§21.2 Definitions

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

For numerical purposes we use the scaled Riemann theta function $\widehat{\theta }\left(𝐳\right|𝛀)$, defined by (Deconinck et al. (2004)),

$\widehat{\theta }\left(𝐳\right|𝛀)$ is a bounded nonanalytic function of $𝐳$. Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. See also §21.10(i).

Let $𝜶,𝜷\in {ℝ}^g$. Define

This function is referred to as a Riemann theta function with characteristics $\left[\begin{array}{c}𝜶\\ 𝜷\end{array}\right]$. 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 $\frac{1}{2}$ are called half-period characteristics. For given $𝛀$, there are $2^{2g}$ $g$-dimensional Riemann theta functions with half-period characteristics.

For $g=1$, and with the notation of §20.2(i),