Fundamentals of Real and Complex Analysis

The primary aim of this text is to help transition undergraduates to study graduate level mathematics. It unites real and complex analysis after developing the basic techniques and aims at a larger readership than that of similar textbooks that have been published, as fewer mathematical requisites are required. The idea is to present analysis as a whole and emphasize the strong connections between various branches of the field. Ample examples and exercises reinforce concepts, and a helpful bibliography guides those wishing to delve deeper into particular topics. Graduate students who are studying for their qualifying exams in analysis will find use in this text, as well as those looking to advance their mathematical studies or who are moving on to explore another quantitative science.

Chapter 1 contains many tools for higher mathematics; its content is easily accessible, though not elementary. Chapter 2 focuses on topics in real analysis such as p-adic completion, Banach Contraction Mapping Theorem and its applications, Fourier series, Lebesgue measure and integration. One of this chapter’s unique features is its treatment of functional equations. Chapter 3 covers the essential topics in complex analysis: it begins with a geometric introduction to the complex plane, then covers holomorphic functions, complex power series, conformal mappings, and the Riemann mapping theorem. In conjunction with the Bieberbach conjecture, the power and applications of Cauchy’s theorem through the integral formula and residue theorem are presented.

Asuman Güven Aksoy is Crown Professor of Mathematics at Clairmont McKenna College. Her research interests include functional analysis, metric geometry, and operator theory. Professor Aksoy is coauthor of A Problem Book in Real Analysis (c) 2010 from the Problem Books in Mathematics series and Nonstandard Methods in Fixed Point Theory (c) 1990 in the Universitext series. Additionally she is recipient of the MAA’s Tensor Summa Grant 2013, the Fletcher Jones Grant for Summer Research 2009-2011, the MAA Award for Distinguished College or University teaching mathematics 2006, Huntoon Senior Teaching Award 2006, and the Roy P. Crocker Award for Merit, 2009, 2010.