**Abelian functions** (also called *hyperelliptic functions*) are a vast generalization of elliptic functions to more than one complex variable. While elliptic functions are associated with elliptic surfaces (called *Riemann surfaces of genus 1*), Abelian functions are associated to Riemann surfaces of higher genus (Doconinck et al., 2003). Both elliptic functions and Abelian functions can be written as a ratio of homogeneous polynomials of an auxiliary function, the Riemann theta function.

## Historical Notes on Abelian Functions

Abelian functions have four periods (i.e. the values repeat as you move in one of four directions) and two variables. One way to define them is (Baker 1907, p. 21):

**Note**: Baker’s historical texts are freely available on the University of Michigan Historical Math Collection website.

While quite popular in the 19th century, the term “Abelian function” fell out of favor for a long time. The functions were rarely used outside of theoretical physics and applied mathematics, where many solutions of differential equations are written in terms of Abelian functions. However, in 1997, Buchstaber et al. published a review in that “recapitulated and developed” classical Abelian function theory in terms of multi-dimensional sigma-functions. The review is freely available as a pdf here.

## Abelian Integrals and Abelian Functions

Abelian functions are obtained by inverting an arbitrary algebraic integral, or a combination of those types of integrals (Papadopoulos, 2017). Algebraic integrals have the form:

∫ *R*(*x*, *y*)*dx*

Where:

- R is a rational function of x and y,
- x and y satisfy the polynomial equation f(x, y) = (0).

Abelian integrals are any integral of an algebraic function which can’t be reduced to elliptic form. These integrals give rise to Abelian functions, defined as symmetric hyperelliptic functions, composed of multi-variable inverses of Abelian integral.

## References

Baker, H. F. An Introduction to the Theory of Multiply Periodic Functions. London: Cambridge University Press, 1907. Available here.

Deconinck, B. et al. (2003). Computing Riemann Theta Functions. Mathematics of Computation. Volume 73, Number 247.

Derbyshire, J. (2003). Prime Obsession. Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. National Academic Press.

Hazelwinkel, M. (2012). Encyclopaedia of Mathematics, Volume 1. Springer, Netherlands.

Papadopoulos, A. (2017). Looking Backward from Euler to Riemann. Retrieved September 24, 2020 from: http://arxiv-export-lb.library.cornell.edu/pdf/1710.03982

Wells, R. (2015). The Origins of Complex Geometry in the 19th Century. Retrieved September 24, 2020 from: https://arxiv.org/pdf/1504.04405.pdf

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