = FullSimplify[Conjugate[wroots1], Assumptions -> {Γ ∈ Reals, κ1 ∈ Reals, κ2 ∈ Reals, g1 ∈ Reals, g2 ∈ Reals, ω ∈ Reals, 4 (-16 (Γ + κ1 + κ2)^2 + 48 (4 g1^2 + 4 g2^2 + Γ κ1 + (Γ + \ κ1) κ2))^3 + 4096 (-36 g1^2 (Γ + κ1 - 2 κ2) + (2 Γ - κ1 - \ κ2) (36 g2^2 + (Γ + κ1 - 2 κ2) (Γ - 2 κ1 + κ2)))^2 > 0}];

wroots2 = wroots2 /. (-36 g1^2 (\[CapitalGamma] + \[Kappa]1 - 2 \[Kappa]2) + (2 \[CapitalGamma] - \[Kappa]1 - \[Kappa]2) \ (36 g2^2 + (\[CapitalGamma] + \[Kappa]1 - 2 \[Kappa]2) (\[CapitalGamma] - 2 \[Kappa]1 + \[Kappa]2)) + 1/64 \[Sqrt](4 (-16 (\[CapitalGamma] + \[Kappa]1 + \[Kappa]2)^2 \ + 48 (4 g1^2 + 4 g2^2 + \[CapitalGamma] \[Kappa]1 + (\[CapitalGamma] \ + \[Kappa]1) \[Kappa]2))^3 + 4096 (-36 g1^2 (\[CapitalGamma] + \[Kappa]1 - 2 \[Kappa]2) + (2 \[CapitalGamma] - \[Kappa]1 - \ \[Kappa]2) (36 g2^2 + (\[CapitalGamma] + \[Kappa]1 - 2 \[Kappa]2) (\[CapitalGamma] - 2 \[Kappa]1 + \[Kappa]2)))^2))^(1/3) -> cRoot

wroots2 = wroots2 /. 1/(-36 g1^2 (\[CapitalGamma] + \[Kappa]1 - 2 \[Kappa]2) + (2 \[CapitalGamma] - \[Kappa]1 - \[Kappa]2) \ (36 g2^2 + (\[CapitalGamma] + \[Kappa]1 - 2 \[Kappa]2) (\[CapitalGamma] - 2 \[Kappa]1 + \[Kappa]2)) + 1/64 \[Sqrt](4 (-16 (\[CapitalGamma] + \[Kappa]1 + \[Kappa]2)^2 \ + 48 (4 g1^2 + 4 g2^2 + \[CapitalGamma] \[Kappa]1 + (\[CapitalGamma] \ + \[Kappa]1) \[Kappa]2))^3 + 4096 (-36 g1^2 (\[CapitalGamma] + \[Kappa]1 - 2 \[Kappa]2) + (2 \[CapitalGamma] - \[Kappa]1 - \ \[Kappa]2) (36 g2^2 + (\[CapitalGamma] + \[Kappa]1 - 2 \[Kappa]2) (\[CapitalGamma] - 2 \[Kappa]1 + \[Kappa]2)))^2))^(1/3) -> (1/cRoot)

ComplexExpand[wroots2]

You can then substitute the expression for cRoot back into the answer.