The Real and the Complex: A History of Analysis in the 19th Century

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.

Lagrange and foundations for the calculus.- Joseph Fourier.- Legendre.- Cauchy and continuity.- Cauchy: differentiation and integration.- Cauchy and complex functions to 1830.- Abel.- Jacobi.- Gauss.- Cauchy and complex function theory, 1830-1857.- Complex functions and elliptic integrals.- Revision.- Gauss, Green, and potential theory.- Dirichlet, potential theory, and Fourier series.- Riemann.- Riemann and complex function theory.- Riemanns later complex function theory.- Responses to Riemanns work.- Weierstrass.- Weierstrasss foundational results.- Revision { and assessment.- Uniform Convergence.- Integration and trigonometric series.- The fundamental theorem of the calculus.- The construction of the real numbers.- Implicit functions.- Towards Lebesgues theory of integration.- Cantor, set theory, and foundations.- Topology.- Assessment.