Argument Principle: visual statement and proof

We start by recalling the definition of multiplicity of a root of a holomorphic function. Let \( A \subseteq \mathbb{R}^2 \) open, and let \( f: A \to \mathbb{C} \) be a holomorphic function. Let \( z_0 \in A \) be a root of \(f\), i.e., \( f(z_0) = 0 \). Suppose that not all derivatives \( f^{(k)}(z_0)\) are zero. We call the minimum \(k\) with \( f^{(k)}(z_0)\neq 0 \) the multiplicity of the root \( z_0 \). Note that if all these derivatives were zero and if \(A\) was connected, then \(f\) would be identically zero.

Before you go over this article, make sure you read the beginning of the Riemann mapping theorem article. Specifically, read up until Theorem 1.

The argument principle is a spectacular theorem that informally says the following (the formal version will follow momentarily): Let \( \ell:[0,a]\rightarrow \mathbb{C} \) be a Jordan curve. Let \(L:=\ell([0,a])\) be its image and \(A\) its interior. Let \( f \) be a continuous function on \(A\cup L \) that is holomorphic on \( A.\) Suppose also that \( f \) has no roots on \( L \). Then the number of roots of \( f \) in \( A \) (counted with multiplicities) is equal to the number of times \( f(L) \) turns around the origin.

Illustration of the Jordan curve and indicated directions

In this example we have two roots, on which \(f'\) is nonzero. We see that \(f(L)\) turns twice around the origin.

We need to formalize the above statement. Given a loop \( \ell \) that does not hit zero (meaning \( \ell(t) \neq 0 \) for all \( t \)), the phrase “number of times \( \ell \) turns around the origin” is not a rigorous statement. Mathematics is built on set theory, not on our visual intuition (although it is inspired and guided by it). To make the phrase rigorous, we need an i.o theorem which establishes the existence of the angle function for a continuous path (for references containing proofs of the i.o. theorems used in this article, see the final section here).

Theorem 1. For every continuous path \( p:[0,a] \rightarrow \mathbb{C} \) that does not hit zero, there exists a unique continuous function \( \theta_p:[0,a] \rightarrow \mathbb{R} \) such that:

For all \( t \in [0,a] \), we have \[ \theta_p(t) \equiv \text{Arg}(p(t)) \pmod{2\pi} \]
\( \theta_p(0) \in [0, 2\pi) \).

We call \( \theta_p \) the angle function of \( p \). Note that, as illustrated in the figure above, continuity may require \( \theta_p \) to take values outside of \([0, 2\pi)\). Let \( \Delta \theta_p := \theta_p(a) - \theta_p(0) \). This is the total angle swept out by \( p \). In the figure above \( \Delta \theta_p = 9\pi/4 \).

From Theorem 1, it follows that if \( \ell: [0,a] \to \mathbb{C} \) is a loop that does not hit zero, then \( \Delta \theta_\ell \) is an integer multiple of \( 2\pi \). For such a loop, we call the integer \[ W(\ell, 0) := \frac{\Delta \theta_\ell}{2\pi} \] the winding number of \( \ell \) around zero. This is the number of times \( \ell \) “turns around the origin.” Now, for a point \( z_0 \in \mathbb{C} \) and a loop \( \ell \) that does not hit \( z_0 \), we define \[ W(\ell, z_0) := W(\ell - z_0, 0) \] where \( \ell - z_0 \) is the loop \( \ell \) translated by \( -z_0 \).

There is one thing left to formalize the argument principle: we need to define what a positively oriented Jordan curve is. We begin with an observation:

Proposition 1. If \( p:[0,a] \to \mathbb{C} \) is a continuous path that does not hit zero, then the reverse path \( \bar{p}:[0,a] \to \mathbb{C} \), defined as \( \bar{p}(t) := p(a - t) \), satisfies \(\Delta \theta_{\bar{p}} = -\Delta \theta_p.\)

Exercise: use Theorem 1 to prove Proposition 1. Now, we will need the following i.o. theorem. This theorem uses the concept of continuous deformation of loops. If you are unaware of this term, click here.

