We 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 even without knowing any explicit antiderivative of . 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 be an open subset of , and let , be two curves in .
- (i) If have the same initial point and final point , we say that and are homotopic with fixed endpoints in if there exists a continuous map such that and for all , and such that and for all .
- (ii) If are closed (but possibly with different initial points), we say that and are homotopic as closed curves in if there exists a continuous map such that and for all , and such that for all .
- (iii) If and are curves with the same initial point and same final point, we say that and are homotopic with fixed endpoints up to reparameterisation in if there is a reparameterisation of which is homotopic with fixed endpoints in to a reparameterisation of .
- (iv) If and are closed curves, we say that and are homotopic as closed curves up to reparameterisation in if there is a reparameterisation of which is homotopic as closed curves in to a reparameterisation of .
In the first two cases, the map will be referred to as a homotopy from to , and we will also say that can be continously deformed to (either with fixed endpoints, or as closed curves).
For a similar reason, in a convex open set , any two closed curves will be homotopic to each other as closed curves.
- (i) Prove that the property of being homotopic with fixed endpoints in is an equivalence relation.
- (ii) Prove that the property of being homotopic as closed curves in is an equivalence relation.
- (iii) If are closed curves with the same initial point, show that is homotopic to with fixed endpoints if and only if is homotopic to as closed curves.
- (iv) Define a point in to be a curve of the form for some and all . Let be a closed curve in . Show that is homotopic with fixed endpoints to a point in if and only if is homotopic as a closed curve to a point in . (In either case, we will call homotopic to a point, null-homotopic, or contractible to a point in .)
- (v) If are curves with the same initial point and the same terminal point, show that is homotopic to with fixed endpoints in if and only if is homotopic to a point in .
- (vi) If is connected, and are any two curves in , show that there exists a continuous map such that and for all . 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 is a reparameterisation of , then and are homotopic with fixed endpoints in U.
- (viii) Prove that the property of being homotopic with fixed endpoints in up to reparameterisation is an equivalence relation.
- (ix) Prove that the property of being homotopic as closed curves in 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:
- (i) If and are rectifiable curves that are homotopic in with fixed endpoints up to reparameterisation, then
- (ii) If and are closed rectifiable curves that are homotopic in as closed curves up to reparameterisation, then
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 thereof). Setting to be a point, we obtain an important special case of Cauchy’s theorem (which is in fact equivalent to the full theorem):
An important feature to note about Cauchy’s theorem is the global nature of its hypothesis on . The conclusion of Cauchy’s theorem only involves the values of a function on the images of the two curves . However, in order for the hypotheses of Cauchy’s theorem to apply, the function must be holomorphic not only on the images on , but on an open set 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:
As a consequence of this and Cauchy’s theorem, we conclude that the contour is not contractible to a point in ; note that this does not contradict Example 2 because is not convex. Thus we see that the lack of holomorphicity (or singularity) of at the origin can be “blamed” for the non-vanishing of the integral of on the closed contour , even though this contour does not come anywhere near the origin. Thus we see that the global behaviour of , not just the behaviour in the local neighbourhood of , 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 denote the half-circle contours and , then are both contours in from to , but one has
In order for this to be consistent with Cauchy’s theorem, we conclude that and are not homotopic in (even after reparameterisation).
In the specific case of functions of the form , or more generally for some point and some that is holomorphic in some neighbourhood of , 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 is any complex linear function of , then has an antiderivative , and hence
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 be an open subset of , and be complex numbers such that the solid (and closed) triangle spanned by (or more precisely, the convex hull of ) is contained in . (We allow the triangle to degenerate in that we allow the to be collinear, or even coincident.) Then for any holomorphic function , one has
where is the closed polygonal path that traverses the vertices of the solid triangle in order.
for some . We now run the following “divide and conquer” strategy. We let , , be the midpoints of . Then from the basic properties of contour integration (see Exercise 16 of Notes 2) we can split the triangular integral as the sum of four integrals on smaller triangles, namely
(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
where is one of the four triangular contours , , , or . Regardless of which of the four contours is, observe that the triangular region enclosed by is contained in that of . Furthermore, the diameter of is precisely half that of , where the diameter of a curve is defined by the formula
similarly, the perimeter of is precisely half that of . If we iterate the above process, we can find a nested sequence of triangular contours, each of which is contained in the previous one with half the diameter and perimeter, such that
for all . If we let be any point enclosed by , then from the decreasing diameters it is clear that the are a Cauchy sequence and thus converge to some limit , which is then contained in all of the closed triangles enclosed by any of the .
