Introduction to Analysis with Complex Numbers, Lecture notes of Calculus

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Introduction to Analysis

with Complex Numbers

Irena Swanson

Purdue University

 - Preface - The briefest overview, motivation, notation 

• Chapter 1: How we will do mathematics
  • Section 1.1: Statements and proof methods
  • Section 1.2: Statements with quantifiers
  • Section 1.3: More proof methods
  • Section 1.4: Logical negation
  • Section 1.5: Summation
  • Section 1.6: Proofs by (mathematical) induction
  • Section 1.7: Pascal’s triangle
• Chapter 2: Concepts with which we will do mathematics
  • Section 2.1: Sets
  • Section 2.2: Cartesian product
  • Section 2.3: Relations, equivalence relations
  • Section 2.4: Functions
  • Section 2.5: Binary operations
  • Section 2.6: Fields
  • Section 2.7: Order on sets, ordered fields
  • Section 2.8: What are the integers and the rational numbers?
  • Section 2.9: Increasing and decreasing functions
  • Section 2.10: The Least upper bound property of R
  • Section 2.11: Absolute values
• Chapter 3: The field of complex numbers, and topology
  • Section 3.1: Complex numbers
  • Section 3.2: Functions related to complex numbers
  • Section 3.3: Absolute value in C
  • Section 3.4: Polar coordinates
  • Section 3.5: Topology on the fields of real and complex numbers
  • Section 3.6: The Heine-Borel theorem
• Chapter 4: Limits of functions
  • Section 4.1: Limit of a function
  • Section 4.2: When a number is not a limit
  • Section 4.3: More on the definition of a limit
  • Section 4.4: Limit theorems
  • Section 4.5: Infinite limits (for real-valued functions)
  • Section 4.6: Limits at infinity
• Chapter 5: Continuity
  • Section 5.1: Continuous functions
  • Section 5.2: Topology and the Extreme value theorem
  • Section 5.3: Intermediate value theorem
  • Section 5.4: Radical functions
  • Section 5.5: Uniform continuity
• Chapter 6: Differentiation
  • Section 6.1: Definition of derivatives
  • Section 6.2: Basic properties of derivatives
  • Section 6.3: The Mean value theorem
  • Section 6.4: L’Hˆopital’s rule
  • Section 6.5: Higher-order derivatives, Taylor polynomials
• Chapter 7: Integration
  • Section 7.1: Approximating areas
  • Section 7.2: Computing integrals from upper and lower sums
  • Section 7.3: What functions are integrable?
  • Section 7.4: The Fundamental theorem of calculus
  • Section 7.5: Integration of complex-valued functions
  • Section 7.6: Natural logarithm and the exponential functions
  • Section 7.7: Applications of integration
• Chapter 8: Sequences
  • Section 8.1: Introduction to sequences
  • Section 8.2: Convergence of infinite sequences
  • Section 8.3: Divergence of infinite sequences and infinite limits
  • Section 8.4: Convergence theorems via epsilon-N proofs
  • Section 8.5: Convergence theorems via functions
  • Section 8.6: Bounded sequences, monotone sequences, ratio test
  • Section 8.7: Cauchy sequences, completeness of R, C
  • Section 8.8: Subsequences
  • Section 8.9: Liminf, limsup for real-valued sequences
• Chapter 9: Infinite series and power series
  • Section 9.1: Infinite series
  • Section 9.2: Convergence and divergence theorems for series
  • Section 9.3: Power series
  • Section 9.4: Differentiation of power series
  • Section 9.5: Numerical evaluations of some series
  • Section 9.6: Some technical aspects of power series
  • Section 9.7: Taylor series
• Chapter 10: Exponential and trigonometric functions
  • Section 10.1: The exponential function
  • Section 10.2: The exponential function, continued
  • Section 10.3: Trigonometry
  • Section 10.4: Examples of L’Hˆopital’s rule
  • Section 10.5: Trigonometry for the computation of some series
• Appendix A: Advice on writing mathematics
• Appendix B: What one should never forget
• Index

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 standard topics of a course in introductory analysis, that handles complex-valued functions, 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 shorter, latter 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 particular, we assume that the set of familiar real numbers forms an ordered field with the Least upper bound property. In Chapter 3 we construct the very useful field of complex numbers, and introduce topology which is indispensable for the rigorous treatment of limits. I cover topology more lightly than what is in the written notes. Subsequent chapters cover standard material for introduction to analysis: limits, continuity, differentiation, integration, sequences, series, ending with the development of the power series

k=0 x

k k! , the exponential and the trigonometric func- tions. Since students have seen limits, continuity, differentiation and integration before, I go through chapters 4 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 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

8 Preface

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 irena@purdue.edu.

