(a) Consider the map f: ℂ^* ⟶ℂ with f(z)=1 / z̅ Give a geometrical construction (with ruler an

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(a) Consider the map f: ℂ^* ⟶ℂ with f(z)=1 / z̅ Give a...

(a) Consider the map $$ f: \mathbb{C}^{*} \longrightarrow \mathbb{C} \text { with } f(z)=1 / \bar{z} $$ Give a geometrical construction (with ruler and compass) for the image $f(z)$ and justify calling this map "reflection at the unit circle". Find the image under $f$ of each of $(\alpha) \quad D_{1}:=\{z \in \mathbb{C} ; \quad 0<|z|<1\}$ $(\beta) \quad D_{2}:=\{z \in \mathbb{C} ; \quad|z|>1\}$ (\gamma) $D_{3}:=\{z \in \mathbb{C} ; \quad|z|=1\}$. (b) Now consider the map $$ g: \mathbb{C}^{*} \longrightarrow C \text { with } g(z)=1 / z(=\overline{f(z)}) $$ and give a geometrical construction for the image $g(z)$ of $z .$ Why is this map called "inversion at the unit circle"? What are the fixed points of $g$, i.e. for which $z \in \mathbb{C}^{\bullet}$ is it true that $g(z)=z ?$

(a) Consider the map
$$
f: \mathbb{C}^{*} \longrightarrow \mathbb{C} \text { with } f(z)=1 / \bar{z}
$$
Give a geometrical construction (with ruler and compass) for the image $f(z)$ and justify calling this map "reflection at the unit circle". Find the image under $f$ of each of
$(\alpha) \quad D_{1}:=\{z \in \mathbb{C} ; \quad 0<|z|<1\}$
$(\beta) \quad D_{2}:=\{z \in \mathbb{C} ; \quad|z|>1\}$
(\gamma) $D_{3}:=\{z \in \mathbb{C} ; \quad|z|=1\}$.
(b) Now consider the map
$$
g: \mathbb{C}^{*} \longrightarrow C \text { with } g(z)=1 / z(=\overline{f(z)})
$$
and give a geometrical construction for the image $g(z)$ of $z .$ Why is this map called "inversion at the unit circle"? What are the fixed points of $g$, i.e. for which $z \in \mathbb{C}^{\bullet}$ is it true that $g(z)=z ?$

Key Concepts

Fixed Points of Inversion Mappings

Fixed points of such mappings are those points that are unchanged by the transformation. In the context of inversion in the unit circle, for instance, solving g(z) = z (with g(z) = 1/z) leads to the identification of points that lie on the circle, since these are precisely the points that satisfy |z| = 1. The study of fixed points is important in determining the symmetry and invariance properties of conformal and Möbius transformations.

Mapping of Regions under Inversion

When applying an inversion or reflection with respect to a circle, regions of the complex plane are transformed in a specific way: the interior of the circle (|z| < 1) is mapped to the exterior (|z| > 1), the exterior is mapped to the interior, and points on the circle remain fixed. This property is crucial in many areas of geometry and complex analysis, particularly in understanding how various shapes and domains behave under conformal maps.

Inversion in the Unit Circle

Inversion in the unit circle is a transformation that maps a nonzero complex number z to 1/z (or its variations, such as 1/¯z). It has the characteristic of interchanging points inside the circle (|z| < 1) with points outside (|z| > 1), while points on the unit circle remain invariant under the inversion. This transformation is a particular case of Möbius transformations, where circles and lines are mapped to circles and lines.

Reflection in the Unit Circle

Reflection at the unit circle refers to the mapping of a point to its image under inversion combined with conjugation, as seen in the map f(z) = 1/¯z. The term 'reflection' is used because this transformation, when interpreted geometrically, creates a symmetry with respect to the unit circle; points are 'reflected' across the circle such that the line joining the point and its image is orthogonal to the circle at the point of intersection.

Geometric Construction with Ruler and Compass

The geometric construction of these transformations involves using basic ruler and compass techniques to locate the inversion of a point with respect to a circle. Generally, one constructs a line through the given point and the center of the circle, then finds the intersection of this line with the circle and uses properties of similar triangles or circle power to locate the reflection of the point. This process is fundamental in classical Euclidean geometry and provides an intuitive understanding of the inversion or reflection process.

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