Customer Reviews

A First Course in Mathematical Analysis

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It is amazing that, despite the changing fads in the curriculum, this book has the quality to remain a steadfast bridge from School Mathematics on the first steps into the realm of the Maths specialist for over 40 years now.

The recipe is simple: keep it short, keep it sweet, keep it simple! Mr Burkill has produced a fine little book that gently guides the new student embarking on a specialism in Maths. The author has struck a good balance between the problem solving so familiar at school and introducing the rigour of Mathematical Analysis.

Familiar concepts like differentiation and integration are brought into play right after a quick refresher on numbers and then introducing the notion of limit within the framework of sequences. The delta-epsilon construct is used to great effect to give meaning to the ideas of convergence of sequences and the continuity of functions.

These then lead naturally to the Differential calculus where previously learnt ideas like the rules of differentiation are placed in a rigorous setting and interesting elementary analytical results such as the Mean Value Theorem and Taylor's theorem are discussed. The chapter on Infinite series together with the elementary rules for testing for convergence is followed by a chapter on the special functions of analysis as defined in terms of series - e.g. exp, log, sin, cos, etc.

The chapter on the Integral Calculus makes a natural next step utilising the ideas of an integral as a limit and of infinite series. Specific techniques such as the integral to infinity and approximation methods are placed on a rigorous footing. The final chapter introduces functions of several variables.

The book has lots of worked examples within the text, which aid understanding of new material. As a bonus, there are also several end of section with notes/hints at the end of the book.

Overall, this is a gentle introduction to Analysis and will help anyone who is overawed by the subject on first encounter.

This book is entirely appropriate as an introduction to analysis for students already familiar with the Calculus. It is on the reading list at Cambridge and Oxford for their undergraduate analysis courses and its reputation is well established. It is not too abstruse but is exceptionally clear and straightforward.

Good book. Lacking in certain important mathematical concepts like lim sup and lim inf, but otherwise good so far.