In particular, lies in and so is differentiable at . This implies, for any , that there exists a such that
whenever . We can rearrange this as
on . In particular, for large enough, this bound holds on the image on . In this case we can bound by , and hence by Exercise 16(v) of Notes 2,
From (1), the second integral vanishes. As each has half the diameter and perimeter of the previous, we thus have
But if one chooses small enough depending on and , we contradict (3).
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:
Proof: We induct on the number of vertices . The cases are trivial, and the case follows directly from Goursat’s theorem (using the convexity of to ensure that the interior of the polygon lies in ). If , we can split
The second integral on the right-hand side vanishes by Goursat’s theorem. The claim then follows from induction.
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 is a rectangle and the derivative of 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 be a convex open subset of , and let be a holomorphic function. Then has an antiderivative . Also, for any closed rectifiable curve in , and whenever are two rectifiable curves in with the same initial point and same terminal point. In other words, is conservative on .
Proof: The first claim follows from Corollary 11 and the first fundamental theorem of calculus (Theorem 27 from Notes 2). The remaining claims then follow from the second fundamental theorem of calculus (Theorem 30 from Notes 2).
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 and are homotopic with fixed endpoints, and not merely homotopic with fixed endpoints up to reparameterisation.
Let be a homotopy from to . Note that for any and , lies in the open set . From compactness, there must exist a radius such that for all and . Next, as is continuous on a compact set, it is uniformly continuous. In particular, there exists such that
whenever and are such that and .
Now partition and as and in such a way that and for all and . For each such and , let denote the closed polygonal contour
(the reader is encouraged here to draw a picture of the situation; we are using polygonal contours here rather than the homotopy because we did not require any rectifiability properties on the homotopy). By construction, the diameter of this contour is at most , so the contour is contained entirely in the disk . This disk is convex and contained in . Applying Corollary 11 or Corollary 13, we conclude that
for all and . If we sum this over all and , and noting that the homotopy fixes the endpoints, we conclude after a lot of cancelling that
(again, the reader is encouraged to draw a picture to see this cancellation). However, from a further application of Corollary 11 we have
for , where is the restriction of to , and similarly for . Putting all this together we conclude that
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 is a curve in , then for a sufficiently fine partition , the polygonal (and hence rectifiable) path will be contained in , and furthermore be homotopic to with fixed endpoints. One can then define when is holomorphic in and is non-rectifiable by declaring
where is any rectifiable curve that is homotopic (with fixed endpoints) to . This is a well defined definition thanks to the above discussion as well as Cauchy’s theorem; also observe that the exact open set in which the homotopy lives is not relevant, since given any two open sets containing the image of one can find a rectifiable curve which is homotopic to with fixed endpoints in , and hence in and 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 ; if one assumes more regularity on the function , one can get away with worse regularity on the curve , and vice versa.
A special case of Cauchy’s theorem is worth recording explicitly. We say that an open set in the complex plane is simply connected if it is non-empty, connected, and if every closed curve in is contractible in 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 be a simply connected subset of , and let be holomorphic. Then for any closed curve in . In particular (by Exercise 31 of Notes 2), 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 be an open subset of , let be holomorphic, and let be a point in . Let be such that the closed disk is contained in . Let be a closed curve in that is homotopic (as a closed curve, and up to reparameterisation) in to in . Then
Here we are already taking advantage of the ability to integrate holomorphic functions (such as , which is holomorphic on ) on curves that are not necessarily rectifiable.
