Introduction to Calculus and Classical Analysis

ISBN-13: 9783319803456 / Angielski / Miękka / 2018 / 427 str.

Omar Hijab

Opis

This text is intended for an honors calculus course or for an introduction to analysis. Involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate majors. This third 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; There is a heavy emphasis on computational problems; There are applications from many parts of analysis, e.g., convexity, the Cantor set, continued fractions, the AGM, the theta and zeta 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; There are 385 problems with all the solutions at the back of the text.

Spis treści

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 π.- 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. 

O autorze

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.

Blurb

This completely self-contained text is intended either for a course in honors calculus or for an introduction to analysis. Beginning with the real number axioms, and involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate math majors. This fourth edition includes an additional chapter on the fundamental theorems in their full Lebesgue generality, based on the Sunrise Lemma.

Key features of this text include:

• Applications from several 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;

• A heavy emphasis on computational problems, from the high-school quadratic formula to the formula for the derivative of the zeta function at zero;

• Traditionally transcendentally presented material, such as infinite products, the Bernoulli series, and the zeta functional equation, is developed over the reals;

• A self-contained treatment of the fundamental theorems of calculus in the general case using the Sunrise Lemma;

• The integral is defined as the area under the graph, while the area is defined for every subset of the plane;

• 450 problems with all the solutions presented at the back of the text.

Reviews:

"Chapter 5 is…an astonishing tour de force…"

—Steven G. Krantz, American Math. Monthly

"For a treatment…[of infinite products and Bernoulli series] that is very close to Euler’s and even more elementary…"

—V. S. Varadarajan, Bulletin AMS

"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