Method of Steepest Descent

An asymptotic technique for integrals, typically in the complex plane, that evaluates dominant contributions by deforming the integration contour onto paths of steepest descent through saddle points where the integrand decays most rapidly.

An asymptotic technique for evaluating integrals with a large parameter by deforming complex integration contours to pass through saddle (critical) points along paths of steepest descent so that integrand decay is maximized and dominant contributions are captured.

An asymptotic technique that evaluates integrals of the form ∫_Γ e^{λ f(z)} g(z) dz for large λ by deforming the integration contour Γ in the complex plane to pass through saddle points of f along paths of steepest descent, where the real part of f decreases most rapidly.

An asymptotic technique for evaluating integrals containing a large parameter by locating complex critical points (saddle points) of the phase or exponent, deforming the integration path to follow trajectories of steepest descent through those points, and approximating the integral by local (typically quadratic) expansions about the dominating critical points.