Introduction to Analysis by Mattuck

In summary, "Introduction to Analysis" by Arthur Mattuck is a comprehensive and challenging textbook for undergraduate students. The book covers topics such as real numbers, limits, sequences, series, functions, differentiation, integration, and more. The author's teaching style is effective, but some may find the subject difficult to grasp. Overall, this book is a valuable resource for learning analysis, but there may be alternative textbooks that are more suitable for some readers.

For those who have used this book

  • Lightly don't Recommend

    Votes: 0 0.0%
  • Stronly don't Recommend

    Votes: 0 0.0%

  • Total voters
    2
  • #1
micromass
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
22,183
3,324

Table of Contents:
Code:
[LIST]
[*] Preface
[*] Real Numbers and Monotone Sequences
[LIST]
[*] Introduction; Real numbers
[*] Increasing sequences
[*] Limit of an increasing sequence
[*] Example: the number e
[*] Example: the harmonic sum and Euler's number
[*] Decreasing sequences; Completeness property
[/LIST]
[*] Estimations and Approximations
[LIST]
[*] Introduction; Inequalities
[*] Estimations
[*] Proving boundedness
[*] Absolute values; estimating size
[*] Approximations
[*] The terminology "for n large"
[/LIST]
[*] The Limit of a Sequence
[LIST]
[*] Definition of limit
[*] Uniqueness of limits; the K-\epsilon principle
[*] Infinite limits
[*] Limit of a^n
[*] Writing limit proofs
[*] Some limits involving integrals
[*] Another limit involving an integral
[/LIST]
[*] The Error Term
[LIST]
[*] The error term
[*] The error in the geometric series; Applications
[*] A sequence converging to \sqrt{2}: Newton's method
[*] The sequence of Fibonacci fractions
[/LIST]
[*] Limit Theorems for Sequences
[LIST]
[*] Limits of sums, products, and quotients
[*] Comparison theorems
[*] Location theorems
[*] Subsequences; Non-existence of limits
[*] Two common mistakes
[/LIST]
[*] The Completeness Property
[LIST]
[*] Introduction; Nested intervals
[*] Cluster points of sequences
[*] The Bolzano-Weierstrass theorem
[*] Cauchy sequences
[*] Completeness property for sets
[/LIST]
[*] Infinite Series
[LIST]
[*] Series and sequences
[*] Elementary convergence tests 
[*] The convergence of series with negative terms
[*] Convergence tests: ratio and n-th root tests
[*] The integral and asymptotic comparison tests
[*] Series with alternating signs: Cauchy's test
[*] Rearranging the terms of a series
[/LIST]
[*] Power Series
[LIST]
[*] Introduction; Radius of convergence
[*] Convergence at the endpoints; Abel summation
[*] Operations on power series: addition
[*] Multiplication of power series
[/LIST]
[*] Functions of One Variable
[LIST]
[*] Functions
[*] Algebraic operations on functions
[*] Some properties of functions
[*] Inverse functions
[*] The elementary functions
[/LIST]
[*] Local and Global Behavior
[LIST]
[*] Intervals; estimating functions
[*] Approximating functions
[*] Local behavior
[*] Local and global properties of functions
[/LIST]
[*] Continuity and Limits of Functions
[LIST]
[*] Continuous functions
[*] Limits of functions
[*] Limit theorems for functions
[*] Limits and continuous functions
[*] Continuity and sequences
[/LIST]
[*] The Intermediate Value Theorem
[LIST]
[*] The existence of zeros
[*] Applications of Bolzano's theorem
[*] Graphical continuity
[*] Inverse functions
[/LIST]
[*] Continuous Functions on Compact Intervals
[LIST]
[*] Compact intervals
[*] Bounded continuous functions
[*] Extremal points of continuous functions
[*] The mapping viewpoint
[*] Uniform continuity
[/LIST]
[*] Differentiation: Local Properties
[LIST]
[*] The derivative
[*] Differentiation formulas
[*] Derivatives and local properties
[/LIST]
[*] Differentiation: Global Properties
[LIST]
[*] The mean-value theorem
[*] Applications of the mean-value theorem
[*] Extension of the mean-value theorem
[*] L'Hospital's rule for indeterminate forms
[/LIST]
[*] Linearization and Convexity
[LIST]
[*] Linearization
[*] Applications to convexity
[/LIST]
[*] Taylor Approximation
[LIST]
[*] Taylor polynomials
[*] Taylor's theorem with Lagrange remainder
[*] Estimating error in Taylor approximation
[*] Taylor series
[/LIST]
[*] Integrability
[LIST]
[*] Introduction; Partitions
[*] Integrability
[*] Integrability of monotone and continuous functions
[*] Basic properties of integrable functions
[/LIST]
[*] The Riemann Integral
[LIST]
[*] Refinement of partitions
[*] Definition of the Riemann integral
[*] Riemann sums
[*] Basic properties of integrals
[*] The interval addition property
[*] Piecewise continuous and monotone functions
[/LIST]
[*] Derivatives and Integrals
[LIST]
[*] The first fundamental theorem of calculus
[*] Existence and uniqueness of antiderivatives
[*] Other relations between derivatives and integrals
[*] The logarithm and exponential functions
[*] Stirling's formula
[*] Growth rate of functions
[/LIST]
[*] Improper Integrals
[LIST]
[*] Basic definitions
[*] Comparison theorems
[*] The gamma function
[*] Absolute and conditional convergence
[/LIST]
[*] Sequences and Series of Functions
[LIST]
[*] Pointwise and uniform convergence
[*] Criteria for uniform convergence
[*] Continuity and uniform convergence
[*] Integration term-by-term
[*] Differentiation term-by-term
[*] Power series and analytic functions
[/LIST]
[*] Infinite Sets and the Lebesgue Integral
[LIST]
[*] Introduction; infinite sets
[*] Sets of measure zero
[*] Measure zero and Riemann-integrability
[*] Lebesgue integration
[/LIST]
[*] Continuous Functions on the Plane
[LIST]
[*] Introduction; Norms and distances in R^2
[*] Convergence of sequences
[*] Functions on R^2
[*] Continuous functions
[*] Limits and continuity
[*] Compact sets in R^2
[*] Continuous functions on compact sets in R^2
[/LIST]
[*] Point-sets in the Plane
[LIST]
[*] Closed sets in R^2
[*] Compactness theorem in R^2
[*] Open sets
[/LIST]
[*] Integrals with a Parameter
[LIST]
[*] Integrals depending on a parameter
[*] Differentiating under the integral sign
[*] Changing the order of integration
[/LIST]
[*] Differentiating Improper Integrals
[LIST]
[*] Introduction
[*] Pointwise vs. uniform convergence of integrals
[*] Continuity theorem for improper integrals
[*] Integrating and differentiating improper integrals
[*] Differentiating the Laplace transform
[/LIST]
[*] Appendix 
[LIST]
[*] Sets, Numbers, and Logic
[LIST]
[*] Sets and numbers
[*] If-then statements
[*] Contraposition and indirect proof
[*] Counterexamples
[*] Mathematical induction
[/LIST]
[*] Quantifiers and Negation
[LIST]
[*] Introduction; Quantifiers
[*] Negation
[*] Examples involving functions
[/LIST]
[*] Picard's Method
[LIST]
[*] Introduction
[*] The Picard iteration theorems
[*] Fixed points
[/LIST]
[*] Applications to Differential Equations
[LIST]
[*] Introduction
[*] Discreteness of the zeros
[*] Alternation of zeros
[*] Reduction to normal form
[*] Comparison theorems for zeros
[/LIST]
[*] Existence and Uniqueness of ODE Solutions
[LIST]
[*] Picard's method of successive approximations
[*] Local existence of solutions to y' = f(x,y)
[*] The uniqueness of solutions
[*] Extending the existence and uniqueness theorems
[/LIST]
[/LIST]
[*] Index
[/LIST]
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
Arthur is a really good teacher, but this is not my favorite intro book on analysis. Actually I don't know what is better though, off hand. this is maybe the hardest subject to teach.
 

