Simple-saddle contribution in steepest descent

ϕ^{'} (z_{0}) = 0

ϕ^{''} (z_{0}) \neq = 0

, and the oriented steepest-descent tangent has angle

α

, then

\int_{C} f (z) e^{x ϕ (z)} d z \sim f (z_{0}) e^{x ϕ (z_{0}) + i α} \frac{2 π}{x ∣ ϕ ^{''} ( z _{0} ) ∣} .

The tangent condition is

ar g ϕ^{''} (z_{0}) + 2 α = π (mod 2 π) .

 Ancestors (6)