A Formal Background to Mathematics 2a: A Critical Approach to Elementary Analysis: Universitext

VII: Convergence of Sequences.- Hidden hypotheses.- VII.1 Sequences convergent inR.- VII.2 Infinite limits.- VII.3 Subsequences.- VII.4 The Monotone Convergence Principle again.- VII.5 Suprema and infima of sets of real numbers.- VII.6 Exponential and logarithmic functions.- VII.7 The General Principle of Convergence.- VIII: Continuity and Limits of Functions.- and hidden hypotheses.- VIII.1 Continuous functions.- VIII.2 Properties of continuous functions.- VIII.3 General exponential, logarithmic and power functions.- VIII.4 Limit of a function at a point.- VIII.5 Uniform continuity.- VIII.6 Convergence of sequences of functions.- VIII.7 Polynomial approximation.- VIII.8 Another approach to expa.- IX: Convergence of Series.- and hidden hypotheses.- IX.1 Series and their convergence.- IX.2 Absolute and conditional convergence.- IX.3 Decimal expansions.- IX.4 Convergence of series of functions.- X: Differentiation.- and hidden hypotheses.- X.1 Derivatives.- X.2 Rules for differentiation.- X.3 The mean value theorem and its corollaries.- X.4 Primitives.- X.5 Higher order derivatives.- X.6 Extrema and derivatives.- X.7 A differential equation and the exponential function again.- X.8 Calculus in several variables.- XI: Integration.- XI.1 Integration and area.- XI.2 Analytic definition and study of integration.- XI.3 Integrals and primitives.- XI.4 Integration by parts.- XI.5 Integration by change of variable (or by substitution).- XI.6 Termwise integration of sequences of functions.- XI.7 Improper integrals.- XI.8 First order linear differential equations.- XI.9 Integrals in several variables.- XII: Complex Numbers: Complex Exponential and Trigonometric Functions.- XII.1 Definition of complex numbers.- XII.2 Groups, subgroups and homomorphisms.- XII.3 Homomorphisms ofRinto?; complex exponentials.- XII.4 The exponential function with domainC.- XII.5 The trigonometric functions cosine and sine.- XII.6 Further inverse trigonometric functions.- XII.7 The simple harmonic equation.- XII.8 Another differential equation.- XII.9 Matrices and complex numbers.- XII.10 A glance at Fourier series.- XII.11 Linear differential equations with constant coefficients.- XIII: Concerning Approximate Integration.- XIII.1 Quotes from syllabus notes.- XIII.2 Notation and preliminaries.- XIII.3 Precise formulation of statements XIII.1.1 – XIII.1.3.- XIII.4 Some corrected versions.- XIII.5 Falsity of statements XIII.3.1 – XIII.3.3.- XIII.6 The formulas applied to tabulated data.- XIV: Differential Coefficients.- XIV.1 The d-notation and differential coefficients.- XIV.2 The simple harmonic equation.- XV: Lengths of Curves.- XV.1 Quotes and criticisms.- XV.2 Paths.- XV.3 Lengths of paths.- XV.4 Path length as an integral.- XV.5 Ratio of arc length to chord length.- XV.6 Additivity of arc length.- XV.7 Equivalent paths; simple paths.- XV.8 Circular arcs; application to complex exponential and trigonometric functions.- XV.9 Angles and arguments.- XV.10 General remarks about curves.