Define **conformal at z _{0}.**

A (real differentiable) map f : 𝑅 ➝ ℂ on a region 𝑅 ⊆ ℂ is **conformal at z _{0} **if it is angle and orientation preserving at z

_{0.}

Finish the following proposition

Finish the corollary: Conformal maps take orthogonal grids (in the x-y plane) to ... ?

Orthogonal grids (in the u-v plane).

Let f = u + iv be a holomorphic function and let Ɣ and Ɣ ̃ be two paths given by the level curves u(x, y) = C_{1} and v(x, y) = C_{2} respectively (for C_{1}, C_{2} real constants). Then what sets (whenever f′(z) ̸= 0)do the paths Ɣ and Ɣ ̃ form?

Orthogonal set in the x-y plane

Define a **biholomorphic map from 𝑅 to 𝑅'.**

We call a holomorphic map f on a region 𝑅 a **biholomorphic map from 𝑅 to 𝑅'** = f(𝑅) if:

- 𝑅' is a region
- f is one-to-one and the inverse map f
^{-1}:𝑅' ➝ 𝑅 is also holomorphic

What symbol do you use to show two regions 𝑅 and 𝑅' are biholomorphic.

f : 𝑅 ⥲ 𝑅'

What is the Lemma about the **automorphism group of 𝑅.**

Define **simply connected.**

A region 𝑅 is **simply-connected **if every closed path Ɣ in 𝑅 (that is Ɣ(0) = Ɣ(1)) can be "continuosuly shrunk" to a point in 𝑅. Heuristically this means 𝑅 has no holes.

Is ℂ٭ or ℂ\ℝ_{≤0} simply connected?

ℂ\ℝ_{≤0}

What is the Riemann mapping theorem.

Every simply connected region 𝑅 ≠ ℂ, there exists a biholomorphic map f:𝑅 ⥲ B_{1}(0)

Define a **Möbius transformation.**

Why do we exclude matrices for which det(T) = 0 in Möbius transformations?

M_{T} is constant if det(T) = 0

If k = sqrt(|det(T)|) show that we can scale any Möbius transformation so that det(T) = ± 1.

What is the Lemma about the set of Möbius transformation forming a group under composition.

Prove the following Lemma.

Need to do - see sheet 5

What is the symbol for the extended complex plane?

What are three algebraic operations on ℂ that are extended to ℂ^{∼} ?

What do these four properties show.

That the

What is the propositiong about circles, line and Möbius transformations?

Möbius transformations map circles or lines to circles or lines

What are the three different maps between circles and lines with Möbius transformations?

###
- Circles not through the origin go to tcircles not through the origin
- Circles through the origin go to lines not through the origin
- Lines through the origin go to lines through the origin

How can you find a map from ℍ to ⅅ by writing ℍ in a clever way?

What is the Mob transformation to take the upper half plane to the unit disc?

What is the inverse of this map called?

Caley Map

How is the Caley Map denoted?

Finish the Lemma: If a Möbius transformation has three or more fixed points in ... ?

What is the proposition about how three points determine a Möbius transformation?

Prove the following proposition.

What is the corollary about find the Möbius transformation using their cross-ratios?

What is the general strategy to find a Mob trans from how it acts on three points?

Rearrange the following equation.

What is the general strategy to find the imagine of a region 𝑅 under a Mob trans M_{T}?

What does Mob(𝑅) stand for?

The set of all Mob trans from a region 𝑅 to itself.

Finish the following Lemma.

What is the H2H proposition?

Prove the following proposition.

What is the D2D proposition?

What is the corollary D2D٭?

What is the H2D proposition?