3 Contour integration

3.1 Complex-valued integrals

Definition 3.1 Integral of a complex-valued function

Let \(f \colon [a,b] \to \mathbb {C}\) be a complex-valued continuous function defined on a closed interval \([a,b] \subset \mathbb {R}\). We define the integral of \(f\) as

\begin{align*} \int _a^b f(t) \, \mathrm{d}t \; = \; \int _a^b \Re \mathfrak {e}\big( f(t) \big) \, \mathrm{d}t \; + \; \mathfrak {i}\int _a^b \Im \mathfrak {m}\big( f(t) \big) \, \mathrm{d}t . \end{align*}

(Note that on the right hand side we have just Riemann integrals of the continuous real-valued functions \(t \mapsto \Re \mathfrak {e}\big( f(t) \big)\) and \(t \mapsto \Im \mathfrak {m}\big( f(t) \big)\).)

Lemma 3.2 Complex linearity of complex-valued integrals

If \(f, g \colon [a,b] \to \mathbb {C}\) are complex-valued continuous functions defined on a closed interval \([a,b] \subset \mathbb {R}\), then the integral of their sum is

\begin{align*} \int _a^b \big( f(t) + g(t) \big) \, \mathrm{d}t \; = \; \int _a^b f(t) \, \mathrm{d}t + \int _a^b g(t) \, \mathrm{d}t . \end{align*}

If \(f \colon [a,b] \to \mathbb {C}\) is a complex-valued continuous function defined on a closed interval \([a,b] \subset \mathbb {R}\), and \(\lambda \in \mathbb {C}\) is a complex number, then the integral of the scalar multiple of \(f\) is

\begin{align*} \int _a^b \lambda \, f(t) \, \mathrm{d}t \; = \; \lambda \int _a^b f(t) \, \mathrm{d}t . \end{align*}

Proof ▶

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Lemma 3.3 Fundamental theorem of calculus for complex-valued integrals

Suppose that \(f \colon [a,b] \to \mathbb {C}\) is a continuously differentiable complex-valued function on a closed interval \([a,b] \subset \mathbb {R}\). Denote its derivative by \(\dot{f}(t) = \frac{\mathrm{d}}{\mathrm{d}t} f(t)\). Then for the integral of the derivative of \(f\) we have

\begin{align*} \int _a^b \dot{f}(t) \, \mathrm{d}t \; = \; f(b) - f(a) . \end{align*}

Proof ▶

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3.2 Paths in the complex plane

Definition 3.4 Path [Palka1991, Sec. IV.1.1]

A path in the complex plane is a continuous function \(\gamma \colon [a,b] \to \mathbb {C}\) from a closed interval \([a,b] \subset \mathbb {R}\) to \(\mathbb {C}\).

When \(A \subset \mathbb {C}\) is a subset of the complex plane, we say that \(\gamma \) is a path in \(A\) if \(\gamma (t) \in A\) for all \(t \in [a,b]\).

If the starting point and the end point of the path \(\gamma \) are the same, \(\gamma (a) = \gamma (b)\), then we say that \(\gamma \) is a closed path.

We sometimes want to disregard the parametrization, and view a path \(\gamma \colon [a,b] \to \mathbb {C}\) as a subset of the complex plane. This subset is the range \(\big\{ \gamma (t) \, \big| \, t \in [a,b] \big\} \subset \mathbb {C}\) of the parametrizing function, but with a slight abuse of notation we often just write \(\gamma \subset \mathbb {C}\) also for this subset.

(Note that the subset \(\gamma \subset \mathbb {C}\) is compact, by continuity of \(\gamma \colon [a,b] \to \mathbb {C}\) and compactness of \([a,b]\).)

