Simple-saddle contribution in steepest descent
= Simple-saddle contribution in steepest descent
If $\phi'(z_0)=0$, $\phi''(z_0)\ne0$, and the oriented steepest-descent tangent has angle $\alpha$, then
$$
\int_C f(z)e^{x\phi(z)}\,dz
\sim
f(z_0)e^{x\phi(z_0)+i\alpha}
\sqrt{\frac{2\pi}{x|\phi''(z_0)|}}.
$$
The tangent condition is
$$
\arg\phi''(z_0)+2\alpha=\pi\pmod{2\pi}.
$$