Introduction to Calculus and Classical Analysis

by Omar Hijab (Author)
# Hardcover: 375 pages
# Publisher: Springer Third Edition (April 2011)
# Language: English
# ISBN-10: 1441994874
# ISBN-13: 9781441994875 [Hardcover]

Intended for an honors calculus course or for an introduction to analysis, this is an ideal text for undergraduate majors since it covers rigorous analysis, computational dexterity, and a breadth of applications. The book contains many remarkable features: * complete avoidance of /epsilon-/delta arguments by using sequences instead * definition of the integral as the area under the graph, while area is defined for every subset of the plane * complete avoidance of complex numbers * heavy emphasis on computational problems * applications from many parts of analysis, e.g. convex conjugates, Cantor set, continued fractions, Bessel functions, the zeta functions, and many more * 344 problems with solutions in the back of the book.

ISSN 0172-6056
ISBN 978-1-4419-9487-5 e-ISBN 978-1-4419-9488-2
DOI 10.1007/978-1-4419-9488-2
Library of Congress Control Number: 2011923653
Mathematics Subject Classification (2010): 41-XX, 40-XX, 33-XX, 05-XX


Contents

Preface

1 The Set of Real Numbers
1.1 Sets and Mappings
1.2 The Set R
1.3 The Subset N and the Principle of Induction
1.4 The Completeness Property
1.5 Sequences and Limits
1.6 Nonnegative Series and Decimal Expansions
1.7 Signed Series and Cauchy Sequences

2 Continuity
2.1 Compactness
2.2 Continuous Limits
2.3 Continuous Functions

3 Differentiation
3.1 Derivatives
3.2 Mapping Properties
3.3 Graphing Techniques
3.4 Power Series
3.5 Trigonometry
3.6 Primitives

4 Integration
4.1 The Cantor Set
4.2 Area
4.3 The Integral
4.4 The Fundamental Theorem of Calculus
4.5 The Method of Exhaustion

Applications
5.1 Euler’s Gamma Function
5.2 The Number π
5.3 Gauss’ Arithmetic–Geometric Mean (AGM)
5.4 The Gaussian Integral
5.5 Stirling’s Approximation of n!
5.6 Infinite Products
5.7 Jacobi’s Theta Functions
5.8 Riemann’s Zeta Function
5.9 The Euler–Maclaurin Formula

A Solutions
A.1 Solutions to Chapter 1
A.2 Solutions to Chapter 2
A.3 Solutions to Chapter 3
A.4 Solutions to Chapter 4
A.5 Solutions to Chapter 5

References

Index

DOWNLOAD THIS BOOK

Posted in: