Having discussed differentiation of complex mappings in the preceding notes, we now turn to the integration of complex maps. We first briefly review the situation of integration of (suitably regular) real functions {f: [a,b] \rightarrow {\bf R}} of one variable. Actually there are three closely related concepts of integration that arise in this setting:

  • (i) The signed definite integral {\int_a^b f(x)\ dx}, which is usually interpreted as the Riemann integral (or equivalently, the Darboux integral), which can be defined as the limit (if it exists) of the Riemann sums

    \displaystyle \sum_{j=1}^n f(x_j^*) (x_j - x_{j-1}) \ \ \ \ \ (1)

    where {a = x_0 < x_1 < \dots < x_n = b} is some partition of {[a,b]}, {x_j^*} is an element of the interval {[x_{j-1},x_j]}, and the limit is taken as the maximum mesh size {\max_{1 \leq j \leq n} |x_j - x_{j-1}|} goes to zero. It is convenient to adopt the convention that {\int_b^a f(x)\ dx := - \int_a^b f(x)\ dx} for {a < b}; alternatively one can interpret {\int_b^a f(x)\ dx} as the limit of the Riemann sums (1), where now the (reversed) partition {b = x_0 > x_1 > \dots > x_n = a} goes leftwards from {b} to {a}, rather than rightwards from {a} to {b}.

  • (ii) The unsigned definite integral {\int_{[a,b]} f(x)\ dx}, usually interpreted as the Lebesgue integral. The precise definition of this integral is a little complicated (see e.g. this previous post), but roughly speaking the idea is to approximate {f} by simple functions {\sum_{i=1}^n c_i 1_{E_i}} for some coefficients {c_i \in {\bf R}} and sets {E_i \subset [a,b]}, and then approximate the integral {\int_{[a,b]} f(x)\ dx} by the quantities {\sum_{i=1}^n c_i m(E_i)}, where {E_i} is the Lebesgue measure of {E_i}. In contrast to the signed definite integral, no orientation is imposed or used on the underlying domain of integration, which is viewed as an “undirected” set {[a,b]}.
  • (iii) The indefinite integral or antiderivative {\int f(x)\ dx}, defined as any function {F: [a,b] \rightarrow {\bf R}} whose derivative {F'} exists and is equal to {f} on {[a,b]}. Famously, the antiderivative is only defined up to the addition of an arbitrary constant {C}, thus for instance {\int x\ dx = \frac{1}{2} x^2 + C}.

There are some other variants of the above integrals (e.g. the Henstock-Kurzweil integral, discussed for instance in this previous post), which can handle slightly different classes of functions and have slightly different properties than the standard integrals listed here, but we will not need to discuss such alternative integrals in this course (with the exception of some improper and principal value integrals, which we will encounter in later notes).

The above three notions of integration are closely related to each other. For instance, if {f: [a,b] \rightarrow {\bf R}} is a Riemann integrable function, then the signed definite integral and unsigned definite integral coincide (when the former is oriented correctly), thus

\displaystyle \int_a^b f(x)\ dx = \int_{[a,b]} f(x)\ dx

and

\displaystyle \int_b^a f(x)\ dx = -\int_{[a,b]} f(x)\ dx

If {f: [a,b] \rightarrow {\bf R}} is continuous, then by the fundamental theorem of calculus, it possesses an antiderivative {F = \int f(x)\ dx}, which is well defined up to an additive constant {C}, and

\displaystyle \int_c^d f(x)\ dx = F(d) - F(c)

for any {c,d \in [a,b]}, thus for instance {\int_a^b F(x)\ dx = F(b) - F(a)} and {\int_b^a F(x)\ dx = F(a) - F(b)}.

All three of the above integration concepts have analogues in complex analysis. By far the most important notion will be the complex analogue of the signed definite integral, namely the contour integral {\int_\gamma f(z)\ dz}, in which the directed line segment from one real number {a} to another {b} is now replaced by a type of curve in the complex plane known as a contour. The contour integral can be viewed as the special case of the more general line integral {\int_\gamma f(z) dx + g(z) dy}, that is of particular relevance in complex analysis. There are also analogues of the Lebesgue integral, namely the arclength measure integrals {\int_\gamma f(z)\ |dz|} and the area integrals {\int_\Omega f(x+iy)\ dx dy}, but these play only an auxiliary role in the subject. Finally, we still have the notion of an antiderivative {F(z)} (also known as a primitive) of a complex function {f(z)}.

As it turns out, the fundamental theorem of calculus continues to hold in the complex plane: under suitable regularity assumptions on a complex function {f} and a primitive {F} of that function, one has

\displaystyle \int_\gamma f(z)\ dz = F(z_1) - F(z_0)

whenever {\gamma} is a contour from {z_0} to {z_1} that lies in the domain of {f}. In particular, functions {f} that possess a primitive must be conservative in the sense that {\int_\gamma f(z)\ dz = 0} for any closed contour. This property of being conservative is not typical, in that “most” functions {f} will not be conservative. However, there is a remarkable and far-reaching theorem, the Cauchy integral theorem (also known as the Cauchy-Goursat theorem), which asserts that any holomorphic function is conservative, so long as the domain is simply connected (or if one restricts attention to contractible closed contours). We will explore this theorem and several of its consequences the next set of notes.

— 1. Integration along a contour —

The notion of a curve is a very intuitive one. However, the precise mathematical definition of what a curve actually is depends a little bit on what type of mathematics one wishes to do. If one is mostly interested in topology, then a good notion is that of a continuous (parameterised) curve. If one wants to do analysis in somewhat irregular domains, it is convenient to restrict the notion of curve somewhat, to the rectifiable curves. If one is doing analysis in “nice” domains (such as the complex plane {{\bf C}}, a half-plane, a punctured plane, a disk, or an annulus), then it is convenient to restrict the notion further, to the piecewise smooth curves, also known as contours. If one wished to get to the main theorems of complex analysis as quickly as possible, then one would restrict attention only to contours and skip much of this section; however we shall take a more leisurely approach here, discussing curves and rectifiable curves as well, as these concepts are also useful outside of complex analysis. In fact, we will structure our notes so that most of our theorems in fact apply to rectifiable curves (and several will in fact be applicable to arbitrary continuous curves).

We begin by defining the notion of a continuous curve.

Definition 1 (Continuous curves) A continuous parameterised curve, or curve for short, is a continuous map {\gamma: [a,b] \rightarrow {\bf C}} from a compact interval {[a,b] \subset {\bf R}} to the complex plane {{\bf C}}. We call the curve trivial if {a=b}, and non-trivial otherwise. We refer to the complex numbers {\gamma(a), \gamma(b)} as the initial point and terminal point (or final point) of the curve respectively, and refer to these two points collectively as the endpoints of the curve. We say that the curve is closed if {\gamma(a) = \gamma(b)}. We say that the curve is simple if one has {\gamma(t) \neq \gamma(t')} for any distinct {t,t' \in [a,b]}, with the possible exception of the endpoint cases {t=a, t'=b} or {t=b, t'=a} (thus we allow closed curves to be simple). We refer to the subset {\gamma([a,b]) := \{ \gamma(t): t \in [a,b]\}} of the complex plane as the image of the curve.

We caution that the term “closed” here does not refer to the topological notion of closure: for any curve {\gamma} (closed or otherwise), the image {\gamma([a,b])} of the curve, being the continuous image of a compact set, is necessarily a compact subset of {{\bf C}} and is thus always topologically closed.

