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
Author(s): J. C. Burkill
Edition: 1
Publisher: Cambridge University Press
Year: 1978
Language: English
Pages: 198
Tags: Mathematical Analysis, Calculus
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
