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Title:A First Course in Mathematical Analysis
Author:David Alexander Brannan
Cambridge University Press,2006
Mathematical Analysis (often called Advanced Calculus) is generally found by students to be one of their hardest courses in Mathematics. This text uses the so-called sequential approach to continuity, differentiability and integration to make it easier to understand the subject.
Topics that are generally glossed over in the standard Calculus courses are given careful study here. For example, what exactly is a ‘continuous’ function? And how exactly can one give a careful definition of ‘integral’? This latter is often one of the mysterious points in a Calculus course – and it is quite tricky to give a rigorous treatment of integration!
The text has a large number of diagrams and helpful margin notes, and uses many graded examples and exercises, often with complete solutions, to guide students through the tricky points. It is suitable for self study or use in parallel with a standard university course on the subject.
Contents:
1 Numbers 1
1.1 Real numbers 2
1.2 Inequalities 9
1.3 Proving inequalities 14
1.4 Least upper bounds and greatest lower bounds 22
1.5 Manipulating real numbers 30
1.6 Exercises 35
2 Sequences 37
2.1 Introducing sequences 38
2.2 Null sequences 43
2.3 Convergent sequences 52
2.4 Divergent sequences 61
2.5 The Monotone Convergence Theorem 68
2.6 Exercises 79
3 Series 83
3.1 Introducing series 84
3.2 Series with non-negative terms 92
3.3 Series with positive and negative terms 103
3.4 The exponential function x 7! ex 122
3.5 Exercises 127
4 Continuity 130
4.1 Continuous functions 131
4.2 Properties of continuous functions 143
4.3 Inverse functions 151
4.4 Defining exponential functions 161
4.5 Exercises 164
5 Limits and continuity 167
5.1 Limits of functions 168
5.2 Asymptotic behaviour of functions 176
5.3 Limits of functions – using " and 181
5.4 Continuity – using " and 193
5.5 Uniform continuity 200
5.6 Exercises 203
6 Differentiation 205
6.1 Differentiable functions 206
6.2 Rules for differentiation 216
6.3 Rolle’s Theorem 228
6.4 The Mean Value Theorem 232
6.5 L’Hoˆpital’s Rule 238
6.6 The Blancmange function 244
6.7 Exercises 252
7 Integration 255
7.1 The Riemann integral 256
7.2 Properties of integrals 272
7.3 Fundamental Theorem of Calculus 282
7.4 Inequalities for integrals and their applications 288
7.5 Stirling’s Formula for n! 303
7.6 Exercises 309
8 Power series 313
8.1 Taylor polynomials 314
8.2 Taylor’s Theorem 320
8.3 Convergence of power series 329
8.4 Manipulating power series 338
8.5 Numerical estimates for p 346
8.6 Exercises 350
Appendix 1 Sets, functions and proofs 354
Appendix 2 Standard derivatives and primitives 359
Appendix 3 p 361
Appendix 4 Solutions to the problems 363
Chapter 1 363
Chapter 2 371
Chapter 3 382
Chapter 4 393
Chapter 5 402
Chapter 6 413
Chapter 7 426
Chapter 8 443
Index 457
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