Introduction to Calculus and Classical Analysis
Springer International Publishing (Verlag)
978-3-319-28399-9 (ISBN)
- Approaches calculus and introductory analysis in a nonstandard way
- New edition extensively revised and updated
- Completely self-contained text
Involving rigorous analysis, computational dexterity, and a breadth of applications, this text is ideal for an undergraduate honors calculus course or for an introduction to analysis. This fourth edition includes corrections as well as some additional material.
Some features of the text:
• The text is completely self-contained and starts with the real number axioms;
• The integral is defined as the area under the graph, while the area is defined for every subset of the plane;
• There is a heavy emphasis on computational problems, from the high-school quadratic formula to the formula for the derivative of the zeta function at zero;
• There are applications from many parts of analysis, e.g., convexity, the Cantor set, continued fractions, the AGM, the theta and zeta functions, transcendental numbers, the Bessel and gamma functions, and many more;
• Traditionally transcendentally presented material, such as infinite products, the Bernoulli series, and the zeta functional equation, is developed over the reals;
• self-contained treatment of the fundamental theorems of calculus in the general case using the Sunrise Lemma
• There are 450 problems with all the solutions at the back of the text.
Omar Hijab is Professor of Mathematics and Associate Dean for Faculty Affairs, Information Technology, and Operations in the College of Science and Technology at Temple University. He received his Ph.D. in Mathematics from the University of California, Berkeley, and has served previously as Chair of the Department of Mathematics at Temple University. His research interests include systems theory and control; probability theory and stochastic processes; differential equations; mathematical physics; and optimization.
Preface
The Set of Real Numbers
Sets and Mappings
The Set R
The Subset N and the Principle of Induction
The Completeness Property
Sequences and Limits
Nonnegative Series and Decimal Expansions
Signed Series and Cauchy Sequences
Continuity
Compactness
Continuous Limits
Continuous Functions
Differentiation
Derivatives
Mapping Properties
Graphing Techniques
Power Series
Taylor Series
Trigonometry
Primitives
Integration
The Cantor Set
Area
The Integral
The Fundamental Theorems of Calculus
The Method of Exhaustion
Applications
Euler's Gamma Function
The Number pi
Gauss' Arithmetic-Geometric Mean (AGM)
The Gaussian Integral
Stirling's Approximation
Infinite Products
Jacobi's Theta Functions
Riemann's Zeta Function
The Euler-Maclaurin Formula
Generalizations
Measurable Functions and Linearity
Limit Theorems
The Fundamental Theorems of Calculus
The Sunrise Lemma
Absolute Continuity
The Lebesgue Differentiation Theorem
Solutions
References
Index.
"This is a very intriguing, decidedly unusual, and very satisfying treatment of calculus and introductory analysis. It's full of quirky little approaches to standard topics that make one wonder over and over again, 'Why is it never done like this?'" —John Allen Paulos, author of Innumeracy and A Mathematician Reads the Newspaper zur Vorauflage
"For a treatment [of infinite products and the Bernoulli series] that is very close to Euler’s and even more elementary..." — V.S. Varadarajan, Bulletin of the American Mathematical Society zur Vorauflage
"Chapter 5 is... an astonishing tour de force" —Steven G. Krantz, The American Mathematical Monthly zur Vorauflage
| Erscheinungsdatum | 26.02.2016 |
|---|---|
| Reihe/Serie | Undergraduate Texts in Mathematics |
| Zusatzinfo | 69 illus., 1 illus. in color. |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Einbandart | gebunden |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Graphentheorie | |
| Schlagworte | Approximations and Expansions • Calculus • classical analysis • combinatorics • Continuity • Differentiation • Integration • mathematics and statistics • Sequences, Series, Summability • Special Functions |
| ISBN-10 | 3-319-28399-5 / 3319283995 |
| ISBN-13 | 978-3-319-28399-9 / 9783319283999 |
| Zustand | Neuware |
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