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I am having the following trouble with definitions I have copied down below regarding conformal mapping from John B. Connway's Functions of One Complex Variable I. I find a number of confusions in the definitions.

Def 1: A smooth path in a region (open connected subset) $G \subseteq \Bbb C$ is a continuous function $\gamma: [a,b] \rightarrow G$ for some closed interval $[a,b]$ in $\Bbb R$ where $\gamma'(t)$ exists for each $t \in [a,b]$ and $\gamma'(t)$ is continuous on $[a,b]$.

Def 2: Suppose $\gamma_1$ and $\gamma_2$ are two smooth paths in $G$ where $\gamma_1(t_1) = \gamma_2(t_2) = z_0$ and that $\gamma_1'(t_1) \not= 0 $ and $\gamma_2'(t_2) \not= 0$. The angle between the paths $\gamma_1$ and $\gamma_2$ at $z_0$ is given by $$ \arg \gamma_2'(t_2) - \arg \gamma_1'(t_1) $$

Def 3: A map $f: U \rightarrow \Bbb C$ is conformal at $z_0 \in G$ if angles is preserved and the limit, $$ \lim _{z \rightarrow z_0} \frac{|f(z) - f(a)|}{|z-a|}$$ exists. It is angle preserving, if, given two smooth paths $\gamma_1$ and $\gamma_2$ which intersect at $z_0 = \gamma_1(t_1) = \gamma_2(t_2)$, with nonzero derivatives at the point, $$ \arg ( f \circ \gamma_2)'(t_2) - \arg ( f \circ \gamma_1)'(t_1) = \arg \gamma_2'(t_2) - \arg \gamma_2 '(t_2). $$

I have questions regarding just these three definitions.

  1. How does Def 2. make clear of the problem of the ambiguity in angle? If we define $\arg z$ for $z \in \mathbb{C}$ to be in range $( - \pi, \pi] $ then we may have the following possible scenario, $$\arg z_1 - \arg z_2 = ( - \pi/2) - \pi = -\frac{3\pi}{2} $$ $$ \arg z_3 - \arg z_4 = \pi - (\pi/2) = \frac{\pi}{2} $$ Then in terms of our usual notion, the above angles are the same but we cannot place equality sign. So I believe we should add$\pmod {2 \pi}$ to modify Def 2 (?)

  2. For Def 3. what kind of set should $U$ be? If $U$ does not contain any open connected $G$, it may be the case that $f \circ \gamma_1$ is not even well-defined, for a given smooth path $\gamma_1$. Hence, for any point $z_0 \in U$, $f$ satisfies the condition for conformality vacuously? Thus, I believe $U$ should at least be an open set.

  3. For Def 3. is it not implied that $f$ satisfies some additional properties? By Def 2, we talk about "angle between the paths" when we have smooth paths. So for Def 3, is it required that $f \circ \gamma_1 : G \to \Bbb C$ and $f \circ \gamma_2: G \to \Bbb C$ to be smooth paths (?) Or does the existence of the limit guarantee that these compositions are smooth paths? In fact I do not understand the meaning of the condition of limit, all I can see is it guarantees the continuity of $f$ - but does it imply it is analytic (continuously differentiable on the open set $U$).

I hope I am not missing something trivial. Thanks.

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