Theta Function - Theta Functions in Terms of The Nome

Theta Functions in Terms of The Nome

Instead of expressing the Theta functions in terms of and, we may express them in terms of arguments and the nome q, where and . In this form, the functions become

\begin{align} \vartheta_{00}(w, q)& = \sum_{n=-\infty}^\infty (w^2)^n q^{n^2}\quad& \vartheta_{01}(w, q)& = \sum_{n=-\infty}^\infty (-1)^n (w^2)^n q^{n^2}\\ \vartheta_{10}(w, q)& = \sum_{n=-\infty}^\infty (w^2)^{\left(n+1/2\right)} q^{\left(n + 1/2\right)^2}\quad& \vartheta_{11}(w, q)& = i \sum_{n=-\infty}^\infty (-1)^n (w^2)^{\left(n+1/2\right)} q^{\left(n + 1/2\right)^2}. \end{align}

We see that the Theta functions can also be defined in terms of w and q, without a direct reference to the exponential function. These formulas can, therefore, be used to define the Theta functions over other fields where the exponential function might not be everywhere defined, such as fields of p-adic numbers.

Read more about this topic:  Theta Function

Famous quotes containing the words functions and/or terms:

    “Let us stop being afraid. Of our own thoughts, our own minds. Of madness, our own or others’. Stop being afraid of the mind itself, its astonishing functions and fandangos, its complications and simplifications, the wonderful operation of its machinery—more wonderful because it is not machinery at all or predictable.”
    —Kate Millett (b. 1934)

    “The very natural tendency to use terms derived from traditional grammar like verb, noun, adjective, passive voice, in describing languages outside of Indo-European is fraught with grave possibilities of misunderstanding.”
    —Benjamin Lee Whorf (1897–1934)