# Calculus Mathematical Analysis by Apostol

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1. Jan 22, 2013

### micromass

Staff Emeritus

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[LIST]
[*] The Real and Complex Number Systems
[LIST]
[*] Introduction
[*] The field axioms
[*] The order axioms
[*] Geometric representation of real numbers
[*] Intervals
[*] Integers
[*] Tim unique factorization theorem for integers
[*] Rational numbers
[*] Irrational numbers
[*] Upper bounds, maximum element, least upper bound (supremum)
[*] The completeness
[*] Some properties of the supremum
[*] Properties of the integer deduced from the completeness axiom
[*] The Archimedean property of the real-number system
[*] Rational numbers with finite decimal representation
[*] Finite decimal approximations to real mumbers
[*] Infinite decimal representation of real numbers
[*] Absolute values and the triangle inequality
[*] The Cauchy-Schwarz inequality
[*] Plus and minus infinity and the extended real number system R*
[*] Complex numbers
[*] Geometric representation of complex numbers
[*] The imaginary unit
[*] Absolute value of a complex number
[*] Impossibility of ordering the complex numbers
[*] Complex exponentials
[*] Further properties of complex exponentials
[*] The argument of a complex number
[*] Integral powers and roots of complex numbers
[*] Complex logarithms
[*] Complex powers
[*] Complex sines and cosines
[*] Infinity and the extended complex plane C*
[*] Exercises
[/LIST]
[*] Some Basic Notions of Set Theory
[LIST]
[*] Introduction
[*] Notations
[*] Ordered pairs
[*] Certesian product of two sets
[*] Relations and functions
[*] Further terminology concerning functions
[*] One-to-one functions and inverses
[*] Composite functions
[*] Sequences
[*] Similar (equinumerous) sets
[*] Finite and infinite sets
[*] Countable and uncountable sets
[*] Uncountability of the real-number system
[*] Set algebra
[*] Countable collection of countable sets
[*] Exercises
[/LIST]
[*] Elements of Point Set Topology
[LIST]
[*] Introduction
[*] Euclidean space $R^n$
[*] Open balls and open sets in $R^n$
[*] The structure of open sets in $R^1$
[*] Closed sets
[*] Closed sets and adherent points
[*] Bolzano-Weierstrass theorem
[*] The Cantor intersection theorem
[*] The Lindelof covering theorem
[*] The Heine-Borel covering theorem
[*] Compactness in $R^n$
[*] Metric spaces
[*] Point set topology in metric spaces
[*] Compact subsets of a metric space
[*] Boundary of a set
[*] Exercises
[/LIST]
[*] Limits and Continuity
[LIST]
[*] Introduction
[*] Convergent sequences in a metric space
[*] Cauchy sequences
[*] Complete metric spaces
[*] Limit of a function
[*] Limits of complex-valued functions
[*] Limits of vector-valued functions
[*] Continuous functions
[*] Continuity of composite functions
[*] Continuous complex-valued and vector-valued functions
[*] Examples of continuous functions
[*] Continuity and inverse images of open or closed sets
[*] Functions continuous on compact sets
[*] Topological mappings (homeomorphisms)
[*] Bolzano's theorem
[*] Connectedness
[*] Components of a metric space
[*] Arcwise connectedness
[*] Uniform continuity
[*] Uniform continuity and compact sets
[*] Fixed-point theorem for contractions
[*] Discontinuities of real-valued functions
[*] Monotonic functions
[*] Exercises
[/LIST]
[*] Derivatives
[LIST]
[*] Introduction
[*] Definition of derivative
[*] Derivatives and continuity
[*] Algebra of derivatives
[*] The chain rule
[*] One-sided derivatives and infinite derivatives
[*] Functions with nonzero derivative
[*] Zero derivatives and local extrema
[*] Rolle's theorem
[*] The Mean-Value Theorem for derivatives
[*] Intermediate-value theorem for derivatives
[*] Taylor's formula with remainder
[*] Derivatives of vector-valued functions
[*] Partial derivatives
[*] Differentiation of functions of a complex variable
[*] The Cauchy-Riemann equations
[*] Exercises
[/LIST]
[*] Functions of Bounded Variation and Rectifiable Curves
[LIST]
[*] Introduction
[*] Properties of monotonic functions
[*] Functions of bounded variation
[*] Total variation
[*] Additive property of total variation
[*] Total variation on [a,x] as a function of x
[*] Functions of bounded variation expressed as the difference of bounded functions
[*] Continuous functions of bounded variation
[*] Curves and paths
[*] Rectifiable paths and arc length
[*] Additive and continuity properties of arc length
[*] Equivalence of paths. Change of parameter
[*] Exercises
[/LIST]
[*] The Riemann-Stieltjes Integral
