4 Cauchy’s theorem and consequences

4.1 Convex, star-shaped, and simply connected domains

Definition 4.1 Line segment

Given two points \(z_1, z_2 \in \mathbb {C}\) in the complex plane, the line segment between \(z_1\) and \(z_2\) is the path

\begin{align*} \gamma \colon [0,1] & \; \to \mathbb {C}\\ \gamma (t) & \; = z_1 + t \, (z_2 - z_1) . \end{align*}

We also often view the line segment as a subset of \(\mathbb {C}\) rather than a parametrized path, and we then denote it by

\begin{align*} [z_1, z_2] \, = \, \Big\{ z_1 + t \, (z_2 - z_1) \; \Big| \; t \in [0,1] \Big\} \, \subset \, \mathbb {C}. \end{align*}

Definition 4.2 Convex set

A subset \(A \subset \mathbb {C}\) of the complex plane is called convex if for any two points \(z_1, z_2 \in A\), the line segment between them is contained in the subset,

\begin{align*} [z_1, z_2] \, \subset \, A . \end{align*}

Definition 4.3 Star-shaped set

A subset \(A \subset \mathbb {C}\) of the complex plane is called star-shaped if there exists a point \(z_* \in A\) such that for any \(z \in A\), the line segment between \(z_*\) and \(z\) is contained in the subset,

\begin{align*} [z_*, z] \, \subset \, A . \end{align*}

Lemma 4.4 Convex sets are star-shaped

Any nonempty convex set is star-shaped.

Proof ▶

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Lemma 4.5 Star-shaped sets are path connected and simply connected

Any star-shaped set \(U \subset \mathbb {C}\) is path connected and simply connected.

Proof ▶

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4.2 Cauchy’s integral theorem

Lemma 4.6 Goursat’s lemma [Palka1991, Lem V.1.1]

Suppose that a function \(f \colon U \to \mathbb {C}\) is analytic on an open set \(U \subset \mathbb {C}\). Then for any closed triangle \(\triangle \subset U\), we have

\begin{align*} \oint _{\partial \triangle } f(z) \, \mathrm{d}z = 0 . \end{align*}

Proof ▶

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Lemma 4.7 Existence of primitives in star-shaped domains

Every analytic function \(f \colon U \to \mathbb {C}\) on a star-shaped domain \(U \subset \mathbb {C}\) has a primitive in \(U\).

Proof ▶

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Theorem 4.8 Cauchy’s integral theorem for star-shaped domains [Palka1991, Thm V.1.5]

Suppose that a function \(f \colon U \to \mathbb {C}\) is analytic on a star-shaped open subset \(U \subset \mathbb {C}\). Then for any closed contour \(\gamma \) in \(U\) we have

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

Proof ▶

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Corollary 4.9 Local Cauchy’s integral theorem [Palka1991, Thm V.5.1]

Suppose that a function \(f \colon U \to \mathbb {C}\) is analytic on a open set \(U \subset \mathbb {C}\). Then for any disk \(B \subset U\) contained in the domain \(U\) and any closed contour \(\gamma \) in \(B\) we have

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

Proof ▶

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4.3 Cauchy’s integral formula

Theorem 4.10 Cauchy’s integral formula for star-shaped subdomains [Palka1991, Thm V.2.3]

Suppose that a function \(f \colon U \to \mathbb {C}\) is analytic on an open set \(U \subset \mathbb {C}\), and suppose that \(\gamma \) is a closed contour in \(U\) parametrizing the boundary of a star-shaped Jordan subdomain \(V \subset U\) in a counterclockwise orientation. Then for any point \(z \in V\) we have

\begin{align*} f(z) = \frac{1}{2\pi \mathfrak {i}} \oint _\gamma \frac{f(\zeta )}{\zeta - z} \, \mathrm{d}\zeta . \end{align*}

Proof ▶

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By far the most commonly used special case of Theorem 4.10 is when the contour \(\gamma \) is a circle, encircling a disk whose closure is contained in the domain of the analytic function (recall that disks are convex and therefore star-shaped).