A basic example of a curve is the directed line segment {\gamma_{z_1 \rightarrow z_2}: [0,1] \rightarrow {\bf C}} from one complex point {z_1} to another {z_2}, defined by

\displaystyle \gamma_{z_1 \rightarrow z_2}(t) := (1-t) z_1 + t z_2

for {0 \leq t \leq 1}. (Thus, contrary to the informal English meaning of the terms, we consider line segments to be examples of curves, despite having zero curvature; in general, it is convenient in mathematics to admit such “degenerate” objects into one’s definitions, in order to obtain good closure properties for these objects, and to maximise the generality of the definition.) If {z_1 \neq z_2}, this is a simple curve, while for {z_1 = z_2} it is (a rather degenerate, but still non-trivial) closed curve. Another important example of a curve is the anti-clockwise circle {\gamma_{z_0,r,\circlearrowleft}: [0,2\pi] \rightarrow {\bf C}} of some radius {r>0} around a complex centre {z_0 \in {\bf C}}, defined by

\displaystyle \gamma_{z_0,r,\circlearrowleft}(\theta) := z_0 + r e^{i\theta}. \ \ \ \ \ (2)

This is a simple closed non-trivial curve. If we extended the domain here from {[0,2\pi]} to (say) {[0,4\pi]}, the curve would remain closed, but would no longer be simple (every point in the image is now traversed twice by the curve).

Note that it is technically possible for two distinct curves to have the same image. For instance, the anti-clockwise circle {\tilde \gamma_{z_0,r,\circlearrowleft}: [0,1] \rightarrow {\bf C}} of some radius {r>0} around a complex centre {z_0 \in {\bf C}} defined by

\displaystyle \tilde \gamma_{z_0,r,\circlearrowleft}(\theta) := z_0 + r e^{2\pi i \theta}

traverses the same image as the previous curve (2), but is considered a distinct curve from {\gamma_{z_0,r,\circlearrowleft}}. Nevertheless the two curves are closely related to each other, and we formalise this as follows. We say that one curve {\gamma_2: [a_2,b_2] \rightarrow {\bf C}} is a continuous reparameterisation of another {\gamma_1: [a_1, b_1] \rightarrow {\bf C}}, if there is a homeomorphism {\phi: [a_1,b_1] \rightarrow [a_2,b_2]} (that is to say, a continuous invertible map whose inverse {\phi^{-1}: [a_2,b_2] \rightarrow [a_1,b_1]} is also continuous) which is endpoint preserving (i.e., {\phi(a_1)=a_2} and {\phi(b_1)=b_2}) such that {\gamma_2(\phi(t)) = \gamma_1(t)} for all {t \in [a_1,b_1]} (that is to say, {\gamma_1 = \gamma_2 \circ \phi}, or equivalently {\gamma_2 = \gamma_1 \circ \phi^{-1}}), in which case we write {\gamma_1 \equiv \gamma_2}. Thus for instance {\gamma_{z_0,r,\circlearrowleft} \equiv \tilde \gamma_{z_0,r,\circlearrowleft}}. The relation of being a continuous reparameterisation is an equivalence relation, so one can talk about the notion of a curve “up to continuous reparameterisation”, by which we mean an equivalence class of a curve under this relation. Thus for instance the image of a curve, as well as its initial point and end point, are well defined up to continuous reparameterisation, since if {\gamma_1 \equiv \gamma_2} then {\gamma_1} and {\gamma_2} have the same image, the same initial point, and the same terminal point. It is common to depict an equivalence class of a curve {\gamma} graphically, by drawing its image together with an arrow depicting the direction of motion from the initial point to its endpoint. (In the case of a non-simple curve, one may need multiple arrows in order to clarify the direction of motion, and also the possible multiplicity of the curve.)

Exercise 2 Let {\phi: [a_1,b_1] \rightarrow [a_2,b_2]} be a continuous invertible map.

  • (i) Show that {\phi^{-1}: [a_2,b_2] \rightarrow [a_1,b_1]} is continuous, so that {\phi} is a homeomorphism. (Hint: use the fact that a continuous image of a compact set is compact, and that a subset of an interval is topologically closed if and only if it is compact.)
  • (ii) If {\phi(a_1)=a_2}, show that {\phi(b_1) = b_2} and that {\phi} is monotone increasing. (Hint: use the intermediate value theorem.)
  • (iii) Conversely, if {\psi: [a_1,b_1] \rightarrow [a_2,b_2]} is a continuous monotone increasing map with {\psi(a_1) = a_2} and {\psi(b_1) = b_2}, show that {\psi} is a homeomorphism.

It will be important for us that we do not allow reparameterisations to reverse the endpoints. For instance, if {z_1,z_2} are distinct points in the complex plane, the directed line segment {\gamma_{z_1 \rightarrow z_2}} is not a reparameterisation of the directed line segment {\gamma_{z_2 \rightarrow z_1}} since they do not have the same initial point (or same terminal point); the map {t \mapsto 1-t} is a homeomorphism from {[0,1]} to {[0,1]} but it does not preserve the initial point or the terminal point. In general, given a curve {\gamma: [a,b] \rightarrow {\bf C}}, we define its reversal {-\gamma: [-b,-a] \rightarrow {\bf C}} to be the curve {(-\gamma)(t) := \gamma(-t)}, thus for instance {\gamma_{z_2 \rightarrow z_1}} is (up to reparameterisation) the reversal of {\gamma_{z_1 \rightarrow z_2}}, thus

\displaystyle \gamma_{z_2 \rightarrow z_1} \equiv - \gamma_{z_1 \rightarrow z_2}.

Another basic operation on curves is that of concatenation. Suppose we have two curves {\gamma_1: [a_1,b_1] \rightarrow {\bf C}} and {\gamma_2: [a_2,b_2] \rightarrow {\bf C}} with the property that the terminal point {\gamma_1(b_1)} of {\gamma_1} equals the initial point {\gamma_2(a_2)} of {\gamma_2}. We can reparameterise {\gamma_2} by translation to {\tilde \gamma_2: [b_1, b_2+b_1-a_2] \rightarrow {\bf C}}, defined by {\tilde \gamma_2(t) := \gamma_2(t - b_1 + a_2)}. We then define the concatenation or sum {\gamma_1 + \gamma_2: [a_1, b_2+b_1-a_2] \rightarrow {\bf C}} by setting

\displaystyle (\gamma_1 + \gamma_2)(t) := \gamma_1(t)

for {a_1 \leq t \leq b_1} and

\displaystyle (\gamma_1 + \gamma_2)(t) := \tilde \gamma_2(t)

for {b_1 \leq t \leq b_2+b_1-a_2} (note that these two definitions agree on their common domain point {t=b_1} by the hypothesis {\gamma_1(b_1) = \gamma_2(a_2)}. It is easy to see that this concatenation {\gamma_1+\gamma_2} is still a continuous curve. The reader at this point is encouraged to draw a picture to understand what the concatenation operation is doing; it is much simpler to grasp it visually than the above lengthy definition may suggest. If the terminal point of {\gamma_1} does not equal the initial point of {\gamma_2}, we leave the sum {\gamma_1+\gamma_2} undefined. (One can define more general spaces than the space of curves in which such an addition can make sense, such as the space of {1}-currents if one assumes some rectifiability on the curves, or the space of {1}-chains, but we will not need such general spaces here.)

Concatenation is well behaved with respect to equivalence and reversal:

Exercise 3 Let {\gamma_1, \gamma_2, \gamma_3, \tilde \gamma_1, \tilde \gamma_2} be continuous curves. Suppose that the terminal point of {\gamma_1} equals the initial point of {\gamma_2}, and the terminal point of {\gamma_2} equals the initial point of {\gamma_3}.

