A Course in Analysis

Volume I: Introductory Calculus, Analysis of Functions of One Real Variable

Nonfiction, Science & Nature, Mathematics, Functional Analysis, Calculus
Cover of the book A Course in Analysis by Niels Jacob, Kristian P Evans, World Scientific Publishing Company
View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart
Author: Niels Jacob, Kristian P Evans ISBN: 9789814689106
Publisher: World Scientific Publishing Company Publication: August 18, 2015
Imprint: WSPC Language: English
Author: Niels Jacob, Kristian P Evans
ISBN: 9789814689106
Publisher: World Scientific Publishing Company
Publication: August 18, 2015
Imprint: WSPC
Language: English

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.

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 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 Variable

Readership: Undergraduate students in mathematics.

View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart

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.

Contents:

Readership: Undergraduate students in mathematics.

More books from World Scientific Publishing Company

Cover of the book Women's Cancers by Niels Jacob, Kristian P Evans
Cover of the book An Introduction to Hydraulics of Fine Sediment Transport by Niels Jacob, Kristian P Evans
Cover of the book Race for Sustainability by Niels Jacob, Kristian P Evans
Cover of the book An Introduction to Bioenergy by Niels Jacob, Kristian P Evans
Cover of the book Method of Lines Analysis of Turing Models by Niels Jacob, Kristian P Evans
Cover of the book The Role of Theory in Computer Science by Niels Jacob, Kristian P Evans
Cover of the book Computational Intelligence and Its Applications by Niels Jacob, Kristian P Evans
Cover of the book Introduction to Stochastic Calculus with Applications by Niels Jacob, Kristian P Evans
Cover of the book Charles Darwin in Cambridge by Niels Jacob, Kristian P Evans
Cover of the book Computational Modeling of Gene Regulatory Networks — A Primer by Niels Jacob, Kristian P Evans
Cover of the book Thermodynamics by Niels Jacob, Kristian P Evans
Cover of the book Global Financial Crisis and Challenges for China by Niels Jacob, Kristian P Evans
Cover of the book Recent Progress in Orbital-free Density Functional Theory by Niels Jacob, Kristian P Evans
Cover of the book Soft, Hard, and Hybrid Janus Structures by Niels Jacob, Kristian P Evans
Cover of the book Peyresq Lectures on Nonlinear Phenomena by Niels Jacob, Kristian P Evans
We use our own "cookies" and third party cookies to improve services and to see statistical information. By using this website, you agree to our Privacy Policy