(a) Let ℍ:={z ∈ℂ ; Im z>0} be the upper half-plane. Show: z ∈H ⟺-1 / z ∈ℍ . (b) Assume z, a ∈ℂ

step 1 All right, we're given here the relation in terms of x. Here we have Z1, Z2 are the complex numb

(a) Let ℍ:={z ∈ℂ ; Im z>0} be the upper half-plane. Show:...

(a) Let $\mathbb{H}:=\{z \in \mathbb{C} ; \quad \operatorname{Im} z>0\}$ be the upper half-plane. Show: $z \in \mathrm{H} \Longleftrightarrow-1 / z \in \mathbb{H} .$ (b) Assume $z, a \in \mathbb{C}$. Show: $\quad|1-z \bar{a}|^{2}-|z-a|^{2}=\left(1-|z|^{2}\right)\left(1-|a|^{2}\right) .$ Deduce: If $|a|<1$, then $$ |z|<1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|<1 \quad \text { and } \quad|z|=1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|=1 $$

(a) Let $\mathbb{H}:=\{z \in \mathbb{C} ; \quad \operatorname{Im} z>0\}$ be the upper half-plane.
Show: $z \in \mathrm{H} \Longleftrightarrow-1 / z \in \mathbb{H} .$
(b) Assume $z, a \in \mathbb{C}$.
Show: $\quad|1-z \bar{a}|^{2}-|z-a|^{2}=\left(1-|z|^{2}\right)\left(1-|a|^{2}\right) .$
Deduce: If $|a|<1$, then
$$
|z|<1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|<1 \quad \text { and } \quad|z|=1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|=1
$$

Key Concepts

Complex Modulus Identity

A complex modulus identity involves equalities where the modulus (or absolute value) of complex expressions is manipulated algebraically. Such identities often reveal invariant quantities under certain transformations and are used to relate geometric properties of complex numbers, as well as to establish equivalences between inequalities or equalities in different regions of the complex plane.

Upper Half-Plane

The upper half-plane is the subset of the complex plane consisting of all points whose imaginary parts are positive. It is a fundamental example in complex analysis and hyperbolic geometry, often used as a model for non-Euclidean geometries and as a domain on which many transformation properties are studied.

Möbius Transformation

A Möbius transformation is a function of the form (az + b)/(cz + d) where ad ? bc ? 0. These transformations are conformal maps of the extended complex plane, preserving circles and lines, and are instrumental in understanding invariance properties as well as symmetry in various complex domains such as the upper half-plane and the unit disk.

Complex Inversion

Complex inversion refers to a mapping in the complex plane, typically of the form z ? -1/z or similar variants, which 'inverts' points relative to a circle or line. This kind of transformation is a special case of Möbius transformations and plays a key role in understanding how certain regions (for example, the upper half-plane) transform under inversion.

Automorphisms of the Unit Disk

Automorphisms of the unit disk are Möbius transformations that map the open unit disk onto itself. They preserve the circular structure and are characterized by specific symmetry conditions. Understanding these mappings is essential in the study of hyperbolic geometry and complex function theory, especially regarding the dynamics and fixed points of such transformations.

Transcript

00:01 All right, we're given here the relation in terms of x.

00:06 Here we have z1, z2 are the complex numbers here we have.

00:11 And so x is also a complex number.

00:13 So now i want to define the value of x from here, this relation.

00:17 So it's expanded.

00:18 So the first part here we have that is given as more z1, z2 as it is.

00:24 Next, if you look at it here, we have real time, real z1, z2.

00:28 So we understand here, if we have complex number, that is given as z and take a conjugate, re conjugate.

00:35 I add now z plus the conjugate.

00:39 Z is x plus iota y, z conjugate, x negative y aorta y and then we get two times real value.

00:46 That's coming out to be two times we get the real value for z plus j conjugate.

00:53 So this way we can write here negative here, we just divide by two, multiply by two.

00:59 So we get it as, let's say, negative half.

01:03 Times not two times real z1 z2 so two times will that one z2 so that will give that one that two we're taking plus it's conjugate so that one conjugate z to conjugate so if we just add the complex number for this conjugate we get two times real value we divide by two so it becomes two times real value next we expand this now so we have negative half and we get z one conjugate knot square plus z2 mod square, then negative we get z1 conjugate, z2 conjugate, negative, z1 conjugate times we have z2.

01:53 And next, expand this, so we get here plus 1 over 2.

01:59 So if expand this, that's coming out to be, we get more z2 square plus more z1 square.

02:08 And next we get it as next one.

02:11 Negative two times mod z2, mod z 1.

02:19 So now we just keep on expanding this expression we have here, mod z1, z2...

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