In mathematics, the steepest descent method or saddle-point approximation is a method used to approximate integrals of the form
As such, significant contributions to the integral of this function will come only from points x in a neighborhood of x0, which can then be estimated.
To state and prove the method, we need several assumptions. We will assume that x0 is not an endpoint of the interval of integration, that the values f(x) cannot be very close to f(x0) unless x is close to x0, and that .
We can expand f(x) around x0 by Taylor's theorem,
Since f has a global maximum at x0, and since x0 is not an endpoint, it is a stationary point, the derivative of f vanishes at x0. Therefore, the function f(x) may be approximated to quadratic order
for x close to x0 (recall that the second derivative is negative at the global maximum f(x0)). The assumptions made ensure the accuracy of the approximation
where the integral is taken in a neighborhood of x0. This latter integral is a Gaussian integral if the limits of integration go from −∞ to +∞ (which can be assumed so because the exponential decays very fast away from x0), and thus it can be calculated. We find
A generalization of this method and extension to arbitrary precision is provided by Fog (2008).
In extensions of Laplace's method, complex analysis, and in particular Cauchy's integral formula, is used to find a contour of steepest descent for an (asymptotically with large M) equivalent integral, expressed as a line integral. In particular, if no point x0 where the derivative of f vanishes exists on the real line, it may be necessary to deform the integration contour to an optimal one, where the above analysis will be possible. Again the main idea is to reduce, at least asymptotically, the calculation of the given integral to that of a simpler integral that can be explicitly evaluated. See the book of Erdelyi (1956) for a simple discussion (where the method is termed steepest descents).
An extension of the steepest descent method is the so-called nonlinear stationary phase/steepest descent method. Here, instead of integrals, one needs to evaluate asymptotically solutions of Riemann-Hilbert factorization problems.
Given a contour C in the complex sphere, a function f defined on that contour and a special point, say infinity, one seeks a function M holomorphic away from the contour C, with prescribed jump across C, and with a given normalization at infinity. If f and hence M are matrices rather than scalars this is a problem that in general does not admit an explicit solution.
An asymptotic evaluation is then possible along the lines of the linear stationary phase/steepest descent method. The idea is to reduce asymptotically the solution of the given Riemann-Hilbert problem to that of a simpler, explicitly solvable, Riemann-Hilbert problem. Cauchy's theorem is used to justify deformations of the jump contour.
The nonlinear stationary phase was introduced by Deift and Zhou in 1993, based on earlier work of Its. A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou.
For complex integrals in the form:
with t >> 1, we make the substitution t = iu and the change of variable s = c + ix to get the Laplace bilateral transform:
Laplace's method can be used to derive Stirling's approximation
From the definition of the Gamma function, we have
Now we change variables, letting