Theorem 2. Let \( \ell:[0,a] \rightarrow \mathbb{C} \) be a Jordan curve. Then, \(W(\ell, z_0)\) is the same for all \(z_0 \in A_\ell\), and is equal to \(\pm 1\). If it is \(+1\), then we say that \(\ell\) is positively oriented, otherwise we say it is negatively oriented. Furthermore, let \( \ell_0:[0,b] \rightarrow \mathbb{C} \) be another Jordan curve lying entirely within \( A_\ell \) and having the same orientation as \(\ell\). Let \( L_0 := \ell_0([0,b]) \). Then, \( \ell_0 \) can be continuously deformed into \( \ell \) in a way that all intermediate loops are Jordan curves lying in \( A_\ell \setminus (A_{\ell_0} \cup L_0 ) \ \) and having \(\ell\)'s orientation.

Click to repeat. Example of deformation from Theorem 2.

Note that by taking the reverse of a Jordan curve we flip its orientation.

Theorem 3 (Argument Principle). Let \( \ell:[0,a] \rightarrow \mathbb{C} \) be a positively oriented Jordan curve, let \(L:=\ell([0,a ]) \) be its image and \( A \) its interior. Let \( f: A \cup L \rightarrow \mathbb{C} \) be a continuous function that is holomorphic on \( A \). Suppose \( f \) has no roots on \( L \). Then,

W(f \circ \ell, 0) = \text{number of roots of } f, \text{counting multiplicities}

Proof

We will first need one more i.o. theorem:

Theorem 4. Let \( \ell_1 : [0,a] \rightarrow \mathbb{C} \) and \( \ell_2 : [0,b] \rightarrow \mathbb{C} \) be two loops, and \( z_0 \) a point in \( \mathbb{C} \). Assume that neither \( \ell_1 \) nor \( \ell_2 \) hit \( z_0 \). Then, the following are equivalent:

\( \ell_1 \) can be continuously deformed to \( \ell_2 \) without hitting \( z_0 \) in the process.
\( W(\ell_1, z_0) = W(\ell_2, z_0) \).

Draw some examples to intuitively convince yourself about the validity of the above theorem.

Now, observe that \(f\) has finitely many roots. Here is why: a standard result in introductory complex analysis is that since \(f\) is not identically zero, its roots have no limit point in \(A\) (for a proof, see [1]). Now, \(f\) is continuous and nonzero on \(L\), so the roots also have no limit point in \(L\). Since \(A \cup L\) is compact, the roots are finitely many. Let's call them \(z_1, z_2, \dots, z_n\) and their respective multiplicities \(m_1, m_2, \dots, m_n\).

We will break the proof into three steps. In each step, I first give an informal argument clarifying the underlying idea, and then I formalize it.

Step 1: No roots

Suppose \(f\) has no roots in \(A\), and assume that \(W(f \circ \ell, 0)\neq 0\). Let \(\bar{A}:= A \cup L\). From Theorem 1 here, \(\bar{A}\) is simply connected and thus \(\ell \) can be continuously deformed to a point \(z_0 \in \bar{A} \) while staying inside \(\bar{A}\) (note that \(z_0\) can be trivially expressed as a loop \(\ell_0(t)=z_0\)). Consider the images of these deformations under \(f\). Since \(f\) is continuous, these images will comprise a continuous deformation of \(f \circ \ell\) to \(f(z_0)\). Since \(f\) has no roots, none of these loops hits the origin. This implies (by Theorem 4) that all of them turn around the origin \(W(f \circ \ell,0)\neq 0\) times. Since these loops shrink down to \(f(z_0)\), we must have \(f(z_0)=0\) (as in the animation below), contradiction.

Click to play/pause.

Step 2: Local argument principle

We will prove the argument principle for the restriction of \(f\) on a small disc of radius \(\varepsilon>0\) centered on some root; suppose it is \(z_1\).

