I searched all internet and I can't get my answer. Please, can you explain how $\mathrm{sgn}(x)$ found its way to solution of this problem ?
$$\mathrm{Arg}[ \frac{w_0^2}{wi (w_0 + iw)}]$$
I need to calculate this expression, where $w_0$ is a parameter. I know the result. It includes $\mathrm{sgn}(w)$, but I can't figure out how it was calculated. Please help :) I know the basic formula, $$\arg(z)=\arctan(\mathrm{Im}(z)/\mathrm{Re}(z)).$$
Definitely wrong. If $w = w_0\ne 0$, $$ \frac{w_0^2}{wi(w_0 + iw)} = \frac{w_0^2}{w_0 i(w_0 + iw_0)} = \frac{1}{i(1 + i)} = -\frac{1 + i}2 $$ has constant argument while $$ (\pi/2)\mathrm{sgn}(w) - \arctan(w/w_0) = (\pi/2)\mathrm{sgn}(w_0) - \arctan(1) = (\pi/2)(w_0/|w_0|) - \pi/4 $$ depends of the argument of $w_0$!
Idea for the search of a (correct) solution: as the (multivalued) argument of the product is sum of arguments: $$ \mathrm{Arg}\left(\frac{w_0^2}{wi(w_0 + iw)}\right) = 2\mathrm{Arg}(w_0) - \mathrm{Arg}(w) - \mathrm{Arg}(i) - \mathrm{Arg}(w_0 + iw). $$