Note the remarkable feature here that the value of at some point other than that on is completely determined by the value of on the curve , 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 available in that setting).
Proof: Observe that for any , the circles and are homotopic (as closed curves) in , and hence in . Since the function is holomorphic on , we conclude from Cauchy's theorem that
As is complex differentiable at , there exists a finite such that
for all in , and all sufficiently small . The length of this circle is of course . Applying Exercise 16(v) of Notes 2 we have
On the other hand, from explicit computation (cf. Example 7) we have
putting all this together, we see that
Sending to zero, we obtain the claim.
Note the same argument would give
if were homotopic to the curve : rather than , for some integer . In particular, if were homotopic to a point in , then the right-hand side would vanish.
Remark 16 For various explicit examples of closed contours , 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 be an open subset of , and let be a closed disk contained in .
- (i) If is holomorphic, show that
. Use this to give an alternate proof of Exercise 26 from Notes 1.
- (ii) If is harmonic, show that
. Use this to give an alternate proof of Theorem 25 from Notes 1.
- (iii) If is harmonic, show that
for any and , where the Poisson kernel is defined by the formula
(Hint: it simplifies the calculations somewhat if one reduces to the case , , and for some . Then compute the integral in two different ways, where is holomorphic with real part .)
The first important consequence of the Cauchy integral formula is the analyticity of holomorphic functions:
Corollary 18 (Holomorphic functions are analytic) Let be an open subset of , let be holomorphic, and let be a point in . Let be such that the closed disk is contained in . For each natural number , let denote the complex number
Proof: By continuity, there exists a finite such that for all on the circle , which of course has length . From Exercise 16(v) of Notes 2 we conclude that
From this and Proposition 7 of Notes 1, we see that the radius of convergence of is indeed at least .
Next, for any , the circle is homotopic (as a closed curve) in (and hence in to to small enough that lies in . Applying the Cauchy integral formula, we conclude that
On the other hand, from the geometric series formula (Exercise 12 of Notes 1) one has
for all , and thus
If we could interchange the sum and integral, we would conclude from (4) that
by the geometric series formula and the hypothesis , the sum is finite, and so the -test applies and we are done.
Remark 19 A function on an open set is said to be complex analytic on if, for every , there is a power series with a positive radius of convergence that converges to on some neighbourhood of . Combining the above corollary with Theorem 15 of Notes 1, we see that is holomorphic on if and only if is complex analytic on ; 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 (i.e., a function that, around every point 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 , we conclude
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.
Corollary 21 (Elliptic regularity) Let be an open subset of , and let be a harmonic function. Then 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 be an open subset of , and let be a point in . Let be a complex analytic function that vanishes at . Then there exists a unique complex analytic function such that for all .
Proof: For , we can simply define , and this is clearly the unique choice here. Uniqueness at follows from continuity. For equal to or near , we can expand as a Taylor series (noting that the constant term vanishes since ) and then set . One can check that these two definitions of agree on their common domain, and that is complex differentiable (and hence analytic) on .
Yet another consequence is the important property of analytic continuation:
Corollary 23 (Analytic continuation) Let be a connected non-empty open subset of , and let , be complex analytic functions. If and agree on some non-empty open subset of , then they in fact agree on all of .
Proof: Let denote the set of all points in where and agree to all orders, that is to say that
for all . By hypothesis, is non-empty; by the continuity of the is closed; and from analyticity and Taylor expansion (Exercise 17 of Notes 1) is open. As is connected, must therefore be all of , and the claim follows.
There is also a variant of the above corollary:
Corollary 24 (Non-trivial analytic functions have isolated zeroes) Let be a connected non-empty open subset of , and let be a function which vanishes at some point but is not identically zero. Then there exists a disk in on which does not vanish except at ; in other words, all the zeroes of are isolated points.
