The development of the concept of uniform convergence in Karl Weierstrass’s lectures and publications between 1861 and 1886

Abstract

The history of uniform convergence is typically focused on the contributions of Cauchy, Seidel, Stokes, and Björling. While the mathematical contributions of these individuals to the concept of uniform convergence have been much discussed, Weierstrass is considered to be the actual inventor of today’s concept. This view is often based on his well-known article from 1841. However, Weierstrass’s works on a rigorous foundation of analytic and elliptic functions date primarily from his lecture courses at the University of Berlin up to the mid-1880s. For the history of uniform convergence, these lectures open up an independent branch of development that is disconnected from the approaches of the previously mentioned authors; to my knowledge, Weierstraß never explicitly referred to Cauchy’s continuity theorem (1821 or 1853) or to Seidel’s or Stokes’s contributions (1847). In the present article, Weierstrass’s contributions to the development of uniform convergence will be discussed, mainly based on lecture notes made by Weierstrass’s students between 1861 and the mid-1880s. The emphasis is on the notation and the mathematical rigor of the introductions to the concept, leading to the proposal to re-date the famous 1841 article and thus Weierstrass’s first introduction of uniform convergence.

A method to test the uniform convergence of function series

“…Cauchy firstly came out with the theory of uniform convergence. Later, Seidel and Stokes pointed out Cauchy's limitations [5]. Cauchy then acknowledged their advice and reached the Stokes' conclusions [6].…”

Optimization Problems in Spanish Differential Calculus Books Published in the 18th Century

“…In this sense, as Maz, Torralbo and Rico [2] note, research on the history of mathematics education shows and explains the process of incorporating new mathematical advances into the teaching of the subject and identifies how the social, political and academic context influences the way in which they are approached and disseminated within the current educational system. Thus, in this field of study, in addition to those that address the state of the art [1] and the different research methodologies [3][4][5], there have been many studies that examine curricula, educational legislation, teaching resources and textbooks [6,7], identify the influence of philosophical, political, social, economic and cultural tendencies on mathematics education [8,9], analyse the evolution of the teaching role [10] and determine the contribution of historical character or educational institutions to mathematics education [11][12][13][14].…”