Saminathan Ponnusamy

Mathematical analysis is fundamental to the undergraduate curriculum not only because it is the stepping stone for the study of advanced analysis, but also because of its applications to other branches of mathematics, physics, and engineering at both the undergraduate and graduate levels. This self-contained textbook consists of eleven chapters, which are further divided into sections and subsections. Each section includes a careful selection of special topics covered that will serve to illustrate the scope and power of various methods in real analysis. The exposition is developed with thorough explanations, motivating examples, exercises, and illustrations conveying geometric intuition in a pleasant and informal style to help readers grasp difficult concepts. Foundations of Mathematical Analysis is intended for undergraduate students and beginning graduate students interested in a fundamental introduction to the subject. It may be used in the classroom or as a self-study guide without any required prerequisites.

Complex numbers can be viewed in several ways: as an element in a field, as a point in the plane, and as a two-dimensional vector. Examined properly, each perspective provides crucial insight into the interrelations between the complex number system and its parent, the real number system. The authors explore these relationships by adopting both generalization and specialization methods to move from real variables to complex variables, and vice versa, while simultaneously examining their analytic and geometric characteristics. The engaging exposition is replete with discussions, remarks, questions, and exercises, motivating not only understanding on the part of the reader, but also developing the tools needed to think critically about mathematical problems. The material includes numerous examples and applications relevant to engineering students, along with some techniques to evaluate various types of integrals. The book may serve as a text for an undergraduate coursein complex variables. The only prerequisite is a basic knowledge of advanced calculus. The presentation is also ideally suited for self-study.