Definition 3.5 Smooth path [Palka1991, Sec. IV.1.2]

A path \(\gamma \colon [a,b] \to \mathbb {C}\) is smooth if it is continuously differentiable, i.e., the derivative

\begin{align*} \dot{\gamma }(t) = \frac{\mathrm{d}}{\mathrm{d}t} \gamma (t) \end{align*}

with respect to the parameter \(t\) exists for all \(t \in [a,b]\) (one-sided derivatives at the interval end points \(a\) and \(b\)), and defines a continuous complex-valued function \(t \mapsto \dot{\gamma }(t)\) on \([a,b]\).

Definition 3.6 Contour / piecewise smooth path [Palka1991, Sec. IV.1.2]

A contour (also called a piecewise smooth path) is a continuous function \(\gamma \colon [a,b] \to \mathbb {C}\) such that for some finite subdivision \(a = t_0 {\lt} t_1 {\lt} \ldots {\lt} t_n = b\), the restrictions \(\gamma |_{[t_{j-1},t_j]}\) to the subintervals \([t_{j-1},t_j] \subset [a,b]\) are smooth paths for each \(j = 1, \ldots , n\).

If the starting point and the end point of the contour \(\gamma \) are the same, \(\gamma (a) = \gamma (b)\), then we say that \(\gamma \) is a closed contour.

Definition 3.7 Reverse path [Palka1991, Sec. IV.1.4]

Given a path \(\gamma \colon [a,b] \to \mathbb {C}\), the reverse path \(\overleftarrow {\gamma } \colon [a,b] \to \mathbb {C}\) is the path defined by

\begin{align*} \overleftarrow {\gamma }(t) = \gamma (a+b-t) \qquad \text{ for } t \in [a,b] . \end{align*}

Definition 3.8 Concatenation of paths [Palka1991, Sec. IV.1.4]

Given path \(\gamma \colon [a,b] \to \mathbb {C}\) and \(\eta \colon [c,d] \to \mathbb {C}\) with \(\gamma (b) = \eta (c)\) (the starting point of \(\eta \) coincides with the end point of \(\gamma \)), the concatenation of \(\gamma \) and \(\eta \) is the path \(\gamma \oplus \eta \colon [a,b+d-c] \to \mathbb {C}\) defined by

\begin{align*} (\gamma \oplus \eta )(t) = \begin{cases} \gamma (t) & \text{ for } t \in [a,b] , \\ \eta (c+t-b)) & \text{ for } t \in [b,b+d-c] . \end{cases}\end{align*}

(The slightly cumbersome formula in the second case is due to the fact that we need to attach the two parameter intervals of lengths \(b-a\) and \(d-c\) to each other, and we have, somewhat arbitrarily, chosen to glue them to form the interval \([a,b+d-c]\).)

Definition 3.9 Reparametrization of paths [Palka1991, Sec. IV.1.5]

Given a path \(\gamma \colon [a,b] \to \mathbb {C}\) and a continuous increasing bijection \(\phi \colon [c,d] \to [a,b]\), we define the reparametrization of \(\gamma \) by \(\phi \) as the path

\begin{align*} \gamma \circ \phi \, \colon \, [c,d] \, \to \; & \mathbb {C}\\ t \, \mapsto \; & \gamma (\phi (t)) . \end{align*}

Note that

\(\phi ^{-1} \colon [a,b] \to [c,d]\) is also a continuous increasing bijection (a continuous bijection from the compact \([a,b]\) is automatically a homeomorphism; see Lemma A.33) and reparametrization can be undone by rereparametrizing by \(\phi ^{-1}\);
If both \(\gamma \) and the reparametrization function \(\phi \) are smooth (continuously differentiable), then the reparametrized path \(\gamma \circ \phi \) is also smooth;
If both \(\gamma \) and the reparametrization function \(\phi \) are piecewise smooth, then the reparametrized path \(\gamma \circ \phi \) is also piecewise smooth, i.e., a contour.

