## COROLLARY TO EULER´S COROLLARY

27 08 2009

It is sure that, after a very short fraction of a second, the first time Euler saw (despite he was one-eyed and partially blind) Gauss´s Gamma Function Multiplication Formula:

$\displaystyle \prod _{k=0}^{n-1} \Gamma \left(\frac{k}{n}+z\right)=(2 \pi )^{\frac{n-1}{2}} n^{\frac{1}{2}-n z}\Gamma (n z)$

Euler tested the expresion with $\displaystyle z=\frac{1}{n}$ to get his corollary:

$\displaystyle \prod _{k=1}^{n-1} \Gamma \left(\frac{k}{n}\right)=\frac{(2 \pi )^{\frac{n-1}{2}}}{\sqrt{n}}$

Or maybe was Gauss who generalized, Legendre´s Gamma Duplication Formula with Euler´s ideas, I haven´t found anything about the real history.

Anyway, if we multiply Euler´s corollary by the Gamma Formula with $\displaystyle z=1$, and if we practice the “good habit” of multiplying things by $1$:

$\displaystyle \Gamma \left(\frac{n}{n}\right)=\Gamma(1)=\Gamma \left(\frac{2n}{n}\right)=\Gamma(2)=1$

Then we get:

$\displaystyle \prod_{k=1}^{2n} \Gamma \left(\frac{k}{n}\right)=\frac{(2 \pi )^{n-1}}{n^n}\Gamma(n)$

References:

[1]-Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function
[2]-Gamma Function @ Wikipedia Gamma Function