A Concise Approach to Mathematical Analysis introduces the undergraduate student to the more abstract concepts of advanced calculus. The main aim of the book is to smooth the transition from the problem-solving approach of standard calculus to the more rigorous approach of proof-writing and a deeper understanding of mathematical analysis. The first half of the textbook deals with the basic foundation of analysis on the real line; the second half introduces more abstract notions in mathematical analysis. Each topic begins with a brief introduction followed by detailed examples. A selection of exercises, ranging from the routine to the more challenging, then gives students the opportunity to practise writing proofs. The book is designed to be accessible to students with appropriate backgrounds from standard calculus courses but with limited or no previous experience in rigorous proofs. It is written primarily for advanced students of mathematics - in the 3rd or 4th year of their degree - who wish to specialise in pure and applied mathematics, but it will also prove useful to students of physics, engineering and computer science who also use advanced mathematical techniques.
Author(s): Mangatiana A. Robdera
Edition: Softcover reprint of the original 1st ed. 2003
Publisher: Springer
Year: 2002
Language: English
Pages: C, xiv, 366, B
Cover
S Title
A Concise Approach to Mathematical Analysis
© Springer-Verlag London 2003
ISBN 978-1-85233-552-6
ISBN 978-0-85729-347-3 (eBook)
DOI 10.1007/978-0-85729-347-3
QA300 R56 2002 515-dc21
LCCN 2001049366
Dedication
Preface
Contents
1 Numbers and Functions
1.1 Real Numbers
1.2 Subsets of R
1.3 Variables and Functions
EXERCISES
2 Sequences
2.1 Definition of a Sequence
2.2 Convergence and Limits
2.3 Subsequences
2.4 Upper and Lower Limits
2.5 Cauchy Criterion
EXERCISES
3 Series
3.1 Infinite Series
3.2 Conditional Convergence
3.3 Comparison Tests
3.4 Root and Ratio Tests
3.5 Further Tests
EXERCISES
4 Limits and Continuity
4.1 Limits of Functions
4.2 Continuity of Functions
4.3 Properties of Continuous Functions
4.4 Uniform Continuity
EXERCISES
5 Differentiation
5.1 Derivatives
5.2 Mean Value Theorem
5.3 L'Hospital's Rule
5.4 Inverse Function Theorems
5.5 Taylor's Theorem
EXERCISES
6 Elements of Integration
6.1 Step Functions
6.2 Riemann Integral
6.3 Functions of Bounded Variation
6.4 Riemann-Stieltjes Integral
EXERCISES
7 Sequences and Series of Functions
7.1 Sequences of Functions
7.2 Series of Functions
7.3 Power Series
7.4 Taylor Series
EXERCISES
8 Local Structure on the Real Line
8.1 Open and Closed Sets in R
8.2 Neighborhoods and Interior Points
8.3 Closure Point and Closure
8.4 Completeness and Compactness
EXERCISES
9 Continuous Functions
9.1 Global Continuity
9.2 Functions Continuous on a Compact Set
9.3 Stone-Weierstrass Theorem
9.4 Fixed-point Theorem
9.5 Ascoli-ArzeUl Theorem
EXERCISES
10 Introduction to the Lebesgue Integral
10.1 Null Sets
10.2 Lebesgue Integral
10.3 Improper Integral
10.4 Important Inequalities
EXERCISES
11 Elements of Fourier Analysis
11.1 Fourier Series
11.2 Convergent Trigonometric Series
11.3 Convergence in 2-mean
11.4 Pointwise Convergence
EXERCISES
A Appendix
A.1 Theorems and Proofs
A.2 Set Notations
A.3 Cantor's Ternary Set
A.4 Bernstein's Approximation Theorem
B Hints for Selected Exercises
Bibliography
Index
Back Cover
