On Those Analytic Functions of One Variable which possess an Algebraic Addition Theorem

TL;DR

This paper provides a detailed proof, in translation, of Weierstrass's theorem classifying all analytic functions with an algebraic addition theorem, clarifying their structure and properties.

Contribution

It offers a complete translation and proof of Weierstrass's theorem on analytic functions with algebraic addition theorems, enhancing understanding of their classification.

Findings

Characterization of all analytic functions with algebraic addition theorems

Proof of Weierstrass's theorem in a translated form

Clarification of the structure of such functions

Abstract

This is a literal word-for-word translation from the German of the article by Paul Koebe which contains a proof of Weierstrass's famous theorem characterizing all analytic functions which possess an algebraic addition theorem.