SPECIAL VALUES OF JACOBI THETA FUNCTION

In this post we well prove the following results



\begin{aligned} &\sum_{n=-\infty}^{\infty} e^{-\pi n^{2}}=\frac{\sqrt[4]{\pi}}{\Gamma\left(\frac{3}{4}\right)} \\ &\sum_{n=-\infty}^{\infty} e^{-2 \pi n^{2}}=\frac{1}{\sqrt{4-2 \sqrt{2}}} \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac{3}{4}\right)} \\ &\sum_{n=-\infty}^{\infty} e^{-\pi n^{2} / 2}=\frac{1}{\sqrt{2-\sqrt{2}}} \frac{\sqrt[4]{\pi}}{\Gamma\left(\frac{3}{4}\right)} \end{aligned}



Click here to see the proof.

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