A First Course in Mathematical Analysis

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Paperback | 12 February 1979

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196 Pages

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21.59 x 13.97 x 1.14

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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.

Industry Reviews

'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

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Preface	p. vii
Numbers
The branches of pure mathematics	p. 1
The scope of mathematical analysis	p. 2
Numbers	p. 3
Irrational numbers	p. 7
Cuts of the rationals	p. 8
The field of real numbers	p. 10
Bounded sets of numbers	p. 13
The least upper bound (supremum)	p. 15
Complex numbers	p. 18
Modulus and phase	p. 19
Sequences
Sequences	p. 23
Null sequences	p. 23
Sequence tending to a limit	p. 25
Sequences tending to infinity	p. 26
Sum and product of sequences	p. 28
Increasing sequences	p. 31
An important sequence a[superscript n]	p. 32
Recurrence relations	p. 34
Infinite series	p. 38
The geometric series [Sigma]x[superscript n]	p. 39
The series [Sigma]n[superscript -k]	p. 40
Properties of infinite scries	p. 43
Continuous Functions
Functions	p. 47
Behaviour of f(x) for large values of x	p. 49
Sketching of curves	p. 49
Continuous functions	p. 51
Examples of continuous and discontinuous functions	p. 53
The intermediate-value property	p. 56
Bounds of a continuous function	p. 57
Uniform continuity	p. 60
Inverse functions	p. 62
The Differential Calculus
The derivative	p. 65
Differentiation of sum, product, etc.	p. 67
Differentiation of elementary functions	p. 69
Repeated differentiation	p. 72
The sign of f'(x)	p. 73
The mean value theorem	p. 75
Maxima and minima	p. 77
Approximation by polynomials. Taylor's theorem	p. 78
Indeterminate forms	p. 82
Infinite Series
Series of positive terms	p. 88
Series of positive and negative terms	p. 90
Conditional convergence	p. 92
Series of complex terms	p. 94
Power series	p. 96
The circle of convergence of a power series	p. 97
Multiplication of series	p. 99
Taylor's series	p. 101
The Special Functions of Analysis
The special functions of analysis	p. 104
The exponential function	p. 105
Repeated limits	p. 105
Rate of increase of exp x	p. 106
Exp x as a power	p. 107
The logarithmic function	p. 110
Trigonometric functions	p. 112
Exponential and trigonometric functions	p. 113
The inverse trigonometric functions	p. 116
The hyperbolic functions and their inverses	p. 117
The Integral Calculus
Area and the integral	p. 119
The upper and lower integrals	p. 121
The integral as a limit	p. 123
Continuous or monotonic functions are integrable	p. 124
Properties of the integral	p. 125
Integration as the inverse of differentiation	p. 128
Integration by parts and by substitution	p. 129
The technique of integration	p. 131
The constant [pi]	p. 136
Infinite integrals	p. 137
Scries and integrals	p. 140
Approximations to definite integrals	p. 144
Approximations by subdivision. Simpson's rule	p. 145
Functions of Several Variables
Functions of x and y	p. 151
Limits and continuity	p. 152
Partial differentiation	p. 153
Differentiability	p. 156
Composite functions	p. 158
Changes of variable. Homogeneous functions	p. 159
Taylor's theorem	p. 163
Maxima and minima	p. 164
Implicit functions	p. 166
Notes on the Exercises	p. 170
Index	p. 185
Table of Contents provided by Syndetics. All Rights Reserved.

A First Course in Mathematical Analysis

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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