Editing Theta function (section)

=== Combination of the integral identities with the nome ===

The elliptic nome function has these important values:
:<math>q(\tfrac{1}{2}\sqrt{2}) = \exp(-\pi)</math>

:<math>q[\tfrac{1}{4}(\sqrt{6} - \sqrt{2})] = \exp(-\sqrt{3}\,\pi)</math>

:<math>q(\sqrt{2} - 1) = \exp(-\sqrt{2}\,\pi)</math>

For the proof of the correctness of these nome values, see the article [[Nome (mathematics)]]!

On the basis of these integral identities and the above-mentioned '''Definition and identities to the theta functions''' in the same section of this article, exemplary theta zero values shall be determined now:
:{| class="wikitable"
|<math>\theta_{3}[q(k)] = \sqrt{2\pi^{-1} K(k)}</math>

|}
:<math>\theta_{3}[\exp(-\pi)] = \theta_{3}[q(\tfrac{1}{2}\sqrt{2})] = \sqrt{2\pi^ {-1}K(\tfrac{1}{2}\sqrt{2})} = 2^{-1/2}\pi^{-1/2}\beta(\tfrac{1}{4} )^{1/2} = 2^{-1/4}\sqrt[4]{\pi}\,{\Gamma\bigl(\tfrac{3}{4}\bigr)}^{-1} </math>

:<math>\theta _{3}[\exp(-\sqrt{3}\,\pi )] = \theta _{3}\bigl\{q\bigl[\tfrac{1}{4}(\sqrt {6} - \sqrt{2})\bigr]\bigr\} = \sqrt{2\pi^{-1}K\bigl[\tfrac{1}{4}(\sqrt{6} - \sqrt {2})\bigr]} = 2^{-1/6}3^{-1/8}\pi^{-1/2}\beta(\tfrac{1}{3})^{1/ 2}</math>

:<math>\theta _{3}[\exp(-\sqrt{2}\,\pi )] = \theta _{3}[q(\sqrt{2} - 1)] = \sqrt{2\pi ^{-1}K(\sqrt{2} - 1)} = 2^{-1/8}\cos(\tfrac{1}{8}\pi)\,\pi^{-1/2} \beta(\tfrac{3}{8})^{1/2}</math>

:{| class="wikitable"
|<math>\theta_{4}[q(k)] = \sqrt[4]{1 - k^2}\,\sqrt{2\pi^{-1} K(k)}</math>
|}

:<math>\theta_{4}[\exp(-\sqrt{2}\,\pi)] = \theta_{4}[q(\sqrt{2} - 1)] = \sqrt[4]{ 2\sqrt{2} - 2}\,\sqrt{2\pi^{-1}K(\sqrt{2} - 1)} = 2^{-1/4}\cos(\tfrac{1} {8}\pi)^{1/2}\,\pi^{-1/2}\beta(\tfrac{3}{8})^{1/2}</math>