Manuals/calci/BASETHETA

BASETHETA (Theta)

Description

This function shows the Base theta value.
The basic theta function is defined to be the function $\theta$ : $\mathbb {C} \rightarrow \mathbb {C}$ given by $\theta (z):=\theta (\tau )(z):=\sum _{n\in z}exp(\pi in^{2}\tau )exp(2\pi inz)$
The function $\theta$ depends on $\tau$ .
So for each $\tau \in \mathbb {C}$ with $Im\tau >0$ we get a (not necessarily different) basic theta function.
Hence there is a whole family of basic theta functions Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\theta (\tau )},\tau \in \mathbb {C} } ,Im Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau > 0} .
But here we assume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} to be fixed, so we have only one basic theta function.
The basic theta function is quasi-periodic.
Base Theta is locally uniformly unordered convergent also basic theta function is an entire function.