[LIST]
[*] Introduction
[*] Notation
[*] The definition of the Riemann-Stieltjes integral
[*] Linear properties
[*] Integration by parts
[*] Change of variable in a Riemann-Stieltjes integral
[*] Reduction to a Riemann integral
[*] Step functions as integrators
[*] Reduction of a Riemann-Stieltjes integral to a finite sum
[*] Euler's summation formula
[*] Monotonically increasing integrators. Upper and lower integrals
[*] Additive and linearity properties of upper and lower integrals
[*] Riemann's condition
[*] Comparison theorems
[*] Integrators of bounded variation
[*] Sufficient conditions for existence of Riemann-Stieltjes integrals
[*] Necessary conditions for existence of Riemann-Stieltjes integrals
[*] Mean Value Theorems for Riemann-Stieltjes integrals
[*] The integral as a function of the interval
[*] Second fundamental theorem of integral calculus
[*] Change of variable in a Riemann integral
[*] Second Mean-Value Theorem for Riemann integrals
[*] Riemann-Stieltjes integrals depending on a parameter
[*] Differentiation under the integral sign
[*] Interchanging the order of integration
[*] Lebesgue's criterion for existence of Riemann integrals
[*] Complex-valued Riemann-Stieltjes integrals
[*] Exercises
[/LIST]
[*] Infinite Series and Infinite Products
[LIST]
[*] Introduction
[*] Convergent and divergent sequences of complex numbers
[*] Limit superior and limit inferior of a real-valued sequence
[*] Monotonic sequences of real numbers
[*] Infinite series
[*] Inserting and removing parentheses
[*] Alternating series
[*] Absolute and conditional convergence
[*] Real and imaginary parts of a complex series
[*] Tests for convergence of series with positive terms
[*] The geometric series
[*] The integral test
[*] The big oh and little oh notation
[*] The ratio test and the root test
[*] Dirichlet's test and Abel's test
[*] Partial sums of the geometric series $\sum z^n$ on the unit circle |z|=1
[*] Rearrangements of series
[*] Riemann's theorem on conditionally convergent series
[*] Subseries
[*] Double sequences
[*] Double series
[*] Rearrangement theorem for double series
[*] A sufficient condition for equality of iterated series
[*] Multiplication of series
[*] Cesaro summability
[*] Infinite products
[*] Euler's product for the Riemann zeta function
[*] Exercises
[/LIST]
[*] Sequences of Functions
[LIST]
[*] Pointwise convergence of sequences of functions
[*] Examples of sequences of real-values functions
[*] Definition of uniform convergence
[*] Uniform concergence and continuity
[*] The Cauchy condition for uniform convergene
[*] Uniform convergence of infinite series of functions
[*] A space-filling curve
[*] Uniform convergence and Riemann-Stieltjes integration
[*] Nonuniformly convergent sequences that can be integrated term by
term
[*] Uniform convergence and differentiation
[*] Sufficient conditions for uniform convergence of a series
[*] Uniform convergence and double sequences
[*] Mean convergence
[*] Power series
[*] Multiplication of power series
[*] The substitution theorem
[*] Reciprocal of a power series
[*] Real power series
[*] The Taylor's series generated by a function
[*] Bernstein's theorem
[*] The binomial series
[*] Abel's limit theorem
[*] Tauber's theorem
[*] Exercises
[/LIST]
[*] The Lebesgue Integral
[LIST]
[*] Introduction
[*] The integral of a step function
[*] Monotonic sequences of step function
[*] Upper function and their integrals
[*] Riemann-integrable functions as eexamples of upper functions
[*] The class of Lebesgue-integrable functions on a general interval
[*] Basic properties of the Lebesgue integral
[*] Lebesgue integration and sets of measure zero
[*] The Levi monotone convergence theorems
[*] Applicatiom of Lebesgue's dominated convergence theorem
[*] Lebesgue integrals on unbounded intervals as limits of integrals on bounded intervals
[*] Improper Riemann integrals
[*] Measurable functions
[*] Continuity of functions defined by Lebesgue integrals
[*] Differentiation under the integral sign
[*] Interchanging the order of integration
[*] Measurable sets on the real line
[*] The Lebesgue integral over arbitrary subsets of R
[*] Lebesgue integrals of complex-valued functions
[*] Inner products and norms
[*] The set $L^2(I)$ of square-integrable functions
[*] The set $L^2(I)$ as a semimetric space
[*] A convergence theorem for series of functions in $L^2(I)$
[*] The Riesz-Fischer theorem
[*] Exercises
[/LIST]
[*] Fourier Series and Fourier Integrals
[LIST]
[*] Introduction
[*] Orthogonal systems of functions
[*] The theorem on best approximation
