Essential Real Analysis

Description

This book provides a rigorous introduction to the techniques and results of real analysis, metric spaces and multivariate differentiation, suitable for undergraduate courses. Starting from the very foundations of analysis, it offers a complete first course in real analysis, including topics rarely found in such detail in an undergraduate textbook such as the construction of non-analytic smooth functions, applications of the Euler-Maclaurin formula to estimates, and fractal geometry. Drawing on the author’s extensive teaching and research experience, the exposition is guided by carefully chosen examples and counter-examples, with the emphasis placed on the key ideas underlying the theory. Much of the content is informed by its applicability: Fourier analysis is developed to the point where it can be rigorously applied to partial differential equations or computation, and the theory of metric spaces includes applications to ordinary differential equations and fractals. Essential Real Analysis will appeal to students in pure and applied mathematics, as well as scientists looking to acquire a firm footing in mathematical analysis. Numerous exercises of varying difficulty, including some suitable for group work or class discussion, make this book suitable for self-study as well as lecture courses. Front Matter ....Pages i-xvii Sets, Functions and the Real Numbers (Michael Field)....Pages 1-29 Basic Properties of Real Numbers, Sequences and Continuous Functions (Michael Field)....Pages 31-90 Infinite Series (Michael Field)....Pages 91-127 Uniform Convergence (Michael Field)....Pages 129-159 Functions (Michael Field)....Pages 161-210 Topics from Classical Analysis: The Gamma-Function and the Euler–Maclaurin Formula (Michael Field)....Pages 211-243 Metric Spaces (Michael Field)....Pages 245-328 Fractals and Iterated Function Systems (Michael Field)....Pages 329-347 Differential Calculus on \(\mathbb{R}^{m}\) (Michael Field)....Pages 349-442 Back Matter ....Pages 443-450