Consider the following integrals-
$Z(\lambda) = \frac{1}{\sqrt{\lambda}}\int_{c_x} dzg(z)e^{-f(z)/\lambda}$, $c_x$ is the real line contour.
This kind of integral is done by the Method of steepest descent. If the saddle points of $f(z)$ are isolated and non-degenerate then the contour of steepest descent through the saddle point $z_{\sigma}$ is determined by the flow equation-
$\frac{dz}{du} = \eta \frac{dF(z)^{*}}{dz^{*}}$,and
$\frac{dz^{*}}{du} = \eta \frac{dF(z)}{dz}$
where $F(z) = -f(z)/\lambda$, (*) corresponds to complex conjugate and $u$ is a real line parameter and $\eta$ = +1 or -1 gives the steepest ascent and descent contours ( called Thimbles in Higher dimensions).
Can anyone explain where these equations come from-? as in how to derive these equations and how to solve them to obtain the contours of steepest descent in the case of the following illustrative function-
$F(z)$=$\frac{-f(z)}{\lambda}$ = -$\frac{z^2}{2\lambda}$ - $\frac{z^4}{4\lambda}$, where $\lambda = 1 + ia$, $a$ is a very small number.