The Winding Number

By Sébastien Boisgérault, Mines ParisTech, under CC BY-NC-SA 4.0

September 30, 2019

Contents

Definitions

The argument of a non-zero complex number is only defined modulo \(2\pi.\) A convenient way to describe mathematically this relationship is to associate to any such number the set of admissible values of its argument:

Definition – The Argument Function.

‌ The set-valued (or multi-valued) function \(\mathrm{Arg},\) defined on \(\mathbb{C}^{*}\) by \[ \mathrm{Arg} \, z = \left\{ \theta \in \mathbb{R} \; \left| \; e^{i\theta} = \frac{z}{|z|} \right. \right\}, \] is called the argument function.

If we need a classic single-valued function instead, we have for example:

Definition – Principal Value of the Argument.

‌ The principal value of the argument is the unique continuous function \[ \mathrm{arg}: \mathbb{C} \setminus \mathbb{R}_- \to \mathbb{R} \] such that \[ \arg \, 1 = 0 \] which is a choice of the argument on its domain: \[ \forall \, z \in \mathbb{C} \setminus \mathbb{R}_-, \; \mathrm{arg}\, z \in \mathrm{Arg} \, z. \]

Proof (existence and uniqueness).

‌ Define \(\mathrm{arg}\) on \(\mathbb{C} \setminus \mathbb{R}_- \to \mathbb{R}\) by: \[ \arg (x + i y) = \left| \begin{array}{cl} \arctan y/x & \mbox{if } x > 0, \\ +\pi/2 - \arctan x/y & \mbox{if } y > 0, \\ -\pi/2 - \arctan x/y & \mbox{if } y < 0. \\ \end{array} \right. \] This definition is non-ambiguous: if \(x>0\) and \(y>0,\) we have \[ \arctan x/y + \arctan y/x = \pi/2 \] and a similar equality holds when \(x>0\) and \(y<0.\) As each of the three expressions used to define \(\mathrm{arg}\) has an open domain and is continuous, the function itself is continuous. It is a choice of the argument thanks to the definition of \(\arctan\): for example, if \(x>0,\) with \(\theta=\arg(x+iy),\) we have \[ \frac{\sin\theta}{\cos\theta}=\tan \theta = \tan (\arctan y/x) = \frac{y}{x}, \] hence, as \(\cos \theta > 0\) and \(x>0,\) there is a \(\lambda > 0\) such that \[ x + i y = \lambda (\cos \theta + i \sin \theta) = \lambda e^{i\theta}, \] This equation yields \(\arg x+ i y \in \mathrm{Arg} \, x + iy.\) The proof for the half-planes \(y>0\) and \(y<0\) is similar.

If \(f\) is another continuous choice of the argument on \(\mathbb{C} \setminus \mathbb{R}_-\) such that \(f(1) = 0,\) the image of \(\mathbb{C} \setminus \mathbb{R}_-\) by the difference \(f - \arg\) is a subset of \(2\pi \mathbb{Z}\) that contains \(0,\) and it’s also path-connected as the image of a path-connected set by a continuous function. Consequently, it is the singleton \(\{0\}\): \(f\) and \(\arg\) are equal. \(\blacksquare\)

We cannot avoid the introduction of a cut in the complex plane when we search for a continuous choice of the argument: there is no continuous choice of the argument on \(\mathbb{C}^*.\) However, for a continuous choice of the argument along a path of \(\mathbb{C}^*,\) there is no such restriction:

The following theorem is a special case of the path lifting property (in the context of covering spaces; refer to (Hatcher 2002) for details).

Theorem – Continuous Choice of the Argument.

‌ Let \(a \in \mathbb{C}\) and \(\gamma\) be a path of \(\mathbb{C} \setminus \{a\}.\) Let \(\theta_0 \in \mathbb{R}\) be a value of the argument of \(\gamma(0) - a\): \[ \theta_0 \in \mathrm{Arg}(\gamma(0) - a). \] There is a unique continous function \(\theta:[0,1] \mapsto \mathbb{R}\) such that \(\theta(0) = \theta_0\) which is a choice of \(z \mapsto \mathrm{Arg}(z-a)\) on \(\gamma\): \[ \forall \, t \in [0,1], \; \theta(t) \in \mathrm{Arg}(\gamma(t) - a). \]

Proof.

