Here is another inordinately silly function:

\[ \Gamma(x)=\int_0^\infty y^{x-1}e^{-y}\,\textrm{d} y,\quad x>0. \]

Prove that \(\Gamma(x+1)=x\Gamma(x)\).

Hint

Start on the left-hand side by writing out \(\Gamma(x+1)\) and evaluating the integral by parts.