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  • Hidden Harmony—Geometric Fantasies: The Rise of Complex Function Theory (1st Edition)

Hidden Harmony—Geometric Fantasies: The Rise of Complex Function Theory (1st Edition)

Book Info
Year: 2013
Pages: 848
Language: English
Series: English
DOI: 10.1007/978-1-4614-5725-1
ISBN: 9781461457244,9781461457251

​This book is a history of complex function theory from its origins to 1914, when the essential features of the modern theory were in place. It is the first history of mathematics devoted to complex function theory, and it draws on a wide range of published and unpublished sources. In addition to an extensive and detailed coverage of the three founders of the subject – Cauchy, Riemann, and Weierstrass – it looks at the contributions of authors from d’Alembert to Hilbert, and Laplace to Weyl.

Particular chapters examine the rise and importance of elliptic function theory, differential equations in the complex domain, geometric function theory, and the early years of complex function theory in several variables. Unique emphasis has been devoted to the creation of a textbook tradition in complex analysis by considering some seventy textbooks in nine different languages. The book is not a mere sequence of disembodied results and theories, but offers a comprehensive picture of the broad cultural and social context in which the main actors lived and worked by paying attention to the rise of mathematical schools and of contrasting national traditions.

The book is unrivaled for its breadth and depth, both in the core theory and its implications for other fields of mathematics. It documents the motivations for the early ideas and their gradual refinement into a rigorous theory.​



Front Matter....Pages i-xvii
Introduction....Pages 1-13
Chapter 1 Elliptic Functions....Pages 15-79
Chapter 2 From Real to Complex Analysis....Pages 81-130
Chapter 3 Cauchy’s “Modern Analysis”....Pages 131-216
Chapter 4 Complex Functions and Elliptic Integrals....Pages 217-258
Chapter 5 Riemann’s Geometric Function Theory....Pages 259-341
Chapter 6 Weierstrass’s Analytic Function Theory....Pages 343-486
Chapter 7 Complex Function Theory and Differential Equations....Pages 487-566
Chapter 8 Advanced Topics in the Theory of Functions....Pages 567-664
Chapter 9 Several Complex Variables....Pages 665-689
Chapter 10 The Textbook Tradition....Pages 691-759
Back Matter....Pages 761-848