The Origins of Complex Geometry in the 19th Century

TL;DR

This paper reviews the historical development of complex geometry in the 19th century, highlighting key innovations like the complex plane, elliptic functions, and Riemann surfaces that shaped 20th-century mathematics.

Contribution

It provides a detailed historical overview of foundational 19th-century mathematical ideas that led to the development of complex geometry in the 20th century.

Findings

Analysis of original papers on complex plane and elliptic functions

Identification of Riemann's creation of Riemann surfaces

Historical connections between 19th-century innovations and modern complex geometry

Abstract

This paper gives an overview of several key innovations in the 19th century which led to complex geometry in the 20th century. This includes the creation of the complex plane, the work of Abel on addition theorems for generalized elliptic integrals, the theory of elliptic functions, holomorphic functions, and the creation of Riemann surfaces by Riemann in the mid 19th century. A number of the original papers which contain these new ideas are looked at in some detail and a detailed set of references is included.