Sigma Function of Half

Theorem

\(\ds \map {\sigma_1} {\dfrac 1 2}\)

\(=\)

\(\ds \dfrac 1 2 \prod_{\substack {m, n \mathop \in \N^2 \\ \tuple {m, n} \mathop \ne \tuple {0, 0} } } \paren {1 - \dfrac 1 {2 \paren {m + n i} } } \map \exp {\dfrac 1 {2 \paren {m + n i} } + \dfrac 1 {8 \paren {m + n i}^2} }\)

\(\ds \)

\(=\)

\(\ds 2^{5/4} \pi^{1/2} e^{\pi/8} \map \Gamma {\dfrac 1 4}^2\)

\(\ds \)

\(\approx\)

\(\ds 0 \cdotp 47494 \, 93802 \ldots\)

This page needs the help of a knowledgeable authority. In particular: 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. If you are knowledgeable in this area, then you can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Help}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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): $0,47494 93802 \ldots$