Introduction to Analysis in One Variable (Pure and Applied Undergraduate Texts)

This is a text for students who have had a three-course calculus sequence and who are ready to explore the logical structure of analysis as the backbone of calculus. It begins with a development of the real numbers, building this system from more basic objects (natural numbers, integers, rational numbers, Cauchy sequences), and it produces basic algebraic and metric properties of the real number line as propositions, rather than axioms. The text also makes use of the complex numbers and incorporates this into the development of differential and integral calculus. For example, it develops the theory of the exponential function for both real and complex arguments, and it makes a geometrical study of the curve $(\mathrm{exp}\thinspace it)$, for real $t$, leading to a self-contained development of the trigonometric functions and to a derivation of the Euler identity that is very different from what one typically sees. Further topics include metric spaces, the Stone–Weierstrass theorem, and Fourier series. Preface Chapter 1. Numbers §1.1. Peano arithmetic §1.2. The integers §1.3. Prime factorization and the fundamental theorem of arithmetic 14 §1.4. The rational numbers §1.5. Sequences §1.6. The real numbers §1.7. Irrational numbers §1.8. Cardinal numbers §1.9. Metric properties of R §1.10. Complex numbers Chapter 2. Spaces §2.1. Euclidean spaces §2.2. Metric spaces §2.3. Compactness §2.4. The Baire category theorem Chapter 3. Functions §3.1. Continuous functions §3.2. Sequences and series of functions §3.3. Power series §3.4. Spaces of functions §3.5. Absolutely convergent series Chapter 4. Calculus §4.1. The derivative §4.2. The integral §4.3. Power series §4.4. Curves and arc length §4.5. The exponential and trigonometric functions §4.6. Unbounded integrable functions Chapter 5. Further Topics in Analysis §5.1. Convolutions and bump functions §5.2. The Weierstrass approximation theorem §5.3. The Stone-Weierstrass theorem §5.4. Fourier series §5.5. Newton’s method §5.6. Inner product spaces Appendix A. Complementary results §A.1. The fundamental theorem of algebra §A.2. More on the power series of (1 − x)b §A.3. π2 is irrational §A.4. Archimedes’ approximation to π §A.5. Computing π using arctangents §A.6. Power series for tan x §A.7. Abel’s power series theorem §A.8. Continuous but nowhere-differentiable functions Bibliography Index