# Introduction to Analysis by Mattuck

• Analysis

• Total voters
2

## Main Question or Discussion Point

Code:
[LIST]
[*] Preface
[*] Real Numbers and Monotone Sequences
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[*] 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
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[*] Estimations and Approximations
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[*] Introduction; Inequalities
[*] Estimations
[*] Proving boundedness
[*] Absolute values; estimating size
[*] Approximations
[*] The terminology "for n large"
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[*] The Limit of a Sequence
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[*] 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
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[*] The Error Term
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[*] The error term
[*] The error in the geometric series; Applications
[*] A sequence converging to \sqrt{2}: Newton's method
[*] The sequence of Fibonacci fractions
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[*] Limit Theorems for Sequences
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[*] Limits of sums, products, and quotients
[*] Comparison theorems
[*] Location theorems
[*] Subsequences; Non-existence of limits
[*] Two common mistakes
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[*] The Completeness Property
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[*] Introduction; Nested intervals
[*] Cluster points of sequences
[*] The Bolzano-Weierstrass theorem
[*] Cauchy sequences
[*] Completeness property for sets
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[*] Infinite Series
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[*] 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
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[*] Power Series
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[*] Convergence at the endpoints; Abel summation
[*] Operations on power series: addition
[*] Multiplication of power series
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[*] Functions of One Variable
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[*] Functions
[*] Algebraic operations on functions
[*] Some properties of functions
[*] Inverse functions
[*] The elementary functions
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[*] Local and Global Behavior
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[*] Intervals; estimating functions
[*] Approximating functions
[*] Local behavior
[*] Local and global properties of functions
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[*] Continuity and Limits of Functions
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[*] Continuous functions
[*] Limits of functions
[*] Limit theorems for functions
[*] Limits and continuous functions
[*] Continuity and sequences
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[*] The Intermediate Value Theorem
[LIST]
[*] The existence of zeros
[*] Applications of Bolzano's theorem
[*] Graphical continuity
[*] Inverse functions
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[*] Continuous Functions on Compact Intervals
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[*] Compact intervals
[*] Bounded continuous functions
[*] Extremal points of continuous functions
[*] The mapping viewpoint
[*] Uniform continuity
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[*] Differentiation: Local Properties
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[*] The derivative
[*] Differentiation formulas
[*] Derivatives and local properties
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[*] Differentiation: Global Properties
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[*] The mean-value theorem
[*] Applications of the mean-value theorem
[*] Extension of the mean-value theorem
[*] L'Hospital's rule for indeterminate forms
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[*] Linearization and Convexity
[LIST]
[*] Linearization
[*] Applications to convexity
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[*] Taylor Approximation
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[*] Taylor polynomials
[*] Taylor's theorem with Lagrange remainder
[*] Estimating error in Taylor approximation
[*] Taylor series
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[*] Integrability
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[*] Introduction; Partitions
[*] Integrability
[*] Integrability of monotone and continuous functions
[*] Basic properties of integrable functions
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[*] The Riemann Integral
[LIST]
[*] Refinement of partitions
[*] Definition of the Riemann integral
[*] Riemann sums
[*] Basic properties of integrals
[*] Piecewise continuous and monotone functions
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[*] 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
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[*] Improper Integrals
[LIST]
[*] Basic definitions
[*] Comparison theorems
[*] The gamma function
[*] Absolute and conditional convergence
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[*] Sequences and Series of Functions
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[*] 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
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[*] Infinite Sets and the Lebesgue Integral
[LIST]
[*] Introduction; infinite sets
[*] Sets of measure zero
[*] Measure zero and Riemann-integrability
[*] Lebesgue integration
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[*] Continuous Functions on the Plane
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[*] 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
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[*] Point-sets in the Plane
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[*] Closed sets in R^2
[*] Compactness theorem in R^2
[*] Open sets
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[*] Integrals with a Parameter
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[*] Integrals depending on a parameter
[*] Differentiating under the integral sign
[*] Changing the order of integration
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[*] Differentiating Improper Integrals
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[*] Introduction
[*] Pointwise vs. uniform convergence of integrals
[*] Continuity theorem for improper integrals
[*] Integrating and differentiating improper integrals
[*] Differentiating the Laplace transform
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[*] Appendix
[LIST]
[*] Sets, Numbers, and Logic
[LIST]
[*] Sets and numbers
[*] If-then statements
[*] Contraposition and indirect proof
[*] Counterexamples
[*] Mathematical induction
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[*] 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
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[*] Existence and Uniqueness of ODE Solutions
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[*] 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
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[/LIST]
[*] Index
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