<s>[INST] How do I find the real or imaginary part of this complex equation? I am trying to solve the below equation for surface plasmon polariton(SPP) wave 
$ \sqrt{\dfrac{1}{\beta^2-(\frac{\omega^2}{c^2})}} + \sqrt{\dfrac{\epsilon_r}{\beta^2-(\frac{\omega^2}{c^2}\epsilon_r)}} = -j \dfrac{\sigma(w)}{\omega \epsilon_0}$
Because i can't solve the problem otherwise I have solved the equation assuming ${\dfrac{1}\epsilon_r} = 1$ present in the second term of the equation $ \frac{\omega^2}{c^2}\epsilon_r $
which looks like these,
$ \dfrac{1}{\beta^2-(\frac{\omega^2}{c^2})} +2\sqrt{\dfrac{1}{\beta^2-(\frac{\omega^2}{c^2})}} \sqrt{\dfrac{\epsilon_r}{\beta^2-(\frac{\omega^2}{c^2})}} + \dfrac{\epsilon_r^2}{\beta^2-(\frac{\omega^2}{c^2})} = - \dfrac{\sigma^2(w)}{\omega^2 \epsilon_0^2}$
Then after some doing some calculation, I have come to this solution
$ \beta = \sqrt{\dfrac{\omega^2}{c^2} + \dfrac{(1 + (\epsilon_r)^2)\omega^2 \epsilon_0) ^2}{{\sigma^2(w)}} }$
But the problem is that it's not been matched with my simulated result, I am thinking it's because of that  ${\epsilon_r} $ that I have not taken into account.
Is there any branch of mathematics I should be aware of, Any kind of help will be highly appreciated. Thanks in Advance. 
 [/INST] You replaced $\epsilon_r$ with $1$ in some but by no means all places, but more to the point, you committed a fallacy of the form $a+b=c\implies a^2+b^2=c^2$. Squaring should give $$2\sqrt{\frac{\epsilon_r}{(\beta^2-\omega^2/c^2)(\beta^2-\epsilon_r\omega^2/c^2)}}=-\frac{\sigma^2}{\omega^2\epsilon_0^2}-\frac{1}{\beta^2-\omega^2/c^2}-\frac{\epsilon_r}{\beta^2-\epsilon_r\omega^2/c^2}.$$Squaring again gives$$\frac{2\epsilon_r+\frac{2\sigma^2}{2\epsilon_r\omega^2/c^2-\omega^2\epsilon_0^2}\left((1+\epsilon_r)\beta^2\right)}{(\beta^2-\omega^2/c^2)(\beta^2-\epsilon_r\omega^2/c^2)}=\frac{\sigma^4}{\omega^4\epsilon_0^4}+\frac{1}{(\beta^2-\omega^2/c^2)^2}+\frac{\epsilon_r^2}{(\beta^2-\epsilon_r\omega^2/c^2)^2}.$$Multiplying both sides by $(\beta^2-\omega^2/c^2)^2(\beta^2-\epsilon_r\omega^2/c^2)^2$,$$\left[2\epsilon_r+\frac{2\sigma^2}{2\epsilon_r\omega^2/c^2-\omega^2\epsilon_0^2}\left((1+\epsilon_r)\beta^2\right)\right](\beta^2-\omega^2/c^2)(\beta^2-\epsilon_r\omega^2/c^2)=\frac{\sigma^4}{\omega^4\epsilon_0^4}(\beta^2-\omega^2/c^2)^2(\beta^2-\epsilon_r\omega^2/c^2)^2+(\beta^2-\epsilon_r\omega^2/c^2)^2+\epsilon_r^2(\beta^2-\omega^2/c^2)^2.$$This is a quartic in $\beta^2$, so you can solve it but not easily. Then we have to get rid of spurious roots from the repeated squaring.
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