‌ Let \((x(t), y(t))\) be the cartesian coordinates of \(\gamma(t)\) in the system with origin \(a\) and basis \((e^{i\theta_0}, i e^{i\theta_0}).\) As long as \(x(t) > 0,\) the function \[t \mapsto \theta_0 + \arg(x(t)+iy(t))\] is a continuous choice of the argument of \(\gamma(t) - a.\) Let \(d\) be the distance between \(a\) and \(\gamma([0,1])\) and let \(n \in \mathbb{N}\) such that \[ |t - s| \leq 2^{-n} \, \Rightarrow \, |\gamma(t) - \gamma(s)| < d. \] The condition \(x(t) > 0\) is ensured for any \(t\) in \([0, 2^{-n}].\) This construction of a continuous choice may be iterated locally on every interval \([k2^{-n}, (k+1)2^{-n}]\) with a new coordinate system to provide a global continuous choice of the argument on \([0,1].\)

The uniqueness of a continuous choice is a consequence of the intermediate value theorem: if we assume that there are two such functions \(\theta_1\) and \(\theta_2\) with the same initial value \(\theta_0,\) as \(\theta_1(0) - \theta_2(0)=0,\) if \(\theta_1(t) - \theta_2(t) \neq 0\) for some \(t\in[0,1],\) then either \(|\theta_1(t) - \theta_2(t)| < \pi,\) or there is a \(\tau \in \left]0,t\right[\) such that \(\theta_1(\tau) - \theta_2(\tau) \neq 0\) and \(|\theta_1(\tau) -\theta_2(\tau)| < \pi.\) In any case, there is a contradiction since all values of the argument differ of a multiple of \(2\pi.\) \(\blacksquare\)

Definition – Variation of the Argument.

‌ Let \(a \in \mathbb{C}\) and \(\gamma\) be a path of \(\mathbb{C} \setminus \{a\}.\) The variation of \(z \mapsto \mathrm{Arg}(z-a)\) on \(\gamma\) is defined as \[ [z \mapsto \mathrm{Arg}(z-a)]_{\gamma} = \theta(1) - \theta(0) \] where \(\theta\) is a continous choice of \(z \mapsto \mathrm{Arg}(z-a)\) on \(\gamma.\)

Proof (unambiguous definition).

‌ If \(\theta_1\) and \(\theta_2\) are two continuous choices of \(z \mapsto \mathrm{Arg}(z-a)\) on \(\gamma,\) for any \(t \in [0,1],\) they differ of a multiple of \(2\pi.\) As the function \(\theta_1 - \theta_2\) is continuous, by the intermediate value theorem, it is constant. Hence \[(\theta_1 - \theta_2)(1) = (\theta_1 - \theta_2)(0),\] and \(\theta_1(1) - \theta_1(0) = \theta_2(1) - \theta_2(0).\) \(\blacksquare\)

Definition – Winding Number / Index.

‌ Let \(a \in \mathbb{C}\) and \(\gamma\) be a closed path of \(\mathbb{C} \setminus \{a\}.\) The winding number – or index – of \(\gamma\) around \(a\) is the integer \[ \mathrm{ind}(\gamma,a) = \frac{1}{2\pi}[z\mapsto \mathrm{Arg}(z-a)]_{\gamma}. \]

Proof – The Winding Number is an Integer.

‌ Let \(\theta\) be a continuous choice function of \(z \mapsto \mathrm{Arg}(z-a)\) on \(\gamma;\) as the path \(\gamma\) is closed, \(\theta(0)\) and \(\theta(1),\) which are values of the argument of \(\gamma(0)-a = \gamma(1)-a,\) are equal modulo \(2\pi,\) hence \((\theta(1) - \theta(0))/2\pi\) is an integer. \(\blacksquare\)

Definition – Path Exterior & Interior.

‌ The exterior and interior of a closed path \(\gamma\) are the subsets of the complex plane defined by \[ \mathrm{Ext} \, \gamma = \{ z \in \mathbb{C} \setminus \gamma([0,1]) \; | \; \mathrm{ind}(\gamma, z) = 0 \}. \] and \[ \mathrm{Int} \, \gamma = \mathbb{C} \setminus (\gamma([0,1]) \cup \mathrm{Ext} \, \gamma) = \{ z \in \mathbb{C} \setminus \gamma([0,1]) \; | \; \mathrm{ind}(\gamma, z) \neq 0 \}. \]

Properties

Theorem – The Winding Number is Locally Constant.

