TWe now come to perhaps the most central theorem in complex analysis (save possibly for the fundamental theorem of calculus), namely Cauchy’s theorem, which allows one to compute (or at least transform) a large number of contour integrals ${\int_\gamma f(z)\ dz}$ even without knowing any explicit antiderivative of ${f}$. There are many forms and variants of Cauchy’s theorem. To give one such version, we need the basic topological notion of a homotopy:

Definition 1 (Homotopy) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${\gamma_0: [a,b] \rightarrow U}$, ${\gamma_1: [a,b] \rightarrow U}$ be two curves in ${U}$.

• (i) If ${\gamma_0, \gamma_1}$ have the same initial point ${z_0}$ and final point ${z_1}$, we say that ${\gamma_0}$ and ${\gamma_1}$ are homotopic with fixed endpoints in ${U}$ if there exists a continuous map ${\gamma: [0,1] \times [a,b] \rightarrow U}$ such that ${\gamma(0,t) = \gamma_0(t)}$ and ${\gamma(1,t) = \gamma_1(t)}$ for all ${t \in [a,b]}$, and such that ${\gamma(s,a) = z_0}$ and ${\gamma(s,b) = z_1}$ for all ${s \in [0,1]}$.
• (ii) If ${\gamma_0, \gamma_1}$ are closed (but possibly with different initial points), we say that ${\gamma_0}$ and ${\gamma_1}$ are homotopic as closed curves in ${U}$ if there exists a continuous map ${\gamma: [0,1] \times [a,b] \rightarrow U}$ such that ${\gamma(0,t) = \gamma_0(t)}$ and ${\gamma(1,t) = \gamma_1(t)}$ for all ${t \in [a,b]}$, and such that ${\gamma(s,a) = \gamma(s,b)}$ for all ${s \in [0,1]}$.
• (iii) If ${\gamma_2: [c,d] \rightarrow U}$ and ${\gamma_3: [e,f] \rightarrow U}$ are curves with the same initial point and same final point, we say that ${\gamma_2}$ and ${\gamma_3}$ are homotopic with fixed endpoints up to reparameterisation in ${U}$ if there is a reparameterisation ${\tilde \gamma_2: [a,b] \rightarrow U}$ of ${\gamma_2}$ which is homotopic with fixed endpoints in ${U}$ to a reparameterisation ${\tilde \gamma_3: [a,b] \rightarrow U}$ of ${\gamma_3}$.
• (iv) If ${\gamma_2: [c,d] \rightarrow U}$ and ${\gamma_3: [e,f] \rightarrow U}$ are closed curves, we say that ${\gamma_2}$ and ${\gamma_3}$ are homotopic as closed curves up to reparameterisation in ${U}$ if there is a reparameterisation ${\tilde \gamma_2: [a,b] \rightarrow U}$ of ${\gamma_2}$ which is homotopic as closed curves in ${U}$ to a reparameterisation ${\tilde \gamma_3: [a,b] \rightarrow U}$ of ${\gamma_3}$.

In the first two cases, the map ${\gamma}$ will be referred to as a homotopy from ${\gamma_0}$ to ${\gamma_1}$, and we will also say that ${\gamma_0}$ can be continously deformed to ${\gamma_1}$ (either with fixed endpoints, or as closed curves).

Example 2 If ${U}$ is a convex set, that is to say that ${(1-s) z_0 + s z_1 \in U}$ whenever ${z_0,z_1 \in U}$ and ${0 \leq s \leq 1}$, then any two curves ${\gamma_0, \gamma_1: [0,1] \rightarrow U}$ from one point ${z_0}$ to another ${z_1}$ are homotopic, by using the homotopy

$\displaystyle \gamma(s,t) := (1-s) \gamma_0(t) + s \gamma_1(t).$

For a similar reason, in a convex open set ${U}$, any two closed curves will be homotopic to each other as closed curves.

Exercise 3 Let ${U}$ be an open subset of ${{\bf C}}$.

• (i) Prove that the property of being homotopic with fixed endpoints in ${U}$ is an equivalence relation.
• (ii) Prove that the property of being homotopic as closed curves in ${U}$ is an equivalence relation.
• (iii) If ${\gamma_0, \gamma_1: [a,b] \rightarrow U}$ are closed curves with the same initial point, show that ${\gamma_0}$ is homotopic to ${\gamma_1}$ as closed curves if and only if ${\gamma_0}$ is homotopic to ${\gamma_2 + \gamma_1 + (-\gamma_2)}$ with fixed endpoints for some closed curve ${\gamma_2}$ with the same initial point as ${\gamma_0}$ or ${\gamma_1}$.
• (iv) Define a point in ${U}$ to be a curve ${\gamma_1: [a,b] \rightarrow U}$ of the form ${\gamma_1(t) = z_0}$ for some ${z_0 \in U}$ and all ${t \in [a,b]}$. Let ${\gamma_0: [a,b] \rightarrow U}$ be a closed curve in ${U}$. Show that ${\gamma_0}$ is homotopic with fixed endpoints to a point in ${U}$ if and only if ${\gamma_0}$ is homotopic as a closed curve to a point in ${U}$. (In either case, we will call ${\gamma_0}$ homotopic to a point, null-homotopic, or contractible to a point in ${U}$.)
• (v) If ${\gamma_0, \gamma_1: [a,b] \rightarrow U}$ are curves with the same initial point and the same terminal point, show that ${\gamma_0}$ is homotopic to ${\gamma_1}$ with fixed endpoints in ${U}$ if and only if ${\gamma_0 + (-\gamma_1)}$ is homotopic to a point in ${U}$.
• (vi) If ${U}$ is connected, and ${\gamma_0, \gamma_1: [a,b] \rightarrow U}$ are any two curves in ${U}$, show that there exists a continuous map ${\gamma: [0,1] \times [a,b] \rightarrow U}$ such that ${\gamma(0,t) = \gamma_0(t)}$ and ${\gamma(1,t) = \gamma_1(t)}$ for all ${t \in [a,b]}$. Thus the notion of homotopy becomes rather trivial if one does not fix the endpoints or require the curve to be closed.
• (vii) Show that if ${\gamma_1: [a,b] \rightarrow U}$ is a reparameterisation of ${\gamma_0: [a,b] \rightarrow U}$, then ${\gamma_0}$ and ${\gamma_1}$ are homotopic with fixed endpoints in U.
• (viii) Prove that the property of being homotopic with fixed endpoints in ${U}$ up to reparameterisation is an equivalence relation.
• (ix) Prove that the property of being homotopic as closed curves in ${U}$ up to reparameterisation is an equivalence relation.

We can then phrase Cauchy’s theorem as an assertion that contour integration on holomorphic functions is a homotopy invariant. More precisely:

Theorem 4 (Cauchy’s theorem) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be holomorphic.

• (i) If ${\gamma_0: [a,b] \rightarrow U}$ and ${\gamma_1: [c,d] \rightarrow U}$ are rectifiable curves that are homotopic in ${U}$ with fixed endpoints up to reparameterisation, then

$\displaystyle \int_{\gamma_0} f(z)\ dz = \int_{\gamma_1} f(z)\ dz.$

• (ii) If ${\gamma_0: [a,b] \rightarrow U}$ and ${\gamma_1: [c,d] \rightarrow U}$ are closed rectifiable curves that are homotopic in ${U}$ as closed curves up to reparameterisation, then

$\displaystyle \int_{\gamma_0} f(z)\ dz = \int_{\gamma_1} f(z)\ dz.$

This version of Cauchy’s theorem is particularly useful for applications, as it explicitly brings into play the powerful technique of contour shifting, which allows one to compute a contour integral by replacing the contour with a homotopic contour on which the integral is easier to either compute or integrate. This formulation of Cauchy’s theorem also highlights the close relationship between contour integrals and the algebraic topology of the complex plane (and open subsets ${U}$ thereof). Setting ${\gamma_1}$ to be a point, we obtain an important special case of Cauchy’s theorem (which is in fact equivalent to the full theorem):

Corollary 5 (Cauchy’s theorem, again) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be holomorphic. Then for any closed rectifiable curve ${\gamma}$ in ${U}$ that is contractible in ${U}$ to a point, one has ${\int_\gamma f(z)\ dz = 0}$.

Exercise 6 Show that Theorem 4 and Corollary 5 are logically equivalent.

An important feature to note about Cauchy’s theorem is the global nature of its hypothesis on ${f}$. The conclusion of Cauchy’s theorem only involves the values of a function ${f}$ on the images of the two curves ${\gamma_0, \gamma_1}$. However, in order for the hypotheses of Cauchy’s theorem to apply, the function ${f}$ must be holomorphic not only on the images on ${\gamma_0, \gamma_1}$, but on an open set ${U}$ that is large enough (and sufficiently free of “holes”) to support a homotopy between the two curves. This point can be emphasised through the following fundamental near-counterexample to Cauchy’s theorem:

Example 7 Let ${U := {\bf C} \backslash \{0\}}$, and let ${f: U \rightarrow {\bf C}}$ be the holomorphic function ${f(z) := \frac{1}{z}}$. Let ${\gamma_{0,1,\circlearrowleft}: [0,2\pi] \rightarrow {\bf C}}$ be the closed unit circle contour ${\gamma_{0,1,\circlearrowleft}(t) := e^{it}}$. Direct calculation shows that

$\displaystyle \int_{\gamma_{0,1,\circlearrowleft}} f(z)\ dz = 2\pi i \neq 0.$

As a consequence of this and Cauchy’s theorem, we conclude that the contour ${\gamma_{0,1,\circlearrowleft}}$ is not contractible to a point in ${U}$; note that this does not contradict Example 2 because ${U}$ is not convex. Thus we see that the lack of holomorphicity (or singularity) of ${f}$ at the origin can be “blamed” for the non-vanishing of the integral of ${f}$ on the closed contour ${\gamma_{0,1,\circlearrowleft}}$, even though this contour does not come anywhere near the origin. Thus we see that the global behaviour of ${f}$, not just the behaviour in the local neighbourhood of ${\gamma_{0,1,\circlearrowleft}}$, has an impact on the contour integral.

One can of course rewrite this example to involve non-closed contours instead of closed ones. For instance, if we let ${\gamma_0, \gamma_1: [0,\pi] \rightarrow U}$ denote the half-circle contours ${\gamma_0(t) := e^{it}}$ and ${\gamma_1(t) := e^{-it}}$, then ${\gamma_0,\gamma_1}$ are both contours in ${U}$ from ${+1}$ to ${-1}$, but one has

$\displaystyle \int_{\gamma_0} f(z)\ dz = +\pi i$

whereas

$\displaystyle \int_{\gamma_1} f(z)\ dz = -\pi i.$

In order for this to be consistent with Cauchy’s theorem, we conclude that ${\gamma_0}$ and ${\gamma_1}$ are not homotopic in ${U}$ (even after reparameterisation).

In the specific case of functions of the form ${\frac{1}{z}}$, or more generally ${\frac{f(z)}{z-z_0}}$ for some point ${z_0}$ and some ${f}$ that is holomorphic in some neighbourhood of ${z_0}$, we can quantify the precise failure of Cauchy’s theorem through the Cauchy integral formula, and through the concept of a winding number. These turn out to be extremely powerful tools for understanding both the nature of holomorphic functions and the topology of open subsets of the complex plane, as we shall see in this and later notes.

— 1. Proof of Cauchy’s theorem —

The underlying reason for the truth of Cauchy’s theorem can be explained in one sentence: complex differentiable functions behave locally like complex linear functions, which are conservative thanks to the fundamental theorem of calculus. More precisely, if ${f(z) = a + bz}$ is any complex linear function of ${z}$, then ${f}$ has an antiderivative ${az + \frac{1}{2} bz^2}$, and hence

$\displaystyle \int_\gamma a+bz\ dz = 0 \ \ \ \ \ (1)$

for any rectifiable closed curve ${\gamma}$ in the complex plane.

Perhaps the slickest way to make this intuition rigorous is through the following special case of Cauchy’s theorem.

Theorem 8 (Goursat’s theorem) Let ${U}$ be an open subset of ${{\bf C}}$, and ${z_1,z_2,z_3}$ be complex numbers such that the solid (and closed) triangle spanned by ${z_1,z_2,z_3}$ (or more precisely, the convex hull of ${\{z_1,z_2,z_3\}}$) is contained in ${U}$. (We allow the triangle to degenerate in that we allow the ${z_1,z_2,z_3}$ to be collinear, or even coincident.) Then for any holomorphic function ${f: U \rightarrow {\bf C}}$, one has

$\displaystyle \int_{\gamma_{z_1 \rightarrow z_2 \rightarrow z_3 \rightarrow z_1}} f(z)\ dz = 0,$

where ${\gamma_{z_1 \rightarrow z_2 \rightarrow z_3 \rightarrow z_1}}$ is the closed polygonal path that traverses the vertices ${z_1, z_2, z_3}$ of the solid triangle in order.

