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What is a Conformal Mapping in Complex Analysis? 📂Complex Anaylsis

What is a Conformal Mapping in Complex Analysis?

Definition 1

A function f:ACCf: A \subset \mathbb{C} \to \mathbb{C} that is analytic at RA\mathscr{R} \subset A and for all zRz \in \mathscr{R} satisfies f(z)0f ' (z) \ne 0 is called a Conformal Mapping if ff. Meanwhile, if there exists a point α\alpha that satisfies f(α)=0f ' (\alpha) = 0, then α\alpha is referred to as the Critical Point of ff.

Description

As the Chinese characters for conformal (等角) directly imply, angles between figures are preserved under a conformal transformation.

It’s essential to know that the composition of conformal mappings is still a conformal mapping. Proof of this can be sufficiently demonstrated by verifying the following contrapositive. (fg)=f(g)g=0    g=0f=0 (f \circ g) ' = f '(g) g' = 0 \iff g' = 0 \lor f ' = 0

Such conformal transformations are very important in complex analysis, which often deals with simple closed paths, as they are helpful when handling integration paths. Geometrically, critical points can be considered points where the direction changes completely, i.e., where it pivots. On the other hand, analytic functions that are injective have the following two crucial properties.

Fundamental Properties 1

  • [1]: If a function ff is analytic and injective in R\mathscr{R}, then for all zRz \in \mathscr{R}, f(z)0f ' (z) \ne 0 is satisfied. In other words, ff is a conformal mapping.
  • [2]: If a function ff is analytic and injective in R\mathscr{R} and maps a simple closed path C\mathscr{C} to C\mathscr{C} ' , then ff maps points inside C\mathscr{C} to either the inside or outside of C\mathscr{C} ' only.

  • Note that the condition in [1] is not sufficient and necessary. [2] Is particularly important because, by checking just one point inside C\mathscr{C}, one can determine whether the rest of the points are mapped to the inside or outside of C\mathscr{C} ' .

  1. Osborne (1999). Complex variables and their applications: p193, 196. ↩︎ ↩︎