A First Course in Mathematical Analysis

✍ Scribed by David Alexander Brannan

• Publisher: Cambridge University Press
• Year: 2006
• Tongue: English
• Leaves: 470
• Category: Library

✦ Table of Contents

Cover
Half-title
Title
Copyright
Dedication
Contents
Preface
Outline of the content of the book
Study guide
Acknowledgements
1 Numbers
1.1 Real numbers
1.1.1 Rational numbers
1.1.2 Decimal representation of rational numbers
1.1.3 Irrational numbers
1.1.4 The real number system
1.1.5 Arithmetic in R
1.2 Inequalities
1.2.1 Rearranging inequalities
1.2.2 Solving inequalities
1.2.3 Inequalities involving modulus signs
1.3 Proving inequalities
1.3.1 The Triangle Inequality
1.3.2 Inequalities involving n
1.3.3 More on inequalities
Three important inequalities in Analysis
1.4 Least upper bounds and greatest lower bounds
1.4.1 Upper and lower bounds
1.4.2 Least upper bounds and greatest lower bounds
Least upper bounds and greatest lower bounds of functions
1.4.3 The Least Upper Bound Property
1.4.4 Proof of the Least Upper Bound Property
1.5 Manipulating real numbers
1.5.1 Arithmetic in R
1.5.2 The existence of roots
1.5.3 Rational powers
1.5.4 Real powers
1.6 Exercises
Section 1.1
Section 1.2
Section 1.3
Section 1.4
2 Sequences
2.1 Introducing sequences
2.1.1 What is a sequence?
Sequence diagrams
2.1.2 Monotonic sequences
2.2 Null sequences
2.2.1 What is a null sequence?
2.2.2 Proofs
Basic null sequences
2.3 Convergent sequences
2.3.1 What is a convergent sequence?
2.3.2 Combination Rules for convergent sequences
Applying the Combination Rules
2.3.3 Further properties of convergent sequences
2.4 Divergent sequences
2.4.1 What is a divergent sequence?
2.4.2 Bounded and unbounded sequences
2.4.3 Sequences which tend to infinity
2.4.4 Subsequences
2.5 The Monotone Convergence Theorem
2.5.1 Monotonic sequences
The Bolzano–Weierstrass Theorem
2.5.2 Sequences defined by recursion formulas
2.5.3 The number e
2.5.4 The number phi
2.5.5 Proofs
2.6 Exercises
Section 2.1
Section 2.2
Section 2.3
Section 2.4
Section 2.5
3 Series
3.1 Introducing series
3.1.1 What is a convergent series?
Sigma notation
3.1.2 Geometric series
Decimal representation of rational numbers
3.1.3 Telescoping series
3.1.4 Combination Rules for convergent series
3.1.5 The Non-null Test
3.2 Series with non-negative terms
3.2.1 Tests for convergence
Another test for convergence
3.2.2 Proofs
3.3 Series with positive and negative terms
3.3.1 Absolute convergence
3.3.2 The Alternating Test
3.3.3 Rearrangement of a series
3.3.4 Multiplication of series
3.3.5 Overall strategy for testing for convergence
3.3.6 Proofs
3.4 The exponential function x…
3.4.1 The definition of ex as a power series, for x > 0
3.4.2 Calculating e
3.4.3 The definition of ex as a power series, for all real x
3.5 Exercises
Section 3.1
Section 3.2
Section 3.3
4 Continuity
4.1 Continuous functions
4.1.1 What is continuity?
4.1.2 Rules for continuous functions
4.1.3 Trigonometric functions and the exponential function
Trigonometric functions
The exponential function x…
4.2 Properties of continuous functions
4.2.1 The Intermediate Value Theorem
Antipodal points
4.2.2 Zeros of polynomials
4.2.3 The Extreme Values Theorem
4.3 Inverse functions
4.3.1 Existence of an inverse function
Proving that a function f is one–one
4.3.2 The Inverse Function Rule
4.3.3 Inverses of standard functions
The nth root function
Inverse trigonometric functions
The function loge
Inverse hyperbolic functions
4.3.4 Proof of the Inverse Function Rule
4.4 Defining exponential functions
4.4.1 The definition of ax
4.4.2 Further properties of exponentials
4.5 Exercises
Section 4.1
Section 4.2
Section 4.3
5 Limits and continuity
5.1 Limits of functions
5.1.1 What is a limit of a function?
5.1.2 Limits and continuity