Proof: If all the derivatives of at vanish, then by Taylor expansion vanishes in some open neighbourhood of , and then by Corollary 23 vanishes everywhere, a contradiction. Thus at least one of the is non-zero. If is the first natural number for which , then by iterating the factor theorem (Corollary 22) we see that for some analytic function which is non-vanishing at . By continuity, is also non-vanishing in some disk in , and the claim follows.
One particular consequence of the above corollary is that if two entire functions agree on the real line (or even on an infinite bounded subset of the complex plane), then they must agree everywhere, since otherwise would have a non-isolated zero, contradicting Corollary 24. This strengthens Corollary 23, and helps explain why real-variable identities such as automatically extend to their complex counterparts . Another consequence is that if an entire function is real-valued on the real axis, then one has the identity
for all complex , because this identity already holds on the real line, and both sides are complex analytic. Thus for instance
Theorem 25 (Higher order Cauchy integral formula, special case) Let be an open subset of , let be holomorphic, and let be a point in . Let be such that the closed disk is contained in . Let be a closed curve in that is homotopic (as a closed curve, up to reparameterisation) in to in . Then for any natural number , the derivative of at is given by the formula
Exercise 26 Give an alternate proof of Theorem 25 by rigorously differentiating the Cauchy integral formula with respect to the 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 be an open subset of , let be holomorphic, and let be a point in . Let be such that the closed disk is contained in . Suppose that there is an such that on the circle . Then for any natural number , we have
Note that the case of this corollary is compatible with the maximum principle (Exercise 26 of Notes 1).
Theorem 28 (Liouville’s theorem) Let be an entire function that is bounded. Then is constant.
Proof: By hypothesis, there is a finite such that for all . Applying the Cauchy inequalities with and any disk , we conclude that
for any and . Sending to infinity, we conclude that vanishes identically. The claim then follows from the fundamental theorem of calculus.
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 that are differentiable (and even smooth and analytic) on the real line and bounded, but definitely not constant. Note that the complex analogue of the sine function is not a counterexample to Liouville’s theorem, since 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 be an entire function which is of polynomial growth in the sense that there exists a finite quantity and some exponent such that for all . Show that is, in fact, a polynomial.
be a polynomial of degree for some with non-zero. Then there exist complex numbers such that
Proof: This is trivial for , so suppose inductively that and the claim has already been proven for . Suppose first that the equation has no roots in the complex plane, then the function is entire. Also, this function goes to zero as , 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, is constant, which implies that is constant, which is absurd (for instance, the derivative of is the non-zero function ). Hence has at least one root . By the factor theorem (which works in any field, including the complex numbers) we can then write for some polynomial , which by the long division algorithm (or by comparing coefficients) must take the form
for some complex numbers . The claim then follows from the induction hypothesis.
The following exercises show that can be alternatively defined as an algebraic closure of the reals (together with a designated square root of ), and that extending using a different irreducible polynomial than 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 be a field containing which is a finite extension of , in the sense that is a finite-dimensional vector space over . Show that is isomorphic (as a field) to either or . (Hint: if is some element of not in , show that for some irreducible polynomial with real coefficients but no real roots. Use this to set up an isomorphism between the field generated by and with . If there is an element of not in this field , show that there for some irreducible polynomial with coefficients in and no roots in , and contradict the fundamental theorem of algebra.)
Exercise 32 A field is said to be algebraically closed if the conclusion of Theorem 30 with replaced by . Show that any algebraically closed field containing , contains a subfield that is isomorphic to (and which contains as a subfield, isomorphic to the copy of inside ). Thus, up to isomorphism, is the unique algebraic closure of , that is to say a minimal algebraically closed field containing .
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 be an open subset of , and let be a continuous function. Suppose that is conservative in the sense that for any closed polygonal path in . Then is holomorphic on .
Proof: By working locally with small balls in we may assume that is a ball (and in particular connected). By Exercise 31 of Notes 2, has an antiderivative . By definition, is complex differentiable at every point of (with derivative ), so by Corollary 20, is smooth, which implies in particular that is holomorphic on as claimed.