Proof: Let us denote the triangular contour ${\gamma_{z_1 \rightarrow z_2 \rightarrow z_3 \rightarrow z_1}}$ as ${T_0}$. It is convenient (though odd-looking at first sight) to prove this theorem by contradiction. That is to say, suppose for contradiction that we had

$\displaystyle |\int_{T_0} f(z)\ dz| \geq \varepsilon \ \ \ \ \ (2)$

for some ${\varepsilon > 0}$. We now run the following “divide and conquer” strategy. We let ${z_{12} := \frac{z_1+z_2}{2}}$, ${z_{23} := \frac{z_2+z_3}{2}}$, ${z_{31} := \frac{z_3 + z_1}{2}}$ be the midpoints of ${z_1, z_2, z_3}$. Then from the basic properties of contour integration (see Exercise 16 of Notes 2) we can split the triangular integral ${\int_{T_1} f(z)\ dz}$ as the sum of four integrals on smaller triangles, namely

$\displaystyle \int_{\gamma_{z_1 \rightarrow z_{12} \rightarrow z_{31} \rightarrow z_1}} f(z)\ dz$

$\displaystyle \int_{\gamma_{z_{12} \rightarrow z_2 \rightarrow z_{23} \rightarrow z_{12}}} f(z)\ dz$

$\displaystyle \int_{\gamma_{z_{23} \rightarrow z_3 \rightarrow z_{31} \rightarrow z_{23}}} f(z)\ dz$

$\displaystyle \int_{\gamma_{z_{12} \rightarrow z_{23} \rightarrow z_{31} \rightarrow z_{12}}} f(z)\ dz.$

(The reader is encouraged to draw a picture to visualise this decomposition.) By (2) and the triangle inequality (or, if one prefers, the pigeonhole principle), we must therefore have

$\displaystyle |\int_{T_1} f(z)\ dz| \geq \frac{\varepsilon}{4}$

where ${T_1}$ is one of the four triangular contours ${\gamma_{z_1 \rightarrow z_{12} \rightarrow z_{31} \rightarrow z_1}}$, ${\gamma_{z_{12} \rightarrow z_2 \rightarrow z_{23} \rightarrow z_{12}}}$, ${\gamma_{z_{23} \rightarrow z_3 \rightarrow z_{31} \rightarrow z_{23}}}$, or ${\gamma_{z_{12} \rightarrow z_{23} \rightarrow z_{31} \rightarrow z_{12}}}$. Regardless of which of the four contours ${T_1}$ is, observe that the triangular region enclosed by ${T_1}$ is contained in that of ${T_0}$. Furthermore, the diameter of ${T_1}$ is precisely half that of ${T_0}$, where the diameter ${\mathrm{diam}(\gamma)}$ of a curve ${\gamma: [a,b] \rightarrow {\bf C}}$ is defined by the formula

$\displaystyle \mathrm{diam}(\gamma) := \sup_{t, t' \in [a,b]} |\gamma(t) - \gamma(t')|;$

similarly, the perimeter ${|T_1|}$ of ${T_1}$ is precisely half that of ${T_0}$. If we iterate the above process, we can find a nested sequence ${T_0, T_1, T_2, \dots}$ of triangular contours, each of which is contained in the previous one with half the diameter and perimeter, such that

$\displaystyle |\int_{T_n} f(z)\ dz| \geq \frac{\varepsilon}{4^n} \ \ \ \ \ (3)$

for all ${n=0,1,2,\dots}$. If we let ${z_n}$ be any point enclosed by ${T_n}$, then from the decreasing diameters it is clear that the ${z_n}$ are a Cauchy sequence and thus converge to some limit ${z_*}$, which is then contained in all of the closed triangles enclosed by any of the ${T_n}$.

In particular, ${z_*}$ lies in ${U}$ and so ${f}$ is differentiable at ${z_*}$. This implies, for any ${\varepsilon'>0}$, that there exists a ${\delta > 0}$ such that

$\displaystyle |\frac{f(z) - f(z_*)}{z-z_*} - f'(z_*)| \leq \varepsilon'$

whenever ${z \in D(z_*,\delta) \backslash \{z_*\}}$. We can rearrange this as

$\displaystyle |f(z) - (f(z_*) + (z-z_*) f'(z_*))| \leq \varepsilon' |z-z_*|$

on ${D(z_*,\delta)}$. In particular, for ${n}$ large enough, this bound holds on the image on ${T_n}$. In this case we can bound ${|z-z_*|}$ by ${\mathrm{diam}(T_n)}$, and hence by Exercise 16(v) of Notes 2,

$\displaystyle |\int_{T_n} f(z)\ dz - \int_{T_n} (f(z_*) + (z-z_*) f'(z_*))\ dz| \leq \varepsilon' \hbox{diam}(T_n) |T_n|.$

From (1), the second integral vanishes. As each ${T_n}$ has half the diameter and perimeter of the previous, we thus have

$\displaystyle |\int_{T_n} f(z)\ dz| \leq \frac{\varepsilon' \hbox{diam}(T_0) |T_0|}{4^n}.$

But if one chooses ${\varepsilon'}$ small enough depending on ${\varepsilon}$ and ${T_0}$, we contradict (3). $\Box$

Remark 9 This is a rare example of an argument in which a hypothesis of differentiability, rather than continuous differentiability, is used, because one can localise any failure of the conclusion all the way down to a single point. Another instance of such an argument is the standard proof of Rolle’s theorem.

Exercise 10 Find a proof of Goursat’s theorem that avoids explicit use of proof by contradiction. (Hint: use the fact that a solid triangle is compact, in the sense that every open cover has a finite subcover. For the purposes of this question, ignore the possibility that the proof of this latter fact might also use proof by contradiction.)

Goursat’s theorem only directly handles triangular contours, but as long as one works “locally”, or more precisely in a convex domain, we can quickly generalise:

Corollary 11 (Local Cauchy’s theorem for polygonal paths) Let ${U}$ be a convex open subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be a holomorphic function. Then for any closed polygonal path ${\gamma = \gamma_{z_1 \rightarrow \dots \rightarrow z_n \rightarrow z_1}}$ in ${U}$, we have ${\int_\gamma f(z)\ dz = 0}$.

Proof: We induct on the number of vertices ${n}$. The cases ${n=1,2}$ are trivial, and the ${n=3}$ case follows directly from Goursat’s theorem (using the convexity of ${U}$ to ensure that the interior of the polygon lies in ${U}$). If ${n > 3}$, we can split

$\displaystyle \int_{\gamma_{z_1 \rightarrow \dots \rightarrow z_n \rightarrow z_1}} f(z)\ dz = \int_{\gamma_{z_1 \rightarrow \dots \rightarrow z_{n-1} \rightarrow z_1}} f(z)\ dz + \int_{z_{n-1} \rightarrow z_n \rightarrow z_1 \rightarrow z_{n-1}} f(z)\ dz.$

The second integral on the right-hand side vanishes by Goursat’s theorem. The claim then follows from induction. $\Box$

Exercise 12 By using the (real-variable) fundamental theorem of calculus and Fubini’s theorem in place of Goursat’s theorem, give an alternate proof of Corollary 11 in the case that ${\gamma}$ is a rectangle ${\gamma = \gamma_{a+bi \rightarrow c+bi \rightarrow c+di \rightarrow a+di \to a+bi}}$ and the derivative ${f'}$ of ${f}$ is continuous. (One can also use Stokes’ theorem in place of the fundamental theorem of calculus and Fubini’s theorem.)

We can amplify Corollary 11 using the fundamental theorem of calculus again:

Corollary 13 (Local Cauchy’s theorem) Let ${U}$ be a convex open subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be a holomorphic function. Then ${f}$ has an antiderivative ${F: U \rightarrow {\bf C}}$. Also, ${\int_\gamma f(z)\ dz = 0}$ for any closed rectifiable curve ${\gamma}$ in ${U}$, and ${\int_{\gamma_1} f(z)\ dz = \int_{\gamma_2} f(z)\ dz}$ whenever ${\gamma_1, \gamma_2}$ are two rectifiable curves in ${U}$ with the same initial point and same terminal point. In other words, ${f}$ is conservative on ${U}$.

Proof: The first claim follows from Corollary 11 and the second fundamental theorem of calculus (Theorem 30 from Notes 2). The remaining claims then follow from the first fundamental theorem of calculus (Theorem 27 from Notes 2). $\Box$

We can now prove Cauchy’s theorem in the form of Theorem 4.

Proof: We will just prove part (i), as part (ii) is similar (and in any event it follows from part (i)). Since reparameterisation does not affect the integral, we may assume without loss of generality that ${\gamma_0: [a,b] \rightarrow U}$ and ${\gamma_1: [a,b] \rightarrow U}$ are homotopic with fixed endpoints, and not merely homotopic with fixed endpoints up to reparameterisation.

Let ${\gamma: [0,1] \times [a,b] \rightarrow U}$ be a homotopy from ${\gamma_0}$ to ${\gamma_1}$. Note that for any ${s \in [0,1]}$ and ${t \in [a,b]}$, ${\gamma(s,t)}$ lies in the open set ${U}$. From compactness, there must exist a radius ${r>0}$ such that ${D(\gamma(s,t),r) \subset U}$ for all ${s \in [0,1]}$ and ${t \in [a,b]}$. Next, as ${\gamma}$ is continuous on a compact set, it is uniformly continuous. In particular, there exists ${\delta > 0}$ such that

$\displaystyle |\gamma(s',t') - \gamma(s,t)| \leq \frac{r}{4}$

whenever ${s,s' \in [0,1]}$ and ${t,t' \in [a,b]}$ are such that ${|s-s'| \leq \delta}$ and ${|t-t'| \leq \delta}$.

Now partition ${[0,1]}$ and ${[a,b]}$ as ${0 = s_0 < \dots < s_n = 1}$ and ${a = t_0 < \dots < t_m = b}$ in such a way that ${|s_i - s_{i-1}| \leq \delta}$ and ${|t_j-t_{j-1}| \leq \delta}$ for all ${1 \leq i \leq n}$ and ${1 \leq j \leq m}$. For each such ${i}$ and ${j}$, let ${C_{i,j}}$ denote the closed polygonal contour

$\displaystyle C_{i,j} := \gamma_{\gamma(s_i,t_{j-1}) \rightarrow \gamma(s_i,t_j) \rightarrow \gamma(s_{i-1},t_j) \rightarrow \gamma(s_{i-1},t_{j-1}) \rightarrow \gamma(s_i,t_{j-1})}.$

(the reader is encouraged here to draw a picture of the situation; we are using polygonal contours here rather than the homotopy ${\gamma}$ because we did not require any rectifiability properties on the homotopy). By construction, the diameter of this contour is at most ${\frac{r}{4}+\frac{4}{4}+\frac{r}{4}+\frac{r}{4} = r}$, so the contour is contained entirely in the disk ${D( \gamma(s_i,t_i), r)}$. This disk is convex and contained in ${U}$. Applying Corollary 11 or Corollary 13, we conclude that

$\displaystyle \int_{C_{i,j}} f(z)\ dz = 0$

for all ${1 \leq i \leq n}$ and ${1 \leq j \leq m}$. If we sum this over all ${i}$ and ${j}$, and noting that the homotopy fixes the endpoints, we conclude after a lot of cancelling that

$\displaystyle \int_{\gamma(0,t_0) \rightarrow \gamma(0, t_1) \rightarrow \dots \rightarrow \gamma(0, t_n)} f(z)\ dz = \int_{\gamma(1,t_0) \rightarrow \gamma(1, t_1) \rightarrow \dots \rightarrow \gamma(1, t_n)} f(z)\ dz$

(again, the reader is encouraged to draw a picture to see this cancellation). However, from a further application of Corollary 13 we have

$\displaystyle \int_{\gamma(0,t_{i-1}) \rightarrow \gamma_{0,t_i}} f(z)\ dz = \int_{\gamma_{0,[t_{i-1},t_i]}} f(z)\ dz$

for ${i=1,\dots,n}$, where ${\gamma_{0,[t_{i-1},t_i]}: [t_{i-1},t_i] \rightarrow U}$ is the restriction of ${\gamma: [a,b] \rightarrow U}$ to ${[t_{i-1},t_i]}$, and similarly for ${\gamma_1}$. Putting all this together we conclude that

$\displaystyle \int_{\gamma_0} f(z)\ dz = \int_{\gamma_1} f(z)\ dz$

as required. $\Box$

One nice feature of Cauchy’s theorem is that it allows one to integrate holomorphic functions on curves that are not necessarily rectifiable. Indeed, if ${\gamma: [a,b] \rightarrow U}$ is a curve in ${U}$, then for a sufficiently fine partition ${a = t_0 < t_1 < \dots < t_n = b}$, the polygonal (and hence rectifiable) path ${\gamma_{t_0 \rightarrow t_1 \rightarrow \dots \rightarrow t_n}}$ will be contained in ${U}$, and furthermore be homotopic to ${\gamma}$ with fixed endpoints. One can then define ${\int_\gamma f(z)\ dz}$ when ${f}$ is holomorphic in ${U}$ and ${\gamma}$ is non-rectifiable by declaring

$\displaystyle \int_\gamma f(z)\ dz := \int_{\tilde \gamma} f(z)\ dz$

where ${\tilde \gamma}$ is any rectifiable curve that is homotopic (with fixed endpoints) to ${\gamma}$. This is a well defined definition thanks to the above discussion as well as Cauchy’s theorem; also observe that the exact open set ${U}$ in which the homotopy lives is not relevant, since given any two open sets ${U,U'}$ containing the image of ${\gamma}$ one can find a rectifiable curve ${\tilde \gamma}$ which is homotopic to ${\gamma}$ with fixed endpoints in ${U \cap U'}$, and hence in ${U}$ and ${U'}$ separately. With this extended notion of the contour integral, one can then remove the hypothesis of rectifiability from many theorems involving integration of holomorphic functions. In particular, Cauchy’s theorem itself now holds for non-rectifiable curves. This reflects some duality in the integration concept ${\int_\gamma f(z)\ dz}$; if one assumes more regularity on the function ${f}$, one can get away with worse regularity on the curve ${\gamma}$, and vice versa.

A special case of Cauchy’s theorem is worth recording explicitly. We say that an open set ${U}$ in the complex plane is simply connected if it is non-empty, connected, and if every closed curve in ${U}$ is contractible in ${U}$ to a point. For instance, from Example 2 we see that any convex non-empty open set is simply connected. From Theorem 4 we then have

Theorem 14 (Cauchy’s theorem, simply connected case) Let ${U}$ be a simply connected subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be holomorphic. Then ${\int_\gamma f(z)\ dz = 0}$ for any closed curve in ${U}$. In particular (by Exercise 31 of Notes 2), ${f}$ is conservative and has an antiderivative.

