Propagation constant equation for plane waves in a good conductor

Question

I came across this equation/expression from a microwave textbook by Pozar. I can't figure out how this came about:

How is the $(1+j)$ formulated? What I am getting is $(j \omega \mu \sigma)^{\frac{1}{2}}$.

Answer 1

I guess that $j$ is (a) square root of $-1$. There are two complex numbers that square to $j$:

$$ \left( \pm \frac{1 + j}{\sqrt{2}} \right)^2 = \frac{1 + 2j + j^2}{2} = j. $$

If you take the square root $\frac{1}{j}$ to be $\frac{\sqrt{2}}{1 + j}$ (as opposed to $-\frac{\sqrt{2}}{1+j}$) then you get

$$ j \frac{1}{\sqrt{j}} = \frac{2j}{1 + j} = \frac{2j(1 - j)}{(1+j)(1-j)} = \frac{2 + 2j}{2} = 1 + j.$$