Corollary 4.11 Cauchy’s integral formula for circles

Suppose that a function \(f \colon U \to \mathbb {C}\) is analytic on an open set \(U \subset \mathbb {C}\). Let \(\overline{\mathcal{B}}(z_0; r) \subset U\) be a closed disk contained in \(U\). Then for any point \(z \in \mathcal{B}(z_0; r)\) we have

\begin{align*} f(z) = \frac{1}{2\pi \mathfrak {i}} \oint _{\partial \mathcal{B}(z_0; r)} \frac{f(\zeta )}{\zeta - z} \, \mathrm{d}\zeta \end{align*}

where the circle \(\partial \mathcal{B}(z_0; r)\) is parametrized in the counterclockwise orientation.

Proof ▶

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4.4 Ideas underlying the generalizations

The generalizations of Cauchy’s integral theorem and Cauchy’s integral formula are based on the following homotopy invariance property of contour integrals (whose proof we do not do in detail in this course).

Lemma 4.12 Homotopy invariance of contour integrals

Let \(f \colon U \to \mathbb {C}\) be an analytic function on an open set \(U \subset \mathbb {C}\), and let \(\gamma _0\) and \(\gamma _1\) be two closed contours in \(U\) which are homotopic to each other in \(U\). Then we have

\begin{align*} \oint _{\gamma _0} f(z) \, \mathrm{d}z = \oint _{\gamma _1} f(z) \, \mathrm{d}z . \end{align*}

This readily implies the following generalization of Cauchy’s integral theorem.

Theorem 4.13 Cauchy’s integral theorem [Palka1991, Thm V.5.1]

Suppose that a function \(f \colon U \to \mathbb {C}\) is analytic on a open set \(U \subset \mathbb {C}\). Then for any contractible closed contour \(\gamma \) we have

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

In particular, if \(U\) is simply connected, then for any closed contour \(\gamma \) in \(U\) we have

\begin{align*} \oint _\gamma f(z) \, \mathrm{d}z = 0 , \end{align*}

and the analytic function \(f\) has a primitive in \(U\).

Proof ▶

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The other ingredient of generalization of Cauchy’s integral formula to arbitrary contours and points not lying on those contours is the winding number of a contour around a point.

Definition 4.14 Winding number

Let \(z \in \mathbb {C}\), and let \(\gamma \) be a closed contour in \(\mathbb {C}\setminus \left\{ z \right\} \). The winding number of \(\gamma \) around \(z\) is defined as

\begin{align*} \mathfrak {n}_{{\gamma }}(z) = \frac{1}{2\pi \mathfrak {i}} \oint _\gamma \frac{\mathrm{d}\zeta }{\zeta - z} . \end{align*}

Lemma 4.15 Winding number concatenation and reversal

Let \(\gamma \) and \(\eta \) be closed contours in \(\mathbb {C}\) both starting and ending at the same point \(z_0 \in \mathbb {C}\). Then for any point \(z \in \mathbb {C}\setminus (\gamma \cup \eta )\) we have

\begin{align*} \mathfrak {n}_{{\gamma \oplus \eta }}(z) \; = \; \mathfrak {n}_{{\gamma }}(z) + \mathfrak {n}_{{\eta }}(z) \end{align*}

and for any point \(z \in \mathbb {C}\setminus \gamma \) we have

\begin{align*} \mathfrak {n}_{{\overleftarrow {\gamma }}}(z) \; = \; -\mathfrak {n}_{{\gamma }}(z) . \end{align*}

Proof ▶

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Lemma 4.16 Winding number properties

Let \(\gamma \) be a closed contour in \(\mathbb {C}\). Then the winding numbers \(\mathfrak {n}_{{\gamma }}(z)\) of points \(z \in \mathbb {C}\setminus \gamma \) satisfy:

\(z \mapsto \mathfrak {n}_{{\gamma }}(z)\) is constant on each connected component of \(\mathbb {C}\setminus \gamma \);
\(\mathfrak {n}_{{\gamma }}(z) = 0\) for all \(z\) in the unbounded connected component of \(\mathbb {C}\setminus \gamma \);
If \(\gamma \) is a Jordan contour and \(V \subset \mathbb {C}\setminus \gamma \) is the bounded connected component of \(\mathbb {C}\setminus \gamma \), then either \(\mathfrak {n}_{{\gamma }}(z) = 1\) for all \(z \in V\) or \(\mathfrak {n}_{{\gamma }}(z) = -1\) for all \(z \in V\).

