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Calculus Mathematical Analysis by Apostol

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  1. Jan 22, 2013 #1


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    Table of Contents:
    Code (Text):

    [*] The Real and Complex Number Systems
    [*] 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
    [*] Some Basic Notions of Set Theory
    [*] 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
    [*] Elements of Point Set Topology
    [*] Introduction
    [*] Euclidean space [itex]R^n[/itex]
    [*] Open balls and open sets in [itex]R^n[/itex]
    [*] The structure of open sets in [itex]R^1[/itex]
    [*] Closed sets
    [*] Adherent points. Accumulation points
    [*] Closed sets and adherent points
    [*] Bolzano-Weierstrass theorem
    [*] The Cantor intersection theorem
    [*] The Lindelof covering theorem
    [*] The Heine-Borel covering theorem
    [*] Compactness in [itex]R^n[/itex]
    [*] Metric spaces
    [*] Point set topology in metric spaces
    [*] Compact subsets of a metric space
    [*] Boundary of a set
    [*] Exercises
    [*] Limits and Continuity
    [*] 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
    [*] Derivatives
    [*] 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
    [*] Functions of Bounded Variation and Rectifiable Curves
    [*] 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
    [*] The Riemann-Stieltjes Integral
    [*] 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
    [*] Infinite Series and Infinite Products
    [*] 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 [itex]\sum z^n[/itex] 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
    [*] Sequences of Functions
    [*] 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
    [*] 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
    [*] The Lebesgue Integral
    [*] 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 [itex]L^2(I)[/itex] of square-integrable functions
    [*] The set [itex]L^2(I)[/itex] as a semimetric space
    [*] A convergence theorem for series of functions in [itex]L^2(I)[/itex]
    [*] The Riesz-Fischer theorem
    [*] Exercises
    [*] Fourier Series and Fourier Integrals
    [*] 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
    [*] Multivariable Differential Calculus
    [*] 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 [itex]R^n[/itex] to [itex]R^1[/itex]
    [*] Exercises
    [*] Implicit Functions and Extremum Problems
    [*] 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
    [*] Multiple Riemann Integrals
    [*] Introduction
    [*] The measure of a bounded interval in [itex]R^n[/itex]
    [*] The Riemann integral of a bounded function defined on a compact interval in [itex]R^n[/itex]
    [*] 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 [itex]R^n[/itex]
    [*] Multiple integration over Jordan-measurable sets
    [*] Additive property of the Riemann integral
    [*] Mean-Value Theorem for multiple integrals
    [*] Exercises
    [*] Multiple Lebesgue Integrals
    [*] Introduction
    [*] Step functions and their integrals
    [*] Upper functions end Lebesgue-integrable functions
    [*] Measurable functions and measurable sets in [itex]R^n[/itex]
    [*] 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
    [*] Cauchy's Theorem and the Residue Calculus
    [*] 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
    [*] Index of Special Symbols
    [*] Index
    Last edited by a moderator: May 6, 2017
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