Saddle point

For a holomorphic phase $\varphi$, a saddle point is a critical point $z_0$ with ${\varphi }^{\prime }(z_0)=0$. At a simple saddle, ${\varphi }^{\prime \prime }(z_0)\ne 0$, and the quadratic Taylor term determines the local descent directions.

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.

When a simple pole approaches a saddle within its Gaussian width, separate residue and saddle approximations cease to be uniform. A local change of variable reduces the transition to a complementary error function whose multiplier smoothly switches the pole residue across the deformed contour.

Let $G^{\prime }(z_0)=0$, $G^{\prime \prime }(z_0)\ne 0$, and let $F,G$ be analytic. Assume a contributing contour can be deformed onto two descent arms of the method of steepest descent with one clockwise semicircular avoidance of the pole at $z_0$, and that its other portions are negligible. For the integrand $F(z)e^{-i\lambda G(z)}/(z-z_0)$, a local coordinate with $G-G_0=-i{w}^2/2$ givesThe odd pole term has zero Cauchy principal value on the real arms and integral $-i\pi F_0$ on the clockwise indentation. The constant regular term integrates to $\sqrt{2\pi /\lambda }$ times its coefficient, proving the expansion and generic correction order. A degenerate saddle has a different local width and possibly a different indentation angle; analyticity and a single critical point alone do not imply this result. If $F_0=0$, retain the additive expansion rather than writing asymptotic equivalence to zero.

For a simple saddle point, a nearby pole coalesces with its contributing region when their separation is of order the inverse square root of the large parameter. A local quadratic phase coordinate changes the singular part to a Gaussian pole integral. Adding any residue crossed by the contour deformation gives a Faddeeva function rather than two separately singular approximations. The regular part of the transformed amplitude remains smaller by the ordinary saddle width.