‌ Let \(a \in \mathbb{C}\) and \(\gamma\) be a closed path of \(\mathbb{C} \setminus \{a\}.\) There is a \(\epsilon > 0\) such that, for any \(b \in \mathbb{C}\) and any closed path \(\beta,\) if \[ |b - a| < \epsilon \; \mbox{ and } \; (\forall \, t \in [0,1], \; |\beta(t) - \gamma(t)| < \epsilon) \] then \(\beta\) is a path of \(\mathbb{C} \setminus \{b\}\) and \[ \mathrm{ind}(\gamma,a) = \mathrm{ind}(\beta,b). \]

Proof.

‌ Let \(\epsilon = d(a, \gamma([0,1])) /2.\) If \(|b - a| < \epsilon\) and for any \(t \in [0,1],\) \(|\gamma(t) - \beta(t)| < \epsilon,\) then clearly \(b \in \mathbb{C} \setminus \beta([0,1]).\) Additionally, for any \(t \in [0,1]\) there are values \(\theta_1\) of \(\mathrm{Arg} (\gamma(t)-a)\) and \(\theta_2\) of \(\mathrm{Arg}(\beta(t)-b)\) such that \(|\theta_1 - \theta_2| < \pi/2.\) If we select some values \(\theta_{1,0}\) of \(\mathrm{Arg} (\gamma(0) - a)\) and \(\theta_{2,0}\) of \(\mathrm{Arg} (\beta(0) -b)\) such that \(|\theta_{1,0} - \theta_{2,0}| < \pi/2,\) then the corresponding continuous choices \(\theta_1\) and \(\theta_2\) satisfy \(|\theta_1(t) - \theta_2(t)| < \pi/2\) for any \(t \in [0,1]\)(1). Consequently \[ |\mathrm{ind}(\gamma,a) - \mathrm{ind}(\beta,b)| = \left| \frac{\theta_1(1) - \theta_1(0)}{2\pi} - \frac{\theta_2(1) - \theta_2(0)}{2\pi} \right| < \frac{1}{2}. \] As both winding numbers are integers, they are equal. \(\blacksquare\)

Corollary – The Winding Number is Constant on Components.

‌ Let \(\gamma\) be a closed path. The function \[ z \in \mathbb{C} \setminus \gamma([0,1]) \mapsto \mathrm{ind}(\gamma,z) \] is constant on each component of \(\mathbb{C} \setminus \gamma([0,1])\). If additionally the component is unbounded, the value of the winding number is zero.

Proof.

‌ The mapping \(z \mapsto \mathrm{ind}(\gamma, z)\) is locally constant – and hence constant – on every connected component of \(\mathbb{C} \setminus \gamma([0,1]).\) If \(a\) belongs to some unbounded component of this set, there is a \(b\) in the same component such that \(|b|> r = \max_{t \in [0,1]} |\gamma(t)|.\) It is possible to connect \(b\) to any point \(c\) such that \(|c| = r\) by a circular path in \(\mathbb{C} \setminus \gamma([0,1]),\) thus we may assume that \(b \in \mathbb{R}_-.\) The function \[ \theta: t \in [0,1] \mapsto \mathrm{arg}(\gamma(t) - b) \] is a continuous choice of \(z \mapsto \mathrm{Arg}(z-b)\) along \(\gamma\) and it satisfies \[ \forall \, t \in [0,1], \; |\theta(t)| = \left|\arctan \frac{\mathrm{Im} (\gamma(t) - b)}{\mathrm{Re} (\gamma(t) -b)} \right| < \arctan \frac{r}{|b| - r} < \frac{\pi}{2}. \] As \(\gamma\) is a closed path, \(\theta(0)\) and \(\theta(1)\) – which are equal modulo \(2\pi\) – are actually equal and \[ \mathrm{ind}(\gamma, a) = \mathrm{ind}(\gamma, b) = \frac{\theta(1) - \theta(0)}{2\pi} =0 \] as expected. \(\blacksquare\)

Simply Connected Sets

Definition – Simply/Multiply Connected Set & Holes.‌

‌ Let \(\Omega\) be an open subset of the plane. A hole of \(\Omega\) is a bounded component of its complement \(\mathbb{C} \setminus \Omega.\) The set \(\Omega\) is simply connected if it has no hole (if every component of its complement is unbounded) and multiply connected otherwise.
  1. The open set \(\Omega = \{(x,y) \in \mathbb{R}^2 \; | \; x < -1 \, \mbox{ or }\, x>1\}\) is not connected but it is simply connected: its complement has a unique component which is unbounded, hence it has no holes.