The power of Morera’s theorem comes from the fact that there are no differentiability requirements in the hypotheses on , and yet the conclusion is that 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 be an open subset of , and let be a sequence of holomorphic functions that converge uniformly on compact sets to a limit . Then is also holomorphic. Furthermore, for each natural number , the derivatives also converge uniformly on compact sets to for any natural number (In particular, converges pointwise to on .)
Proof: Again we may work locally and assume that is a ball (and in partiular is convex and simply connected). The are continuous, hence their locally uniform limit is also continuous. From Corollary 11 (or Corollary 14), we have on any closed polygonal path in , hence on taking locally uniform limits we also have for such paths. The holomorphicity of then follows from Morera’s theorem. The uniform convergence of to on compact sets follows from applying Theorem 25 to circular contours for and small enough that these contours lie in (note from compactness that one can take independent of ).
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 will remain harmonic. However, we will not need these weaker versions of this theorem here.
Exercise 35 (Riemann’s theorem on removable singularities) Let be an open subset of , let be a point in , and let be a holomorphic function on which is bounded near , in the sense that it is bounded on some punctured disk contained in . Show that has a removable singularity at , in the sense that is the restriction to of a holomorphic function on . (Hint: show that is conservative near , find an antiderivative, extend it to , 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 be an open subset of , and let be a continuous function such that, for each , the function is holomorphic on . Show that the function is also holomorphic on . (Hint: work locally and use Cauchy’s theorem, Morera’s theorem, and Fubini’s theorem.)
Exercise 37 (Schwarz reflection principle) Let be an open subset of that is symmetric around the real axis, that is to say whenever . Let be a continuous function on the set that is holomorphic in the open subset . Similarly, let be continuous on that is holomorphic in the open subset . Suppose further that and agree on . Show that and are both restrictions of a single holomorphic function .
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 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 of , the picture is different:
Now, very few continuous functions are conservative, and only slightly more functions are complex differentiable (and for simply connected domains , 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 involved has to be homotopic (as a closed curve, up to reparameterisation) to a circular arc , or at least to a curve of the form , for some integer . We now investigate what happens when this hypothesis is removed. A key notion is that of a winding number.
Definition 38 (Winding number) Let be a closed curve, and let be a complex number that is not in the image of . The winding number of around is defined by the integral
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 by any equivalent curve, and if one replaces the curve with its reversal , 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
when is homotopic in (as a closed curve, up to reparameterisation) to a circle , and more generally that
if is homotopic in (as a closed curve, up to reparameterisation) to a curve of the form , . Thus we see, intuitively at least, that measures the number of times winds counterclockwise about , which explains the term “winding number”.
We can now state a more general form of the Cauchy integral formula:
The claim then follows from (6).
Exercise 40 (Higher order general Cauchy integral formula) With as in the above theorem, show that
for every natural number . (Hint: instead of approximating by , use a partial Taylor expansion of . 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
The following specific corollary of this lemma will be useful for us.
Corollary 42 can be used to compute the winding number near infinity as follows. Given a curve and a point , define the distance
and the diameter
Proof: Apply Corollary 42 with equal to and equal to any point in the image of .
Corollary 42 also gives local constancy of the winding number:
Proof: From Corollary 42, we see that if is small enough and , then
where is the translation of by . But by a translation change of variables we see that
and the claim follows.
Exercise 45 Give an alternate proof of Lemma 44 based on differentiation under the integral sign and using the fact that has an antiderivative away from .
As confirmation of the interpretation of as a winding number, we can now establish integrality:
Proof: By Corollary 42 we may assume without loss of generality that is a closed polygonal path. By partitioning a polygon into triangles (and using Lemma 44 to move slightly out of the way of any new edges formed by this partition) it suffices to verify this for triangular . But this follows from the Cauchy integral formula (if is in the interior of the triangle) or Cauchy’s theorem (if is in the exterior).
