Difference between revisions of "Manuals/calci/BASETHETA"
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==Description== | ==Description== | ||
*This function shows the Base theta value. | *This function shows the Base theta value. | ||
| − | *The basic theta function is defined to be the function <math> \theta</math>:<math>\mathbb{C} \rarr \mathbb{C} </math> given by | + | *The basic theta function is defined to be the function <math> \theta</math>:<math>\mathbb{C} \rarr \mathbb{C} </math> given by <math>\theta(z):= \theta(\tau)(z):=\sum_{n\in z} exp(\pi in^2 \tau)exp(2\pi inz)</math> |
*The function <math>\theta</math> depends on <math>\tau</math>τ . | *The function <math>\theta</math> depends on <math>\tau</math>τ . | ||
*So for each <math>\tau \in \mathbb{C}</math> with <math>Im \tau \ge 0</math> we get a (not necessarily different) basic theta function. | *So for each <math>\tau \in \mathbb{C}</math> with <math>Im \tau \ge 0</math> we get a (not necessarily different) basic theta function. | ||
Revision as of 17:08, 21 December 2016
BASETHETA (Theta)
- 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 Theta } is the angle value.
Description
- This function shows the Base theta value.
- The basic theta function is defined to be the function 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 \theta} :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 \mathbb{C} \rarr \mathbb{C} } given by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \theta (z):=\theta (\tau )(z):=\sum _{n\in z}exp(\pi in^{2}\tau )exp(2\pi inz)}
- The function 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 \theta} depends on 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} τ .
- So for each 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 \in \mathbb{C}} with 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 Im \tau \ge 0} we get a (not necessarily different) basic theta function.
- Hence there is a whole family of basic theta functions {\theta(\tau )}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 \in \mathbb{C}} τ∈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\ge 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.