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

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.

Set $t=\log (1/x)$, so $x=e^{-t}$ and $dx=-e^{-t}dt$. The improper integral becomesThe Gamma function recurrence givesThe exponential-transformed integrand also proves convergence at the logarithmically singular endpoint.

In the Gamma function integral at $p=1/2$, use $x=t^2$, $dx=2tdt$, with $t>0$. Then $x^{-1/2}dx=2dt$, soThe substitution can first be made on finite positive intervals and then extended by monotone limits, justifying the endpoint passage.

The integrand in the preceding expression is nonnegative. Tonelli's theorem permits squaring as a quadrant integral; then polar coordinates giveThe defining integral is positive, fixing the sign of the square root. ThereforeThis is the positive-quadrant proof of the Gaussian integral.