Exercise 47 Give another proof of Lemma 46 by restricting again to closed polygonal paths , and showing that the function is constant on by establishing that it is continuous and has vanishing derivative at all but finitely many points. (Note that exists for all but finitely many , 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.
Furthermore the exterior region is connected and unbounded, and the interior region is connected, non-empty and bounded. Finally, if is any open set that contains and its interior, then is contractible to a point in .
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 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 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 in the above theorem is equal to , we say that the simple closed curve has an anticlockwise orientation; if instead we say that has a clockwise orientation. Thus for instance, has an anticlockwise orientation, while its reversal has the clockwise orientation.
- (i) If have disjoint image, show that either lies entirely in the interior of , or in the exterior.
- (ii) If avoids the exterior of , show that the interior of is contained in the interior of , and the exterior of contains the exterior of .
- (iii) If avoids the interior of , and avoids the interior of , show that the interior of is contained in the exterior of , and the exterior of contains the interior of .
(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 be a non-trivial simple closed curve. Show that the interior of is simply connected. (Hint: first show that any simple closed polygonal path in 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 there is a homeomorphism that maps to the unit circle , the interior of to the unit disk , and the exterior to the exterior region . 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 be a simple closed curve, and let be an open set containing and its interior. Let be a holomorphic function.
- (i) (Cauchy’s theorem) One has .
- (ii) (Cauchy integral formula) If lies outside of the image of , then the expression vanishes if lies in the exterior of , equals if lies in the interior of and is oriented anti-clockwise, and equals if lies in the interior of and is oriented clockwise.
Exercise 53 Let be a polynomial with complex coefficients and . For any , let denote the closed contour .
- (i) Show that if is sufficiently large, then .
- (ii) Show that if does not vanish on the closed disk , then .
- (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 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 be a simple closed contour formed by concatenating smooth curves together. Let be an interior point of one of these curves , thus for some . Set for some and . Recall from Exercise 24 of Notes 2 that for sufficiently small , the set can be expressed as a graph of the form
for some interval and some continuously differentiable function with . Show that if is oriented anticlockwise, and is sufficiently small then the interior of contains all points in of the form for some and , and the exterior of contains all points in of the form for some and . Similarly if oriented clockwise, with the conditions and swapped.
Exercise 55 (Alexander numbering rule) Let be a simple closed contour oriented anticlockwise formed by concatenating smooth curves together. Let be a contour formed by concatenating smooth curves , with initial point and final point . Assume that there are only finitely many points where the images of and of intersect. Furthermore, assume at each of the points , , that one has a “smooth simple transverse intersection” in the sense that the following axioms are obeyed:
- (i) lies in the interior of one of the smooth curves that make up , thus for some .
- (ii) lies in the interior of one of the smooth curves that make up , thus for some .
- (iii) is only traversed once by , thus there do not exist in such that .
- (iv) The derivatives and are linearly independent over . In other words, we either have a crossing from the right in which for some and , or else we have a crossing from the left in which for some and .
Show that is equal to the total number of crossings from the left, minus the total number of crossings from the right.
Exercise 56 Let be a non-empty connected open subset of . Show that is simply connected if and only if every holomorphic function on 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 is only required to have image close to the image of , rather than be close to in a pointwise (and uniform) sense. For any curve and any , let denote the -neighbourhood of .
Proposition 57 Let be a non-trivial simple closed curve, and let . Suppose that is sufficiently small depending on and . Let be a closed curve (not necessarily simple) whose image lies in . Then is homotopic (as a closed curve, up to reparameterisation) to in , where is defined as the concatenation of copies of if is positive, the trivial curve at the initial point of if is zero, and the concatenation of copies of if is negative. In particular, from Lemma 41 one has
for all .
Proof: After reparameterisation, we can take to have domain on the unit interval , and then by periodic extension we can view as a continuous -periodic function on .
