Abelian functions

Abelian functions are a class of complex functions that generalize elliptic functions, associated with multi-dimensional shapes called algebraic curves of higher genus. They are periodic with respect to several independent periods, meaning they repeat values in multiple directions within a complex space. These functions arise naturally in advanced areas of mathematics like algebraic geometry and complex analysis, often describing solutions to complex equations or relating to the properties of multi-dimensional geometric objects. In essence, Abelian functions extend the concept of well-known periodic functions (like sine and cosine) to multiple variables and more complex symmetries.