Sigma Function of Half
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This page has a problem that needs the help of a knowledgeable authority, as follows:
Haven't got a clue what this is all about. This is what it says in Le Lionnais and Brette:
Theorem
| \(\displaystyle \map \sigma {\dfrac 1 2}\) | \(=\) | \(\displaystyle \dfrac 1 2 \prod_{\substack {m, n \mathop \in \N^2 \\ \tuple {m, n} \mathop \ne \tuple {0, 0} } } \paren {1 - \dfrac 1 {2 \paren {m + n i} } } \map \exp {\dfrac 1 {2 \paren {m + n i} } + \dfrac 1 {8 \paren {m + n i}^2} }\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle 2^{5/4} \pi^{1/2} e^{\pi/8} \map \Gamma {\dfrac 1 4}^2\) | |||||||||||
| \(\displaystyle \) | \(\approx\) | \(\displaystyle 0 \cdotp 47494 \, 93802 \ldots\) |
Haven't got a clue what this is all about. This is what it says in Le Lionnais and Brette:
"Point value $1/2$, of the Weierstrass product associated with the network of Gauss integers.
This suggests that the sigma function could be an instrument for determining whether the numbers $\pi$, $e^x$, $\map \Gamma {\dfrac 1 4}$ are algebraically independent."
If anyone can put this into context, please do.
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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): $0,47494 93802 \ldots$
