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The Springer Undergraduate Mathematics Series (SUMS) is a series designed for undergraduates in mathematics and the sciences worldwide. From core foundational material to final year topics, SUMS books take a fresh and modern approach. Textual explanations are supported by a wealth of examples, problems and fully-worked solutions, with particular attention paid to universal areas of difficulty. These practical and concise texts are designed for a one- or two-semester course but the self-study approach makes them ideal for independent use.
Asuman Guren Aksoy
Fundamentals of Real and Complex Analysis
\(\underline{\epsilon}\)) SpringerAsuman Guven Aksoy (c)The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024
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## References
* [1] A. Abrah
## Preface
Real and complex analysis are fundamental topics in understanding mathematics at both undergraduate and graduate levels. While real and complex analysis are traditionally separated, this book unites these subjects after developing the basic techniques to understand them. The few texts which do unify real and complex analysis, such as Rudin’s excellent “Real and Complex Analysis,” often presume readers begin with a high level of mathematical sophistication. In contrast, this book aims to accommodate a larger population of readers, especially advanced undergraduates, and emphasizes the strong connection between various branches of analysis.
We aim to present different subareas of analysis as interconnected, rather than separate disciplines in order to give the reader understanding of analysis as a whole. Furthermore, this book is geared toward those interested in continuing further in mathematics, whether you are an advancing math major or a quantitative scientist, or simply an interested person wanting to become well-versed in advanced mathematics beyond the level of popularizations. We include ample examples, exercises, and copious references in order to guide those who wish to go deeper into particular topics.
Many of the properties of the field of real numbers \(\mathbb{R}\) hold in the field of complex numbers \(\mathbb{C}\) as well. There are some truly remarkable differences. For example, many results in real analysis make use of the fact that \(\mathbb{R}\) is totally ordered, \(\mathbb{C}\) loses order but gains a lot of properties such as well-behaved derivatives. A real valued function could be differentiated once, but there is no guarantee it can be differentiated twice; in contrast, if we have a complex valued differentiable function, then it can be differentiated many times. Although exponential function \(e^{x}\) is one-to-one, \(e^{z}\) is not, in fact it is a periodic function, \(\log_{e}x\) is a single valued function, but \(\ln z\) is multi-valued. Writing \(\sin x=5\) makes no sense, but if we replace \(x\) by \(z\) and write \(\sin z=5\) makes perfect sense.
The book is organized as follows. In Chapter 1 we begin our exploration of advanced mathematics with foundational topics. Starting with set theory, we look at the differences between finite and infinite sets and lay the groundwork for analysis and topology. We introduce equivalence relations and quotient sets and congruences, prompting a discussion of the field axioms. We meet fields, both finite and infinite, before turning our attention to the properties of the field of real numbers, including their completeness and topology. We also present in this chapter a review of some important facts about infinitesequences and series, building on the reader’s experience of calculus to develop the beginning of analysis. With the topological concepts developed earlier, we study continuity, differentiability, and integrability of real valued functions. Properties of continuous functions on compact and connected sets, the mean value theorem, the fundamental theorem of calculus, and the inverse function theorem are all covered. We also touch upon the concept of sets of measure zero and Lebesgue’s idea of integrability, which will be studied in detail in the next chapter. Thus, Chapter 1 can be seen as a “toolbox” for higher mathematics. Its contents are easily accessible but not elementary.
In Chapter 2, we focus on topics in real analysis. We start with the concepts of metric, inner product, and normed spaces. Since research in \(p\)-adic analysis is very prominent not only in analysis but also in number theory and mathematical physics, we examine the \(p\)-adic completion of the rational numbers relative to a prime \(p\). We also study Baire’s theorem which is central to functional analysis. We prove Baire’s theorem for the reals, completing the discussion from Chapter 1 about the size of the real numbers. The second section in this chapter is on the Banach fixed point theorem, whose utility in solving systems of linear equations as well as its applications to differential and integral equations is illustrated with numerous examples. The topics chosen here are not only of theoretical value, they provide essential knowledge for anyone interested in applied mathematics. We then cover modes of convergence and approximation by polynomials. We emphasize the differences between uniform and pointwise convergence and their effect on the continuity, differentiability, and integrability of sequences or series of functions. Next, we ask the question whether there are any “useful” dense subspaces of the space of continuous functions on a closed and bounded interval, leading us to the Weierstrass approximation theorem, which we prove using Bernstein polynomials. Another proof of the Weierstrass approximation theorem for trigonometric polynomials is mentioned in a subsequent section on Fourier series.
One of the unique features of Chapter 2 is its treatment of functional equations, which is often neglected in most analysis texts. In the section on Fourier series, we start with historical background, then quickly move on to orthogonality relations and emphasize convergence issues. Finally, our discussion shifts to the continuous analog, then to the Fourier Transform, and its inversion, including convolution and the relationship between the Fourier transform and the Laplacian. We introduce Lebesgue measurable sets and functions while stating Littlewood’s three principles. Here we will also encounter fantastic examples that stretch the imagination, like the Cantor set and the Cantor-Lebesgue function. The Lebesgue measure will allow us to define the correct notion of integration, through which we will be able to integrate far more functions as well as prove that the integral of the limit of functions is equal to limit of the integrals (the Lebesgue-dominated convergence theorem). We conclude this section with the Banach-Tarski paradox, which roughly shows us a way to take a ball, decompose it into a finite number of pieces, then reassemble it into two balls identical to the original.
