Talk:Product Formula for Sine

Also see section

I'm trying to see what direct relevance the entries in the Also see section has to this result, but I got nothing. --prime mover (talk) 06:05, 13 December 2023 (UTC)

I see a pattern and maybe in a few months I'll stumble on the work that someone else did 100 years ago explaining the common thread:

\(\ds \map \sin {n z}\)

\(=\)

\(\ds 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \frac {k \pi} n}\)

Product Formula for Sine

\(\ds \map \Gamma {n z}\)

\(=\)

\(\ds \paren {2 \pi}^{\paren {1 - n} / 2} n^{n z - 1 / 2} \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n}\)

Gauss Multiplication Formula

\(\ds \map \psi {n z}\)

\(=\)

\(\ds \frac 1 n \sum_{k \mathop = 0}^{n - 1} \map \psi {z + \frac k n} + \ln n\)

Digamma Additive Formula

\(\ds \map f {n z}\)

\(=\)

\(\ds \map g {n } \prod_{k \mathop = 0}^{n - 1} \map f {z + \frac k n}\)

--Robkahn131 (talk) 02:52, 14 December 2023 (UTC)