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

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