Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 16 / a

For $p>0$, integration by parts gives the Gamma function recurrence:The endpoint terms vanish because $x^p\to 0$ at zero and the exponential dominates the power at infinity. Also $\Gamma (1)={\int }_0^{\infty }{e}^{-x}dx=1$. Iterating for nonnegative integers $n$ therefore yieldsNegative integers are outside this integral definition and factorial conclusion.