In Chapter 3, we cover the most essential topics in complex analysis. Starting with a geometric introduction to the complex plane, we study holomorphic functions, complex power series, conformal mappings, and the Riemann mapping theorem. We focus on the power and significance of Cauchy’s theorem, the centerpiece of complex analysis. We emphasize applications of this theorem through Cauchy’s integral formula and residue theorem. Even though these concepts date back to the nineteenth century, we illustrate their power when we briefly glimpse into the Bieberbach conjecture. In the decades that the Bieberbach conjecture stood unsolved, mathematicians discovered many properties of univalent functions. In fact, most of the theory of univalent functions and geometric function theory arose from these efforts. The significance of the Bieberbach conjecture lies not only with its solution, but also the theory developed to solve it.
My hope is this book will be useful to anyone interested in upper division mathematics. Textbook uses aside, this book is meant as a coherent response to a motivated student who, wanting to truly understand analysis, asks “Can you please get to the point?”
Auman Guven Aksoy
Claremont, CA, USA
## Acknowledgments
The results in this book belong to the common heritage of mathematics. Needless to say, I am indebted intellectually to the many mathematicians who contributed to the creation of real and complex analysis. Even though certain theorems are designated with the usual proper names, I have otherwise made no special effort to attribute theorems and proofs.
My special thanks goes to my colleague and dear friend Sam Nelson from Claremont McKenna College. He has been involved with this project since its inception and has created nearly all illustrations in this book. Many many thanks! I also like to thank two graduate students: Chris Donnay of Ohio State University and Daniel Akech of Claremont Graduate University. They both read the entire text and made many valuable corrections. Finally, I thank Elizabeth Loew for her wonderful work as an editor. I am grateful to my best friend Ercument G. Aksoy for the constant support.
Asuman G. Aksoy
Claremont, California
December 2023
###### Contents
* 1 Introductory Analysis
* 1.1 Set Theory
* 1.2 Number Systems
* 1.3 Completeness and the Real Number System
* 1.4 Sequences and Series
* 1.5 Topology of the Real Line
* 1.6 Continuous Functions
* 1.7 Differentiability on \(\mathbb{R}\)
* 1.8 The Riemann Integral
* 2 Real Analysis
* 2.1 Metric, Normed, and Inner Product Spaces
* 2.2 Fixed Point Theorems and Applications
* 2.3 Modes of Convergence
* 2.4 Approximation by Polynomials
* 2.5 Functional Equations
* 2.6 Fourier Series
* 2.7 Lebesgue Measure and Integration
* 2.8 Banach-Tarski Paradox
* 3 Complex Analysis
* 3.1 The Complex Plane
* 3.2 Holomorphic Functions
* 3.3 Power Series
* 3.4 Some Holomorphic Functions
* 3.5 Conformal Mappings
* 3.6 Integration in the Complex Plane
* 3.7 Cauchy’s Theorem
* 3.8 Cauchy’s Formulae
* 3.9 Laurent Expansion and Singularities
* 3.10 The Bieberbach Conjecture
* 4
## Chapter 1 Introductory Analysis
If now to any \(x\) there corresponds a unique, finite \(y\)…Then \(y\) is called a function of \(x\) for this interval…This definition does not require a common rule for different parts of the curve; one can imagine the curve as being composed of the most heterogeneous components or as being drawn without following any law.
_Dirichlet 1837_
### 1.1 Set Theory
Some ancient mathematicians did not consider zero to be a number. It was not that they had not thought of zero; rather, they considered numbers to be lengths of line segments, i.e., geometric extensions in space, and a line segment of length zero seems like hardly a line segment at all. This lack of zero meant that ancient mathematics lacked the co ncept of an _empty set_, i.e., a set without any elements. This in turn had consequences for ancient logic, leading to certain invalid arguments being accepted as valid. Modern logic arises from _set theory_, and indeed all of mathematics has set theory as its fundamental base. For further reading on the topics of this chapter, we recommend the following references: [3, 9, 17, 28, 29, 42, 43, 46, 52].
### Sets
A _set_ is an unordered list or collection \(X\) of things called _elements_. We can define a set by explicitly listing its elements, like
\[X=\{1,2,3,\pi\},\]
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The _power set_ of a set \(A\) is the set of all subsets of \(A\), i.e.,
\[\mathcal{P}(A)=\{S\ |\ S\subset A\}.\]
The power set of the set \(A=\{1,2,3\}\) is the set
\[\mathcal{P}(A)=\{\{1,2,3\}\,\{1,2\},\{1,3\},\{2,3\},\{1\},\{2\},\{3\},\emptyset\}.\]
One useful way to avoid the issues that can arise when dealing with proper classes is to select an appropriately large set to be our “universe” and restrict ourselves to subsets of this universal set, i.e., elements of the power set of our universal set. For instance, we might only be interested in integer solutions for a given problem, in which case we could declare the set of integers to be our universal set.
### Set Operations
Given two sets \(A\) and \(B\), we can combine \(A\) and \(B\) into new sets in several ways:
* **Union.** The _union_ of \(A\) and \(B\) denoted \(A\cup B\) is the set in Figure 1.1. \(A\cup B=\{x\ |\ x\in A\ \text{or}\ x\in B\}\) For example, \[\{1,2,3,4\}\cup\{3,4,5\}=\{1,2,3,4,5\}.\]
* **Intersection.** The _intersection_ of \(A\) and \(B\) denoted \(A\cap B\) is the set in Figure 1.2. \(A\cap B=\{x\ |\ x\in A\ \text{and}\ x\in B\}\) For example, \[\{1,2,3,4\}\cap\{3,4,5\}=\{3,4\}.\]Two sets \(A\) and \(B\) are called _disjoint_ if their intersection is empty, i.e., if \(A\cap B=\emptyset\). For example, \(\{1,2,3\}\) and \(\{4,5,6\}\) are disjoint sets.
* **Setwise Difference.** The _setwise difference_ of \(A\) and \(B\) denoted \(A\setminus B\) and read “\(A\) minus \(B\)” is shown in Figure 1.3.
For example,
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