— 2. Consequences of Cauchy’s theorem —

Now that we have Cauchy’s theorem, we use it to quickly give a large number of striking consequences. We begin with a special case of the Cauchy integral formula.

Theorem 15 (Cauchy integral formula, special case) Let ${U}$ be an open subset of ${{\bf C}}$, let ${f: U \rightarrow {\bf C}}$ be holomorphic, and let ${z_0}$ be a point in ${U}$. Let ${r>0}$ be such that the closed disk ${\overline{D(z_0,r)} := \{ z \in {\bf C}: |z-z_0| \leq r \}}$ is contained in ${U}$. Let ${\gamma}$ be a closed curve in ${U \backslash \{z_0\}}$ that is homotopic (as a closed curve, and up to reparameterisation) in ${U \backslash \{z_0\}}$ to ${\gamma_{z_0,r,\circlearrowleft}}$ in ${U}$. Then

$\displaystyle f(z_0) = \frac{1}{2\pi i} \int_\gamma \frac{f(z)}{z-z_0}\ dz.$

Here we are already taking advantage of the ability to integrate holomorphic functions (such as ${\frac{f(z)}{z-z_0}}$, which is holomorphic on ${U \backslash \{z_0\}}$) on curves ${\gamma}$ that are not necessarily rectifiable.

Note the remarkable feature here that the value of ${f}$ at some point other than that on ${\gamma}$ is completely determined by the value of ${f}$ on the curve ${\gamma}$, which is a strong manifestation of the “rigid” or “global” nature of holomorphic functions. Such a formula is certainly not available in the real case (Cauchy’s theorem is technically true on the real line, but there is no analogue of the circular contours ${\gamma_{z_0,r,\circlearrowleft}}$ available in that setting).

Proof: Observe that for any ${0 < \varepsilon < r}$, the circles ${\gamma_{z_0,r,\circlearrowleft}}$ and ${\gamma_{z_0,\varepsilon,\circlearrowleft}}$ are homotopic (as closed curves) in ${\overline{D(z_0,r)}}$, and hence in ${U}$. Since the function ${z \mapsto \frac{f(z)}{z-z_0}}$ is holomorphic on ${U \backslash \{z_0\}}$, we conclude from Cauchy’s theorem that

$\displaystyle \int_\gamma \frac{f(z)}{z-z_0}\ dz = \int_{\gamma_{z_0,\varepsilon,\circlearrowleft}} \frac{f(z)}{z-z_0}\ dz$

As ${f}$ is complex differentiable at ${z_0}$, there exists a finite ${M}$ such that

$\displaystyle |\frac{f(z)-f(z_0)}{z-z_0}| \leq M$

for all ${z}$ in ${\gamma_{z_0,\varepsilon,\circlearrowleft}}$, and all sufficiently small ${\varepsilon}$. The length of this circle is of course ${2\pi \varepsilon}$. Applying Exercise 16(v) of Notes 2 we have

$\displaystyle | \int_{\gamma_{z_0,\varepsilon,\circlearrowleft}} \frac{f(z)-f(z_0)}{z-z_0}\ dz| \leq 2 \pi \varepsilon M.$

On the other hand, from explicit computation (cf. Example 7) we have

$\displaystyle \int_{\gamma_{z_0,\varepsilon,\circlearrowleft}} \frac{1}{z-z_0}\ dz = 2\pi i;$

putting all this together, we see that

$\displaystyle |\frac{1}{2\pi i} \int_\gamma \frac{f(z)}{z-z_0}\ dz - f(z_0)| \leq \varepsilon M.$

Sending ${\varepsilon}$ to zero, we obtain the claim. $\Box$

Note the same argument would give

$\displaystyle m f(z_0) = \frac{1}{2\pi i} \int_\gamma \frac{f(z)}{z-z_0}\ dz$

if ${\gamma}$ were homotopic to the curve ${t \mapsto z_0 + r e^{i m t}}$: ${0 \leq t \leq 2\pi}$ rather than ${\gamma_{z_0,r,\circlearrowleft}}$, for some integer ${m}$. In particular, if ${\gamma}$ were homotopic to a point in ${U \backslash \{z_0\}}$, then the right-hand side would vanish.

Remark 16 For various explicit examples of closed contours ${\gamma}$, it is also possible to prove the Cauchy integral formula by applying Cauchy’s theorem to various “keyhole contours”. We will not pursue this approach here, but see for instance Chapter 2 of Stein-Shakarchi.

Exercise 17 (Mean value property and Poisson kernel) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${\overline{D(z_0,r)}}$ be a closed disk contained in ${U}$.

• (i) If ${f: U \rightarrow {\bf C}}$ is holomorphic, show that

$\displaystyle f(z_0) = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + re^{i\theta})\ d\theta.$.  Use this to give an alternate proof of Exercise 26 from Notes 1.

• (ii) If ${u: U \rightarrow {\bf R}}$ is harmonic, show that

$\displaystyle u(z_0) = \frac{1}{2\pi} \int_0^{2\pi} u(z_0 + re^{i\theta})\ d\theta.$.  Use this to give an alternate proof of Theorem 25 from Notes 1.

• (iii) If ${u: U \rightarrow {\bf R}}$ is harmonic, show that

$\displaystyle u(z) = \frac{1}{2\pi} \int_0^{2\pi} P( \frac{z - z_0}{re^{i\theta}} ) u(z_0 + re^{i\theta})\ d\theta$

for any ${z \in D(z_0,r)}$, where the Poisson kernel ${P: D(0,1) \rightarrow {\bf R}}$ is defined by the formula

$\displaystyle P(z) := \mathrm{Re} \frac{1 + z}{1-z}.$

(Hint: it simplifies the calculations somewhat if one reduces to the case ${z_0=0}$, ${r=1}$, and ${z = s}$ for some ${0 < s < 1}$. Then compute the integral ${\frac{1}{2\pi i} \int_{\gamma_{0,1,\circlearrowleft}} f(w) \frac{1}{2} (\frac{1+s/w}{1-s/w} + \frac{1+sw}{1-sw})\ dw}$ in two different ways, where ${f=u+iv}$ is holomorphic with real part ${u}$.)

The first important consequence of the Cauchy integral formula is the analyticity of holomorphic functions:

Corollary 18 (Holomorphic functions are analytic) Let ${U}$ be an open subset of ${{\bf C}}$, let ${f: U \rightarrow {\bf C}}$ be holomorphic, and let ${z_0}$ be a point in ${U}$. Let ${r>0}$ be such that the closed disk ${\overline{D(z_0,r)} := \{ z \in {\bf C}: |z-z_0| \leq r \}}$ is contained in ${U}$. For each natural number ${n}$, let ${a_n}$ denote the complex number

$\displaystyle a_n := \frac{1}{2\pi i} \int_{\gamma_{z_0,r,\circlearrowleft}} \frac{f(z)}{(z-z_0)^{n+1}}\ dz. \ \ \ \ \ (4)$

Then the power series ${\sum_{n=0}^\infty a_n (z-z_0)^n}$ has radius of convergence at least ${r}$, and converges to ${f(z)}$ inside the disk.

Proof: By continuity, there exists a finite ${M}$ such that ${|f(z)| \leq M}$ for all ${z}$ on the circle ${\gamma_{z_0,r,\circlearrowleft}}$, which of course has length ${2\pi r}$. From Exercise 16(v) of Notes 2 we conclude that

$\displaystyle |a_n| \leq \frac{1}{2\pi} 2\pi r \frac{M}{r^{n+1}}.$

From this and Proposition 7 of Notes 1, we see that the radius of convergence of ${a_n}$ is indeed at least ${r}$.

Next, for any ${w \in D(z_0,r)}$, the circle ${\gamma_{z_0,r,\circlearrowleft}}$ is homotopic (as a closed curve) in ${\overline{D(z_0,r)} \backslash \{w\}}$ (and hence in ${U \backslash \{w\})}$ to ${\gamma_{w,\varepsilon,\circlearrowleft}}$ to ${\varepsilon}$ small enough that ${D(w,\varepsilon)}$ lies in ${D(z_0,r)}$. Applying the Cauchy integral formula, we conclude that

$\displaystyle f(w) = \frac{1}{2\pi i} \int_{\gamma_{z_0,r,\circlearrowleft}} \frac{f(z)}{z-w}\ dz.$

On the other hand, from the geometric series formula (Exercise 12 of Notes 1) one has

$\displaystyle \frac{1}{z-w} = \sum_{n=0}^\infty \frac{1}{(z-z_0)^{n+1}} (w-z_0)^n$

for all ${w \in D(z_0,r)}$, and thus

$\displaystyle f(w) = \frac{1}{2\pi i} \int_{\gamma_{z_0,r,\circlearrowleft}} (\sum_{n=0}^\infty \frac{f(z)}{(z-z_0)^{n+1}} (w-z_0)^n)\ dz.$

If we could interchange the sum and integral, we would conclude from (4) that

$\displaystyle f(w) = \sum_{n=0}^\infty a_n (w-z_0)^n$

which would give the claim. To justify the interchange, we will use the Weierstrass ${M}$-test (the dominated convergence theorem would also work here). We have the pointwise bound

$\displaystyle |\frac{f(z)}{(z-z_0)^{n+1}} (w-z_0)^n| \leq \frac{M}{r^{n+1}} |w-z_0|^n;$

by the geometric series formula and the hypothesis ${w \in D(z_0,r)}$, the sum ${\sum_{n=0}^\infty \frac{M}{r^{n+1}} |w-z_0|^n}$ is finite, and so the ${M}$-test applies and we are done. $\Box$

Remark 19 A function ${f: U \rightarrow {\bf C}}$ on an open set ${U \subset {\bf C}}$ is said to be complex analytic on ${U}$ if, for every ${z_0 \in U}$, there is a power series ${\sum_{n=0}^\infty a_n(z-z_0)^n}$ with a positive radius of convergence that converges to ${f}$ on some neighbourhood of ${z_0}$. Combining the above corollary with Theorem 15 of Notes 1, we see that ${f}$ is holomorphic on ${U}$ if and only if ${f}$ is complex analytic on ${U}$; thus the terms “complex differentiable”, “holomorphic”, and “complex analytic” may be used interchangeably. This can be contrasted with real variable case: there is a completely parallel notion of a real analytic function ${f: (a,b) \rightarrow {\bf R}}$ (i.e., a function that, around every point ${x_0}$ in the domain, can be expanded as a convergent power series in some neighbourhood of that point), and real analytic functions are automatically smooth and differentiable, but the converse is quite false.

Recalling (see Remark 21 of Notes 1) that power series are infinitely differentiable (in both the real and complex senses) inside their disk of convergence, and working locally in various small disks in ${U}$, we conclude

Corollary 20 Let ${U}$ be an open subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be a holomorphic function. Then ${f': U \rightarrow {\bf C}}$ is also holomorphic, and ${f}$ is smooth (i.e. infinitely differentiable in the real sense).

In view of this corollary, we may now drop hypotheses of continuous first or second differentiability from several of the theorems in Notes 1, such as Exercise 26 from that set of notes.

Combining Corollary 20 with Proposition 28 of Notes 1 (with ${{\bf C}}$ replaced by various rectangles in ${U}$), we obtain a form of elliptic regularity:

Corollary 21 (Elliptic regularity) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${u: U \rightarrow {\bf R}}$ be a harmonic function. Then ${u}$ is smooth.

In fact one can even omit the hypothesis of continuous twice differentiability in the definition of harmonicity if one works with the notion of weak harmonicity, but this is a topic for a PDE or distribution theory course and will not be pursued further here.

Another immediate consequence of Corollary 18 is a version of the factor theorem:

Corollary 22 (Factor theorem for analytic functions) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${z_0}$ be a point in ${{\bf C}}$. Let ${f: U \rightarrow {\bf C}}$ be a complex analytic function that vanishes at ${z_0}$. Then there exists a unique complex analytic function ${g: U \rightarrow {\bf C}}$ such that ${f(z) = g(z) (z-z_0)}$ for all ${z \in U}$.

Proof:  For ${z \neq z_0}$, we can simply define ${g(z) := f(z)/(z-z_0)}$, and this is clearly the unique choice here.  For ${z}$ equal to or near ${z_0}$, we can expand ${f}$ as a Taylor series ${f(z) = \sum_{n=1}^\infty a_n (z-z_0)^n}$ (noting that the constant term vanishes since ${f(z_0)=0}$) and then set ${g(z) := \sum_{n=0}^\infty a_{n+1} (z-z_0)^n}$. One can check that these two definitions of ${g}$ agree on their common domain, so on gluing the two functions together one obtains a function ${g}$ that is complex differentiable (and hence analytic) on ${U}$ with the desired factorisation. Uniqueness at ${z_0}$ then follows from uniqueness at ${z \neq z_0}$ and continuity. $\Box$

Yet another consequence is the important property of analytic continuation:

Corollary 23 (Analytic continuation) Let ${U}$ be a connected non-empty open subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$, ${g: U \rightarrow {\bf C}}$ be complex analytic functions. If ${f}$ and ${g}$ agree on some non-empty open subset of ${U}$, then they in fact agree on all of ${U}$.

