Complex Integration¶

Job: contour integrals, Cauchy–Goursat theory, integral formulas, and homotopy invariance — the engine that upgrades differentiability to analyticity.

Core¶

Contour integrals¶

For piecewise $C^1$ path $\gamma:[a,b]\to\mathbb{C}$ and continuous $f$ along $\gamma$, $$ \int_\gamma f(z)\,dz:=\int_a^b f(\gamma(t))\gamma'(t)\,dt. $$ Linear in $f$; reverses under orientation reverse; ML-inequality: $\bigl|\int_\gamma f\bigr|\le \mathrm{length}(\gamma)\cdot\sup_\gamma\|f\|$.

Primitive: $F'=f$ on domain $U$ $\Rightarrow$ $\int_\gamma f=F(\gamma(b))-F(\gamma(a))$ (path independent). Existence of global primitive $\Leftrightarrow$ integrals over all closed curves vanish.

Cauchy–Goursat¶

Theorem (Goursat). $f$ holomorphic on a domain containing a triangle $T$ and its interior $\Rightarrow$ $\int_{\partial T}f=0$. (No a-priori continuity of $f'$ needed — historical improvement over Cauchy.)

Theorem (Cauchy integral theorem, local). Holo $f$ $\Rightarrow$ locally has primitives $\Rightarrow$ $\int_\gamma f=0$ for closed $\gamma$ in a disk.

Theorem (Homotopy version). $f$ holo on $U$; closed curves $\gamma_0\simeq\gamma_1$ in $U$ $\Rightarrow$ $\int_{\gamma_0}f=\int_{\gamma_1}f$. Simply connected $U$ $\Rightarrow$ all closed integrals vanish $\Rightarrow$ global primitives.

Deformation: integrals unchanged under continuous deformation of contour within holo region (away from singularities when meromorphic).

Cauchy integral formula¶

Theorem. $f$ holo on and inside simple positively oriented contour $\gamma$, $a$ inside: $$ f(a)=\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{z-a}\,dz,\qquad f^{(n)}(a)=\frac{n!}{2\pi i}\int_\gamma\frac{f(z)}{(z-a)^{n+1}}\,dz. $$

Corollaries: power series expansion; Cauchy estimates; Morera’s theorem (converse of Cauchy: continuous + vanishing triangle integrals $\Rightarrow$ holo); mean value property over circles.

Winding numbers and general Cauchy¶

For closed rectifiable $\gamma$ and $a\notin\gamma$, $$ n(\gamma,a)=\frac{1}{2\pi i}\int_\gamma\frac{dz}{z-a}\in\mathbb{Z}. $$ Global Cauchy formula: $f$ holo on $U$, $\gamma$ homologous to $0$ in $U$ (winding $0$ about all points of $U^c$): $$ n(\gamma,a)f(a)=\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{z-a}\,dz. $$ Homology version packages multiply connected domains (annuli, keyholes).

Estimates and applications¶

Tool	Use
ML / length estimates	vanishing of arcs as $R\to\infty$, $\varepsilon\to 0$
Jordan lemma	semicircle integrals of $e^{iaz}Q(z)$
Differentiation under integral	parametric integrals remain holo

See it¶

Poisson kernel / Schwarz formula. Recover holo function on $\mathbb{D}$ from boundary values of real part: Cauchy formula + conjugate. Links to harmonic functions ($u=\operatorname{Re}f$ is harmonic; local converse on simply connected domains).

Holomorphic functional calculus. For Banach algebra element $a$, $f(a)=\frac{1}{2\pi i}\int_\Gamma f(z)(z-a)^{-1}dz$ — same Cauchy formula, operator-valued.

Check¶

Why $\int_{\|z\|=1}z^n\,dz=0$ for $n\neq -1$ integer, and $2\pi i$ for $n=-1$?

Parametrize $z=e^{it}$, $dz=ie^{it}dt$; integrate $e^{i(n+1)t}$ over $[0,2\pi]$. Only $n=-1$ survives. Laurent coefficient reading.

Show Morera $\Rightarrow$ holomorphicity: outline.

Vanishing on triangles $\Rightarrow$ local primitives via integrating from basepoint in a disk (Goursat/Cauchy construction) $\Rightarrow$ $F$ holo with $F'=f$ $\Rightarrow$ $f$ holo (derivative of holo is holo — from CIF).

Pitfalls¶

Simple connectedness is essential for “all closed integrals vanish”; on $\mathbb{C}\setminus\{0\}$, $\int dz/z=2\pi i$.
Continuity of $f$ on the contour needed for the integral; poles on the path are undefined without principal values.
Orientation: standard positive orientation is counterclockwise (region on the left).
Homotopy vs homology: in planar domains they are closely related for integration of holo forms; higher dim differs.

Refs¶

Ahlfors, Complex Analysis
Conway, Functions of One Complex Variable
Rudin, Real and Complex Analysis

Next¶

Residue theory · Analytic functions · Foundations · Complex analysis overview