  • (i) (Concatenation well defined up to equivalence) If {\gamma_1 \equiv \tilde \gamma_1} and {\gamma_2 \equiv \tilde \gamma_2}, show that {\gamma_1+\gamma_2 \equiv \tilde \gamma_1 + \tilde \gamma_2} and {-\gamma_1 \equiv - \tilde \gamma_1}.
  • (ii) (Concatenation associative) Show that {(\gamma_1+\gamma_2)+\gamma_3 = \gamma_1 + (\gamma_2 + \gamma_3)}. In particular, we certainly have {(\gamma_1+\gamma_2)+\gamma_3 \equiv \gamma_1 + (\gamma_2 + \gamma_3)}
  • (iii) (Concatenation and reversal) Show that {-(\gamma_1+\gamma_2) \equiv (-\gamma_2) + (-\gamma_1)}.
  • (iv) (Non-commutativity) Give an example in which {\gamma_1+\gamma_2} and {\gamma_2+\gamma_1} are both well-defined, but not equivalent to each other.
  • (v) (Identity) If {z_1} and {z_2} denote the initial and terminal points of {\gamma_1} respectively, and {\delta_z: [0,0] \rightarrow {\bf C}} is the trivial curve {\delta_z: 0 \rightarrow z} defined for any {z \in {\bf C}}, show that {\delta_{z_1} + \gamma_1 \equiv \gamma_1} and {\gamma_1 + \delta_{z_2} \equiv \gamma_1}.
  • (vi) (Non-invertibility) Give an example in which {\gamma_1 + (-\gamma_1)} is not equivalent to a trivial curve. (It will however be homologous to a trivial curve, as we will discuss in later notes.)

Remark 4 The above exercise allows one to view the space of curves up to equivalence as a category, with the points in the complex plane being the objects of the category, and each equivalence class of curves being a single morphism from the initial point to the terminal point (and with the equivalence class of trivial curves being the identity morphisms). This point of view can be useful in topology, particularly when relating to concepts such as the fundamental group (and fundamental groupoid), monodromy, and holonomy. However, we will not need to use any advanced category-theoretic concepts in this course.

Exercise 5 Let {z_0 \in {\bf C}} and {r>0}. For any integer {m}, let {\gamma_m: [0,2\pi] \rightarrow {\bf C}} denote the curve

\displaystyle \gamma_m(t) := z_0 + r e^{i m t}

(thus for instance {\gamma_1 = \gamma_{z_0,r,\circlearrowright}}).

  • (i) Show that for any integer {m,m'}, we have {-\gamma_m \equiv \gamma_{-m}}.
  • (ii) Show that for any non-negative integers {m,m'}, we have {\gamma_m + \gamma_{m'} \equiv \gamma_{m+m'}}. What happens for other values of {m,m'}?
  • (ii) If {m,m'} are distinct integers, show that {\gamma_m \not \equiv \gamma_{m'}}.

Given a sequence of complex numbers {z_0,z_1,\dots,z_n}, we define the polygonal path {\gamma_{z_0 \rightarrow z_1 \rightarrow \dots \rightarrow z_n}} traversing these numbers in order to be the curve

\displaystyle \gamma_{z_0 \rightarrow z_1 \rightarrow \dots \rightarrow z_n} := \gamma_{z_0 \rightarrow z_1} + \gamma_{z_1 \rightarrow z_2} + \dots + \gamma_{z_{n-1} \rightarrow z_n}.

This is well-defined thanks to Exercise 3(ii) (actually all we really need in applications is being well-defined up to equivalence). Thus for instance {\gamma_{z_0 \rightarrow z_1 \rightarrow z_2 \rightarrow z_0}} would traverse a closed triangular path connecting {z_0}, {z_1}, and {z_2} (this path may end up being non-simple if the points {z_0,z_1,z_2} are collinear).

In order to do analysis, we need to restrict our attention to those curves which are rectifiable:

Definition 6 Let {\gamma: [a,b] \rightarrow {\bf C}} be a curve. The arc length {|\gamma|} of the curve is defined to be the supremum of the quantities

\displaystyle \sum_{j=1}^n |\gamma(t_{j+1})-\gamma(t_j)|

where {n} ranges over the natural numbers and {a = t_0 < t_1 < \dots < t_n = b} ranges over the partitions of {[a,b]}. We say that the curve {\gamma} is rectifiable if its arc length is finite.

The concept is best understood visually: a curve is rectifiable if there is some finite bound on the length of polygonal paths one can form while traversing the curve in order. From Exercise 2 we see that equivalent curves have the same arclength, so the concepts of arclength and rectifiability are well defined for curves that are only given up to continuous reparameterisation.

Exercise 7 Let {\gamma_1,\gamma_2} be curves, with the terminal point of {\gamma_1} equal to the initial point of {\gamma_2}. Show that

\displaystyle |\gamma_1+\gamma_2| = |\gamma_1| + |\gamma_2|.

In particular, {\gamma_1+\gamma_2} is rectifiable if and only if {\gamma_1, \gamma_2} are both individually rectifiable.

It is not immediately obvious that any reasonable curve (e.g. the line segments {\gamma_{z_1 \rightarrow z_2}} or the circles {\gamma_{z_0,r,\circlearrowleft}}) are rectifiable. To verify this, we need two preliminary results.

Lemma 8 (Triangle inequality) Let {f: [a,b] \rightarrow {\bf C}} be a continuous function. Then

\displaystyle |\int_a^b f(t)\ dt| \leq \int_a^b |f(t)|\ dt.

Here we interpret {\int_a^b f(t)\ dt} as the Riemann integral (or equivalently, {\int_a^b f(t)\ dt = \int_a^b \mathrm{Re} f(t)\ dt + i \int_a^b \mathrm{Im} f(t)\ dt}).

Proof: We first attempt to prove this inequality by considering the real and imaginary parts separately. From the real-valued triangle inequality (and basic properties of the Riemann integral) we have

\displaystyle | \mathrm{Re} \int_a^b f(t)\ dt | = |\int_a^b \mathrm{Re} f(t)\ dt |

\displaystyle \leq \int_a^b |\mathrm{Re} f(t)|\ dt

\displaystyle \leq \int_a^b |f(t)|\ dt

and similarly

\displaystyle |\mathrm{Im} \int_a^b f(t)\ dt| \leq \int_a^b |f(t)|\ dt

but these two bounds only yield the weaker estimate

\displaystyle |\int_a^b f(t)\ dt| \leq \sqrt{2} \int_a^b |f(t)|\ dt.

To eliminate this {\sqrt{2}} loss we can amplify the above argument by exploiting phase rotation. For any real {\theta}, we can repeat the above arguments (using the complex linearity of the Riemann integral, which is easily verified) to give

\displaystyle | \mathrm{Re} e^{i\theta} \int_a^b f(t)\ dt | = |\int_a^b \mathrm{Re} e^{i\theta} f(t)\ dt |

\displaystyle \leq \int_a^b |\mathrm{Re} e^{i\theta} f(t)|\ dt

\displaystyle \leq \int_a^b |f(t)|\ dt.

But we have {|z| = \sup_\theta \mathrm{Re} e^{i\theta} z} for any complex number {z}, so taking the supremum of both sides in {\theta} we obtain the claim. \Box

Exercise 9 Let {a \leq b} be real numbers. Show that the interval {[a,b]} is topologically connected, that is to say the only two subsets of {[a,b]} that are both open and closed relative to {[a,b]} are the empty set and all of {[a,b]}. (Hint: if {E} is a non-empty set that is both open and closed in {[a,b]} and contains {a}, consider the supremum of all {T_* \in [a,b]} such that {[a,T_*] \subset E}.)

