Stirling's Formula, Two Routes

Lesson 11 · a real peak and a complex saddle, same critical point, same formula

Laplace's method  (real line)

tne−t normalized to peak height 1
t* t

steepest descent  (complex plane)

z*
|e^(h(z)−h(z*))| along the line steepest descent
cos 2θ = —
n 10
n! ~ √(2πn) (n/e)n  =  —    rel. err —
n!  =  —    —
n! ≈ √(2π) (n+1)n+½ e−(n+1)  =  —    rel. err —
In the right-hand formula the exponent is −(n+1), not −n+1: drag the line off θ=π/2 and the circle's own decaying direction inverts into growth.