Krantz Steven G. A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis

Birkhauser, 2003. — 217 p. — ISBN10: 081764329X, ISBN13: 978-0817643294This concise, well-written handbook provides a distillation of real variable theory with a particular focus on the subject's significant applications to differential equations and Fourier analysis. Ample examples and brief explanations-with very few proofs and little axiomatic machinery-are used to highlight all the major results of real analysis, from the basics of sequences and series to the more advanced concepts of Taylor and Fourier series, Baire Category, and the Weierstrass Approximation Theorem. Replete with realistic, meaningful applications to differential equations, boundary value problems, and Fourier analysis, this unique work is a practical, hands-on manual of real analysis that is ideal for physicists, engineers, economists, and others who wish to use the fruits of real analysis but who do not necessarily have the time to appreciate all of the theory. Valuable as a comprehensive reference, a study guide for students, or a quick review, "A Handbook of Real Variables" will benefit a wide audience.Basics
Sets
Operations on Sets
Functions
Operations on Functions
Number Systems
The Real Numbers
Countable and Uncountable Sets
Sequences
Introduction to Sequences
The Definition and Convergence
The Cauchy Criterion
Monotonicity
The Pinching Principle
Subsequences
The Bolzano-Weierstrass Theorem ,
Limsup and Liminf
Some Special Sequences
Series
Introduction to Series
The Definition and Convergence
Partial Sums
Elementary Convergence Tests
The Comparison Test ,
The Cauchy Condensation Test
Geometric Series
The Root Test
The Ratio Test
Root and Ratio Tests for Divergence
Advanced Convergence Tests
Summation by Parts
Abel's Test
Absolute and Conditional Convergence
Rearrangements of Series
Some Particular Series
The Series for e
Other Representations for e
Sums of Powers
Operations on Series
Sums and Scalar Products of Series
Products of Series
The Cauchy Product
The Topology of the Real Line
Open and Closed Sets
OpenSets
ClosedSets
Characterization of Open and Closed Sets
In Terms of Sequences
Further Properties of Open and Closed Sets
Other Distinguished Points
Interior Points and Isolated Points
Accumulation Points
Bounded Sets
Compact SetsThe Heine-Borel Theorem, Part I
The Heine-Borel Theorem, Part II
The Cantor Set
Connected and Disconnected Sets
Connectivity
PerfectSets
Limits and the Continuity of Functions
Definitions and Basic Properties
Limits
A Limit that Does Not Exist
Uniqueness of Limits
Properties of Limits
Characterization of Limits Using Sequences
Continuous Functions
Continuity at a Point
The Topological Approach to Continuity
Topological Properties and Continuity
The Image of a Function
Uniform Continuity
Continuity and Connectedness
The Intermediate Value Property
Classifying Discontinuities and Monotonicity
Left and Right Limits
Types of Discontinuities
Monotonic Functions
The Derivative
The Concept of Derivative
TheDefinition
Properties of the Derivative
The Weierstrass Nowhere Differentiable Function
The Chain Rule
The Mean Value Theorem and Applications
Local MAXIMA and Minima
Fermat'sTest
Darboux's Theorem
The Mean Value Theorem
Examples of the Mean Value Theorem
Further Results on the Theory of Differentiation
UHopital'sRule
The Derivative of an Inverse Function
Higher-Order Derivatives
Continuous Differentiability
The Integral
The Concept of Integral
Partitions
Refinements of Partitions
Existence of the Riemann Integral
Integrability of Continuous Functions
Properties of the Riemann Integral
Existence Theorems
Inequalities for Integrals
Preservation of Integrable Functions under Composition
The Fundamental Theorem of Calculus
Further Results on the Riemann Integral
The Riemann-Stieltjes Integral
Riemann's Lemma
Advanced Results on Integration Theory
Existence of the Riemann-Stieltjes Integral
ntegration by Parts
Linearity Properties
Bounded Variation
Sequences and Series of Functions
Partial Sums and Pointwise Convergence
Sequences of Functions
Uniform Convergence
More on Uniform Convergence
Commutation of Limits
The Uniform Cauchy Condition
Limits of Derivatives
Series of Functions
Series and Partial Sums
Uniform Convergence of a Series
The Weierstrass M-Test
The Weierstrass Approximation Theorem
Weierstrass's Main Result Ill
Some Special Functions
Power Series
Convergence
Interval of Convergence
Real Analytic Functions
Multiplication of Real Analytic Functions
Division of Real Analytic Functions
More on Power Series: Convergence Issues
The Hadamard Formula
The Derived Series
Formula for the Coefficients of a Power Series
Taylor's Expansion
The Exponential and Trigonometric Functions
The Series Definition
The Trigonometric Functions
Euler's Formula
The Trigonometric Functions
Logarithms and Powers of Real Numbers
The Logarithmic Function
Characterization of the Logarithm
The Gamma Function and Stirling's Formula
Stirling's Formula
An Introduction to Fourier Series
Trigonometric Series
Formula for the Fourier Coefficients
Bessel's Inequality
The Dirichlet Kernel
Advanced Topics
Metric Spaces
The Concept of a Metric
Examples of Metric Spaces
Convergence in a Metric Space
The Cauchy Criterion
Completeness
Isolated Points
Topology in a Metric Space
Balls in a Metric Space
Accumulation Points
Compactness
The Baire Category Theorem
Density
Closure
Baire's Theorem
The Ascoli-Arzela Theorem
Equicontinuity
Equiboundedness
The Ascoli-Arzela Theorem
Differential Equations
Picard's Existence and Uniqueness Theorem
The Form of a Differential Equation
Picard's Iteration Technique
Some Illustrative Examples
Estimation of the Picard Iterates
The Method of Characteristics
Power Series Methods
Fourier Analytic Methods
Remarks on Different Fourier Notations
The Dirichlet Problem on the Disc
The Poisson Integral
The Wave Equation
Glossary of Terms from Real Variable Theory
List of Notation
Guide to the Literature