# §21.2 Definitions

## §21.2(i) Riemann Theta Functions

21.2.1

This -tuple Fourier series converges absolutely and uniformly on compact sets of the and spaces; hence is an analytic function of (each element of) and (each element of) . is also referred to as a theta function with components, a -dimensional theta function or as a genus theta function.

For numerical purposes we use the scaled Riemann theta function , defined by (Deconinck et al. (2004)),

21.2.2

is a bounded nonanalytic function of . Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. See also §21.10(i).

## §21.2(ii) Riemann Theta Functions with Characteristics

Let . Define

This function is referred to as a Riemann theta function with characteristics . It is a translation of the Riemann theta function (21.2.1), multiplied by an exponential factor:

and

Characteristics whose elements are either 0 or are called half-period characteristics. For given , there are -dimensional Riemann theta functions with half-period characteristics.