Next, we say that a non-trivial curve {\gamma: [a,b] \rightarrow {\bf C}} is continuously differentiable if the derivative

\displaystyle \gamma'(t) := \lim_{t' \rightarrow t: t' \in [a,b] \backslash \{t\}} \frac{\gamma(t') - \gamma(t)}{t'-t}

exists and is continuous for all {t \in [a,b]} (note that we are only taking right-derivatives at {t=a} and left-derivatives at {t=b}).

Proposition 10 (Arclength formula) If {\gamma: [a,b] \rightarrow {\bf C}} is a continuously differentiable curve, then it is rectifiable, and

\displaystyle |\gamma| = \int_a^b |\gamma'(t)|\ dt.

Proof: We first prove the upper bound

\displaystyle |\gamma| \leq \int_a^b |\gamma'(t)|\ dt \ \ \ \ \ (3)

which in particular implies the rectifiability of {\gamma} since the right-hand side of (3) is finite. Let {a = t_0 < \dots < t_n = b} be any partition of {[a,b]}. By the fundamental theorem of calculus (applied to the real and imaginary parts of {\gamma}) we have

\displaystyle \gamma(t_j) - \gamma(t_{j-1}) = \int_{t_{j-1}}^{t_j} \gamma'(t)\ dt

for any {1 \leq i \leq n}, and hence by Lemma 8 we have

\displaystyle |\gamma(t_j) - \gamma(t_{j-1})| \leq \int_{t_{j-1}}^{t_j} |\gamma'(t)|\ dt.

Summing in {j} we obtain

\displaystyle \sum_{j=1}^n |\gamma(t_i) - \gamma(t_{i-1})| \leq \int_a^b |\gamma'(t)|\ dt

and taking suprema over all partitions we obtain (3).

Now we need to show the matching lower bound. Let {\varepsilon>0} be a small quantity, and for any {a \leq T \leq b}, let {\gamma_{[a,T]}: [a,T] \rightarrow {\bf C}} denote the restriction of {\gamma: [a,b] \rightarrow {\bf C}} to {[a,T]}. We will show the bound

\displaystyle |\gamma_{[a,T]}| \geq \int_a^T |\gamma'(t)|\ dt - \varepsilon (T-a) \ \ \ \ \ (4)

for all {a \leq T \leq b}; specialising to {T=b} and then sending {\varepsilon \rightarrow 0} will give the claim.

It remains to prove (4) for a given choice of {\varepsilon}. We will use a continuous version of induction known as the continuity method, which exploits Exercise 9.

Let {\Omega_\varepsilon \subset [a,b]} denote the set of {T_* \in [a,b]} such that (4) holds for all {a \leq T \leq T_*}. It is clear (using Exercise 7, which implies that {|\gamma_{[a,T]}|} is an increasing function of {T}) that this set {\Omega_\varepsilon} is topologically closed, and also contains the left endpoint {a} of {[a,b]}. If {T_* \in \Omega_\varepsilon} and {T_* < b}, then from the differentiability of {\gamma} at {T_*}, we have some interval {[T_*, T_*+\delta] \subset [a,b]} such that

\displaystyle |\frac{\gamma(T) - \gamma(T_*)}{T-T_*} - \gamma'(T_*)| \leq \varepsilon/2

for all {T \in [T_*, T_*+\delta]}. Rearranging this using the triangle inequality, we have

\displaystyle |\gamma(T) - \gamma(T_*)| \geq |\gamma'(T_*)| (T-T_*) - \frac{\varepsilon}{2} (T-T_*).

Also, from the continuity of {|\gamma'|} we have

\displaystyle \int_{T_*}^T |\gamma'(t)\ dt| \leq |\gamma'(T_*)| (T-T_*) + \frac{\varepsilon}{2} (t-T_*)

for all {T \in [T_*, T_*+\delta]}, if {\delta} is small enough. We conclude that

\displaystyle |\gamma(T) - \gamma(T_*)| \geq \int_{T_*}^T \gamma'(t)\ dt - \varepsilon (T-T_*),

and hence

\displaystyle |\gamma_{[T_*,T]}| \geq \int_{T_*}^T \gamma'(t)\ dt - \varepsilon (T-T_*)

where {\gamma_{[T_*,T]}: [T_*,T] \rightarrow {\bf C}} is the restriction of {\gamma: [a,b] \rightarrow {\bf C}} to {[T_*,T]}. Adding this to the {T=T_*} case of (4) using (7) we conclude that (4) also holds for all {T \in [T_*,T_*+\delta]}. From this we see that {\Omega_\varepsilon} is (relatively) open in {[a,b]}; from the connectedness of {[a,b]} we conclude that {\Omega_\varepsilon = [a,b]}, and we are done. \Box

It is now easy to verify that the line segment {\gamma_{z_1 \rightarrow z_2}} is rectifiable with arclength {|z_2-z_1|}, and that the circle {\gamma_{z_0,r,\circlearrowleft}} is rectifiable with arclength {2\pi r}, exactly as one would expect from elementary geometry. Finally, from Lemma 7, a polygonal path {\gamma_{z_0 \rightarrow z_1 \rightarrow \dots \rightarrow z_n}} will be rectifiable with arclength {|z_0-z_1| + \dots + |z_{n-1}-z_n|}, again exactly as one would expect.

Exercise 11 Show that the curve {\gamma: [0,1] \rightarrow {\bf C}} defined by setting {\gamma(t) := t + i t \sin(\frac{1}{t})} for {0 < t \leq 1} and {\gamma(0) := 0} is continuous but not rectifiable. (Hint: it is not necessary to compute the arclength precisely; a lower bound that goes to infinity will suffice. Graph the curve to discover some convenient partitions with which to generate such lower bounds. Alternatively, one can apply the arclength formula to some subcurves of {\gamma}.)

Exercise 12 (This exercise presumes familiarity with Lebesgue measure.) Show that the image of a rectifable curve is necessarily of measure zero in the complex plane. (In particular, space-filling curves such as the Peano curve or the Hilbert curve cannot be rectifiable.)

Remark 13 As the above exercise suggests, many fractal curves will fail to be rectifiable; for instance the Koch snowflake is a famous example of an unrectifiable curve. (The situation is clarified once one develops the theory of Hausdorff dimension, as is done for instance in this previous post: any curve of Hausdorff dimension strictly greater than one will be unrectifiable.)

Much as continuous functions {f: [a,b] \rightarrow {\bf R}} on an interval may be integrated by taking limits of Riemann sums, we may also integrate continuous functions {f: \gamma([a,b]) \rightarrow {\bf C}} on the image of a rectifiable curve {\gamma: [a,b] \rightarrow {\bf C}}:

Proposition 14 (Integration in rectifiable curves) Let {\gamma: [a,b] \rightarrow {\bf C}} be a rectifiable curve, and let {f: \gamma([a,b]) \rightarrow {\bf C}} be a continuous function on the image {\gamma([a,b])} of {\gamma}. Then the “Riemann sums”

\displaystyle \sum_{j=1}^n f(\gamma(t^*_j)) (\gamma(t_j) - \gamma(t_{j-1})), \ \ \ \ \ (5)

where {a = t_0 < \dots < t_n = b} ranges over the partitions of {[a,b]}, and for each {1 \leq j \leq n}, {t^*_j} is an element of {[t_{j-1},t_j]}, converge as the maximum mesh size {\max_{1 \leq j \leq n} |t_j - t_{j-1}|} goes to zero to some complex limit, which we will denote as {\int_\gamma f(z)\ dz}. In other words, for every {\varepsilon>0} there exists a {\delta} such that

\displaystyle |\sum_{j=1}^n f(\gamma(t^*_j)) (\gamma(t_j) - \gamma(t_{j-1})) - \int_\gamma f(z)\ dz| \leq \varepsilon

whenever {\max_{1 \leq j \leq n} |t_j - t_{j-1}| \leq \delta}.