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.

Introduction to Analysis

with Complex Numbers

Chapter 1: How we will do mathematics

1.1 Statements and proof methods

Definition 1.1.1. A statement is a reasonably grammatical and unambiguous sentence that can be declared either true or false.

Why do we specify “reasonably grammatical”? We do not disqualify a statement just because of poor grammar, nevertheless, we strive to use correct grammar and to express the meaning clearly. And what do we mean by true or false? For our purposes, a statement is false if there is at least one counterexample to it, and a statement is true if it has been proved so, or if we assume it to be true.

Examples and non-examples 1.1.2. (i) The sum of 1 and 2 equals 3. (This is a true statement.) (ii) Seventeen. (This is not a statement.) (iii) Seventeen is the seventh prime number. (This is a true statement.) (iv) Is x positive? (This is not a statement.) (v) 1 = 2.* (This is a false statement.) (vi) For every real number  > 0 there exists a real number δ > 0 such that for all x, if 0 < |x − a| < δ then x is in the domain of f and |f (x) − L| < . (This is a statement, and it is (a part of) the definition of the limit of a (special) function f at a being L. Out of context, this statement is neither true or false, but we can prove it or assume it for various functions f .) (vii) Every even number greater than 4 can be written as a sum of two odd primes. (This statement is known as Goldbach’s conjecture. No counterexample is known, and no proof has been devised, so it is currently not known if it is true or false.) These examples show that not all statements have a definitive truth value. What makes them statements is that after possibly arbitrarily assigning them truth values, differ- ent consequences follow. For example, if we assume that (vi) above is true, then the graph of f near a is close to the graph of the constant function L. If instead we assume that (vi) above is false, then the graph of f near a has infinitely many values at some vertical distance away from L no matter how much we zoom in at a. With this in mind, even “I am good” is a statement: if I am good, then I get a cookie, but if I am not good, then you

• This statement can also be written in plain English as “One equals two.” In mathematics it is acceptable to use symbolic notation to some extent, but keep in mind that too many symbols can make a sentence hard to read. In general we avoid starting sentences with a symbol. In particular, do not make the following sentence. “=” is a verb. Instead make a sentence such as the following one. Note that “=” is a verb.

14 Chapter 1: How we will do mathematics

get the cookie. On the other hand, if “Hello” were to be true or false, I would not be able to make any further deductions about the world or my next action, so that “Hello” is not a statement, but only a sentence.

A useful tool for manipulating statements is a truth table: it is a table in which the first few columns may set up a situation, and the subsequent columns record truth values of statements applying in those particular situations. Here are two examples of truth tables, where “T ” of course stands for “true” and “F ” for “false”:

f constant continuous differentiable everywhere f (x) = x^2 F T T f (x) = |x| F T F f (x) = 7 T T T x y xy > 0 xy ≤ 0 xy < 0 x > 0 y > 0 T F F x > 0 y ≤ 0 F T F x < 0 y > 0 F T T x < 0 y ≤ 0 F F F Note that in the second row of the last table, in the exceptional case y = 0, the statement xy < 0 is false, but in “the majority” of the cases in that row xy < 0 is true. The one counterexample is enough to declare xy < 0 not true, i.e., false.

Statements can be manipulated just like numbers and variables can be manipulated, and rather than adding or multiplying statements, we connect them (by compounding the sentences in grammatical ways) with connectors such as “not”, “and”, “or”, and so on.

