Ebook: Real Analysis

From the point of view of strict logic, a rigorous course on real analysis should precede a course on calculus. Strict logic, is, however, overruled by both history and practicality. Historically, calculus, with its origins in the 17th century, came first, and made rapid progress on the basis of informal intuition. Not until well through the 19th century was it possible to claim that the edifice was constructed on sound logical foundations. As for practicality, every university teacher knows that students are not ready for even a semi-rigorous course on analysis until they have acquired the intuitions and the sheer technical skills that come from a traditional calculus course. 1 Real analysis, I have always thought, is the pons asinorv.m of modern mathematics. This shows, I suppose, how much progress we have made in two thousand years, for it is a great deal more sophisticated than the Theorem of Pythagoras, which once received that title. All who have taught the subject know how patient one has to be, for the ideas take root gradually, even in students of good ability. This is not too surprising, since it took more than two centuries for calculus to evolve into what we now call analysis, and even a gifted student, guided by an expert teacher, cannot be expected to grasp all of the issues immediately.

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Understanding the concepts and methods of real analysis is an essential skill for every undergraduate mathematics student. Written in an easy-to-read style, Real Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. Combining an informal style with precision mathematics, the book covers all the key topics with fully worked examples and exercises with solutions.

The book features: sequences and series - considering the central notion of a limit; continuous functions; differentiation; integration; logarithmic and exponential functions; uniform convergence; and circular functions.

All these concepts and techniques are deployed in examples in the final chapter to provide the student with a thorough understanding of this challenging subject.

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Understanding the concepts and methods of real analysis is an essential skill for every undergraduate mathematics student. Written in an easy-to-read style, Real Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. Combining an informal style with precision mathematics, the book covers all the key topics with fully worked examples and exercises with solutions.

The book features: sequences and series - considering the central notion of a limit; continuous functions; differentiation; integration; logarithmic and exponential functions; uniform convergence; and circular functions.

All these concepts and techniques are deployed in examples in the final chapter to provide the student with a thorough understanding of this challenging subject.

Content:
Front Matter....Pages i-x
Introductory Ideas....Pages 1-26
Sequences and Series....Pages 27-62
Functions and Continuity....Pages 63-98
Differentiation....Pages 99-118
Integration....Pages 119-163
The Logarithmic and Exponential Functions....Pages 165-179
Sequences and Series of Functions....Pages 181-216
The Circular Functions....Pages 217-228
Miscellaneous Examples....Pages 229-235
Back Matter....Pages 237-276

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Understanding the concepts and methods of real analysis is an essential skill for every undergraduate mathematics student. Written in an easy-to-read style, Real Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. Combining an informal style with precision mathematics, the book covers all the key topics with fully worked examples and exercises with solutions.

The book features: sequences and series - considering the central notion of a limit; continuous functions; differentiation; integration; logarithmic and exponential functions; uniform convergence; and circular functions.

All these concepts and techniques are deployed in examples in the final chapter to provide the student with a thorough understanding of this challenging subject.

Content:
Front Matter....Pages i-x
Introductory Ideas....Pages 1-26
Sequences and Series....Pages 27-62
Functions and Continuity....Pages 63-98
Differentiation....Pages 99-118
Integration....Pages 119-163
The Logarithmic and Exponential Functions....Pages 165-179
Sequences and Series of Functions....Pages 181-216
The Circular Functions....Pages 217-228
Miscellaneous Examples....Pages 229-235
Back Matter....Pages 237-276
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