Proof: In real analysis courses, one often uses the order properties of the real line to replace the rather complicated looking Riemann sums with the simpler Darboux sums, en route to proving the real-variable analogue of the above proposition. However, in our complex setting the ordering of the real line is not available, so we will tackle the Riemann sums directly rather than try to compare them with Darboux sums.

It suffices to prove that the “Riemann sums” (5) are a Cauchy sequence, in the sense that the difference

\displaystyle \sum_{j=1}^n f(\gamma(t^*_j)) (\gamma(t_j) - \gamma(t_{j-1})) - \sum_{k=1}^m f(\gamma(s^*_k)) (\gamma(s_k) - \gamma(s_{k-1}))

between two sums of the form (5) is smaller than any specified {\varepsilon>0} if the maximum mesh sizes of the two partitions {a = t_0 < \dots < t_n = b} and {a = s_0 < \dots < s_m = b} are both small enough. From the triangle inequality, and from the fact that any two partitions have a common refinement, it suffices to prove this under the additional assumption that the second partition {a = s_0 < \dots < s_m = b} is a refinement of {a = t_0 < \dots < t_n = b}. This means that there is an increasing sequence {0 = m_0 < m_1 < \dots < m_n = b} of natural numbers such that {s_{m_j} = t_j} for {j=0,\dots,n}. In that case, the above difference may be rearranged as

\displaystyle \sum_{j=1}^n E_j

where

\displaystyle E_j := f(\gamma(t^*_j)) (\gamma(t_j) - \gamma(t_{j-1})) - \sum_{k=m_{j-1}+1}^{m_j} f(\gamma(s^*_k)) (\gamma(s_k) - \gamma(s_{k-1})).

By telescoping series, we may rearrange {E_j} further as

\displaystyle E_j := \sum_{k=m_{j-1}+1}^{m_j} (f(\gamma(t^*_j)) - f(\gamma(s^*_k))) (\gamma(s_k) - \gamma(s_{k-1})).

As {\gamma: [a,b] \rightarrow {\bf C}} is continuous and {[a,b]} is compact, {\gamma} is uniformly continuous. In particular, if the maximum mesh sizes are small enough, we have

\displaystyle |f(\gamma(t^*_j)) - f(\gamma(s^*_k))| \leq \varepsilon

for all {1 \leq j \leq n} and {m_{j-1} \leq k \leq m_j}. From the triangle inequality we conclude that

\displaystyle |E_j| \leq \varepsilon \sum_{k=m_{j-1}+1}^{m_j} |\gamma(s_k) - \gamma(s_{k-1})|

and hence on summing in {j} and using the triangle inequality, we can bound

\displaystyle |\sum_{j=1}^n E_j| \leq \varepsilon |\gamma|.

Since {|\gamma|} is finite, and {\varepsilon} can be made arbitrarily small, we obtain the required Cauchy sequence property. \Box

One cannot simply omit the rectifiability hypothesis from the above proposition:

Exercise 15 Give an example of a curve {\gamma: [a,b] \rightarrow {\bf C}} such that the Riemann sums

\displaystyle \sum_{j=1}^n \gamma(t^*_j) (\gamma(t_j) - \gamma(t_{j-1}))

fail to converge to a limit as the maximum mesh size goes to zero, so the integral {\int_\gamma z\ dz} does not exist in the sense of convergent Riemann sums even though the integrand {z} is extremely smooth. (Of course, such a curve {\gamma} cannot be rectifiable, thanks to Proposition 14.) Hint: the non-rectifiable curve in Exercise 11 is a good place to start, but it turns out that this curve does not oscillate wildly enough to make the Riemann sums here diverge, because of the decay of the function {z \mapsto z} near the origin. Come up with a variant of this curve which oscillates more. Nevertheless, it is still possible to assign a meaning to integrals such as {\int_\gamma z\ dz}, as we shall see in the next set of notes once Cauchy’s theorem is established.

By abuse of notation, we will refer to the quantity {\int_\gamma f(z)\ dz} as the contour integral of {f} along {\gamma}, even though {\gamma} is not necessarily a contour (we will define this concept shortly). We have some easy properties of this integral:

Exercise 16 Let {\gamma: [a,b] \rightarrow {\bf C}} be a rectifiable curve, and {f: \gamma([a,b]) \rightarrow {\bf C}} be a continuous function.

  • (i) (Independence of parameterisation) If {\tilde \gamma \equiv \gamma} is another curve equivalent to {\gamma}, show that {\int_{\tilde \gamma} f(z)\ dz = \int_\gamma f(z)\ dz}.
  • (ii) (Reversal) Show that {\int_{-\gamma} f(z)\ dz = - \int_\gamma f(z)\ dz}.
  • (iii) (Concatenation) If {\gamma = \gamma_1 + \gamma_2} for some curves {\gamma_1, \gamma_2}, show that

    \displaystyle \int_\gamma f(z)\ dz = \int_{\gamma_1} f(z)\ dz + \int_{\gamma_2} f(z)\ dz.

  • (iv) (Change of variables) If {\gamma} is continuously differentiable, show that

    \displaystyle \int_\gamma f(z)\ dz = \int_a^b f(\gamma(t)) \gamma'(t)\ dt.

  • (v) (Upper bound) If there is a pointwise bound of the form {|f(z)| \leq M} for all {z \in \gamma([a,b])} and some {M>0}, show that

    \displaystyle |\int_\gamma f(z)\ dz| \leq M |\gamma|.

  • (vi) (Linearity) If {c} is a complex number and {g: \gamma([a,b]) \rightarrow {\bf C}} is a continuous function, show that

    \displaystyle \int_\gamma c f(z)\ dz = c \int_\gamma f(z)\ dz

    and

    \displaystyle \int_\gamma f(z) + g(z)\ dz = \int_\gamma f(z)\ dz + \int_\gamma g(z)\ dz.

  • (vii) (Integrating a constant) If {\gamma} has initial point {z_1} and terminal point {z_2}, show that

    \displaystyle \int_\gamma\ dz = z_2 - z_1.

  • (viii) (Uniform convergence) If {f_n: \gamma([a,b]) \rightarrow {\bf C}}, {n \geq 1} is a sequence of continuous functions converging uniformly to {f} as {n \rightarrow \infty}, show that {\int_\gamma f_n(z)\ dz} converges to {\int_\gamma f(z)\ dz} as {n \rightarrow \infty}. (The requirement of uniform convergence can be relaxed substantially, thanks to tools such as the dominated convergence theorem, but this weaker convergence theorem will suffice for most of our applications.)
  • (ix) (Change of variables, II) Let {U} be an open neighbourhood of {\gamma([a,b])}, let {\phi: U \rightarrow {\bf C}} be a holomorphic function, and let {g: \phi(\gamma([a,b])) \rightarrow {\bf C}} be a continuous function. Show that

    \displaystyle \int_{\phi \circ \gamma} g(w)\ dw = \int_\gamma g(\phi(z)) \phi'(z)\ dz.

    (For this question, you may assume without proof that holomorphic functions are continuously differentiable; this fact will be proven (without reliance on this part of the exercise) in the next set of notes.)

