Editing Laplace's method (section)

==Steepest descent extension==
{{main|Method of steepest descent}}

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 ''x''<sub>0</sub> where the derivative of <math>f</math> 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'').

The appropriate formulation for the complex ''z''-plane is

:<math>\int_a^b e^{M f(z)}\, dz \approx \sqrt{\frac{2\pi}{-Mf''(z_0)}}e^{M f(z_0)} \text{ as } M\to\infty.</math>

for a path passing through the saddle point at ''z''<sub>0</sub>. 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'').