Sigma Function of Half
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Theorem
\(\ds \map \sigma {\dfrac 1 2}\) | \(=\) | \(\ds \dfrac 1 2 \prod_{\substack {m, n \mathop \in \N^2 \\ \tuple {m, n} \mathop \ne \tuple {0, 0} } } \paren {1 - \dfrac 1 {2 \paren {m + n i} } } \map \exp {\dfrac 1 {2 \paren {m + n i} } + \dfrac 1 {8 \paren {m + n i}^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^{5/4} \pi^{1/2} e^{\pi/8} \map \Gamma {\dfrac 1 4}^2\) | ||||||||||||
\(\ds \) | \(\approx\) | \(\ds 0 \cdotp 47494 \, 93802 \ldots\) |
Sources
- 1978: Michel Waldschmidt: Fonctions entieres et nombres transcendants (Congr. Nat. Soc. Sav. Nancy Vol. 104)
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,47494 93802 \ldots$