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.
Author(s): Jeremy Gray
Series: Springer Undergraduate Mathematics Series
Edition: 1
Publisher: Springer International Publishing
Year: 2015
Language: English
Pages: 350
Tags: Functions of a Complex Variable; Real Functions; History of Mathematical Sciences
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
