information:filtering
Complex-analytic version

A complex-analysis version of this method is to consider 1 n ! {\displaystyle {\frac {1}{n!}}} as a Taylor coefficient of the exponential function e z = ∑ n = 0 ∞ z n n ! {\displaystyle e^{z}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}}, computed by Cauchy's integral formula as 1 n ! = 1 2 π i ∮ | z | = r e z z n + 1 d z. {\displaystyle {\frac {1}{n!}}={\frac {1}{2\pi i}}\oint \limits _{|z|=r}{\frac {e^{z}}{z^{n+1}}}\,\mathrm {d} z.}

This line integral can then be approximated using the saddle-point method with an appropriate choice of contour radius r = r n {\displaystyle r=r_{n}}. The dominant portion of the integral near the saddle point is then approximated by a real integral and Laplace's method, while the remaining portion of the integral can be bounded above to give an error term.