Statement connecting: (1) Negation of a statement P is a statement whose truth values are exactly opposite from the truth values of P (under any specific circumstance). The negation of P is denoted “ not P ” (or “¬P ”). Some simple examples: the negation of “A = B” is “A 6 = B”; the negation of “A ≤ B” is “A > B”; the negation of “I am here” is “I am not here” or “It is not the case that I am here”. Now go back to the last truth table. Note that in the last line, the truth values of “xy > 0” and “xy ≤ 0” are both false. But one should think that “xy > 0” and “xy ≤ 0” are negations of each other! So what is going on, why are the two truth values not opposites of each other? The problem is of course that the circumstances x < 0 and y ≤ 0 are not specific enough. The statement “xy > 0” is under these circumstances false precisely when y = 0, but then “xy ≤ 0” is true. Similarly, the statement “xy ≤ 0” is under the given circumstances false precisely

16 Chapter 1: How we will do mathematics

“Given P , Q follows,” “Q whenever P ”. P is called the antecedent and Q the consequent. A symbolic abbreviation is “P ⇒ Q.” An implication is true when a true conclusion follows a true assumption, or when- ever the assumption is false. In other words, P ⇒ Q is false exactly when P is true and Q is false. P Q P ⇒ Q T T T T F F F T T F F T It may be counterintuitive that a false antecedent always makes the implication true. Bertrand Russell once lectured on this and claimed that if 1 = 2 then he (Bertrand Russell) was the pope. An audience member challenged him to prove it. So Russell reasoned somewhat like this: “If I am the pope, then the consequent is true. If the consequent is false, then I am not the pope. But if I am not the pope, then the pope and I are two different people. By assumption 1 = 2, so we two people are one, so I am the pope. Thus no matter what, I am the pope.” Furthermore, if 1 = 2, then Bertrand Russell is also not the pope. Namely, if he is not the pope, the consequent is true, but if he is the pope, then the pope and he are one, and since one equals two, then the pope and he are two people, so Russell cannot be the pope. A further discussion about why false antecedent makes the implication true is in the next discussion (5). Unfortunately, the implication statement is not used consistently in informal spo- ken language. For example, your grandmother may say: “You may have ice cream if you eat your broccoli” when she means “You may have ice cream only if you eat your broccoli.” Be nice to your grandmother and eat that broccoli even if she does not express herself precisely because you know precisely what she means. But in mathematics you do have to express yourself precisely! (Well, read the next paragraph.) Even in mathematics some shortcuts in precise expressions are acceptable. Here is an example. The statements “An object x has property P if somethingorother holds” and “An object x has property P if and only if somethingorother holds” (see (5) below for “if and only if”) in general have different truth values and the proof of the second is longer. However, the definition of what it means for an object to have property P in terms of somethingorother is usually phrased with “if”, but

Section 1.1: Statements and proof methods 17

“if and only if” is meant. For example, the following is standard: “Definition: A positive integer strictly bigger than 1 is prime if whenever it can be written as a product of two positive integers, one of the two factors must be 1.” The given definition, if read logically precisely, since it said nothing about numbers such as 4 = 2 · 2, would allow us to call 4 prime. However, it is an understood shortcut that only the numbers with the stated property are called prime. (5) Equivalence or the logical biconditional of P and Q stands for the compound statement (P ⇒ Q) and (Q ⇒ P ). It is abbreviated “P ⇔ Q” or “P iff Q”, and is true precisely when P and Q have the same truth values. For example, for real numbers x and y, the statement “x ≤ y + 1” is equivalent to “x − 1 ≤ y.” Another example: “2x = 4x^2 ” is equivalent to “x = 2x^2 ,” but it is not equivalent to “1 = 2x.” (Say why!)

We now backtrack on the truth values of P ⇒ Q. We can certainly fill in some parts without qualms, leaving some unknown truth values x and y: P Q P ⇒ Q Q ⇒ P P ⇔ Q T T T T T T F F x F F T x F F F F y y T Since the last column above is the conjunction of the previous two, the last line forces the value of y to be T. If x equals F , then the truth values of P ⇒ Q are the same as the truth values of P ⇔ Q, which would say that the statements P ⇒ Q and P ⇔ Q are logically the same. But this cannot be: “If r > 0 then r ≥ 0” is true whereas “If r ≥ 0 then r > 0” is false. So this may convince you that the truth values for the third and the fifth column have to be distinct, and this is only possible if x is T.

