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!