Path Independence and Primitives

A central question in complex integration is: when does the integral $\int_\gamma f(z)\,dz$ depend only on the endpoints and not on the particular path? The answer connects analytic functions, simply connected domains, and the existence of antiderivatives.

Path Independence

Definition3.4Path independence

A continuous function $f: D \to \mathbb{C}$ on a domain $D$ is said to have path-independent integrals if for any two contours $\gamma_1, \gamma_2$ in $D$ with the same initial and terminal points, $\int_{\gamma_1} f\,dz = \int_{\gamma_2} f\,dz$ .

Equivalently, $f$ has path-independent integrals if and only if $\oint_\gamma f\,dz = 0$ for every closed contour $\gamma$ in $D$ .

Theorem3.3Equivalence of path independence and existence of primitive

Let $f: D \to \mathbb{C}$ be continuous on a domain $D$ . The following are equivalent:

$f$ has path-independent integrals in $D$ .
$\oint_\gamma f(z)\,dz = 0$ for every closed contour $\gamma$ in $D$ .
$f$ has a primitive (antiderivative) on $D$ : there exists holomorphic $F: D \to \mathbb{C}$ with $F' = f$ .

RemarkConstructing the primitive

When path independence holds, the primitive is constructed explicitly by fixing $z_0 \in D$ and defining

$F(z) = \int_{z_0}^{z} f(w)\,dw$

where the integral is taken along any contour in $D$ from $z_0$ to $z$ . The holomorphicity of $F$ follows from the definition of the complex derivative.

Simply Connected Domains

Definition3.5Simply connected domain

A domain $D \subseteq \mathbb{C}$ is simply connected if every closed curve in $D$ can be continuously deformed to a point within $D$ . Equivalently, $D$ is simply connected if $\mathbb{C} \setminus D$ is connected (in the Riemann sphere $\hat{\mathbb{C}}$ ).

Informally, $D$ is simply connected if it has "no holes."

ExampleSimply connected and non-simply connected domains

$\mathbb{C}$ , any open disk $D(z_0, r)$ , any half-plane, and any convex domain are simply connected.
$\mathbb{C} \setminus \{0\}$ is not simply connected: a circle around the origin cannot be shrunk to a point.
An annulus $\{z : r < |z| < R\}$ is not simply connected.
A slit plane $\mathbb{C} \setminus (-\infty, 0]$ is simply connected, enabling a single-valued branch of $\log z$ .

Cauchy's Theorem for Simply Connected Domains

Theorem3.4Cauchy's theorem (simply connected version)

If $f$ is holomorphic on a simply connected domain $D$ , then

$\oint_\gamma f(z)\,dz = 0$

for every closed contour $\gamma$ in $D$ . Consequently, $f$ has a primitive on $D$ .

RemarkThe role of topology

Cauchy's theorem reveals a deep interplay between analysis (holomorphicity) and topology (simple connectivity). The failure of Cauchy's theorem on non-simply connected domains is precisely measured by residue theory: $\oint_\gamma f\,dz = 2\pi i \sum \text{Res}(f, z_k)$ .

ExampleLogarithm on a slit plane

On the slit plane $D = \mathbb{C} \setminus (-\infty, 0]$ , the function $1/z$ is holomorphic and $D$ is simply connected. By Cauchy's theorem, $1/z$ has a primitive on $D$ , namely the principal branch of the logarithm:

$\text{Log}(z) = \ln|z| + i\text{Arg}(z), \quad -\pi < \text{Arg}(z) < \pi.$

This is well-defined and holomorphic on $D$ with $(\text{Log}\,z)' = 1/z$ .

Deformation of Contours

Definition3.6Homotopy of curves

Two closed curves $\gamma_0, \gamma_1$ in a domain $D$ are homotopic if there exists a continuous map $H: [0,1] \times [0,1] \to D$ such that $H(t,0) = \gamma_0(t)$ , $H(t,1) = \gamma_1(t)$ , and $H(0,s) = H(1,s)$ for all $s$ .

Theorem3.5Deformation invariance

If $f$ is holomorphic on a domain $D$ and $\gamma_0, \gamma_1$ are homotopic closed curves in $D$ , then

$\oint_{\gamma_0} f(z)\,dz = \oint_{\gamma_1} f(z)\,dz.$

ExampleDeforming around a singularity

Let $f(z) = 1/z$ and let $\gamma_R$ be the circle $|z| = R$ for any $R > 0$ . All these circles are homotopic in $\mathbb{C} \setminus \{0\}$ , so

$\oint_{|z|=R} \frac{dz}{z} = 2\pi i$

for all $R > 0$ . However, a circle not enclosing the origin is homotopic to a point, giving integral zero.

Winding Number

RemarkWinding number

For a closed curve $\gamma$ in $\mathbb{C} \setminus \{z_0\}$ , the winding number (or index) of $\gamma$ around $z_0$ is

$n(\gamma, z_0) = \frac{1}{2\pi i} \oint_\gamma \frac{dz}{z - z_0}.$

This is always an integer and counts the number of times $\gamma$ winds around $z_0$ (with sign indicating direction). The winding number is a homotopy invariant and plays a central role in the general form of Cauchy's integral formula.