FAQ: Introduction to Analysis by Mattuck

1. What is the purpose of "Introduction to Analysis" by Mattuck?

The purpose of "Introduction to Analysis" by Mattuck is to provide an introduction to the fundamental concepts and techniques of mathematical analysis, including limits, continuity, differentiation, integration, and series. It is intended for students who have completed a course in calculus and have a strong foundation in algebra and trigonometry.

2. What level of mathematics is required to understand "Introduction to Analysis"?

As stated in the title, "Introduction to Analysis" is an introductory course, therefore a strong foundation in calculus, algebra, and trigonometry is necessary. Some familiarity with basic concepts in real analysis and linear algebra may also be helpful.

3. What are the main topics covered in "Introduction to Analysis" by Mattuck?

The main topics covered in "Introduction to Analysis" include limits, continuity, differentiation, integration, and series. Other topics such as sequences, convergence, and theorems related to these topics are also covered. The book also includes exercises and examples to help students better understand the concepts.

4. Is "Introduction to Analysis" by Mattuck suitable for self-study?

Yes, "Introduction to Analysis" by Mattuck is suitable for self-study. The book is written in a clear and concise manner, making it easy for students to follow along. It also includes exercises and examples for students to practice on their own.

5. How does "Introduction to Analysis" by Mattuck differ from other books on the same subject?

One of the main differences between "Introduction to Analysis" by Mattuck and other books on the same subject is its focus on conceptual understanding rather than just problem-solving. The book also includes a variety of examples and exercises, making it suitable for both theoretical and applied mathematics students.

Similar threads

Replies
5
Views
5K
Replies
2
Views
7K
Replies
22
Views
16K
Replies
11
Views
2K
Replies
4
Views
6K
Replies
12
Views
11K
Replies
1
Views
6K
Replies
8
Views
8K
Back
Top