Introduction to Calculus and Classical Analysis (Undergraduate Texts in Mathematics) - Tapa dura

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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.

Contraportada

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