Abelian Functions

Abstract

As the theory of elliptic integrals leads naturally to the theory of elliptic functions, the theory of general algebraic integrals leads to the theory of Riemann surfaces and then to the theory of abelian functions. These are meromorphic functions on a higherdimensional complex torus. The case n > 1 is much more involved than the case of the elliptic functions (n = 1). The reason for this is that the Weierstrass approach using the ℘-function does not work, since the zeros and poles of meromorphic functions of several variables are no longer discrete.