Saddle-Point Approximation — Steepest Descent

The integral ∫ e^{N·f(z)} dz is dominated, for large N, by the saddle point z* where f'(z*)=0. The Gaussian integral around z* gives the approximation e^{N·f(z*)}·√(2π/N|f''(z*)|). Applied to n!, this yields Stirling's formula. The steepest descent contour passes through the saddle where Im(f) is constant.

∫ e^{N·f(z)} dz ≈ e^{N·f(z*)} · √(2π / N|f''(z*)|)

N (large parameter)10

Function

x range2.0

Exact I(N): —

Saddle approx: —

Relative error: —

Stirling N!: —