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Twenty-One Lectures on Complex Analysis

A First Course

Author: Alexander Isaev

Publisher: Springer International Publishing

Book Series : Springer Undergraduate Mathematics Series

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

About this book

At its core, this concise textbook presents standard material for a first course in complex analysis at the advanced undergraduate level. This distinctive text will prove most rewarding for students who have a genuine passion for mathematics as well as certain mathematical maturity. Primarily aimed at undergraduates with working knowledge of real analysis and metric spaces, this book can also be used to instruct a graduate course. The text uses a conversational style with topics purposefully apportioned into 21 lectures, providing a suitable format for either independent study or lecture-based teaching. Instructors are invited to rearrange the order of topics according to their own vision. A clear and rigorous exposition is supported by engaging examples and exercises unique to each lecture; a large number of exercises contain useful calculation problems. Hints are given for a selection of the more difficult exercises. This text furnishes the reader with a means of learning complex analysis as well as a subtle introduction to careful mathematical reasoning. To guarantee a student’s progression, more advanced topics are spread out over several lectures.

This text is based on a one-semester (12 week) undergraduate course in complex analysis that the author has taught at the Australian National University for over twenty years. Most of the principal facts are deduced from Cauchy’s Independence of Homotopy Theorem allowing us to obtain a clean derivation of Cauchy’s Integral Theorem and Cauchy’s Integral Formula. Setting the tone for the entire book, the material begins with a proof of the Fundamental Theorem of Algebra to demonstrate the power of complex numbers and concludes with a proof of another major milestone, the Riemann Mapping Theorem, which is rarely part of a one-semester undergraduate course.

Frontmatter

Lecture 1. Complex Numbers. The Fundamental Theorem of Algebra

Abstract

The field of complex numbers, or the complex plane, denoted by $ \mathbb{C} $, is just the usual Euclidean plane $ \mathbb{R}^{2} $ endowed with the additional operation of multiplication of vectors defined as follows: for (x₁,y₁) and (x₂,y₂) in $ \mathbb{R}^{2} $ let

$$ (x_{1},y_{1})\cdot (x_{2},y_{2}):=(x_{1}x_{2}-y_{1}y_{2},x_{1}y_{2}+x_{2}y_{1}). $$

Notice that if y₁ = 0, the above operation is simply the scaling of the vector (x₂,y₂) by x₁.

Alexander Isaev

Lecture 2. ℝ- and ℂ-Differentiability

Abstract

Let z₀ = x₀+iy₀ = (x₀,y₀) be a point in $ \mathbb{C} $ and f a function defined on a neighbourhood of z₀ (e.g., on an open disk $ \varDelta(z_{0},\ r) $ for some r > 0) with values in $ \mathbb{C} $. Write $ f(z) = \mathrm{Re}\ f(z)+i\mathrm{Im}\ f(z) = u(z)+iv(z) = u(x,y)+iv(x,y) $.

Alexander Isaev

Lecture 3. The Stereographic Projection. Conformal Maps. The Open Mapping Theorem

Abstract

In the previous lecture we introduced functions holomorphic on domains in $ \mathbb{C} $. Here we will allow domains to include “the infinity” and look at the so-called conformal maps on such extended domains. As we shall see, maps of this kind are analogous to one-to-one holomorphic functions.

Alexander Isaev

Lecture 4. Conformal Maps (Continued). Möbius Transformations

Abstract

We will now accumulate some examples of conformal maps between domains in $ \mathbb{C} $ and, more generally, in $ \mathbb{\overline{C}} $. Let us start with domains in $ \mathbb{C} $ and first explore the exponential function e ^z = e ^x(cos y+i sin y). As this function is entire, in order to see where it is conformal, by Corollary 3.1 we only need to understand on what domains in $ \mathbb{C} $ it is 1-to-1.

