Introduction to Analysis with Complex Numbers

388 Pages·2020·2.416 MB·English

by  Irena Swanson

Introduction to Analysis
with Complex Numbers
Irena Swanson
Purdue University

Table of Contents
Preface 7
The briefest overview, motivation, notation 9
Chapter 1: How we will do mathematics 13
Section 1.1: Statements and proof methods 13
Section 1.2: Statements with quantifiers 25
Section 1.3: More proof methods 28
Section 1.4: Logical negation 32
Section 1.5: Summation 34
Section 1.6: Proofs by (mathematical) induction 36
Section 1.7: Pascal’s triangle 45
Chapter 2: Concepts with which we will do mathematics 49
Section 2.1: Sets 49
Section 2.2: Cartesian product 58
Section 2.3: Relations, equivalence relations 60
Section 2.4: Functions 66
Section 2.5: Binary operations 76
Section 2.6: Fields 83
Section 2.7: Order on sets, ordered fields 87
Section 2.8: What are the integers and the rational numbers? 93
Section 2.9: Increasing and decreasing functions 96
Section 2.10: Absolute values 98
Chapter 3: Construction of the number systems 102
Section 3.1: Inductive sets, a construction of natural numbers 102
Section 3.2: Arithmetic on N0 107
Section 3.3: Order on N0 110
Section 3.4: Cancellation in N0 113
Section 3.5: Construction of Z, arithmetic, and order on Z 114
4
Section 3.6: Construction of the ordered field Q 120
Section 3.7: Construction of the field R of real numbers 125
Section 3.8: Order on R, the Least upper bound theorem 133
Section 3.9: Complex numbers 139
Section 3.10: Functions related to complex numbers 142
Section 3.11: Absolute value in C 144
Section 3.12: Polar coordinates 148
Section 3.13: Topology on the fields of real and complex numbers 153
Section 3.14: The Heine-Borel theorem 158
Chapter 4: Limits of functions 161
Section 4.1: Limit of a function 161
Section 4.2: When a number is not a limit 172
Section 4.3: More on the definition of a limit 176
Section 4.4: Limit theorems 179
Section 4.5: Infinite limits (for real-valued functions) 187
Section 4.6: Limits at infinity 190
Chapter 5: Continuity 193
Section 5.1: Continuous functions 193
Section 5.2: Topology and the Extreme value theorem 198
Section 5.3: Intermediate value theorem 201
Section 5.4: Radical functions 205
Section 5.5: Uniform continuity 210
Chapter 6: Differentiation 214
Section 6.1: Definition of derivatives 214
Section 6.2: Basic properties of derivatives 218
Section 6.3: The Mean value theorem 226
Section 6.4: L’Hoˆpital’s rule 231
Section 6.5: Higher-order derivatives, Taylor polynomials 234
Chapter 7: Integration 239
Section 7.1: Approximating areas 239
Section 7.2: Computing integrals from upper and lower sums 250
Section 7.3: What functions are integrable? 253
Section 7.4: The Fundamental theorem of calculus 258
Section 7.5: Integration of complex-valued functions 265
Section 7.6: Natural logarithm and the exponential functions 266
5
Section 7.7: Applications of integration 272
Chapter 8: Sequences 278
Section 8.1: Introduction to sequences 278
Section 8.2: Convergence of infinite sequences 283
Section 8.3: Divergence of infinite sequences and infinite limits 290
Section 8.4: Convergence theorems via epsilon-N proofs 294
Section 8.5: Convergence theorems via functions 300
Section 8.6: Bounded sequences, monotone sequences, ratio test 303
Section 8.7: Cauchy sequences, completeness of R, C 307
Section 8.8: Subsequences 311
Section 8.9: Liminf, limsup for real-valued sequences 314
Chapter 9: Infinite series and power series 319
Section 9.1: Infinite series 319
Section 9.2: Convergence and divergence theorems for series 323
Section 9.3: Power series 329
Section 9.4: Differentiation of power series 334
Section 9.5: Numerical evaluations of some series 338
Section 9.6: Some technical aspects of power series 340
Section 9.7: Taylor series 344
Chapter 10: Exponential and trigonometric functions 349
Section 10.1: The exponential function 349
Section 10.2: The exponential function, continued 351
Section 10.3: Trigonometry 358
Section 10.4: Examples of L’Hoˆpital’s rule 363
Section 10.5: Trigonometry for the computation of some series 365
Appendix A: Advice on writing mathematics 371
Appendix B: What one should never forget 375
Index 379