Restriction for the local argument principle

The corresponding circle is \(C_{z_1,\varepsilon}(t)= \varepsilon e^{i 2\pi t} + z_1\), \(t\in[0,1]\). We also denote by \(D_{z_1,\varepsilon}\) and \(\bar{D}_{z_1,\varepsilon}\) the corresponding open and closed discs. Now, from Taylor's theorem we have that for \(z\) close to \(z_1\), \[ f(z)=a(z-z_1)^{m_1}+E(z-z_1) \] where \(a= f^{(m_1)}(z_1)/m_1!\) and \(E\) is an error-function that is much smaller compared the leading term \(a(z-z_1)^{m_1}\). Specifically, \(E(z-z_1)\) is of the order of \(|z-z_1|^{m_1+1} \ll |a| |z-z_1|^{m_1}\), for \(z\) close to \(z_1\). Now, note that if we apply the function \(a(z-z_1)^{m_1}\) to the circle \(C_{z_1,\varepsilon}\), we get the circle centered at the origin and of radius \(|a|\varepsilon^{m_1}\), wrapped on itself \(m_1\) times. Since \(f\circ C_{z_1,\varepsilon}\) is a small perturbation of that loop (see figure below), we have \(W(f\circ C_{z_1,\varepsilon},0)=m_1\).

Illustration of the local argument principle

We will show that there is an \( \varepsilon_0 > 0 \) such that for all \( \varepsilon \in (0, \varepsilon_0) \) we have \( W(f \circ C_{z_1, \varepsilon}, 0) = m_1 \). Let \( R > 0 \) such that not only the disc \( D_{z_1, R} \) is inside \(A\), but also \( \bar{D}_{z_1, 2R} \subseteq A \). The reason why we take the double radius and the closure will become apparent momentarily.

Let's express \( f \) as a power series in \( D_{z_1, 2R} \). We have \(f(z) = a(z - z_1)^{m_1} + E(z - z_1)\), where \[ E(z - z_1) = \sum_{n \ge m_1 + 1} \frac{f^{(n)}(z_1)}{n!}(z - z_1)^n. \]

There is a complex-analytic version of Taylor's theorem that bounds \( |E(z - z_1)| \). However, it is not part of the standard curriculum in complex analysis courses. Thus, we will prove the bound here. We start with

\[ |E(z - z_1)| \le \sum_{n \ge m_1 + 1} \frac{|f^{(n)}(z_1)|}{n!} |z - z_1|^n. \]

Cauchy's inequality [3] tells us that \(|f^{(n)}(z_1)| \le \frac{n!}{(2R)^n} M,\) where \( M \) is the maximum of \( f \) on the circle \( C_{z_1, 2R} \). Observe that to apply Cauchy's inequality (as it is stated in [3]), we needed the closure of the disc to lie inside \( A \). Using the inequality,

\[ |E(z - z_1)| \le M \sum_{n \ge m_1 + 1} \left( \frac{|z - z_1|}{2R} \right)^n. \]

Now we can see why we are working with the double radius \( 2R \). It is because we can show that on the smaller disc \( D_{z_1, R} \), the error \( |E(z - z_1)| \) is bounded by a constant times \( |z - z_1|^{m_1 + 1}. \) Let's prove this. For \( z \in D_{z_1, R} \),

\[ |E(z - z_1)| \le M \left( \frac{|z - z_1|}{2R} \right)^{m_1 + 1} \cdot \frac{1}{1 - \frac{|z - z_1|}{2R}} \le \frac{2M}{(2R)^{m_1 + 1}} |z - z_1|^{m_1 + 1}. \]

Letting \( C := \frac{2M}{(2R)^{m_1 + 1}} \), we get \(|E(z - z_1)| \le C |z - z_1|^{m_1 + 1}\), on \( D_{z_1, R} \).

