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 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.

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.

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.