Method of steepest descent

A contour integral with a large exponential parameter is deformed through saddle points along curves on which the real part of the phase decreases most rapidly and the imaginary part is locally constant.

For an exponential integral with phase $\varphi$, a constant-phase contour holds the imaginary part of $\varphi$ fixed. Along it, decreasing real part of $\varphi$ gives steepest descent, while increasing real part gives steepest ascent. At a cubic stationary point, six rays alternate between descent and ascent.

For an integrand $e^{kf(z)}$, a contour along which $\mathrm{Im}f$ is constant and $\mathrm{Re}f$ decreases away from the contributing endpoint or saddle. This suppresses oscillation while localizing the large-parameter integral.

For $e^{-\lambda z^2}$, the saddle point descent is the real axis and ascent is the imaginary axis. A descent path from a nonzero endpoint $b$ has $z(s{)}^2=b^2+s$, $s\ge 0$, with the continuous square-root branch from $b$. It ends at positive or negative real infinity according to the sign of $\mathrm{Re}b$. The oriented outward endpoint contribution in steepest descent is $e^{-\lambda b^2}/(2\lambda b)$.

For a Gaussian integral from $-1/\sqrt{2}$ to $e^{i\theta }$, a saddle point contribution dominates for $0\le \theta \le \pi /4$. The moving endpoint contribution in steepest descent dominates for $\pi /4<\theta <5\pi /6$. At $5\pi /6$ the two endpoints have equal exponential rates and both are needed; for larger angles the fixed endpoint dominates. Saddle connectivity switches at $\pi /2$, but that change is subdominant in the leading endpoint regime.

If ${\varphi }^{\prime }(z_0)=0$, ${\varphi }^{\prime \prime }(z_0)\ne 0$, and the oriented steepest-descent tangent has angle $\alpha$, thenThe tangent condition is

If a deformed contour starts at an endpoint $z_0$ with ${\varphi }^{\prime }(z_0)\ne 0$ and initially enters a decaying sector, local integration gives the leading contribution $-a(z_0)e^{\lambda \varphi (z_0)}/[\lambda {\varphi }^{\prime }(z_0)]$, with the sign adjusted to the contour orientation.

An endpoint at infinity contributes when the region controlling an asymptotic integral recedes without bound as the large parameter changes. Expanding the phase and amplitude at infinity reveals the rescaling that converts the distant tail into a finite limiting integral.