Let \(\varepsilon_0:=\frac{|a|}{2C}\) and consider an \(\varepsilon\in(0,\varepsilon_0)\). If we apply \(a(z-z_1)^{m_1}\) to the circle \(C_{z_1,\varepsilon}\), we get \(\tilde{C}(t):= a \varepsilon^{m_1}e^{i2\pi m_1 t}\ \), \(t\in[0,1]\). The following is a continuous deformation of \(\tilde{C}\) to \(f\circ C_{z_1,\varepsilon}\): \[ \gamma(s,t):=\tilde{C}(t) + s E(\varepsilon e^{i2\pi t}),\ \ s,t \in[0,1] \]

Now note that by triangle inequality, \(|\gamma(s,t)|\ge |a| \varepsilon^{m_1}-C\varepsilon^{m_1+1}= \varepsilon^{m_1}(|a|-C\varepsilon)>0.\) Thus, \(\gamma\) does not hit the origin. Since \(W(\tilde{C},0)=m_1\), by Theorem 4, \(W(f\circ C_{z_1,\varepsilon},0)=m_1\).

Step 3: Global argument principle

First, suppose \(f\) has only one root: \(z_1\in A\). Consider a small disc \(\bar{D}_{z_1,\varepsilon}\) contained in \(A\). We proved that \(W(f\circ C_{z_1,\varepsilon},0)=m_1\). From Theorem 2, we can continuously deform \(C_{z_1,\varepsilon}\) to \(\ell\) while staying in \(\bar{A}\setminus D_{z_1,\varepsilon}\). The images of these deformations do not touch the origin, and thus they all have the same winding number, as in the animation below.

Click to play/pause.

Suppose \(f\) has exactly two distinct roots \(z_1,z_2\). The idea here is to split \(\bar{A}\) into two pieces - each containing exactly one root - as in the figure below:

Let \(\ell_1:[0,1]\rightarrow \mathbb{C}\) be the positively oriented Jordan curve bounding the left piece and \(\ell_2:[0,1]\rightarrow \mathbb{C}\) the corresponding for the right piece. Let \(B\) be the point where both curves start and finish, i.e., \(\ell_1(0)=\ell_1(1)=\ell_2(0)=\ell_2(1)=B\). As we see in the figure, both \(\ell_1\) and \(\ell_2\) share the segment \(BC\), but they traverse it in opposite directions. From the case of one root, we know that \(W(f\circ \ell_1,0)=m_1\) and \(W(f\circ \ell_2,0)=m_2\). I claim that \(\Delta \theta_{f\circ \ell}= \Delta \theta_{f\circ \ell_1}+ \Delta \theta_{f\circ \ell_2} \). To see why, consider the loop \(\ell_1 * \ell_2\) that does this: starts from \(B\), goes around \(\ell_1\) and returns to \(B\), and then goes around \(\ell_2\). Let's study \(\Delta \theta_{f\circ (\ell_1 * \ell_2)}\). Since \(f\circ (\ell_1 * \ell_2)\) first traverses \(f\circ \ell_1\) and then \(f\circ \ell_2,\) we have \(\Delta \theta_{f\circ (\ell_1 * \ell_2)}= \Delta \theta_{f\circ \ell_1} + \Delta \theta_{f\circ \ell_2}\). On the other hand, the back-and-forth of \(f\circ (\ell_1 * \ell_2)\) on \(f(BC)\) contributes nothing on the total angle swept, and thus \(\Delta \theta_{f\circ (\ell_1 * \ell_2)}=\Delta \theta_{f\circ \ell}\). Having shown that \(\Delta \theta_{f\circ \ell}= \Delta \theta_{f\circ \ell_1}+ \Delta \theta_{f\circ \ell_2} \), we get that \(W(f\circ \ell,0)=m_1+m_2\).

If we have \(n\) distinct roots, we simply extend the above cancellation idea as in the figure below:

We first rigorously define path concatenation.

