I'm attempting to solve the following system of equations for k and $\kappa$ $$\mu\epsilon\omega^2=k^2-\kappa^2$$ $$\mu\sigma\omega=2k\kappa$$ but when I type into Mathematica
Simplify[Solve[{\[Mu] \[Epsilon] \[Omega]^2 == k^2 - \[Kappa]^2, \[Mu] \[Sigma] \[Omega] == 2 k \[Kappa]}, {k, \[Kappa]}]
I get the following output, which is pretty gnarly
The solution I was hoping to get (from a textbook) is $$k\equiv \omega\sqrt{\frac{\epsilon\mu}{2}}\left[\sqrt{1+\left(\frac{\sigma}{\epsilon\omega}\right)^2}+1\right]^{1/2}$$ $$\kappa\equiv \omega\sqrt{\frac{\epsilon\mu}{2}}\left[\sqrt{1+\left(\frac{\sigma}{\epsilon\omega}\right)^2}-1\right]^{1/2}$$ I'm not sure how to simplify the output from Mathematica to get it into that specific from I'm looking for. Are there extra assumptions I need to pass? I've tried the same command as Simplify only changing it to Reduce but it's still not quite clear the direction to take. For some context, this is looking at the complex wave number from Griffiths' Introduction to Electrodynamics.
PowerExpandbefore or afterSolve? $\endgroup$ – Bo Johnson 12 mins agoSimplifyyou can useAssumptionslike this:sol = Solve[{μ ϵ ω^2 == k^2 - κ^2, μ σ ω == 2 k κ}, {k, κ}]; Simplify[sol, Assumptions -> {μ > 0, ϵ > 0, ω > 0}]. And then select the solution you need, I guess. $\endgroup$ – Anjan Kumar 4 mins ago