Can we proof

$$\vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$

By Mellin transform ?

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## JAcobi Theta

This can be derived using the fact that

\(\displaystyle \varphi(q) = \sqrt{\frac{2K(k)}{\pi}}\)

where \(\displaystyle \varphi (q) = \sum_{n=-\infty}^\infty q^{-n^2}\)

It is well known that when \(q=e^{-\pi}\), \(k\) becomes \(\frac{1}{\sqrt{2}}\). Now,

\(\displaystyle \begin{align*}

K \left(\frac{1}{\sqrt{2}} \right) &= \sqrt{2} \int_0^1 \frac{dt}{\sqrt{(1-t^2)(2-t^2)}} \\

&= \sqrt{2} \int_0^1 \frac{dx}{\sqrt{1-x^4}} \quad x^2 :=t^2/(2-t^2) \\

&= \frac{\sqrt{2}}{4} \beta \left(\frac{1}{4},\frac{1}{2} \right) \\

&= \frac{1}{2\sqrt{2}} \frac{\Gamma \left(\frac{1}{4} \right)\Gamma \left(\frac{1}{2} \right)}{\Gamma \left(\frac{3}{4} \right)} \\

&= \frac{\pi^{\frac{3}{2}}}{2\Gamma^2 \left(\frac{3}{4} \right)}

\end{align*}\)

So, we get

\(\displaystyle \varphi(e^{-\pi}) = \frac{\sqrt[4]{\pi}}{\Gamma \left(\frac{3}{4} \right)}\)

Hey Z, you should also see about Elliptic Integral Evaluation at Singular values.

Here >>. I evaluated these for \(\displaystyle k_1,k_2,k_3,k_4\) and \(\displaystyle k_{25}\) but the others are really tough. There are some sophisticated number theoretic evaluations available but they are hard to understand. I believe Ramanujan's modular identities will prove to be useful for proving them all.

**Moderator:** Shobhit

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Can we proof

$$\vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$

By Mellin transform ?

$$\vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$

By Mellin transform ?

Wanna learn what we discuss , see Book

\(\displaystyle \varphi(q) = \sqrt{\frac{2K(k)}{\pi}}\)

where \(\displaystyle \varphi (q) = \sum_{n=-\infty}^\infty q^{-n^2}\)

It is well known that when \(q=e^{-\pi}\), \(k\) becomes \(\frac{1}{\sqrt{2}}\). Now,

\(\displaystyle \begin{align*}

K \left(\frac{1}{\sqrt{2}} \right) &= \sqrt{2} \int_0^1 \frac{dt}{\sqrt{(1-t^2)(2-t^2)}} \\

&= \sqrt{2} \int_0^1 \frac{dx}{\sqrt{1-x^4}} \quad x^2 :=t^2/(2-t^2) \\

&= \frac{\sqrt{2}}{4} \beta \left(\frac{1}{4},\frac{1}{2} \right) \\

&= \frac{1}{2\sqrt{2}} \frac{\Gamma \left(\frac{1}{4} \right)\Gamma \left(\frac{1}{2} \right)}{\Gamma \left(\frac{3}{4} \right)} \\

&= \frac{\pi^{\frac{3}{2}}}{2\Gamma^2 \left(\frac{3}{4} \right)}

\end{align*}\)

So, we get

\(\displaystyle \varphi(e^{-\pi}) = \frac{\sqrt[4]{\pi}}{\Gamma \left(\frac{3}{4} \right)}\)

Here >>. I evaluated these for \(\displaystyle k_1,k_2,k_3,k_4\) and \(\displaystyle k_{25}\) but the others are really tough. There are some sophisticated number theoretic evaluations available but they are hard to understand. I believe Ramanujan's modular identities will prove to be useful for proving them all.

3 posts
• Page **1** of **1**