Proof: Let ${V}$ denote the set of all points ${z_0}$ in ${U}$ where ${f}$ and ${g}$ agree to all orders, that is to say that

$\displaystyle f^{(n)}(z_0) = g^{(n)}(z_0)$

for all ${n=0,1,\dots}$. By hypothesis, ${V}$ is non-empty; by the continuity of the ${f^{(n)}}$, ${V}$ is closed; and from analyticity and Taylor expansion (Exercise 17 of Notes 1) ${V}$ is open. As ${U}$ is connected, ${V}$ must therefore be all of ${U}$, and the claim follows. (This is an example of the continuity method.) $\Box$

There is also a variant of the above corollary:

Corollary 24 (Non-trivial analytic functions have isolated zeroes) Let ${U}$ be a connected non-empty open subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be a function which vanishes at some point ${z_0 \in {\bf C}}$ but is not identically zero. Then there exists a disk ${D(z_0,r)}$ in ${U}$ on which ${f}$ does not vanish except at ${z_0}$; in other words, all the zeroes of ${f}$ are isolated points.

Proof: If all the derivatives ${f^{(n)}(z_0)}$ of ${f}$ at ${z_0}$ vanish, then by Taylor expansion ${f}$ vanishes in some open neighbourhood of ${z_0}$, and then by Corollary 23 ${f}$ vanishes everywhere, a contradiction. Thus at least one of the ${f^{(n)}(z_0)}$ is non-zero. If ${n_0}$ is the first natural number for which ${f^{(n_0)}(z_0) \neq 0}$, then by iterating the factor theorem (Corollary 22) we see that ${f(z) = (z-z_0)^{n_0} g(z)}$ for some analytic function ${g:U \rightarrow {\bf C}}$ which is non-vanishing at ${z_0}$. By continuity, ${g}$ is also non-vanishing in some disk ${D(z_0,r)}$ in ${U}$, and the claim follows. $\Box$

One particular consequence of the above corollary is that if two entire functions ${f,g}$ agree on the real line (or even on an infinite bounded subset of the complex plane), then they must agree everywhere, since otherwise ${f-g}$ would have a non-isolated zero, contradicting Corollary 24. This strengthens Corollary 23, and helps explain why real-variable identities such as ${\sin^2(x)+\cos^2(x)=1}$ automatically extend to their complex counterparts ${\sin^2(z) + \cos^2(z) = 1}$. Another consequence is that if an entire function ${f: {\bf C} \rightarrow {\bf C}}$ is real-valued on the real axis, then one has the identity

$\displaystyle f(z) = \overline{f(\overline{z})}$

for all complex ${z}$, because this identity already holds on the real line, and both sides are complex analytic. Thus for instance

$\displaystyle \sin(z) = \overline{\sin(\overline{z})}.$

Next, if we combine Corollary 18 with Exercise 17 of Notes 1, as well as Cauchy’s theorem, we obtain

Theorem 25 (Higher order Cauchy integral formula, special case) Let ${U}$ be an open subset of ${{\bf C}}$, let ${f: U \rightarrow {\bf C}}$ be holomorphic, and let ${z_0}$ be a point in ${U}$. Let ${r>0}$ be such that the closed disk ${\overline{D(z_0,r)} := \{ z \in {\bf C}: |z-z_0| \leq r \}}$ is contained in ${U}$. Let ${\gamma}$ be a closed curve in ${U \backslash \{z_0\}}$ that is homotopic (as a closed curve, up to reparameterisation) in ${U \backslash \{z_0\}}$ to ${\gamma_{z_0,r,\circlearrowleft}}$ in ${U}$. Then for any natural number ${n}$, the ${n^{\mathrm{th}}}$ derivative ${f^{(n)}(z_0)}$ of ${f}$ at ${z_0}$ is given by the formula

$\displaystyle f^{(n)}(z_0) = \frac{n!}{2\pi i} \int_\gamma \frac{f(z)}{(z-z_0)^{n+1}}\ dz.$

Exercise 26 Give an alternate proof of Theorem 25 by rigorously differentiating the Cauchy integral formula with respect to the ${z_0}$ parameter.

Combining Theorem 25 with Exercise 16(v) of Notes 2, we obtain a more quantitative form of Corollary 20, which asserts not only that the higher derivatives of a holomorphic function exist, but also places a bound on them:

Corollary 27 (Cauchy inequalities) Let ${U}$ be an open subset of ${{\bf C}}$, let ${f: U \rightarrow {\bf C}}$ be holomorphic, and let ${z_0}$ be a point in ${U}$. Let ${r>0}$ be such that the closed disk ${\overline{D(z_0,r)} := \{ z \in {\bf C}: |z-z_0| \leq r \}}$ is contained in ${U}$. Suppose that there is an ${M}$ such that ${|f(z)| \leq M}$ on the circle ${\{ z \in {\bf C}: |z-z_0| = r\}}$. Then for any natural number ${n}$, we have

$\displaystyle |f^{(n)}(z_0)| \leq \frac{n!}{r^n} M. \ \ \ \ \ (5)$

Note that the ${n=0}$ case of this corollary is compatible with the maximum principle (Exercise 26 of Notes 1).

The right-hand side of (5) has a denominator ${r^n}$ that improves when ${r}$ gets large. In particular we have the remarkable theorem of Liouville:

Theorem 28 (Liouville’s theorem) Let ${f: {\bf C} \rightarrow {\bf C}}$ be an entire function that is bounded. Then ${f}$ is constant.

Proof: By hypothesis, there is a finite ${M}$ such that ${|f(z)| \leq M}$ for all ${M}$. Applying the Cauchy inequalities with ${n=1}$ and any disk ${D(z_0,r)}$, we conclude that

$\displaystyle |f'(z_0)| \leq \frac{M}{r}$

for any ${z_0 \in {\bf C}}$ and ${r>0}$. Sending ${r}$ to infinity, we conclude that ${f'}$ vanishes identically. The claim then follows from the fundamental theorem of calculus. $\Box$

This theorem displays a strong “rigidity” property for entire functions; if such a function is even vaguely close to being constant (by being bounded), then it almost magically “snaps into place” and actually is forced to be a constant! This is in stark contrast to the real case, in which there are functions such as ${\sin(x)}$ that are differentiable (and even smooth and analytic) on the real line and bounded, but definitely not constant. Note that the complex analogue ${\sin(z)}$ of the sine function is not a counterexample to Liouville’s theorem, since ${\sin(z)}$ becomes quite unbounded away from the real axis (Exercise 16 of Notes 0). This also fits well with the intuition of harmonic functions (and hence also holomorphic functions) being “balanced” in that any convexity in one direction has to be balanced by concavity in the orthogonal direction, and vice versa (as discussed before Theorem 25 of Notes 1): any attempt to create an entire function that is bounded and oscillating in one direction will naturally force that function to become unbounded in the orthogonal direction.

Exercise 29 Let ${f: {\bf C} \rightarrow {\bf C}}$ be an entire function which is of polynomial growth in the sense that there exists a finite quantity ${M>0}$ and some exponent ${A \geq 0}$ such that ${|f(z)| \leq M (1+|z|)^A}$ for all ${z \in {\bf C}}$. Show that ${f}$ is, in fact, a polynomial.

Now we can prove the fundamental theorem of algebra discussed back in Notes 0.

Theorem 30 (Fundamental theorem of algebra) Let

$\displaystyle P(z) = a_n z^n + \dots + a_0$

be a polynomial of degree ${n \geq 0}$ for some ${a_0,\dots,a_n \in {\bf C}}$ with ${a_n}$ non-zero. Then there exist complex numbers ${z_1,\dots,z_n}$ such that

$\displaystyle P(z) = a_n (z-z_1) \dots (z-z_n).$

Proof: This is trivial for ${n=0,1}$, so suppose inductively that ${n \geq 2}$ and the claim has already been proven for ${n-1}$. Suppose first that the equation ${P(z)=0}$ has no roots in the complex plane, then the function ${1/P(z)}$ is entire. Also, this function goes to zero as ${|z| \rightarrow \infty}$, and so is bounded on the exterior of any sufficiently large disk; as it is also continuous, it is bounded on any disk and is thus bounded everywhere. By Liouville’s theorem, ${1/P(z)}$ is constant, which implies that ${P(z)}$ is constant, which is absurd (for instance, the ${n^{\mathrm{th}}}$ derivative of ${P}$ is the non-zero function ${n! a_n}$). Hence ${P(z)}$ has at least one root ${z_n}$. By the factor theorem (which works in any field, including the complex numbers) we can then write ${P(z) = Q(z) (z-z_n)}$ for some polynomial ${Q(z)}$, which by the long division algorithm (or by comparing coefficients) must take the form

$\displaystyle Q(z) = a_n z^{n-1} + b_{n-2} z^{n-2} + \dots + b_0$

for some complex numbers ${b_0,\dots,b_{n-2}}$. The claim then follows from the induction hypothesis. $\Box$

The following exercises show that ${{\bf C}}$ can be alternatively defined as an algebraic closure of the reals ${{\bf R}}$ (together with a designated square root ${i}$ of ${-1}$), and that extending ${{\bf R}}$ using a different irreducible polynomial than ${x^2+1}$ would still give a field isomorphic to the complex numbers, thus supporting the notion that the complex numbers are not an arbitrary extension of the reals, but rather a quite natural and canonical one.

Exercise 31 Let ${k}$ be a field containing ${{\bf R}}$ which is a finite extension of ${{\bf R}}$, in the sense that ${k}$ is a finite-dimensional vector space over ${{\bf R}}$. Show that ${k}$ is isomorphic (as a field) to either ${{\bf R}}$ or ${{\bf C}}$. (Hint: if ${\alpha}$ is some element of ${k}$ not in ${{\bf R}}$, show that ${P(\alpha)=0}$ for some irreducible polynomial ${P}$ with real coefficients but no real roots. Use this to set up an isomorphism between the field ${\tilde k}$ generated by ${{\bf R}}$ and ${\alpha}$ with ${{\bf C}}$. If there is an element ${\beta}$ of ${k}$ not in this field ${\tilde k}$, show that there ${Q(\beta)=0}$ for some irreducible polynomial ${Q}$ with coefficients in ${\tilde k}$ and no roots in ${\tilde k}$, and contradict the fundamental theorem of algebra.)

Exercise 32 A field ${k}$ is said to be algebraically closed if the conclusion of Theorem 30 with ${{\bf C}}$ replaced by ${k}$. Show that any algebraically closed field ${k}$ containing ${{\bf R}}$, contains a subfield that is isomorphic to ${{\bf C}}$ (and which contains ${{\bf R}}$ as a subfield, isomorphic to the copy of ${{\bf R}}$ inside ${{\bf C}}$). Thus, up to isomorphism, ${{\bf C}}$ is the unique algebraic closure of ${{\bf R}}$, that is to say a minimal algebraically closed field containing ${{\bf R}}$.

Another nice consequence of the Cauchy integral formula is a converse to Cauchy’s theorem known as Morera’s theorem.

Theorem 33 (Morera’s theorem) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be a continuous function. Suppose that ${f}$ is conservative in the sense that ${\int_\gamma f(z)\ dz = 0}$ for any closed polygonal path in ${U}$. Then ${f}$ is holomorphic on ${U}$.

Proof: By working locally with small balls in ${U}$ we may assume that ${U}$ is a ball (and in particular connected). By Exercise 31 of Notes 2, ${f}$ has an antiderivative ${F: U \rightarrow {\bf R}}$. By definition, ${F}$ is complex differentiable at every point of ${U}$ (with derivative ${f}$), so by Corollary 20, ${F}$ is smooth, which implies in particular that ${f' = F}$ is holomorphic on ${U}$ as claimed. $\Box$

The power of Morera’s theorem comes from the fact that there are no differentiability requirements in the hypotheses on ${f}$, and yet the conclusion is that ${f}$ is differentiable (and hence smooth, by Corollary 20); it can be viewed as another manifestation of “elliptic regularity”. Here is one basic application of Morera’s theorem:

Theorem 34 (Uniform limit of holomorphic functions is holomorphic) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${f_n: U \rightarrow {\bf C}}$ be a sequence of holomorphic functions that converge uniformly on compact sets to a limit ${f: U \rightarrow {\bf C}}$. Then ${f}$ is also holomorphic. Furthermore, for each natural number ${k}$, the derivatives ${f_n^{(k)}: U \rightarrow {\bf C}}$ also converge uniformly on compact sets to ${f^{(k)}: U \rightarrow {\bf C}}$ for any natural number  (In particular, ${f_n^{(k)}}$ converges pointwise to ${f^{(k)}}$ on ${U}$.)

Proof: Again we may work locally and assume that ${U}$ is a ball (and in partiular is convex and simply connected). The ${f_n}$ are continuous, hence their locally uniform limit ${f}$ is also continuous. From Corollary 11 (or Corollary 14), we have ${\int_\gamma f_n(z)\ dz = 0}$ on any closed polygonal path in ${U}$, hence on taking locally uniform limits we also have ${\int_\gamma f(z)\ dz = 0}$ for such paths. The holomorphicity of ${f}$ then follows from Morera’s theorem. The uniform convergence of ${f_n^{(k)}}$ to ${f^{(k)}}$ on compact sets ${K}$ follows from applying Theorem 25 to circular contours ${\gamma_{z_0,\varepsilon,\circlearrowleft}}$ for ${z_0 \in K}$ and ${\varepsilon>0}$ small enough that these contours lie in ${U}$ (note from compactness that one can take ${\varepsilon}$ independent of ${z_0}$). $\Box$

Actually, one can weaken the uniform nature of the convergence in Theorem 34 substantially; even the weak limit of holomorphic functions in the space of locally integrable functions on ${U}$ will remain harmonic. However, we will not need these weaker versions of this theorem here.

