Notes in Geometric Complex Analysis

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Status	Last Update	Fields
Published	05/05/2025	What does it mean that $f$ be a Schlicht map?
Published	05/05/2025	Which map maps $\mathbb{D}$ to the cardioid?
Published	05/05/2025	List as many transformations of a schlicht map $f$ as you can which give a new Schlicht map.
Published	05/05/2025	What is $\Sigma$? What is a notable subset thereof?
Published	05/05/2025	What is the relation between maps from $\Sigma'$ and $\mathcal{S}$? Why do we not use $\Sigma$?
Published	05/05/2025	What are full mappings?
Published	05/05/2025	What is the area theorem?
Published	05/05/2025	For which values of $n$ is $\|b_n\| \leq \frac{1}{\sqrt{n}}$ sharp? Why? Here $b_n$ is a coefficient in the series at 0 for a map in the class $\Sigma$.
Published	05/05/2025	State the distortion theorem in words and precisely.
Published	05/05/2025	Remind me of the growth theorem
Published	05/05/2025	How do you prove theorem 6.10, the combined growth and distortion theorem?$$\frac{1-\|z\|}{1+\|z\|}\leq \left\| \frac{zf'(z)}{f(z)}\right\| \leq \frac{1+\|z\|…
Published	05/05/2025	What is the radial distortion theorem?
Published	05/08/2025	What is interesting about the map $\frac{z}{(1-z)^2}$
Published	05/05/2025	Name some interesting conformal maps and their domains and codomains.
Published	05/05/2025	State the Schwarz lemma in its entirety:
Published	05/05/2025	State Hurwitz's theorem and give an indication for the proof.
Published	05/08/2025	What is the dilatation quotient?
Published	05/05/2025	What is Weyl's lemma?
Published	05/05/2025	What is the general form of an automorphism of $\mathbb{D}$?
Published	05/05/2025	If $f: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ is a map with $f(\infty) = \infty$, then under what condition is $f$ continuous/holomorphic at $\infty$?
Published	05/05/2025	If $f: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ is a map with $f(\infty) \neq \infty$, then under what condition is $f$ continuous/holomorphic at $\inft…
Published	05/05/2025	If $f: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ is a map with $f(a) = \infty$, with $a \neq \infty$, then under what condition is $f$ continuous/holomor…
Published	05/05/2025	If $f: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ is holomorphic, then what form must it take?
Published	05/05/2025	What is the Poincare metric on $\mathbb{D}$?
Published	05/05/2025	For what conformal metric $\rho$ is every automoprhism of $\mathbb{D}$ an isometry of?
Published	05/05/2025	What is the closed form of the metric $d_{\rho}$ induced by the Poincare metric on $\mathbb{D}$?
Published	05/05/2025	What is the Farkas-Ritt theorem?
Published	05/05/2025	What is the Koebe map?
Published	05/05/2025	If $f_{n} \to f$ uniformly on compact subsets as $n \to \infty$, then what can we say about $f'_{n}$ as $n \to \infty$? Why do we not discuss fur…
Published	05/05/2025	What does it mean for a family of holomorphic maps to be uniformly bounded on compact subsets?
Published	05/05/2025	What does it mean for a family of holomorphic maps to be equicontinuous on a subset?
Published	05/05/2025	What is Montels Theorem?
Published	05/05/2025	Let $\Omega \subseteq \mathbb{C}$ be open. What is an exhaustion of $\Omega$ by compact sets? Do they always exist?
Published	05/05/2025	If $\mathcal{F}$ is a family of holomorphic maps defined on an open subset $\Omega \in \mathbb{C}$, and $(E_{i})_{i \in \mathbb{N}}$ is an exhastion o…
Published	05/05/2025	When is a family of holomorphic maps $\mathcal{F}$ compact with respect to the $d''$ metric?
Published	05/05/2025	What are the conditions on $\Omega \subset \mathbb{C}$ for there to be a biholomorphism between $\mathbb{D}$ and $\Omega$?
Published	05/05/2025	Exercise 2.2 offers two results known as the Schwarz-Pick lemma about maps $f:\mathbb{D}\to \mathbb{D}$ which are holomorphic. What are they?
Published	05/05/2025	Exercise 2.3 was a useful resulr for CW 1 about holomorphic endomorphisms of $\mathbb{H}$.
Published	05/05/2025	Suppose that $g:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$ is such that $g(\mathbb{C})\subseteq \mathbb{C}$. What can we say about $g$?
Published	05/05/2025	What strategy ought one to use when considering how circles and lines change under mobius transformations$?$
Published	05/05/2025	Name a metric on $\mathbb{D}$ which is invariant under $\text{Aut}(\mathbb{D})$ that is \emph{not} the Poincare metric.
Published	05/05/2025	Exercise 6.6 was an intriguing result about convex subsets. Do you recall what it was?
Published	05/05/2025	Exercise 6.5 was a result proven on the CW2 about the conformal radius. Define this concept and state the related result.
Published	05/05/2025	State the complex chain rules on exercise 7.2
Published	05/05/2025	Can you do exercise 7.6?
Published	05/05/2025	What is a $K$ quasi conformal map?
Published	05/05/2025	What is ACL on lines?
Published	05/05/2025	What si the Pompeiu formula?
Published	05/05/2025	What is a stble point in the riemann sphere for a holomorphic map thereon?
Published	05/05/2025	What is a wandering component?
Published	05/08/2025	What do we know about $z$'s stability?
Published	05/08/2025	True or false $R^{-1}(\mathcal{F}(R)) = \mathcal{F}(R)$?
Status	Last Update	Fields