3.3 Integrals along paths

Definition 3.10 Contour integral along a smooth path [Palka1991, Sec. IV.2.1]

Let \(f : A \to \mathbb {C}\) be a continuous function defined on a subset \(A \subset \mathbb {C}\). Let \(\gamma \colon [a,b] \to A\) be a smooth path in \(A\). We define the integral of \(f\) along \(\gamma \) as

\begin{align*} \int _\gamma f(z) \, dz \; = \; \int _a^b f \big( \gamma (t) \big) \, \dot{\gamma }(t) \; \mathrm{d}t . \end{align*}

(Here \(\dot{\gamma }(t) = \frac{\mathrm{d}}{\mathrm{d}t} \gamma (t)\) denotes the derivative of the smooth path \(\gamma \) with respect to its parameter \(t\).)

Sometimes it is appropriate to integrate functions with respect to the arc length in the following sense.

Definition 3.11 Arc length integral along a smooth path [Palka1991, Sec. IV.2.1]

\begin{align*} \int _\gamma f(z) \, |dz| \; = \; \int _a^b f \big( \gamma (t) \big) \, |\dot{\gamma }(t)| \; \mathrm{d}t . \end{align*}

(Here \(\dot{\gamma }(t) = \frac{\mathrm{d}}{\mathrm{d}t} \gamma (t) \in \mathbb {C}\) denotes the derivative of the smooth path \(\gamma \) with respect to its parameter \(t\), and \(|\dot{\gamma }(t)| \ge 0\) denotes the absolute value of this derivative.)

In order to extend the definition of contour integrals to piecewise smooth paths, we note that the definition behaves additively under path concatenation.

Lemma 3.12 Contour integrals and smooth path concatenation [Palka1991, Lem IV.2.1(iv)]

If a smooth path \(\gamma \) in \(A\) is a concatenation of smooth paths \(\eta _1, \ldots , \eta _n\), and \(f : A \to \mathbb {C}\) is a continuous function defined on \(A \subset \mathbb {C}\), then we have

\begin{align*} \int _\gamma f(z) \, \mathrm{d}z \; = \; \sum _{j=1}^n \int _{\eta _j} f(z) \, \mathrm{d}z \end{align*}

and

\begin{align*} \int _\gamma f(z) \, |\mathrm{d}z| \; = \; \sum _{j=1}^n \int _{\eta _j} f(z) \, |\mathrm{d}z| . \end{align*}

Proof ▶

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By virtue of the above, the following gives a well-defined meaning to integrals along piecewise smooth paths.

Definition 3.13 Contour integral [Palka1991, Sec. IV.2.1]

Let \(f : A \to \mathbb {C}\) be a continuous function defined on a subset \(A \subset \mathbb {C}\). Let \(\gamma \colon [a,b] \to A\) be a piecewise smooth path in \(A\), which is a concatenation of smooth paths \(\eta _1, \ldots , \eta _n\). We define the integral of \(f\) along \(\gamma \) as

\begin{align*} \int _\gamma f(z) \, dz \; = \; \sum _{j=1}^n \int _{\eta _j} f(z) \, dz . \end{align*}

Definition 3.14 Arc-length integral

Let \(f : A \to \mathbb {C}\) be a continuous function defined on a subset \(A \subset \mathbb {C}\). Let \(\gamma \colon [a,b] \to A\) be a piecewise smooth path (i.e., a contour) in \(A\), which is a concatenation of smooth paths \(\eta _1, \ldots , \eta _n\). We define the integral of \(f\) along \(\gamma \) as

\begin{align*} \int _\gamma f(z) \, dz \; = \; \sum _{j=1}^n \int _{\eta _j} f(z) \, dz . \end{align*}

Definition 3.15 Length of a path or a contour

Let \(\gamma \colon [a,b] \to \mathbb {C}\) be a piecewise smooth path (i.e., a contour) in \(\mathbb {C}\). The length \(\ell (\gamma )\) of \(\gamma \) is defined as

\begin{align*} \ell (\gamma ) = \int _\gamma |\mathrm{d}z| . \end{align*}

Lemma 3.16 Reparametrization invariance of integrals [Palka1991, Lem IV.2.1(v)]