  2. The open set \(\Omega = \mathbb{C} \setminus \overline{\{2^{-n} \, | \, n \in \mathbb{N}\}}\) is multiply connected: its holes are exactly the singletons of its complement.

Intuitively, we should be able to circle around any hole of \(\Omega\) without leaving the set; this idea leads to an alternate characterization of simply connected sets.

Theorem – Simply Connected Sets & The Winding Number.

‌ An open subset \(\Omega\) of the complex plane is simply connected if and only if the interior of any closed path \(\gamma\) of \(\Omega\) is included in \(\Omega\): \[ \forall \, z \in \mathbb{C} \setminus \gamma([0,1]),\; \mathrm{ind}(\gamma, z) \neq 0 \, \Rightarrow \, z \in \Omega, \] or equivalently, if the complement of \(\Omega\) is included in the exterior of \(\gamma\): \[ \forall \, z \in \mathbb{C} \setminus \Omega, \; \mathrm{ind}(\gamma, z) = 0. \]
  1. If \(\gamma\) is a closed path of \(\Omega = \{(x,y) \in \mathbb{R}^2 \; | \; x < -1 \, \mbox{ or }\, x>1\}\) and \(z \in \mathbb{C} \setminus \Omega\), since \(\mathbb{C} \setminus \Omega\) is connected and unbounded, \(z\) belongs to an unbounded component of \(\mathbb{C} \setminus \gamma([0,1]).\) Thus \(\mathrm{ind}(\gamma, z)=0\) for any \(z \in \mathbb{C} \setminus \Omega.\)

  2. The open set \(\Omega = \mathbb{C} \setminus \overline{\{2^{-n} \, | \, n \in \mathbb{N}\}}\) is open and multiply connected: for example \(\gamma = 1 + 1/4 [\circlearrowleft]\) is a path of \(\Omega\), \(z=1\) is a point of \(\mathbb{C} \setminus \Omega\) and \(\mathrm{ind}(\gamma, 1) = 1.\)

Remark.

‌Note that we may not always be able to encircle only one hole at a time. For example, in the case of the set \(\Omega = \mathbb{C} \setminus \overline{\{2^{-n} \, | \, n \in \mathbb{N}\}},\) we can find a closed path \(\gamma\) of \(\mathbb{C} \setminus \Omega\) such that \(\mathrm{ind}(\gamma, 0)=1,\) but then we also have \(\mathrm{ind}(\gamma, 2^{-n})=1\) for \(n\) large enough: we cannot encircle the hole \(\{0\}\) of \(\Omega\) unless we also encircle an infinity of extra holes.

Lemma.

‌ If the compact set \(K\) is a hole of the open set \(\Omega\), there is a compact subset \(L\) of \(\mathbb{C} \setminus \Omega\) such that \(K\subset L\) and \(d(L, (\mathbb{C} \setminus \Omega) \setminus L) > 0.\)

Proof of the Lemma.

‌ The set \(C = \mathbb{C} \setminus \Omega\) is closed in \(\mathbb{C}\) which is locally compact, thus it is locally compact. Since \(\mathbb{C}\) is Hausdorff, its subspace \(C\) is also Hausdorff. By the Šura-Bura theorem (Remmert 1998, 304), \(K\) has a neighbourhood base in \(C\) consisting in open compact subsets of \(C\). Since \(C\) is a neighbourhood of \(K\) in \(C\), there is a compact set \(L\) such that \(K \subset L \subset C\) which is open in \(C\). Thus, its complement \((\mathbb{C} \setminus \Omega) \setminus L\) is also closed in \(C\). Since \(C\) is closed, both sets are also closed in \(\mathbb{C}\). They are also disjoint by construction, and thus \(d(L, (\mathbb{C} \setminus \Omega) \setminus L) > 0.\) \(\blacksquare\)

Proof – Simply Connected Sets & The Winding Number.