Exercise 17 (This exercise assumes familiarity with the Riemann-Stieltjes integral.) Let {\gamma: [a,b] \rightarrow {\bf C}} be a rectifiable curve. Let {g: [a,b] \rightarrow {\bf C}} denote the monotone non-decreasing function

\displaystyle g(T) := |\gamma_{[a,T]}|

for {a \leq T \leq b}, where {\gamma_{[a,T]}: [a,T] \rightarrow {\bf C}} is the restriction of {\gamma: [a,b] \rightarrow {\bf C}} to {[a,T]}. For any continuous function {f: \gamma([a,b]) \rightarrow {\bf C}}, define the arclength measure integral {\int_\gamma f(z)\ |dz|} by the formula

\displaystyle \int_\gamma f(z)\ |dz| := \int_a^b f(\gamma(t))\ dg(t)

where the right-hand side is a Riemann-Stieltjes integral. Establish the triangle inequality

\displaystyle |\int_\gamma f(z)\ dz| \leq \int_\gamma |f(z)|\ |dz|

for any continuous {f: \gamma([a,b]) \rightarrow {\bf C}}. Also establish the identity

\displaystyle \int_\gamma\ |dz| = |\gamma|

and obtain an alternate proof of Exercise 16(v).

The change of variables formula (iv) lets one compute many contour integrals using the familiar Riemann integral. For instance, if {a < b} are real numbers and {f: [a,b] \rightarrow {\bf C}} is continuous, then the contour integral along {\gamma_{a \rightarrow b}} coincides with the Riemann integral,

\displaystyle \int_{\gamma_{a \rightarrow b}} f(z)\ dz = \int_a^b f(x)\ dx

and on reversal we also have

\displaystyle \int_{\gamma_{b \rightarrow a}} f(z)\ dz = \int_b^a f(x)\ dx = - \int_a^b f(x)\ dx.

Similarly, if {f} is continuous on the circle {\{ z \in {\bf C}: |z-z_0| = r \}}, we have

\displaystyle \int_{\gamma_{z_0,r,\circlearrowleft}} f(z)\ dz = i \int_0^{2\pi} f(z_0+re^{i\theta}) r e^{i\theta}\ d\theta \ \ \ \ \ (6)

\displaystyle = 2\pi i r \int_0^1 f(z_0 + re^{2\pi i \theta}) e^{2\pi i \theta}\ d\theta.

Remark 18 We caution that if {f(z)} is real-valued, we cannot conclude that {\int_\gamma f(z)\ dz} is also real valued, unless the contour {\gamma} lies in the real line. This is because the complex line element {dz} may introduce some non-trivial imaginary part, as is the case for instance in (6). For similar reasons, we have

\displaystyle \mathrm{Re} \int_\gamma f(z)\ dz \neq \int_\gamma \mathrm{Re} f(z)\ dz

\displaystyle \mathrm{Im} \int_\gamma f(z)\ dz \neq \int_\gamma \mathrm{Im} f(z)\ dz

and

\displaystyle \overline{\int_\gamma f(z)\ dz} \neq \int_\gamma \overline{f(z)}\ dz.

in general. If one wishes to mix line integrals with real and imaginary parts, it is recommended to replace the contour integrals above with the line integrals

\displaystyle \int_\gamma f(x+iy) dx + g(x+iy) dy,

which are defined as in Proposition 14 but where the expression

\displaystyle f(\gamma(t^*_j)) (\gamma(t_j) - \gamma(t_{j-1}))

appearing in (5) is replaced by

\displaystyle f(\gamma(t^*_j)) \mathrm{Re}(\gamma(t_j) - \gamma(t_{j-1})) + g(\gamma(t^*_j)) \mathrm{Im}(\gamma(t_j) - \gamma(t_{j-1})).

The contour integral corresponds to the special case {g=if} (or more informally, {dz = dx + idy}). Line integrals are in turn special cases of the more general concept of integration of differential forms, discussed for instance in this article of mine, and which are used extensively in differential geometry and geometric topology. However we will not use these more general line integrals or differential form integrals much in this course.

In later notes it will be convenient to restrict to a more regular class of curves than the rectifiable curves. We thus give the definitions here:

Definition 19 (Smooth curves and contours) A smooth curve is a curve {\gamma: [a,b] \rightarrow {\bf C}} that is continuously differentiable, and such that {\gamma'(t) \neq 0} for all {t \in [a,b]}. A contour is a curve that is equivalent to the concatenation of finitely many smooth curves (that is to say, a piecewise smooth curve).

Example 20 The line segments {\gamma_{z_1 \rightarrow z_2}} and circles {\gamma_{z_0, r, \circlearrowright}, \gamma_{z_0,r,\circlearrowleft}} are smooth curves and hence contours. Polygonal paths are usually not smooth, but they are contours. Any sum of finitely many contours is again a contour, and the reversal of a contour is also a contour.

Note here that the term “smooth” differs somewhat here from the real-variable notion of smoothness, which is defined to be “infinitely differentiable”. Smooth curves are still only assumed to just be continuously differentiable; we do not assume that the second derivative of {\gamma} exists. (In particular, smooth curves may have infinite curvature at some points.) In practice, this distinction tends to be minor, though, as the smooth curves that one actually uses in complex analysis do tend to be infinitely differentiable. On the other hand, for most applications one does not need to control any derivative of a contour beyond the first.

The following examples and exercises may help explain why the non-vanishing condition {\gamma'(t) \neq 0} imbues curves with a certain degree of “smoothness”.

Example 21 (Cuspidal curve) Consider the curve {\gamma: [-1,1] \rightarrow {\bf C}} defined by {\gamma(t) := t^2 + it^3}. Clearly {\gamma} is continously differentiable (and even infinitely differentiable), but we do not view this curve as smooth, because {\gamma'(t) = 2t + 3it^2} vanishes at the origin. Indeed, the image of the curve is {\{ x + i x^{3/2}: 0 \leq x \leq 1 \} \cup \{ x - i x^{3/2}: 0 \leq x \leq 1 \}}, which looks visibly non-smooth at the origin if one plots it, due to the presence of a cusp.

Example 22 (Absolute value function) Consider the curve {\gamma: [-1,1] \rightarrow {\bf C}} defined by {\gamma(t) := t^5 + i |t|^5}. This curve is certainly continuously differentiable, and in fact is four times continuously differentiable, but is not smooth because {\gamma'(t) = 5 t^4 + i 4 |t|^3 t} vanishes at the origin. The image of this curve is {\{ x + i |x|: -1 \leq x \leq 1\}}, which looks visibly non-smooth at the origin (in particular, there is no unique tangent line to this curve here).

Example 23 (Spiral) Consider the curve {\gamma: [-1,1] \rightarrow {\bf C}} defined by {\gamma(t) := t^4 e^{i/t}} for {t \neq 0}, and {\gamma(0) := 0}. One can check (exercise!) that {\gamma} is continuously differentiable, even at the origin {t=0}; but it is not a smooth curve because {\gamma'(0)} vanishes. The image of {\gamma} has some rather complicated behaviour at the origin, for instance it intersects itself multiple times (try to sketch it!).

Exercise 24 (Local behaviour of smooth curves) Let {\gamma: [a,b] \rightarrow {\bf C}} be a simple smooth curve, and let {t_0} be an interior point of {(a,b)}. Let {\theta \in {\bf R}} be a phase of {\gamma'(t_0)}, thus {\gamma'(t_0)= re^{i\theta}} for some {r>0}. Show that for all sufficiently small {\varepsilon>0}, the portion {\gamma([a,b]) \cap D(\gamma(t_0),\varepsilon)} of the image of {\gamma} near {\gamma(t_0)} looks like a rotated graph, in the sense that

\displaystyle \gamma([a,b]) \cap D(\gamma(t_0),\varepsilon) = \{ \gamma(t_0) + e^{i\theta} ( s + i f(s) ): s \in I_\varepsilon \}

for some interval {I_\varepsilon = (-c_-(\varepsilon), c_+(\varepsilon))} containing the origin, and some continuously differentiable function {f: I_\varepsilon \rightarrow {\bf R}} with {f(0) = f'(0) = 0}. Furthermore, show that {c_-(\varepsilon), c_+\varepsilon = \varepsilon + o(\varepsilon)} as {\varepsilon \rightarrow 0} (or equivalently, that {\frac{c_-(\varepsilon)}{\varepsilon}, \frac{c_+(\varepsilon)}{\varepsilon} \rightarrow 1} as {\varepsilon \rightarrow 0}). (Hint: you may find it easier to first work with the model case where {\gamma(t_0) = 0} and {\gamma'(t_0) = 1}. The real-variable inverse function theorem will also be helpful.)

