The Real and the Complex: A History of Analysis in the 19th Century (Springer Undergraduate Mathematics Series)

This book contains a history of real and complex analysis in the nineteenth century, from the work of Lagrange and Fourier to the origins of set theory and the modern foundations of analysis. It studies the works of many contributors including Gauss, Cauchy, Riemann, and Weierstrass. This book is unique owing to the treatment of real and complex analysis as overlapping, inter-related subjects, in keeping with how they were seen at the time. It is suitable as a course in the history of mathematics for students who have studied an introductory course in analysis, and will enrich any course in undergraduate real or complex analysis. Front Matter....Pages i-xvi Lagrange and Foundations for the Calculus....Pages 1-11 Joseph Fourier....Pages 13-19 Legendre and Elliptic Integrals....Pages 21-31 Cauchy and Continuity....Pages 33-48 Cauchy: Differentiation and Integration....Pages 49-57 Cauchy and Complex Functions to 1830....Pages 59-68 Abel....Pages 69-77 Jacobi....Pages 79-93 Gauss....Pages 95-104 Cauchy and Complex Function Theory, 1830–1857....Pages 105-113 Complex Functions and Elliptic Integrals....Pages 115-122 Revision....Pages 123-129 Gauss, Green, and Potential Theory....Pages 131-141 Dirichlet, Potential Theory, and Fourier Series....Pages 143-151 Riemann....Pages 153-163 Riemann and Complex Function Theory....Pages 165-174 Riemann’s Later Complex Function Theory....Pages 175-184 Responses to Riemann’s Work....Pages 185-194 Weierstrass....Pages 195-206 Weierstrass’s Foundational Results....Pages 207-216 Revision—and Assessment....Pages 217-218 Uniform Convergence....Pages 219-226 Integration and Trigonometric Series....Pages 227-237 The Fundamental Theorem of the Calculus....Pages 239-251 The Construction of the Real Numbers....Pages 253-258 Implicit Functions....Pages 259-269 Towards Lebesgue’s Theory of Integration....Pages 271-281 Cantor, Set Theory, and Foundations....Pages 283-288 Topology....Pages 289-294 Assessment....Pages 295-296 Back Matter....Pages 297-350 " ... Contains a history of real and complex analysis in the nineteenth century, from the work of Lagrange and Fourier to the origins of set theory and the modern foundations of analysis. It studies the works of many contributors including Gauss, Cauchy, Riemann, and Weierstrass. This book is unique owing to the treatment of real and complex analysis as overlapping, inter-related subjects, in keeping with how they were seen at the time. It is suitable as a course in the history of mathematics for students who have studied an introductory course in analysis, and will enrich any course in undergraduate real or complex analysis."--Back cover