[*] The Fourier series of a function relative to an orthonormal system
[*] Properties of the Fourier coefficients
[*] The Riesz-Fischer theorem
[*] The convergence and representation problems for trigonometric series
[*] The Riemann-Lebesgue lemma
[*] The Dirichlet integrals
[*] An integral representation for the partial sums of a Fourier series
[*] Riemann's localization theorem
[*] Sufficient conditions for convergence of a Fourier series at a particular point
[*] Cesaro summability of Fourier series
[*] Consequence, of Fejer's theorem
[*] The Weierstrass approximation theorem
[*] Other forms of Fourier series
[*] The Fourier integral theorem
[*] The exponential form of the Fourier integral theorem
[*] Integral transforms
[*] Convolutions
[*] The convolution theorem for Fourier transforms
[*] The Poisson summation formula
[*] Exercises
[/LIST]
[*] Multivariable Differential Calculus
[LIST]
[*] Introduction
[*] The directional derivative
[*] Directional derivatives and continuity
[*] The total derivative
[*] The total derivative expressed in terms of partial derivatives
[*] An application to complex-values functions
[*] The matrix of linear function
[*] The Jacobian matrix
[*] The chain rule
[*] Matrix form of the chain rule
[*] The Mean-Value Theorem for differentiable functions
[*] A sufficient condition for differentiability
[*] A sufficient condition for equality of mixed partial derivatives
[*] Taylor's formula for functions from $R^n$ to $R^1$
[*] Exercises
[/LIST]
[*] Implicit Functions and Extremum Problems
[LIST]
[*] Introduction
[*] Functions with nonzero Jacobian determinant
[*] The inverse function theorem
[*] The implicit function theorem
[*] Extrema of real-valued functions of one variable
[*] Extrema of real-valued functions of several variables
[*] Extremum problems with side conditions
[*] Exercises
[/LIST]
[*] Multiple Riemann Integrals
[LIST]
[*] Introduction
[*] The measure of a bounded interval in $R^n$
[*] The Riemann integral of a bounded function defined on a compact interval in $R^n$
[*] Sets of measure zero and Lebesgue's criterion for existence of a multiple Riemann integral
[*] Evaluation of a multiple integral by iterated integration
[*] Jordan-measurable sets in $R^n$
[*] Multiple integration over Jordan-measurable sets
[*] Additive property of the Riemann integral
[*] Mean-Value Theorem for multiple integrals
[*] Exercises
[/LIST]
[*] Multiple Lebesgue Integrals
[LIST]
[*] Introduction
[*] Step functions and their integrals
[*] Upper functions end Lebesgue-integrable functions
[*] Measurable functions and measurable sets in $R^n$
[*] Fubini's reduction theorem for double integrals
[*] Some properties of sets of measure zero
[*] Fubini's reduction theorem for double integrals
[*] The Tonelli-Hobson test for integrability
[*] Coordinate transformations
[*] The transformation formula for multiple integrals
[*] Proof of the transformation formula for linear coordinate transformations
[*] Proof of the transformation formula for the characteristic function of a compact cube
[*] Completion of the proof of the transformation formula
[*] Exercises
[/LIST]
[*] Cauchy's Theorem and the Residue Calculus
[LIST]
[*] Analytic functions
[*] Paths and curves in the complex plane
[*] Contour integrals
[*] The integral along a circular path as a function of the radius
[*] Cauchy's integral theorem for a circle
[*] Homotopic curves
[*] Invariance of contour integrals under homotopy
[*] General form of Cauchy's integral theorem
[*] Cauchy's integral formula
[*] The winding number of a circuit with respect to a point
[*] The unboundedness of the set of points with winding number zero
[*] Analytic functions defined by contour integrals
[*] Power-series expansions for analytic functions
[*] Cauchy's inequalities. Liouville's theorem
[*] Isolation of the zeros of an analytic function
[*] The identity theorem for analytic functions
[*] The maximum and minimum modulus of an analytic function
[*] The open mapping theorem
[*] Laurent expansions for functions analytic in an annulus
[*] Isolated singularities
[*] The residue of a function at an isolated singular point
[*] The Cauchy residue theorem
[*] Counting zeros and poles in a region
[*] Evaluation of real-valued integrals by means of residues
[*] Evaluation of Gauss's sum by residue calculus
[*] Application of the residue theorem to the inversion formula for Laplace transforms
[*] Conformal mappings
[*] Exercises
[/LIST]
[*] Index of Special Symbols
[*] Index
[/LIST]

Last edited by a moderator: May 6, 2017