Alexander Isaev

Lecture 5. Möbius Transformations (Continued). Generalised Circles. Symmetry

Abstract

Notice that every Möbius transformation

$$ \lambda(z)=\frac{az+b}{cz+d} $$

is given by the non-degenerate matrix $ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) $, which is defined uniquely up to a complex non-zero multiple.

Alexander Isaev

Lecture 6. Domains Bounded by Pairs of Generalised Circles. Integration

Abstract

So far, we have accumulated the following examples of conformal maps:

(1)

$ z^{n},\ n= 2,3,\ldots\!, $ conformal on any angle $ A_{\alpha,\beta} $, with $ \beta - \alpha \leq 2\pi/n; $

Alexander Isaev

Lecture 7. Primitives Along Paths. Holomorphic Primitives. The Existence of a Holomorphic Primitive of a Function Holomorphic on a Disk. Goursat’s Lemma

Abstract

We will now look at methods for computing integrals.

Definition 7.1. Let $ D \subset \mathbb{C} $ be a domain, $ f \in C(D) $ and $ \gamma $ a C ¹-path in D.

Alexander Isaev

Lecture 8. Proof of Lemma 7.2. Constructible Primitives of Holomorphic Functions along Paths. Integration of Holomorphic Functions over Arbitrary Paths. Homotopy. Simply-Connected Domains. The Riemann Mapping Theorem

Abstract

We will now prove Goursat’s Lemma stated at the end of Lecture 7. Proof (Lemma 7.2). Fix a triangle $ T \subset D $. Using the mid-points of the sides, we split T into four subtriangles $ T^{\prime},\ T^{\prime\prime},\ T^{\prime\prime\prime},\ T^{\prime\prime\prime\prime} $ (see Fig. 8.1).

Alexander Isaev

Lecture 9. Cauchy’s Independence of Homotopy Theorem. Integration over Piecewise C 1-paths. Jordan Domains and Integration over their Boundaries

Abstract

We will now state and prove one of the central theorems of the course.

Theorem 9.1. (Cauchy’s Independence of Homotopy Theorem) Let $ D \subset \mathbb{C} $ be a domain and $ f \in H(D) $.

Alexander Isaev

Lecture 10. Cauchy’s Integral Theorem. Proof of Theorem 3.1. Cauchy’s Integral Formula

Abstract

We will now give further applications of Theorem 9.1. In what follows we often consider functions holomorphic on domains containing the closure $ \overline{D} $ of a domain D. For such a function f we write $ f \in H(\overline{D})$.

Alexander Isaev

Lecture 11. Morera’s Theorem. Sequences and Series of Functions. Uniform Convergence Inside a Domain. Power Series. Abel’s Theorem. Disk of Convergence. Radius of Convergence

Abstract

Now that we have proved Theorem 3.1, we can establish the converse to Lemma 7.2, which will be useful for us in what follows.

Alexander Isaev

Lecture 12. Proof of Theorem 11.9. Power Series (Continued). Taylor Series. Local Power Series Expansion of a Holomorphic Function. Cauchy’s Inequalities. The Uniqueness Theorem

Abstract

Clearly, a power series centred at a converges on its disk of convergence $ \varDelta(a,R) $ and diverges at every point z satisfying $ |z-a|>R.\ \mathrm{For}\neq0, \infty $ it is therefore natural to investigate convergence properties of the series at the points of the circle $ \{z \in \mathbb{C} : |z-a| =R \} $, which is the boundary $ \partial \varDelta (a,R) $ of the disk of convergence.

Alexander Isaev

Lecture 13. Liouville’s Theorem. Laurent Series. Annulus of Convergence. Laurent Series Expansion of a Function Holomorphic on an Annulus. Cauchy’s Inequalities. Isolated Singularities of Holomorphic Functions

Abstract

Another application of Theorem 12.1 is the following important fact:

Theorem 13.1. (Liouville’s Theorem)Let f be an entire function (i.e., $ f \in H (\mathbb{C}) $).

