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!

  • $\begingroup$ Do you understand the geometry of the $n$th power mapping $re^{i\theta} \mapsto r^{n} e^{in\theta}$ at the origin? $\endgroup$ – Andrew D. Hwang May 2 '17 at 23:10
  • $\begingroup$ @AndrewD.Hwang Is this the length enlarged by n times and rotated by $nθ$? $\endgroup$ – Parting May 2 '17 at 23:12
  • $\begingroup$ Not quite: The length is raised to the $n$th power, and the angle is multiplied by $n$. This is the local model for a root of order $n$, and what your book means by "angles ... get enlarged $n$ times". You might enjoy this simple interactive web program. $\endgroup$ – Andrew D. Hwang May 2 '17 at 23:21

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.

enter image description here

  • $\begingroup$ I understand this case, this mapping, $z \to z^2$ is like a mapping from a $z$-plane to a $w$-plane, so that, after mapping, the angle of two curves in the $z$-plane is enlarged 2 times in $w$-plane, right? But what is "at $z_0$" meaning in the theorem? And why specially the angle is enlarged k times only at $z_0$? $\endgroup$ – Parting May 3 '17 at 18:13

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