Editing Theta function (section)

==Theta functions in terms of the nome==
Instead of expressing the Theta functions in terms of {{mvar|z}} and {{mvar|τ}}, we may express them in terms of arguments {{mvar|w}} and the [[nome (mathematics)|nome]] {{mvar|q}}, where {{math|''w'' {{=}} ''e''<sup>''πiz''</sup>}} and {{math|''q'' {{=}} ''e''<sup>''πiτ''</sup>}}. In this form, the functions become

:<math>\begin{align}
\vartheta_{00}(w, q)& = \sum_{n=-\infty}^\infty  \left(w^2\right)^n q^{n^2}\quad&
\vartheta_{01}(w, q)& = \sum_{n=-\infty}^\infty (-1)^n \left(w^2\right)^n q^{n^2}\\[3pt]
\vartheta_{10}(w, q)& = \sum_{n=-\infty}^\infty \left(w^2\right)^{n+\frac12} q^{\left(n + \frac12\right)^2}\quad&
\vartheta_{11}(w, q)& = i \sum_{n=-\infty}^\infty (-1)^n \left(w^2\right)^{n+\frac12} q^{\left(n + \frac12\right)^2}.
\end{align}</math>

We see that the theta functions can also be defined in terms of {{mvar|w}} and {{mvar|q}}, without a direct reference to the exponential function. These formulas can, therefore, be used to define the Theta functions over other [[field (mathematics)|fields]] where the exponential function might not be everywhere defined, such as fields of [[p-adic number|{{mvar|p}}-adic numbers]].