Let \(\gamma \) be a piecewise smooth path in \(A\), and let \(\widetilde{\gamma }\) be obtained from \(\gamma \) by an orientation-preserving reparametrization. Then for any continuous function \(f : A \to \mathbb {C}\) we have

\begin{align*} \int _{\widetilde{\gamma }} f(z) \, dz \; = \; \int _{\gamma } f(z) \, dz \end{align*}

and

\begin{align*} \int _{\widetilde{\gamma }} f(z) \, |dz| \; = \; \int _{\gamma } f(z) \, |dz| . \end{align*}

Lemma 3.17 Contour integrals and path reversal [Palka1991, Lem IV.2.1(iii)]

If \(f : A \to \mathbb {C}\) is a continuous function defined on \(A \subset \mathbb {C}\), and \(\gamma \) is a piecewise path in \(A\), then for the contour integral and the arc length integral behave as follows under path reversal: then we have

\begin{align*} \int _{\overleftarrow {\gamma }} f(z) \, \mathrm{d}z \; = \; - \int _{\gamma } f(z) \, \mathrm{d}z \end{align*}

and

\begin{align*} \int _{\overleftarrow {\gamma }} f(z) \, |\mathrm{d}z| \; = \; \int _{\gamma } f(z) \, |\mathrm{d}z| \end{align*}

Proof ▶

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Lemma 3.18 Linearity of integrals [Palka1991, Lem IV.2.1(i-ii)]

Let \(A \subset \mathbb {C}\) be a subset of the complex plane let and \(\gamma \colon [a,b] \to A\) be a contour in \(A\).

If \(f, g \colon A \to \mathbb {C}\) are continuous functions defined on \(A\), then the contour integral and the arc length integral of their sum are

\begin{align*} \int _\gamma \big( f(z) + g(z) \big) \, \mathrm{d}z \; = \; & \int _\gamma f(z) \, \mathrm{d}z + \int _\gamma g(z) \, \mathrm{d}z \\ \int _\gamma \big( f(z) + g(z) \big) \, |\mathrm{d}z| \; = \; & \int _\gamma f(z) \, |\mathrm{d}z| + \int _\gamma g(z) \, |\mathrm{d}z| . \end{align*}

If \(f \colon A \to \mathbb {C}\) is a complex-valued continuous function defined on \(A\), and \(\lambda \in \mathbb {C}\) is a complex number, then the contour integral and the arc length integral of the scalar multiple of \(f\) are

\begin{align*} \int _\gamma \lambda \, f(z) \, \mathrm{d}z \; = \; & \lambda \int _\gamma f(z) \, \mathrm{d}z \\ \int _\gamma \lambda \, f(z) \, |\mathrm{d}z| \; = \; & \lambda \int _\gamma f(z) \, |\mathrm{d}z| . \end{align*}

Proof ▶

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Lemma 3.19 Triangle inequality for contour integrals [Palka1991, Lem IV.2.1(vi)]

Let \(f : A \to \mathbb {C}\) be a continuous function defined on \(A \subset \mathbb {C}\), and let \(\gamma \) be a contour in \(A\). Then we have

\begin{align*} \left| \int _\gamma f(z) \, \mathrm{d}z \right| \; \le \; \int _\gamma |f(z)| \, |dz| . \end{align*}

Proof ▶

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Corollary 3.20 An a priori bound for contour integrals

Let \(f : A \to \mathbb {C}\) be a continuous function defined on \(A \subset \mathbb {C}\), and let \(\gamma \) be a contour in \(A\). Assume that \(|f(z)| \le M\) for all points \(z\) on the contour \(\gamma \). Then we have

\begin{align*} \left| \int _\gamma f(z) \, \mathrm{d}z \right| \; \le \; M \, \ell (\gamma ) , \end{align*}

where \(\ell (\gamma ) = \int _\gamma |\mathrm{d}z|\) denotes the length of the contour \(\gamma \).

