mathimatikoi.orghttps://www.mathimatikoi.org/forum/ Gamma function and producthttps://www.mathimatikoi.org/forum/viewtopic.php?f=3&t=928 Page 1 of 1

 Author: Tolaso J Kos [ Thu Jul 14, 2016 10:14 am ] Post subject: Gamma function and product Evaluate the product:$$\Gamma \left ( \frac{1}{n} \right )\Gamma \left ( \frac{2}{n} \right )\cdots \Gamma \left ( \frac{n-1}{n} \right )$$

 Author: Grigorios Kostakos [ Thu Jul 14, 2016 10:15 am ] Post subject: Re: Gamma function and product Hello Tolis.Using Gauss Multiplication Formula $\displaystyle\prod_{k \mathop = 0}^{n - 1} \Gamma \Bigl({z + \frac k n}\Bigr) = ({2 \pi})^{\frac{n - 1}{2}} n^{\frac{1}{2} - n z} \Gamma({n z})$ for $$z=0$$, we have that \begin{align*}\prod_{k \mathop = 1}^{n - 1} \Gamma \Bigl({\frac k n}\Bigr)&=\prod_{k \mathop = 1}^{n - 1} \Gamma \Bigl({0 + \frac k n}\Bigr) \\&= ({2 \pi})^{\frac{n - 1}{2}} n^{\frac{1}{2} - n \cdot0} \Gamma({n \cdot0})\\&= ({2 \pi})^{\frac{n - 1}{2}} \,\sqrt{n}\,.\end{align*}A proof of the Gauss Multiplication Formula can be found in here.

 Page 1 of 1 All times are UTC [ DST ] Powered by phpBB® Forum Software © phpBB Grouphttps://www.phpbb.com/