Complex Analysis: Foundations¶

Job: field structure of $\mathbb{C}$, complex differentiability / Cauchy–Riemann, elementary maps, and domain topology used by Cauchy theory and residues.

Core¶

The field $\mathbb{C}$ and the sphere¶

$\mathbb{C}\cong\mathbb{R}^2$ as real vector spaces; multiplication $(x+iy)(u+iv)=(xu-yv)+i(xv+yu)$ makes $\mathbb{C}$ an algebraically closed field (FTA). Modulus $|z|=\sqrt{z\bar z}$; polar form $z=re^{i\theta}$ with $\arg$ multi-valued mod $2\pi$.

Object	Notes
Open set / domain	open connected $U\subseteq\mathbb{C}$
Extended plane $\widehat{\mathbb{C}}$	$\mathbb{C}\cup\{\infty\}\cong S^2$ (Riemann sphere); charts $z$, $1/z$
Chordal metric	neighborhoods of $\infty$; spherical continuity of rational maps
Curves	piecewise $C^1$ paths $\gamma:[a,b]\to\mathbb{C}$; reverse $-\gamma$; cycles

Stereographic projection identifies $\widehat{\mathbb{C}}$ with $S^2$; meromorphic on $\mathbb{C}$ + behavior at $\infty$ $\Leftrightarrow$ meromorphic on $\widehat{\mathbb{C}}$.

Complex derivative and Cauchy–Riemann¶

$f:U\to\mathbb{C}$ is complex differentiable at $z_0$ if $$ f'(z_0)=\lim_{h\to 0}\frac{f(z_0+h)-f(z_0)}{h}\in\mathbb{C} $$ exists independently of how $h\to 0$ in $\mathbb{C}$. Holomorphic on $U$: complex differentiable on a neighborhood of every point of $U$ (some authors reserve “holomorphic” for the $C^1$+CR package and prove analyticity later).

Write $f=u+iv$, $z=x+iy$. Identifying $\mathbb{C}\cong\mathbb{R}^2$:

Theorem (Cauchy–Riemann). If $f'(z_0)$ exists, then $u_x=v_y$, $u_y=-v_x$ at $z_0$, and $f'=u_x+iv_x$. Conversely, if $u,v$ are $C^1$ near $z_0$ and CR holds on a nbhd, then $f$ is holomorphic there. (Weaker pointwise results exist — Looman–Menchoff — but $C^1$+CR is the working hypothesis.)

Real differential: $Df_{z_0}$ is $\mathbb{C}$-linear iff CR. Equiv. $\partial f/\partial\bar z=0$ in Wirtinger calculus $$ \frac{\partial}{\partial z}=\frac12\bigl(\partial_x-i\partial_y\bigr),\qquad \frac{\partial}{\partial\bar z}=\frac12\bigl(\partial_x+i\partial_y\bigr), $$ so $df=(\partial f/\partial z)\,dz+(\partial f/\partial\bar z)\,d\bar z$.

Consequences of holomorphy (preview of later pages). Local power series, Cauchy estimates, identity theorem, open mapping, maximum modulus — all flow from the integral formula once Goursat/Cauchy is available; see analytic functions, integration.

Elementary holomorphic functions¶

Class	Construction / caveat
Polynomials, rationals	poles where denominator vanishes (after canceling)
$\exp$, $\sin$, $\cos$	entire via power series; Euler $e^{iz}=\cos z+i\sin z$
$\operatorname{Log}$	$e^w=z\Rightarrow w=\log
$z^\alpha$	$\exp(\alpha\operatorname{Log} z)$ on a branch
Möbius	$z\mapsto(az+b)/(cz+d)$, $\operatorname{PSL}_2(\mathbb{C})=\operatorname{Aut}(\widehat{\mathbb{C}})$

No entire branch of $\arg$ or $\log$ on $\mathbb{C}\setminus\{0\}$: the covering $\exp:\mathbb{C}\to\mathbb{C}\setminus\{0\}$ is nontrivial; $d\theta$ is closed but not exact.

Local conformal geometry¶

If $f'(z_0)\neq 0$, then $f$ is locally biholomorphic and conformal (angle- and orientation-preserving) at $z_0$. Critical points $f'=0$ are where conformality fails (local $k$-to-$1$ branched behavior if zero of order $k$).

