Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. Quotes Le développement naturel de cette étude conduisit bientôt les géomètres à embrasser dans leurs recherches les valeurs imaginaires de la variable aussi bien que les valeurs réelles. La théorie de la série de Taylor, celle des fonctions elliptiques, la vaste doctrine de Cauchy firent éclater la fécondité de cette généralisation. Il apparut que, entre deux vérités du domaine réel, le chemin le plus facile et le plus court passe bien souvent par le domaine complexe. The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain. Paul Painlevé Analyse des travaux scientifiques 1967 1900 Gauthier-Villars (reprinted by Librairie Scientifique et Technique, Albert Blanchard) 2853671313 1-2 Reproduced in Oeuvres de Paul Painlevé, Éditions du CNRS, Paris, 1972-1975, vol. 1, pp. 72-73. This is often quoted in English in the following form: It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one. Jacques Hadamard, in An Essay on the Psychology of Invention in the Mathematical Field (1945), page 123. At the beginning of the new millennium the most famous unsolved problem in complex analysis, if not in all of mathematics, is to determine whether the Riemann hypothesis holds. Theodore Gamelin Complex Analysis http://books.google.com/books?id=61BO2UY9QN4C&pg=PA370 17 July 2003 Springer Science & Business Media 978-0-387-95069-3 370 The result has caught the imagination of most mathematicians because it is so unexpected, connecting two seemingly unrelated areas in mathematics; namely, number theory, which is the study of the discrete, and complex analysis, which deals with continuous processes. David M. Burton Elementary Number Theory http://books.google.com/books?id=XMQjuoTqqRMC 1 May 2006 Tata McGraw-Hill Publishing Company Limited 978-0-07-061607-3 376 External links Category:Mathematical analysis