Proof ▶

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Lemma 4.17 Homotopy invariance of winding numbers

Let \(z \in \mathbb {C}\) be a point and let \(\gamma \) and \(\eta \) be two closed contours in \(\mathbb {C}\setminus \left\{ z \right\} \) which are homotopic to each other in \(\mathbb {C}\setminus \left\{ z \right\} \). Then we have

\begin{align*} \mathfrak {n}_{{\gamma }}(z) = \mathfrak {n}_{{\eta }}(z) . \end{align*}

Proof ▶

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The following is then a version of Cauchy’s integral formula which has no restrictions on the closed contour and no restrictions on the position of the point with respect to the contour, except that the point must not lie on the contour (for otherwise there is a singularity in the integrand).

Theorem 4.18 Cauchy’s integral formula [Palka1991, Thm V.2.3]

Suppose that a function \(f \colon U \to \mathbb {C}\) is analytic on an open simply connected subset \(U \subset \mathbb {C}\) of the complex plane. Then for any closed contour \(\gamma \) in \(U\) we have

\begin{align*} \oint _\gamma \frac{f(\zeta )}{\zeta - z} \, \mathrm{d}\zeta \; = \; 2\pi \mathfrak {i}\, \mathfrak {n}_{{\gamma }}(z) \, f(z) . \end{align*}

Proof ▶

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4.5 Analyticity of derivatives

Lemma 4.19 Analyticity of derivatives [Palka1991, Thm V.3.1]

If a function \(f \colon U \to \mathbb {C}\) is analytic on an open set \(U \subset \mathbb {C}\), then its derivative \(f'\) is also analytic on \(U\). In particular, then \(f\) is continuously differentiable, \(f \in \mathcal{C}^{{1}}(U)\).

Proof ▶

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Corollary 4.20 Analyticity of higher derivatives [Palka1991, Cor V.3.2]

If a function \(f \colon U \to \mathbb {C}\) is analytic on an open set \(U \subset \mathbb {C}\), then its derivatives \(f', f'', \ldots , f^{(k)}, \ldots \) of all orders are also analytic on \(U\). In particular, then \(f\) is infinitely differentiable, \(f \in \mathcal{C}^{{\infty }}(U)\).

Proof ▶

Straightforward induction using Lemma 4.19.

Theorem 4.21 Morera’s theorem [Palka1991, Thm V.3.3]

Let \(f \colon U \to \mathbb {C}\) be a continuous function on an open set \(U \subset \mathbb {C}\). If \(f\) has the property that

\begin{align*} \oint _{\partial \triangle } f(z) \, \mathrm{d}z = 0 \end{align*}

for any closed triangle \(\triangle \subset U\), then \(f\) is analytic on \(U\).

Proof ▶

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Theorem 4.22 Cauchy’s integral formula for derivatives

\begin{align*} \mathfrak {n}_{{\gamma }}(z) \, f^{(n)}(z) = \frac{n!}{2\pi \mathfrak {i}} \oint _\gamma \frac{f(\zeta )}{(\zeta - z)^{n+1}} \, \mathrm{d}\zeta . \end{align*}

Proof ▶

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Lemma 4.23 Cauchy’s estimate for derivatives [Palka1991, Thm V.3.6]

Suppose that a function \(f \colon U \to \mathbb {C}\) is analytic on an open set \(U \subset \mathbb {C}\) containing the disk \(\mathcal{B}(z_0; r) \subset U\), and suppose that there exists a constant \(M{\gt}0\) such that \(|f(z)| \le M\) for all \(z \in \mathcal{B}(z_0; r)\). Then for any \(n \in \mathbb {N}\) and any \(z \in \mathcal{B}(z_0; r)\) we have the following bound for the \(n\)th derivative \(f^{(n)}\) of \(f\):

\begin{align*} \big| f^{(n)}(z) \big| \le \frac{n! \, M \, r}{( r - |z-z_0| )^{n+1}} . \end{align*}

In particular, for the center point \(z_0\) of the disk, we have

\begin{align*} \big| f^{(n)}(z_0) \big| \le n! \, M \, r^{-n} . \end{align*}

Proof ▶

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4.6 Liouville’s theorem

Theorem 4.24 Liouville’s theorem [Palka1991, Thm V.3.7]

If a function \(f \colon \mathbb {C}\to \mathbb {C}\) on the entire complex plane is analytic and bounded, then \(f\) is a constant function.