Proof ▶

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The following slightly technical auxiliary result will only be used later (for winding number properties and for Cauchy’s formula for the derivative). But since the result only requires contour integration, the natural logical place for it is here. Also, strictly speaking, we only need the cases \(k = 1\) and \(k = 2\) in this lemma; but including general \(k \in \mathbb {N}\) gives the quickest route to Cauchy’s formula for higher order derivatives.

Lemma 3.21 [Palka1991, Lemma V.1.6]

Let \(\gamma \) be a contour in \(\mathbb {C}\), and let \(h \colon \gamma \to \mathbb {C}\) be a continuous function on the contour (we slightly abuse the notation here to identify the contour as a subset \(\gamma \subset \mathbb {C}\)). Let \(k \in \mathbb {N}\) be a positive integer. Define \(H \colon \mathbb {C}\setminus \gamma \to \mathbb {C}\) by

\begin{align*} H(z) = \int _\gamma \frac{h(\zeta )}{(\zeta - z)^{k}} \, \mathrm{d}\zeta . \end{align*}

Then \(H\) is analytic on \(\mathbb {C}\setminus \gamma \), and its derivative at \(z \in \mathbb {C}\setminus \gamma \) is given by

\begin{align*} H’(z) = k \, \int _\gamma \frac{h(\zeta )}{(\zeta - z)^{k+1}} \, \mathrm{d}\zeta . \end{align*}

Proof ▶

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3.4 Primitives

Definition 3.22 Primitive of a function [Palka1991, Sec. IV.2.3]

Let \(f : U \to \mathbb {C}\) be a function defined on an open subset \(U \subset \mathbb {C}\). A primitive of \(f\) is a function \(F : U \to \mathbb {C}\) such that \(F\) is analytic (i.e., complex differentiable) on \(U\),

\begin{align*} F’(z) = f(z) \qquad \text{ for all } z \in U . \end{align*}

Theorem 3.23 Fundamental theorem of calculus for contour integrals [Palka1991, Thm IV.2.2]

Suppose that \(f \colon U \to \mathbb {C}\) is a continuous function on an open set \(U \subset \mathbb {C}\), and that \(f\) has a primitive \(F \colon U \to \mathbb {C}\). Then for any contour \(\gamma \colon [a,b] \to U\) we have

\begin{align*} \int _\gamma f(z) \, dz \; = \; F \big( \gamma (b) \big) - F \big( \gamma (a) \big) . \end{align*}

In particular for any closed contour \(\gamma \) in \(U\), we have

\begin{align*} \oint _\gamma f(z) \, dz \; = \; 0 . \end{align*}

Proof ▶

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Lemma 3.24 Existence of primitives for monomials

For \(n \in \left\{ 0,1,2,\ldots \right\} \), the monomial function \(f(z) = z^n\) has a primitive \(F(z) = \frac{1}{n+1} z^{n+1} + c\) (with \(c \in \mathbb {C}\) arbitrary) in the whole complex plane \(\mathbb {C}\).

For \(n \in \left\{ -2,-3,-4,\ldots \right\} \), the monomial function \(f(z) = z^n\) has a primitive \(F(z) = \frac{1}{n+1} z^{n+1} + c\) (with \(c \in \mathbb {C}\) arbitrary) in the punctured complex plane \(\mathbb {C}\setminus \left\{ 0 \right\} \).

The monomial function \(f(z) = z^{-1} = \frac{1}{z}\) does not have a primitive in the punctured complex plane \(\mathbb {C}\setminus \left\{ 0 \right\} \).

Proof ▶

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Theorem 3.25 Characterization of the existence of primitives

Let \(f \colon U \to \mathbb {C}\) be a continuous function on an open set \(U \subset \mathbb {C}\). Then the following conditions are equivalent:

\(f\) has a primitive on \(U\);
the contour integrals \(\int _\gamma f(z) \, \mathrm{d}z\) of \(f\) along contours \(\gamma \) in \(U\) only depend on the starting point and the end point of \(\gamma \);
for all closed contours \(\gamma \) in \(U\) we have \(\oint _\gamma f(z) \, \mathrm{d}z = 0\).

Proof ▶

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