The Complex Plane: Field, Metric, and Limits

Before differentiating anything we need the object being differentiated: the field $\mathbb{C}$, its geometry as the plane ${\mathbb{R}}^2$, and the notions of limit and completeness on which every later page silently relies. This page constructs $\mathbb{C}$, fixes the metric and topology used throughout the folder, grounds the complex limit in the real analysis spine, and adds the point at infinity — the Riemann sphere — that the global theory needs. It opens the Complex Analysis folder; everything downstream cites it for "open set", "limit", and "$\infty$".

The complex field

Take ${\mathbb{R}}^2$ with its usual addition and define a multiplication $$(a,b)\cdot (c,d)=(ac-bd,ad+bc).$$ Writing $1=(1,0)$ and $i=(0,1)$ gives $i^2=-1$ and the familiar $z=a+bi$. The result is a field $\mathbb{C}$: multiplication is commutative, associative, distributes over addition, and every $z\ne 0$ has an inverse. Two equivalent constructions worth keeping in mind:

• As ${\mathbb{R}}^2$ — a two-dimensional real vector space with a compatible product; this is the geometric picture (the Argand plane).
• As a quotient ring $\mathbb{C}\cong \mathbb{R}[x]/(x^2+1)$ — adjoin a formal root of $x^2+1$ to $\mathbb{R}$. This makes manifest that $\mathbb{C}$ is the algebraic closure of $\mathbb{R}$ (the fundamental theorem of algebra, proved later, says one root suffices to close $\mathbb{R}$ completely).

$\mathbb{C}$ is a field but not an ordered field: no order makes it compatible with the arithmetic (an ordered field has $x^2\ge 0$, yet $i^2=-1<0$). Order is replaced by the size function below.

Modulus and conjugate

The conjugate and modulus of $z=a+bi$ are $$\bar{z}=a-bi,|z|=\sqrt{a^2+b^2}=\sqrt{z\bar{z}}.$$ Conjugation is the field automorphism fixing $\mathbb{R}$; it satisfies $\overline{z+w}=\bar{z}+\bar{w}$, $\overline{zw}=\bar{z}\bar{w}$, and $\mathrm{Re}z=\frac{1}{2}(z+\bar{z})$, $\mathrm{Im}z=\frac{1}{2i}(z-\bar{z})$. The modulus is multiplicative, $|zw|=|z||w|$, gives inverses $z^{-1}=\bar{z}/|z{|}^2$, and satisfies the triangle inequality $$|z+w|\le |z|+|w|,||z|-|w||\le |z-w|.$$ Thus $d(z,w)=|z-w|$ is a metric — and it is exactly the Euclidean metric of ${\mathbb{R}}^2$. Every metric-space fact the folder uses is therefore imported, not re-proved, from metric-spaces.md.

Polar form and roots of unity

Writing $z=r{e}^{i\theta }$ with $r=|z|$ and $\theta =\arg z$ (defined mod $2\pi$) turns multiplication into rotate-and-scale: $$z_1{z}_2=r_1{r}_2{e}^{i({\theta }_1+{\theta }_2)},z^n=r^n{e}^{in\theta }\text{(de Moivre)}.$$ Every $z\ne 0$ has exactly $n$ $n$-th roots, the vertices of a regular $n$-gon; the $n$-th roots of unity ${\zeta }_k=e^{2\pi ik/n}$ form a cyclic group under multiplication. The multivaluedness of $\arg z$ — and hence of $\log z$ and $z^{\alpha }$ — is previewed in holomorphic.md and made geometric in riemann-surfaces.md.

Topology and completeness of the plane

Because $(\mathbb{C},|\cdot |)$ is $({\mathbb{R}}^2,\Vert \cdot \Vert )$ as a metric space, its point-set topology is imported wholesale from metric-spaces.md:

• open / closed sets, the open disc $D(z_0,r)=\{z:|z-z_0|<r\}$ as the basic neighbourhood, and the induced notions of interior, closure, and boundary;
• connectedness — a domain is an open connected set (the standing hypothesis of nearly every theorem in the folder); path-connectedness coincides with connectedness for open subsets of $\mathbb{C}$;
• compactness — by Heine–Borel, a subset of $\mathbb{C}$ is compact iff it is closed and bounded;
• completeness — $\mathbb{C}$ is a complete metric space: every Cauchy sequence converges (inherited from the completeness of $\mathbb{R}$, real-numbers.md).

Definition (domain). A domain $U\subseteq \mathbb{C}$ is a non-empty open connected set. "Holomorphic on $U$" always presupposes $U$ is open; most global theorems add connectedness, and the strongest add simple connectivity (${\pi }_1(U)=0$, from topology-manifolds.md).

Sequences and limits

A sequence $z_n\to z$ iff $|z_n-z|\to 0$, equivalently iff $\mathrm{Re}{z}_n$ and $\mathrm{Im}{z}_n$ converge separately — so complex convergence is just paired real convergence, and the tests of sequences-series.md transfer. For a function, ${\lim }_{z\to z_0}f(z)=L$ has the usual $\epsilon$–$\delta$ meaning of continuity.md, with one crucial difference: $z$ may approach $z_0$ from any direction in the plane, and the limit must be the same along all of them. This two-dimensional approach is weak as a continuity condition but, applied to the difference quotient, becomes the severe constraint that defines holomorphic functions. A series $\sum z_n$ converges absolutely iff $\sum |z_n|<\infty$; absolute convergence implies convergence (completeness), underwriting every power series in the folder.

The extended plane and the Riemann sphere

Adjoining a single point at infinity gives the extended complex plane $\hat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}$. Stereographic projection identifies it with the unit sphere $S^2$ — the Riemann sphere — sending $\infty$ to the north pole; the induced chordal metric makes $\hat{\mathbb{C}}$ a compact metric space (the one-point compactification of $\mathbb{C}$).

Convention. $f$ is holomorphic at $\infty$ if $f(1/w)$ is holomorphic at $w=0$; it has a pole/zero at $\infty$ according to $f(1/w)$ at $0$. A function meromorphic on all of $\hat{\mathbb{C}}$ is exactly a rational function.

The Riemann sphere is the natural home of Möbius transformations (its conformal automorphisms), the simplest compact Riemann surface, and the target that lets us speak of poles as honest values $\infty$ rather than singularities. It is used throughout Sections C–F.

References

• Ahlfors, Complex Analysis, Ch. 1 (the complex field, the sphere).
• Stein & Shakarchi, Complex Analysis, Ch. 1.
• Needham, Visual Complex Analysis, Ch. 1 (geometry of $\mathbb{C}$ and the sphere).