Synopsis
Preface.- The Set of Real Numbers.- Sets and Mappings.- The Set R.- The Subset N and the Principle of Induction.- The Completeness Property.- Sequences and Limits.- Nonnegative Series and Decimal Expansions.- Signed Series and Cauchy Sequences.- Continuity.- Compactness.- Continuous Limits.- Continuous Functions.- Differentiation.- Derivatives.- Mapping Properties.- Graphing Techniques.- Power Series.- Taylor Series.- Trigonometry.- Primitives.- Integration.- The Cantor Set.- Area.- The Integral.- The Fundamental Theorems of Calculus.- The Method of Exhaustion.- Applications.- Euler's Gamma Function.- The Number π.- Gauss' Arithmetic-Geometric Mean (AGM).- The Gaussian Integral.- Stirling's Approximation.- Infinite Products.- Jacobi's Theta Functions.- Riemann's Zeta Function.- The Euler-Maclaurin Formula.- Generalizations.- Measurable Functions and Linearity.- Limit Theorems.- The Fundamental Theorems of Calculus.- The Sunrise Lemma.- Absolute Continuity.- The Lebesgue Differentiation Theorem.- Solutions.- References.- Index.
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