Laplace's Method Extension: 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 x0 where the derivative of ƒ 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
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).
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