Exercise 25 Show that a curve {\gamma} is a contour if and only if it is equivalent to the concatenation of finitely many simple smooth curves.

Exercise 26 Show that the cuspidal curve and absolute value curves in Examples 21, 22 are contours, but the curve in Exercise 23 is not.

— 2. The fundamental theorem of calculus —

Now we establish the complex analogues of the fundamental theorem of calculus. As in the real-variable case, there are two useful formulations of this theorem. Here is the first:

Theorem 27 (First fundamental theorem of calculus) Let {U} be an open subset of {{\bf C}}, let {f: U \rightarrow {\bf C}} be a continuous function, and suppose that {f} has an antiderivative {F: U \rightarrow {\bf C}}, that is to say a holomorphic function with {f(z) = F'(z)} for all {z \in U}. Let {\gamma: [a,b] \rightarrow {\bf C}} be a rectifiable curve in {U} with initial point {z_0} and terminal point {z_1}. Then

\displaystyle \int_\gamma f(z)\ dz = F(z_1) - F(z_0).

Proof: If {\gamma} were continuously differentiable, or at least piecewise continuously differentiable (the concatenation of finitely many continuously differentiable curves), we could establish this theorem by using Exercise 16 to rewrite everything in terms of real-variable Riemann integrals, at which point one can use the real-variable fundamental theorem of calculus (and the chain rule). But actually we can just give a direct proof that does not need any differentiability hypothesis whatsoever.

We again use the continuity method. Let {\varepsilon > 0}, and for each {a \leq T \leq b}, let {\gamma_{[a,T]}: [a,T] \rightarrow {\bf C}} be the restriction of {\gamma: [a,b] \rightarrow {\bf C}} to {[a,T]}. It will suffice to show that

\displaystyle |\int_{\gamma_{[a,T]}} f(z)\ dz - (F(\gamma(T)) - F(\gamma(a)))| \leq \varepsilon |\gamma_{[a,T]}| \ \ \ \ \ (7)

for all {a \leq T \leq b}, as the claim then follows by setting {T=b} and sending {\varepsilon} to zero.

Let {\Omega_\varepsilon} denote the set of all {T_* \in [a,b]} such that (7) holds for all {a \leq T \leq T_*}. As before, {\Omega_\varepsilon} is clearly closed and contains {a}; as {[a,b]} is connected, the only remaining task is to show that {\Omega_\varepsilon} is open in {[a,b]}. Let {T_* \in \Omega_\varepsilon} be such that {T_* < b}. As {F} is differentiable at {\gamma(T_*)} with derivative {f(\gamma(T_*))}, and {\gamma} is continuous, there exists {\delta>0} with {[T_*,T_*+\delta] \subset [a,b]} such that

\displaystyle |\frac{F(\gamma(T)) - F(\gamma(T_*))}{\gamma(T) - \gamma(T_*)} - f(\gamma(T_*))| \leq \varepsilon/2

for any {T_* < T \leq T_*+\delta}, and hence

\displaystyle |F(\gamma(T)) - F(\gamma(T_*)) - f(\gamma(T_*)) (\gamma(T)-\gamma(T_*))| \leq \varepsilon/2 |\gamma_{[T_*,T]}| \ \ \ \ \ (8)

for any {T \in [T_*, T_*+\delta]}. On the other hand, if {\delta} is small enough, we have that

\displaystyle |f(\gamma(t)) - f(\gamma(T_*))| \leq \varepsilon/2

for all {t \in [T_*, T_*+\delta]}, and hence by Exercise 16(v) we have

\displaystyle |\int_{\gamma_{[T_*,T]}} (f(z) - f(\gamma(T_*)))\ dz| \leq \frac{\varepsilon}{2} |\gamma_{[T_*, T]}|

where {\gamma_{[T_*,T]}: [T_*,T] \rightarrow {\bf C}} is the restriction of {\gamma: [a,b] \rightarrow {\bf C}} to {[T_*,T]}. Applying Exercise 16(vi), (vii) we thus have

\displaystyle |\int_{\gamma_{[T_*,T]}} f(z)\ dz - f(\gamma(T_*)) (\gamma(T)-\gamma(T_*))| \leq \frac{\varepsilon}{2} |\gamma_{[T_*, T]}|.

Combining this with (8) and the triangle inequality, we conclude that

\displaystyle |\int_{\gamma_{[T_*,T]}} f(z)\ dz - (F(\gamma(T)) - F(\gamma(T_*)))| \leq \varepsilon |\gamma_{[T_*,T]}|

and on adding this to the {T=T_*} case of (7) and again using the triangle inequality, we conclude that (7) holds for all {T \in [T_*,T_*+\delta]}. This ensures that {\Omega_\varepsilon} is open, as desired. \Box

One can use this theorem to quickly evaluate many integrals by using an antiderivative for the integrand as in the real-variable case. For instance, for any rectifiable curve {\gamma} with initial point {z_1} and terminal point {z_2}, we have

\displaystyle \int_\gamma z\ dz = \frac{1}{2} z_2^2 - \frac{1}{2} z_1^2,

\displaystyle \int_\gamma e^z\ dz = e^{z_2} - e^{z_1},

\displaystyle \int_\gamma \cos(z)\ dz = \sin(z_2) - \sin(z_1),

and so forth. If the curve {\gamma} avoids the origin, we also have

\displaystyle \int_\gamma \frac{1}{z^2}\ dz = -\frac{1}{z_2} + \frac{1}{z_1}

since {-\frac{1}{z}} is an antiderivative of {\frac{1}{z^2}} on {{\bf C} \backslash \{0\}}. If {\sum_{n=0}^\infty a_n (z-z_0)^n} is a power series with radius of convergence {R}, and {\gamma} is a curve in {D(z_0,R)} with initial point {z_1} and terminal point {z_2}, we similarly have

\displaystyle \int_\gamma \sum_{n=0}^\infty a_n (z-z_0)^n\ dz = \sum_{n=0}^\infty a_n ( \frac{(z_2-z_0)^{n+1}}{n+1} - \frac{(z_1-z_0)^{n+1}}{n+1} ).

For the second fundamental theorem of calculus, we need a topological preliminary result.

Exercise 28 Let {U} be a non-empty open subset of {{\bf C}}. Show that the following statements are equivalent:

  • (i) {U} is topologically connected (that is to say, the only two subsets of {U} that are both open and closed relative to {U} are the empty set and {U} itself).
  • (ii) {U} is path connected (that is to say, for any {z_1,z_2 \in U} there exists a curve {\gamma} with image in {U} whose initial point is {z_1} and terminal point is {z_2}).
  • (iii) {U} is polygonally path connected (the same as (ii), except that {\gamma} is now required to also be a polygonal path).

(Hint: to show that (i) implies (iii), pick a base point {z_0} in {U} and consider the set of all {z_1} in {U} that can be reached from {z_0} by a polygonal path.)

We remark that the relationship between path connectedness and connectedness is more delicate when one does not assume that the space {U} is open; every path-connected space is still connected, but the converse need not be true.

Remark 29 There is some debate as to whether to view the empty set {\emptyset} as connected, disconnected, or neither. I view this as analogous to the debate as to whether the natural number {1} should be viewed as prime, composite, or neither. In both cases I personally prefer the convention of “neither” (and like to use the term “unconnected” to describe the empty set, and “unit” to describe {1}), but to avoid any confusion I will restrict the discussion of connectedness to non-empty sets in this course (which is all we will need in applications).