‌ Assume that \(\Omega\) is simply connected and let \(\gamma\) be a closed path of \(\Omega\). Let \(z \in \mathbb{C} \setminus \Omega;\) this point belongs to an unbounded connected component of \(\mathbb{C} \setminus \Omega\) and therefore to an unbounded connected component of \(\mathbb{C} \setminus \gamma([0,1]),\) thus \(\mathrm{ind}(\gamma, z)=0.\)

Conversely, if \(\Omega\) is not simply connected, the set \(\mathbb{C} \setminus \Omega\) has a hole \(K\) which is contained in some compact subset \(L\) of \(\mathbb{C} \setminus \Omega\) such that the distance \(\epsilon\) between \(L\) and \((\mathbb{C} \setminus \Omega) \setminus L\) is positive. Let \(r < \epsilon/\sqrt{2};\) Define for any pair \((k, l)\) of integers the node \(n_{k,l} = (k + i l)r\) and \(S_{k,l}\) as the closed square with vertices \(n_{k,l},\) \(n_{k+1,l},\) \(n_{k+1,l+1}\) and \(n_{k,l+1}.\) The (positively) oriented boundary of the square \(S_{k,l}\) is the polyline \[ [n_{k,l} \to n_{k+1,l} \to n_{k+1,l+1} \to n_{k,l+1} \to n_{k,l}] \] The collection of squares that intersect \(L\) is finite and covers \(L.\) Additionally, all of its squares are included in \(\Omega \cup L.\)

For any square \(S\) in the cover of \(L\) and any interior point \(a\) of \(S\) if \(\gamma\) is the oriented boundary of \(S,\) then \(\mathrm{ind}(\gamma, a) = 1.\) Additionally, \(\mathrm{ind}(\mu, a) = 0\) for the oriented boundary \(\mu\) of any other square in the collection. Consequently, if \(\Gamma\) denotes the collection of oriented line segments that composes the oriented boundaries of all squares of the cover of \(L,\) we have \[ \sum_{\gamma \in \Gamma}\frac{1}{2\pi} [z \mapsto \mathrm{Arg}(z-a)]_{\gamma} = 1. \] Now if the line segment \(\gamma\) belongs to \(\Gamma\) and \(\gamma([0,1]) \cap L \neq \varnothing,\) then \(\gamma^{\leftarrow}\) also belongs to \(\Gamma;\) if we remove all such pairs from \(\Gamma,\) the resulting collection \(\Gamma'\) also satisfies \[ \sum_{\gamma \in \Gamma'}\frac{1}{2\pi} [z \mapsto \mathrm{Arg}(z-a)]_{\gamma} = 1. \] and by construction the image of any \(\gamma\) in \(\Gamma'\) is included in \(\Omega.\) The original collection \(\Gamma\) is balanced: for any square vertice \(n,\) the number of line segments with \(n\) as an initial point and with \(n\) as a terminal point is the same. The collection \(\Gamma'\) has the same property. Consequently, the line segments of \(\Gamma'\) may be assembled in a finite sequence of closed paths \(\gamma_1,\) \(\dots,\) \(\gamma_n\) and \[ \sum_{k=1}^n \mathrm{ind}(\gamma_k, a) = 1. \] Every point of \(L\) is either an interior point of some square of the collection, or the limit of such point; anyway, that means that \[ \forall \, z \in L, \; \sum_{k=1}^n \mathrm{ind}(\gamma_k, z) = 1 \] and thus that there is at least one path \(\gamma_k\) such that \(\mathrm{ind}(\gamma_k, z) \neq 0.\) \(\blacksquare\)

A Complex Analytic Approach

If a closed path is rectifiable, we may compute its winding number as a line integral; to prove this, we need the:

Lemma.

‌ Let \(a \in \mathbb{C}\) and \(\gamma\) be a rectifiable path of \(\mathbb{C} \setminus \{a\}.\) For any \(t \in [0,1],\) let \(\gamma_t\) be the path such that for any \(s\in[0,1],\) \(\gamma_t(s) = \gamma(t s).\) The function \(\mu:[0,1]\to \mathbb{C},\) defined by \[ \mu(t) = \int_{\gamma_t} \frac{dz}{z-a} \] satisfies \[ \exists \, \lambda \in \mathbb{C}^*, \, \forall \, t \in [0,1], \, e^{\mu(t)} = \lambda \times (\gamma(t) - a). \]

Proof.