Exercise 35 (Riemann’s theorem on removable singularities) Let ${U}$ be an open subset of ${{\bf C}}$, let ${z_0}$ be a point in ${U}$, and let ${f: U \backslash \{z_0\} \rightarrow {\bf C}}$ be a holomorphic function on ${U \backslash \{z_0\}}$ which is bounded near ${z_0}$, in the sense that it is bounded on some punctured disk ${D(z_0,r) \backslash \{z_0\}}$ contained in ${z_0}$. Show that ${f}$ has a removable singularity at ${z_0}$, in the sense that ${f}$ is the restriction to ${U \backslash \{z_0\}}$ of a holomorphic function ${\tilde f: U \rightarrow {\bf C}}$ on ${U}$. (Hint: show that ${f}$ is conservative near ${z_0}$, find an antiderivative, extend it to ${U}$, and use Morera’s theorem to show that this extension is holomorphic. Alternatively, one can also proceed by some version of the Cauchy integral formula.)

Exercise 36 (Integrals of holomorphic functions) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${f: [0,1] \times U \rightarrow {\bf C}}$ be a continuous function such that, for each ${t \in [0,1]}$, the function ${z \mapsto f(t,z)}$ is holomorphic on ${U}$. Show that the function ${z \mapsto \int_0^1 f(t,z)\ dt}$ is also holomorphic on ${U}$. (Hint: work locally and use Cauchy’s theorem, Morera’s theorem, and Fubini’s theorem.)

Exercise 37 (Schwarz reflection principle) Let ${U}$ be an open subset of ${{\bf C}}$ that is symmetric around the real axis, that is to say ${\overline{z} \in U}$ whenever ${z \in U}$. Let ${f_+: \overline{U_+} \rightarrow {\bf C}}$ be a continuous function on the set ${\overline{U_+} := \{ z \in U: \mathrm{Im}(z) \geq 0\}}$ that is holomorphic in the open subset ${U_+ := \{ z \in U: \mathrm{Im}(z) > 0 \}}$. Similarly, let ${f_-: \overline{U_-} \rightarrow {\bf C}}$ be continuous on ${\overline{U_-} := \{ z \in U: \mathrm{Im}(z) \leq 0\}}$ that is holomorphic in the open subset ${U_- := \{ z \in U: \mathrm{Im}(z) < 0 \}}$. Suppose further that ${f_+}$ and ${f_-}$ agree on ${U \cap {\bf R}}$. Show that ${f_+}$ and ${f_-}$ are both restrictions of a single holomorphic function ${f: U \rightarrow {\bf C}}$.

The following two Venn diagrams (or more precisely, Euler diagrams) summarise the relationships between different types of regularity amongst continuous functions over both the reals and the complexes. The first diagram

describes the class of continuous functions on some interval ${(a,b)}$ in the real line; such functions are automatically conservative, but not necessarily differentiable, while differentiable functions are not necessarily smooth, and smooth functions are not necessarily analytic. On the other hand, when considering the class of continuous functions on an open subset ${U}$ of ${{\bf C}}$, the picture is different:

Now, very few continuous functions are conservative, and only slightly more functions are complex differentiable (and for simply connected domains ${U}$, these two classes in fact coincide). Whereas in the real case, differentiable functions were considerably less regular than analytic functions, in the complex case the two classes in fact coincide.

— 3. Winding number —

One defect of the current formulation of the Cauchy integral formula (see Theorem 15 and the ensuing discussion) is that the curve ${\gamma}$ involved has to be homotopic (as a closed curve, up to reparameterisation) to a circular arc ${\gamma_{z_0,r,\circlearrowleft}}$, or at least to a curve of the form ${t \mapsto z_0 + re^{imt}}$, ${t \in [0,2\pi]}$ for some integer ${m}$. We now investigate what happens when this hypothesis is removed. A key notion is that of a winding number.

Definition 38 (Winding number) Let ${\gamma}$ be a closed curve, and let ${z_0}$ be a complex number that is not in the image of ${\gamma}$. The winding number ${W_\gamma(z_0)}$ of ${\gamma}$ around ${z_0}$ is defined by the integral

$\displaystyle W_\gamma(z_0) := \frac{1}{2\pi i} \int_\gamma \frac{dz}{z-z_0}. \ \ \ \ \ (6)$

Here we again take advantage of the ability to integrate holomorphic functions on curves that are not necessarily rectifiable. Clearly the winding number is unchanged if we replace ${\gamma}$ by any equivalent curve, and if one replaces the curve ${\gamma}$ with its reversal ${-\gamma}$, then the winding number is similarly negated. In some texts, the winding number is also referred to as the index or degree.

From the Cauchy integral formula we see that

$\displaystyle W_{\gamma}(z_0) = 1$

when ${\gamma}$ is homotopic in ${{\bf C} \backslash \{z_0\}}$ (as a closed curve, up to reparameterisation) to a circle ${\gamma_{z_0,r,\circlearrowleft}}$, and more generally that

$\displaystyle W_{\gamma}(z_0) = m$

if ${\gamma}$ is homotopic in ${{\bf C} \backslash \{z_0\}}$ (as a closed curve, up to reparameterisation) to a curve of the form ${t \mapsto z_0 + r e^{imt}}$, ${t \in [0,2\pi]}$. Thus we see, intuitively at least, that ${W_\gamma(z_0)}$ measures the number of times ${\gamma}$ winds counterclockwise about ${z_0}$, which explains the term “winding number”.

We can now state a more general form of the Cauchy integral formula:

Theorem 39 (General Cauchy integral formula) Let ${U}$ be a simply connected subset of ${{\bf C}}$, let ${\gamma}$ be a closed curve in ${U}$, and let ${f: U \rightarrow {\bf C}}$ be holomorphic. Then for any ${z_0}$ that lies in ${U}$ but not in the image of ${\gamma}$, we have

$\displaystyle \frac{1}{2\pi i} \int_\gamma \frac{f(z)}{z-z_0}\ dz = W_\gamma(z_0) f(z_0).$

Proof: By Corollary 22 (or Exercise 35), we have ${f(z) - f(z_0) = (z-z_0) g(z)}$ for some holomorphic function ${g: U \rightarrow {\bf C}}$. Hence by Theorem 14 we have

$\displaystyle \int_\gamma \frac{f(z)-f(z_0)}{z-z_0}\ dz = \int_\gamma g(z)\ dz = 0.$

The claim then follows from (6). $\Box$

Exercise 40 (Higher order general Cauchy integral formula) With ${U, \gamma, f, z_0}$ as in the above theorem, show that

$\displaystyle W_\gamma(z_0) f^{(n)}(z_0) = \frac{n!}{2\pi i} \int_\gamma \frac{f(z)}{(z-z_0)^{n+1}}\ dz$

for every natural number ${n}$. (Hint: instead of approximating ${f(z)}$ by ${f(z_0)}$, use a partial Taylor expansion of ${f}$. Many of the terms that arise can be handled using the fundamental theorem of calculus. Alternatively, one can use differentiation under the integral sign and Lemma 44 below.)

To use Theorem 39, it becomes of interest to obtain more properties on the winding number. From Cauchy’s theorem we hav

Lemma 41 (Homotopy invariance) Let ${z_0 \in {\bf C}}$, and let ${\gamma_0, \gamma_1}$ be two closed curves in ${{\bf C} \backslash \{z_0\}}$ that are homotopic as closed curves up to reparameterisation in ${{\bf C} \backslash \{z_0\}}$. Then ${W_{\gamma_0}(z_0) = W_{\gamma_1}(z_0)}$.

The following specific corollary of this lemma will be useful for us.

Corollary 42 (Rouche’s theorem for winding number) Let ${\gamma_0: [a,b] \rightarrow {\bf C}}$ be a closed curve, and let ${z_0}$ lie outside of the image of ${\gamma_0}$. Let ${\gamma_1: [a,b] \rightarrow {\bf C}}$ be a closed curve such that

$\displaystyle |\gamma_1(t) -\gamma_0(t)| < |\gamma_0(t) - z_0| \ \ \ \ \ (7)$

for all ${t \in [a,b]}$. Then ${W_{\gamma_0}(z_0) = W_{\gamma}(z_1)}$.

Proof: The map ${\gamma: [0,1] \times [a,b] \rightarrow {\bf C}}$ defined by ${\gamma(s,t) := (1-s) \gamma_0(t) + s \gamma_1(t)}$ is a homotopy from ${\gamma_0}$ to ${\gamma_1}$; by (7) and the triangle inequality, it avoids ${z_0}$. The claim then follows from Lemma 41. $\Box$

Corollary 42 can be used to compute the winding number near infinity as follows. Given a curve ${\gamma: [a,b] \rightarrow {\bf C}}$ and a point ${z_0}$, define the distance

$\displaystyle \mathrm{dist}(z_0,\gamma) := \inf_{t \in [a,b]} |z_0-\gamma(t)|$

and the diameter

$\displaystyle \mathrm{diam}(\gamma) := \sup |\gamma(t) - \gamma(t')|.$

Corollary 43 (Vanishing near infinity) Let ${\gamma}$ be a closed curve. Then ${W_\gamma(z_0) = 0}$ whenever ${z_0 \in {\bf C}}$ is such that ${\mathrm{dist}(z_0,\gamma) > \mathrm{diam}(\gamma)}$.

Proof: Apply Corollary 42 with ${\gamma_0}$ equal to ${\gamma}$ and ${\gamma_1}$ equal to any point in the image of ${\gamma_0}$. $\Box$

Corollary 42 also gives local constancy of the winding number:

Lemma 44 (Local constancy in ${z_0}$) Let ${\gamma}$ be a closed curve. Then ${W_\gamma}$ is locally constant. That is to say, if ${z_0}$ does not lie in the image of ${\gamma}$, then there exists a disk ${D(z_0,r)}$ outside of the image of ${\gamma}$ such that ${W_\gamma(z) = W_\gamma(z_0)}$ for all ${z \in D(z_0,r)}$.

Proof: From Corollary 42, we see that if ${r}$ is small enough and ${h \in D(0,r)}$, then

$\displaystyle W_{\gamma-h}(z_0) = W_\gamma(z_0),$

where ${\gamma-h: t \mapsto \gamma(t)-h}$ is the translation of ${\gamma}$ by ${h}$. But by a translation change of variables we see that

$\displaystyle W_{\gamma-h}(z_0) = W_\gamma(z_0+h)$

and the claim follows. $\Box$

Exercise 45 Give an alternate proof of Lemma 44 based on differentiation under the integral sign and using the fact that ${\frac{1}{(z-z_0)^2}}$ has an antiderivative away from ${z_0}$.

As confirmation of the interpretation of ${W_\gamma(z_0)}$ as a winding number, we can now establish integrality:

Lemma 46 (Integrality) Let ${\gamma}$ be a closed curve, and let ${z_0}$ lie outside of the image of ${\gamma}$. Then ${W_\gamma(z_0)}$ is an integer.

Proof: By Corollary 42 we may assume without loss of generality that ${\gamma}$ is a closed polygonal path. By partitioning a polygon into triangles (and using Lemma 44 to move ${z_0}$ slightly out of the way of any new edges formed by this partition) it suffices to verify this for triangular ${\gamma}$. But this follows from the Cauchy integral formula (if ${z_0}$ is in the interior of the triangle) or Cauchy’s theorem (if ${z_0}$ is in the exterior). $\Box$

Exercise 47 Give another proof of Lemma 46 by restricting again to closed polygonal paths ${\gamma: [a,b] \rightarrow {\bf C}}$, and showing that the function ${t \mapsto \exp( \int_a^t \frac{\gamma'(s)}{\gamma(s)-z_0} )\ ds / (\gamma(t) - z_0)}$ is constant on ${[a,b]}$ by establishing that it is continuous and has vanishing derivative at all but finitely many points. (Note that ${\gamma'(s)}$ exists for all but finitely many ${s}$, so the integral here can be well defined.)

We now come to a fundamental and well known theorem about simple closed curves, namely the Jordan curve theorem.

Theorem 48 (Jordan curve theorem) Let ${\gamma: [a,b] \rightarrow {\bf C}}$ be a non-trivial simple closed curve. Then there is an orientation ${\sigma \in \{-1,+1\}}$ such that the complex plane ${{\bf C}}$ is partitioned into the boundary region ${\gamma([a,b])}$, the exterior region

$\displaystyle \{ z_0 \not \in \gamma([a,b]): W_\gamma(z_0) = 0 \}, \ \ \ \ \ (8)$

and the interior region

$\displaystyle \{ z_0 \not \in \gamma([a,b]): W_\gamma(z_0) = \sigma \}. \ \ \ \ \ (9)$

Furthermore the exterior region is connected and unbounded, and the interior region is connected, non-empty and bounded. Finally, if ${U}$ is any open set that contains ${\gamma}$ and its interior, then ${\gamma}$ is contractible to a point in ${U}$.

This theorem is relatively easy to prove for “nice” curves, such as polygons, but is surprisingly delicate to prove in general. Some idea of the subtlety involved can be seen by considering pathological examples such as the lakes of Wada, which are three disjoint open connected subsets of ${{\bf C}}$ which all happen to have exactly the same boundary! This does not contradict the Jordan curve theorem, because the boundary set in this example is not given by a simple closed curve. However it does indicate that one has to carefully use the hypothesis of being a simple closed curve in order to prove Theorem 48. Another indication of the difficulty of the theorem is its global nature; the claim does not hold if one replaces the complex plane ${{\bf C}}$ by other surfaces such as the torus, the projective plane, or the Klein bottle, so the global topological structure of the complex plane must come into play at some point. For the sake of completeness, we give a proof of this theorem in an appendix to these notes.

If the quantity ${\sigma}$ in the above theorem is equal to ${+1}$, we say that the simple closed curve ${\gamma}$ has an anticlockwise orientation; if instead ${\sigma=-1}$ we say that ${\gamma}$ has a clockwise orientation. Thus for instance, ${\gamma_{z_0,r,\circlearrowleft}}$ has an anticlockwise orientation, while its reversal ${-\gamma_{z_0,r,\circlearrowleft}}$ has the clockwise orientation.

