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)),
is a bounded nonanalytic function of . Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. See also §21.10(i).
With , ,
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:
Characteristics whose elements are either 0 or are called half-period characteristics. For given , there are -dimensional Riemann theta functions with half-period characteristics.
For , and with the notation of §20.2(i),