= Simple saddle coinciding with a simple pole
{title2=$J=e^{-i\lambda G_0}[-i\pi F_0+O(\lambda^{-1/2})]$}
Let $G'(z_0)=0$, $G''(z_0)\ne0$, 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 $G-G_0=-iw^2/2$ gives
$$
z-z_0=cw+dw^2+\cdots,\qquad c^2=-i/G''(z_0),\qquad
\frac{F(z)}{z-z_0}\frac{dz}{dw}=\frac{F_0}w+c\left(F'_0-\frac{F_0G'''_0}{6G''_0}\right)+O(w).
$$
The 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.
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