Introduction to calculus and classical analysis
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Bibliographic Information
Introduction to calculus and classical analysis
（Undergraduate texts in mathematics）
Springer, 2016
4th ed
- : softcover
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"Softcover reprint of the hardcover 4th edition 2016"--T.p. verso
Includes bibliographical references (p. 419) and index
Description and Table of Contents
Description
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.
Table of Contents
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 .- 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.
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