Where did the examples of Abel's continuity theorem go?

TL;DR

This paper revisits Abel's continuity theorem for power series, exploring its historical context, applications, and reasons for its diminished presence in modern analysis education.

Contribution

It provides a comprehensive overview of Abel's theorem, highlighting its significance and encouraging renewed interest in its applications and historical development.

Findings

Historical analysis of Abel's theorem

Examples of applications in literature

Discussion on its educational decline

Abstract

Abel's continuity theorem for power series is a great tool to compute sums and prove properties of special functions. However, apart from two basic examples, this no longer occupies a place in Analysis courses. This note is an invitation to get acquainted with this theorem, its history, and many of its applications dispersed in the literature.