Hijab Introduction to Calculus and Classical Analysis

4th Auflage 2016
ISBN: 978-3-319-28400-2
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 427 Seiten, eBook

Reihe: Undergraduate Texts in Mathematics

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 include: 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; and There are 385 problems with all the solutions at the back of the text.

Zielgruppe

Lower undergraduate

Autoren/Hrsg.

Hijab, Omar

Inhaltsverzeichnis

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 p.- 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. 

Über Autor(innen)

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.