Integration and Trigonometric Series

Abstract

Riemann’s previously unpublished Habilitation essay on the integration of trigonometric series was published for the first time in 1867, in an issue of the Göttingen Nachrichten that came out shortly after his death and carried several of his papers that he had left in almost a fit state to print. By then his reputation as a remarkable, but difficult, mathematician had spread, and what he wrote was read. This chapter considers some of the responses to his ideas about real functions and looks at the changes wrought in the relationship between continuity and differentiability, and between the implications of convergence and uniform convergence. This will lead us to look at the world of functions defined by series that do not converge uniformly, and were often considered in the late 19th century to be the most general kind of function, which is why they were often called ‘assumptionless functions’.