Fix this . Observe that the function is continuous and nowhere vanishing on the region . Thus, if is small enough depending on , we have the lower bound
whenever are such that . Using the -periodicity of , we conclude that if are such that
Using (11), we conclude that for each , there is an integer such that
for all . In particular, from (10) we have
whenever . Also, as is simple, we have from (14) that
for some integer . (Note that by enforcing (15), we no longer have the freedom to individually move or by an integer, so we cannot assume without loss of generality that vanishes.)
For , let denote the curve
from to ; similarly let denote the curve
for all and . Thus and are homotopic as closed curves in . But by Exercise 3, is homotopic up to reparameterisation as closed curves to in , and is similarly homotopic up to reparameterisation as closed curves to in , and the claim follows.
We can now prove Theorem 48. We first verify the claim in the easy (and visually intuitive) case that is a non-trivial simple closed polygonal curve. Removing the polygon from 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 is constant. Since each component has a non-empty boundary that is contained in , this constant value of must also be attained arbitrarily close to .
Now, a routine application of the Cauchy integral formula (see Exercise 59) shows that as crosses one of the edges of the polygon , the winding number is shifted by either or . Hence at each point on , the winding number will take two values in a sufficiently small neighbourhood of (excluding ). By a continuity argument, the integer is independent of . On the other hand, from Corollary 43 the winding number must be able to attain the value of zero. Thus we have for some . Dividing a small neighbourhood of (excluding itself) into the regions where the winding numbers are or , 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 . 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 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 outside of in which 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, contains at least two points; by an appropriate rotation, translation, and dilation we may assume that contains the points and , with being both the initial point and the final point. Then we can decompose , where is a curve from to , and is a curve from to .
separating from whenever , are such that (17) holds.
for all . Although it is not particularly necessary, we can ensure that and . By perturbing the edges of the polygonal path slightly, we may assume that none of the vertices of lie on the real axis, and that none of the self-crossings of (if any exist) lie on the real axis; thus, whenever crosses the real axis, it does so at an interior point of an edge, with no other edge of passing through that point. Note that we do not assert that the curve is simple; with some more effort one could “prune” by deleting short loops to make it simple, but this turns out to be unnecessary for the parity argument we give below.
Let be the points on the real axis where crosses. By Exercise 59 below, the winding number changes by or as crosses each of the ; by Lemma 44, this winding number is constant otherwise, and by Corollary 43 it vanishes near infinity. Thus is even, and the winding number is odd between and for any odd .
Next, observe that each point belongs to exactly one of the polygonal paths or . 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 can be grouped into pairs with odd. We conclude that there must exist an odd such that one of the lies in and the other lies in .
Fix such a . For sake of discussion let suppose that lies in and lies in . From (19) we have
and from (18) we have
for any . By the intermediate value theorem, we can thus (for large enough) find such that
We arrive at the same conclusion in the opposite case when lies in and lies in .
in particular does not lie in . By construction of , we know that is odd for all ; using Lemma 44 and Lemma 42 we conclude that 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 be a non-trivial simple closed curve. By the preceding discussion, we can find a point outside of where the winding number is non-zero. Let be a sufficiently small parameter, and let be sufficiently small depending on . By compactness, one can cover the region by a finite number of (solid) squares of sidelength 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 , and in particular will not contain if 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 , contains the point . This region cannot contain any interior point of a square , since otherwise would be trapped inside a square of sidelength and hence not contain . In particular, avoids . The region cannot be unbounded, since one could then continuously move to infinity without ever meeting , contradicting Lemma 44, Corollary 43, and the non-vanishing nature of . The boundary of 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 would be disconnected, a contradiction. Let denote the path that bounds (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 crossing an edge of another square at a point that is not on the corner of either of the two squares; as avoids both and , 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 , we conclude that there is such that on , and for all outside of and . On the other hand, by Proposition 57 there is an integer such that is homotopic (as closed curves, up to reparameterisation) in to , so in particular
We now define the interior and exterior regions by (9), (8), then we have partitioned into the interior, exterior, and . 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 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 lie in the interior region. Then for small enough, lie outside of . From (20), (21) we conclude that lie in . As is connected, we can thus join to by a path in . As the region avoids , we see from Lemma 44 that the winding number 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 in any open set that contains and its interior. Once again, we begin with the simpler case when is a simple closed polygonal path. We induct on the number of edges in . The cases can be handled by direct calculation, so suppose that and the claim has been proven for all smaller values of . 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 (see Exercise 11 below), thus there are two adjacent edges whose interior angle exceeds . If one extends in the interior until it meets the polygon again, this wil divide the polygon into two subpolygons , each of which can be verified to have fewer than edges. By Exercise 49 (which does not use the contractibility part of the Jordan curve theorem), the interiors of and are contained in the interior of , and so by the induction hypothesis they are contractible to a point in . Using Exercise 3 we conclude that is contractible to a point in also.
