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Main Question or Discussion Point
- Author: Arthur Mattuck
- Title: Introduction to Analysis
- Amazon Link: https://www.amazon.com/dp/0130811327/?tag=pfamazon01-20
- Prerequisities:
- Level: Undergrad
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]
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