Gamma, Gamma, Gamma!

Calculus Level 2

$\left(\frac{1}{2} \right)!$

What is true about the above expression?

Note: $n!= n \times (n-1) \times (n-2) \times \; ... \; \times \; 3 \times 2 \times 1, \: \: n \in \mathbb{N}$

Factorials only apply to positive integers! There is no such number! It's equal to

\frac{\pi^2}{6}

, Euler said so! It's equal to

\frac{\sqrt{\pi}}{2}

1 solution

Jason Simmons
Dec 21, 2015

The Gamma function can also evaluate the factorials of fractional numbers as well as complex numbers.

$\Gamma (t) = \int_{0}^{\infty} x^{t-1} e^{-x} \; dx, \textrm{where} \: \: \Gamma (n) = (n-1)!$

$n \in \mathbb{N}$

From this function we can conclude that

$\Gamma \left (\frac{3}{2} \right ) =\frac{1}{2}!=\frac{\sqrt{\pi}}{2}$