Preface
These notes were written expressly for Mathematics 112 at Reed College, with first
use in the spring of 2013. The title of the course is “Introduction to Analysis”. The
prerequisite is calculus. Recently used textbooks have been Steven R. Lay’s “Analysis,
With an Introduction to Proof” (Prentice Hall, Inc., Englewood Cliffs, NJ, 1986, 4th
edition), and Ray Mayer’s in-house notes “Introduction to Analysis” (2006, available at
http://www.reed.edu/~mayer/math112.html/index.html). Ray Mayer’s notes strongly
influenced the coverage in this book.
In Math 112 at Reed College, students learn to write proofs while at the same time
learning about binary operations, orders, fields, ordered fields, complete fields, complex
numbers, sequences, and series. We also review limits, continuity, differentiation, and
integration. My aim for these notes is to constitute a self-contained book that covers the
standardtopicsofacourseinintroductoryanalysis, thathandlescomplex-valuedfunctions,
sequences, and series, that has enough examples and exercises, that is rigorous, and is
accessible to undergraduates. I maintain two versions of these notes, one in which the
natural, rational and real numbers are constructed and the Least upper bound theorem
is proved for the ordered field of real numbers, and one version in which the Least upper
bound property is assumed for the ordered field of real numbers. You are reading the
longer, former version.
Chapter 1 is about how we do mathematics: basic logic, proof methods, and Pascal’s
triangle for practicing proofs. Chapter 2 introduces foundational concepts: sets, Carte-
sian products, relations, functions, binary operations, fields, ordered fields, Archimedean
property for the set of real numbers. In Chapters 1 and 2 we assume knowledge of high
school mathematics so that we do not practice abstract concepts and methods in a vac-
uum. Chapter 3 through Section 3.8 takes a step back: we “forget” most previously learned
mathematics, and we use the newly learned abstract tools to construct natural numbers,
integers, rational numbers, real numbers, with all the arithmetic and order. I do not teach
theseconstructionsingreatdetail; myaimistogiveasenseofthemandtopracticeabstract
logical thinking. The remaining sections in Chapter 3 are new material for most students:
the field of complex numbers, and some topology. I cover the last section of Chapter 3 very
lightly. Subsequent chapters cover standard material for introduction to analysis: limits,
continuity, differentiation, integration, sequences, series, ending with the development of
the power series ∞ xk, the exponential and the trigonometric functions. Since students
k=0 k!
have seen limits, continuity, differentiation and integration before, I go through chapters 4
P
through 7 quickly. I slow down for sequences and series (the last three chapters).
An effort is made throughout to use only what had been proved. For this reason, the
chapters on differentiation and integration do not have the usual palette of trigonometric
8 Preface
and exponential examples of other books. The final chapter makes up for it and works out
much trigonometry in great detail and depth.
I acknowledge and thank the support from the Dean of Faculty of Reed College
to fund exercise and proofreading support in the summer of 2012 for Maddie Brandt,
Munyo Frey-Edwards, and Kelsey Houston-Edwards. I also thank the following people
for their valuable feedback: Mark Angeles, Josie Baker, Marcus Bamberger, Anji Bodony,
Zachary Campbell, Nick Chaiyachakorn, Safia Chettih, Laura Dallago, Andrew Erlanger,
Joel Franklin, Darij Grinberg, Rohr Hautala, Palak Jain, Ya Jiang, Albyn Jones, Wil-
low Kelleigh, Mason Kennedy, Christopher Keane, Michael Keppler, Ryan Kobler, Oleks
Lushchyk, Molly Maguire, Benjamin Morrison, Samuel Olson, Kyle Ormsby, Ang´elica Os-
orno, Shannon Pearson, David Perkinson, Jeremy Rachels, Ezra Schwartz, Jacob Sharkan-
sky, Marika Swanberg, Simon Swanson, Matyas Szabo, Ruth Valsquier, Xingyi Wang,
Emerson Webb, Livia Xu, Qiaoyu Yang, Dean Young, Eric Zhang, Jialun Zhao, and two
anonymous reviewers. If you have further comments or corrections, please send them to
[email protected].
The briefest overview, motivation, notation
What are the meanings of the following:
5+6
7 9
·
8 4
−
4/5
√2
4 8
−
1/3 = 0.333... 1 = 3 1/3 = 0.999...
·
(a b)2 = a2 b2
· ·
(a+b) (c+d) = ac+ad+bc+bd
·
(a+b) (a b) = a2 b2
· − −
(a+b)2 = a2 +2ab+b2
√a √b = √ab (for which a,b?)
·
What is going on: √ 4 √ 9 = ( 4)( 9) = √36 = 6,
− · − − −
√ 4 √ 9 = 2i 3i =p6
− · − · −
You know all of the above except possibly the complex numbers in the last two rows,
where obviously something went wrong. We will not resolve this last issue until later in
the semester, but the point for now is that we do need to reason carefully.
The main goal of this class is to learn to reason carefully, rigorously. Since one
cannot reason in a vacuum, we will (but of course) be learning a lot of mathematics as
well: sets, logic, various number systems, fields, the field of real numbers, the field of
complex numbers, sequences, series, some calculus, and that eix = cosx+isinx.
We will make it all rigorous, i.e., we will be doing proofs. A proof is a sequence of
steps that logically follow from previously accepted knowledge.
But no matter what you do, never divide by 0. For further wise advice, turn to
Appendix A.
10 The briefest overview, motivation, notation
[Notational convention: Text between square brackets in this font and in
red color should be read as a possible reasoning going on in the background
in your head, and not as part of formal writing.]
1. Exercises with a dagger are invoked later in the text.
†
*2. Exercises with a star are more difficult.