A Course in Analysis Volume I Introductory Calculus Analysis of Functions of One Real Variable 1st Edition by Niels Jacob, Kristian P Evans ISBN 9789814689083 9814689084

A Course in Analysis Volume I Introductory Calculus Analysis of Functions of One Real Variable 1st Edition by Niels Jacob, Kristian P Evans- Ebook PDF Instant Download/Delivery: 9789814689083, 9814689084

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Product details:

ISBN 10:  9814689084

ISBN 13: 9789814689083

Author:  Niels Jacob, Kristian P Evans

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.

Table of contents:

 Part 1: Introductory Calculus 

1 Numbers Revision 

2 The Absolute Value, Inequalities and Intervals 

3 Mathematical Induction 

4 Functions and Mappings 

5 Functions and Mappings Continued 

6 Derivatives 

7 Derivatives Continued 

8 The Derivative as a Tool to Investigate Functions 

9 The Exponential and Logarithmic Functions 

10 Trigonometric Functions and Their Inverses 

11 Investigating Functions 

12 Integrating Functions

13 Rules for Integration 

Part 2: Analysis in One Dimension 

14 Problems with the Real Line 

15 Sequences and their Limits 

16 A First Encounter with Series 

17 The Completeness of the Real Numbers 

18 Convergence Criteria for Series, b-adic Fractions 

19 Point Sets in R 

20 Continuous Functions 

21 Differentiation 

22 Applications of the Derivative 

23 Convex Functions and some Norms on R” 

24 Uniform Convergence and Interchanging Limits 

25 The Riemann Integral 

26 The Fundamental Theorem of Calculus 

27 A First Encounter with Differential Equations 

28 Improper Integrals and the I-Function 

29 Power Series and Taylor Series 

30 Infinite Products and the Gauss Integral 

31 More on the I-Function 

32 Selected Topics on Functions of a Real Variable

Appendices 

Appendix I: Elementary Aspects of Mathematical Logic 

Appendix II: Sets and Mappings. A Collection of Formulae

Appendix III: The Peano Axioms 

Appendix IV: Results from Elementary Geometry 

Appendix V: Trigonometric and Hyperbolic Functions 

Appendix VI: More on the Completeness of R 

Appendix VII: Limes Superior and Limes Inferior 

Appendix VIII: Connected Sets in R 

Solutions to Problems of Part 1 

Solutions to Problems of Part 2.

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Tags: Niels Jacob, Kristian P Evans, Course, Analysis, Introductory, Calculus, Functions, Real Variable