Exercise 49 Let ${\gamma_1}$, ${\gamma_2}$ be non-trivial simple closed curves.

• (i) If ${\gamma_1,\gamma_2}$ have disjoint image, show that ${\gamma_2}$ either lies entirely in the interior of ${\gamma_1}$, or in the exterior.
• (ii) If ${\gamma_2}$ avoids the exterior of ${\gamma_1}$, show that the interior of ${\gamma_2}$ is contained in the interior of ${\gamma_1}$, and the exterior of ${\gamma_2}$ contains the exterior of ${\gamma_1}$.
• (iii) If ${\gamma_2}$ avoids the interior of ${\gamma_1}$, and ${\gamma_1}$ avoids the interior of ${\gamma_2}$, and the two curves have disjoint images, show that the interior of ${\gamma_2}$ is contained in the exterior of ${\gamma_1}$, and the exterior of ${\gamma_2}$ contains the interior of ${\gamma_1}$.

(This is all visually “obvious” as soon as one draws a picture, but the challenge is to provide a rigorous proof. One should of course use the Jordan curve theorem extensively to do so. You will not need to use the final part of the Jordan curve theorem concerning contractibility.)

Exercise 50 Let ${\gamma}$ be a non-trivial simple closed curve. Show that the interior of ${\gamma}$ is simply connected. (Hint: first show that any simple closed polygonal path in ${\gamma}$ is contractible to a point in the interior; then extend this to closed polygonal paths that are not necessarily simple by an induction on the number of edges in the path; then handle general closed curves.)

Remark 51 There is a refinement of the Jordan curve theorem known as the Jordan-Schoenflies theorem, that asserts that for non-trivial simple closed curve ${\gamma}$ there is a homeomorphism ${\phi: {\bf C} \rightarrow {\bf C}}$ that maps ${\gamma}$ to the unit circle ${S^1}$, the interior of ${\gamma}$ to the unit disk ${D(0,1)}$, and the exterior to the exterior region ${\{ z \in {\bf C}: |z| > 1 \}}$. The proof of this improved version of the Jordan curve theorem will have to wait until we have the Riemann mapping theorem (as well as a refinement of this theorem due to Carathéodory). The Jordan-Schoenflies theorem may seem self-evident, but it is worth pointing out that the analogous result in three dimensions fails without additional regularity assumptions on the boundary surface, thanks to the counterexample of the Alexander horned sphere.

From the Jordan curve theorem we have yet another form of the Cauchy theorem and Cauchy integral formula:

Theorem 52 (Cauchy’s theorem and Cauchy integral formula for simple curves) Let ${\gamma}$ be a simple closed curve, and let ${U}$ be an open set containing ${\gamma}$ and its interior. Let ${f: U \rightarrow {\bf C}}$ be a holomorphic function.

• (i) (Cauchy’s theorem) One has ${\int_\gamma f(z)\ dz = 0}$.
• (ii) (Cauchy integral formula) If ${z_0 \in U}$ lies outside of the image of ${\gamma}$, then the expression ${\frac{1}{2\pi i} \int_\gamma \frac{f(z)}{z-z_0}\ dz}$ vanishes if ${z_0}$ lies in the exterior of ${\gamma}$, equals ${f(z_0)}$ if ${z_0}$ lies in the interior of ${\gamma}$ and ${\gamma}$ is oriented anti-clockwise, and equals ${-f(z_0)}$ if ${z_0}$ lies in the interior of ${\gamma}$ and ${\gamma}$ is oriented clockwise.

Exercise 53 Let ${P(z) = a_n z^n + \dots + a_0}$ be a polynomial with complex coefficients ${a_0,\dots,a_n}$ and ${a_n \neq 0}$. For any ${R>0}$, let ${\gamma_R: [0,2\pi] \rightarrow {\bf C}}$ denote the closed contour ${\gamma_R(t) := P(R e^{it})}$.

• (i) Show that if ${R}$ is sufficiently large, then ${W_{\gamma_R}(0) = n}$.
• (ii) Show that if ${P}$ does not vanish on the closed disk ${\overline{D(0,R)}}$, then ${W_{\gamma_R}(0)=0}$.
• (iii) Use these facts to give an alternate proof of the fundamental theorem of algebra that does not invoke Liouville’s theorem.

In the case when the closed curve ${\gamma}$ is a contour (which includes of course the case of closed polygonal paths), one can describe the interior and exterior regions, as well as the winding number, more explicitly.

Exercise 54 (Local structure of interior and exterior) Let ${\gamma = \gamma_1 + \dots + \gamma_n: [a,b] \rightarrow {\bf C}}$ be a simple closed contour formed by concatenating smooth curves ${\gamma_1,\dots,\gamma_n}$ together. Let ${z_0}$ be an interior point of one of these curves ${\gamma_i: [a_i,b_i] \rightarrow {\bf C}}$, thus ${z_0 = \gamma_i(t_i)}$ for some $latex {a_i < t_i 0}&fg=000000$ and ${\theta \in {\bf R}}$. Recall from Exercise 24 of Notes 2 that for sufficiently small ${\varepsilon}$, the set ${\gamma([a,b]) \cap D(z_0,\varepsilon)}$ can be expressed as a graph of the form

$\displaystyle \gamma([a,b] \cap D(z_0,\varepsilon)) = \{ z_0 + e^{i\theta} (s + i f(s)): s \in I_\varepsilon \}$

for some interval ${I_\varepsilon}$ and some continuously differentiable function ${f: I_\varepsilon \rightarrow {\bf R}}$ with ${f(0) =f'(0) = \varepsilon}$. Show that if ${\gamma}$ is oriented anticlockwise, and ${\varepsilon}$ is sufficiently small then the interior of ${\gamma}$ contains all points in ${D(z_0,\varepsilon)}$ of the form ${z_0 + e^{i\theta} (s + i (f(s)+u))}$ for some ${s \in I_\varepsilon}$ and ${u>0}$, and the exterior of ${\gamma}$ contains all points in ${D(z_0,\varepsilon)}$ of the form ${z_0 + e^{i\theta} (s + i (f(s)+u))}$ for some ${s \in I_\varepsilon}$ and ${u0}$ and ${u<0}$ swapped.

Exercise 55 (Alexander numbering rule) Let ${\gamma = \gamma_1 + \dots + \gamma_n: [a,b] \rightarrow {\bf C}}$ be a simple closed contour oriented anticlockwise formed by concatenating smooth curves ${\gamma_1,\dots,\gamma_n}$ together. Let ${\sigma = \sigma_1 + \dots + \sigma_m: [c,d] \rightarrow {\bf C}}$ be a contour formed by concatenating smooth curves ${\sigma_1,\dots,\sigma_m}$, with initial point ${z_0}$ and final point ${z_1}$. Assume that there are only finitely many points ${w_1,\dots,w_k}$ where the images of ${\gamma}$ and of ${\sigma}$ intersect. Furthermore, assume at each of the points ${w_l}$, ${l=1,\dots,k}$, that one has a “smooth simple transverse intersection” in the sense that the following axioms are obeyed:

• (i) ${w_l}$ lies in the interior of one of the smooth curves ${\gamma_i: [a_i,b_i] \rightarrow {\bf C}}$ that make up ${\gamma}$, thus ${w_l = \gamma_i(t_i)}$ for some ${a_i < t_i < b_i}$.
• (ii) ${w_l}$ lies in the interior of one of the smooth curves ${\sigma_j: [c_j,d_j] \rightarrow {\bf C}}$ that make up ${\sigma}$, thus ${w_l = \sigma_j(s_j)}$ for some ${c_j < t_j < b_j}$.
• (iii) ${w_l}$ is only traversed once by ${\sigma}$, thus there do not exist ${t \neq t'}$ in ${[c,d]}$ such that ${\sigma(t)=\sigma(t')=w_l}$.
• (iv) The derivatives ${\gamma'_i(t_i)}$ and ${\sigma'_j(s_j)}$ are linearly independent over ${{\bf R}}$. In other words, we either have a crossing from the right in which ${\sigma'_j(s_j) = \lambda e^{i\theta} \gamma'_i(t_i)}$ for some ${\lambda > 0}$ and ${0 < \theta < \pi}$, or else we have a crossing from the left in which ${\sigma'_j(s_j) = \lambda e^{i\theta} \gamma'_i(t_i)}$ for some ${\lambda > 0}$ and ${-\pi < \theta < 0}$.

Show that ${W_\gamma(z_0) - W_\gamma(z_1)}$ is equal to the total number of crossings from the left, minus the total number of crossings from the right.

Exercise 56 Let ${U}$ be a non-empty connected open subset of ${{\bf C}}$. Show that ${U}$ is simply connected if and only if every holomorphic function on ${U}$ is conservative.

— 4. Appendix: proof of the Jordan curve theorem (optional) —

We now prove the Jordan curve theorem. We begin with a variant of Corollary 42 in which the curve ${\gamma_1}$ is only required to have image close to the image of ${\gamma_0}$, rather than be close to ${\gamma_0}$ in a pointwise (and uniform) sense. For any curve ${\gamma}$ and any ${r>0}$, let ${N_r(\gamma) := \{ z \in {\bf C}: \mathrm{dist}(z,\gamma) < r \}}$ denote the ${r}$-neighbourhood of ${\gamma}$.

Proposition 57 Let ${\gamma_0}$ be a non-trivial simple closed curve, and let ${\delta>0}$. Suppose that ${\varepsilon>0}$ is sufficiently small depending on ${\gamma_0}$ and ${\delta}$. Let ${\gamma_1}$ be a closed curve (not necessarily simple) whose image lies in ${N_{\varepsilon}(\gamma_0)}$. Then ${\gamma_1}$ is homotopic (as a closed curve, up to reparameterisation) to ${m\gamma_0}$ in ${N_\delta(\gamma_0)}$, where ${m\gamma_0}$ is defined as the concatenation of ${m}$ copies of ${\gamma_0}$ if ${m}$ is positive, the trivial curve at the initial point of ${\gamma_0}$ if ${m}$ is zero, and the concatenation of ${-m}$ copies of ${-\gamma_0}$ if ${m}$ is negative. In particular, from Lemma 41 one has

$\displaystyle W_{\gamma_1}(z_0) = m W_{\gamma_0}(z_0)$

for all ${z_0 \in {\bf C} \backslash N_{\delta}(\gamma_0)}$.

Proof: After reparameterisation, we can take ${\gamma_0: [0,1] \rightarrow {\bf C}}$ to have domain on the unit interval ${[0,1]}$, and then by periodic extension we can view ${\gamma_0: {\bf R} \rightarrow {\bf C}}$ as a continuous ${1}$-periodic function on ${{\bf R}}$.

As ${[0,1]}$ is compact, ${\gamma_0}$ is uniformly continuous on ${[0,1]}$, and hence also on ${{\bf R}}$. In particular, there exists ${0 < \kappa < \frac{1}{10}}$ such that

$\displaystyle |\gamma_0(t_1) - \gamma_0(t_2)| \leq \frac{\delta}{2} \ \ \ \ \ (10)$

whenever ${t_1,t_2 \in {\bf R}}$ are such that ${|t_1-t_2| \leq \kappa}$.

Fix this ${\kappa}$. Observe that the function ${(t_1,t_2) \mapsto |\gamma_0(t_1)-\gamma_0(t_2)|}$ is continuous and nowhere vanishing on the region ${\{ (t_1,t_2) \in [0,2] \times [0,2]: \kappa \leq |t_1-t_2| \leq 1-\kappa \}}$. Thus, if ${\varepsilon}$ is small enough depending on ${\gamma_0,\kappa}$, we have the lower bound

$\displaystyle |\gamma_0(t_1)-\gamma_0(t_2)| \geq 3\varepsilon$

whenever ${t_1,t_2 \in [0,2]}$ are such that ${\kappa \leq |t_1-t_2| \leq 1-\kappa}$. Using the ${1}$-periodicity of ${\gamma_0}$, we conclude that if ${t_1,t_2 \in {\bf R}}$ are such that

$\displaystyle |\gamma_0(t_1)-\gamma_0(t_2)| < 3\varepsilon$

the there must be an integer ${m_{t_1,t_2}}$ such that

$\displaystyle |t_2 - (t_1+m_{t_1,t_2})| < \kappa. \ \ \ \ \ (11)$

Note that this integer ${m_{t_1,t_2}}$ is uniquely determined by ${t_1}$ and ${t_2}$.

Let ${[a,b]}$ be the domain of ${\gamma_1: [a,b] \rightarrow {\bf C}}$. By the uniform continuity of ${\gamma_1}$, we can find a partition ${a = s_0 < \dots < s_n = b}$ of ${[a,b]}$ such that

$\displaystyle |\gamma_1(s) - \gamma_1(s')| < \varepsilon \ \ \ \ \ (12)$

for all ${1 \leq j \leq n}$ and ${s_{j-1} \leq s, s' \leq s_j}$. Since the image of ${\gamma_1}$ lies in ${N_{\varepsilon}(\gamma_0)}$, we can find, for each ${0 \leq j \leq n}$, a real number ${t_j}$ such that

$\displaystyle |\gamma_1(s_j) - \gamma_0(t_j)| < \varepsilon. \ \ \ \ \ (13)$

Since ${\gamma_1}$ is closed, we may arrange matters so that

$\displaystyle \gamma_0(t_0) = \gamma_0(t_n). \ \ \ \ \ (14)$

From the triangle inequality and (12), (13) we have

$\displaystyle |\gamma_0(t_j) - \gamma_0(t_{j-1})| < 3\varepsilon.$

Using (11), we conclude that for each ${1 \leq j \leq n}$, there is an integer ${m_j}$ such that

$\displaystyle |t_j - (t_{j-1}+m_j)| < \kappa.$

As ${\gamma_0}$ is ${1}$-periodic, we have the freedom to shift each of the ${t_j}$ by an arbitrary integer, and by doing this for ${t_1, \dots, t_n}$ in turn, we may assume without loss of generality that all the ${m_j}$ vanish, thus

$\displaystyle |t_j - t_{j-1}| < \kappa \ \ \ \ \ (15)$

for all ${j=1,\dots,n}$. In particular, from (10) we have

$\displaystyle |\gamma_0(t) - \gamma_0(t')| < \frac{\delta}{2} \ \ \ \ \ (16)$

whenever ${t_{j-1} \leq t, t' \leq t_j}$. Also, as ${\gamma_0}$ is simple, we have from (14) that

$\displaystyle t_n = t_0 + m$

for some integer ${m}$. (Note that by enforcing (15), we no longer have the freedom to individually move ${t_0}$ or ${t_n}$ by an integer, so we cannot assume without loss of generality that ${m}$ vanishes.)

