Saddle Point / Method of Steepest Descent

Integrand e^{Nf(z)} concentrates at saddle as N→∞. Compare exact n! vs Stirling.

∫ e^{Nf(z)} dz ≈ e^{Nf(z₀)} √(2π/N|f″(z₀)|)

The saddle point z₀ satisfies f′(z₀)=0. As N grows, the integrand becomes exponentially sharper — a Gaussian of width 1/√N around z₀.


For n! = ∫₀^∞ t^n e^{-t} dt = ∫ e^{N(ln t − t/N)} dt, saddle at t*=n gives Stirling's formula:

n! ≈ √(2πn) (n/e)^n

N = 10

Exact n!: -

Stirling: -

Error: -%


The steepest descent contour passes through the saddle where Im(f) = const, so the phase doesn't oscillate.