Why not $f(z)=z^2$ conformal at $z=0$? $$f(z)=z^2$$ is not conformal at $z=0$ Why? * * * Conformal definition: $f$ is conformal at z if f preserves angles there.

The angle between the rays $t$ and $it$ with $t\in[0,\infty)$ is $90^\circ$. The angle between thier images $t^2$ and $-t^2$ is $180^\circ$