An Abelian function is a -fold periodic, meromorphic function of complex variables. In consequence, Abelian functions are generalizations of elliptic functions (§23.2(iii)) to more than one complex variable. For every Abelian function, there is a positive integer , such that the Abelian function can be expressed as a ratio of linear combinations of products with factors of Riemann theta functions with characteristics that share a common period lattice. For further information see Igusa (1972, pp. 132–135) and Markushevich (1992).