Method of Steepest Descent: Complex Saddle Points

Method of steepest descent (saddle-point approximation): For integrals I(λ) = ∫ e^{λf(z)} dz with λ → ∞, the dominant contribution comes from the saddle point z* where f'(z*) = 0.

The contour of integration is deformed through z* along the path of steepest descent — where Im(f) is constant (so oscillations vanish) and Re(f) decreases as fast as possible. Near z*:

I(λ) ≈ e^{λf(z*)} √(2π / (−λf''(z*))) × (1 + O(1/λ))

The surface shows Re(f) on the complex plane. Saddle points are cols (passes) where Re(f) has a maximum in one direction and minimum in another. The steepest descent path passes through the saddle like a mountain pass. Example: Stirling's formula ln(n!) ≈ n ln n − n follows from saddle-point evaluation of Γ(n+1) = ∫ t^n e^{−t} dt.