What Is a Conformal Mapping in Complex Analysis?
Definition 1
If a function $f: A \subset \mathbb{C} \to \mathbb{C}$ is analytic on $\mathscr{R} \subset A$ and $f ' (z) \ne 0$ for all $z \in \mathscr{R}$, then $f$ is called a conformal mapping or conformal transform. On the other hand, if there exists a point $\alpha$ satisfying $f ' (\alpha) = 0$, then $\alpha$ is called a critical point of $f$.
Explanation
As the Chinese characters 等角 (equal angle) suggest, applying a conformal transform preserves the angles formed by figures.
Living up to its name, let us note the fact that the composition of conformal mappings is itself a conformal mapping. For the proof, it suffices to check the following contrapositive. $$ (f \circ g) ' = f '(g) g' = 0 \iff g' = 0 \lor f ' = 0 $$
Such conformal transforms are very important in complex analysis, where simple closed paths are dealt with frequently, and they are useful when handling integration paths. Geometrically speaking, a critical point can be described as a point where the mapping comes to a complete stop in order to change direction, that is, a point where it bends. On the other hand, a function that is analytic and injective has the following two important properties.
Basic Properties 1
- [1]: If a function $f$ is analytic and injective on $\mathscr{R}$, then $f ' (z) \ne 0$ at every $z \in \mathscr{R}$. In other words, $f$ is a conformal mapping.
- [2]: Suppose a function $f$ is analytic and injective on $\mathscr{R}$ and maps a simple closed path $\mathscr{C}$ to $\mathscr{C} ' $. Then $f$ maps points inside $\mathscr{C}$ only to either the interior or the exterior of $\mathscr{C} ' $.
- Note that in [1] this is not a necessary and sufficient condition. [2] is an especially important property: by checking just a single point inside $\mathscr{C}$, we can tell whether the other points go to the interior or the exterior of $\mathscr{C} ' $.