Automorphic Function Theory Development And Evolution Of The Nineteenth Century

In the second half of the 19th century, group theory entered the center of mathematics study—It has applied to analysis, algebra, geometry and number theory, which indicates the trend to unify all mathematical branches. The application of group theory in analysis is the study of automorphic function.The theory of automorphic function is the result of the intersection of geometry, algebra, complex analysis and differential equation, which manifest the unity of mathematics. This thesis analyzes the rudiments of automorphic function theory from three subjects—analysis, differential equation and algebra, and systematically surveys its development in the 19th century with emphasis on the work of H.Poincare and F.Klein. Finally, this thesis briefly concludes its development in the 20th century.