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. The book contains many remarkable features: - complete avoidance of /epsilon-/delta arguments by instead using sequences, - definition of the integral as the area under the graph, while area is defined for EVERY subset of the plane, - complete avoidance of complex numbers, - heavy emphasis on computational problems, - applications from many parts of analysis, e.g. convex conjugates, Cantor set, continued fractions, Bessel functions, the zeta functions, and many more, - 344 problems with solutions in the back of the book, - interesting applications, many of which are not usually found in advanced calculus booksAbout the first edition:"The treatment, however, is far from standard, and many topics are included, especially in the last chapter, that are not available in other modern elementary texts…This is a rigorous book with good motivational material and explanatory remarks. It includes 347 problems, with solutions given in a 73-page appendix…some of the most attractive material requires considerable skill in manipulation." D.H. Armitage (MathSciNet).New in the second edition:For the new edition, the author has corrected errors and rewritten large portions of the text. In addition, the author has introduced new topics, such as a combinatorial proof that the radius of convergence of the Bernoulli series is 2p."ICCA is beautifully conceived and carefully executed...[this book] has much to teach, both about mathematics and how to write mathematics." Marvin Knopp, American Mathematical Monthly.
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 second 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;
* there are 366 problems.
About the first edition:
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
