Sigma Function of Half

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Theorem

\(\displaystyle \map \sigma {\dfrac 1 2}\) \(=\) \(\displaystyle \dfrac 1 2 \prod_{\substack {m, n \mathop \in \N^2 \\ \tuple {m, n} \mathop \ne \tuple {0, 0} } } \paren {1 - \dfrac 1 {2 \paren {m + n i} } } \map \exp {\dfrac 1 {2 \paren {m + n i} } + \dfrac 1 {8 \paren {m + n i}^2} }\)
\(\displaystyle \) \(=\) \(\displaystyle 2^{5/4} \pi^{1/2} e^{\pi/8} \map \Gamma {\dfrac 1 4}^2\)
\(\displaystyle \) \(\approx\) \(\displaystyle 0 \cdotp 47494 \, 93802 \ldots\)



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