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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.