ガンマ関数の性質

\[\Gamma(p)=\int^{\infty}_{0}x^{p-1}e^{-x}\,dt\qquad(\Re{p}>0)\]であるから，\[\begin{eqnarray}\Gamma(p+1)&=&\int^{\infty}_{0}x^{p}e^{-x}dx\\&=&\int^{\infty}_{0}x^{p}(-e^{-x})'dx\\&=&lim_{a \to \infty}[-x^{p}e^{-x}]_{0}^{a}-\int_{0}^{\infty}px^{p-1}\cdot(-e^{-x})dx\\&=&lim_{a \to \infty}(-a^{p}e^{-a}+0^{p}e^{-0})-\int_{0}^{\infty}px^{p-1}\cdot(-e^{-x})dx\\&=&p\Gamma(p)\end{eqnarray}\]

参照：ガンマ関数

参照：ガンマ関数$\Gamma(p)$と階乗

Mathematics is the language with which God has written the universe.