Let \( p_1 : [0,a] \rightarrow \mathbb{C} \) and \( p_2 : [0,b] \rightarrow \mathbb{C} \) be continuous paths such that \( p_1(a) = p_2(0) \). We define their concatenation \( p_1 * p_2 : [0, a + b] \rightarrow \mathbb{C} \) by:

\[ p_1 * p_2(t) = \begin{cases} p_1(t), & \text{if } t \in [0, a] \\ p_2(t - a), & \text{if } t \in [a, a + b] \end{cases} \]

We inductively define the concatenation \( p_1 * p_2*\dots p_n\) of continuous paths with matching endpoints.

Proposition 2. Let \( p_i : [0, a_i] \rightarrow \mathbb{C} \), for \( 1 \leq i \leq k \), be \( k \) continuous paths such that \( p_{i+1}(0) = p_i(a_i) \) for all \( 1 \leq i \leq k-1 \). Suppose that no \( p_i \) hits the origin. Then

\Delta \theta_{p_1 * p_2 * \dots * p_k} = \sum_{i=1}^k \Delta \theta_{p_i}

Proposition 2 follows from Theorem 1 (why?).

Also, note that if \( p_1, p_2, \dots, p_k \) are continuous paths with matching endpoints such that they all lie inside \( \bar{A} \), then

f \circ (p_1 * p_2 * \dots * p_k) = (f \circ p_1) * (f \circ p_2) * \dots * (f \circ p_k) \tag{1}

Furthermore, if \( p \) is a continuous path inside \( \bar{A} \), then

f \circ \bar{p} = \overline{f \circ p} \tag{2}

where, remember, \( \bar{p} \) denotes the reverse of \( p \).

We now use an i.o. theorem that tells us that the splitting in the above figure is indeed possible.

Theorem 5. Let \( \ell : [0,a] \to \mathbb{C} \) be a positively oriented Jordan curve and \( A \) its interior. Let \( z_1, z_2, \dots, z_n \) be distinct points in \( A \). Then, there exist \( 0 < t_1 < t_2 < \dots < t_{n-1} < a \) and continuous paths \( p_1, p_2, \dots, p_{n-1} : [0,1] \to \mathbb{C} \) that satisfy the following:

For all \( i \), \( p_i(0) = \ell(0) \) and \( p_i(1) = \ell(t_i) \).
For all \( i \), \( \ell_i((0,1)) \subseteq A \).
Let \[ \ell_1 := \ell \big|_{[0,t_1]} * \bar{p}_1, \quad \ell_2 := p_1 * \ell \big|_{[t_1,t_2]} * \bar{p}_2, \quad \dots, \quad \ell_n := p_{n-1} * \ell \big|_{[t_{n-1},a]}. \] Each \( \ell_i \) is a positively oriented Jordan curve. Furthermore, each \( z_j \) belongs to the interior of exactly one \( \ell_i \), and to the exterior of all the others.

We are now ready to complete the proof of the argument principle. We apply Theorem 5 to our setting. Let \( \ell_{\pi(i)} \) be the Jordan curve corresponding to the root \( z_i \). We previously saw that Theorems 2 and 4 imply that \( W(f \circ \ell_{\pi(i)}, 0) = m_i \). Thus,

\[ \sum_{i=1}^n m_i = \sum_{i=1}^n W(f \circ \ell_i, 0) = \frac{1}{2\pi} \sum_{i=1}^n \Delta \theta_{f \circ \ell_i} \]

Using Propositions 1 and 2 and equations (1) and (2), we get

\[ \sum_{i=1}^n \Delta \theta_{f \circ \ell_i} = \sum_{i=1}^n \Delta \theta_{f \circ \ell|_{[t_{i-1}, t_i]}} \]

where \( t_0 = 0 \), \( t_n = a \). The last sum is equal to \( \Delta \theta_{f \circ \ell} = 2\pi W(f \circ \ell, 0) \).

References

Complex Analysis, Elias M. Stein and Rami Shakarchi. Chapter 2, Theorem 4.8.
Taylor's Theorem in Complex Analysis (Wikipedia)
Complex Analysis, Elias M. Stein and Rami Shakarchi. Chapter 2, Corollary 4.3.

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