Hidden Harmony—Geometric Fantasies: The Rise of Complex Function Theory

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Springer Science & Business Media, Jun 21, 2013 - Mathematics - 848 pages

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is the first history of mathematics devoted to complex function theory, and it draws on a wide range of published and unpublished sources.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.

 

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Contents

Introduction
1
1 Elliptic Functions
14
2 From Real to Complex Analysis
81
3 Cauchys Modern Analysis
131
4 Complex Functions and Elliptic Integrals
217
5 Riemanns Geometric Function Theory
259
6 Weierstrasss Analytic Function Theory
342
7 Complex Function Theory and Differential Equations
487
8 Advanced Topics in the Theory of Functions
567
9 Several Complex Variables
665
10 The Textbook Tradition
691
Bibliography
760
Author Index
825
Subject Index
839
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About the author (2013)

Jeremy Gray is professor of the history of mathematics at the Open University, and an honorary professor at the University of Warwick. His most recent book is "Plato's Ghost: The Modernist Transformation of Mathematics" (Princeton).

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