Laplace's method

is large this creates only a small error because the exponential decays very fast away from

Computing this Gaussian integral we obtain: A generalization of this method and extension to arbitrary precision is provided by the book Fog (2008).

Then we have the following lower bound: where the last equality was obtained by a change of variables Remember

Then we can calculate the following upper bound: If we divide both sides of the above inequality by and take the limit we get: Since

The proof proceeds otherwise as above, but with a slightly different approximation of integrals: When we divide by we get for this term whose limit as

is a large number obviously and the relative error will be Now, let us separate this integral into two parts:

From this equation you will find that the terms higher than second derivative in this Taylor expansion is suppressed as the order of

Relying on the 3rd concept, even if we choose a very large Dy, sDy will finally be a very small number when

as shown in the figure: If the interval of the integration of this method is finite, we will find that no matter

Importantly, the accuracy of the approximation depends on the variable of integration, that is, on what stays in

; however, you can find that the remainders of these two expansions are both inversely proportional to the square root of

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.

See the book of Erdelyi (1956) for a simple discussion (where the method is termed steepest descents).

The appropriate formulation for the complex z-plane is for a path passing through the saddle point at z0.

Note the explicit appearance of a minus sign to indicate the direction of the second derivative: one must not take the modulus.

Also note that if the integrand is meromorphic, one may have to add residues corresponding to poles traversed while deforming the contour (see for example section 3 of Okounkov's paper Symmetric functions and random partitions).

Here, instead of integrals, one needs to evaluate asymptotically solutions of Riemann–Hilbert factorization problems.

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 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.

As in the linear case, "steepest descent contours" solve a min-max problem.

In the nonlinear case they turn out to be "S-curves" (defined in a different context back in the 80s by Stahl, Gonchar and Rakhmanov).

The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and integrable models, random matrices and combinatorics.

Matching the logarithm of the density functions and their derivatives at the median point up to a given order yields a system of equations that determine the approximate values of

The approximation was introduced in 2019 by D. Makogon and C. Morais Smith primarily in the context of partition function evaluation for a system of interacting fermions.

to get the bilateral Laplace transform: We then split g(c + ix) in its real and complex part, after which we recover u = t/i.

This is useful for inverse Laplace transforms, the Perron formula and complex integration.

Laplace's method can be used to derive Stirling's approximation for a large integer N. From the definition of the Gamma function, we have Now we change variables, letting

Plug these values back in to obtain This integral has the form necessary for Laplace's method with which is twice-differentiable: The maximum of