I am learning complex variable and encounter a theorem as below:
Let $z_0$ be a root of multiplicity k $\ge 2$ of equation $P(z)=P(z_0)$. Then, under the mapping $w=P(z)$, every angle between curves at $z_0$ is enlarged k times.
I am confused that what are those "curves" mean in this theorem? Mapping just map a point in $z$-plane to $w$-plane. So where are these curves come from? And what is the "every angle" means? Shouldn't it be only one angle between two curves?
Can any one explain explicitly to me? Thank you guys so much!
The simplest way to understand this is to visualize the image of a polar domain under the action of $z\to z^2$, as shown below. Note that, if $z=re^{i\theta}$, then $z^2 = r^2e^{2i\theta}$. In particular, the angles between the radial lines shown in the polar domain double under the action of the squaring function.