Proof ▶

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4.7 The fundamental theorem of algebra

Theorem 4.25 Fundamental theorem of algebra [Palka1991, Thm V.3.8]

Every non-constant polynomial function \(p \colon \mathbb {C}\to \mathbb {C}\) has a root, i.e., there exists a \(z_0 \in \mathbb {C}\) such that \(p(z_0) = 0\).

Proof ▶

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Corollary 4.26 Polynomial factorization [Palka1991, Thm V.3.9]

A complex-coefficient polynomial \(p(z) = a_0 + a_1 z + a_2 z^2 + \cdots + a_n z^n\) of degree \(n \in \mathbb {N}\) can be factored as

\begin{align*} p(z) = c \, (z - z_1) \, (z - z_2) \, \cdots \, (z - z_n) \end{align*}

where \(c = a_n \ne 0\), and \(z_1 , \ldots , z_n \in \mathbb {C}\) are the roots of \(p\) (with repetition according to the multiplicities of the roots).

Proof ▶

This follows from Theorem ?? by induction on the degree of the polynomial, using the polynomial division (Euclidean algorithm in the ring of univariate polynomials, see MS-C1081 Abstract Algebra).

4.8 Maximum principle

Theorem 4.27 Maximum principle for analytic functions [Palka1991, Thm V.3.10]

Let \(f \colon \mathcal{D}\to \mathbb {C}\) be an analytic function on a connected open set \(\mathcal{D}\subset \mathbb {C}\). Suppose that there exists a point \(z_0 \in \mathcal{D}\) such that

\begin{align*} |f(z)| \le |f(z_0)| \qquad \text{ for all } z \in \mathcal{D}. \end{align*}

Then \(f\) is a constant function.

Proof ▶

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Corollary 4.28 Maximum principle for analytic functions continuous up to the boundary [Palka1991, Cor V.3.11]

Let \(\mathcal{D}\subset \mathbb {C}\) be a bounded connected open set. Let \(f \colon \overline{{\mathcal{D}}} \to \mathbb {C}\) be a continuous function on its closure which is analytic in \(\mathcal{D}\). Then \(z \mapsto |f(z)|\) attains its maximum in \(\overline{{\mathcal{D}}}\) at some point of the boundary \(\partial \mathcal{D}\).

Proof ▶

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Lemma 4.29 Schwarz’s lemma [Palka1991, Thm V.3.14]

Let \(f \colon \mathcal{B}(0; 1) \to \mathbb {C}\) be an analytic function on the open unit disk such that \(|f(z)| \le 1\) for all \(z \in \mathcal{B}(0; 1)\) and \(f(0) = 0\). Then we have

\begin{align*} |f’(0)| \le 1 \qquad \text{ and } \qquad |f(z)| \le |z| \quad \text{ for all } z \in \mathcal{B}(0; 1) . \end{align*}

Furthermore, unless \(f\) is of the form \(f(z) = \lambda z\) for some \(\lambda \in \mathbb {C}\) with \(|\lambda | = 1\), then we have

\begin{align*} |f’(0)| {\lt} 1 \qquad \text{ and } \qquad |f(z)| {\lt} |z| \quad \text{ for all } z \in \mathcal{B}(0; 1) \setminus \left\{ 0 \right\} . \end{align*}

Proof ▶

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4.9 The mean value property

Theorem 4.30 Mean value property for analytic functions

Suppose that a function \(f \colon U \to \mathbb {C}\) is analytic on an open set \(U \subset \mathbb {C}\) containing the closed disk \(\overline{\mathcal{B}}(z; r) \subset U\). Then we have

\begin{align*} f(z) = \frac{1}{2 \pi r} \oint _{\partial \mathcal{B}(z; r)} \frac{f(\zeta )}{\zeta - z} \, |\mathrm{d}\zeta | . \end{align*}

Proof ▶

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