Conformal Mapping $w=z^2$

Question

I have to map $|argz≤\frac{π}{4}|$ under the transformation $w=z^2$ ,the answer mentioned is right side of $w$ plane how ever Iam getting $u≤\frac{π}{4}$ could someone help me out with this

Answer 1

Use polar. $z=re^{i\theta}\implies w=z^2=r^2e^{2i\theta}$. Thus $|\operatorname{arg}z|\le\pi/4$ gets mapped to $|\operatorname{arg}z|\leπ/2$.