Alexander Isaev

Lecture 14. Isolated Singularities of Holomorphic Functions (Continued). Characterisation of an Isolated Singularity via the Laurent Series Expansion. Orders of Poles and Zeroes. Casorati-Weierstrass’ Theorem. Isolated Singularities of Holomorphic Functions at ∞ and their Characterisation via Laurent Series Expansions

Abstract

Definition 14.1. Let $ a \in \mathbb{C} $ be an isolated singularity of a function f, so we have $ f \in H(\varDelta(a,0,r)) $ for some $ 0 < r \leq \infty $. By Theorem 13.2, the function f expands into a (uniquely determined) Laurent series centred at a:

Alexander Isaev

Lecture 15. Isolated Singularities of Holomorphic Functions at ∞ (Continued). Orders of Poles at ∞. Casorati-Weierstrass’ Theorem for an Isolated Singularity at ∞. Residues. Cauchy’s Residue Theorem. Computing Residues

Abstract

We continue studying isolated singularities at ∞. Analogously to Proposition 14.1 we have:

Proposition 15.1. Let ∞ be an isolated singularity of a function $ f\ \in\ H(\varDelta(0,r,\infty)) $, with $ 0 \leq r < \infty $.

Alexander Isaev

Lecture 16. Computing Residues (Continued). Computing Integrals over the Real Line Using Contour Integration. The Argument Principle

Abstract

The field of complex numbers, or the complex plane, denoted by $$ \mathbb{C} $$, is just the usual Euclidean plane $$ \mathbb{R}^{2} $$ endowed with the additional operation of multiplication of vectors defined as follows: for (x₁;y₁) and (x₂,y₂) in $$ \mathbb{R}^{2} $$ let $$ $$ Notice that if y₁ = 0, the above operation is simply the scaling of the vector (x₂,y₂) by x1.

Alexander Isaev

Lecture 17. Index of a Path. The Argument Principle (Continued). Rouché’s Theorem. Theorem 1.1 Revisited. Proof of Theorem 3.2. The Maximum Modulus Principle. Proof of Theorem 3.3

Abstract

We continue to progress towards the standard version of the Argument Principle.

By Example 6.1, Theorem 9.1, and Proposition 16.2, for a closed path $ \gamma\ \mathrm{in}\ \mathbb{C}\backslash\{0\} $ the expression

$$ \frac{1}{2\pi i} \int\nolimits_{\gamma} \frac{1}{z} dz $$

(17.1)

is an integer, namely, the integer m supplied by Proposition 16.2.

Alexander Isaev

Lecture 18. Schwarz’s Lemma. Conformal Maps of the Unit Disk and the Upper Half-Plane. (Pre)-Compact Subsets of a Metric Space. Continuous Linear Functionals on H(D). Arzelà-Ascoli’s Theorem. Montel’s Theorem. Hurwitz’s Theorem

Abstract

We will now derive another very useful corollary from Theorem 17.3.

Lemma 18.1. (Schwarz’s Lemma) Let $ f \in\ H(\varDelta)\ and\ |f(z)| \leq 1\ for\ all\ z \in \varDelta $

Alexander Isaev

Lecture 19. Analytic Continuation

Abstract

The second group of results that we need to obtain Theorem 8.3 concerns analytic continuation. In fact, as explained in Remark 20.1, we could avoid using analytic continuation for the purposes of proving Theorem 8.3. However, this is an important concept that is widely used in complex analysis in general, and we feel that it should be part of our presentation.

Alexander Isaev

Lecture 20. Analytic Continuation (Continued). The Monodromy Theorem

Abstract

We will now show that the “discrete” and “continuous” variants of analytic continuation given in Definitions 19.3 and 19.4, respectively, are closely related. The fact stated below is not required for our proof of Theorem 8.3 later in the course, but we have chosen to include it in the lecture in order to further clarify the concept of analytic continuation.

