Saddle-point method - Knowlpedia

The saddle-point method is a complex-analytic method for estimating integrals with a large parameter,

I(n)=\int_{\Gamma} e^{n\phi(z)}\,\psi(z)\,dz

as $n\to\infty$ . Here $\phi$ and $\psi$ are analytic near the relevant part of the contour $\Gamma$ . One deforms $\Gamma$ , without crossing singularities, through stationary points of $\phi$ along directions on which $\operatorname{Re}\phi$ decreases.

Nondegenerate saddle

A point $z_0$ is a nondegenerate saddle of $\phi$ if

\phi'(z_0)=0 \quad\text{and}\quad \phi''(z_0)\neq 0.

Near such a point,

\phi(z)=\phi(z_0)+\tfrac12\phi''(z_0)(z-z_0)^2+\cdots,

so integration along a steepest-descent direction reduces locally to a Gaussian integral. The resulting leading factor has order

e^{n\phi(z_0)}n^{-1/2}.

Its complex phase and square-root branch depend on the orientation of the deformed contour, so there is no contour-independent formula obtained merely by inserting $\phi''(z_0)$ .

Coefficient asymptotics

For an analytic generating function

F(z)=\sum_{n\ge 0} a_n z^n.

Cauchy's formula gives

a_n=\frac{1}{2\pi i}\oint \frac{F(z)}{z^{n+1}}\,dz,

and a positive saddle radius $r=r_n$ is commonly chosen to satisfy

\frac{r F'(r)}{F(r)} = n.

Define

a(r)=\frac{rF'(r)}{F(r)},\qquad b(r)=r\,a'(r),

and let $a(r_n)=n$ . Under additional admissibility hypotheses that ensure this saddle dominates the rest of the circle,

a_n \sim \frac{F(r_n)}{r_n^{\,n}\sqrt{2\pi\,b(r_n)}}.

Analyticity and the saddle equation alone do not imply this formula; one also needs uniform local approximation and decay away from the saddle.

Relation to Laplace's method

The method is the complex analogue of Laplace's method. Both rely on:

identifying the dominant point (maximum/saddle),
making a local quadratic approximation,
evaluating a Gaussian integral.

Practical notes

Choosing the correct contour (often a steepest descent path) is part of the method; it ensures the contribution away from the saddle is negligible.
Multiple saddle points can contribute, in which case the leading term may be a sum of their contributions.
Degenerate saddles, where $\phi''(z_0)=0$ , require higher-order expansions and have different powers of $n$ .