In real analysis, the second fundamental theorem of calculus asserts that if a function {f: [a,b] \rightarrow {\bf R}} is continuous, then the function {F: x \mapsto \int_a^x f(y)\ dy} is an antiderivative of {f}. In the complex case, there is an analogous result, but one needs the additional requirement that the function is conservative:

Theorem 30 (Second fundamental theorem of calculus) Let {U \subset {\bf C}} be a non-empty open connected subset of the complex numbers. Let {f: U \rightarrow {\bf C}} be a continuous function which is conservative in the sense that

\displaystyle \int_\gamma f(z)\ dz = 0 \ \ \ \ \ (9)

whenever {\gamma} is a closed polygonal path in {U}. Fix a base point {z_0 \in U}, and define the function {F: U \rightarrow {\bf C}} by the formula

\displaystyle F(z_1) := \int_{\gamma_{z_0 \rightsquigarrow z_1}} f(z)\ dz

for all {z_1 \in U}, where {\gamma_{z_0 \rightsquigarrow z_1}} is any polygonal path from {z_0} to {z_1} in {U} (the existence of such a path follows from Exercise 28 and the hypothesis that {U} is connected, and the independence of the choice of path for the purposes of defining {F(z_1)} follows from Exercise 16 and the conservative hypothesis (9)). Then {F} is holomorphic on {U} and is an antiderivative of {f}, thus {f(z_1) = F'(z_1)} for all {z_1 \in U}.

Proof: We mimic the proof of the real-variable second fundamental theorem of calculus. Let {z_1} be any point in {U}. As {U} is open, it contains some disk {D(z_1,r)} centred at {z_1}. In particular, if {z_2} lies in this disk, then the line segment {\gamma_{z_1 \rightarrow z_2}} will have image in {U}. If {\gamma_{z_0 \rightsquigarrow z_1}} is any polygonal path from {z_0} to {z_1}, then we have

\displaystyle F(z_1) = \int_{\gamma_{z_0 \rightsquigarrow z_1}} f(z)\ dz

and

\displaystyle F(z_2) = \int_{\gamma_{z_0 \rightsquigarrow z_1} + \gamma_{z_1 \rightarrow z_2}} f(z)\ dz

and hence by Exercise 16

\displaystyle F(z_2) - F(z_1) - f(z_1) (z_2-z_1) = \int_{\gamma_{z_1 \rightarrow z_2}} (f(z) - f(z_1))\ dz.

(The reader is strongly advised to draw a picture depicting the situation here.) Let {\varepsilon>0}, then for {z_2} sufficiently close to {z_1}, we have {|f(z)-f(z_1)| \leq \varepsilon} for all {z} in the image of {\gamma_{z_1 \rightarrow z_2}}. Thus by Exercise 16(v) we have

\displaystyle |F(z_2) - F(z_1) - f(z_1) (z_2-z_1)| \leq \varepsilon |z_2-z_1|

for {z_2} sufficiently close to {z_1}, which implies that

\displaystyle \lim_{z_2 \rightarrow z_1; z_2 \in U \backslash \{z_1\}} \frac{F(z_2)-F(z_1)}{z_2-z_1} = f(z_1)

for any {z_1 \in U}, and thus {F} is an antiderivative of {f} as required. \Box

The notion of a non-empty open connected subset {U} of the complex plane comes up so frequently in complex analysis that many texts assign a special term to this notion; for instance, Stein-Shakarchi refers to such sets as regions, and in other texts they may be called domains. We will stick to just “non-empty open connected subset of {{\bf C}}” in this course.

The requirement that {f} be conservative is necessary, as the following exercise shows. Actually it is also necessary in the real-variable case, but it is redundant in that case due to the topological triviality of closed polygonal paths in one dimension: see Exercise 33 below.

Exercise 31 Let {U} be a non-empty connected subset of {{\bf C}}, and let {f: U \rightarrow {\bf C}} be continuous. Show that the following are equivalent:

  • (i) {f} possesses at least one antiderivative {F}.
  • (ii) {f} is conservative in the sense that (9) holds for all closed polygonal paths in {U}.
  • (iii) {f} is conservative in the sense that (9) holds for all simple closed polygonal paths in {U}.
  • (iv) {f} is conservative in the sense that (9) holds for all closed contours in {U}.
  • (v) {f} is conservative in the sense that (9) holds for all closed rectifiable curves in {U}.

(Hint: to show that (iii) implies (ii), induct on the number of edges in the closed polygonal path, and find a way to decompose non-simple closed polygonal paths into paths with fewer edges. One should avoid non-rigorous “hand-waving” arguments, and make sure that one actually has covered all possible cases, e.g. paths that include some backtracking.) Furthermore, show that if {f} has two antiderivatives {F_1, F_2}, then there exists a constant {C \in {\bf C}} such that {F_2 = F_1 + C}.

Exercise 32 Show that the function {\frac{1}{z}} does not have an antiderivative on {{\bf C} \backslash \{0\}}. (Hint: integrate {\frac{1}{z}} on {\gamma_{0,1,\circlearrowleft}}.) In later notes we will see that {\frac{1}{z}} nevertheless does have antiderivatives on many subsets of {{\bf C} \backslash \{0\}}, formed by various branches of the complex logarithm.

Exercise 33 If {f: [a,b] \rightarrow {\bf C}} is a continuous function on an interval, show that {f} is conservative in the sense that (9) holds for any closed polygonal path in {[a,b]}. What happens for closed rectifiable paths?

Exercise 34 Let {U} be an open subset of {{\bf C}} (not necessarily connected).

  • (i) Show that there is a unique collection {{\mathcal U}} of non-empty subsets of {U} that are open, connected, disjoint, and partition {U}: {U = \biguplus_{V \in {\mathcal U}} V}. (The elements of {{\mathcal U}} are known as the connected components of {U}.)
  • (ii) Show that the number of connected components of {U} is at most countable. (Hint: show that each connected component contains at least one complex number with rational real and imaginary parts.)
  • (iii) If {f: U \rightarrow {\bf C}} is a continuous conservative function on {U}, show that {f} has at least one antiderivative {F}.
  • (iv) If {U} has more than one connected component, show that it is possible for a function {f: U \rightarrow {\bf C}} to have two antiderivatives {F_1, F_2} which do not differ by a constant (i.e. there is no complex number {C} such that {F_1 = F_2 + C}).

Exercise 35 (Integration by parts) Let {f,g: U \rightarrow {\bf C}} be holomorphic functions on an open set {U}, and let {\gamma} be a rectifiable curve in {U} with initial point {z_1} and terminal point {z_2}. Prove that

\displaystyle \int_\gamma f(z) g'(z)\ dz = f(z_2) g(z_2) - f(z_1) g(z_1) - \int_\gamma f'(z) g(z)\ dz.

(The exercise below was added after the notes was originally written; it will be moved to a more appropriate location later, but is placed here so as not to disrupt the numbering of existing exercises.)

Exercise 36 (Arclength parameterisation) Let {\gamma: [a,b] \rightarrow {\bf C}} be a rectifiable curve. Show that there exists a unique reparameterisation {\tilde \gamma: [0,|\gamma|] \rightarrow {\bf C}} with the property that for each {0 \leq t \leq |\gamma|}, the restriction of {\tilde \gamma} to {[0,t]} has arclength exactly {t}. (Hint: one may wish to first establish the preliminary statement that the arclength of the restriction of {\gamma} to {[a,t]} for {a \leq t \leq b} varies continuously in {t}.)