Riemann mapping theorem. Any simply connected domain $U\subsetneq\mathbb{C}$ is biholomorphic to the unit disc $\mathbb{D}$. Carathéodory: extends continuously to the boundary under prime-end / Jordan curve hypotheses. $\operatorname{Aut}(\mathbb{D})\cong\operatorname{PSU}(1,1)$; classification of simply connected Riemann surfaces is the start of uniformization.

Topology of domains¶

Notion	Working meaning for open $U\subseteq\mathbb{C}$
Connected	path-connected (open sets in $\mathbb{R}^2$)
Simply connected	every closed curve null-homotopic; equiv. $\widehat{\mathbb{C}}\setminus U$ connected
Winding number	$n(\gamma,a)=\frac{1}{2\pi i}\int_\gamma\frac{dz}{z-a}\in\mathbb{Z}$
Homology	$\gamma\sim 0$ in $U$ if $n(\gamma,a)=0$ all $a\notin U$ (Cauchy’s theorem in homology form)

Winding numbers package orientation and multiplicity for the residue theorem and argument principle — residue theory.

See it¶

Möbius geometry. $\operatorname{Aut}(\widehat{\mathbb{C}})=\operatorname{PSL}_2(\mathbb{C})$ acts triply transitively on $\widehat{\mathbb{C}}$. Cross-ratio is the fundamental invariant. $\operatorname{Aut}(\mathbb{D})$ and $\operatorname{Aut}(\mathbb{H})$ are conjugate; hyperbolic geometry on $\mathbb{D}$ is complex analysis with the Poincaré metric. Uniformization of Riemann surfaces begins with these model geometries ($\widehat{\mathbb{C}}$, $\mathbb{C}$, $\mathbb{D}$).

CR as integrability. In one variable, CR is a single real PDE pair; in several variables, $\bar\partial f=0$ is overdetermined and drives SCV. Already here, writing $f_{\bar z}=0$ is the right notation for generalization.

Check¶

Verify CR for $f(z)=z^2$ and failure for $f(z)=\bar z$.

$z^2=(x^2-y^2)+2ixy$: $u_x=2x=v_y$, $u_y=-2y=-v_x$. For $\bar z=x-iy$: $u_x=1$, $v_y=-1$, CR fails everywhere; equivalently $f_{\bar z}=1\neq 0$.

Why is there no continuous branch of $\arg$ on $\mathbb{C}\setminus\{0\}$?

A continuous $\arg$ would be a continuous section of $z\mapsto z/|z|$ into $\mathbb{R}/2\pi\mathbb{Z}$, lifting through the covering $\mathbb{R}\to S^1$. No global continuous section exists; equivalently $\int_{|z|=1}d\theta=2\pi\neq 0$ so $d\theta$ is not exact.

If $f$ is holomorphic on a domain and $|f|$ is constant, show $f$ is constant.

$|f|^2=f\bar f$ constant $\Rightarrow$ differentiate: if $f\not\equiv 0$, CR / Wirtinger gives $f'\bar f=0$ hence $f'=0$ on the open set where $f\neq 0$, so $f$ constant there; identity theorem (or continuity) finishes. Alternatively maximum modulus applied to $f$ and $1/f$ when $f\neq 0$.

Pitfalls¶

Partials exist + CR at a single point $\not\Rightarrow$ complex differentiable there without extra assumptions (continuity of partials, or Looman–Menchoff-type hyp.).
$|f|$ or $\bar f$ holomorphic on a domain $\Rightarrow$ $f$ constant (if $f$ itself is holomorphic for the $|f|$ statement).
Infinity: a polynomial of degree $d$ has a pole of order $d$ at $\infty$; $e^z$ has an essential singularity at $\infty$.
Simply connected characterizations differ in general topology; stick to the plane equivalences above for contour integration.
Real $C^1$ $\neq$ holomorphic: $f(z)=\bar z$ is smooth as a map $\mathbb{R}^2\to\mathbb{R}^2$ but nowhere complex differentiable.

Refs¶

Ahlfors, Complex Analysis
Conway, Functions of One Complex Variable I
Stein–Shakarchi, Complex Analysis
Needham, Visual Complex Analysis (geometric intuition)
Rudin, Real and Complex Analysis

Next¶

Analytic functions · Complex integration · Residue theory · Complex analysis overview · SCV · Topology foundations