For ${j=1,\dots,n}$, let ${\gamma_{0,j}: [j-1,j] \rightarrow {\bf C}}$ denote the curve

$\displaystyle \gamma_{0,j}(t) := \gamma_0( t_{j-1} + (t-j+1) (t_j - t_{j-1}) )$

from ${\gamma_0(t_{j-1})}$ to ${\gamma_0(t_j)}$; similarly let ${\gamma_{1,j}: [j-1,j] \rightarrow {\bf C}}$ denote the curve

$\displaystyle \gamma_{1,j}(t) := \gamma_1( s_{j-1} + (t-j+1) (s_j - s_{j-1}) )$

from ${\gamma_1(s_{j-1})}$ to ${\gamma_1(s_j)}$. Observe from (16), (12), (13) that for each ${j=1,\dots,n}$, the images of ${\gamma_{0,j}}$ and ${\gamma_{1,j}}$ both lie in ${D( \gamma_0(t_j), \frac{\delta}{2} + 3\varepsilon)}$, which will lie in ${N_\delta(\gamma_0([a,b]))}$ if ${\varepsilon}$ is small enough. We can thus form a homotopy ${\gamma: [0,1] \times [0,n] \rightarrow N_\delta(\gamma_0([a,b]))}$ from ${\gamma_{0,1} + \dots + \gamma_{0,n}}$ to ${\gamma_{1,1} + \dots + \gamma_{1,n}}$ by defining

$\displaystyle \gamma( s, t ) = (1-s) \gamma_{0,j}(t) + s \gamma_{1,j}(t)$

for all ${1 \leq j \leq n}$ and ${j-1 \leq t \leq j}$. Thus ${\gamma_{0,1} + \dots + \gamma_{0,n}}$ and ${\gamma_{1,1} + \dots + \gamma_{1,n}}$ are homotopic as closed curves in ${N_\delta(\gamma_0([a,b]))}$. But by Exercise 3, ${\gamma_{0,1} + \dots + \gamma_{0,n}}$ is homotopic up to reparameterisation as closed curves to ${m \gamma_0}$ in ${N_\delta(\gamma_0([a,b]))}$, and ${\gamma_{1,1} + \dots + \gamma_{1,n}}$ is similarly homotopic up to reparameterisation as closed curves to ${\gamma_1}$ in ${N_\delta(\gamma_0([a,b]))}$, and the claim follows. $\Box$

We can now prove Theorem 48. We first verify the claim in the easy (and visually intuitive) case that ${\gamma}$ is a non-trivial simple closed polygonal curve. Removing the polygon ${\gamma([a,b])}$ from ${{\bf C}}$ leaves an open set, which we may decompose into connected components as per Exercise 34 of Notes 2. On each of these components, the winding number ${W_\gamma}$ is constant. Since each component has a non-empty boundary that is contained in ${\gamma([a,b])}$, this constant value of ${W_\gamma}$ must also be attained arbitrarily close to ${\gamma([a,b])}$.

Now, a routine application of the Cauchy integral formula (see Exercise 59) shows that as ${z_0}$ crosses one of the edges of the polygon ${\gamma([a,b])}$, the winding number ${W_\gamma(z_0)}$ is shifted by either ${+1}$ or ${-1}$. Hence at each point ${z}$ on ${\gamma([a,b])}$, the winding number will take two values ${\{ k, k+1\}}$ in a sufficiently small neighbourhood of ${z}$ (excluding ${\gamma([a,b])}$). By a continuity argument, the integer ${k}$ is independent of ${z}$. On the other hand, from Corollary 43 the winding number must be able to attain the value of zero. Thus we have ${\{k,k+1\} = \{0,\sigma\}}$ for some ${\sigma = \pm 1}$. Dividing a small neighbourhood of ${\gamma([a,b])}$ (excluding ${\gamma([a,b])}$ itself) into the regions where the winding numbers are ${0}$ or ${\sigma}$, a further continuity argument shows that each of these regions lie in a single connected component. Thus there are only two connected components, one where the winding number is zero and one where the winding number is ${\sigma}$. From (43) the latter component is bounded, hence the former is unbounded, and the claim follows.

Now we handle the significantly more difficult case when ${\gamma}$ is just a non-trivial simple closed curve. As one may expect, the strategy will be to approximate this curve by a polygonal path, but some care has to be taken when performing a limit, in order to prevent the interior region from collapsing into nothingness, or becoming disconnected, in the limit.

The first challenge is to ensure that there is at least one point ${z_0}$ outside of ${\gamma([a,b])}$ in which ${W_f(z_0)}$ is non-zero. This is actually rather tricky; we will achieve this by a parity argument (loosely inspired by a nonstandard version of this argument from this paper of Kanovei and Reeken). Clearly, ${\gamma([a,b])}$ contains at least two points; by an appropriate rotation, translation, and dilation we may assume that ${\gamma([a,b])}$ contains the points ${+i}$ and ${-i}$, with ${i}$ being both the initial point and the final point. Then we can decompose ${\gamma = \gamma_1 + \gamma_2}$, where ${\gamma_1: [a,c] \rightarrow {\bf C}}$ is a curve from ${i}$ to ${-i}$, and ${\gamma_2: [c,b] \rightarrow {\bf C}}$ is a curve from ${-i}$ to ${i}$.

Observe from the simplicity of ${\gamma}$ that ${|\gamma_1(t_1) - \gamma_2(t_2)| > 0}$ whenever ${t_1 \in [a,c]}$ and ${t_2 \in [c,b]}$ are such that

$\displaystyle |\mathrm{Im}(\gamma_1(t))|, |\mathrm{Im}(\gamma_2(t_2))| \leq \frac{1}{2}. \ \ \ \ \ (17)$

Thus, by compactness, there exists ${0 < \delta < \frac{1}{10}}$ such that one has the lower bound

$\displaystyle |\gamma_1(t_1) - \gamma_2(t_2)| \geq \delta \ \ \ \ \ (18)$

separating ${\gamma_1}$ from ${\gamma_2}$ whenever ${t_1 \in [a,c]}$, ${t_2 \in [c,b]}$ are such that (17) holds.

Next, for any natural number ${N}$, we may approximate ${\gamma: [a,b] \rightarrow {\bf C}}$ by a polygonal closed path ${\gamma^{(N)}: [a,b] \rightarrow {\bf C}}$ with

$\displaystyle |\gamma^{(N)}(t) - \gamma(t)| < \frac{1}{N} \ \ \ \ \ (19)$

for all ${t \in [a,b]}$. Although it is not particularly necessary, we can ensure that ${\gamma^{(N)}(a) = \gamma(a) = i}$ and ${\gamma^{(N)}(c) = \gamma(c) = -i}$. By perturbing the edges of the polygonal path ${\gamma^{(N)}}$ slightly, we may assume that none of the vertices of ${\gamma^{(N)}}$ lie on the real axis, and that none of the self-crossings of ${\gamma^{(N)}}$ (if any exist) lie on the real axis; thus, whenever ${\gamma^{(N)}}$ crosses the real axis, it does so at an interior point of an edge, with no other edge of ${\gamma^{(N)}}$ passing through that point. Note that we do not assert that the curve ${\gamma^{(N)}}$ is simple; with some more effort one could “prune” ${\gamma^{(N)}}$ by deleting short loops to make it simple, but this turns out to be unnecessary for the parity argument we give below.

Let ${x^{(N)}_1 < x^{(N)}_2 < \dots < x^{(N)}_{n^{(N)}}}$ be the points on the real axis where ${\gamma^{(N)}}$ crosses. By Exercise 59 below, the winding number ${W_{\gamma^{(N)}}(x)}$ changes by ${+1}$ or ${-1}$ as ${x}$ crosses each of the ${x^{(N)}_j}$; by Lemma 44, this winding number is constant otherwise, and by Corollary 43 it vanishes near infinity. Thus ${n^{(N)}}$ is even, and the winding number is odd between ${x^{(N)}_j}$ and ${x^{(N)}_{j+1}}$ for any odd ${j}$.

Next, observe that each point ${x^{(N)}_j}$ belongs to exactly one of the polygonal paths ${\gamma^{(N)}([a,c])}$ or ${\gamma^{(N)}([c,b])}$. Since each of these curves starts on one side of the real axis and ends up on the other, they must both cross the real axis an odd number of times. On the other hand, the crossing points ${x^{(N)}_1,\dots,x^{(N)}_n}$ can be grouped into pairs ${\{x^{(N)}_j,x^{(N)}_{j+1}\}}$ with ${j}$ odd. We conclude that there must exist an odd ${j}$ such that one of the ${x^{(N)}_j,x^{(N)}_{j+1}}$ lies in ${\gamma^{(N)}([a,c])}$ and the other lies in ${\tilde \gamma([c,b])}$.

Fix such a ${j}$. For sake of discussion let suppose that ${x^{(N)}_j}$ lies in ${\gamma^{(N)}([a,c])}$ and ${x^{(N)}_{j+1}}$ lies in ${\gamma^{(N)}([c,b])}$. From (19) we have

$\displaystyle \mathrm{dist}( x^{(N)}_j, \gamma([a,c]) ) \leq \frac{1}{N}; \quad \mathrm{dist}( x^{(N)}_{j+1}, \gamma([c,b]) ) \leq \frac{1}{N}$

and from (18) we have

$\displaystyle \mathrm{dist}( x, \gamma([a,c]) ) + \mathrm{dist}( x, \gamma([c,b]) ) \geq \delta.$

for any ${x \in [x^{(N)}_j, x^{(N)}_{j+1}]}$. By the intermediate value theorem, we can thus (for ${N}$ large enough) find ${x^{(N)}_j < x^{(N)}_* < x^{(N)}_{j+1}}$ such that

$\displaystyle \mathrm{dist}( x^{(N)}_*, \gamma([a,c]) ) = \mathrm{dist}( x^{(N)}_*, \gamma([c,b]) )$

and thus

$\displaystyle \mathrm{dist}( x^{(N)}_*, \gamma([a,c]) ), \mathrm{dist}( x^{(N)}_*, \gamma([c,b]) ) \geq \frac{\delta}{2}$

or equivalently

$\displaystyle \mathrm{dist}( x^{(N)}_*, \gamma([a,b]) ) \geq \frac{\delta}{2}.$

We arrive at the same conclusion in the opposite case when ${x^{(N)}_j}$ lies in ${\gamma^{(N)}([c,b])}$ and ${x^{(N)}_{j+1}}$ lies in ${\gamma^{(N)}([a,c])}$.

By Corollary 43 (and (19)), the ${x^{(N)}_*}$ are bounded in ${N}$. By the Bolzano-Weierstrass theorem, we can thus extract a subsequence of the ${x^{(N)}_*}$ that converges to some limit ${x_*}$. By continuity we then have

$\displaystyle \mathrm{dist}( x_*, \gamma([a,b]) ) \geq \frac{\delta}{2},$

in particular ${x_*}$ does not lie in ${\gamma([a,b])}$. By construction of ${x^{(N)}_*}$, we know that ${W_{\gamma^{(N)}}( x^{(N)}_* )}$ is odd for all ${N}$; using Lemma 44 and Lemma 42 we conclude that ${W_\gamma(x_*)}$ is also odd. Thus we have found at least one point where the winding number is non-zero.

Now we can finish the proof of the Jordan curve theorem. Let ${\gamma: [a,b] \rightarrow {\bf C}}$ be a non-trivial simple closed curve. By the preceding discussion, we can find a point ${z_*}$ outside of ${\gamma([a,b])}$ where the winding number ${W_\gamma}$ is non-zero. Let ${\delta > 0}$ be a sufficiently small parameter, and let ${0 < \varepsilon = \varepsilon(\delta) < \delta}$ be sufficiently small depending on ${\delta}$. By compactness, one can cover the region ${N_{\varepsilon/10}(\gamma([a,b]))}$ by a finite number of (solid) squares ${S}$ of sidelength ${\varepsilon/10}$ and sides parallel to the real and imaginary axes; by perturbation we may assume that no edge of one square is collinear to an edge of any other square. These squares all lie in ${N_\varepsilon(\gamma([a,b]))}$, and in particular will not contain ${z_*}$ if ${\delta}$ is small enough; their union can easily be seen to be connected. The boundaries of these squares divide the complex plane into a finite number of polygonal regions (one of whom is unbounded). One of these regions, call it ${\Omega_\delta}$, contains the point ${z_*}$. This region cannot contain any interior point of a square ${S}$, since otherwise ${\Omega_\delta}$ would be trapped inside a square of sidelength ${\varepsilon/10}$ and hence not contain ${z_*}$. In particular, ${\Omega_\delta}$ avoids ${N_{\varepsilon/10}(\gamma([a,b]))}$. The region ${\Omega_\delta}$ cannot be unbounded, since one could then continuously move ${z_*}$ to infinity without ever meeting ${\gamma([a,b])}$, contradicting Lemma 44, Corollary 43, and the non-vanishing nature of ${W_\gamma(z_*)}$. The boundary of ${\Omega_\delta}$ consists of one or more disjoint closed polygonal paths, whose edges consist of horizontal and vertical line segments. Actually, the boundary must consist of just one closed path, since otherwise the union of the squares ${S}$ would be disconnected, a contradiction. Let ${\gamma_\delta}$ denote the path that bounds ${\Omega_\delta}$ (traversed in either of the two possible directions). This path must be simple, because a crossing can only be formed by an edge of one square ${S}$ crossing an edge of another square ${S'}$ at a point that is not on the corner of either of the two squares; as ${\Omega_\delta}$ avoids both ${S}$ and ${S'}$, it can thus only occupy one quadrant of a neighbourhood of this crossing and so cannot bound all four edges of the crossing.

