Course In Analysis, A - Volume I: Introductory Calculus, Analysis Of Functions Of One Real Variable
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.
- Paperback | 768 pages
- 165.1 x 246.38 x 38.1mm | 1,292.74g
- 22 Nov 2015
- World Scientific Publishing Co Pte Ltd
- Singapore, Singapore
Volume 1 covers the content of two typical modules in undergraduate course for mathematics: Part 1: Introductory calculus Part 2: Analysis of functions of one variable These two parts are divided into 32 chapters. Part 1 begins with an overview of a set of real numbers: rational and irrational. This is accompanied by a discussion of arithmetic rules, inequalities, absolute values. Doing this, the authors, in a clever way, introduce elements of a set theory using Cantor's approach. This is followed by the definition of a function and discussion of operations on functions. From this they move to differential and integral calculus and their applications. Part 2 contains rigorous proofs of theorems from Part 1. In particular, they introduce the order structure on a real line using axioms of an ordered field. It is worth mentioning that Part 2 contains elements of topology obviously on real line and a good insight on convex functions. The convex functions relate Analysis to linear spaces equipped with norm. Finally, to close Part 2 the authors supply a number of Appendices, among them Appendices on logic, set theory and Peano axioms. Remarkably, Volume 1 contains 360 problems with complete solutions.
Table of contents
Introductory 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 Integration; Analysis 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; Point Sets in Continuous Functions; Differentiation; Applications of the Derivative; Convex Functions and some Norms on n; Uniform Convergence and Interchanging Limits; The Riemann Integral; The Fundamental Theorem of Calculus; A First Encounter with Differential Equations; Improper Integrals and the GAMMA-Function; Power Series and Taylor Series; Infinite Products and the Gauss Integral; More on the GAMMA-Function; Selected Topics on Functions of a Real Variable;