Introduction to Calculus and Classical Analysis (Undergraduate Texts in Mathematics) [9783319803456]

Overview

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.

Reviews from previous editions:

"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

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

—Steven G. Krantz, The American Mathematical Monthly

"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