How to determine if $f(z)= Ln(z)$ is confromal at $z=0$?

Question

The function $$f(z)=Ln(z)$$ has at $z=0$ a singularity and therefore the usual analysis to determine if it is conformal (i.e. determining if $f'(z)=0$ at the point of interest) fails. How can we therefore determine weather such a function is conformal at it's singularity?

Answer 1

First thing to consider is that $\ln z$ is not defined at $z=0$, and that $z=0$ is not an isolated singularity, but a branching point.

I will answer your question considering curves that approach $z=0$, but never pass trough it.

Consider a ray $\{r\,e^{i\theta}:r>0\}$, with fixed $\theta\in(-\pi,\pi)$. Its image under the function $\ln z$ (main branch) is the horizontal line $\{\ln|r|+i\,\theta:r>0\}$. All rays are transformed into horizontal lines; the angle between two rays is not conserved.