Saddle-point approximation

A saddle-point approximation expands an exponent around a stationary configuration and integrates its leading Gaussian fluctuations. A large parameter multiplying the exponent suppresses higher fluctuations.

When two quadratic saddles coalesce and the first nonzero phase derivative is cubic, the contributing neighborhood has width $O({\lambda }^{-1/3})$ rather than $O({\lambda }^{-1/2})$. The leading contour integral reduces to a Gamma-function or Airy-function constant.