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:

    “Those things which now most engage the attention of men, as politics and the daily routine, are, it is true, vital functions of human society, but should be unconsciously performed, like the corresponding functions of the physical body.”
    —Henry David Thoreau (1817–1862)

    “A radical is one of whom people say “He goes too far.” A conservative, on the other hand, is one who “doesn’t go far enough.” Then there is the reactionary, “one who doesn’t go at all.” All these terms are more or less objectionable, wherefore we have coined the term “progressive.” I should say that a progressive is one who insists upon recognizing new facts as they present themselves—one who adjusts legislation to these new facts.”
    —Woodrow Wilson (1856–1924)