I need to solve a few equations and inequalities for my homework. I did most of them, but there are three I have problem with. Can someone give me a hint for each of them?
- $|\frac{z+i}{z^2+i}|>1$
I tried to switch it to an algebraic form and then solve but I ended up with: $a^2+b^2+2b > a^4 +2a^2b^2 + b^4 + 4ab$ Which still gives me nothing.
- $z^3=-\overline{z}$
I came up with $(|z|(\cos\alpha + i\sin \alpha))^3=-|z|(\cos-\alpha + i\sin -\alpha)$
$(|z|(\cos2\alpha + i\sin 2\alpha))^2 = -1$
$|z|(\cos2\alpha + i\sin 2\alpha) = \left\lbrace -i,i \right\rbrace$
And can't move on. But it's probably wrong way of thinking anyway.
- $arg(-\overline{z}) \ge \pi$
I got $arg(z) \le 0$ so what? The only answer will be $\Im z=0\land \Re z>0$?
What should I do with $arg(z)=500\pi$? Transfrom it to $Arg(z)=0$ and then solve?