A first course in complex differentiation can give the misleading impression that complex analysis is a local theory: a derivative is tested by zooming in near a point, and the [Cauchy-Riemann equations](/page/Cauchy-Riemann%20Equations) are checked by comparing infinitesimal directions. Complex integration reveals the opposite force. Once a function is holomorphic, the value of an integral around a closed curve remembers holes, singularities, winding, and global geometry.

The basic surprise is that integrating around a loop often gives zero, but not always. The function

\begin{align*} f:\mathbb C\setminus\{0\} &\to \mathbb C \\ z &\mapsto \frac{1}{z} \end{align*}

is holomorphic, and yet the unit circle detects the missing origin:

[example: The Unit Circle Detects a Hole] Let $\gamma:[0,2\pi]\to \mathbb C\setminus\{0\}$ be the unit-circle path \begin{align*} \gamma(t)=e^{it}, \end{align*} and let \begin{align*} f:\mathbb C\setminus\{0\}&\to\mathbb C,\\ z&\mapsto \frac{1}{z}. \end{align*} Since $\gamma'(t)=ie^{it}$, the definition of the complex line integral gives \begin{align*} \int_\gamma f(z)\,dz &=\int_0^{2\pi} f(\gamma(t))\gamma'(t)\,dt\\ &=\int_0^{2\pi} \frac{1}{e^{it}}\,ie^{it}\,dt\\ &=\int_0^{2\pi} i\,\frac{e^{it}}{e^{it}}\,dt\\ &=\int_0^{2\pi} i\,dt\\ &=i(2\pi-0)\\ &=2\pi i. \end{align*} The nonzero value shows that this closed integral is not determined only by the local holomorphicity of $f$ along the curve: the path winds once around the missing point $0$, where $f$ is not defined. [/example]

This computation is the gateway to the subject. Complex integrals are defined by parametrising curves, but the main theorems explain why many such integrals depend only on endpoints, singularities, or winding numbers. The theory turns path integrals into a method for proving [power series](/page/Power%20Series) expansions, evaluating real integrals, counting zeros, and measuring [analytic continuation](/page/Analytic%20Continuation) around holes.

## Curves and Orientation

### Parametrized Curves

Before we can integrate, we need a controlled class of curves. A completely arbitrary continuous curve may move too wildly for a derivative vector to exist. The line integral needs a velocity along the path, but we still want to allow curves made from finitely many smooth arcs, such as rectangles and polygonal contours.

[definition: Piecewise $C^1$ Path] Let $a,b\in\mathbb R$ with $a<b$. A piecewise $C^1$ path in $\mathbb C$ is a continuous map $\gamma:[a,b]\to\mathbb C$ for which there exist points \begin{align*} a=t_0<t_1<\cdots<t_n=b \end{align*} such that $\gamma|_{[t_{j-1},t_j]}$ is $C^1$ for each $j\in\{1,\ldots,n\}$. [/definition]

The word path keeps track of parametrisation, not merely the geometric trace. Two different parametrisations can travel over the same set at different speeds, and a closed path is the first case where the integral has no endpoint displacement to measure. We therefore name the closed case separately before studying what a loop can detect.

[definition: Closed Path] A piecewise $C^1$ path $\gamma:[a,b]\to\mathbb C$ is closed if \begin{align*} \gamma(a)=\gamma(b). \end{align*} [/definition]

### Closed Loops and Reversal

Closed paths are often called contours when they are used as integration paths. Since a contour may be travelled clockwise or counterclockwise, we need a formal operation that changes the orientation while keeping the same trace. This operation is essential because contour integrals must remember direction.

[definition: Reverse Path] Let $\mathcal P_{a,b}$ be the set of piecewise $C^1$ paths $\gamma:[a,b]\to\mathbb C$. The path reversal map is \begin{align*} R:\mathcal P_{a,b} &\to \mathcal P_{a,b} \\ \gamma &\mapsto \left(t\mapsto \gamma(a+b-t)\right) \end{align*} defined as follows. If $\gamma\in\mathcal P_{a,b}$, then the reverse path $-\gamma=R(\gamma):[a,b]\to\mathbb C$ is given by \begin{align*} (-\gamma)(t)=\gamma(a+b-t). \end{align*} [/definition]

## Definition

The actual integral is now just the real-variable integral of the pullback of $f(z)\,dz$ along the path. This definition is computational, but it is carefully chosen so that the chain rule drives all later structure.

[definition: Complex Line Integral] Let $\Omega\subset\mathbb C$ be open, let $f:\Omega\to\mathbb C$ be continuous, and let $\gamma:[a,b]\to\Omega$ be a piecewise $C^1$ path. The complex line integral of $f$ over $\gamma$ is \begin{align*} I_\Omega:\bigcup_{a<b}\{(f,\gamma): f\in C(\Omega;\mathbb C),\ \gamma:[a,b]\to\Omega\text{ is piecewise }C^1\} &\to \mathbb C \\ (f,\gamma) &\mapsto \int_\gamma f(z)\,dz, \end{align*} with value \begin{align*} \int_\gamma f(z)\,dz = \sum_{j=1}^n \int_{t_{j-1}}^{t_j} f(\gamma(t))\gamma'(t)\,dt, \end{align*} where $a=t_0<t_1<\cdots<t_n=b$ is any subdivision on which $\gamma$ is $C^1$ on each subinterval. [/definition]

Here $dt$ denotes integration with respect to the ordinary real parameter. Some theorem cards write the same one-dimensional integral as $d\mathcal L^1(t)$, where $\mathcal L^1$ is one-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on the parameter interval; for the piecewise continuous integrands in this page, this is the same parameter integral written in measure notation.

The notation hides the parametrisation, but the definition uses it. A useful contour integral must be independent of the speed at which the contour is traversed, otherwise geometric contour methods would depend on arbitrary bookkeeping. The next foundational fact supplies that invariance and records the effect of reversing orientation.