Mathematics of the 19th Century
Subtitle: Geometry, Analytic Function Theory
Contributors: A. N. Kolmogorov (Editor); A. P. Iushkevich (Editor)
Publisher: Birkhauser
Published: May 1996
Language: eng
Format: Hardcover
Number of pages: 291
ISBN-13: 9780817650483
Subjects: Geometry - General; Science/Mathematics; Geometry; Mathematics; Analytic functions; History
First sentence: Although the main achievements of eighteenth-century mathematics were connected with the development of mathematical analysis, important discoveries were made in the course of the century in geometry also.
Description: This book is the second volume of a study of the history of mathematics in the nineteenth century.

The first part of the book describes the development of geometry. The many varieties of geometry are considered and three main themes are traced: the development of a theory of invariants and forms that determine certain geometric structures such as curves or surfaces; the enlargement of conceptions of space which led to non-Euclidean geometry; and the penetration of algebraic methods into geometry in connection with algebraic geometry and the geometry of transformation groups.

The second part, on analytic function theory, shows how the work of mathematicians like Cauchy, Riemann and Weierstrass led to new ways of understanding functions. Drawing much of their inspiration from the study of algebraic functions and their integrals, these mathematicians and others created a unified, yet comprehensive theory in which the original algebraic problems were subsumed in special areas devoted to elliptic, algebraic, Abelian and automorphic functions.

The use of power series expansions made it possible to include completely general transcendental functions in the same theory and opened up the study of the very fertile subject of entire functions.