Jacob N., Evans K.P. Course in Analysis. Volume I: Introductory Calculus, Analysis of Functions of One Real Variable

World Scientific Publishing Co, 2016. — xxiv, 744 p. — ISBN: 978-981-4689-08-3,978-981-4689-09-0.
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Part 1 begins with an overview of properties of the real numbers and starts to introduce the notions of set theory. The absolute value and in particular inequalities are considered in great detail before functions and their basic properties are handled. From this the authors move to differential and integral calculus. Many examples are discussed. Proofs not depending on a deeper understanding of the completeness of the real numbers are provided. As a typical calculus module, this part is thought as an interface from school to university analysis.
Part 2 returns to the structure of the real numbers, most of all to the problem of their completeness which is discussed in great depth. Once the completeness of the real line is settled the authors revisit the main results of Part 1 and provide complete proofs. Moreover they develop differential and integral calculus on a rigorous basis much further by discussing uniform convergence and the interchanging of limits, infinite series (including Taylor series) and infinite products, improper integrals and the gamma function. In addition they discussed in more detail as usual monotone and convex functions.
Finally, the authors supply a number of Appendices, among them Appendices on basic mathematical logic, more on set theory, the Peano axioms and mathematical induction, and on further discussions of the completeness of the real numbers.
Remarkably, Volume I contains ca. 360 problems with complete, detailed solutions.Acknowledgements and Apologies
List of Symbols
The Greek AlphabetIntroductory Calculus
Numbers - Revision
The Absolute Value, Inequalities and Intervals
Mathematical Induction
Functions and Mappings
Functions and Mappings Continued
Derivatives
Derivatives Continued
The Derivative as a Tool to Investigate Functions
The Exponential and Logarithmic Functions
Trigonometric Functions and Their Inverses
Investigating Functions
Integrating Functions
Rules for IntegrationAnalysis in One Dimension
Problems with the Real Line
Sequences and their Limits
A First Encounter with Series
The Completeness of the Real Numbers
Convergence Criteria for Series, b-adic Fractions 243
Point Sets in R
Continuous Functions
Differentiation
Applications of the Derivative
Convex Functions and some Norms on Rⁿ
Uniform Convergence and Interchanging Limits
The Riemann Integral
The Fundamental Theorem of Calculus
A First Encounter with Differential Equations
Improper Integrals and the Γ-Function
Power Series and Taylor Series
Infinite Products and the Gauss Integral
More on the Γ-Function
Selected Topics on Functions of a Real VariableAppendices
Elementary Aspects of Mathematical Logic
Sets and Mappings. A Collection of Formulae
The Peano Axioms
Results from Elementary Geometry
Trigonometric and Hyperbolic Functions
More on the Completeness of R
Limes Superior and Limes Inferior
Connected Sets in R
Solutions to Problems of Part 1
Solutions to Problems of Part 2
References
Mathematicians Contributing to Analysis
Subject Index