Now suppose that is an arbitrary simple closed curve. Let be a small parameter. As before, we can find a simple polygonal path whose interior lies in the interior of , and such that is homotopic to in , and hence in if is small enough, for some . From the previous discussion we see that is contractible to a point in , and so is also. The claim then follows (after reversing the contour if necessary). This concludes the proof of the Jordan curve theorem.
Exercise 58 Let be a simple closed polygonal path with all edges of positive length. Suppose that all interior angles of (that is, the angle that two adjacent edges make in the interior of the polygon) are less than or equal to . Show that the interior of is convex. (Hint: use a continuity argument to show that every line meets the interior of in at most one interval.)
Exercise 59 Let be a non-trivial simple closed polygonal curve, and let be a point in the interior of an edge of (i.e., is not one of the two vertices of ). Let be two points sufficiently close to that lie on opposite sides of . Without using the Jordan curve theorem, show that . (Hint: replace by a “local” closed contour that is quite short, and a “global” closed contour which avoids the line segment connecting and . Then use the Cauchy integral formula.)
Exercise 60 (Jordan arc theorem) Let be a simple non-closed curve. Show that the complement of in is connected. (Hint: first establish a variant of Proposition 57 for non-closed curves, in which is now set to zero. Then adapt the proof of the Jordan curve theorem.
Exercise 61 Let be a bounded connected non-empty open subset of . Show that is simply connected if and only if the complement is connected. (Hint: suppose that there is a point in that is separated from infinity by . Show that there is some compact subset of that also separates from infinity. Then cover by small squares as in the proof of the Jordan curve theorem to locate a simple closed polygonal path in that separates 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 be a simply connected subset of , and let be a harmonic function. Show that has a harmonic conjugate , 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 be an open subset of . Define a -chain in to be a formal linear combination
of points (which we enclose in brackets to avoid confusion with the arithmetic operations on , in particular is not identified with ), where is a natural number and the are integers; these form an additive abelian group in the usual fashion. Similarly, define a -chain in to be a formal linear combination
of curves in , which (for very minor notational reasons) we will fix to have domain in the unit interval . Finally, define a -chain in to be a formal linear combination
of -simplices , defined as continuous maps from the solid triangle .
Given a -chain in , we define its boundary to be the -chain
and call a -cycle if . Similarly, given a -chain , we define its boundary to be the -chain
where is the curve on that maps to , and similarly for and . If is a -cycle, and is holomorphic, define the integral by
If lies outside of a -cycle , define the winding number
- (i) Show that if is a -chain in , then .
- (ii) Show that if is a -chain in and is holomorphic, then .
- (iii) If is a -cycle in , and for all , show that . (Hint: first perturb to be the union of line segments coming from a grid of some small sidelength . Observe that the winding number is constant whenever ranges in the interior of one of the squares in this grid. Then find another -cycle coming from summing boundaries of such squares such that for all in the interior of grid squares. Then show that and .)
- (iv) If is a -cycle, and has an antiderivative, show that .
- (v) If is a -cycle, for all , is holomorphic, and is a point lying outside of any of the , show that
Exercise 64 Let be a subset of the complex plane which is star-shaped, which means that there exists such that for any , the line segment is also contained in . Show that every star-shaped set is simply connected.