Imagine you are watching a satellite orbit the Earth. From your vantage point at the centre, the satellite traces a closed path in the sky. After one full orbit, it has gone around you exactly once. But what if the orbit is more complicated — perhaps the satellite weaves in and out, crossing its own path, doubling back? How many times has it truly gone around you? This question, deceptively simple to ask, leads to one of the most fundamental invariants in complex analysis: the *winding number*.

The winding number formalises the intuitive notion of "how many times a closed curve encircles a point." Its power becomes apparent when you realise that this purely topological-looking quantity is computable via a contour integral, and that its value governs which functions can be integrated over which contours, whether a given point lies inside or outside a curve, and — through the argument principle — how many zeros and poles a meromorphic function has inside a region. The winding number is the invisible integer that organises all of residue theory.

[example: A Figure-Eight Does Not Enclose Points Uniformly] Consider the figure-eight curve $\gamma: [0, 2\pi] \to \mathbb{C}$ given by $\gamma(t) = \sin(t) + i\sin(2t)/2$. This curve crosses itself at the origin and traces two loops, one traversed counterclockwise and one clockwise. A point inside the upper loop, say $z_0 = 0.3i$, is encircled once counterclockwise. A point inside the lower loop, say $z_1 = -0.3i$, is encircled once clockwise — that is, with winding number $-1$. But the origin itself, through which the curve passes, is neither strictly inside nor outside either loop in the naive sense. This example already shows that "inside" and "outside" are not binary concepts for general closed curves. Different points can be surrounded different numbers of times, and the sign carries meaning: counterclockwise is positive, clockwise is negative. To verify: as $t$ increases from $0$ to $\pi$, the curve $\gamma$ traces the upper loop counterclockwise. For $z_0 = 0.3i$, the vector $\gamma(t) - z_0$ starts pointing roughly rightward, rotates counterclockwise through a full turn, and returns to its starting direction — one full positive rotation, so $n(\gamma, 0.3i) = +1$. As $t$ increases from $\pi$ to $2\pi$, the curve traces the lower loop clockwise. For $z_1 = -0.3i$, the vector $\gamma(t) - z_1$ rotates clockwise through a full turn, giving $n(\gamma, -0.3i) = -1$. Neither loop winds around the other loop's interior point, so the outside of each loop contributes zero net rotation. [/example]

## Definition

Before giving the formal definition, let us think about what we need to measure. Given a closed curve $\gamma$ and a point $z_0$ not on $\gamma$, we want to count the net number of times $\gamma$ winds around $z_0$. The key idea is to track the argument of $\gamma(t) - z_0$ as $t$ runs from the start to the end of $\gamma$. Since $\gamma$ is closed, this argument returns to its starting value modulo $2\pi$, and the integer multiple of $2\pi$ that it has accumulated is the winding number.

To make this rigorous, we use the contour integral. The expression $\frac{1}{z - z_0}$ is holomorphic away from $z_0$, and its integral over a closed curve detects how many times that curve goes around $z_0$ — this is precisely what the residue theorem will later tell us, but we can define the winding number directly from the integral without invoking residues.

[definition: Winding Number] Let $\gamma: [a, b] \to \mathbb{C}$ be a piecewise $C^1$ closed curve, so that $\gamma(a) = \gamma(b)$. Let $z_0 \in \mathbb{C} \setminus \operatorname{im}(\gamma)$. The **winding number** of $\gamma$ around $z_0$ is the integer \begin{align*} n(\gamma, z_0) &= \frac{1}{2\pi i} \oint_\gamma \frac{dz}{z - z_0}. \end{align*} [/definition]

That $n(\gamma, z_0)$ is always an integer requires proof. Writing $\gamma(t) - z_0 = r(t) e^{i\theta(t)}$ for a continuous choice of argument $\theta(t)$, the integral becomes

\begin{align*} \frac{1}{2\pi i} \int_a^b \frac{\gamma'(t)}{\gamma(t) - z_0}\, dt &= \frac{1}{2\pi i} \int_a^b \left(\frac{r'(t)}{r(t)} + i\theta'(t)\right) dt \\ &= \frac{1}{2\pi i}\left[\log r(t)\right]_a^b + \frac{\theta(b) - \theta(a)}{2\pi}. \end{align*}

Since $\gamma$ is closed, $r(a) = r(b)$, so the first term vanishes. The second term is $(\theta(b) - \theta(a))/(2\pi)$, which counts the net number of full rotations of the argument. This is an integer because the curve is closed and the argument returns to the same value modulo $2\pi$.

[remark: Orientation Convention] Throughout this page, curves are traversed with their specified orientation. The standard orientation for a circle is counterclockwise, which gives winding number $+1$ for points in the interior. Reversing the curve — writing $-\gamma$ for the curve traversed in the opposite direction — negates the winding number: $n(-\gamma, z_0) = -n(\gamma, z_0)$. [/remark]

The simplest and most important case to compute directly is the unit circle: it confirms that the integral definition matches the intuitive count of one full revolution.

[example: Winding Number of a Circle] Let $\gamma_r: [0, 2\pi] \to \mathbb{C}$ be the circle of radius $r > 0$ centred at $a \in \mathbb{C}$, parametrised counterclockwise: \begin{align*} \gamma_r(t) &= a + r e^{it}, \quad t \in [0, 2\pi]. \end{align*} For $z_0 = a$ (the centre), we compute: \begin{align*} n(\gamma_r, a) &= \frac{1}{2\pi i} \oint_{\gamma_r} \frac{dz}{z - a} = \frac{1}{2\pi i} \int_0^{2\pi} \frac{i r e^{it}}{r e^{it}}\, dt = \frac{1}{2\pi i} \int_0^{2\pi} i\, dt = \frac{1}{2\pi i} \cdot 2\pi i = 1. \end{align*} For $z_0$ with $|z_0 - a| > r$ (outside the circle), the function $\frac{1}{z - z_0}$ is holomorphic on the closed disk $\overline{B}(a, r)$, so by Cauchy's theorem the integral is zero: $n(\gamma_r, z_0) = 0$. These two computations — winding number $1$ inside, winding number $0$ outside — are the prototype for the general theory. [/example]

## Topological Nature of the Winding Number

The winding number is not just an analytic quantity — it is a topological invariant of the pair $(\gamma, z_0)$. This means it does not change under deformations of the curve that avoid the point $z_0$. Understanding why this is so explains why the winding number is such a robust concept.

The key topological fact is that the function $z_0 \mapsto n(\gamma, z_0)$ is locally constant on $\mathbb{C} \setminus \operatorname{im}(\gamma)$. This is what allows us to speak of the winding number of a curve around a *region*, not just a single point.

[quotetheorem:3360]

This theorem formalises the idea that winding number changes only when the point crosses the curve. Within a connected region of the complement, every point sees the same net encircling behavior. The unbounded component is the outside region: points very far away are not enclosed by the curve, so their winding number is zero throughout that whole outside component.

[illustration:winding-number-local-constancy]

[explanation: Why the Outside Region Has Winding Number Zero]