Theta Function - Explicit Values

Explicit Values

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\varphi(e^{-\pi x}) = \vartheta(0; {\mathrm{i}}x) = \theta_3(0;e^{-\pi x}) = \sum_{n=-\infty}^\infty e^{-x \pi n^2}
\varphi\left(e^{-\pi} \right) = \frac{\sqrt{\pi}}{\Gamma(\frac{3}{4})}
\varphi\left(e^{-2\pi} \right) = \frac{\sqrt{6\pi+4\sqrt2\pi}}{2\Gamma(\frac{3}{4})}
\varphi\left(e^{-3\pi}\right) = \frac{\sqrt{27\pi+18\sqrt3\pi}}{3\Gamma(\frac{3}{4})}
\varphi\left(e^{-4\pi}\right) =\frac{\sqrt{8\pi}+2\sqrt{\pi}}{4\Gamma(\frac{3}{4})}
\varphi\left(e^{-5\pi} \right) =\frac{\sqrt{225\pi+ 100\sqrt5 \pi}}{5\Gamma(\frac{3}{4})}
\varphi\left(e^{-6\pi}\right) = \frac{\sqrt{3\sqrt{2}+3\sqrt{3}+2\sqrt{3}-\sqrt{27}+\sqrt{1728}-4}\cdot \sqrt{243{\pi}^2}}{6\sqrt{1+\sqrt6-\sqrt2-\sqrt3}{\Gamma(\frac{3}{4})}}

Read more about this topic:  Theta Function

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