5.1.3 Rules for limits
5.1.4 One-sided limits
5.2 Asymptotic behaviour of functions
5.2.1 Functions which tend to infinity
5.2.2 Behaviour of f(x) as x tends to…
5.2.3 Composing asymptotic behaviour
5.3 Limits of functions – using Epsilon and Delta
5.3.1 The Epsilon – Delta definition of limit of a sequence
5.3.2 The Epsilon – Delta definition of limit of a function
5.4 Continuity – using Epsilon and Delta
5.4.1 The Epsilon – Delta definition of continuity
5.4.2 The Dirichlet and Riemann functions
5.4.3 Proofs
5.5 Uniform continuity
5.6 Exercises
Section 5.1
Section 5.2
Section 5.3
Section 5.4
Section 5.5
6 Differentiation
6.1 Differentiable functions
6.1.1 What is differentiability?
6.1.2 Differentiability and continuity
6.1.3 The sine, cosine and exponential functions
6.1.4 Higher-order derivatives
6.2 Rules for differentiation
6.2.1 The Combination Rules
6.2.2 The Composition Rule
6.2.3 The Inverse Function Rule
Exponential functions
6.2.4 Proofs
6.3 Rolle’s Theorem
6.3.1 The Local Extremum Theorem
6.3.2 Rolle’s Theorem
6.4 The Mean Value Theorem
6.4.1 The Mean Value Theorem
6.4.2 Positive, negative and zero derivatives
Inequalities
6.5 L’Hopital’s Rule
6.5.1 Cauchy’s Mean Value Theorem
6.5.2 l’Hopital’s Rule
6.6 The Blancmange function
6.6.1 What is the Blancmange function?
6.6.2 Continuity of the Blancmange function
6.6.3 The Blancmange function is differentiable nowhere
6.7 Exercises
Section 6.1
Section 6.2
Section 6.3
Section 6.4
Section 6.5
7 Integration
7.1 The Riemann integral
7.1.1 The Riemann integral and integrability
Properties of Riemann sums
7.1.2 Criteria for integrability
7.2 Properties of integrals
7.2.1 Infimum and Supremum of functions (revisited)
7.2.2 Monotonic and continuous functions
7.2.3 Rules for integration
Sum Rule
Multiple Rule
Product Rule
Modulus Rule
7.3 Fundamental Theorem of Calculus
7.3.1 Fundamental Theorem of Calculus
7.3.2 Finding primitives
7.3.3 Techniques of integration
7.4 Inequalities for integrals and their applications
7.4.1 The key inequalities
7.4.2 Wallis’s Formula
Reduction of Order method
Wallis’s Formula
7.4.3 Maclaurin Integral Test
Sequences revisited
7.5 Stirling’s Formula for n!
7.5.1 The tilda notation
7.5.2 Stirling’s Formula
7.5.3 Proof of Stirling’s Formula
7.6 Exercises
Section 7.1
Section 7.2
Section 7.3
Section 7.4
Section 7.5
8 Power series
8.1 Taylor polynomials
8.1.1 What are Taylor polynomials?
8.1.2 Approximation by Taylor polynomials
The function f (x)…
The function f(x)=sin x
The function f(x)…
Which functions can be approximated by Taylor polynomials?
8.2 Taylor’s Theorem
8.2.1 Taylor’s Theorem and approximation
8.2.2 Taylor’s Theorem and power series
8.3 Convergence of power series
8.3.1 The radius of convergence
8.3.2 Abel’s Limit Theorem
8.3.3 Proofs
8.4 Manipulating power series
8.4.1 Rules for power series
8.4.2 General Binomial Theorem
8.4.3 Proofs
8.5 Numerical estimates for Phi
8.5.1 Calculating Phi
8.5.2 Proof that is Phi irrational
8.6 Exercises
Section 8.1
Section 8.2
Section 8.3
Section 8.4
Appendix 1: Sets, functions and proofs
Sets
Functions
Principle of Mathematical Induction
Appendix 2: Standard derivatives and primitives
Appendix 3: The first 1000 decimal places of…
Appendix 4: Solutions to the problems
Chapter 1
Section 1.1
Section 1.2
Section 1.3
Section 1.4
Chapter 2
Section 2.1
Section 2.2
Section 2.3
Section 2.4
Section 2.5
Chapter 3
Section 3.1
Section 3.2
Section 3.3
Section 3.4
Chapter 4
Section 4.1
Section 4.2
Section 4.3
Section 4.4
Chapter 5
Section 5.1
Section 5.2
Section 5.3
Section 5.4
Section 5.5
Chapter 6
Section 6.1
Section 6.2
Section 6.3
Section 6.4
Section 6.5
Chapter 7
Section 7.1
Section 7.2
Section 7.3
Section 7.4
Section 7.5
Chapter 8
Section 8.1
Section 8.2
Section 8.3
Section 8.4
Section 8.5
Index