A First Course in Mathematical Analysis

Description

Description This straightforward course based on the idea of a limit is intended for students who have acquired a working knowledge of the calculus and are ready for a more systematic treatment which also brings in other limiting processes, such as the summation of infinite series and the expansion of trigonometric functions as power series. Particular attention is given to clarity of exposition and the logical development of the subject matter. A large number of examples is included, with hints for the solution of many of them. Reviews & endorsements 'Books of this quality are rare enough to be hailed enthusiastically … Essentially an introductory book for the mathematics specialist. But it is so fresh in conception and so lucid in style that it will appeal to anyone who has a general interest in mathematics.' The Times Educational Supplement 'This is an excellent book … If I were teaching a course for honours students of the type described, this book would rank high as a possible choice of text.' Canadian Mathematical Bulletin ' … it is a pleasure to be able to welcome a book on analysis written by an author who has a sense of style and who avoids the excessive use of symbolism which can make the subject unnecessarily difficult for the student.' Proceedings of the Edinburgh Mathematical Society Title Copyright © Cambridge University Press 1962 ISBN 0 521 04381 6 ISBN 0 521 29468 1 Contents Preface 1. Numbers 1.1. The branches of pure mathematics 1.2. The scope of mathematical analysis 1.3. Numbers Exercises 1 (a) 1.4. Irrational numbers Exercises 1 (b) 1.5. Cuts of the rationals Exercises 1 (c) 1.6. The field of real numbers 1.7. Bounded sets of numbers 1.8. The least upper bound (supremum) Exercises 1 (d) 1.9. Complex numbers Exercises 1 (e) 1.10. Modulus and phase Exercises 1 (f ) 2. Sequences 2.1. Sequen 2.2. Null sequences 2.3. Sequence tending to a limit Exercises 2 (a) 2.4. Sequences tending to infinity Exercises 2 (b) 2.5. Sum and product of sequences 2.6. Increasing sequences 2.7. An important sequence a^n Exercises 2 (d) 2.8. Recurrence relations Exercises 2 (e) 2.9. Infinite series 2.10. The geometric series \Sigma x^n 2.11. The series \Sigma n^-k Exercises 2 (f) 2.12. Properties of infinite series Exercises 2 (g) 3. Continuous Functions 3.1. Functions 3.2. Behaviour of f(x) for large values of x 3.3. Sketching of curves Exercises 3 (a) 3.4. Continuous functions 3.5. Examples of continuous and discontinuous functions Exercises 3 (b) 3.6. The intermediate-value property 3.7. Bounds of a continuous function 3.8. Uniform continuity 3.9. Inverse functions Exercises 3 (c) 4. The Differential Calculus 4.1. The derivative Exercises 4 (a) 4.2. Differentiation of sum, product, et 4.3. Differentiation of elementary functions Exercises 4 (b) 4.4. Repeated differentiation Exercises 4 (c) 4.5. The sign of f'(x) Exercises 4 (d) 4.6. The mean value theorem Exercises 4 (e) 4.7. Maxima and minima 4.8. Approximation by polynomials. Taylor's theorem 4.9. Indeterminate forms Exercises 4 (f) 5. Infinite Series 5.1. Series of positive terms Exercises 5 (a) 5.2. Series of positive and negative terms Exercises 5 (b) 5.3. Conditional convergence 5.4. Series of complex terms Exercises 5 (c) 5.5. Power series 5.6. The circle of convergence of a power series Exercises 5 (d) 5.7. Multiplication of series 5.8. Taylor's series Exercises 5 (e) 6. The Special Functions of Analysis 6.1. The special functions of analysis 6.2. The exponential function 6.3. Repeated limits 6.4. Rate of increase of exp x 6.5. exp x as a power Exercises 6 (a) 6.6. The logarithmic function Exercises 6 (b) 6.7. Trigonometric functions 6.8. Exponential and trigonometric functions Exercises 6 (c) 6.9. The inverse trigonometric functions 6.10. The hyperbolic functions and their inverses Exercises 6 (d) 7. The Integral Calculus 7.1. Area and the integral 7.2. The upper and lower integrals 7.3. The integral as a limit 7.4. Continuous or monotonic functions are integrable 7.5. Properties of the integral 7.6. Integration as the inverse of differentiation 7.7. Integration by parts and by substitution 7.8. The technique of integration Exercises 7 (b) 7.9. The constant n 7.10. Infinite integrals Exercises 7 (c) 7.11. Series and integrals Exercises 7 (d) 7.12. Approximations to definite integrals 7.13. Approximations by subdivision. Simpson's rule Exercises 7 (e) (Approximations) Exercises 7 (f) (Miscellaneous) 8. Functions of Several Variables 8.1. Functions of x and y 8.2. Limits and continuity 8.3. Partial differentiation Exercises 8 (a) 8.4. Differentiability Exercises 8 (b) 8.5. Composite functions 8.6. Changes of variable. Homogeneous functions Exercises 8 (c) 8.7. Taylor's theorem 8.8. Maxima and minima 8.9. Implicit functions Exercises 8 (d) Notes on the Exercises Index Back Cover