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Analysis Introduction to Analysis by Mattuck

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  1. Jan 24, 2013 #1

    micromass

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    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award


    Table of Contents:
    Code (Text):

    [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: May 6, 2017
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  3. Jan 24, 2013 #2

    mathwonk

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    Science Advisor
    Homework Helper

    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.
     
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