Alexander Isaev

Lecture 21. Proof of Theorem 8.3. Conformal Transformations of the Canonical Simply-Connected Domains

Abstract

We are now ready to prove the Riemann Mapping Theorem.

Proof (Theorem 8.3). First of all, by applying a suitable Möbius transformation we can assume that one of the points in $ \overline{\mathbb{C}} \backslash D\ \mathrm{is}\ \infty,\ \mathrm{i.e}. $, that D lies in $ \mathbb{C} $.

Alexander Isaev

Backmatter

Title: Twenty-One Lectures on Complex Analysis
Author: Alexander Isaev
Publisher: Springer International Publishing
Electronic ISBN: 978-3-319-68170-2
Print ISBN: 978-3-319-68169-6
DOI: https://doi.org/10.1007/978-3-319-68170-2

Springer Professional

Twenty-One Lectures on Complex Analysis

A First Course

About this book

Table of Contents

Frontmatter

Lecture 1. Complex Numbers. The Fundamental Theorem of Algebra

Lecture 2. ℝ- and ℂ-Differentiability

Lecture 3. The Stereographic Projection. Conformal Maps. The Open Mapping Theorem

Lecture 4. Conformal Maps (Continued). Möbius Transformations

Lecture 5. Möbius Transformations (Continued). Generalised Circles. Symmetry

Lecture 6. Domains Bounded by Pairs of Generalised Circles. Integration

Lecture 7. Primitives Along Paths. Holomorphic Primitives. The Existence of a Holomorphic Primitive of a Function Holomorphic on a Disk. Goursat’s Lemma

Lecture 8. Proof of Lemma 7.2. Constructible Primitives of Holomorphic Functions along Paths. Integration of Holomorphic Functions over Arbitrary Paths. Homotopy. Simply-Connected Domains. The Riemann Mapping Theorem

Lecture 9. Cauchy’s Independence of Homotopy Theorem. Integration over Piecewise C 1-paths. Jordan Domains and Integration over their Boundaries

Lecture 10. Cauchy’s Integral Theorem. Proof of Theorem 3.1. Cauchy’s Integral Formula

Lecture 11. Morera’s Theorem. Sequences and Series of Functions. Uniform Convergence Inside a Domain. Power Series. Abel’s Theorem. Disk of Convergence. Radius of Convergence

Lecture 12. Proof of Theorem 11.9. Power Series (Continued). Taylor Series. Local Power Series Expansion of a Holomorphic Function. Cauchy’s Inequalities. The Uniqueness Theorem

Lecture 13. Liouville’s Theorem. Laurent Series. Annulus of Convergence. Laurent Series Expansion of a Function Holomorphic on an Annulus. Cauchy’s Inequalities. Isolated Singularities of Holomorphic Functions

Lecture 15. Isolated Singularities of Holomorphic Functions at ∞ (Continued). Orders of Poles at ∞. Casorati-Weierstrass’ Theorem for an Isolated Singularity at ∞. Residues. Cauchy’s Residue Theorem. Computing Residues

Lecture 16. Computing Residues (Continued). Computing Integrals over the Real Line Using Contour Integration. The Argument Principle

Lecture 17. Index of a Path. The Argument Principle (Continued). Rouché’s Theorem. Theorem 1.1 Revisited. Proof of Theorem 3.2. The Maximum Modulus Principle. Proof of Theorem 3.3

Lecture 18. Schwarz’s Lemma. Conformal Maps of the Unit Disk and the Upper Half-Plane. (Pre)-Compact Subsets of a Metric Space. Continuous Linear Functionals on H(D). Arzelà-Ascoli’s Theorem. Montel’s Theorem. Hurwitz’s Theorem

Lecture 19. Analytic Continuation

Lecture 20. Analytic Continuation (Continued). The Monodromy Theorem

Lecture 21. Proof of Theorem 8.3. Conformal Transformations of the Canonical Simply-Connected Domains

Backmatter

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