Applying the Jordan curve theorem to the polygonal path ${\gamma_\delta}$, we conclude that there is ${\sigma_\delta \in \{-1,+1\}}$ such that ${W_{\gamma_\delta}(z) = \sigma_\delta}$ on ${\Omega_\delta}$, and ${W_{\gamma_\delta}(z) = 0}$ for all ${z}$ outside of ${\Omega_\delta}$ and ${\tilde \gamma([a,b])}$. On the other hand, by Proposition 57 there is an integer ${m_\delta}$ such that ${\gamma_\delta}$ is homotopic (as closed curves, up to reparameterisation) in ${N_\delta([a,b])}$ to ${m_\delta \gamma}$, so in particular

$\displaystyle W_{\gamma_\delta}(z) = m_\delta W_\gamma(z)$

for all ${z \not \in N_\delta(\gamma([a,b]))}$. Applying this to ${z = z_*}$, we conclude that ${m_\delta}$ is either ${+1}$ or ${-1}$. If we write ${\sigma_\delta = m_\delta \sigma}$ (where ${\sigma}$, a priori, may depend on ${\delta}$), then ${\sigma \in \{-1,+1\}}$, and we have

$\displaystyle W_\gamma(z) = \sigma \ \ \ \ \ (20)$

for ${z \in \Omega_\delta \backslash N_\delta(\gamma([a,b]))}$, and

$\displaystyle W_\gamma(z) = 0 \ \ \ \ \ (21)$

for ${z \in ({\bf C} \backslash \Omega) \backslash N_\delta(\gamma([a,b]))}$. Thus ${W_\gamma}$ takes only two values outside of ${N_\delta(\gamma([a,b]))}$. Sending ${\delta \rightarrow 0}$, we conclude that ${\sigma}$ is in fact independent of ${\delta}$, and ${W_\gamma}$ takes only the two values ${0, \sigma}$ outside of ${\gamma([a,b])}$.

We now define the interior and exterior regions by (9), (8), then we have partitioned ${{\bf C}}$ into the interior, exterior, and ${\gamma([a,b])}$. From Lemma 44 the interior and exterior are open, and from Lemma 43 the interior is bounded, and hence the exterior is unbounded. The point ${z_*}$ lies in the interior, so the interior is non-empty. The only remaining task to show is that the interior and exterior are connected. Suppose for instance that ${z_1, z_2}$ lie in the interior region. Then for ${\delta}$ small enough, ${z_1, z_2}$ lie outside of ${N_\delta(\gamma([a,b]))}$. From (20), (21) we conclude that ${z_1,z_2}$ lie in ${\Omega_\delta}$. As ${\Omega_\delta}$ is connected, we can thus join ${z_1}$ to ${z_2}$ by a path in ${\Omega_\delta}$. As the region ${\Omega_\delta}$ avoids ${N_{\varepsilon(\delta)/10}(\gamma([a,b]))}$, we see from Lemma 44 that the winding number ${W_\gamma}$ stays constant on this path, and so the path remains in the interior region (9). This establishes the connectedness of the interior region; the connectedness of the exterior is proven similarly.

It remains to prove the contractibility of ${\gamma}$ in any open set ${U}$ that contains ${\gamma}$ and its interior. Once again, we begin with the simpler case when ${\gamma}$ is a simple closed polygonal path. We induct on the number ${n}$ of edges in ${\gamma}$. The cases ${n \leq 3}$ can be handled by direct calculation, so suppose that ${n>3}$ and the claim has been proven for all smaller values of ${n}$. We may remove any edges of zero length from the polygon. If the interior of the polygon is convex, then the claim follows from Example 2, so we may assume that the interior is non-convex. This implies that one of the interior angles in the polygon exceeds ${\pi}$ (see Exercise 11 below), thus there are two adjacent edges ${e,f}$ whose interior angle exceeds ${\pi}$. If one extends ${e}$ in the interior until it meets the polygon again, this wil divide the polygon into two subpolygons ${\gamma_1,\gamma_2}$, each of which can be verified to have fewer than ${n}$ edges. By Exercise 49 (which does not use the contractibility part of the Jordan curve theorem), the interiors of ${\gamma_1}$ and ${\gamma_2}$ are contained in the interior of ${\gamma}$, and so by the induction hypothesis they are contractible to a point in ${U}$. Using Exercise 3 we conclude that ${\gamma}$ is contractible to a point in ${U}$ also.

Now suppose that ${\gamma: [a,b] \rightarrow {\bf C}}$ is an arbitrary simple closed curve. Let ${\delta > 0}$ be a small parameter. As before, we can find a simple polygonal path ${\gamma_\delta}$ whose interior ${\Omega_\delta}$ lies in the interior of ${\gamma}$, and such that ${\gamma_\delta}$ is homotopic to ${m_\delta \gamma}$ in ${N_\delta(\gamma([a,b]))}$, and hence in ${U}$ if ${\delta}$ is small enough, for some ${m_\delta = \pm 1}$. From the previous discussion we see that ${\gamma_\delta}$ is contractible to a point in ${U}$, and so ${m_\delta \gamma}$ is also. The claim then follows (after reversing the contour ${m_\delta \gamma}$ if necessary). This concludes the proof of the Jordan curve theorem.

Exercise 58 Let ${\gamma}$ be a simple closed polygonal path with all edges of positive length. Suppose that all interior angles of ${\gamma}$ (that is, the angle that two adjacent edges make in the interior of the polygon) are less than or equal to ${\pi}$. Show that the interior of ${\gamma}$ is convex. (Hint: use a continuity argument to show that every line meets the interior of ${\gamma}$ in at most one interval.)

Exercise 59 Let ${\gamma}$ be a non-trivial simple closed polygonal curve, and let ${z_0}$ be a point in the interior of an edge ${e}$ of ${\gamma}$ (i.e., ${z_0}$ is not one of the two vertices of ${e}$). Let ${z, z'}$ be two points sufficiently close to ${z_0}$ that lie on opposite sides of ${e}$. Without using the Jordan curve theorem, show that ${|W_\gamma(z) - W_\gamma(z')| = 1}$. (Hint: replace ${\gamma}$ by a “local” closed contour that is quite short, and a “global” closed contour which avoids the line segment connecting ${z}$ and ${z'}$. Then use the Cauchy integral formula.)

Exercise 60 (Jordan arc theorem) Let ${\gamma: [a,b] \rightarrow {\bf C}}$ be a simple non-closed curve. Show that the complement of ${\gamma([a,b])}$ in ${{\bf C}}$ is connected. (Hint: first establish a variant of Proposition 57 for non-closed curves, in which ${m}$ is now set to zero. Then adapt the proof of the Jordan curve theorem.

Exercise 61 Let ${U}$ be a bounded connected non-empty open subset of ${{\bf C}}$. Show that ${U}$ is simply connected if and only if the complement ${{\bf C} \backslash U}$ is connected. (Hint: suppose that there is a point ${z_*}$ in ${{\bf C} \backslash U}$ that is separated from infinity by ${U}$. Show that there is some compact subset ${K}$ of ${U}$ that also separates ${z_*}$ from infinity. Then cover ${K}$ by small squares as in the proof of the Jordan curve theorem to locate a simple closed polygonal path in ${U}$ that separates ${z_*}$ from infinity.)

(The exercises below were added after the notes were first released; they will ultimately be moved to a more appropriate location, but are being placed here for now in order to not disrupt existing numbering.)

Exercise 62 Let ${U}$ be a simply connected subset of ${{\bf C}}$, and let ${u: U \rightarrow {\bf R}}$ be a harmonic function. Show that ${u}$ has a harmonic conjugate ${v: U \rightarrow {\bf R}}$, which is unique up to additive constants.

One can interpret Cauchy’s theorem through the lens of algebraic topology, and particularly through the machinery of homology and cohomology. We will not develop this perspective in depth in these notes, but the following exercise will give a brief glimpse of the connections to homology and cohomology.

Exercise 63 Let ${U}$ be an open subset of ${{\bf C}}$. Define a ${0}$-chain in ${U}$ to be a formal linear combination

$\displaystyle \sum_{i=1}^n m_i [z_i]$

of points ${z_i \in U}$ (which we enclose in brackets to avoid confusion with the arithmetic operations on ${{\bf C}}$, in particular ${[z]+[w]}$ is not identified with ${[z+w]}$), where ${n}$ is a natural number and the ${m_i}$ are integers; these form an additive abelian group in the usual fashion. Similarly, define a ${1}$-chain in ${U}$ to be a formal linear combination

$\displaystyle \sum_{i=1}^n m_i [\gamma_i]$

of curves ${\gamma_i: [0,1] \rightarrow U}$ in ${U}$, which (for very minor notational reasons) we will fix to have domain in the unit interval ${[0,1]}$. Finally, define a ${2}$-chain in ${U}$ to be a formal linear combination

$\displaystyle \sum_{i=1}^n m_i [T_i]$

of ${2}$-simplices ${T_i: \Delta_2 \rightarrow U}$, defined as continuous maps from the solid triangle ${\Delta_2 := \{ (x,y) \in {\bf R}^2: x,y \geq 0; x+y \leq 1 \}}$.

Given a ${1}$-chain ${c = \sum_{i=1}^n m_i [\gamma_i]}$ in ${U}$, we define its boundary ${\partial c}$ to be the ${0}$-chain

$\displaystyle \partial c := \sum_{i=1}^n m_i ( [\gamma_i(1)] - [\gamma_i(0)] )$

and call ${c}$ a ${1}$-cycle if ${\partial c = 0}$. Similarly, given a ${2}$-chain ${c = \sum_{i=1}^2 m_i [T_i]}$, we define its boundary ${\partial c}$ to be the ${1}$-chain

$\displaystyle \partial c := \sum_{i=1}^n m_i ( [t \mapsto T_i(0,t)] + [t \mapsto T_i(t,1-t)]$

$\displaystyle + [t \mapsto T_i(1-t,0)] )$

where ${t \mapsto T_i(0,t)}$ is the curve on ${[0,1]}$ that maps ${t}$ to ${T_i(0,t)}$, and similarly for ${t \mapsto T_i(t,1-t)}$ and ${t \mapsto T_i(1-t,0)}$. If ${c := \sum_{i=1}^n m_i [\gamma_i]}$ is a ${1}$-cycle, and ${f: U \rightarrow {\bf C}}$ is holomorphic, define the integral ${\int_c f}$ by

$\displaystyle \int_c f := \sum_{i=1}^n m_i \int_{\gamma_i} f(z)\ dz.$

If ${z_0}$ lies outside of a ${1}$-cycle ${c}$, define the winding number

$\displaystyle W_c(z_0) := \frac{1}{2\pi i} \int_c \frac{1}{z-z_0}.$

• (i) Show that if ${c}$ is a ${2}$-chain in ${U}$, then ${\partial \partial c = 0}$.
• (ii) Show that if ${c}$ is a ${2}$-chain in ${U}$ and ${f: U \rightarrow {\bf C}}$ is holomorphic, then ${\int_{\partial c} f = 0}$.
• (iii) If ${c}$ is a ${1}$-cycle in ${U}$, and ${W_c(z) = 0}$ for all ${z \in {\bf C} \backslash U}$, show that ${\int_c f = 0}$. (Hint: first perturb ${c}$ to be the union of line segments coming from a grid of some small sidelength ${\varepsilon}$. Observe that the winding number ${W_c(z)}$ is constant whenever ${z}$ ranges in the interior of one of the squares in this grid. Then find another ${1}$-cycle ${c'}$ coming from summing boundaries of such squares such that ${W_c(z) = W_{c'}(z)}$ for all ${z}$ in the interior of grid squares. Then show that ${\int_{c-c'} f = 0}$ and ${\int_{c'} f = 0}$.)
• (iv) If ${c}$ is a ${1}$-cycle, and ${f: U \rightarrow {\bf C}}$ has an antiderivative, show that ${\int_c f = 0}$.
• (v) If ${c = \sum_{i=1}^n m_i [\gamma_i]}$ is a ${1}$-cycle, ${W_c(z) = 0}$ for all ${z \in {\bf C} \backslash U}$, ${f: U \rightarrow {\bf C}}$ is holomorphic, and ${z_0}$ is a point lying outside of any of the ${\gamma_i}$, show that

$\displaystyle \frac{1}{2\pi i} \int_c \frac{f(z)}{z-z_0} = W_c( z_0) f(z_0).$

Exercise 64 Let ${U}$ be a subset of the complex plane which is star-shaped, which means that there exists ${z_0 \in U}$ such that for any ${z \in U}$, the line segment ${\{ (1-t) z_0 + tz: t \in [0,1]\}}$ is also contained in ${U}$. Show that every star-shaped set is simply connected.

1