Chapter 1 • Theory & Derivations

Metric Topology of the Complex Plane, Stereographic Projection & Power Series

Point-set topology of C, open disks, connectedness, stereographic projection onto the Riemann sphere, chordal metric, complex sequences, and power series convergence.

§1.1Complex Plane C, Metric Topology, Open/Connected Domains & Jordan Curve Theorem

1. The Complex Field and Metric Structure

The set of complex numbers $\mathbb{C} = \{z = x + iy : x, y \in \mathbb{R}, i^2 = -1\}$ forms an algebraically closed complete field. Endowed with the Euclidean modulus metric:

$$d(z_1, z_2) = |z_1 - z_2| = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$

$\mathbb{C}$ is isometric to $\mathbb{R}^2$ as a real metric space, but possesses the richer structure of complex multiplication.

An open $\epsilon$-disk centered at $z_0 \in \mathbb{C}$ is denoted by $D(z_0, \epsilon) = \{z \in \mathbb{C} : |z - z_0| < \epsilon\}$. A subset $\Omega \subseteq \mathbb{C}$ is called:

Open: if for every $z \in \Omega$, there exists $\epsilon > 0$ such that $D(z, \epsilon) \subseteq \Omega$.
Connected: if it cannot be partitioned into two disjoint, non-empty open sets. In $\mathbb{C}$, an open set is connected if and only if it is path-connected (any two points can be connected by a polygonal path lying entirely within $\Omega$).
Domain: an open, connected non-empty subset $D \subseteq \mathbb{C}$.
Simply Connected Domain: a domain $D$ without holes; every closed continuous loop $\gamma \subset D$ can be continuously contracted to a point within $D$.

2. The Jordan Curve Theorem

Theorem 1.1 (Jordan Curve Theorem): Let $\gamma: [0, 1] \to \mathbb{C}$ be a simple (non-self-intersecting) closed continuous curve (a Jordan curve). Then its complement $\mathbb{C} \setminus \gamma$ consists of exactly two connected components:

A bounded open domain called the Interior $\text{Int}(\gamma)$.
An unbounded open domain called the Exterior $\text{Ext}(\gamma)$.

The curve $\gamma$ is the common topological boundary of both components.

§1.2Stereographic Projection, The Riemann Sphere Ĉ and Chordal Metric

1. The Extended Complex Plane & The Riemann Sphere

To analyze limits at infinity in complex analysis, we compactify $\mathbb{C}$ by adjoining a single point at infinity, $\infty$, creating the extended complex plane:

$$\widehat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$$

Topologically, $\widehat{\mathbb{C}}$ is homeomorphic to the unit 2-sphere $S^2 = \{(X, Y, Z) \in \mathbb{R}^3 : X^2 + Y^2 + Z^2 = 1\}$ via stereographic projection from the North Pole $N = (0, 0, 1)$.

2. Explicit Coordinate Transformations

A ray joining the North Pole $N(0, 0, 1)$ to a point $P(X, Y, Z) \in S^2$ intersects the equatorial plane $Z = 0$ (identified with $\mathbb{C}$ via $z = x + iy$) at:

$$x = \frac{X}{1 - Z}, \quad y = \frac{Y}{1 - Z} \implies z = \frac{X + iY}{1 - Z}$$

Conversely, for any $z = x + iy \in \mathbb{C}$, the unique point on the sphere $S^2$ is:

$$X = \frac{2x}{|z|^2 + 1} = \frac{z + \bar{z}}{|z|^2 + 1}, \quad Y = \frac{2y}{|z|^2 + 1} = \frac{z - \bar{z}}{i(|z|^2 + 1)}, \quad Z = \frac{|z|^2 - 1}{|z|^2 + 1}$$

As $|z| \to \infty$, $Z \to 1$, mapping the point at infinity $\infty$ directly to the North Pole $N(0, 0, 1)$.

3. The Chordal Metric

The chordal distance $\chi(z_1, z_2)$ is the 3D Euclidean distance between their spherical projections:

$$\chi(z_1, z_2) = \frac{2|z_1 - z_2|}{\sqrt{1 + |z_1|^2}\sqrt{1 + |z_2|^2}}, \quad \chi(z, \infty) = \frac{2}{\sqrt{1 + |z|^2}}$$

Under $\chi$, $\widehat{\mathbb{C}}$ is a compact metric space.

§1.3Complex Sequences, Infinite Series, Radius of Convergence & Cauchy-Hadamard Formula

1. Complex Sequences and Series

A sequence $\{z_n\} \subset \mathbb{C}$ converges to $L = \alpha + i\beta$ if and only if $\text{Re}(z_n) \to \alpha$ and $\text{Im}(z_n) \to \beta$ as $n \to \infty$. Completeness of $\mathbb{R}^2$ guarantees that every Cauchy sequence in $\mathbb{C}$ converges.

2. Complex Power Series and The Cauchy-Hadamard Theorem

Consider the formal power series centered at $z_0 \in \mathbb{C}$:

$$\sum_{n=0}^\infty a_n (z - z_0)^n$$

Theorem 1.2 (Cauchy-Hadamard Formula): The radius of convergence $R \in [0, \infty]$ of the power series is given by: $$\frac{1}{R} = \limsup_{n \to \infty} \sqrt[n]{|a_n|}$$

If $|z - z_0| < R$, the series converges absolutely and locally uniformly to an analytic function.
If $|z - z_0| > R$, the series diverges.
On the boundary circle $|z - z_0| = R$, the series may converge at some points and diverge at others.

TIERED UNIVERSITY HONORS PROBLEMS

Step-by-Step Solved Examination Problems

Comprehensive analytical derivations, multi-tier solutions (Foundational, Intermediate Exam, and Honors/Proof Challenge) with complete line-by-line verification.

Tier 1 • Foundational Radius of Convergence of Complex Power Series

Find the exact radius of convergence of the complex power series:
$$\sum_{n=1}^\infty \frac{(3 + 4i)^n}{n^2} z^n$$

Tier 1 • Foundational Taylor Series & Exact Radius of Convergence

Find the Taylor series expansion of $f(z) = \frac{1}{z^2 - 4}$ about $z_0 = 0$ and determine its exact radius of convergence.

Tier 3 • Honors Challenge Analytic Continuation & Euler's Reflection Formula for Gamma Function

Derive Euler's reflection formula $\Gamma(z)\Gamma(1 - z) = \frac{\pi}{\sin(\pi z)}$ via contour integration.