A First Course in Mathematical Analysis

✍ Scribed by J. C. Burkill

• Publisher: Cambridge University Press
• Year: 1978
• Tongue: English
• Leaves: 197
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover......Page 1
Title......Page 3
ISBN 0 521 29468 1......Page 4
Contents......Page 6
Preface......Page 8
1.1. The branches of pure mathematics......Page 10
1.2. The scope of mathematical analysis......Page 11
1.3. Numbers......Page 12
Exercises 1 (a)......Page 15
1.4. Irrational numbers......Page 16
1.5. Cuts of the rationals......Page 17
1.6. The field of real numbers......Page 19
1.7. Bounded sets of numbers......Page 22
1.8. The least upper bound (supremum)......Page 24
Exercises 1 (d)......Page 26
1.9. Complex numbers......Page 27
1.10. Modulus and phase......Page 28
Exercises 1 (f )......Page 30
2.2. Null sequences......Page 32
2.3. Sequence tending to a limit......Page 34
2.4. Sequences tending to infinity......Page 35
2.5. Sum and product of sequences......Page 37
2.6. Increasing sequences......Page 40
2.7. An important sequence a^n......Page 41
Exercises 2 (d)......Page 42
2.8. Recurrence relations......Page 43
Exercises 2 (e)......Page 45
2.9. Infinite series......Page 47
2.10. The geometric series \Sigma x^n......Page 48
2.11. The series \Sigma n^-k......Page 49
Exercises 2 (f)......Page 51
2.12. Properties of infinite series......Page 52
Exercises 2 (g)......Page 53
3.1. Functions......Page 56
3.3. Sketching of curves......Page 58
3.4. Continuous functions......Page 60
3.5. Examples of continuous and discontinuous functions......Page 62
Exercises 3 (b)......Page 64
3.6. The intermediate-value property......Page 65
3.7. Bounds of a continuous function......Page 66
3.8. Uniform continuity......Page 69
3.9. Inverse functions......Page 71
Exercises 3 (c)......Page 72
4.1. The derivative......Page 74
4.2. Differentiation of sum, product, et......Page 76
4.3. Differentiation of elementary functions......Page 78
Exercises 4 (b)......Page 80
Exercises 4 (c)......Page 81
4.5. The sign of f'(x)......Page 82
Exercises 4 (d)......Page 83
4.6. The mean value theorem......Page 84
4.7. Maxima and minima......Page 86
4.8. Approximation by polynomials. Taylor's theorem......Page 87
4.9. Indeterminate forms......Page 91
Exercises 4 (f)......Page 93
5.1. Series of positive terms......Page 97
Exercises 5 (a)......Page 98
5.2. Series of positive and negative terms......Page 99
5.3. Conditional convergence......Page 101
5.4. Series of complex terms......Page 103
Exercises 5 (c)......Page 104
5.5. Power series......Page 105
5.6. The circle of convergence of a power series......Page 106
Exercises 5 (d)......Page 107
5.7. Multiplication of series......Page 108
5.8. Taylor's series......Page 110
Exercises 5 (e)......Page 112
6.1. The special functions of analysis......Page 113
6.3. Repeated limits......Page 114
6.4. Rate of increase of exp x......Page 115
6.5. exp x as a power......Page 116
Exercises 6 (a)......Page 118
6.6. The logarithmic function......Page 119
Exercises 6 (b)......Page 120
6.7. Trigonometric functions......Page 121
6.8. Exponential and trigonometric functions......Page 122
Exercises 6 (c)......Page 124
6.9. The inverse trigonometric functions......Page 125
6.10. The hyperbolic functions and their inverses......Page 126
Exercises 6 (d)......Page 127
7.1. Area and the integral......Page 128
7.2. The upper and lower integrals......Page 130
7.3. The integral as a limit......Page 132
7.4. Continuous or monotonic functions are integrable......Page 133
7.5. Properties of the integral......Page 134
7.6. Integration as the inverse of differentiation......Page 137
7.7. Integration by parts and by substitution......Page 138
7.8. The technique of integration......Page 140
Exercises 7 (b)......Page 143
7.9. The constant n......Page 145
7.10. Infinite integrals......Page 146
Exercises 7 (c)......Page 148
7.11. Series and integrals......Page 149
Exercises 7 (d)......Page 152
7.12. Approximations to definite integrals......Page 153
7.13. Approximations by subdivision. Simpson's rule......Page 154
Exercises 7 (e) (Approximations)......Page 157
Exercises 7 (f) (Miscellaneous)......Page 158
8.1. Functions of x and y......Page 160
8.2. Limits and continuity......Page 161
8.3. Partial differentiation......Page 162
Exercises 8 (a)......Page 164
8.4. Differentiability......Page 165
Exercises 8 (b)......Page 166
8.5. Composite functions......Page 167
8.6. Changes of variable. Homogeneous functions......Page 168
Exercises 8 (c)......Page 170
8.7. Taylor's theorem......Page 172
8.8. Maxima and minima......Page 173
8.9. Implicit functions......Page 175
Exercises 8 (d)......Page 178
Notes on the Exercises......Page 179
Index......Page 194
Back Cover......Page 197