[math] \newcommand{\mathds}{\mathbb}[/math]

Develop the theory of the gamma function, defined as

[[math]] \Gamma(z)=\int_0^\infty x^{z-1}e^{-x}dx [[/math]]

notably by establishing the following formula, for any [math]N\in\mathbb N[/math],

[[math]] \Gamma(N)=(N-1)! [[/math]]

and then comment on the formulae for the volumes and areas of spheres.