= Stirling correction by Gaussian moments
{c}
{title2=$\Gamma(x)\sim\sqrt{2\pi}\,x^{x-1/2}e^{-x}(1+1/(12x))$}
In <Laplace method> for the <Gamma function>, scaling $t=x(1+s/\sqrt x)$ gives a standard <Gaussian integral>. The first even correction polynomial is $s^2-7s^4/12+s^6/18$. Its expectation under a unit-variance centered Gaussian is $1-7\cdot3/12+15/18=1/12$. The amplitude expansion contributes to this coefficient as well as the phase expansion; expanding only the latter gives the wrong <Stirling formula> correction.
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