Mathematics of the 19th Century: Geometry, Analytic Function Theory / Edition 1

Mathematics of the 19th Century: Geometry, Analytic Function Theory / Edition 1

by Andrei N. Kolmogorov
     
 

ISBN-10: 3764350482

ISBN-13: 9783764350482

Pub. Date: 04/30/1996

Publisher: Birkhauser Basel

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

Overview

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 integral, 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. This book will be a valuable source of information for the general reader, as well as historians of science. It provides the reader with a good understanding of the overall picture of these two areas in the nineteenth century and their significance today.

Product Details

ISBN-13:
9783764350482
Publisher:
Birkhauser Basel
Publication date:
04/30/1996
Edition description:
1996
Pages:
291
Product dimensions:
6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1. Geometry.- 1. Analytic and Differential Geometry.- Analytic Geometry.- The Differential Geometry of Monge’s Students.- Gauss’ Disquisitiones generales circa superficies curvas.- Minding and the Formulation of the Problems of Intrinsic Geometry.- The French School of Differential Geometry.- Differential Geometry at Midcentury.- Differential Geometry in Russia.- The Theory of Linear Congruences.- 2. Projective Geometry.- The Rise of Projective Geometry.- Poncelet’s Traité des propriéte's projectives des figures.- The Analytic Projective Geometry of Möbius and Plücker.- The Synthetic Projective Geometry of Steiner and Chasles.- Staudt and the Foundation of Projective Geometry.- Cayley’s Projective Geometry.- 3. Algebraic Geometry and Geometric Algebra.- Algebraic Curves.- Algebraic Surfaces.- Geometric Computations Connected with Algebraic Geometry.- Grassmann’s Lineale Ausdehnungslehre.- Hamilton’s Vectors.- 4. Non-Euclidean Geometry.- Nikola? Ivanovich Lobachevski? and the Discovery of Non-Euclidean Geometry.- Gauss’ Research in Non-Euclidean Geometry.- János Bólyai.- Hyperbolic Geometry.- J. Bóyai’s “Absolute Geometry”.- The Consistency of Hyperbolic Geometry.- Propagation of the Ideas of Hyperbolic Geometry.- Beltrami’s Interpretation.- Cayley’s Interpretation.- Klein’s Interpretation.- Elliptic Geometry.- 5. Multi-Dimensional Geometry.- Jacobi’s Formulas for Multi-dimensional Geometry.- Cayley’s Analytic Geometry of n Dimensions.- Grassmann’s Multi-dimensional Geometry.- Plücker’s Neue. Geometrie des Raumes.- Schläfli’s Theorie der vielfachen Kontinuilät.- The Multi-dimensional Geometry of Klein and Jordan.- Riemannian Geometry.- Riemann’s Idea of Complex Parameters of Euclidean Motions.- Riemann’s Ideas on Physical Space.- The Work of Christoffel, Lipschitz. and Suvorov on Riemannian Geometry.- The Multi-dimensional Theory of Curves.- Multi-dimensional Surface Theory.- Multi-dimensional Projective Geometry.- The Terminology of Multi-dimensional Geometry.- 6. Topology.- Gauss’ Topology.- Generalizations of Euler’s Theorem on Polyhedra in the Early Nineteenth Century.- Listing’s Vorstudien zur Topologie.- Möbius’ “Theorie der elementaren Verwandschaft”.- The Topology of Surfaces in Riemann’s “Theorie der Abel’schen Funktionen”.- The Multi-dimensional Topology of Riemann and Betti.- Jordan’s Topological Theorems.- The “Klein Bottle”.- 7. Geometric Transformations.- Geometrie Transformations in the Work of Möbius.- Helmholtz’ Paper “Über die Thatsachen, die der Geometrie zu Grunde liegen”.- Klein’s “Erlanger Programm”.- Transference Principles.- Cremona Transformations.- Conclusion.- 2. Analytic Function.- Results Achieved in Analytic Function Theory in the Eighteenth Century.- Development of the Concept of a Complex Number.- Complex Integration.- The Cauchy Integral Theorem. Residues.- Elliptic Functions in the Work of Gauss.- Hypergeometric Functions.- The First Approach to Modular Functions.- Power Series. The Method of Majorants.- Elliptic Functions in the Work of Abel.- C.G.J. Jacobi. Fundamenta nova functionum ellipticarum.- The Jacobi Theta Functions.- Elliptic Functions in the Work of Eisenstein and Liouville. The First Textbooks.- Abelian Integrals. Abel’s Theorem.- Quadruply Periodic Functions.- Summary of the Development of Analytic Function Theory over the First Half of the Nineteenth Century.- V. Puiseux. Algebraic Functions.- Bernhard Riemann.- Riemann’s Doctoral Dissertation. The Dirichlet Principle.- Conformal Mappings.- Karl Weierstrass.- Analytic Function Theory in Russia. Yu.V. Sokhotski? and the Sokhotski?-Casorati-Weierstrass Theorem.- Entire and Meromorphic Functions. Picard’s Theorem.- Abelian Functions.- Abelian Functions (Continuation).- Automorphic Functions. Uniformization.- Sequences and Series of Analytic Functions.- Conclusion.- Literature.- (F. A. Medvedev).- General Works.- Collected Works and Other Original Sources.- Auxiliary Literature to Chapter 1.- Auxiliary Literature to Chapter 2.- Index of Names (A. F. Lapko).

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