‌ We only prove the lemma under the assumption that \(\gamma\) is continuously differentiable; the rectifiable case is a straightforward extension.

We have for any \(t\in [0,1]\) \[ \mu(t) = \int_{\gamma_t} \frac{dz}{z-a} = \int_0^1 \frac{\gamma'(ts) \times t}{\gamma(t s) - a} ds = \int_0^t \frac{\gamma'(s)}{\gamma(s) - a} ds, \] hence \[ \mu'(t) = \frac{\gamma'(t)}{\gamma(t) - a} \] and the derivative of the quotient \(\phi(t) = {e^{\mu(t)}}/{(\gamma(t)}-a)\) satisfies \[ \phi'(t) = \mu'(t) \phi(t) - \frac{\gamma'(t)}{\gamma(t) - a} \phi(t) = 0 \] which yields the result. \(\blacksquare\)

Theorem – The Winding Number as a Line Integral.

‌ Let \(a \in \mathbb{C}\) and \(\gamma\) be a rectifiable path of \(\mathbb{C} \setminus \{a\}.\) Then \[ [z \mapsto \mathrm{Arg} (z-a)]_{\gamma} = \mathrm{Im} \left( \int_{\gamma} \frac{dz}{z - a} \right). \] If the path \(\gamma\) is closed, then \[ \mathrm{ind}(\gamma,a) = \frac{1}{i2\pi} \int_{\gamma} \frac{dz}{z - a}. \]

Proof.

‌We use the function \(\mu\) of the previous lemma. Applying the modulus to both sides of the equation \(e^{\mu(t)} = \lambda \times (\gamma(t) - a)\) provides \(e^{\mathrm{Re}(\mu(t))} = |\lambda| \times |\gamma(t) - a|,\) hence \[ e^{i \mathrm{Im}(\mu(t))} = \frac{\lambda}{|\lambda|} \frac{\gamma(t) - a}{|\gamma(t) - a|}. \] The function \(t \in [0,1] \mapsto \mathrm{Im}(\mu(t))\) is – up to a constant – a continuous choice of \(z \mapsto \mathrm{Arg}\, (z-a)\) on \(\gamma.\) Consequently, \[[z \mapsto \mathrm{Arg} \, (z-a)]_{\gamma} = \mathrm{Im}(\mu(1)) - \mathrm{Im}(\mu(0)) = \mathrm{Im}(\mu(1)),\] which is the desired result.

If additionally \(\gamma\) is a closed path, the equations \[ \gamma(0) = \gamma(1) \; \mbox{ and } \; e^{\mathrm{Re}(\mu(t))} = |\lambda| \times |\gamma(t) - a| \] yield \(e^{\mathrm{Re}(\mu(0))}= e^{\mathrm{Re}(\mu(1))}\) and hence \(\mathrm{Re}(\mu(1)) = \mathrm{Re}(\mu(0)) = 0.\) Thus, \[ \mathrm{ind}(\gamma,a) = \frac{1}{2\pi} \mathrm{Im}(\mu(1)) = \frac{1}{i2\pi} \mu(1), \] which concludes the proof. \(\blacksquare\)

References

  1. Algebraic Topology
    Allen Hatcher, 2002.
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  2. Classical Topics in Complex Function Theory
    Reinhold Remmert, 1998.
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Notes


  1. Otherwise, by the intermediate value theorem, we could find some \(t \in \left]0,1\right]\) such that \(|\theta_1(t) - \theta_2(t)| = \pi/2,\) but then, for every value \(\theta_{1,t}\) of \(\mathrm{Arg} (\gamma(t)-a)\) and \(\theta_{2,t}\) of \(\mathrm{Arg}(\beta(t)-b),\) we would have \[\theta_{1,t}- \theta_{2,t} = \theta_1(t) - \theta_2(t) + 2\pi k\] for some \(k \in \mathbb{Z}.\) Therefore, the choice of \(\theta_{1,t}\) and \(\theta_{2,t}\) such that \(|\theta_{1,t} - \theta_{2,t}| < \pi/2\) would be impossible.