Here is the truth table for all the connectives so far:

P Q not P P and Q P or Q P xor Q P ⇒ Q P ⇔ Q T T F T T F T T T F F F T T F F F T T F T T T F F F T F F F T T One can form more elaborate truth tables if we start not with two statements P and Q but with three or more. Examples of logically compounding P, Q, and R are: P and Q and R, (P and Q) ⇒ Q, et cetera. For manipulating three statements, we would fill a total of 8 rows of truth values, for four statements there would be 16 rows, and so on.

Section 1.1: Statements and proof methods 19

Therefore, f is continuous.

Another tautology is modus tollens: ((P ⇒ Q) and ( not Q)) ⇒ ( not P ). To prove it, one constructs a truth table as before for modus ponens. — It is a common proof technique to invoke the similarity principle with previous work that allows one to not carry out all the steps, as I just did. However, whenever you invoke the proof-similarity principle, you better be convinced in your mind that the similar proof indeed does the job; if you have any doubts, show all work instead! In this case, I am sure that the truth table does the job, but if you are seeing this for the first time, you may want to do the actual truth table explicitly to get a better grasp on these concepts. Here is a mathematical example of modus tollens: Every differentiable function is continuous. f is not continuous. Therefore, f is not differentiable. Here is another example on more familiar ground: If you are in Oregon, then you are in the USA. You are not in the USA. Therefore, you are not in Oregon.

Some proofs can be pictorial/graphical. Here we prove with this method that for any real numbers x and y, |x| < y if and only if −y < x < y. (We will see many uses of absolute values.) Proof: [For a biconditional P ⇔ Q we need to prove P ⇒ Q and Q ⇒ P .] The assumption |x| < y implies that y must be positive, and the assumption −y < x < y implies that −y < y, which also says that y must be positive. So, with either assumption, we can draw the following part of the real number line:

−y 0 y

Now, by drawing, the real numbers x with |x| < y are precisely those real num- bers x with −y < x < y. A fancier way of saying this is that |x| < y if and only if −y < x < y.

Similarly, for all real numbers x and y, |x| ≤ y if and only if −y ≤ x ≤ y. (Here, the word “similarly” is a clue that I am invoking the proof-similarity principle, and a reader who wants to practice proofs or is not convinced should at this point work through a proof by mimicking the steps in the previous one.)

Some (or actually most) proofs invoke previous results without re-doing the previ- ous work. In this way we prove the triangle inequality, which asserts that for all

20 Chapter 1: How we will do mathematics

real numbers x and y, |x ± y| ≤ |x| + |y|. (By the way, we will use the triangle in- equality intensely, so understand it well.) Proof: Note that always −|x| ≤ x ≤ |x|, −|y| ≤ ±y ≤ |y|. Since the sum of smaller numbers is always less than or equal to the sum of larger numbers, we then get that −|x| − |y| ≤ x ± y ≤ |x| + |y|. But −|x| − |y| = −(|x| + |y|), so that −(|x| + |y|) ≤ x ± y ≤ |x| + |y|. But then by the previous result, |x ± y| ≤ |x| + |y|.

Most proofs require a combination of methods. Here we prove that whenever x is a real number with |x− 5 | < 4, then |x^3 − 3 x| < 900. Proof: The following is standard formatting that you should adopt: first write down the left side of the desired inequality (|x^3 − 3 x|), then start manipulating it algebraically, in intermediate steps add a clever 0 here and there, multiply by a clever 1 here and there, rewrite, simplify, make it less than or equal to something else, and so on, every step should be either obvious or justified on the right, until at the end you get the quantity on the right (900): |x^3 − 3 x| ≤ |x^3 | + | 3 x| (by the triangle inequality) = |x|^3 + 3|x| = |x − 5 + 5|^3 + 3|x − 5 + 5| (by adding a clever 0) ≤ (|x − 5 | + 5)^3 + 3(|x − 5 | + 5) (by the triangle inequality and since a ≤ b implies that a^3 ≤ b^3 ) ≤ (4 + 5)^3 + 3(4 + 5) (since by assumption |x − 5 | < 4) = 9^3 + 3 · 9 = 9(9^2 + 3) < 900.

Here is a pictorial proof establishing the basis of trigonometry and the definition of slope as rise over run: namely that BA = (^) ab.

︸ a ︷︷ ︸ A

b

B